Magnetotransport in two distinct AlGaN/GaN HEMT structures grown by Molecular Beam Epitaxy (MBE) on Fe-doped templates is investigated using Shubnikov de-Haas Oscillations in the temperature range of 1.8–6 K and multicarrier fitting in the temperature range of 1.8–300 K. The temperature dependence of the two dimensional electron gas mobility is extracted from simultaneous multicarrier fitting of transverse and longitudinal resistivity as a function of magnetic field and the data is utilized to estimate contribution of interface roughness to the mobility and the corresponding transport lifetime. The quantum scattering time obtained from the analysis of Shubnikov de Haas Oscillations in transverse magnetoresistance along with the transport lifetime time were used to estimate interface roughness amplitude and lateral correlation length. The results indicate that the insertion of AlN over layer deposited prior to the growth of GaN base layer on Fe doped GaN templates for forming HEMT structures reduced the parallel conduction but resulted in an increase in interface roughness.

## I. INTRODUCTION

Over the last decade or so the AlGaN/GaN based hetero-structures have found wide ranging application in high power microwave devices.^{1,2} Some unique physical properties such as wide band gap tuneable between 3.4 to 6.2 eV, spontaneous polarization, high electron saturation velocities, low lattice mismatch, large band offset etc. render AlGaN/GaN based heterostructures particularly ideal for realizing high electron mobility transistors (HEMT) for high-frequency and high-power applications.^{1,2} The electron mobility and the carrier density of the two dimensional electron gas (2DEG) formed at the AlGaN/GaN heterostructures are generally considered critical parameters indicating their suitability for fabrication of high performance HEMT devices. However, two other parameters namely transport lifetime (*τ*_{t}) and quantum scattering times (*τ*_{q}) can elaborate further on to the predominant scattering mechanisms controlling the electrical characteristics. The *τ*_{t} is basically the relaxation time related with drift mobility (*eτ*_{t} /*m*^{*}) and the *τ*_{q} is the decay time of one particle excitation characterizing the quantum mechanical broadening of electron energy states (h/*τ*_{q}).^{3} The ratio *τ*_{t}/*τ*_{q} is generally considered to be a measure of whether large or small angle scattering is predominant in electronic transport indicating, for example, whether background impurities or interface roughness limit the electrical characteristics. But, Hsu et. al^{4} have theortically predicted that *τ*_{t}/*τ*_{q} ratio alone may not be sufficient for such conclusions and their individual magnitude also matters in this respect. Nevertheless, estimation and analysis *τ*_{t} and *τ*_{q} can provide crucial feedback for optimization of growth process for 2DEG structures.

The lateral carrier transport in AlGaN/GaN HEMT structures is greatly influenced by interface roughness close to the 2DEG.^{5} As reported extensively for III–V semiconductor heterostructures in general, the interface roughness can affect several device characteristics such as carrier mobility, leakage current and dielectric breakdown due to carrier scattering, electric field enhancement effects at the interface etc., which in turn can influence the performance and reliability of devices.^{6} Particularly for GaN/AlGaN heterostructures a large band discontinuity combined with a high effective mass in the 2DEG channel renders interface roughness much more important in controlling the 2DEG mobility. Additionally, the combination of interface roughness and piezoelectric effect can cause the charges at the interface to be distributed non-uniformly, which in turn can induce Anderson localization leading to low lying electron states.^{5,6} These interface states could be responsible for a variety of issues related with device degradation and reliability which have been comprehended by Brianna S. Eller *et al.*^{7} The effect of interface roughness becomes predominant in low temperature magnetotransport in 2DEG heterostructures and hence the same can be utilized to evaluate the interface roughness amplitude and correlation length.^{8}

Shubnikov de Haas (SdH) oscillations observed in low temperature magnetoresistance are an effective means of evaluating the quantum scattering time, electron effective mass (*m*^{*}) and 2DEG carrier density.^{9} However, the effects of parallel conduction due to background doping, surface/interface conduction as well as carriers belonging to different subbands in 2DEG structures can often complicate the magnetoresistance variation with magnetic field. In particular, the structures grown on Fe doped GaN template have been reported to have interfacial impurities^{10} which may also contribute to parallel conduction. However, multicarrier analysis using simultaneous fitting of the magnetic field dependence of transverse and longitudinal sheet resistances, *i.e. R*_{xy} and *R*_{xx} respectively, can separate out the contribution of different carriers in parallel conduction with a reasonable accuracy.^{11}

