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Electron transport in wide energy gap semiconductors Cheekoori, Reddiprasad 2012

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Electron Transport in Wide Energy Gap Semiconductors by Reddiprasad Cheekoori A THESIS SUBMITTED IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF MASTER OF APPLIED SCIENCE in The College of Graduate Studies (Electrical Engineering) THE UNIVERSITY OF BRITISH COLUMBIA (Okanagan) January, 2012 c Reddiprasad Cheekoori 2012Abstract The dependence of the low- eld electron drift mobility on the crystal temperature is determined for a number of wide energy gap semiconductors of interest. The materials considered include gallium nitride, aluminium nitride, indium nitride, and zinc oxide; while indium nitride is not a wide energy gap semiconductor in of itself, alloys of indium nitride with gallium nitride are. For the bulk results, it is found that indium nitride exhibits the highest low- eld electron drift mobility while aluminium nitride exhibits the lowest low- eld electron drift mobility. This is related to the small ef- fective mass of electrons in indium nitride and the large e ective mass of electrons in aluminium nitride. For the case of electrons con ned within a two-dimensional electron gas, it is found that the low- eld electron drift mobility exceeds that corresponding to the bulk material. This is due to the enhanced screening that electron concentrations exceeding the bulk ion- ized impurity concentrations level o er, i.e., surplus electrons act to further screen the ionized impurities; in a two-dimensional electron gas, the electron concentrations may far exceed those found in a bulk material. Recommen- dations for further study are suggested. iiTable of Contents Abstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ii Table of Contents . . . . . . . . . . . . . . . . . . . . . . . . . . . . iii List of Tables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . vi List of Figures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . vii Acknowledgements . . . . . . . . . . . . . . . . . . . . . . . . . . . xx 1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 2 Material properties . . . . . . . . . . . . . . . . . . . . . . . . . 10 2.1 Material properties and electron device performance . . . . . 10 2.2 The distribution of electronic states . . . . . . . . . . . . . . 11 2.3 Direct and indirect energy gaps . . . . . . . . . . . . . . . . 15 2.4 Temperature dependence of the energy gap . . . . . . . . . . 15 2.5 Wide energy gap semiconductors . . . . . . . . . . . . . . . . 17 2.6 Other material properties and their dependence on the energy gap . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19 iiiTable of Contents 3 Electron drift mobility in semiconductors . . . . . . . . . . 27 3.1 Electron drift mobility and scattering mechanisms . . . . . . 27 3.2 The random thermal motion of charge carriers in the absence of an applied electric  eld . . . . . . . . . . . . . . . . . . . . 28 3.3 Electron transport under the action of an applied electric  eld 29 3.4 Scattering mechanisms . . . . . . . . . . . . . . . . . . . . . 33 3.5 Analytical expressions for the contributions to the low- eld electron drift mobility . . . . . . . . . . . . . . . . . . . . . . 36 3.5.1 Ionized impurity scattering . . . . . . . . . . . . . . . 36 3.5.2 Polar optical phonon scattering . . . . . . . . . . . . 38 3.5.3 Piezoelectric scattering . . . . . . . . . . . . . . . . . 38 3.5.4 Acoustic phonon deformation potential scattering . . 40 3.6 Electron transport in a two-dimensional electron gas . . . . . 41 4 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44 4.1 Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44 4.2 Material parameter selections . . . . . . . . . . . . . . . . . . 45 4.3 GaN bulk results . . . . . . . . . . . . . . . . . . . . . . . . . 47 4.4 AlN bulk results . . . . . . . . . . . . . . . . . . . . . . . . . 55 4.5 InN bulk results . . . . . . . . . . . . . . . . . . . . . . . . . 70 4.6 ZnO bulk results . . . . . . . . . . . . . . . . . . . . . . . . . 79 4.7 A comparison of the bulk results . . . . . . . . . . . . . . . . 93 4.8 Two-dimensional electron gas . . . . . . . . . . . . . . . . . . 95 5 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100 ivTable of Contents References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104 vList of Tables 4.1 The material parameters corresponding GaN, AlN, and InN. These parameter selections are from O’Leary et al: [15]. . . . 46 4.2 The material parameters corresponding to ZnO. The ZnO parameter selections are from Albrecht et al: [21]. . . . . . . . 47 viList of Figures 1.1 The intrinsic electron concentrations, for c-Si, c-Ge, and c- GaAs, as a function of the reciprocal temperature. . . . . . . 3 1.2 The energy gap represents the di erence in energy between top of the valence band and the bottom of the conduction band. At zero temperature, the valence band is fully occupied and the conduction band is completely unoccupied. . . . . . . 4 1.3 The room temperature energy gaps associated with a variety of compound semiconductors. The data depicted in this  gure is from Adachi [8]. . . . . . . . . . . . . . . . . . . . . . . . . 6 1.4 The velocity- eld characteristics associated with c-Si and c- GaAs. Logarithmic scales are employed for both axes. For both cases, the crystal temperature is set to 300 K. . . . . . . 8 2.1 The electron energy levels associated with an isolated C atom. This  gure is after Streetman [3]. . . . . . . . . . . . . . . . . 12 2.2 The formation of the energy gap within C. This  gure is after Streetman [3]. This particular image is from Malik [12]. . . . 14 viiList of Figures 2.3 A one-dimensional depiction of the energy band diagrams as- sociated with c-Si and c-GaAs. This  gure is after Streetman [3]. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16 2.4 The energy gaps associated with c-Si and c-GaAs as a func- tion of the crystal temperature. These plots are determined using the Varshni relationship, i.e., Eq. (2.1), for the empiri- cal parameter selections suggested in the text. . . . . . . . . . 18 2.5 The di erences in the energy gap between a semiconductor and a wide energy gap semiconductor . . . . . . . . . . . . . 20 2.6 The electron e ective mass as a function of the energy gap for selected III-V and II-VI semiconductors. The direct en- ergy gap materials are depicted with the solid points, while the indirect energy gap materials are represented with the open points. Most of the data depicted in this  gure is from Adachi [8]. The InN data, however, is from O’Leary et al: [15]. 21 2.7 The breakdown electric  eld as a function of the energy gap for selected III-V and II-VI semiconductors. The direct en- ergy gap materials are depicted with the solid points, while the indirect energy gap materials are depicted with the open points. The data depicted in this  gure is from Adachi [8]. . . 22 viiiList of Figures 2.8 The static relative dielectric constant, ks =  s= o, as a func- tion of the energy gap for selected III-V and II-VI semicon- ductors. The direct energy gap materials are depicted with the solid points, while the indirect energy gap materials are depicted with the open points. The data depicted in this  gure is from Adachi [8]. . . . . . . . . . . . . . . . . . . . . . 24 2.9 The high-frequency relative dielectric constant, k1 =  1= o, as a function of the energy gap for selected III-V and II-VI semiconductors. The direct energy gap materials are depicted with the solid points, while the indirect energy gap materials are depicted with the open points. Most of the data depicted in this  gure is from Adachi [8]. The InN data, however, is from O’Leary et al: [15]. . . . . . . . . . . . . . . . . . . . . . 25 2.10 The electron a nity as a function of the energy gap for se- lected III-V and II-VI semiconductors. The direct energy gap materials are depicted with the solid points, while the indi- rect energy gap materials are depicted with the open points. The data depicted in this  gure is from Adachi [8]. . . . . . . 26 ixList of Figures 3.1 The random thermal motion of a representative electron in a semiconductor crystal in the absence of an applied electric  eld. This particular electron engages in scattering a number of times, the numbers 2 through 12 representing the scatter- ing events that occur; number 1 corresponds to this particular electron’s initial position. Between scattering events, the elec- tron propagates with no acceleration. The scattering events are seen to randomize the direction of the electron transit. This  gure is after Streetman [3]. . . . . . . . . . . . . . . . . 30 3.2 The di erent scattering mechanisms that can occur in semi- conductors. The particular scattering mechanisms that we examine for the purposes of this analysis are presented with the bold font. This  gure is after Adachi [8]. . . . . . . . . . 34 3.3 The energy band diagram associated with c-GaAs. A three- valley representation of this band diagram for the conduction band is depicted beneath a more exact one-dimensional rep- resentation of the energy band diagram corresponding to this material [16, 17]. . . . . . . . . . . . . . . . . . . . . . . . . . 35 3.4 The energy band diagram of an AlGaN/GaN high electron mobility transistor. A quantum well is formed in the GaN layer close to the AlGaN/GaN interface. Mobile electrons, contributed from the AlGaN and GaN layers, are trapped in the quantum well that is present, thus forming a two- dimensional electron gas [20]. . . . . . . . . . . . . . . . . . . 