JOURNAL OF APPLIED PHYSICS 109, 113711 (2011) Electron scattering from polarization charges bound on a rough interface of polar heterostructures Doan Nhat Quang,1,2,a) Nguyen Huyen Tung,3 and Nguyen Thanh Tien4 Institute of Physics, Vietnamese Academy of Science and Technology, 10 Dao Tan Street, Hanoi, Vietnam University of Technology and Management, N3 Ho Tung Mau Road, Hanoi, Vietnam Institute of Engineering Physics, Hanoi University of Technology, Dai Co Viet Road, Hanoi, Vietnam College of Science, Can Tho University, 3-2 Road, Can Tho City, Vietnam (Received 13 November 2010; accepted 14 April 2011; published online June 2011) We present the theory of an ad hoc scattering mechanism for carriers confined in a heterostructure (HS) made of polar materials, such as zinc blends, nitrides, and oxides We show that the carriers in an actual polar HS must be laterally scattered extra from both piezoelectric and spontaneous polarization charges bound on a rough interface of the system This is due to roughness-induced fluctuations in the position of interface polarization charges, so referred to as polarization roughness (PR) scattering The new scattering is combined with the normal surface roughness (SR) scattering, giving rise to an effective roughness-related process, which is referred to as polarization surface roughness (PSR) scattering The PSR scattering is found to be more important for nearly forward events and at small sheet carrier densities, and it is one of the key mechanisms governing transport in polar HSs This enables a successful explanation of the mobility data on polar HSs made, e.g., of AlGaN/GaN, which has not been understood so far, starting only from the traditional C 2011 American Institute of Physics [doi:10.1063/1.3592187] scattering mechanisms V I INTRODUCTION The electrical and optical properties of semiconductors have been intensively investigated for a long time because these are considered to be the figure of merit for material characterization These properties are determined by the decisive factors, such as electronic structure, scattering mechanisms, and confining sources.1 For instance, impurity doping of some sample takes a threefold role This is an indispensable source for carrier supply to the system On the other hand, this is a scattering mechanism for carriers moving along the in-plane, and this is also a carrier confining source in the quantization direction It is well known2 that polarization is a very important property of a great many semiconductors For instance, III-V zinc blends with cubic symmetry exhibit strong piezoelectric (PZ) polarization.3,4 Wurtzites, such as III-V nitrides and IIVI oxides with hexagonal symmetry, possess both strong PZ and spontaneous (SP) polarization.2,5–7 Furthermore, IV-IV compounds, e.g., SiGe and SiC, are also polar materials.8,9 It has been pointed out2–7 that an abrupt change in polarization across an interface of layered systems may induce huge polarization charges bound thereon These charges act as a carrier supply source to the HS For instance, due to PZ polarization in III-V zinc blends, one can achieve a twodimensional electron gas (2DEG) with sheet carrier densities of the order of 1012 cmÀ2 3,4 Meanwhile, due to both PZ and SP polarization in III-V nitrides one can achieve a 2DEG with higher sheet carrier densities of the order of 1013 cmÀ2 2,5–7 Hence, there is generally no need to dope the polar systems in question with impurities The polarization a) Author to whom correspondence should be addressed Electronic mail: dnquang@iop.vast.ac.vn 0021-8979/2011/109(11)/113711/8/$30.00 charges bound on the interface of polar HSs have sheet densities as high as those of 2DEGs.2,5–7 Thus, these interface charges may make an important contribution to the confining potential, so they heavily modify the electronic structure and the electrical and optical properties of polar HSs.3,10–14 However, it is worth mentioning that up to date polarization charges bound on an interface of polar HSs have been considered merely as a carrier supply source and a confining source rather than as a scattering mechanism.2–7 It is clear that the electric field created by a sheet density of polarization charges that is constant on a flat and infinite interface, is uniform and normal to the interface plane However, all realistic HSs always exhibit lateral structures in the form of interface roughness When the flatness condition of the interface is broken, the lateral component of the electric field in question becomes nonvanishing Because the roughness changes at random along the in-plane, the lateral electric field due to the interface polarization charges is subject to random fluctuations, leading to the deformation along the in-plane of the carrier distribution in the quantization direction (wave packet) This means that the polarization charges on a rough interface with a constant sheet density must act as a scattering source for carriers moving along the in-plane, which is referred to as polarization roughness (PR) scattering As known,1 the normal surface roughness (SR) scattering is connected with roughness-induced fluctuations in position of the potential barrier, hence, with the deformation along the in-plane of the wave packet In combination with the normal SR scattering, the PR scattering leads to an effective roughness-related process, which is referred to as polarization surface roughness (PSR) scattering Thus, the PSR scattering implies the deformation of the wave packet due to simultaneous fluctuations in position of the potential barrier as well as of the polarization charges bound thereon So this 109, 113711-1 C 2011 American Institute of Physics V Downloaded 07 Jun 2011 to 139.78.49.186 Redistribution subject to AIP license or copyright; see http://jap.aip.org/about/rights_and_permissions 113711-2 Quang, Tung, and Tien J Appl Phys 109, 113711 (2011) is a scattering mechanism characteristic of all actual polar HSs Therefore, the aim of this paper is to develop the theory of PSR scattering in polar single HSs In Sec II, we derive an analytic expression for the PSR potential and distinguish PSR scattering from the existing polarization charge-related scatterings In Sec III, we formulate the basic equations for transport in polar HSs We estimate, in Sec IV, the magnitude of the PSR process and apply it to explain some recent experiments on polar HSs that have not been understood so far, basing merely on the traditional scattering mechanisms At last, a summary is given in Sec V II ELECTRIC FIELD DUE TO POLARIZATION CHARGES BOUND ON A ROUGH INTERFACE We begin by studying PSR scattering in a polar HS with a derivation of an expression for the electrostatic potential induced by a sheet density r of both PZ and SP polarization charges bound on a rough interface The coordinate system is such that the quantization z axis coincides with the growth direction, and z ¼ defines the interface plane of the HS between the channel (z > 0) and the barrier (z < 0) The dielectric constant is, in general, subject to an abrupt change across the interface Then the Coulomb potential energy of an electron with charge Àe located at (r,z) is provided by two different expressions Namely, assuming the roughness Dðr0 Þ > 0, it holds1 for the electron potential energy in the channel: ( er dr0 Rỵ r; zị ẳ ec ẵr r ị ỵ ðz À Dðr0 ÞÞ2 1=2 ) (1) ec À eb ; ỵ ec ỵ eb ẵr r0 ị2 þ ðz þ Dðr0 ÞÞ2 1=2 and in the barrier: R r; zị ẳ er dr0 : ea ẵr r ị ỵ z Dr0 ÞÞ2 1=2 ð (2) (4) It is clearly seen from the right-hand side of Eqs (3) and (4) that the zero-order term is dependent on z (in the quantization direction), but independent of r (in the in-plane), so it gives a confining potential along the growth direction, while the first-order (dipole) term is dependent on both r and z, so it gives a scattering potential along the in-plane This result can be interpreted as follows Interface roughness induces random fluctuations in the z coordinate of polarization charges on the interface and, hence, in their spatial z distance to a carrier When the carrier moves along the in-plane, it is then scattered by the fluctuating Coulomb potential Hereafter, the potential reference origin is, as usual, put at the interface plane z ¼ Then, from the zero-order term in Eqs (3) and (4), it holds for the confining potential due to interface polarization charges of a sheet density r: Vr ðzÞ ¼ 2per jzj: ea (5) This is the well-known expression for the electrostatic potential induced by an infinite flat interface (plane) charged uniformly with an areal density of r The effect on the confining potential from the difference in dielectric constant between two layers is taken into account with the use of their average ea As to scattering by polarization charges bound on a rough interface, from the first-order term in Eqs (3) and (4), it holds for the potential for polarization roughness scattering: UPR;6 r; zị ẳ Here, the subindices ỵ and stand for the regions with z > and z < 0, respectively ea ¼ eb ỵ ec ị=2 is the average of the dielectric constants of the barrier (eb ) and channel (ec ) The first term on the right-hand side of Eq (1) arises from the interaction of the electron with the polarization charges, and the second term from its interaction with their images Next, by expanding the integrands in Eqs (1) and (2) in small Dðr0 Þ, for the electron potential energy, we may arrive at some approximation As an example, up to the first order we have: ð ( er Rỵ r; zị ẳ dr0 ea ẵr r ị2 ỵ z2 1=2 ) (3) zDr0 ị ; re ẵr r0 ị2 ỵ z2 3=2 where by definition re ¼ ec =eb , and ð ( er R r; zị ẳ dr0 ea ẵr r ị2 ỵ z2 1=2 ) zDr0 ị ; ẵr r0 ị2 ỵ z2 3=2 er zDr0 ị dr0 : ea r6 ẵr r0 ị2 ỵ z2 3=2 (6) Here, the subindex ỵ refers again to the potential in the region with z > (and rỵ ẳ re ), while the subindex À to the one with z < (and rÀ ¼ 1) Last, by expanding the roughness Dðr0 Þ appearing in Eq (6) in the 2D Fourier transform Dq , we may derive the scattering potential from interface polarization charges in 2D wave vector space as follows: UPR;6 ðq; zÞ ¼ 2per Dq eÇqz ; ea r6 (7) where the upper and lower signs refer to the regions with z > and z < 0, respectively It should