Energy Relaxation Rates in AlInN/AlN/GaN Heterostructures

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Journal of Electronic Materials ISSN 0361-5235Volume 41Number 9 Journal of Elec Materi (2012)41:2350-2361DOI 10.1007/s11664-012-2158-7

Energy Relaxation Rates in AlInN/AlN/GaN Heterostructures

E. Tiras, S. Ardali, E. Arslan & E. Ozbay

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Energy Relaxation Rates in AlInN/AlN/GaN Heterostructures

E. TIRAS,1,3 S. ARDALI,1 E. ARSLAN,2 and E. OZBAY2

1.—Department of Physics, Faculty of Science, Anadolu University, Yunus Emre Campus, 26470Eskisehir, Turkey. 2.—Nanotechnology Research Center, Department of Physics, and Departmentof Electrical and Electronics Engineering, Bilkent University, 06800 Ankara, Turkey. 3.—e-mail:[email protected]

The two-dimensional (2D) electron energy relaxation in Al0.83In0.17N/AlN/GaNheterostructures has been investigated experimentally. Shubnikov–de Haas(SdH) effect measurements were employed in the investigations. The electrontemperature (Te) of hot electrons was obtained from the lattice temperature(TL) and the applied electric field dependencies of the amplitude of SdHoscillations. The experimental results for the electron temperature depen-dence of power loss are also compared with current theoretical models forpower loss in 2D semiconductors. The power loss from the electrons was foundto be proportional to (Te

3 � TL3) for electron temperatures in the range

1.8 K< Te < 14 K, indicating that the energy relaxation of electrons is due toacoustic phonon emission via unscreened piezoelectric interaction. The effec-tive mass and quantum lifetime of the 2D electrons have been determinedfrom the temperature and magnetic field dependencies of the amplitude ofSdH oscillations, respectively. The values obtained for quantum lifetimesuggest that remote ionized impurity scattering is the dominant scatteringmechanism in Al0.83In0.17N/AlN/GaN heterostructures.

Key words: GaN heterostructure, electron energy relaxation, power loss,phonon emission, Shubnikov–de Haas, Hall mobility

INTRODUCTION

Group III-nitride materials are very suitable forapplications in high-power, high-frequency, andhigh-temperature electronics.1 Knowledge on fun-damental electron transport properties, such as theeffective mass of two-dimensional (2D) electronsand relaxation times, is important for explorationand optimization of this material system for deviceapplications. Device performance under high elec-tric field is also important for these systems. At highelectric field, the electrons equilibrate at muchhigher temperature than the lattice temperature.Therefore, determination of the temperature ofthose hot electrons is of technological and funda-mental importance.

The energy relaxation of hot carriers in semicon-ductors via electron–phonon interaction has beeninvestigated extensively, both experimentally and

theoretically, in bulk and two-dimensional 2Dstructures (for review see Refs. 2–4). The determi-nation of the temperature of electrons, under elec-tric field heating conditions in the steady state,provides useful information about the electron–phonon interactions involved in the energy relaxa-tion process. Since, at temperatures below approxi-mately 30 K to 40 K, the population of opticalphonons is negligibly small, acoustic phonon scat-tering provides the only inelastic scatteringmechanism.4–7

There are four experimental techniques that havebeen widely employed, successfully, in investigationsof electron relaxation. First, in heavily modulation-doped structures where a highly degenerate electrongas exists, the variation of the amplitude of quantumoscillations, such as in the Shubnikov–de Haas (SdH)effect, with the applied field and lattice temperaturecan be used in the determination of the electron tem-perature–power loss characteristics.3,8,9 Second, in amaterial where the momentum relaxation is domi-nated by ionized impurity, remote impurity, interface

(Received February 6, 2012; accepted May 25, 2012;published online June 27, 2012)

Journal of ELECTRONIC MATERIALS, Vol. 41, No. 9, 2012

DOI: 10.1007/s11664-012-2158-7� 2012 TMS

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roughness, or optical phonon scattering ,8–11 electrontemperatures can be determined as a function of theapplied electric field by a simple comparison of theelectric field-dependent and lattice temperature-dependent mobility curves.8–10,12,13 Third, using thenoise technique, the electron temperature can beestimatedbymeasuring the electromagnetic radiationresults of fluctuations in the electron velocities underhigh electric field.14,15 Fourth, with the pump–probeRaman spectroscopy technique, the energy relaxationtime can be directly determined from the decay of theanti-Stokes line intensity.16

Electron energy relaxation rates in GaN-basedsamples have been investigated using the differenttechniques described above.12–20 In GaN-basedheterostructures, a two-dimensional electron gas(2DEG) is formed at the GaN side of the interfacebetween the barrier and GaN layers. Insertion of athin AlN spacer layer between the barrier and GaNlayers helps to increase electron mobility.21–23

In-depth understanding of the fundamental opti-cal and electronic properties is yet to be establishedfor the design and development of functional devices.Determination of the temperature of electrons, un-der electric field heating conditions in the steadystate, provides useful information about the elec-tron–phonon interactions involved in the energyrelaxation process.2,5 Furthermore, electron–pho-non scattering processes determine the high-fieldtransport phenomena in semiconductors and thusform the basis for many ultrafast electronic andoptoelectronic devices. The field of hot carriers insemiconductors thus provides a link between fun-damental semiconductor physics and high-speeddevices.24 Despite the fact that the energy relaxationtime is a scientifically, technologically, and funda-mentally important parameter for designing opto-electronic devices, it is not yet well known. In thiswork, the temperature of hot electrons (Te) of thesample and the corresponding power loss (P) havebeen determined as a function of the applied electricfield using the SdH effect method in Al0.83In0.17

N/AlN/GaN heterostructures. The experimentalresults determined from SdH measurements arealso compared with a two-dimensional model in theacoustic phonon regime. The results are discussed inthe framework of current theoretical modelsconcerning carrier energy loss rates in dilutesemiconductors.