42 xList of Figures 4.1 The ionized impurity contribution to the low- eld electron drift mobility as a function of the crystal temperature for the case GaN. For all cases, the electron concentration, n, is set to the bulk ionized impurity concentration, NT . . . . . . . . . 48 4.2 The polar optical phonon contribution to the low- eld elec- tron drift mobility as a function of the crystal temperature for the case of GaN. . . . . . . . . . . . . . . . . . . . . . . . 50 4.3 The piezoelectric contribution to the low- eld electron drift mobility as a function of the crystal temperature for various selections of the electron concentration for the case of GaN. . 51 4.4 The acoustic deformation potential contribution to the low-  eld electron drift mobility as a function of the crystal tem- perature for the case of GaN. . . . . . . . . . . . . . . . . . . 53 4.5 The polar optical phonon, piezoelectric, and acoustic defor- mation potential contributions to the low- eld electron drift mobility, and the combined low- eld electron drift mobility, plotted as functions of the crystal temperature for the case of GaN. For the purposes of this analysis, the ionized impu- rity contribution is neglected, i.e., it is set to in nity. The electron concentration, n, is set to 1017 cm 3 for all cases. . . 54 xiList of Figures 4.6 The combined low- eld electron drift mobility as a function of the crystal temperature with the polar optical phonon, piezo- electric, and acoustic deformation potential scattering mech- anisms taken into account for various electron concentrations for the case of GaN. For all cases, the ionized impurity con- tribution is neglected, i.e., it is set to in nity. . . . . . . . . . 56 4.7 The ionized impurity, polar optical phonon, piezoelectric, and acoustic deformation potential contributions to the low- eld electron drift mobility as a function of the crystal temperature for the case of GaN. The bulk ionized impurity concentration, NT , is set to 1017 cm 3. The electron concentration, n, is set to the bulk ionized impurity concentration, NT . . . . . . . . . 57 4.8 The combined low- eld electron drift mobility as a function of the crystal temperature for the case of GaN. The contribu- tions to the low- eld electron drift mobility considered include those related to the ionized impurity, polar optical phonon, piezoelectric, and acoustic deformation potential scattering mechanisms. For all cases, the electron concentration, n, is set to the bulk ionized impurity concentration, NT . . . . . . . 58 4.9 The ionized impurity contribution to the low- eld electron drift mobility as a function of the crystal temperature for the case AlN. For all cases, the electron concentration, n, is set to the bulk ionized impurity concentration, NT . . . . . . . . . 60 xiiList of Figures 4.10 The polar optical phonon contribution to the low- eld elec- tron drift mobility as a function of the crystal temperature for the case of AlN. . . . . . . . . . . . . . . . . . . . . . . . . 61 4.11 The piezoelectric contribution to the low- eld electron drift mobility as a function of the crystal temperature for various selections of the electron concentration for the case of AlN. . 62 4.12 The acoustic deformation potential contribution to the low-  eld electron drift mobility as a function of the crystal tem- perature for the case of AlN. . . . . . . . . . . . . . . . . . . 64 4.13 The polar optical phonon, piezoelectric, and acoustic defor- mation potential contributions to the low- eld electron drift mobility, and the combined low- eld electron drift mobility, plotted as functions of the crystal temperature for the case of AlN. For the purposes of this analysis, the ionized impurity contribution is neglected, i.e., it is set to in nity. The electron concentration, n, is set to 1017 cm 3 for all cases. . . . . . . . 65 4.14 The combined low- eld electron drift mobility as a function of the crystal temperature with the polar optical phonon, piezo- electric, and acoustic deformation potential scattering mech- anisms taken into account for various electron concentrations for the case of AlN. For all cases, the ionized impurity con- tribution is neglected, i.e., it is set to in nity. . . . . . . . . . 67 xiiiList of Figures 4.15 The ionized impurity, polar optical phonon, piezoelectric, and acoustic deformation potential contributions to the low- eld electron drift mobility as a function of the crystal temperature for the case of AlN. The bulk ionized impurity concentration, NT , is set to 1017 cm 3. The electron concentration, n, is set to the bulk ionized impurity concentration, NT . . . . . . . . . 68 4.16 The combined low- eld electron drift mobility as a function of the crystal temperature for the case of AlN. The contribu- tions to the low- eld electron drift mobility considered include those related to the ionized impurity, polar optical phonon, piezoelectric, and acoustic deformation potential scattering mechanisms. For all cases, the electron concentration, n, is set to the bulk ionized impurity concentration, NT . . . . . . . 69 4.17 The ionized impurity contribution to the low- eld electron drift mobility as a function of the crystal temperature for the case InN. For all cases, the electron concentration, n, is set to the bulk ionized impurity concentration, NT . . . . . . . . . 71 4.18 The polar optical phonon contribution to the low- eld elec- tron drift mobility as a function of the crystal temperature for the case of InN. . . . . . . . . . . . . . . . . . . . . . . . . 72 4.19 The piezoelectric contribution to the low- eld electron drift mobility as a function of the crystal temperature for various selections of the electron concentration for the case of InN. . 74 xivList of Figures 4.20 The acoustic deformation potential contribution to the low-  eld electron drift mobility as a function of the crystal tem- perature for the case of InN. . . . . . . . . . . . . . . . . . . . 75 4.21 The polar optical phonon, piezoelectric, and acoustic defor- mation potential contributions to the low- eld electron drift mobility, and the combined low- eld electron drift mobility, plotted as functions of the crystal temperature for the case of InN. For the purposes of this analysis, the ionized impu- rity contribution is neglected, i.e., it is set to in nity. The electron concentration, n, is set to 1017 cm 3 for all cases. . . 77 4.22 The combined low- eld electron drift mobility as a function of the crystal temperature with the polar optical phonon, piezo- electric, and acoustic deformation potential scattering mech- anisms taken into account for various electron concentrations for the case of InN. For all cases, the ionized impurity contri- bution is neglected, i.e., it is set to in nity. . . . . . . . . . . 78 4.23 The ionized impurity, polar optical phonon, piezoelectric, and acoustic deformation potential contributions to the low- eld electron drift mobility as a function of the crystal temperature for the case of InN. The bulk ionized impurity concentration, NT , is set to 1017 cm 3. The electron concentration, n, is set to the bulk ionized impurity concentration, NT . . . . . . . . . 80 xvList of Figures 4.24 The combined low- eld electron drift mobility as a function of the crystal temperature for the case of InN. The contribu- tions to the low- eld electron drift mobility considered include those related to the ionized impurity, polar optical phonon, piezoelectric, and acoustic deformation potential scattering mechanisms. For all cases, the electron concentration, n, is set to the bulk ionized impurity concentration, NT . . . . . . . 81 4.25 The ionized impurity contribution to the low- eld electron drift mobility as a function of the crystal temperature for the case ZnO. For all cases, the electron concentration, n, is set to the bulk ionized impurity concentration, NT . . . . . . . . . 82 4.26 The polar optical phonon contribution to the low- eld elec- tron drift mobility as a function of the crystal temperature for the case of ZnO. . . . . . . . . . . . . . . . . . . . . . . . 84 4.27 The piezoelectric contribution to the low- eld electron drift mobility as a function of the crystal temperature for various selections of the electron concentration for the case of ZnO. . 85 4.28 The acoustic deformation potential contribution to the low-  eld electron drift mobility as a function of the crystal tem- perature for the case of ZnO. . . . . . . . . . . . . . . . . . . 87 xviList of Figures 4.29 The polar optical phonon, piezoelectric, and acoustic defor- mation potential contributions to the low- eld electron drift mobility, and the combined low- eld electron drift mobility, plotted as functions of the crystal temperature for the case of ZnO. For the purposes of this analysis, the ionized impu- rity contribution is neglected, i.e., it is set to in nity. The electron concentration, n, is set to 1017 cm 3 for all cases. . . 88 4.30 The combined low- eld electron drift mobility plotted as a function of the crystal temperature with the polar optical phonon, piezoelectric, and acoustic deformation potential scat- tering mechanisms taken into account for various electron concentrations for the case of ZnO. For all cases, the ionized impurity contribution is neglected, i.e., it is set to in nity. . . 90 4.31 The ionized impurity, polar optical phonon, piezoelectric, and acoustic deformation potential contributions to the low- eld electron drift mobility as a function of the crystal temperature for the case of ZnO. The bulk ionized impurity concentration, NT , is set to 1017 cm 3. The electron concentration, n, is set to the bulk ionized impurity concentration, NT . . . . . . . . . 