In this paper we present a comprehensive analysis of magneto-transport in two distinct GaN/AlGaN HEMT structures using multicarrier fitting and SdH Oscillations. The 2DEG characteristics, mainly *τ*_{t} and *τ*_{q}, so obtained are utilized to evaluate the interface roughness parameters.

## II. EXPERIMENTAL

Two distinct AlGaN/GaN HEMT structures grown by Molecular Beam Epitaxy (MBE) on high resistivity Fe-doped MOCVD grown GaN on sapphire templates have been used in this work. The HEMT structures, hereafter referred to as sample A and B, are schematically depicted in Fig. 1(a) and 1(b). Both the heterostructures had 24 nm AlGaN barrier layer and a 1nm AlN spacer layer. In Sample B, 10 nm of barrier layer was doped with Si to lower the contact resistance and also had a 2 nm GaN cap layer and a 1nm AlN layer over the GaN template. The thickness of base layer in sample A is 350 nm and that in sample B is 200 nm. The AlGaN supply layer of samples A and B had 35% and 32% of AlN respectively. Sample B also had a1nm AlN interfacial layer over GaN template.

The magneto-transport measurements (i.e. *R*_{xx} and *R*_{xy} as a function of magnetic field intensity *B*) were done in the temperature range of 1.8K to 300K and magnetic field up to 8T on 5 × 5 mm^{2} samples in van der Pauw geometry using a Cryogenics UK make cryogen free 8T magnet with a variable temperature insert.

## III. RESULTS AND DISCUSSIONS

Two sets of SdH oscillations in *R*_{xx} *vs. B* plots were observed in both the samples below 6K. Figure 2 shows a higher frequency oscillation superimposed on a lower frequency oscillation overriding the classical *B*^{2} kind of dependence due to parallel conduction, observed in sample A as an example.

We separated the two oscillatory components from the non oscillatory part by using successive Fast Fourier Transform (FFT) smoothing. This method provided a better resolution of the SdH data for both bands compared to another method of double differentiation.^{12} The double differentiation can only eliminate a simple *B*^{2} dependence which may not be strictly valid for cases where multiple carrier types contribute to the parallel conduction. Two distinct oscillatory components of SdH in magnetoresistance, attributed to the carriers in first and second subband according to the frequency of oscillations, are shown for samples A and B in Fig. 3(a)–3(d) for the temperature range of 1.8 to 6K.

The SdH oscillations in 2DEG are characterized by the following standard expression for the normalized magnetoresistance:^{9}

With |$\chi = \frac{{2\pi ^2 k_b Tm^* }}{{e\hbar B}}$|$\chi =2\pi 2kbTm*e\u210fB$ is a parameter dependent on temperature (*T*), magnetic field intensity (*B*) and the electron effective mass (*m*^{*}), |$\omega _c = \frac{{eB}}{{m^* }}$|$\omega c=eBm*$ is the cyclotron frequency, |$\varepsilon = \frac{{\pi \hbar ^2 n_{2DEG} }}{{m^* }}$|$\u025b=\pi \u210f2n2DEGm*$ is the Fermi Energy of the 2DEG for a particular subband, with *n*_{2DEG} being the 2DEG carrier density. The period of oscillations is governed by the cosine term in Eq. (1) and is constant as function of 1/*B*. The 2DEG concentration can hence be evaluated from the period of oscillations Δ(1/*B*) as

At any peak in the SdH oscillations, cosine term in Eq. (1) becomes unity and the equation reduces to the form