THEORETICAL BACKGROUND

Magnetotransport measurements have beenextensively used in investigations of the electronictransport properties of 2D structures at low temper-ature. SdH oscillations in the magnetoresistanceprovide an accurate and sensitive technique that hasbeen employed successfully in investigations of elec-tron energy relaxation in the acoustic phononregime.3,25,26 In heavily modulation-doped struc-tures, where a highly degenerate electron gas exists,

variations of the amplitude of the SdH oscillationswith applied electric field and lattice temperature canbe used in the determination of the power loss–elec-tron temperature characteristics. The sample used inthe present study is highly degenerate, so that thereduced Fermi energy g = (EF � E1)/kBT � 1(Table I), even at electron temperatures of approxi-mately 30 K, which is well above the range of tem-peratures considered here. Therefore, we employedthe SdH oscillations technique in our investigations.The method is based on the assumption that ionizedimpurity scattering, alloy scattering, and interfaceroughness scattering, which determine the low-tem-perature transport mobility of electrons, are elastic innature. Consequently, the energy that is gained byelectrons in an applied electric field is dissipated viaemission of acoustic phonons.2,5–7,23,27–32

The SdH oscillations in the magnetoresistance ofa 2D electron gas of single-subband occupancy arewell described by the analytical function33,34

Dqxx

q0

/ DðvÞ exp�pxcsq

� �cos

2pðEF � E1Þ�hxc

� p

� �; (1)

where Dqxx; q0;EF;E1;xcð¼ eB=m�Þ; sq; and �h are theoscillatory magnetoresistivity, zero-magnetic-fieldresistivity, Fermi energy, first subband energy,cyclotron frequency, quantum lifetime, and Planck’sconstant, respectively. The exponential term,expð�p=xcsq), describes the damping due to the col-lision broadening of the Landau levels. The temper-ature dependence of the envelope function of the SdHoscillations is totally contained in the term

DðvÞ ¼ vsinh v

(2)

with

v ¼ 2p2kBT

�hxc;

Table I. Electronic transport properties of the AlInN/AlN/GaN heterojunction determined at 1.8 K

Parameter Value

2D carrier density, N2D (1016 m�2) 7.25Sheet carrier density, NH (1016 m�2) 11.88Effective mass, m* (m0) 0.188Fermi energy, EF � E1 (meV) 92.26Hall mobility, lH (cm2 V�1 s�1) 6858Transport mobility, lt (cm2 V�1 s�1) 7820Parallel channel mobility, lB (cm2 V�1 s�1) 3800Transport lifetime, str (10�12 s) 0.836Quantum lifetime, sq (10�12 s) 0.264Lifetime ratio, str/sq 3.2c 3.55A (eV s�1 K�c) 2.9

Energy Relaxation Rates in AlInN/AlN/GaN Heterostructures 2351


where kB is the Boltzmann constant. The 2D carrierdensity (N2D) can be calculated25 using the argu-ments of Eq. (1)

4 1

B

� �¼ e

p�hN2D¼ e�h

m�ðEF � E1Þ: (3)

The thermal damping of the amplitude of the SdHoscillations is, therefore, determined by the tem-perature, magnetic field, and effective mass via

AðT;BnÞAðT0;BnÞ

¼ T sinhð2p2kBT0m�=�heBnÞT0 sinhð2p2kBTm�=�heBnÞ

; (4)

where A(T, Bn) and A(T0, Bn) are the amplitudes ofthe oscillation peaks observed at magnetic field Bn

and temperatures T and T0. In the derivation ofEq. (4) from Eq. (1) it has been assumed that thequantum lifetime (sq) is independent of both thetemperature and the magnetic field. The quantumlifetime can be determined from the magnetic fielddependence of the amplitude of the SdH oscillations(i.e., Dingle plots) at constant temperature providedthat the electron effective mass is known25,26,35

lnAðT;BnÞ � B�1=2

n : sinhðvÞv

" #¼ C� pm�

esq

1

Bn; (5)

where C is a constant.At low temperatures, the contribution to the

energy relaxation rates by elastic scattering mech-anisms, such as ionized impurity scattering, alloydisorder scattering, and interface roughness scat-tering, can be neglected.2 Therefore, inelastic scat-tering mechanisms should be considered in order toexplain the rise of temperature of the 2D electrongas where the applied electric field causes theheating of electrons. Typically, at temperaturesbelow 30 K, longitudinal optical phonon scatteringbecomes negligible and the main source of energyrelaxation is acoustic phonon scattering. Scatteringfrom acoustic phonons includes two independentprocesses: deformation potential (nonpolar acoustic)scattering and piezoelectric (polar acoustic) scatter-ing. At low temperatures, the carrier distribution isoften degenerate, and Pauli exclusion is importantin limiting the scattering that is allowed.2,5–7,28–30 Inthe 2D calculations, the scattering by the absorptionof acoustic phonons was neglected and only sponta-neous emission was considered to be important, theinfinite-well approximation was used in the extremequantum limit, and the phonons were assumed to bebulk phonons.2,3