91 xviiList of Figures 4.32 The combined low- eld electron drift mobility as a function of the crystal temperature for the case of ZnO. The contribu- tions to the low- eld electron drift mobility considered include those related to the ionized impurity, polar optical phonon, piezoelectric, and acoustic deformation potential scattering mechanisms. For all cases, the electron concentration, n, is set to the bulk ionized impurity concentration, NT . . . . . . . 92 4.33 The combined low- eld electron drift mobility dependence on the crystal temperature for the cases of GaN, AlN, InN, and ZnO. For cases, the bulk ionized impurity is set to 1017 cm 3. For all cases, the electron concentration, n, is set to the bulk ionized impurity concentration, NT . . . . . . . . . . . . . . . 94 4.34 The combined low- eld electron drift mobility as a function of the crystal temperature for the case of GaN for various electron concentrations. The bulk ionized impurity concen- tration, NT , is set to 1017cm 3 for all cases. . . . . . . . . . . 96 4.35 The combined low- eld electron drift mobility as a function of the crystal temperature for the case of AlN for various electron concentrations. The bulk ionized impurity concen- tration, NT , is set to 1017cm 3 for all cases. . . . . . . . . . . 97 4.36 The combined low- eld electron drift mobility as a function of the crystal temperature for the case of InN for various electron concentrations. The bulk ionized impurity concentration, NT , is set to 1017cm 3 for all cases. . . . . . . . . . . . . . . . . . 98 xviiiList of Figures 4.37 The combined low- eld electron drift mobility as a function of the crystal temperature for the case of ZnO for various electron concentrations. The bulk ionized impurity concen- tration, NT , is set to 1017cm 3 for all cases. . . . . . . . . . . 99 xixAcknowledgements I would like to thank Dr. Stephen O’Leary, my advisor during the course of my studies, with whom this ambition for my life could not have been ful lled. I would also like to thank my committee members, Drs. Jonathan Holzman and Julian Cheng, for giving invaluable input during the course of my research, and for providing me with support along the way. In addition, I would like to thank my friends for their encouragement and support which made my graduate experience a memorable one. This thesis is dedicated to my parents, Sri. Anand Babu Cheekoori and Smt. Vijaya Kumari Cheekoori, whose love, prayers, and faith in me have led me thus far and will continue to take me through the rest of my life. xxChapter 1 Introduction The electronics age found its genesis with the development of the  rst transistor in 1947 [1]. Much of the progress that has occurred in electron- ics since that time has arisen as a result of making the electron devices within electronic systems smaller, faster, cheaper, and more reliable. These advances have occurred as a result of a detailed and quantitative understand- ing of the material properties of the materials used within electron devices. Electron devices are comprised of conductors, insulators, and semiconduc- tors. While the material properties of conductors and insulators have been well understood for many years, the material properties of semiconductors were relatively unknown until the development of the electronics industry. Prior to that time, semiconductors were considered a laboratory curiosity. Further progress in electronics will undoubtedly require further research into the material properties of the semiconductors used within electron de- vices. Since the 1950s, crystalline silicon (c-Si) has served as the dominant semiconductor within the electronics industry. Although c-Si possesses an indirect energy gap, its popularity stems from the fact that it is inexpensive to manufacture and is widely available [2]. At the present moment, c-Si is used in a variety of electron device con gurations, including recti ers, 1Chapter 1. Introduction metal-oxide-semiconductor  eld e ect transistors, bipolar junction transis- tors, and thyristors [3]. The dominant role that c-Si plays in the electronics industry is unlikely to change in the near-term future. Unfortunately, the energy gap of c-Si is narrow, i.e., 1.12 eV at room temperature [4]. This limits the range of electron device applications that can employ this material. At high temperatures, i.e., temperatures beyond 200  C, the intrinsic electron concentration associated with c-Si exceeds de- sired levels; the intrinsic electron concentrations, corresponding to c-Si, crys- talline germanium (c-Ge), and crystalline gallium arsenide (c-GaAs), are de- picted as a function of the reciprocal temperature in Figure 1.1. At 300  C, for example, it is noted that the intrinsic electron concentration associated with c-Si is found to be in excess of 1015 cm 3. As a consequence, the dop- ing levels required for the realization of p-n junction based electron devices simply can not be achieved. Accordingly, alternate semiconductor materials, with wider energy gaps, and hence, lower intrinsic electron concentrations, must be considered instead for high-temperature electron device applica- tions [5]. The energy gap associated with a semiconductor corresponds to the en- ergy di erence between the top of the valence band and the bottom of the conduction band; the energy gap is depicted in Figure 1.2. At zero temper- ature, the valence band is fully occupied and the conduction band is com- pletely unoccupied. If an electron within the valence band acquires su cient energy, through the absorption of a photon or through thermal processes, it can transition into the conduction band. Accordingly, the energy gap asso- ciated with a semiconductor plays an important role in determining the ma- 2Chapter 1. Introduction 1 1.5 2 2.5 3 3.5 4 106 108 1010 1012 1014 1016 c−GaAs c−Si c−Ge 1000/T (K−1) Intrinsic electron concentration ( c m− 3  ) Figure 1.1: The intrinsic electron concentrations, for c-Si, c-Ge, and c-GaAs, as a function of the reciprocal temperature. 3Chapter 1. Introduction                                                                                                                                                                                                                                                                                                                                                                                                                                          Energy gap         Energ y                                         Conduction band (completely unoccupied) Valence band (fully occupied) Figure 1.2: The energy gap represents the di erence in energy between top of the valence band and the bottom of the conduction band. At zero temperature, the valence band is fully occupied and the conduction band is completely unoccupied. 4Chapter 1. Introduction terial properties of the semiconductor [6]. The energy gaps associated with naturally occurring semiconductors range from 0:17 to 6:2 eV [5, 7]. The energy gaps associated with a number of elemental and compound semicon- ductors are depicted in Figure 1.3 [8]. Wide energy gap semiconductors are generally referred to as those that exhibit energy gaps in excess of 2 eV [5]. Interest in wide energy gap semiconductors has been considerable in re- cent years. In addition to o ering lower intrinsic electron concentrations, wide energy gap semiconductors o er a number of other favourable mate- rial properties, including higher electric breakdown  elds and higher ther- mal conductivities. In addition, many of the wide energy gap semicon- ductors possess direct energy gaps, making them useful for a variety of opto-electronic device applications; as c-Si possesses an indirect energy gap, opto-electronic device applications using this material are not possible. A few of the more important wide energy gap semiconductors include gallium nitride (GaN), with a direct energy gap of 3:39 eV at room temperature [9], aluminium nitride (AlN), with a direct energy gap of 6:2 eV at room temperature [10], zinc oxide (ZnO), with a direct energy gap of 3:4 eV [11] at room temperature, and zinc selenide (ZnSe), with a direct energy gap of 2:7 eV at room temperature. The performance of an electron device is often related to the transport of the charge carriers within the device. The transport of charge carri- ers depends on a variety of factors, including the crystal temperature, the electron concentration, and the impurities that are present. The charge transport that occurs is often characterized in terms of the dependence of the electron drift velocity on the applied electric  eld, i.e., ve(E), where ve 5Chapter 1. Introduction 0  1  2  3  4  5  6  7  InSb  InAs  InN  GaSb  InP  GaAs  CdTe  AlSb  ZnSe  GaP  GaN  ZnO  AlP  SiC  AlN  Ene rgy  g ap  ( e V ) Figure 1.3: The room temperature energy gaps associated with a variety of compound semiconductors. The data depicted in this  gure is from Adachi [8]. 