Dislocation scattering effect on two-dimensional electron gas transport in an GaN/AlGaN modulation-doped heterostructure Sibel Gökden Balikesir University, Department of Physics, Balikesir-Turkey *Corresponding Author: e-mail: sozalp@balikesir.edu.tr PACS 7..Dp, 7.4.Kp Abstract. We present the effect of all standard scattering mechanisms, including scattering by acoustic and optical phonons, remote and background impurities and dislocation, on two-dimensional electron gas (DEG) transport in AlGaN/GaN modulation doped-heterostructures. The most important scattering mechanisms limiting electron transport are identified. From the calculated dependence of mobility on temperature, it is clear that dislocation scattering dominates the low temperature mobility of twodimensional (D) electrons in GaN/AlGaN structures with a high electron density n s > cm - and the maximum DEG mobilities will be in the - 4 cm /V.s range for dislocation density of 4x cm - and carrier densities in the x x cm -. This theoretical calculations are fairly agree with the same mobility value obtained by the experimental for 4x cm - dislocation density. The results are compared to the transport to quantum lifetime ratios due to charge dislocations. We find that the ratio is larger for dislocation scattering than for impurity scattering. INTRODUCTION The wide band gap semiconductors have attracted great interest for possible insertion in future high power, high frequency electronics applications. [-]. GaN and its alloys with AlN are currently under investigation as candidate materials for high power, high temperature microelectronic devices [4-6]. Compared with their technological applications, partly due to poor material quality fundamental research in nitrides, particularly in electronic transport, appears to be in its infancy. For example the rapid advance in fabricating high quality sub-µm III-nitride modulation doped field effect transistors [7] calls for reliable and predictive device simulations. While published
transport studies of nitride compounds so far focused on bulk properties [8-], the prediction of efficient AlGaN/GaN heterostructure devices requires accurate modeling of quantum confinement effects of the carriers in the D channels. This is a prerequisite for understanding, predicting, and optimizing the effects of remote doping, interface roughness, or the temperature dependence [-] of the electron mobility. Thin films of the nitride semiconductors are most commonly grown in the hexagonal wurtzite structure on sapphire substrates. Due to the large lattice mismatch with the present substrate of choice, sapphire, state of the art AlGaN/GaN heterostructure have a DEG which has x 8 cm - x cm - line dislocations [,4] passing through it. The effect of dislocation scattering on transport in DEG has received renewed attention recently owing to its significance in the technologically important AlGaN/GaN MDH. In recent years the effect of dislocation scattering on the mobility of bulk GaN has been studied [5-7]. However, neither there is any existing adequate theory for dislocation scattering in a DEG nor efforts at explaining the observed mobilities in AlGaN/AlGaN DEG have considered the dislocations scattering [8-]. In this paper, we use simple analytical expressions for the determination of low field mobility in DEG confined in a triangular well, taking into consideration all major scattering mechanisms, i.e. dislocation scattering, deformation-potential acoustic phonon scattering, piezoelectric scattering and polar-optical phonon scattering, remote and background impurities, using D degenerate statistics of DEG. From the calculated dependence of mobility on temperature we show that the electron drift mobility in the channel at high temperature is ultimately limited by the interaction of electrons with optical phonons. At intermediate temperatures piezoelectric scattering and deformationpotential acoustic phonon scattering are the dominant scattering mechanisms. The dislocation scattering prevails at low temperatures and DEG mobility is affected strongly by a high density of dislocations. To investigate the validity of this model, we also calculate the ratios of the transport to quantum lifetimes at low temperatures. Thus, the common interpretation of the transport and quantum lifetime ratios by using analytical equation solutions gives us dominant scattering mechanisms at low temperature. Section II discusses the mobility calculations and the quantum and transport scattering rates due to charged dislocations in detail. The results of the calculations as