The power loss from a degenerate electron gasdue to scattering by acoustic phonons has beencalculated in two temperature regimes2,6,7,29: (i) thelow-temperature regime, where the electron tem-perature Te > Te

c, and (ii) the high-temperatureregime, where Te � Te

c, hence the critical electrontemperature is given29 by

Tce ¼

8m�V2SðEF � E1Þ

� �1=2kB

; (6)

where VS is the sound velocity. The regime betweenthese two temperature limits is called the interme-diate regime.2,3

In the low-temperature (Bloch–Gruneisen)regime ð�hxq=kBTe � 1Þ) the phonon distribution isgiven3 by

nðxqÞ ¼1

exp ð�hxq=kBTLÞ � 1ffi exp � �hxq

kBTL

� �; (7)

where �hxq is the acoustic phonon energy at wave-vector q. At low temperatures, the Fermi gas has asharp boundary curve, and consequently momentumchanges that involve the emission of an acousticphonon of energy much greater than kBTe are hin-dered greatly by Pauli exclusion, and hence onlysmall-angle scattering is allowed at very low tem-peratures.28,36 Then, in the case under discussion,low-angle scattering should occur in the first sub-band, which is characterized by the dependenciesPnp � (Te

5 � TL5) for deformation potential scattering

and Pnp � (Te3 � TL

3) for piezoelectric scattering.28,29

Therefore, the total energy loss rate of a 2D electrongas, P = Pnp + Pp, in the low-temperature regime,can be represented2,3 by

P¼Cnp ðkBTeÞ5�ðkBTLÞ5h i

þCp ðkBTeÞ3�ðkBTLÞ3h i

;

(8)

where

Cnp ¼6N2m�2Lz

p3q�h7V4SN2D

(9)

and

Cp ¼e2K2

avm�2

2p2�s�h5kFN2D

; (10)

are the magnitudes of the deformation potential andpiezoelectric interactions, respectively. Here, N isthe acoustic deformation potential, q is the massdensity, �s is the static permittivity, and kF ¼2pN2D½ �1=2 is the Fermi wavevector of 2D electronsin which N2D is the 2D carrier density. The averageelectromechanical coupling constant Kav

2 for cubiccrystal is given3 by

K2av ¼

e214

�s

12

35CLþ 16

35CT

� �: (11)

Here, e14 is the piezoelectric stress constant, and CL

and CT are the average longitudinal and transverseelastic constants, given37 in terms of the compo-nents of the elastic stiffness constants Cij by

CL ¼ C11 þ2

5C12 þ C44 � C11ð Þ (12)

Tiras, Ardali, Arslan, and Ozbay2352


and

CT ¼ C44 �1

5C12 þ 2C44 � C11ð Þ: (13)

The screening of the electron–phonon interaction inthe 2D case, which is not included in the abovecalculations, is predicted to increase the exponent ofthe kBTe and kBTL terms in Eq. (8) by two.2,5

We assumed that the effective well width (Lz) ofthe potential well at the Al0.83In0.17N/AlN interfaceapproximately equals the average distance of theelectrons from the interface derived for onlythe lowest subband occupied structures using theFang–Howard variational wave function34

Lz � z0 ¼9�s�h

2

4m�e2ðNdepl þ 11N2D=32Þ

" #1=3

; (14)

where Ndepl ¼ ð2�sVbðND �NAÞ=eÞ1=2 is the depletionlayer charges per unit area, Vb is the conduction-band energy offset, ND is the donor concentration,and NA is the acceptor concentration.

The variation of the power loss per electron withelectron temperature has been often approximatedby the relationship

P ¼ A Tce � Tc

L0

� ; (15)

where TL0 is the lowest lattice temperature and A isa proportionality constant that depends on theelastic moduli of the matrix, the coupling constants,and the 2D carrier density. Theoretical calculationsof the acoustic phonon-assisted energy loss rates ofhot electrons in a 2D electron gas of single subbandoccupancy predict c = 1 at high temperatures (whenMaxwell–Boltzmann statistics is applicable andequipartition is assumed) and c = 3 (unscreenedpiezoelectric scattering), c = 5 (unscreened defor-mation potential and heavily screened piezoelectricscatterings), and c = 7 (heavily screened deforma-tion potential scattering) at low temperatures (see,for instance, Refs. 2,28–30,38).

EXPERIMENTAL PROCEDURES

Al1�xInxN/AlN/GaN (x = 0.17) heterostructureswere grown on double-polished 2-inch-diametersapphire (Al2O3) substrates in a low-pressure met-alorganic chemical vapor deposition (MOCVD)reactor (Aixtron 200/4 HT-S) by using trimethyl-gallium (TMGa), trimethylaluminum (TMAl), andammonia as Ga, Al, and N precursors, respectively.Prior to the epitaxial growth, Al2O3 substrate wasannealed at 1100�C for 10 min in order to removesurface contamination. The buffer structures con-sisted of a 15-nm-thick, low-temperature (770�C)AlN nucleation layer, and high-temperature(1120�C) 270-nm AlN templates. A 1.16-lm, nomi-nally undoped GaN layer was grown on an AlNtemplate layer at 1060�C, followed by a 1.5-nm-thick high-temperature (1075�C) AlN spike layer.