6Chapter 1. Introduction denotes the electron drift velocity and E represents the applied electric  eld. The velocity- eld characteristic, as this dependence is usually referred to as, plays an important role in determining the transport of the charge carriers within semiconductors, and, as a consequence, its determination is a central goal of studies into the material properties of semiconductors. The 300 K velocity- eld characteristic associated with c-Si is contrasted with that as- sociated with c-GaAs in Figure 1.4. The low- eld electron drift mobility,  e, corresponds to the slope of the electron velocity- eld characteristic at zero applied electric  eld, i.e.,  e = dve(E) dE      E=0 : (1.1) While not as informative as the velocity- eld characteristic itself, the low-  eld electron drift mobility plays an important role in shaping the perfor- mance of many electron devices, as many devices operate at low applied elec- tric  elds. Indeed, for low applied electric  elds, the non-linearities found in the full velocity- eld characteristic do not play a role in shaping the corre- sponding device performance. It is this low- eld electron drift mobility,  e, that will be the focus of the subsequent work presented in this thesis. In this thesis, how the low- eld electron drift mobility,  e, associated with a number of wide energy gap semiconductors, varies with the crystal temperature, the doping concentration, and the charge carrier concentra- tion, will be examined. Initially, the analysis will focus on the contributions to the low- eld electron drift mobility associated with some of the more im- portant underlying scattering mechanisms, including ionized impurity scat- 7Chapter 1. Introduction 102 103 104 105 105 106 107 108 Applied electric field (kV / cm) Electron drift velocity (cm / s ) c−Si c−GaAs Figure 1.4: The velocity- eld characteristics associated with c-Si and c- GaAs. Logarithmic scales are employed for both axes. For both cases, the crystal temperature is set to 300 K. 8Chapter 1. Introduction tering, polar optical phonon scattering, piezoelectric scattering, and acoustic deformation potential scattering. This analysis will be performed for bulk materials. Then, the role that the electron concentration plays in determin- ing the low- eld electron drift mobility associated with electrons con ned within a two-dimensional electron gas will be studied, this con guration being representative of that found in many electron device con gurations. The wide energy gap semiconductors considered in this analysis will include GaN, AlN, and ZnO. As indium nitride (InN), with a direct energy gap of 0:7 eV at room temperature [7], may be alloyed with GaN, and, as it is a semiconductor material of considerable current interest, InN will also be the focus of this analysis; while InN is not a wide energy gap semiconductor in of itself, alloys of InN with GaN are. This thesis is organized in the following manner. In Chapter 2, the mate- rial properties of a number of wide energy gap semiconductors are presented. Then, in Chapter 3, expressions for the various contributions to low- eld electron drift mobility will be provided. The analysis of the low- eld elec- tron drift mobility associated with GaN, AlN, InN, and ZnO will then be discussed in Chapter 4. The impact that electron concentrations play in determining the low- eld electron drift mobility of the electrons con ned within a two-dimensional electron gas will also be examined in Chapter 4. Finally, conclusions and recommendations for further study are presented in Chapter 5. 9Chapter 2 Material properties 2.1 Material properties and electron device performance The performance of an electron device may be determined from the material properties of the semiconductors found within the device. Accord- ingly, interest in the material properties of semiconductors have been the focus of great attention since the dawn of the electronics revolution more than half a century ago. With the dimensions of modern electron devices approaching fundamental quantum limits, researchers are currently seeking means of bypassing these limits. These e orts have, thus far anyway, focused on the material properties of semiconductors at the atomic level. Thus, cur- rent interest in the material properties of semiconductors is intense. This interest seems likely to continue for the foreseeable future. In this chapter, the material properties of semiconductors will be ex- amined. The focus of this examination will be on material properties that directly relate to the low- eld electron drift mobility,  e. The analysis will begin, however, with a description of the energy levels of the electrons within materials. The relationship between the distributions of electronic states 102.2. The distribution of electronic states and the energy gap will then be featured, this allowing one to qualitatively understand the di erences between conductors, insulators, and semiconduc- tors. Following these preliminaries, the dependence of a number of key ma- terial properties on the energy gap will be presented, some of the materials considered being those that will be subsequently studied. This chapter is organized in the following manner. In Section 2.2, the distribution of electronic states within a semiconductor is described. Then, in Section 2.3, the di erence between indirect and direct energy gap semicon- ductors is discussed. The temperature dependence of the energy gap is then examined in Section 2.4. In Section 2.5, wide energy gap semiconductors are introduced. Finally, in Section 2.6, the dependence of the other material properties on the energy gap is examined for a wide variety of compound semiconductors. 2.2 The distribution of electronic states Isolated atoms possess a discrete spectrum of allowed energy lev- els. Consider, for example, the case of the isolated carbon (C) atom. The discrete electron energy levels associated with this atom are depicted in Fig- ure 2.1 [3]. Note that these electron energy levels are associated with the di erent orbital states that are present. The occupancy of these electronic states determines the properties of the material. The electronic structure, i.e., the states that are present and their occupancy, associated with the iso- lated C atom in the ground state, for example, may be succinctly expressed in the usual shorthand chemical nomenclature, 1s22s22p2 [3]. 112.2. The distribution of electronic states                 Energy                                  Figure 2.1: The electron energy levels associated with an isolated C atom. This  gure is after Streetman [3]. 122.2. The distribution of electronic states As isolated atoms are brought together in order to form a solid, these discrete energy levels will broaden into bands. Consider the case of C, for example, as depicted in Figure 2.2 [3, 12]. For the case N C atoms at in nite inter-atomic separation, the isolated C atom energy levels appear. As the N C atoms are brought together, however, this discrete spectrum of energy levels will broaden into bands, the 2s and 2p energy levels being the  rst to broaden; the 1s energy levels will broaden at smaller inter-atomic separation distances, as these electrons correspond to the core electrons of the atom, i.e., their electron wave functions are more tightly bound to the atom than the 2s and 2p electron wave functions. The band associated with the 2p states will be comprised of 6N states, 2N of which are occupied. In contrast, the band associated with the 2s states will be comprised of 2N states, 2N of which are occupied. The band associated with the 1s states will be comprised of 2N states, 2N of which are occupied [3]. At su ciently small inter-atomic spacings, the band associated with the 2s and 2p states will merge into a hybrid band, comprised of 8N electronic states, 4N of which will be occupied with electrons. Further reductions in the inter-atomic spacing will result in a split of this merged band into two bands, both with 4N electronic states, separated by an energy gap, Eg. At zero temperature, the lower energy band, which is referred to as the valence band, is fully occupied, i.e., it will host 4N electrons. In contrast, the upper energy band, which is referred to as the conduction band, is completely unoccupied, i.e., it will host no electrons. The energy di erence between the valence band maximum and the conduction band minimum, at the equilibrium lattice spacing, is the energy gap. The energy gap, Eg, 132.2. The distribution of electronic states Figure 2.2: The formation of the energy gap within C. This  gure is after Streetman [3]. This particular image is from Malik [12]. 142.3. Direct and indirect energy gaps corresponding to c-Si maybe accounted for in a similar manner [3]. 2.3 Direct and indirect energy gaps The character of a band may be speci ed by determining how the elec- tron energy levels vary as a function of the crystal momentum, ~k. These dependencies are usually speci ed through an energy band diagram, which plots the corresponding electron energy levels as a function of the crystal momentum, ~k, along critical lines of symmetry. Semiconductors for which the conduction band minimum does not occur at the same value of k as the valence band maximum, such as c-Si, are referred to as indirect energy gap semiconductors. In contrast, semiconductors for which the conduction band minimum occurs at the same value of k as the valence band maximum, such as c-GaAs, are referred to as direct energy gap semiconductors. One dimen- sional representations of the energy band diagrams associated with c-Si and c-GaAs are depicted in Figure 2.3 [3]. Direct energy gap semiconductors have better optical properties than their indirect counterparts. This is why c-Si is not used for opto-electronic device applications. 2.4 Temperature dependence of the energy gap As the energy gap is an important parameter in the understanding of the material properties of a semiconductor, it has been the focus of much attention over the years. Detailed studies have demonstrated that the en- ergy gap monotonically decreases with increases in the crystal temperature. 152.4. Temperature dependence of the energy gap                             k    Direct energy gap  c- GaAs   Eg  Eg           Indirect energy gap    c- Si  k  Figure 2.3: A one-dimensional depiction of the energy band diagrams asso- ciated with c-Si and c-GaAs. This  gure is after Streetman [3]. 