Where *A* is amplitude and *C* is a temperature independent term *i.e.* a constant for a particular *B*. For |$\chi \mathbin{\lower.3ex\hbox{\buildrel>\over{\smash{\scriptstyle\sim}\vphantom{_x}}}} 1.7$|$\chi \u22731.7$, ln sinh(χ) can be approximated to χ/2 and hence a plot of ln(*A/T*) *vs. T* for a particular *B* would be a straight line with slope being proportional to *m*^{*}. It is a usual practice to evaluate *m*^{*} from the slope of a linear ln(*A/T*) *vs. T* fit.^{9,12} However, for a tentative value of *m*^{*} ∼ 0.2 *m*_{e} for GaN, the value of χ is significantly less than 1.7 at low temperature and in high magnetic field range and hence the ln(*A/T*) vs. *T* is not strictly a straight line.^{13} We, therefore, utilized Eq. (2) as such for a non linear fitting of ln (*A*/*T*) vs. *T* data to estimate *m*^{*} as depicted in Fig. 4 as an example. In our analysis we observed that the *m*^{*} values obtained from linear fitting are consistently higher than those obtained from non-linear curve fit as also reported by Kurakin *et al.*^{13}

The quantum scattering time is contained in the exponential term in Eq. (1) and signifies the growth of the amplitude of oscillations with magnetic field. Once *m*^{*} is evaluated from Eq. (2), the parameter *χ* is known and the Eq. (1) can be rearranged as

Where *c* is a constant. A plot of ln (*A*B sinh χ) *vs.* 1/*B* traditionally known as Dingle plot is a straight line and its slope yields *τ*_{q}.^{9} Figure 5 depicts a Dingle plot corresponding to the curves shown in Fig. 3(a).

The magnetoresistance data, as mentioned earlier, indicated parallel conduction in both the samples making it essential to isolate/extract 2DEG mobility exclusively for estimating *τ*_{t}. Simultaneous multicarrier fitting of *R*_{xx}, *Rxy vs. B* data was performed to extract concentration and mobility values of carriers involved in parallel conduction in a manner similar to the approach described by Murthy *et al.*^{11} The multicarrier fitting revealed the presence of a high mobility and a low mobility electron in both the samples over the entire temperature range. The density of high mobility electrons remained fairly constant at low temperatures and equalled roughly with the sum of carrier densities for two subbands estimated from SdH analysis. The multi-carrier fitting results for both the samples are depicted in Fig. 6(a) and 6(b). Since the carriers in two subbands could not be distinctly resolved in multicarrier fitting it was concluded that they must be having closely matching mobility. We therefore attributed the same mobility to both 2DEG carriers and the *τ*_{t} was estimated using only the different *m*^{*} values obtained from SdH oscillations as described later on.

The 2DEG parameters extracted as above are presented in Table I. The higher 2DEG carrier density observed in sample A compared to the sample B can be attributed to a relatively higher Al concentration resulting in increased polarization charge.^{14} Interestingly the ratio of carriers in 1^{st} subband to those in 2^{nd} subband in both the cases has been found to be around 16.8, similar to a value of around 18 reported by Ikai Lo *et al.*^{15} in their Fe doped samples.

. | Sample A . | Sample B . | ||
---|---|---|---|---|

. | 1^{st} Subband
. | 2^{nd} Subband
. | 1^{st} Subband
. | 2^{nd} Subband
. |

n_{2DEG} (cm^{−2}) | 1.32 × 10^{13} | 7.85 × 10^{11} | 1.26 × 10^{13} | 7.50 × 10^{11} |

m^{*}/m_{e} | 0.216 | 0.209 | 0.235 | 0.207 |

τ_{q} (ps) | 0.158 | 0.398 | 0.068 | 0.295 |

τ_{t} (ps) | 0.413 | 0.399 | 0.677 | 0.600 |

τ_{t} /τ_{q} | 2.6 | ∼ 1 | ∼ 10 | ∼ 2 |

. | Sample A . | Sample B . | ||
---|---|---|---|---|

. | 1^{st} Subband
. | 2^{nd} Subband
. | 1^{st} Subband
. | 2^{nd} Subband
. |

n_{2DEG} (cm^{−2}) | 1.32 × 10^{13} | 7.85 × 10^{11} | 1.26 × 10^{13} | 7.50 × 10^{11} |