The AlN barrier layer was used to reduce the alloydisorder scattering by minimizing the wave functionpenetration from the two-dimensional electron gas(2DEG) channel into the AlInN layer. After deposi-tion of these layers, we used 1-nm AlN (1075�C) and3-nm GaN (1075�C) between the Al0.83In0.17N bar-rier layer and the AlN spike layer. The thickness ofthe Al0.83In0.17N barrier layer was 13 nm, and itwas grown at 830�C. Finally, a 2-nm-thick GaN caplayer growth was carried out at temperature of830�C. After the growth, these parameters weremeasured for each wafer, using standard charac-terization techniques, such as transmission electronmicroscopy (TEM), capacitance–voltage profiling,and high-resolution x-ray analysis.

Measurements of longitudinal resistance alongthe direction of applied current (Rxx) were carriedout as functions of: (i) the applied electric field F atfixed lattice temperature TL0, and (ii) lattice tem-perature TL at a fixed electric field F0 that was lowenough to ensure ohmic conditions and hence toavoid carrier heating. In the experiments, a con-ventional direct-current (DC) technique in combi-nation with a constant-current source (Keithley2400) and a nanovoltmeter (Keithley 2182A) in acryogen-free superconducting magnet system(model J2414; Cryogenics Ltd.) were used. Thecurrent (I) flow was in the plane of the electron gas.Steady magnetic fields up to 11 T were appliedperpendicular to the plane of the samples and,therefore, to the plane of the 2D electron gas. Allmeasurements were taken in the dark. To check the2D nature of the electron gas giving rise to thequantum oscillations in magnetoresistance, mea-surements were also performed as a function of theangle h between the normal to the plane of the 2Delectron gas and the applied magnetic field. It wasfound that the peak position shifted with a factor ofcos h and the oscillations disappeared at h = 90�.This observation is a characteristic of a 2D electrongas.25

For the classical low-magnetic-field temperature-dependent Hall-effect measurements, Rxx and theHall resistance (Rxy) were measured as a function oftemperature from 1.8 K to 275 K. A static magneticfield (B = 1 T) was applied to the sample perpen-dicular to the current plane. The Hall mobility (lH)and the sheet carrier density (NH) were obtainedusing the following equations:

Rxy ¼B

NHe; (16)

lH ¼L

NHeRxxb; (17)

where b (=0.6 mm) and L (=1 mm) are the widthand length of the Hall bar. The applied electric fieldwas also obtained using the longitudinal resistancemeasured at B = 0 T ðRxxðB ¼ 0Þ ¼ 118XÞ in thefollowing equation:



F ¼ RxxðB ¼ 0ÞIL

: (18)

In the applied electric field-dependent magnetore-sistance measurements, current was applied alongthe length of the sample in the range of I = 100 lAto 2000 lA.

The Raman spectra were obtained at room tem-perature using a Bruker Optics FT-RamanScope III system. As an excitation source, a wave-length of 785 nm (1.58 eV) was applied in the sam-ple growth direction (c-axis).

RESULTS AND DISCUSSION

The temperature dependence of the sheet carrierdensity and Hall mobility in the Al0.83In0.17N/AlN/GaN heterostructure is plotted in Fig. 1. At lowtemperatures, the sheet carrier density remainspractically constant up to temperature of 30 K. Athigher temperatures, the sheet carrier densityincreases monotonically with increasing tempera-ture, possibly due to thermally generated carrierslocated outside the channel. We note that a decreasein the sheet carrier density has been observed in thetemperature range from 30 K to 90 K (Fig. 1). Asimilar behavior of carrier density with tempera-ture, although less pronounced, was reported pre-viously for modulation-doped GaAs/Ga1-xAlxAsheterojunctions.39�41

The Hall mobility of electrons in the Al0.83In0.17N/AlN/GaN heterostructure increases monotonicallywith decreasing temperature from room tempera-ture, begins to level off at about 100 K, and satu-rates at about 30 K (Fig. 1). This behavior reflectsthe 2D character of the electrons in the channel.25

In the temperature range below 30 K, the mobilitiesmeasured for the sample are essentially indepen-dent of temperature. A similar behavior for thevariation of Hall mobility with temperature wasreported14 for a lattice-matched AlInN/AlN/GaNsample with a 1-nm spacer layer that helps to

reduce remote alloy scattering and achieve highelectron mobility (see, for instance, Refs. 21–23,42).

Figure 2 shows typical examples of the magneto-resistance Rxx(B) measured at different tempera-tures and applied electric fields for Al0.83In0.17N/AlN/GaN heterostructures. SdH oscillations are clearlyvisible over the magnetic field range between B = 7 Tand 11 T. No higher harmonics are apparent in theoscillations. It is also evident that the oscillatoryeffect is superimposed on a monotonically increasingcomponent, which occurs as a result of positivemagnetoresistance in the barriers.3 This may affectthe accuracy of the determination of the oscillationamplitude, particularly at elevated temperatures.Stradling and Wood43 adopted the following methodto detect the oscillatory component of the magneto-resistance by removing the monotonic component(Rb). Therefore, we used the negative second deriv-ative of the raw magnetoresistance data with respectto the magnetic field, i.e. (�¶2Rxx/¶B2).3,25,26,43,44 TheSdH oscillations have also been obtained by sub-tracting the background magnetoresistance (in theform of a polynomial of second degree) from the rawexperimental data (DR ¼ Rxx � Rb).45 The valuesobtained for effective mass and quantum lifetimefrom the temperature and magnetic field dependenceof the normalized amplitude of the oscillations in

Fig. 1. Temperature dependence of the sheet carrier density (NH)and Hall mobility (lH) of electrons in the Al0.83In0.17N/AlN/GaNheterostructure.