162.5. Wide energy gap semiconductors Varshni [13] suggest that Eg (T) = Ego   T2 T +  ; (2.1) where Ego denotes the energy gap at absolute zero and  and  are empir- ically determined parameters that depend on the particular material be- ing considered. For the case of c-Si, it is found that Ego = 1:169 eV,  = 4:9  10 4 eV=K, and  = 655 K, while for the case of c-GaAs, it is found that Ego = 1:519 eV,  = 5:4  10  4 eV=K, and  = 204 K [14]. The dependence of the energy gaps associated with c-Si and c-GaAs on the crystal temperature, as determined from Eq. (2.1) and the associated pa- rameter values, is depicted in Figure 2.4. It is noted that the decrease in the energy gap corresponding to increased crystal temperatures is quite sig- ni cant, c-GaAs loosing in excess of 30% of its zero-temperature energy gap when the crystal temperature is set to 1000 K. 2.5 Wide energy gap semiconductors Wide energy gap semiconductors are semiconductors that possess wide energy gaps. GaN, AlN, and ZnO, with room temperature energy gaps of 3:39, 6:2, and 3:4 eV, respectively, are referred to as wide energy gap semi- conductors [5]. While the exact use of the term may be the subject of debate, there is general consensus that materials with energy gaps in excess of 2 eV may be referred to as wide energy gap semiconductors [5]. The energy gap di erences, between a semiconductor and a wide energy gap semiconductor, 172.5. Wide energy gap semiconductors 0 200 400 600 800 1000 0.9 1 1.1 1.2 1.3 1.4 1.5 1.6  Energy gap ( eV ) Temperature ( K ) c−Si c−GaAs Figure 2.4: The energy gaps associated with c-Si and c-GaAs as a function of the crystal temperature. These plots are determined using the Varshni relationship, i.e., Eq. (2.1), for the empirical parameter selections suggested in the text. 182.6. Other material properties and their dependence on the energy gap are depicted are Figure 2.5. Recalling the energy gap tabulation provided in Figure 1.3, it is seen that there are a number of wide energy gap semi- conductors that are available, including aluminium antimonide (AlSb), AlN, aluminium phosphide (AlP), GaN, gallium phosphide (GaP), silicon carbide (SiC), ZnO, and ZnSe. 2.6 Other material properties and their dependence on the energy gap The e ective mass of a semiconductor is directly tied to its energy gap. In Figure 2.6, the electron e ective mass is plotted as a function of the corresponding energy gap for a number of compound semiconductors. It is seen that the electron e ective mass essentially scales linearly with the en- ergy gap, both for indirect and direct energy gap semiconductors; deviations from an exact linear dependence are in evidence, however. As the low- eld electron drift mobility scales with the reciprocal of the electron e ective mass, it is clear that wider energy gaps will correspond to lower low- eld electron drift mobilities. This is in accord with experimental observation. The breakdown electric  eld is known to increase with the breadth of the energy gap. In Figure 2.7, the breakdown electric  eld is plotted as function of the energy gap for a number of compound semiconductors. As is expected, wider energy gaps correspond to higher breakdown  elds. The fact that GaN, for example, exhibits a breakdown  eld of 2:6 MV=cm, while c-Si exhibits a breakdown  eld of 300 kV=cm, was one of the major reasons why interest in the wide energy gap semiconductors started in the  rst place. 192.6. Other material properties and their dependence on the energy gap     Wide energy gap                                 semiconductor Eg VB Semiconductor CB Eg VB CB Figure 2.5: The di erences in the energy gap between a semiconductor and a wide energy gap semiconductor 202.6. Other material properties and their dependence on the energy gap 0 0.5 1 1.5 2 2.5 3 3.5 0 0.05 0.1 0.15 0.2 0.25  Electron effective mass ( m e  ) Energy gap( eV )   Direct energy gap Indirect energy gap GaNZnSe CdS CdSe InSb GaAs ZnTe GaP AlAs GaSb ZnO CdTe InP InN InAs AlSb Figure 2.6: The electron e ective mass as a function of the energy gap for selected III-V and II-VI semiconductors. The direct energy gap materials are depicted with the solid points, while the indirect energy gap materials are represented with the open points. Most of the data depicted in this  gure is from Adachi [8]. The InN data, however, is from O’Leary et al: [15]. 212.6. Other material properties and their dependence on the energy gap 0 0.5 1 1.5 2 2.5 3 3.5 103 104 105 106 107    Breakdown field ( V/cm ) Energy gap ( eV ) Direct energy gap Indirect energy gap InSb InN CdTe GaAs AlSb ZnO GaN AlAs GaSb GaPInAs ZnSeZnTe CdSe CdS InP Figure 2.7: The breakdown electric  eld as a function of the energy gap for selected III-V and II-VI semiconductors. The direct energy gap materials are depicted with the solid points, while the indirect energy gap materials are depicted with the open points. The data depicted in this  gure is from Adachi [8]. 222.6. Other material properties and their dependence on the energy gap Thus, GaN devices  nd applications in high- eld settings. The same follows for other wide energy gap semiconductors. The static and high frequency relative dielectric constants also vary with the energy gap. In Figure 2.8, the static relative dielectric constant, ks =  s= o, is plotted as a function of the energy gap for a number of compound semiconductors, where  o denotes the dielectric function associ- ated with free space. In Figure 2.9, the high-frequency relative dielectric constant, k1 =  1= o, is plotted as a function of the energy gap for a number of compound semiconductors. It is seen that, in both cases, these optical constants diminish corresponding to increases in the energy gap. It is interesting to note that these relationships seem to cluster along two par- allel lines, for the di erent compound semiconductors that are considered, both for the case of the static relative dielectric constant and for the case of the high-frequency relative dielectric constant. The reasons for this are unknown at present. The electron a nity is also plotted as a function of the energy gap for the compound semiconductors considered in this analysis. This plot is depicted in Figure 2.10. It is seen that the electron a nity decreases in response to increases in the energy gap. This is likely arising as a consequence of the fact that large energy gap materials correspond to smaller energy di erences between the vacuum level and the conduction band edge, thereby leading to reduced electron a nities. 232.6. Other material properties and their dependence on the energy gap 0 0.5 1 1.5 2 2.5 3 3.5 0 2 4 6 8 10 12 14 16 18 20 Relative dielectric constant ( k s ) Energy gap ( eV )   Direct energy gap Indirect energy gap InSb GaSb InAs InN InP CdTe CdSe AlSb GaP AlAs GaN ZnO ZnTe CdS ZnSe GaAs Figure 2.8: The static relative dielectric constant, ks =  s= o, as a function of the energy gap for selected III-V and II-VI semiconductors. The direct energy gap materials are depicted with the solid points, while the indirect energy gap materials are depicted with the open points. The data depicted in this  gure is from Adachi [8]. 242.6. Other material properties and their dependence on the energy gap 0 0.5 1 1.5 2 2.5 3 3.5 0 2 4 6 8 10 12 14 16 18 20 Relative dielectric constant ( k ∞  ) Energy gap ( eV )   Direct energy gap Indirect energy gap InAs InN InSb GaSb GaAs InP AlSb GaP AlAs GaN ZnO CdTe CdSe ZnTe CdS ZnSe Figure 2.9: The high-frequency relative dielectric constant, k1 =  1= o, as a function of the energy gap for selected III-V and II-VI semiconductors. The direct energy gap materials are depicted with the solid points, while the indirect energy gap materials are depicted with the open points. Most of the data depicted in this  gure is from Adachi [8]. The InN data, however, is from O’Leary et al: [15]. 252.6. Other material properties and their dependence on the energy gap 0 0.5 1 1.5 2 2.5 3 3.5 0 1 2 3 4 5 6 7 Energy gap ( eV )  Electron affinity ( eV )   Direct energy gap Indirect energy gap GaN ZnO GaP AlAs ZnSe GaAs CdTe InP GaSb InAs InSb AlSb & ZnTe Figure 2.10: The electron a nity as a function of the energy gap for se- lected III-V and II-VI semiconductors. The direct energy gap materials are depicted with the solid points, while the indirect energy gap materials are depicted with the open points. The data depicted in this  gure is from Adachi [8]. 26Chapter 3 Electron drift mobility in semiconductors 3.1 Electron drift mobility and scattering mechanisms The low- eld electron drift mobility,  e, depends on the crystal tem- perature, the doping concentration, and the electron concentration. There are a variety of scattering processes that contribute to the low- eld electron drift mobility. These include, but are not limited to, ionized impurity scat- tering, polar optical phonon scattering, piezoelectric scattering, and acoustic deformation potential scattering. In this thesis, the dependence of the low-  eld electron drift mobility,  e, on crystal temperature, the doping concen- tration, and the electron concentration, corresponding to GaN, AlN, InN, and ZnO, will be examined for the case of bulk materials. The low- eld electron drift mobility that occurs for the case of electrons in a con ned two-dimensional electron gas, such as that that is present at the interface of a wide energy gap/narrow energy gap semiconductor discontinuity, is also evaluated for the case of these materials, deviations from the bulk results 273.2. The random thermal motion of charge carriers in the absence of an applied electric  eld being the focus of this particular analysis. This chapter is organized in the following manner. In Section 3.2, the motion of charge carriers within a semiconductor in the absence of an ap- plied electric  eld is discussed. Then, in Section 3.3, how the charge carriers drift under the action of an applied electric  eld is considered. The di er- ent scattering mechanisms that the electrons will encounter during electron transport are then enumerated in Section 3.4. The drift mobilities asso- ciated with each of the di erent scattering mechanisms considered in this analysis, i.e., ionized impurity scattering, polar optical phonon scattering, piezoelectric scattering, and acoustic deformation potential scattering, are then presented, analytical expressions, corresponding to these drift mobil- ities, being provided in Section 3.5. Finally, in Section 3.6, means of ana- lyzing the electron transport that occurs within a two-dimensional electron gas will be discussed. 