be noticed that PR scattering in question, defined by Eq (7), is a dipole scattering the rate of which depends quadratically on the total sheet density of PZ and SP polarization charges, r ẳ rPZ ỵ rSP The preceding dipole scattering is a surface phenomenon that is basically different from the polarization chargerelated scatterings in the literature both on their nature and the magnitude of their effect Indeed, a dipole scattering may Downloaded 07 Jun 2011 to 139.78.49.186 Redistribution subject to AIP license or copyright; see http://jap.aip.org/about/rights_and_permissions 113711-3 Quang, Tung, and Tien J Appl Phys 109, 113711 (2011) stem from interface dipoles due to the mismatch in dielectric constant across the HS interface,1,15,16 so this is significant for several systems, e.g., Si/SiO2, and is normally negligibly weak in many HSs.17 Another dipole scattering stems from alloy disorderinduced fluctuations in magnitude of microscopic dipoles in the region occupied by a ternary (e.g., AlGaN and AlInN) or quaternary (AlInGaN) barrier.2,18 The scattering is a volume phenomenon fixed by the alloy composition So this is rather weak and vanishing in HSs made of binaries only, despite the fact that they are very highly polar For instance, in the binary HSs made of AlN/GaN and AlN/InN the bulk dipole scattering is vanishing, but the PR scattering is very strong because the alloy composition is equal to zero, but the magnitude of the total sheet density of polarization charges is very large, about r=e $  1013 and Â1014 cm À2 , respectively.19 Furthermore, it was indicated10–12 that there exist in some polar HSs a specific polarization charge-related scattering, known as random PZ field This volume phenomenon arises from roughness-induced fluctuations in the bulk density of PZ polarization charges in the system The PZ field scattering occurs merely in systems under strain, so being vanishing in lattice-matched HSs For instance, the Inx Al1Àx N/GaN HS with x ¼ 0:18 is lattice-matched,19 so PZ field scattering is vanishing, but PR scattering is very strong because the sheet SP polarization charge density is large: rSP =e $ 1013 cmÀ2 19 This is also the case of all polar surface quantum wells An example is the ZnO surface quantum well because ZnO is an unstrained system limited by open surfaces between vacuum or air/ZnO, but ZnO is very highly polar with a large sheet density of SP polarization charges rSP =e ¼ 3:6  1013 cmÀ2 20–22 It is worth mentioning that PR and, hence, PSR scattering always takes place with any (PZ and/or SP) polarization charges bound on a rough interface of any HS, irrespective of whether the HS is made of binary or ternary materials and irrespective of whether it is strained or relaxed In other word, PR and, hence, PSR scattering is a process inherent in all actual polar HSs This is a result of a combination of two phenomena of different natures, viz., interface roughness and Coulomb interaction with interface polarization charges III ELECTRON MOBILITY IN POLAR HETEROSTRUCTURES A Low-temperature transport in polar HSs In what follows, we are concerned with samples at very low temperature Then within the relaxation time approximation, the electron mobility may be represented as1: l ¼ es=mà ; (8) where s is the transport lifetime and mà is the in-plane effective mass of the channel electron The electrons moving along the in-plane are scattered by various sources of disorder that are usually characterized by some random fields It is well known1 that scattering by a Gaussian random field is specified by its autocorrelation function (ACF) in wave vector space, hjUðqÞj2 i Here the angular brackets stand for the ensemble average UðqÞ is a 2D Fourier transform of the unscreened scattering potential weighted with the distribution of 2D subband carriers: ỵ1 dzjfzịj2 Uq; zị; (9) Uqị ¼ À1 where fðzÞ is the carrier envelop wave function As noticed in the preceding text, in polar HSs, one may > 1012 achieve 2DEG of high sheet carrier density (ns $ À2 cm ) Then we may adopt the linear transport theory as a good approximation, the multiple scattering effects being negligibly small.23,24 At zero temperature, the scattering rate (1=s) is expressed via the ACF for each disorder as follows:1,25–27 1 ¼ s 2phEF ð 2kF dq hjUðqÞj2 i : ð4kF2 À q2 Þ1=2 e2 ðqÞ q2 (10) Here q ẳ jqj ẳ 2kF sin#=2ị, with # as a scattering angle, EF ¼ h2 kF2 =2mà is the Fermi energy with kF as thepFermi ffiffiffiffiffiffiffiffiffi wave number fixed by the sheet carrier density: kF ¼ 2pns eðqÞ is the dielectric function for screening of the scattering potentials by 2DEG,1 including the many-body exchange effect in the in-plane.11,28 At low temperature, phonon scattering is negligibly weak.29 Therefore the electrons in a doped polar HS are, in general, subject to the following scattering sources: (i) impurity doping (ID), (ii) alloy disorder (AD), and (iii) polarization surface roughness (PSR) The overall transport lifetime is determined by the ones due to the partial scatterings according to the Matthiessen’s rule, 1 1 ẳ ỵ ỵ : stot sID sAD sPSR (11) At