(a)

(b)

Fig. 2. Experimental results showing the effects of (a) temperatureand (b) applied electric field on the magnetoresistance Rxx(B)measured for an Al0.83In0.17N/AlN/GaN heterostructure sample.



�¶2Rxx/¶B2 agree to within 1% with those found fromthat of the oscillations in DRxx ¼ Rxx � Rb: Theoscillations in the second derivative of magnetore-sistance have well-defined envelopes and are sym-metrical about the horizontal line as shown in Fig. 3.The double-differentiation technique does notchange the peak position or the period of theoscillations.25

The period of the SdH oscillations has beenobtained from plots of the reciprocal magnetic field(1/Bn), at which the nth peak occurs, against thepeak number n. If electrons in only one subbandparticipate in the SdH oscillations, the graph of 1/Bn

versus n gives a straight line (Fig. 4), the slope ofwhich yields the oscillation period, D(1/B). TheFourier analysis of the SdH oscillations (see theinsert in Fig. 4) confirms that only the first subbandis populated and that the contribution of higherharmonics is insignificant. The 2D carrier density(N2D) can be calculated using25 Eq. (3). The oscilla-tion period (and hence the carrier density) that isdetermined from the SDH oscillation measurementsis found to be essentially independent of tempera-ture in the range from 1.8 K to 14 K. The Fermienergies with respect to the subband energy

(EF � E1) have been obtained from the oscillationperiod using Eq. (3) together with the in-planeeffective mass m* of 2D electrons as obtained fromthe temperature dependence of SdH oscillations (seebelow). The results found for EF � E1 are given inTable I.

The 2D carrier density in the Al0.83In0.17N/AlN/GaN sample determined from the SdH oscillationsand the sheet carrier density (NH) obtained from thelow-field Hall-effect measurements at 1.8 K are alsoincluded in Table I. It can be seen that, within theexperimental error owing to the SdH oscillations,the sheet carrier density determined is larger thanthe carrier density of 2D electrons. This indicatesthat parallel conduction due to carriers outsidethe 2D channel is effective for this sample at lowtemperatures.

The in-plane effective mass of 2D electrons in theAl0.83In0.17N/AlN/GaN heterostructure can be deter-mined from the temperature dependence of the SdHamplitude at constant magnetic field using Eq. (4). Atypical example for the variation of SdH oscillationamplitude with temperature is shown in Fig. 5. Therelative amplitude A(T, Bn)/A(T0, Bn) decreases withincreasing temperature in accordance with the usualthermal damping factor [see Eq. (2)]. The in-planeeffective mass of 2D electrons determined by fittingthe experimental data for the temperature depen-denceofA(T, Bn)/A(T0, Bn) toEq. (4) isalso included inTable I. Similar analysis for all the oscillation peaksobserved in the magnetic field range from 7 T to 11 Thas established that the in-plane effective mass of the2D electrons is essentially independent of themagnetic field.

Experimentally evaluated electron effective massin AlGaN/GaN as a function of the two-dimensionalelectron gas density formed at the interface of anAlGaN/GaN heterostructure with a different alloy

(a)

(b)

Fig. 3. Effects of temperature (a) and applied electric field (b) on theSdH oscillations arising from the electrons in the subband, asextracted from the Rxx(B) data for an Al0.83In0.17N/AlN/GaN hetero-structure sample (shown in Fig. 1). The solid curves through theexperimental data points are intended as a guide for the eye. Thedouble differentiation removes the background magnetoresis-tance without affecting the position or amplitude of the oscillatorycomponent.

Fig. 4. The reciprocal magnetic field (1/Bn) plotted as a function ofthe oscillation peak number (n) of Al0.83In0.17N/AlN/GaN hetero-structures measured at 1.8 K. Filled squares correspond to the datagiven in Fig. 3. The straight line is the least-squares fit to theexperimental data. The insert shows the fast Fourier spectrum of theoscillations (N = 28 and DT ¼ 4:9 10�2 T�1 is the sampling inter-val). There is no evidence for population of higher subbands or forany contribution from higher harmonics.



composition was summarized by Kurakin et al.46

They found the electron effective mass in AlGaN/GaN to be independent of the electron concentra-tion. This result for the effective mass of 2D elec-trons was in good agreement with the bulk effectivemass in GaN, which was reported46,47 to be0.2 ± 0.02m0 (where m0 is the free electron mass).The effective mass of 2D electrons in theAl0.83In0.17N/AlN/GaN heterostructure is in goodagreement with the bulk effective mass in GaN.46,47

This indicates that both the nonparabolicity of theconduction band of GaN and the wave functionpenetration into the AlN barrier/spacer layer haveno significant effects on the effective mass of 2Delectrons in our samples.

The SdH oscillations and classical Hall-effectmeasurements allow for determination of both thequantum and transport lifetimes of the electrons inthe Al0.83In0.17N/ AlN/GaN heterostructure andhence to investigate the relative importance ofvarious scattering mechanisms including ionizedimpurity scattering, alloy scattering, and interfaceroughness scattering. The quantum lifetime (sq) canbe determined from the magnetic field dependence

of the amplitude of the SdH oscillations usingEq. (5) together with the measured values of m*

(Table I). Figure 6 shows typical examples of theDingle plots for the samples investigated. There isgood agreement between the experimental data andthe straight line described by Eq. (5). The quantumlifetime obtained from the slope of the Dingle plot isalso included in Table I. These values remain con-stant within 2% in the whole temperature andmagnetic field ranges of the measurements. Thequantum lifetime (sq) and quantum mobility (lq) arealso tabulated together with the transport lifetime(str) and Hall mobility (lH) determined using theresults of zero-field resistivity and low-field Hall-effect measurements in Table I.