well as their relation to experimentally obtained mobility are the focus of Section III. Finally, we summarize our results in Section IV. II. SCATTERING MECHANISMS The scattering theories of the D carriers in the III-V heterojunctions system have been well-developed by several authors [-7]. The dominant scattering mechanisms for the D and bulk III-V compounds are now well established [8]. In our calculations of electron mobility in the DEG in GaN/AlGaN heterojunction we included dislocation scattering, impurity scattering by remote donors and due to interface charge, acoustic deformation potential scattering, piezoelectric scattering and polar-optical phonon scattering. We consider the degenerate statistics of DEG for the lowest subband occupation for the structure as shown in Fig.. The analytical expression for the above mentioned scattering mechanisms are briefly summarized below for convenience and material parameters used in the calculation are also listed in Table I. i) Ionized impurity scattering due to remote donors At low temperatures, the electron mobility limited by the ionized remote impurity scattering by remote parent donors in the barrier separated from the channel by a thin spacer layer. The value of the mobility limited by this scattering mechanism is given by [9,]. µ I 64 = π η ε S * e m ( π n ) s ( Z ) ( d + Z ) () where ε is the dielectric constant of the crystal, ηis the reduced Planck constant, m * is the electron effective mass, n s is the density of the D electron gas, Z is the width of the quantum well, d is the width of depletion layer. The screening constant S is a
function of n s and the lattice temperature T L which is given, in the non-degenerate case, by [] S e n s = εkt L and in the degenerate case by S * e m =. πεη () () Since impurity scattering dominates only at low temperatures, the degenerate limit should be used for S. d is given by the approximation as where N d is the donor density in the barrier. ns d =. N d ii) Ionized impurity scattering due to interface charge Since the DEG is formed on the GaN side of the (AlGa)N/GaN heterointerface, there is additional scattering due to background impurities, the density of which is in the order of 4 cm -, as well as due to the interface charge []. The corresponding mobility µ BI is given by [] k F I B * N BI ( β) 8 π η ε µ BI = (4) e m where N BI is the D impurity density in the potential well due to background impurities and/or interface charge and B π ( β) sin θdθ/ ( sinθ β) I = +. (5) where β =S. (6) / k F where k F is the wavevector on Fermi surface. iii) Acoustic deformation potential scattering When the temperature increases, the electron mobility depends on the acoustic-phonon scattering. The mobility limited by this scattering mechanism is given by []:
L B d A T k E m Z u ρ e µ * λ η = (7) where ρ is the density of the crystal, λ u is the longitudinal acoustic phonon velocity, Z and d E are the effective width of DEG and the deformation potential constant, respectively as shown in Fig. and B k is the Boltzman constant. iv) Piezoelectric scattering At intermediate temperatures, the electron mobility is related to piezoelectric scattering in DEG []. ( ) ( ) A A t A t 4 d F PE I I u u 9 eh Z E k µ γ γ + π = µ λ λ (8) h 4 is piezoelectric constant, t u is the velocity of transverse acoustic phonon. ( ) F n s k π = (9) ( ) t t A 4 I + π γ = γ () ( ) A 4 I + π γ = γ λ λ () T k k t u t B F η = γ () T k k u B F λ η λ = γ ()
v) Polar-optical-phonon scattering At high temperatures, the mobility of the carriers is limited by the polar-optical-phonon scattering that is comparable to acoustic deformation potential and piezoelectric scattering. The expression of mobility limited by polar-optical phonon is [5] 4 = e π ε p η µ B PO * ω m Z where ε p = ε ε ηω k T e s ε (4), ε and ε s are the dielectric constants of the semiconductor at high and low frequencies, respectively. η ω is the optical phonon energy. vi) Dislocation Scattering In this model, we consider dislocations with their axes perpendicular to the quantum well plane and the strong screening contribution in the highly degenerate DEG and the system is perfect DEG. The analytical expression for the dislocation scattering rate for a degenerate DEG is given by [,4], * N m e I = dis t. (5) τ 4 t η ε c 4π k F where N dis is the charge dislocation density, c is the lattice constant in the () direction of wurtzite GaN and I t is given as It du = ξ. (6) u ( + ξ u )
* Where ξ is the dimensionless parameters ζ=k F /q TF, q TF = a B is the D Thomas Fermi wave vector, a * B being the effective Bohr radius in the material. The DEG mobility for all scattering mechanisms is determined by eτ µ = * m (7) vii) Transport to quantum lifetime ratios One way to assess the relative importance of different scattering mechanisms is to measure the ratio of the transport and quantum relaxation times τ t and τ q [5-7]. The transport lifetime is typically calculated from Hall mobility measurements and the quantum lifetime can be determined SdH oscillations. The quantum lifetime τ q is a measure of the mean time a carrier remains in a particular state before being scattered to a different state. The transport time τ t is a measure of the amount of time a carrier remains moving in a particular direction. The quantum scattering rate due to charged dislocations is given by the following expression [8]. * N dis e m I q () 4 τ q = (8) dis q η ε c 4ππ F Where I q, an integral dependent on the dimensioless parameters ζ=k F /q TF and u=q/k F =sin(θ/) (θ is the angle of scattering) is given by du Iq = ξ. u u ( + ξ u ) (9) The transport scattering rate due to dislocation scattering is given in Eq. (5).