low temperature, the electrons are assumed to primarily occupy the lowest subband of the HS, and the scattering processes limiting its mobility occur mainly within this subband In the calculation of the disorder-limited mobility by means of Eq (10), the ACF hjUðqÞj2 i takes a key role We ought to determine the ACF for all scattering mechanisms of interest and thus specify the envelop wave function for this subband For actual single HSs, we adopt the realistic model with triangular potential well of a finite height.30 It was shown31,37,42 that for such a potential well, the lowest subband of a HS may be very well described by a modified Fang–Howard wave function proposed by Ando:31 & Aj1=2 expjz=2ị for z < 0; (12) fzị ẳ Bk1=2 kz þ cÞ expðÀkz=2Þ for z > 0; Here, A; B; c; k, and j are variational parameters to be determined k and j are half the wave numbers in the channel and barrier layers, respectively A; B, and c are dimensionless parameters, given in terms of k and j through the boundary conditions at the interface plane z ¼ and the normalization.11,12 As seen in the preceding text, the confining potential and, hence, the total energy of an electron in polar HSs, involve a term from polarization charges, Vr in Downloaded 07 Jun 2011 to 139.78.49.186 Redistribution subject to AIP license or copyright; see http://jap.aip.org/about/rights_and_permissions 113711-4 Quang, Tung, and Tien J Appl Phys 109, 113711 (2011) Eq (5) Thus the wave numbers k and j and the wave function become dependent on the sheet polarization charge density r because k and j are found by minimizing the total energy per electron, e.g., numerically as shown in detail in Ref 12 for uniform doping B Autocorrelation function for scatterings in polar HSs Impurity doping The ACFs for ID and AD scattering were derived from detail in Refs 11 and 12, taking all confining sources in polar HSs Thus in this paper, we list the main results for these ACFs and need to perform the subsequent calculation for the PSR one As pointed out,11,12,32–34 the correlation between ionized impurities can exert a strong influence on scattering by them The effect is due to Coulomb repulsion between the charged impurities in their diffusion during growth This leads to a more homogeneous distribution of impurities along the inplane, hence weakening their field and their lateral scattering For uniform doping, the ACF for correlated impurities was derived12 to be 2 2pe NI FID q=kị; (13) hjUID qịj2 i ẳ ea 2k3 where NI is a bulk density of impurities The relevant form factor is given by FID tị ẳ t2 t ẵd1 tị 12d2 tị 4d3 tị ỵ tc ị (14) ỵ 6d4 tị ỵ d5 tị d5 tị; where t ẳ q=k is the dimensionless momentum transfer The auxiliary functions dn tị (n ẳ 1, 2, 3, 4, 5) are defined in Ref 12 The ionic correlation effect is quantified by a dimensionless parameter: tc ¼ qc =k, where qc is the inverse 2D Debye screening length of the impurity gas at the freezing temperature for impurity diffusion T0, qc ¼ 2pe2 nI ; ea kB T0 (15) with nI as a sheet impurity density of the order of 2=3 nI % NI ,32 and kB is the Boltzmann constant Alloy disorder The alloy is assumed to be located in the barrier layer.2,18 Because the alloy disorder is a short (zero)-range interaction, this affects the tail of the z-direction carrier distribution in the barrier merely from ÀLa , with La as the distance from the layer, which is composed of both host and alloy atoms and which is nearest to the interface ˚ ).6,35,36 With the envelop wave function from ($ 3:3 A Eq (12), the ACF for alloy disorder scattering is given in terms of the barrier wave number j as follows12,31,37,38: hjUAD ðqÞj2 i ẳ x1 xịu2al X0 A4 j À2jLa e À eÀ2jLb : (16) Here, x is the alloy composition in the barrier, Lb is its thickness, ual is the alloy potential, and X0 is the volume occupied by one atom.38,39 The alloy potential is an adjustable parameter for fitting to the experiment40 and assumed37 to be close to the conduction band offset between the two binaries forming the alloy: ual $ DEc ð1Þ The atomic volume X0 is related to the volume of the alloy unit cell Xc For hexagonal wurtzite crystals, there are four per unit cell:41 So that pffiffiffi atoms X0 ¼ Xc =4, where Xc ẳ 3=2ịa xịcxị with a(x) and c(x) as the lattice constants of the alloy.18 Polarization surface roughness We now turn to PSR scattering due to the potential induced by both normal surface roughness, or, strictly speaking, barrier roughness (position fluctuations of the potential barrier), USR , and polarization roughness (position fluctuations of the polarization charges), UPR : UPSR ẳ USR ỵ UPR : (17) As well known,1 the weighted Fourier transform of the normal SR potential is supplied by USR qị ẳ FSR Dq ; (18) where Dq is, as before, the Fourier transform of interface roughness, and FSR is a SR form factor The factor may be presented in terms of a sum of the expectation values of the electric fields of all electrostatic confining sources, e.g., ionized impurities of bulk density NI, 2DEG of sheet density ns, and polarization charges of sheet density r This was derived for uniform