The transport mobilities (lt) of the 2D electrons inthe quantum well have been calculated using thevalues (Table I) determined experimentally for thesheet carrier density (NH), 2D carrier density (N2D),and Hall mobility (lH) by following the analysis forparallel conduction by Kane et al.48 The expressionsfor the effective Hall mobility and carrier densitydue to the two conducting channels (quantum welland bulk carriers outside the 2D channel) may bewritten as

NH ¼ðN2Dlt þNBlBÞ2

N2Dl2t þNBl2

B

; (19)

lH ¼N2Dl2

t þNBl2B

N2Dlt þNBlB

; (20)

where NB and lB are the electron density and themobility of the bulk carriers outside the 2D channel.Here, NB is calculated from the difference betweenNH and N2D. The transport mobility (or transportlifetime) of 2D electrons in the Al0.83In0.17N/AlN/GaN heterostructure sample has been found to beessentially independent of both the lattice temper-ature in the range from 1.8 K to 14 K and theapplied electric field in the range from 11.87 V m�1

to 230 V m�1.

(a)

(b)

Fig. 5. (a) Temperature and (b) electric field dependencies of thenormalized amplitude of the oscillation peak at Bn measured in anAl0.83In0.17N/AlN/GaN heterostructure. The data points representedby the filled circles correspond to the SdH oscillations arising fromthe electrons in the first subband. The solid curve in (a) is the best fitof Eq. (4) to the experimental data. The solid curve in (b) is intendedas a guide for the eye.

Fig. 6. Determination of the quantum lifetime in the Al0.83In0.17

N/AlN/GaN heterostructure sample. The data points are representedby solid squares, and the straight line is the least-squares fit ofEq. (5) to the experimental data.



The ratio of the quantum to transport lifetime,str/sq, in our samples is larger than unity (Table I).Theoretical calculations relating the 2D single-par-ticle scattering time (quantum lifetime) to themomentum relaxation time (transport lifetime)predict a str/sq ratio equal to or less than unity forwide-angle scattering and greater than unity forsmall-angle scattering in the extreme quantumlimit for single subband occupancy.25 This impliesthat, in our sample, electron scattering with small-angle scattering, such as remote ionized impurityscattering and ionized surface states, is on aver-age forward displaced in momentum space. A simi-lar result is also attributed to AlGaN/GaNheterostructures.19

Assuming that the change in the SdH amplitudewith applied electric field can be described in termsof electric field-induced electron heating, the tem-perature T in Eqs. (1–5) can be replaced by theelectron temperature Te.

3,8,9,25 Therefore, Te can bedetermined by comparing the relative amplitudes ofthe SdH oscillations measured as functions of thelattice temperature (T = TL) and the applied electricfield (F) using3,8,9

AðTL;BnÞAðTL0;BnÞ

� �F¼F0

¼ AðF;BnÞAðF0;BnÞ

� �TL¼TL0

: (21)

Here, A(F, Bn) and A(F0,Bn) are the amplitudes ofthe oscillation peaks observed at a magnetic field Bn

and at electric fields F and F0, respectively. In orderto obtain the electron temperature from the latticetemperature and electric field dependencies of theamplitude of the SdH oscillations, the quantumlifetime has to be independent of both the latticetemperature and the applied electric field. Figure 5bshows the amplitudes of the SdH oscillations, nor-malized as described by Eq. (18), as functions of F forthe Al0.83In0.17N/AlN/GaN heterostructure sample.In Fig. 5, only the relative amplitudes at a givenmagnetic field Bn are shown for clarity. A similaranalysis conducted for all the SdH peaks that wereobserved in the magnetic field range from 7 T to

11 T has established that the relative amplitudes ofSdH oscillations (and hence the electron tempera-tures) in our samples are essentially independent ofmagnetic field. This indicates that the magneticfield used in our experiments does not significantlyalter the energy relaxation processes of hotelectrons.

Electron temperatures (Te) for the Al0.83In0.17

N/AlN/GaN heterostructure sample as obtained bydirectly comparing the curves similar to those inFig. 5a, b are plotted as a function of the appliedelectric field in Fig. 7. The SdH oscillations mea-sured for the Al0.83In0.17N/AlN/GaN heterostructuresample decrease rapidly with increasing appliedelectric field and become vanishingly small forF > 230 V m�1 (Figs. 3b, 5b). The electron temper-ature determined for this sample rises quickly withincreasing F.

In the steady state, the power loss from hot elec-trons by the emission of acoustic phonons is equal tothe power supplied by the applied electric field,which can be calculated using the energy balanceequation8,9

P ¼ eltF2; (22)

where P, lt, and F are the energy loss (or energysupply) rate per electron/hole, transport mobility,and applied electric field, respectively. In the cal-culations of power loss, we used the calculatedtransport mobilities as given in Table I. The powerloss versus electron temperature is plotted in Fig. 8.

Comparing our results determined from SdHmeasurements with previous reports on energyrelaxation of hot electrons in GaN/AlGaN hetero-junctions, we find that the magnitude of the powerloss determined in this study varies significantlyfrom the power loss given in literature.17–20 Theobserved variations in power loss may be associatedwith the differences in the mobility of the samples,

Fig. 7. Electron temperature (Te) versus applied electric field (F)for Al0.83In0.17N/AlN/GaN heterostructures.