III. ELECTRON MOBILITIES IN AlGaN/GaN MDH s The sample investigated in this work was grown using MBE on tungsten-backed sapphire substrates. The thickness of Al.5 Ga.85 N barrier is 5Å and has a doping density of x 8 cm -. In order to see the relative importance of the various scattering mechanisms described above in determining the total mobility, we first present the temperature dependence of the DEG mobility and carrier density between at lattice temperatures T L = K and K in Figure.. Fig. shows that the mobility is µ =55.7 cm /V.s at K and µ =48.5 cm /V.s at K. A basic mobility characteristic of the modulation-doped heterostructure, which reveals the importance of the different mechanisms, is temperature dependence of the electron mobility. Fig. depicts the calculated low-field drift mobilities of the electrons in the channel as a function of temperature. The triangles represent the drift mobility calculated from the Matthiessen s rule for the sample with the doping concentration of x 8 cm -. This density yields a sheet charge density of x cm - in the channel. We find the corresponding drift mobility of D electrons in the GaN channel to be 58.7 cm /V.s at room temperature and to 8685.7 cm /V.s at K. It is clear that at room temperature, polar-optical-phonon scattering is the dominant scattering mechanism. On the other hand, at intermediate temperatures, electron mobility is limited by deformation potential acoustic and piezoelectric acoustic scatterings. At low temperatures, the ionized-impurity scattering is independent of temperatures, with the constant mobility value of µ cm I = /V.s. It is clear that the calculated mobilities at room temperature agree well with the experimental results. However, at low temperatures, the calculated values are higher than the experimental mobilities. The reason for this discrepancy may be associated with the dislocation scattering due to the large lattice mismatch with the present substrate of choice, sapphire and interface roughness scattering (IFR). We have reported the effect of the IFR on the electron mobility recently [9]. In this work, we discuss the scattering by dislocations for explaining observed mobility and we prove the validity of the dislocation model.