doping in Ref 12 In the case of large-size uniform doping, with the use of Eqs (9) and (12) we have: FSR ¼ 4pe2 n ns NI ½A2 Lb À 2B2 Lc ðc ỵ 1ị ẵA4 ea o r B ẵcc ỵ 2ị ỵ ỵ À 2A2 Þ ; 2e (19) with Lb and Lc as the barrier and channel thicknesses, respectively It is to be noticed that SR scattering in polar HSs depends on the sheet density of polarization charges r That is interpreted as follows For r > 0, these charges can cause an attraction of electrons toward the potential barrier, thereby in case of infinite confinement raising the 2DEG distribution peak11 so enhancing SR scattering, and for r < 0, a repulsion of them far away therefrom, thereby weakening the scattering As for PR scattering, upon making use of Eqs (7), (9), and (12), it holds for the weighted Fourier transform of the PR potential: UPR qị ẳ FPR tịDq ; (20) where the PR form factor is given by # ( " ) 2per B2 2c c2 a FPR ðtÞ ẳ A2 ỵ ỵ : re t ỵ 1ị3 t ỵ 1ị2 t ỵ ea tỵa (21) Upon inserting Eqs (18) and (20) into Eq (17), we may present the weighted potential for PSR scattering in a simple form: Downloaded 07 Jun 2011 to 139.78.49.186 Redistribution subject to AIP license or copyright; see http://jap.aip.org/about/rights_and_permissions 113711-5 Quang, Tung, and Tien J Appl Phys 109, 113711 (2011) UPSR qịtị ẳ FPSR ðtÞDq ; (22) where the form factor is FPSR tị ẳ FSR ỵ FPR tị; (23) with FSR and FPR ðtÞ given by Eqs (19) and (21), respectively The ACF for the scattering in question is given by hjUPSR qịj2 i ẳ jFPSR tịj2 hjDq j2 i (24) We now discuss the angular distribution in PSR scattering As seen from Eqs (18), (20), and (22), the ACFs for unscreened roughness-related scatterings are composed of a form factor and a Fourier component of interface roughness From Eqs (19), (21), and (23), the former is generally a decreasing function of momentum transfer t (i.e., scattering angle #), viz., the form factor for SR scattering is independent of t, whereas those for PR and PSR scattering are mainly decreased with t The latter is a decreasing function of t.11,12 Thus the unscreened scattering is reduced with t According to Eq (10), the transport property is fixed also by 2DEG screening, which is a decreasing function of t.11,12 Thus owing to the screening, the scattering is enhanced with t As a result of combination of the quoted components of opposite variation with a rise of t, the ACFs for screened roughnessrelated scatterings, viz SR, PR, and PSR, exhibit some pronounced peak in their angular distribution This peak is found12 to be at smaller angles #, i.e., more forward scattering, for larger correlation lengths of the interface profile Thus with the use of the wave function from Eq (12), we may derive the ACF for PSR scattering in an analytic form The ACF is supplied by Eq (24) with the form factor by Eqs (19), (21), and (23) These reveal that PSR scattering depends on the sheet density of polarization charges and the roughness profile The latter is, as usual, is assumed to be Gaussian with some roughness amplitude and correlation length that are adjustable for fitting to experiment FIG (Color online) Ratio R between the ACFs for PSR and SR scattering in the [111]-growth zinc-blend InAs/GaAs HS vs dimensionless momentum transfer t=2tF ¼ sinð#=2Þ (or scattering angle #) for sheet electron densities ns ¼ 1011 ; 1012 ; 1013 cmÀ2 zero for the [001] and [110] growths.4 The ratio R between the ACFs for PSR and SR scattering in the [111] growth InAs/GaAs HS is evaluated following Eq (25), where the relevant form factors are given by Eqs (19) and (21) for infinite confinement.11,12 The result is displayed in Fig 1, where the ratio R is plotted versus the dimensionless momentum transfer t=2tF ẳ sin#=2ị (with tF ẳ kF =k and # as scattering angle) for various sheet carrier densities ns ¼ 1011 , 1012 , 1013 cmÀ2 In Fig 2, the ACF ratio R for forward scattering (t ¼ 0) in the preceding HS is plotted versus the growth direction (polar angle H) for various sheet carrier densities ns ¼ 1011 , 1012 , 1013 cm À2 The dependence of the sheet PZ charge density on the growth direction, rPZ ðHÞ, is inferred by IV NUMERICAL RESULTS AND CONCLUSIONS A Autocorrelation function ratio As mentioned in the preceding text, the transport limited by some disorder is quantified by its ACF Moreover, SR scattering is often one of the key mechanisms limiting transport in many polar HSs.11,12 Thus as a measure of the relative importance of PSR scattering, we introduce the ratio between the ACFs for PSR and SR scattering: Rẳ hjUPSR qịj2 i hjUSR qịj2 i ẳ FPR tị 1ỵ FSR 2 : (25) It is well known3,4 that the interface of many strained III-V zinc-blend HSs possesses a large sheet density of PZ polarization charges, rPZ The density is anisotropic, strongly dependent on the sample growth direction For the pseudomorphic InAs/GaAs HS made of a strained InAs on a relaxed GaAs layer, this shows a clear maximum with rPZ =e ¼  1012 cmÀ2 for the [111] growth and tends to FIG (Color online) Ratio R between the ACFs for PSR and SR scattering for the forward process (t ¼ 0) in the InAs/GaAs HS in Fig vs growth direction