3.2 The random thermal motion of charge carriers in the absence of an applied electric  eld In general, charge carriers within a semiconductor, i.e., electrons and holes, are generated and recombined. In thermal equilibrium, these pro- cesses are exactly balanced, i.e., the generation rate is exactly equal to the recombination rate. Before a charge carrier recombines, in the absence of an electric  eld, this charge carrier will randomly drift through the material. As 283.3. Electron transport under the action of an applied electric  eld it drifts, it will be scattered through interactions with the various scattering mechanisms that are available. Scattering processes tend to randomize the direction that a charge carrier will propagate in. A representative random electron trajectory, corresponding to a particular individual electron in the ensemble of electrons that are available, is depicted in Figure 3.1 [3]. Holes also exhibit random trajectories in the absence of an applied electric  eld. The electron drift velocity at any given time, t, corresponds to the aver- age of the electron velocities over the entire ensemble of electrons. Focussing on the x-direction, if the i{th electron has a velocity, vxi, in the x-direction, then the average velocity over the ensemble of electrons, or drift velocity, may be expressed as vdx  1 N [vx1 + vx2 + vx3 +   +vxN ]: (3.1) Given the randomness of the electron trajectories, as seen in Figure 3.1 [3], this velocity is nil in the absence of an applied electric  eld. 3.3 Electron transport under the action of an applied electric  eld If an electric  eld, E , is applied, a net drift velocity will occur. Be- tween scattering events, each electron will accelerate under the action of this applied electric  eld. Focussing on the motion of the i{th electron in the ensemble, if t = ti is the time of the last scattering event that this particular electron has experienced, and if the applied electric  eld in the x-direction 293.3. Electron transport under the action of an applied electric  eld                                                                                                      2                                                                                                                                                                                   4                                                                                              5                                                                                  1          e -                                                                              3                                                                                                                                                                              11                                                                                                                                                                                                         9                                                                                                                                                                                                                                                           e -                                                                                                                                10                       1 2                   6                                        7                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                    8                                                                                                                                                                                                                                                                                                                                                       Figure 3.1: The random thermal motion of a representative electron in a semiconductor crystal in the absence of an applied electric  eld. This par- ticular electron engages in scattering a number of times, the numbers 2 through 12 representing the scattering events that occur; number 1 cor- responds to this particular electron’s initial position. Between scattering events, the electron propagates with no acceleration. The scattering events are seen to randomize the direction of the electron transit. This  gure is after Streetman [3]. 303.3. Electron transport under the action of an applied electric  eld is Ex, then it can be shown that this particular electron’s velocity at time t is vx (i)(t) =  qEx m (t ti) + vx (i)(ti); (3.2) where q denotes the electron charge, m represents the electron e ective mass, and vx(i)(ti) denotes the velocity of this electron in the x-direction immediately following the last scattering event [14]. Averaging over the ensemble of electrons, i.e., using the de nition in Eq. (3.1), it may be shown that vdx =  q(t ti) m Ex; (3.3) where (t ti) denotes the mean time between scattering events, a term which is henceforth denoted  . The low- eld electron drift mobility,  e, provides for the constant of proportionality between the electron drift ve- locity and the applied electric  eld. That is, vdx =   eEx; (3.4) where  e = q m : (3.5) If  denotes the rate with which scattering processes interact with individual electrons, the mean time since the last scattering event,  = 1  : (3.6) 313.3. Electron transport under the action of an applied electric  eld Hence, the overall low- eld electron drift mobility may be expressed as  e = q m  ; (3.7) where q, m , and  are as previously de ned. It is instructive to introduce a low- eld electron drift mobility,  e, asso- ciated with each scattering process. For the j{th scattering process, with a scattering rate of  (j), the j{th low- eld electron drift mobility may be expressed as  (j) = q m  (j) ; (3.8) following the form presented in Eq. (3.7). As the overall scattering rate,  , may be expressed as a sum over these individual rates, i.e.,  = NX j=1  (j); (3.9) it is thus seen that  e = 1 NP j=1 1  (j)e : (3.10) That is, the overall electron drift mobility corresponds to the reciprocal of the sum over the individual reciprocal low- eld electron drift mobilities corresponding to the di erent scattering mechanisms. This overall electron drift mobility is henceforth referred to as the combined electron drift mo- bility. The remaining analysis presented in this chapter is focussed upon determining these individual contributions to the electron drift mobility. 323.4. Scattering mechanisms 3.4 Scattering mechanisms As was discussed earlier, the low- eld electron drift mobility,  e, de- pends on a variety of di erent scattering mechanisms. The various scattering mechanisms, in their totality, are depicted in Figure 3.2 [8]. The transport of electrons within a semiconductor is usually characterized by an energy band diagram for the conduction band with a number of valleys, i.e., the actual energy band diagram is approximated by a series of valleys, the low- est energy valley corresponding to areas in the energy band diagram in the immediate vicinity of the conduction band minimum. In Figure 3.3 [16, 17], for example, a three-valley representation for the conduction band energy band diagram associated with c-GaAs is contrasted with the corresponding one-dimensional representation of the full energy band diagram. There are important scattering mechanisms that occur, both within val- leys, i.e., intravalley scattering mechanisms, and between valleys, i.e., inter- valley scattering mechanisms. For the purposes of this analysis, the focus will be on the low- eld electron drift mobility,  e. As a result, interval- ley scattering mechanisms will not be considered. The particular scattering processes that will be focus of this subsequent analysis are the ionized im- purity scattering, polar optical phonon scattering, piezoelectric scattering, and acoustic deformation potential scattering mechanisms. 333.4. Scattering mechanisms Ionized  Optical Intervalley Intravalley Impurity   Space-charge  Alloy Acoustic Optical  Acoustic  Polar  Piezoelectric   Non Polar Deformation potential   Neutral Carrier-carrier Defect Phonon Figure 3.2: The di erent scattering mechanisms that can occur in semi- conductors. The particular scattering mechanisms that we examine for the purposes of this analysis are presented with the bold font. This  gure is after Adachi [8]. 343.4. Scattering mechanisms     Figure 3.3: The energy band diagram associated with c-GaAs. A three- valley representation of this band diagram for the conduction band is de- picted beneath a more exact one-dimensional representation of the energy band diagram corresponding to this material [16, 17]. 