The comparison of the experimental results and the dislocation scattering can be clearly seen in Fig.4. Fig.4 shows the mobility dependence of the D carrier density. As D carrier density increases the mobility increases, on the contrary, as dislocation density increases the mobility decreases. i.e, the figure indicates that the strong mobility reduction has been seen at 4x 4 m - dislocation density and DEG mobilities will be in the - 4 cm /V.s range. Fig 4 proves that the theoretical mobility value for N dis 4 = 4x m and n s 7 = x m coincides with the experimental value of mobility ( µ.5m /V.s ) at low temperatures. The pointed dislocation density value is agreement with given values published in the literature [,4]. Consequently, the dislocation scattering is an effective mechanism which limits the mobility and the theory for this mechanism is suitable for modulation-doped structures. Fig.5 shows that the transport to quantum scattering lifetime ratios versus dimensionless parameters. As seen in Eq. (6) and Eq.(9), while I t can be evaluated exactly but I q diverges as u = sin( θ / ), or, in other words, when θ.this is case of scattering that is strongly peaked in the forward direction. The ratio of the quantum and transport scattering times, given by the ratio τ t /τ q thus acquires a singularity at small scattering angles. We define a scattering angle cutoff θ c and evaluate the ratio τ t /τ q by including dislocation scattering restricted to θ θ c only. We evaluate the ratio for the cutoffs θ c =π/, π/, π/, π/. As more small angle scattering contribution is included, the ratio becomes much larger than unity. On the other hand, as the scattering is restricted to be more large angle, the ratio approaches unity since it mimics isotropic scattering. The ratio is much larger than that for impurity scattering as θ c []. One can argue that as opposed to the case of impurity scattering for DEG where the modulation dopants can be placed at any distance from the DEG, a charged dislocation line always has a strong remote impurity nature due to geometry. This causes a strong preference for small angle scattering, causing the ratio τ t /τ q as θ c. As mentioned above, at low temperatures, the charged dislocation scattering is dominant scattering mechanism and is limiting the electron mobility.
IV. Conclusion In this paper we compare experimentally determined Hall mobility of a DEG formed AlGaN/GaN heterointerface with the theory. In the theory, all major scattering mechanisms, including dislocation scattering, acoustic deformation potential scattering, piezoelectric scattering, polar-optical phonon scattering and remote and background impurities have been taken into account. We show that the theory predicts a mobility much higher than the experimental value at low temperatures and the discrepancy can be explained invoking the dislocation scattering, which is well-known to be abundant in GaN/AlGaN, grown on sapphire substrates. Theoretical calculations prove the same mobility value obtained by the experimental for 4x cm - dislocation density. Further comparison between experiment and theoretical results shows that the theoretical calculations show the observed low temperature mobility enhancement in AlGaN/GaN MDH upon reduction of dislocation density. It is seen that the transport to quantum scattering ratio is larger for charged dislocation scattering than for impurity scattering since dislocation scattering is more anisotropic than impurity scattering.
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TABLE I. Values of GaN material constants used in the calculation. Electron effective mass m * =.m High frequency dielectric constant ε = 5.5 Static dielectric constant ε s = 9.7 LO-phonon energy η ω = 9meV The width of the quantum well Longitudinal acoustic phonon velocity Transverse acoustic phonon velocity Density of the crystal Deformation potential Dielectric constant of the crystal Donor density Z u u = λ 65A = 6.56x t =.68x m/s m/s ρ = 6.5x kg/m E d = 8.eV ε = 8.58x N d = x 4 F.m m Density of the DEG n s = x 7 m The width of the depletion layer d =.x Impurity density N BI = x m - Piezoelectric constant h 4 =.75 C/m Electron wavevector k=7.x 8 m - The lattice constant in the () direction of Wurtzite GaN c =5.85 Å 8 The D Thomas Fermi vawe vector q = 8.68x m The effective Bohr radius α *. Å TF B = 8 m
Figure Captions Fig.. Energy and band diagram of a modulation-doped heterojunction. d is the width of the depletion layer, Z is the average distance of the electronic wave function from the heterointerface corresponding to the lowest subband. Fig.. Two-dimensional electron density (open circles) and Hall mobility (filled circles) versus temperature.. Fig.. The calculated D electron mobility versus temperature for GaN/AlGaN. triangles is calculated from Matthiessen s rule. LO: Optic Phonon, AC: Acoustic Phonon, DP: Deformation Potential, PE: Piezoelectric, IMP: Remote and background impurities. Fig. 4. The calculated dislocation scattering limited DEG mobility. Mobilities for three different dislocation densities have been shown. Fig.5. Plot showing τ t /τ q for scattering from charged dislocations for a DEG. The different curves are for different small angle cutoffs θ c for scattering.
Fig.
4 4 T (K) Fig.
5 4 LO AC=DP+PE IMP EXP TOTAL T(K) Fig.
Fig.4
4 π/ π/ π/ θ c =π/.5.5.5.5 4 ζ =k F /q TF Fig.5