m^{*}/m_{e} | 0.216 | 0.209 | 0.235 | 0.207 |

τ_{q} (ps) | 0.158 | 0.398 | 0.068 | 0.295 |

τ_{t} (ps) | 0.413 | 0.399 | 0.677 | 0.600 |

τ_{t} /τ_{q} | 2.6 | ∼ 1 | ∼ 10 | ∼ 2 |

Let us now consider the *m*^{*} values from Table I. It is interesting to note that *m*^{*} values for the 2^{nd} subband (*i.e.* 0.209 *m*_{e} and 0.207 *m*_{e} for sample A and B respectively) are also closer to the theoretically estimated value of 0.2 *m*_{e} as well as within the range of experimentally deduced value of (0.20 ± 0.01)*m*_{e} for band edge effective mass in GaN.^{13} On the other hand the *m*^{*} values for the 1^{st} subband are consistently higher for both the samples and are in the same range as reported in earlier studies such as by Kurakin *et al.*^{13} In this context attention is drawn towards the location of electron wave function for the two sub bands in GaN/AlGaN HEMT Structures. The Electron wave functions for two samples as deducted from the 1D Schrodinger Poisson solver^{16} is depicted in Fig. 7. It is clearly indicated that for both the samples (a) the wave function peak for the 2^{nd} subband is located away from the interface while the same for the 1^{st} subband is located quite close to the interface and (b) since there is not much variation in Al concentration and since both the samples have an AlN spacer layer the wave function overlap is negligible in both the samples. In such a situation states related with interface roughness mainly appear responsible for the increase in the *m*^{*} value for the 1^{st} subband.^{13} We, therefore, predicted a comparatively higher interface roughness contributing to *m*^{*} enhancement in sample B. Similarly, the location of 2^{nd} subband electrons away from the interface explains the closeness of *m*^{*} values to the bulk values in that case.We now shift our attention to the results of multicarrier fitting shown in Fig. 6(a) and 6(b). The 2DEG carrier densities for both the samples remained invariant at low temperatures but changed mildly while approaching 300 K, in opposite manner though. The variation in 2DEG carrier densities at higher temperatures may be due to a combination of reduced conduction band offset and enhanced activation of background donors. The density of the second electron also remained almost constant at low temperatures but showed activated increase at higher temperatures. The donor activation energies of ∼12.4 meV and ∼14.5 meV in sample A and B match well with reported values for Si impurities in GaN.^{17} The sheet concentration of electrons conducting parallel to the 2DEG is about an order of magnitude less in sample B compared to sample A with sheet donor densities of ∼10^{14} cm^{−2} and ∼2 × 10^{13} cm^{−2} estimated by assuming a single donor.^{18} The saturation of sheet carrier densities at low temperature in both the samples indicates the presence of a degenerate electron gas. The results are consistent with earlier reports on electrical characteristics of Fe doped GaN templates where conduction due to the interfacial/surface impurities has been held responsible for this behaviour.^{18,19} In fact, the conduction due to the second electron mentioned above is again from a combination of bulk electrons observable only at high temperatures with sufficient donor ionization and a degenerate electron gas at the GaN template surface as modelled by Morkoc.^{18} The activated parallel conduction at high temperatures appears to be due to Si donors which are either incorporated in the GaN base layer from the template interface during growth or are being injected into the base layers from the donors trapped at interface. Parallel conduction from the bulk of GaN template is ruled out due to an order of magnitude difference in donor concentration of the two samples grown on similar templates. A detailed investigation of this aspect is beyond the scope of this work and is, therefore, not being taken up further. Nevertheless, the sheet carrier densities of ∼10^{13} cm^{−2} and ∼2 × 10^{12} cm^{−2} corresponding to degenerate conduction due to second electron in sample A and B respectively indicates two points (a) The growth recipe employed to grow these layers has effectively controlled the surface impurity concentration to a reasonable limit and (b) the AlN over layer on the template has been further effective in reducing the parallel conduction induced in GaN base layer. Also, the electron mobility in parallel conduction appears to be influenced at high temperature by electron-electron interaction and is lower in sample A having comparatively higher electron density. A detailed analysis of the mobility in parallel conducting channels was not attempted due to complications expected from degenerate electron gas. However, the low temperature mobility values are similar for both the samples indicating similar conduction mechanisms therein.