Fig. 8. Electron temperature dependence of power loss per electrondetermined from SdH measurements. Solid circles correspond toexperimental data. Dashed, dotted, dash-dotted, and solid curvescorrespond to the power loss calculated using Eq. (8), nonpolarcomponent of Eq. (8) (Pnp), polar component of Eq. (8) (Pp), andEq. (15), respectively.



due to its primary role in the calculation of thepower loss [see Eq. (21)]. If experimentally deter-mined electron temperature-dependent power lossdata are normalized to the mobility, it can be shownthat the power loss determined from SdH mea-surements in literature17–20 and our results (Fig. 8)match each other rather well.

We found the exponent 3.55 (Table I) by fittingEq. (15) to the experimental data determined fromSdH measurements (Fig. 8). In all cases, a constantvalue for the exponent c is obtained over the wholetemperature range. This indicates that the experi-ments were carried out in the low-temperatureregime and that the energy relaxation is due toacoustic phonon emission via mixed unscreened pie-zoelectric and deformation potential interactions.2,49

We, therefore, fitted the experimental P(Te) data,obtained from the measurements at TL0 � 1.8 K, tothe analytical expressions for power loss in the low-temperature regime (Fig. 8). The power loss, asgiven by Eq. (8), was calculated using the values(Table I) determined experimentally for the effec-tive mass, carrier density, and Fermi energy of 2Delectrons in the Al0.83In0.17N/AlN/GaN heterostruc-ture sample; other parameters were taken from theliterature (Table II).20,50–52

We attempted to fit the 2D theoretical power lossin the low-temperature regime (Eq. 8) to theexperimental P(Te) results determined from SdHmeasurements. However, the low-temperatureregime model does not offer a satisfactory fit to theexperimental data for the Al0.83In0.17N/AlN/GaNheterostructure sample. It is also instructive tostudy the relative magnitude of the deformationpotential (nonpolar acoustic) Pnp and piezoelectric(polar acoustic) Pp components of the electronenergy loss rates. In the low-temperature regime,the piezoelectric coupling dominates: Pnp/Pp > 1 forthe Al0.83In0.17N/AlN/GaN heterostructure sample.These trends remain at all electron temperaturesin the range from 1.8 K to 14 K. The polar andnonpolar components of the power loss in the

low-temperature regime are also plotted as a func-tion of electron temperature in Fig. 8. It is evidentfrom Fig. 8 that Pp is in agreement with theexperimental power loss data determined above 8 Kfor the Al0.83In0.17N/AlN/GaN heterostructure sam-ple. This result is also in accord with the outcome offitting Eq. (15) and other researchers’ results inAlGaN/GaN heterojunctions.17–20 To our knowl-edge, there is no study to date concerning acousticphonon-assisted energy relaxation of hot electronsin AlInN/AlN/GaN or AlInN/GaN heterostructuresamples. Therefore, we have no possibility to com-pare our experimental results with experimentalresults from literature.

The dominant process for relaxing the hot elec-tron energy is via the interaction with acousticphonons below about 100 K and polar optical pho-nons at 300 K in GaN-based heterostructures.53,54

The acoustic phonon scattering includes deforma-tion potential scattering and piezoelectric scatter-ing. In view of our experimental observations, weconclude that, in the low-temperature regime, thepiezoelectric scattering rates in the 2D case aresomewhat overestimated for the Al0.83In0.17N/AlN/GaN heterostructure and, hence, the theoreticalcalculations of the 2D polar interactions need to bereconsidered. From the theoretical calculation, itwas shown that in the energy relaxation rates foracoustic phonons the lattice temperature deter-mines the number of excited phonons and the elec-tron temperature affects the screening.2,53 Ifscreening were to be included in the 2D theoreticalcalculations for the GaN-based structure, the non-polar component of the interaction would be reducedsignificantly for

kffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi2m�xLO=�h

p < 1; (23)

where k is the electron wavevector and xLO is theoptical phonon frequency,53 whereas, if screeningwere to be included in the 2D theoretical calcula-tions for the GaAs-based structure, the polar com-ponent of the interaction would be reducedsignificantly.2,5,7 Therefore, in order to compare ourexperimental results with the theoretical resultsand the calculation using Eq. 23, we used ourexperimental data to determine the optical phononenergy.

The techniques that are often used to find theoptical phonon (LO) energy are Raman or infraredmeasurements, temperature-dependent Hall mobil-ity measurements, and hot electron power lossmeasurements. While the first technique gives theLO energy directly from the spectra, the two lattertechniques yield this value via use of appropriateenergy and momentum relaxation expressions.

In the temperature range above approximately90 K, where the electron mobility is expected to belimited primarily by polar optical phonon scattering,the temperature dependence of the differential

Table II. Material parameters of the Al0.83In0.17

N/AlN/GaN heterostructure used in the calculation(Refs. 20,50–52)

Parameter Value

N (eV) �7.7eSðe0Þ 10C11 (GPa) 296C12 (GPa) 141C44 (GPa) 94e14 (C m–2) 0.375q (kg m�3) 6150VS (m/s) 6560Vb (eV) 1.923ND (cm�3) 1018



inverse mobility (1/lH � 1/l0) can be approximatedby55–57

1

lLO

¼ 1

lH

� 1

l0

¼ 2m�axLO

eexp � �hxLO

kBT

� �; (24)

where l0 is the low-temperature Hall mobility,which is independent of temperature, lH is thetemperature-dependent Hall mobility measured attemperatures above approximately 90 K, a is thedimensionless polar constant, and xLO is the angu-lar frequency of the optical phonon mode. Therefore,the LO phonon energy can be determined from aplot of the natural logarithm of (1/lH � 1/l0) versus1/T. Figure 9 presents such a plot of the naturallogarithm of (1/lH � 1/l0) versus 1/T. The LO pho-non energy determined from the slope of thestraight line, which is the best fit to the experi-mental data above approximately 170 K, is�hxLO ¼ 91:2 meV.