353.5. Analytical expressions for the contributions to the low- eld electron drift mobility 3.5 Analytical expressions for the contributions to the low- eld electron drift mobility As was discussed earlier, the overall low- eld electron drift mobility associated with a semiconductor may be related to the various scattering mechanisms that are present. For the purposes of this low- eld electron drift mobility analysis, four scattering mechanisms will be considered, these being the dominant scattering mechanisms present at low applied electric  elds, i.e., (1) ionized impurity scattering, (2) polar optical phonon scatter- ing, (3) piezoelectric scattering, and (4) acoustic deformation potential scat- tering. The approach adopted will involve the determination of the electron drift mobility associated with each individual scattering mechanism, and the evaluation of a combined, i.e., overall, form of the electron drift mobility using Eq. (3.10). 3.5.1 Ionized impurity scattering The impurities within a semiconductor play an important role in in u- encing the electron transport within these materials. Impurities are e cient scattering centres, especially when they are charged. Ionized donors or ac- ceptors in a semiconductor are a common example of such impurities. The amount of scattering due to the electrostatic forces between the carriers and the ionized impurities depends on the interaction time and the concentration of impurities that are present. Higher impurity concentrations and greater interaction times result in a lower electron drift mobility. The dependence on the interaction time accounts for the crystal temperature dependence 363.5. Analytical expressions for the contributions to the low- eld electron drift mobility of the low- eld electron drift mobility, i.e., greater crystal temperature re- sults in a higher thermal electron velocity, which in turn leads to a reduced interaction time and therefore an enhanced low- eld electron drift mobility. The low- eld electron drift mobility contribution corresponding to the ionized impurity scattering mechanism was shown by Shur et al: [18] to be  I = 25:5 3 "2s (kBT) 3=2 F3 NTq3m 1=2 n F1 ; (3.11) where F1 = Z 1 0 x1=2 1 + exp (x eF) dx; (3.12) F2 = Z 1 0 x1=2 exp (x eF) [1 + exp (x eF)] 2 dx; (3.13) F3 = Z 1 0 x3 exp (x eF)  i [1 + exp (x eF)] 2 dx; (3.14)  i = ln(1 +  )  1 +  ; (3.15)  = 8mn"s(kBT) 2 F1 q2~2nF2 x; (3.16) and where ~ denotes the reduced Planck’s constant, q is the electron charge, mn is the electron e ective mass, "s is the static dielectric constant, NT is the total concentration of ionized impurities, and n is the electron concentration. 373.5. Analytical expressions for the contributions to the low- eld electron drift mobility 3.5.2 Polar optical phonon scattering Scattering by lattice waves involves the absorption or emission of po- lar optical phonons by the electrons within the ensemble. These phonons represent quanta of mechanical waves that travel through the semiconductor crystal. Since the density of phonons in a solid increases with the crystal temperature, the mean time between scattering events corresponding to this particular mechanism will decrease with increases in the crystal tempera- ture. The low- eld electron drift mobility contribution corresponding to the polar optical phonon scattering mechanism was shown by Shur et al: [18] to be  po = 4 k o~2 qNpmn (1 5Vth=Eg) [2mnqEo (1 + Eo=Eg)] 1=2 ; (3.17) where k = 1=(1=k1  1=ks) is the coupling constant, k1 =  1= o and ks =  s= o denoting the high frequency and static relative dielectric con- stants, respectively,  o being the dielectric constant associated with free space, Np = 1=[exp(Eo=Vth) 1]; (3.18) being Planck’s function, representing the occupancy of the phonon states, Vth = kBT=q is the thermal voltage, Eo is the polar optical phonon energy, and Eg is the energy gap. 3.5.3 Piezoelectric scattering Piezoelectric scattering occurs as a consequence of the acoustic wave vibrations through the piezoelectric e ect. If the atoms in the semiconduc- 383.5. Analytical expressions for the contributions to the low- eld electron drift mobility tor crystal are partially ionized, and the displacement of the atoms due to acoustic vibrations produces an electric potential, piezoelectric scattering will occur. The magnitude of the piezoelectric scattering depends on the structure of the ionized atoms within the crystal. The magnitude of the piezoelectric scattering will be larger in crystals with lower symmetry. The scattering of electrons through the piezoelectric potential is referred to as piezoelectric scattering. Piezoelectric scattering plays an important role in compound semiconductors, particularly at low crystal temperatures. The low- eld electron drift mobility contribution corresponding to the piezoelec- tric scattering mechanism was shown by Shur et al: [18] to be  p = q p m ; (3.19) where  p = 2 ~2 sv1 q2K2avkBT 2Fp 3F1 ; (3.20) Fp = Z 1 0 x2 exp (x eF) Fpz [1 + exp (x eF)] 2 dx; (3.21) and Fpz = 1 + 1 1 +   2ln[1 +  ]  : (3.22) F1 is as de ned in Eq. (3.12) and  is as de ned in Eq. (3.16). Here, the electromechanical coe cient K2av = e214(12=cL + 16=cT) 35 s ; (3.23) 393.5. Analytical expressions for the contributions to the low- eld electron drift mobility where cL and cT denote the longitudinal and transverse elastic constants, respectively, e14 is the piezoelectric constant, "s is the static dielectric con- stant, and v1 is the thermal velocity, i.e., v1 = (2kBT=mn)1=2. 3.5.4 Acoustic phonon deformation potential scattering The thermal energy within a semiconductor material leads to  uctu- ations in the location of the atomic positions with time. These acoustic waves, as such vibrations are commonly referred to as, lead to deforma- tions in the material, an electrostatic potential being associated with these deformations. Shur et al: [18] demonstrate that the contribution to the low-  eld electron drift mobility associated with acoustic deformation potential scattering  ac = p 8 q~4cL 3E21mn 5=2(kBT) 3=2 ; (3.24) where cL is the longitudinal elastic constant of the semiconductor, E1 is the displacement of the band edge per unit dilation of the lattice, i.e., it is the deformation potential, and mn is the electron e ective mass. It is seen from Eq. (3.24) that  ac is inversely proportional to a polynomial function of the electron e ective mass and the crystal temperature. Increased crystal temperatures increase the intensity of the lattice vibrations that occur, and thus, the mean free time between the scattering events decreases, i.e., the mobility decreases. Thus, acoustic deformation potential scattering plays an important role, particularly at high temperatures[19]. 403.6. Electron transport in a two-dimensional electron gas 3.6 Electron transport in a two-dimensional electron gas At a heterojunction, such as the AlGaN/GaN interface in a high- frequency  eld e ect transistor con guration [20], a discontinuity in the po- tential energy associated with the conduction band is found, as is shown in Figure 3.4 [20]. When the band bending that occurs in a real device is taken into account, the resultant potential pro le is as shown in this  gure. Elec- tron from the bulk AlGaN and GaN layers will pour into the two-dimensional GaN quantum well, thus producing a channel within which charge carrier conduction will occur. This GaN channel is thus characterized with an elec- tron concentration that far exceeds that of the corresponding bulk material. These extra free electrons will act to screen the underlying distribution of ionized impurity atoms within the channel, i.e., a two-dimensional electron gas is formed within this channel with an enhanced low- eld electron drift mobility. This screening reduces the e ect of the ionized impurity scat- tering. This in turn impacts upon the overall resultant electron transport characteristics. A detailed and rigorous treatment of this problem is quite complex. The distribution of charge must be evaluated by simultaneously solving both Poisson’s and Schr odinger’s equations. This is beyond the scope of the present analysis. Instead, the strategy adopted here will be to examine how electron concentrations in excess of the bulk ionized impurity concentra- tion, for a  xed bulk ionized impurity concentration, will impact upon the dependence of the low- eld electron drift mobility on the crystal tempera- 413.6. Electron transport in a two-dimensional electron gas AlGaN  GaN  2DEG  EC  EF  EV  Figure 3.4: The energy band diagram of an AlGaN/GaN high electron mo- bility transistor. A quantum well is formed in the GaN layer close to the AlGaN/GaN interface. Mobile electrons, contributed from the AlGaN and GaN layers, are trapped in the quantum well that is present, thus forming a two-dimensional electron gas [20]. 423.6. Electron transport in a two-dimensional electron gas ture. In a bulk material at thermal equilibrium, charge neutrality mandates that the electron concentration, n, is exactly equal to the bulk ionized im- purity concentration, NT . In contrast, in the two-dimensional electron gas con guration, electron concentrations in excess of the bulk ionized impu- rity concentration are expected for the reasons suggested earlier. Thus, the adopted approach allows one to estimate the impact of the presence of a two- dimensional electron gas on the results within the framework of the already developed analysis. The additional electrons that are present in a two- dimensional electron gas contribute to increased screening. The increased screening that accompanies increased electron concentrations should allow for an increase in the resultant low- eld electron drift mobility,  e. The exact manner in which this occurs will be the focus of this analysis. The results of this analysis will be presented in Chapter 4. 