As a next step we simulated the 2DEG mobility in both samples including all important scattering mechanisms *i.e.* polar optical and acoustic phonon, piezoelectric deformation, dislocation and interface roughness.^{20} It is important to mention here that the 2DEG electron mobility at higher temperatures is significantly influenced by the polar optical phonon scattering. In cases where the difference in the 2^{nd} and 1^{st} subband electron states (*ɛ*_{2}-*ɛ*_{1}) is less than the optical phonon energy (h*ω*_{op}), a significant enhancement of the 2DEG mobility is expected due to the contribution of inter-subband transitions.^{21} However, in our samples the estimated^{22} h*ω*_{op} is of ∼ 91 meV which is significantly lower than the estimated values^{23} of *ɛ*_{2}-*ɛ*_{1} of ∼137 meV and ∼119 meV obtained for samples A and B respectively. Hence the scattering due to inter-subband transitions is neglected.

Figure 8 shows a theoretical fit to the experimental 2DEG mobility in sample B. It could be clearly concluded that the interface roughness is the most dominating scattering factor in both the samples. The interface roughness limited mobility (say *μ*_{IR}) values of 4500 cm^{2}V^{−1}s^{−1} and 6100 cm^{2}V^{−1}s^{−1} were hence deduced. The significance of these *μ*_{IR} values would become clear a bit later on.

We now draw attention to the *τ*_{t} and *τ*_{q} values presented in Table I. Since the 2DEG mobility in sample B is higher, it has a higher transport life time. We would also like to mention here that the higher *τ*_{t} values obtained for the 1^{st} subband compared to the 2^{nd} subband may only be an artefact of our earlier mentioned assumption of identical mobility for electrons belonging to both subbands. However, we ignore the marginal inaccuracy in this estimation due to its insignificant effect on the analysis presented later on. More importantly, the *τ*_{q} values for the 1^{st} subband differ significantly for the two samples but for the 2^{nd} subband these are closer to each other. This indicates again that the electrons in 2^{nd} subband are less influenced by the interface due to their farther location. The correlation of *τ*_{t} and *τ*_{q} with interface roughness can now be made as per the treatment given by Hyun-Ick Cho *et al.* as follows.^{8}

The *τ*_{t} and *τ*_{q} in 2DEGs can be related to the interface roughness amplitude (Δ) and correlation length (Λ) through the following expressions:^{8}

Here *θ* is the scattering angle, *ɛ*_{q} = (1 + *q*_{s}/*q*) is the dielectric function in the Thomas Fermi approximation, where|$,q_s = \frac{{{\rm m}^{\rm *} {\rm e}^2 }}{{{2{\rm \pi }\varepsilon_L }\varepsilon_o \hbar ^2 }}$|$,qs=m*e22\pi \u025bL\u025bo\u210f2$ and |$q = 2 k_F sin\frac{\theta }{2}$|$q=2kFsin\theta 2$ with *ɛ*_{L} and *ɛ*_{o} being the permittivity of semiconductor and of free space respectively and *k*_{f} being the Fermi vector. The random potential due to the interface roughness is expressed as