with a fixed azimuthal angle (U ¼ p=4) and a varying polar angle H, for sheet electron densities ns ¼ 1011 ; 1012 ; 1013 cmÀ2 The inset shows the sheet PZ polarization charge density as a function of the growth direction, rPZ ðHÞ Downloaded 07 Jun 2011 to 139.78.49.186 Redistribution subject to AIP license or copyright; see http://jap.aip.org/about/rights_and_permissions 113711-6 Quang, Tung, and Tien J Appl Phys 109, 113711 (2011) (iii) As clearly seen from Figs and 3, the ACF ratio is larger than unity The role of PSR scattering is increased with a decrease of the sheet carrier density For instance, for ns ¼ 1011 cmÀ2 , the ACF ratio can be so large as R $ Thus PSR scattering is considerably stronger than the normal SR one B Comparison with recent experiments FIG (Color online) Ratio R between the ACFs for PSR and SR scattering for the forward process (t ¼ 0) in the Ga(Al)-face [0001]-growth Alx Ga1Àx N/GaN HS vs sheet electron densities ns for various Al composition x ¼ 0:05, 1, 2, 4, means of Eq (5) from the vertical PZ field given in Ref and shown in the inset Next, we are concerned with III-V nitride HSs, where the interface may possess a huge sheet density of polarization charges of both kinds: r ẳ rPZ ỵ rSP This density depends strongly on the alloy composition, r ẳ rxị As an example, for the Ga(Al)-face [0001] growth AlxGa1Àx N/GaN HS, the total sheet density is varying with the Al composition as follows2,5,6: r=e ¼ 2:69, 5.43, 11.0, 22.8, and 35.6 (in units of 1012 cmÀ2 ) for x ¼ 0:05, 0.1, 0.2, 0.4, and 0.6, respectively The ACF ratio for forward scattering (t $ 0) in the polar HS is plotted in Fig versus the sheet electron density for the various Al compositions Therefore, we may conclude the following (i) (ii) Figure reveals that PSR scattering is more important for smaller scattering angles and, in particular, strongest for forward scattering (t ¼ 0) The ACF ratio as a function of the growth direction in III-V zinc-blend HSs has a broad maximum for the [111] growth The broadening is increased with a decrease of the sheet carrier density Next, we examine the effect from PSR scattering on lateral transport in wurtzite III-nitride HSs As is well known, there existed experimental data about the transport made in many polar HSs, e.g., of Si and SiGe43,44 as well as of IIInitrides,45–48 that one has not been able to understand so far, if starting from the traditional scattering mechanisms only In the literature, one reported the observed result with no theoretical analysis45 or one had to invoke a special scattering mechanism from ionized impurities that are assumed to be distributed at random on the (flat) interface with no clarification of their nature The sheet density of interface impurities (rII ) was high and extremely high for III-nitride HSs, up to rII $  1013 cmÀ2 46 It should be stressed that this concept was indicated43,44 to be very suspect even at a much lower interface impurity density, rII $ 1011 cmÀ2 Therefore we try to apply the preceding theory of PSR scattering to explain some recent low-temperature data about the lateral transport in III-nitride HSs We begin with samples of a fixed alloy composition As an example, the Al0:09 Ga0:91 N/GaN HS (x ¼ 0:09), reported in Ref 47 was intentionally undoped and its 2DEG mobility was measured at 10 K as a function of ns, shown in Fig 4(a) Therefore the overall mobility, l ¼ estot =mà , is calculated within a finite confinement by employing the Matthiessen rule, Eq (11), to include all possible scattering mechanisms: (i) uniform (background) doping (ID) with an impurity bulk density NI ¼  1016 cmÀ3 , (ii) alloy disorder (AD) in the barrier with an alloy potential ual ¼ 2:1 eV, and (iii) PSR scattering around the interface with a total sheet density of polarization charges r=e ¼ 4:9  1012 cmÀ2 The roughness profile is taken with a roughness amplitude D and a correlation length ˚ ) For nuK as follows: (D; Kị ẳ 5:6; 298ị (in units of A merical evaluation, we use the material parameters listed in FIG (Color online) Overall 2DEG mobility l in the Ga(Al)-face [1000]growth Al x Ga 1Àx N/GaN HS vs sheet electron density ns for a fixed Al composition The overall mobility calculated on the basis of SR or PSR scattering are marked by tot/SR dashed line or tot/PSR solid one, respectively, and the experimental mobility is marked by solid circles The result is plotted for different Al compositions: (a) x ¼ 0:09 with the 10 K measured mobility,47 and (b) x ¼ 0:2 with the 4.2 K measured mobility.45 Downloaded 07 Jun 2011 to 139.78.49.186 Redistribution subject to AIP license or copyright; see http://jap.aip.org/about/rights_and_permissions 113711-7 Quang, Tung, and Tien Refs 2, and 12 In Fig 4(a), the overall mobility is calculated on the basis of PSR or SR scattering and marked accordingly by tot/PSR solid line or tot/SR dashed line It is interesting to notice that the measured mobility in Ref 47 exhibits a pronounced peak lpeak % 4:4  104 cm2 =Vs at ns; peak % 1:9  1012 cmÀ2 Next, we are dealing with the intentionally undoped nitride HS with a larger Al composition x ¼ 