Figure 10 shows the room-temperature Ramanspectrum for the Al0.83In0.17N/AlN/GaN hetero-structure sample recorded in the grown-axis back-scattering configuration ðzðxxÞ�zÞ. In our sample,GaN crystallizes in a wurtzite structure whosez-axis is perpendicular to the sapphire substrateplane. The space group is C6v

4 , and the A1(LO) andE2 modes are allowed in this configuration.58 Thereis a sharp and strong peak at 575 cm�1, known asthe nonpolar high-frequency E2 mode, whichimplies a strong correlation between Ga and Natoms on the c-plane.58–60 The polar vibrationsA1(LO) observed at 738.5 cm�1 also correspond to acorrelation between Ga and N atoms. Since the lightpenetration depth of 785 nm is longer than thethickness of the coated wafer on the sapphire sub-strate, the sapphire origin A1g and Eg modes wereobserved at approximately 644 cm�1 and 754 cm�1,respectively. The energy of LO phonons in GaNð�hxLO ¼ 91:6 meVÞ is determined using �hxLO ¼ �hc�m;

where c is the speed of light and �m is the wave-number of the A1(LO) mode. The value for �hxLO

determined by this method is in good agreementwith that ð�hxLO ¼ 91:2 meVÞ obtained from ourpresent mobility measurements and that ð�hxLO ¼91:8 meVÞ reported in literature.61,62

We found that the ratio given in Eq. 23 is lessthan and equal to 1 using k< kF and our experi-mentally evaluated optical phonon energy. There-fore, it can be stated that the energy loss rate perelectron for the Al0.83In0.17N/AlN/GaN heterostruc-ture sample in the acoustic phonon regime is inagreement with the theoretical results. In addition,there is a small deviation between the experimentand theory at low electron temperature. This isprobably because the ideal quantum-well approxi-mation in the extreme quantum limit, which wasused in the 2D power loss calculations, predicts anenhancement in the confined electron–acousticphonon interaction, compared with the case in areal triangle quantum well with finite barriers inour Al0.83In0.17N/AlN/GaN heterostructure.2,3,7,49

Furthermore, the existing 2D theories use the bulkphonon approximation, which requires consider-ation of the confinement and folding of the longitu-dinal and transverse acoustic modes.

The energy relaxation time (sE) for intrasubbandprocesses can be obtained from the power lossmeasurements using2

P ¼ < �hx >

sE

ðkBTe � kBTLÞkBTe

; (25)

where < �hx > ¼ 21=2�hVSkF and < �hx > is the acous-tic phonon energy averaged over the Fermi surface.Figure 11 shows the energy relaxation time as afunction of electron temperature for the Al0.83In0.17

N/AlN/GaN heterostructure sample studied. Suchlarge values of sE indicate that the energy lossmechanism in this temperature range is not veryefficient and leads to rapid rise of the electrontemperature when the input power is increased(Fig. 8). However, as can be seen in Fig. 11, the

Fig. 9. The natural logarithm of the differential inverse mobilityas a function of inverse temperature for an Al0.83In0.17N/AlN/GaNheterostructure. For symbols, refer to Fig. 1. The straight line is aleast-squares fit to the experimental data.

Fig. 10. Room-temperature Raman spectrum for the Al0.83In0.17

N/AlN/GaN heterostructure sample.



energy relaxation due to acoustic phonons becomesfaster at higher electron temperatures.

CONCLUSIONS

The carrier density (N2D), effective mass (m*),and quantum lifetime (sq) for electrons in anAl0.83In0.17N/AlN/GaN heterostructure have beendetermined from the Shubnikov–de Haas (SdH)oscillations. The two-dimensional (2D) carrier den-sity and the Fermi energy with respect to the sub-band energy (EF � E1) have been obtained from theperiods of the SdH oscillations. The m* and sq ofelectrons have been extracted from the temperatureand magnetic field dependencies of the SdH ampli-tude, respectively. The results obtained for thetransport-to-quantum lifetime ratios of the respec-tive subbands indicate that the scattering of elec-trons by remote ionized impurities is on averageforward displaced in momentum space.

The energy loss rates, in the acoustic phononregime, of 2D electrons in an Al0.83In0.17N/AlN/GaNheterostructure have also been investigated usingSdH effect measurements. The experimental resultswere compared with the predictions of current the-oretical models for power loss in semiconductors.The energy relaxation of electrons is due to acousticphonon emission via unscreened piezoelectricinteraction. In the low-temperature regime, thepiezoelectric component is significantly greater thanthe deformation potential component of the 2Dpower loss, for electron temperatures lower than14 K.

ACKNOWLEDGEMENTS

We are grateful to TUBITAK Ankara (Project No.110T377) and Anadolu University (Project No. BAP-1001F99) for their financial support.

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Energy Relaxation Rates in AlInN/AlN/GaN Heterostructures

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