It is evident from the above expressions that (a) the (1-cos*θ*) term within the integral in Eq. (3) makes *τ*_{t} less sensitive to small angle scattering events and (b) the angular dependence of the random potential is predominantly influenced by the correlation length rather than by the amplitude of the roughness. We iteratively solved Eq. (3) and (4) to numerically estimate Δ and Λ from the known values of *τ*_{t} and *τ*_{q}. Since the above expressions exclusively deal with interface roughness parameters the *τ*_{t} values were estimated solely from the interface roughness mobility (*μ*_{IR}) and not from the total mobility value which include other contributions as well. The results are compiled in Table II. It is not possible to actually measure the interface roughness parameters, a vague correlation of the same can be done by the surface roughness of the heterostructures though.^{8} Atomic force microscopy and X-ray reflectivity studies of our samples were also attempted but only indicative results consistent with the values in Table II were obtained and hence are not presented here. The high resolution X-ray diffraction studies on these samples indicated similar dislocation densities in both the samples, hence the presence of a highly strained AlN back over-layer in sample B is attributed to be responsible for higher interface roughness close to 2DEG. The higher values of both Δ and Λ in sample B as compared to sample A is also consistent with this proposition. It is further interesting to note that although a high Λ value has resulted in higher mobility in sample B, higher Δ has resulted in a lower quantum scattering time. It would not be out of place to mention again that for the electrons in the 2^{nd} subband the random potential due to the interface roughness would be effectively screened by a high electron density in the 1^{st} subband and therefore even in sample B the *τ*_{q} is reasonably higher in the 2^{nd} subband.

. | . | . | Sample A . | . | Sample B . |
---|---|---|---|---|---|

1.8K μ_{2DEG} (cm^{2}V^{−1}s^{−1}) | 3359 | 5029 | |||

μ_{IR} (cm^{2}V^{−1}s^{−1}) | ∼4500 | ∼6100 | |||

τ_{q} (ps) | Estimated | ∼0.159 | ∼0.067 | ||

Experimental | 0.158 | 0.067 | |||

Roughness | Δ (nm) | 0.82 | 2.91 | ||

parameters | Λ (nm) | 1.36 | 5.50 |

. | . | . | Sample A . | . | Sample B . |
---|---|---|---|---|---|

1.8K μ_{2DEG} (cm^{2}V^{−1}s^{−1}) | 3359 | 5029 | |||

μ_{IR} (cm^{2}V^{−1}s^{−1}) | ∼4500 | ∼6100 | |||

τ_{q} (ps) | Estimated | ∼0.159 | ∼0.067 | ||

Experimental | 0.158 | 0.067 | |||

Roughness | Δ (nm) | 0.82 | 2.91 | ||

parameters | Λ (nm) | 1.36 | 5.50 |

We now consider the ratio *τ*_{t}/*τ*_{q} which respectively for samples A and B is (a) of ∼2.6 and of ∼10 for the 1st subband and (b) of ∼1 and ∼2 for the 2nd subband. A vast variation in *τ*_{t}/*τ*_{q} ratio is observed here in spite of the fact that the dominant scattering mechanism in both cases is interface roughness. This finding further supports the theoretical conclusion of Hsu *et al.*^{4} that the *τ*_{t}/*τ*_{q} ratio alone cannot be used to identify the dominant scattering mechanism in semiconductor heterostructures, particularly in GaN/AlGaN HEMT structures.

## IV. CONCLUSIONS

In conclusion a comprehensive analysis of magnetotransport measurements in GaN/AlGaN heterostructures involving Shubnikov de Haas Oscillations and simultaneous multicarrier fitting of transverse and longitudinal resistivity vs. magnetic field data has been utilized to isolate the effect of parallel conduction and 2DEG parameters. Effective contribution of interface roughness to the 2DEG transport mobility and quantum scattering time estimated from Shubnikov de Haas oscillations can be utilized to evaluate interface roughness amplitude and lateral correlation length as useful parameters for correlation with heterostructure growth conditions. The presence of AlN over-layer on GaN templates in MBE grown AlGaN/GaN HEMT structures has considerably reduced the parallel conduction by passivating the interfacial donors but appears to have resulted in increased interface roughness close to 2DEG thereby reducing the quantum scattering time. It is emphasized that the ratio of quantum scattering time to the transport lifetime cannot in general be used to establish the dominant scattering mechanism in two dimensional electron gas in GaN/AlGaN heterostructures.

## ACKNOWLEDGMENTS

The authors are thankful to Dr. Sushil Lamba for (a) providing the GaN HEMT samples grown by his group at SSPL for the current study and (b) many helpful discussions leading to the major conclusions in this work. One of the authors (MKM) is thankful to the Department of Science and Technology, Govt. of India for the INSPIRE fellowship.

## REFERENCES

*Eds.*