0:2 (r=e ¼ 11:0  1012 cmÀ2 ) Its lateral mobility was measured at 4.2 K as a function of the 2DEG density, reported in Ref 45 with no interpretation This also exhibits a mobility peak with lpeak % 1:1  104 cm2/Vs at larger sheet density ns; peak % 7:2  1012 cmÀ2 The overall mobility is calculated, based on PSR scattering, with a background doping density NI ¼ 1016 cmÀ3 and a roughness profile (D; KÞ ˚ ), and displayed in Fig 4(b) ẳ 5:6; 168ị (A And last, we are concerned with lateral transport in the AlxGa1Àx N/GaN HS, reported in Ref 48 It is noticed that the 2DEG mobility was measured at $ K in case not only the sheet electron density ns but also the alloy composition x were varied Namely, the Al composition (in units of %) and the 2DEG density (in units of 1012 cm À2 ) were given as follows: ðx; ns ị ẳ 5; 1:38ị, (5, 1.48), (6, 1.6), (5, 2.35), (5, 2.4), (5, 2.74), (9, 3.4), (10, 3.9), and (12, 4.36) The doping is unintentional with a background bulk density NI ¼ 8:2  1016 cmÀ3 The roughness profile is with ˚ ) The calculated mobility is plotted (D; Kị ẳ 6:6; 115ị (A in Fig 5, where the K experimental data48 is also shown for comparison Therefore, we may conclude the following (i) Figures and reveal that the measured mobility ($ 104 cm2 /Vs) is found much smaller than that limited by background impurities in the HS or by bulk J Appl Phys 109, 113711 (2011) (ii) (iii) (iv) dipoles in the barrier ($ 106 cm2/Vs),18 thus both of them cannot be transport limiting mechanisms As clearly seen from Figs 4(a) and 4(b), alloy disorder and roughness-related scattering dominate the transport in the nitride HSs under study With a reduction of the sheet carrier density ns, the AD-limited mobility shows an increasing trend, while the SR- and PSR-limited mobilities a decreasing trend Because PSR scattering is remarkably stronger than SR scattering, the overall mobility in Al0:09 Ga0:91 N/ GaN HS, calculated with PSR scattering, exhibits a peak at ns % 1:9  1012 cmÀ2 as observed [Fig 4(a)], while the overall mobility based on SR scattering a monotonic decrease with a rise of ns The mobility peak detected in Al0:2 Ga0:8 N/GaN HS with larger Al composition is explained similarly With PSR scattering, we are able to well reproduce the data about the density dependence of 2DEG mobility in the HSs under study The agreement is quantitative for the HSs of a fixed Al composition,45,47 while qualitative for the HS of a varying composition.48 The deviation is found at large Al compositions (Fig 5) This might be connected with the dependence of the roughness profile (D; K) on the alloy composition as shown in Refs 49 and 50 A quantitative agreement may be achieved if another roughness profile is chosen for large Al compositions as illustrated in Fig In the end, we give a brief discussion of the transport-to-quantum lifetime ratio, Rt=q In accordance with the preceding facts that PSR scattering is dominant and focusing mainly on forward scattering, the calculated lifetime ratio may be rather large, which is in agreement with experiment As an illustrating example, this ratio was reported47 for the Al0:09 Ga0:91 N/GaN HS At a carrier density ns ¼ 1:99  1012 cm À2 , the measured ratio was Rexper t=q ¼ 17:64, while the calculation using a correla˚ gives Rtheor ¼ 11:6 and 17.9, tion length K ¼ 298 A t=q with a doping level of NI ¼  1016 and  1016 cmÀ3 , respectively V SUMMARY FIG (Color online) Overall 2DEG mobility l in the Ga(Al)-face [1000]growth AlxGa1Àx N/GaN HS vs sheet electron density ns for the Al composition x varying as given in the text The empty squares and empty circles refer to the K measured mobility48 and the overall mobility based on SR scatter˚ ) The filled and ing with a fixed roughness profile (D; Kị ẳ 6:6; 115ị (A half-filled circles refer to the overall mobility based on PSR scattering with ˚ ), but a the roughness profiles of the same roughness amplitude D ¼ 6:6 (A ˚ ) for all values of x correlation length with two choices: a) K ẳ 115ị (A ) for x % 0.05, and K ¼ 165, 215, 315 (A ˚) (filled circles), and b) K ¼ 115 (A for x ¼ 0:09, 0.10, 0.12, respectively (half-filled circles) The lines are a guide for the eyes In summary, we have shown that PSR scattering by any PZ and SP polarization charges bound on a rough interface is a phenomenon inherent in all actual polar HSs This is results from a combination of two phenomena of different nature, namely, interface roughness and Coulomb interaction with interface polarization charges We have developed a theory of PSR scattering in an analytically tractable form This process dominates lateral transport in almost all actual polar HSs So the interface polarization charges also take the same three-fold role as ionized impurities in doped systems ACKNOWLEDGMENTS This work was supported by the National Foundation for Science and Technology Development (NAFOSTED, Vietnam) Downloaded 07 Jun 2011 to 139.78.49.186 Redistribution subject to AIP license or copyright; see http://jap.aip.org/about/rights_and_permissions 113711-8 Quang, Tung, and Tien J Appl Phys 109, 113711 (2011) 27 28 T Ando, A B Fowler, and F Stern, Rev Mod Phys 54, 437 (1982) Polarization Effects in Semiconductors: From Ab Initio Theory to Device Applications, edited by C Wood and D Jena (Springer, New York, 2008) D L Smith, C Mailhiot, Rev Mod Phys 62, 173 (1990) M Ilg, K H Ploog, and A Trampert, Phys Rev B 50, 17111 (1994) O Ambacher, J Smart, J R Shealy, N G Weimann, K Chu, M Murphy, W J Schaff, L F Eastman, R Dimitrov, L Wittmer, M Stutzmann, W Rieger, and J Hilsenbeck, J App Phys 85, 3222 (1999) O Ambacher, B Foutz, J Smart, J R Shealy, N G Weimann, K Chu, M Murphy, A J Sierakowski, W J Schaff, L F Eastman, R Dimitrov, A Mitchell, and M Stutzmann, J App Phys 87, 344 (2000) A Tsukazaki, A Ohtomo, T Kita, Y Ohno, and M Kawasaki, Science 315, 1388 (2007) Y H Xie, R People, J C Bean, and K W Wecht, Appl Phys Lett 49, 283 (1986) H Morkoc¸, S Strite, G B Gao, M E Lin, B Sverdlov, and M Burns, J Appl Phys 76, 1363 (1994) 10 D N Quang, V N Tuoc, N H Tung, and T D Huan, Phys Rev Lett 89, 077601 (2002) 11 D N Quang, V N Tuoc, N H Tung, N V Minh, and P N Phong, Phys Rev B 72, 245303 (2005) 12 D N Quang, N H Tung, V N Tuoc, N V Minh, H A Huy, and D T Hien, Phys Rev B 74, 205312 (2006) 13 D N Quang, L Tuan, and N T Tien, Phys Rev B 77, 125326 (2008) 14 D N Quang, L Tuan, and N T Tien, J Appl Phys 107, 123709 (2010) 15 J D Jackson, Classical Electrodynamics, 3rd ed (Wiley, New York, 1999) 16 S Jin, M V Fischetti, and T.-W Tang, IEEE Trans Electron Devices 54, 2191 (2007) 17 A Gold, Phys Rev B 32, 4014 (1985) 18 D Jena, A C Gossard, and U K Mishra, J Appl Phys 88, 4734 (2000) 19 B E Foutz, O Ambacher, M J Murphy, V Tilak, and L F Eastman, Phys Stat Sol B 216, 415 (1999) 20 A Dal Corso, M Posternak, R Resta, and A Baldereschi, Phys Rev B 50, 10715 (1994) 21 F Bernardini, V Fiorentini, and D Vanderbilt, Phys Rev B 56, R10024 (1997) 22 M W Allen, P Miller, R J Reeves, and S M Durbin, Appl Phys Lett 90, 062104 (2007) 23 A Gold and W Goătze, J Phys C 14, 4049 (1981); Phys Rev B 33, 2495 (1986) 24 M J Kearney and A I Horrell, Semicond Sci Technol 14, 211 (1999) 25 F Stern and W E Howard, Phys Rev 163, 816 (1967) 26 S Das Sarma and F Stern, Phys Rev B 32, 8442 (1985) A Gold, Phys Rev B 35, 723 (1987); Phys Rev B 38, 10798 (1988) M Jonson, J Phys C 9, 3055 (1976) 29 L Hsu and W Walukievicz, Phys Rev B 56, 1520 (1997); Appl Phys Lett 73, 339 (1998) 30 M Morkoc¸, F Hamdani, and A Salvador, in Gallium Nitride (GaN) I, Semiconductors and Semimetals, edited by J I Pankove and T D Moustakas (Academic, San Diego, 1998), Vol 50, pp 193 31 T Ando, J Phys Soc Jpn 51, 3893 (1982); J Phys Soc Jpn 51, 3900 (1982) 32 E F Schubert, J E Cunningham, and N T Tang, Solid State Commun 63, 591 (1987) 33 E F Schubert, J M Kuo, R F Kopf, H S Luftman, L.C Hopkins, and N J Sauer, J Appl Phys 67, 1969 (1990) 34 A L Efros, F G Pikus, and G G Samsonidze, Phys Rev B 41, 8295 (1990) 35 N Maeda, T Nishida, N Kobayashi, and M Tomizawa, Appl Phys Lett 73, 1856 (1998) 36 T.-H Yu and K F Brennan, J Appl Phys 89, 3827 (2001) 37 G Bastard, Wave Mechanics Applied to Semiconductor Heterostructures (Les Editions de Physique, Paris, 1988) 38 M J Kearney and A I Horrell, Semicond Sci Technol 13, 174 (1998) 39 V Venktaraman, C W Liu, and J C Sturm, Appl Phys Lett 63, 2795 (1993) 40 D Jena, S Heikman, J S Speck, A Gossard, and U Mishra, Phys Rev B 67, 153306 (2003) 41 K Seeger, Semiconductor Physics: An Introduction (Springer-Verlag, Berlin, 1985) 42 Y Okuyama and N Tokuda, Phys Rev B 40, 9744 (1989) 43 F Schaăffler, Semicond Sci Technol 12, 1515 (1997) 44 R J P Lander, M J Kearney, A I Horrell, E H C Parker, P J Phillips, and T E Whall, Semicond Sci Technol 12, 1604 (1997) 45 R Gaska, M S Shur, A D Bykhovski, A O Orlov, and G L Snider, Appl Phys Lett 74, 287 (1999) 46 J Antoszewski, M Gracey, J M Dell, L Faraone, T A Fisher, G Parish, Y.-F Wu, and U K Mishra, J Appl Phys 87, 3900 (2000) 47 S Elhamri, A Saxter, W C Mitchel, C R Elsass, I P Smorchkova, B Heying, E Haus, P Fini, J P Ibbetson, S Keller, P M Pretroff, S P DenBaars, U K Mishra, and J S Speck, J Appl Phys 88, 6583 (2000) 48 S Syed, J M Manfra, Y J Wang, R J Molnar, and H L Stormer, Appl Phys Lett 84, 1507 (2004) 49 J W P Hsu, E A Fitzgeraldt, Y H Xie, P J Silverman, and M J Cardillo, Appl Phys Lett 61, 1293 (1992) 50 S Arulkumaran, T Egawa, H Ishikawa, and T Jimbo, J Vac Sci Technol B 21, 888 (2003) Downloaded 07 Jun 2011 to 139.78.49.186 Redistribution subject to AIP license or copyright; see http://jap.aip.org/about/rights_and_permissions View publication stats ... interaction with interface polarization charges We have developed a theory of PSR scattering in an analytically tractable form This process dominates lateral transport in almost all actual polar. .. mentioning that PR and, hence, PSR scattering always takes place with any (PZ and/or SP) polarization charges bound on a rough interface of any HS, irrespective of whether the HS is made of binary... PSR scattering in a polar HS with a derivation of an expression for the electrostatic potential induced by a sheet density r of both PZ and SP polarization charges bound on a rough interface