43Chapter 4 Results 4.1 Overview The low- eld electron drift mobility,  e, is an important material prop- erty that provides a measure of the electron transport that occurs within a given material. The low- eld electron drift mobility,  e, provides an elec- tron transport metric with which various materials can be compared and contrasted. At low-applied electric  elds, the performance of a given elec- tron device may be directly related to this parameter. The low- eld electron drift mobility,  e, is known to be a strong function of the crystal tempera- ture, the doping concentration, and the electron concentration. Establishing how this low- eld electron drift mobility depends on these parameters is an important goal in the characterization of any semiconductor. From this de- pendence, the suitability of an electronic material for a particular electron device con guration may be assessed. In this chapter, the dependence of the low- eld electron drift mobility,  e, on the crystal temperature, the doping concentration, and the elec- tron concentration, will be examined for the cases of GaN, AlN, InN, and ZnO. The expressions presented in the previous chapter will be employed for the purposes of this analysis, the individual contributions to the low- 444.2. Material parameter selections  eld electron drift mobility related to the ionized impurity, polar optical phonon, piezoelectric, and acoustic deformation potential scattering mecha- nisms being combined to yield an overall low- eld electron drift mobility in the manner prescribed in Eq. (3.10). Initially, each of the materials will be considered individually. Later, comparisons between the di erent materials will be made. Finally, a contrast between results obtained corresponding to the case of a bulk semiconductor and the case of a con ned two-dimensional electron gas within such a semiconductor will be presented. This chapter is organized in the following manner. In Section 4.2, the material parameters, corresponding to bulk GaN, AlN, InN, and ZnO, are provided. Then, in Sections 4.3, 4.4, 4.5, and 4.6, the contributions to the low- eld electron drift mobility attributed to the ionized impurity, polar optical phonon, piezoelectric, and acoustic deformation potential scattering mechanisms, corresponding to GaN, AlN, InN, and ZnO, respectively, are presented. A comparison between these di erent semiconductor materials is then presented in Section 4.7. Finally, in Section 4.8, the dependence of the low- eld electron drift mobility on the crystal temperature for electrons in a two-dimensional electron gas, for the cases of GaN, AlN, InN, and ZnO, is considered. 4.2 Material parameter selections The low- eld electron drift mobility,  e, is a strong function of the ma- terial properties. Accordingly, in order to perform this analysis, the material parameters, corresponding to GaN, AlN, InN, and ZnO, must be speci ed. 454.2. Material parameter selections In Tables 4.1 and 4.2, the material parameters corresponding to these dif- ferent semiconductor materials are speci ed. The material parameters that are speci ed in these tables correspond to those that are important for low-  eld electron drift mobility determinations. These values are mostly from O’Leary et al: [15] and Albrecht et al: [21]. Table 4.1: The material parameters corresponding GaN, AlN, and InN. These parameter selections are from O’Leary et al: [15]. Parameter GaN AlN InN Longitudinal sound velocity (cm=s) 6:56 105 9:06 105 6:24 105 Transverse sound velocity (cm=s) 2:68 105 3:70 105 2:55 105 Acoustic deformation potential (eV) 8.3 9.5 7.1 Static dielectric constant 8.9 8.5 15.3 High-frequency dielectric constant 5.35 4.77 8.4 E ective mass ( 1valley) 0.20 me 0.48 me 0.045 me Piezoelectric constant, e14  C=cm2  3:75 10 5 9:2 10 5 3:75 10 5 Direct energy gap (eV) 3.39 6.2 0.75a Optical phonon energy (meV) 91.2 99.2 89.0 464.3. GaN bulk results Table 4.2: The material parameters corresponding to ZnO. The ZnO pa- rameter selections are from Albrecht et al: [21]. Parameter ZnO Longitudinal sound velocity (cm=s) 4:0 105 Transverse sound velocity (cm=s) 2:7 105 Acoustic deformation potential (eV) 3.83 Static dielectric constant 8.2 High-frequency dielectric constant 3.7 E ective mass ( 1valley) 0.17 me Piezoelectric constant, e14  C=cm2  3:75 10 5 Direct energy gap (eV) 3.40 Optical phonon energy (meV) 72.0 4.3 GaN bulk results The ionized impurity contribution to the low- eld electron drift mo- bility for the case of GaN is shown as a function of the crystal temperature in Figure 4.1. Three cases are considered for the purposes of this particular analysis; (1) the case of NT set to 1016 cm 3, (2) the case of NT set to 1017 cm 3, and (3) the case of NT set to 1018 cm 3, NT denoting the bulk ionized impurity concentration. In keeping with the general principle of bulk charge neutrality, the electron concentration, n, is set to the bulk ionized 474.3. GaN bulk results 100 200 300 400 500 600 700 800 900 1000 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 x 105  n=NT=10 16 cm−3  n=NT=10 17 cm−3  n=NT=10 18 cm−3 GaN Temperature (K) Mobility (c m2  / Vs ) Figure 4.1: The ionized impurity contribution to the low- eld electron drift mobility as a function of the crystal temperature for the case GaN. For all cases, the electron concentration, n, is set to the bulk ionized impurity concentration, NT . 484.3. GaN bulk results impurity concentration, NT , for all cases. It is seen that this contribution to the electron drift monotonically increases with the crystal temperature for all cases. This monotonic dependence occurs as the thermal velocity increases monotonically with the crystal temperature, and therefore, the in- teraction time with the ionized impurities is reduced with greater crystal temperatures. For a  xed crystal temperature, the electron drift mobility is seen to monotonically increase with a reduction in the ionized impurity con- centration. This dependence arises as a consequence of the increased ionized impurity scattering that occurs at higher ionized impurity concentrations. Polar optical phonon scattering is an important scattering mechanism that often plays a dominant role in shaping the electron transport properties that occur within compound semiconductors and their elemental counter- parts. In Figure 4.2, the contribution to the low- eld electron drift mobility associated with polar optical phonon scattering is plotted as a function of the crystal temperature for the case of GaN. It is seen that this contribu- tion to the low- eld electron drift mobility diminishes with increases in the crystal temperature. Greater crystal temperatures correspond to greater numbers of polar optical phonons. This results in the observed decrease in the low- eld electron drift mobility corresponding to increases in the crystal temperature. In Figure 4.3, the contribution to the low- eld electron drift mobility as- sociated with piezoelectric scattering is plotted as a function of the crystal temperature for various selections of the electron concentration for the case of GaN. As with the polar optical phonon case, it is seen that this contri- bution to the low- eld electron drift mobility monotonically decreases with 494.3. GaN bulk results 100 200 300 400 500 600 700 800 900 1000 101 102 103 104 105 106 Temperature (K) Mobility (c m2  / Vs ) GaN Figure 4.2: The polar optical phonon contribution to the low- eld electron drift mobility as a function of the crystal temperature for the case of GaN. 504.3. GaN bulk results 100 200 300 400 500 600 700 800 900 1000 0 2000 4000 6000 8000 10000 12000 14000  n=1016 cm−3  n=1017 cm−3  n=1018 cm−3 Temperature (K) Mobility (c m2  / Vs ) GaN Figure 4.3: The piezoelectric contribution to the low- eld electron drift mobility as a function of the crystal temperature for various selections of the electron concentration for the case of GaN. 514.3. GaN bulk results increases in the crystal temperature. For a  xed crystal temperature, it is seen that the contribution to the low- eld electron drift mobility increases monotonically with increases in the electron concentration. This trend oc- curs as increases in the electron concentration enhance the screening, thereby diminishing the strength of the piezoelectric scattering that occurs. In Figure 4.4, the contribution to the low- eld electron drift mobility as- sociated with acoustic phonon deformation potential scattering is plotted as a function of the crystal temperature for the case of GaN. It is seen that this contribution to the low- eld electron drift mobility monotonically decreases with increases in the crystal temperature. Greater crystal temperatures cor- respond to a larger number of acoustic deformation potential phonons, i.e., lattice position  uctuations of greater intensity. This results in the observed decrease in the low- eld electron drift mobility. In Figure 4.5, the contributions to the low- eld electron drift mobility associated with the polar optical phonon, the piezoelectric, and the acous- tic deformation potential scattering mechanisms are plotted as a function of the crystal temperature for the case of GaN. Using Eq. (3.10), the re- sultant combined low- eld electron drift mobility is also plotted as function of the crystal temperature. For the purposes of this analysis, ionized im- purity scattering is neglected, i.e., its contribution to the low- eld electron drift mobility is set to in nity. The electron concentration, n, is set to 1017 cm 3 for all cases. It is seen that the combined low- eld electron drift mobility monotonically decreases with increases in the crystal temperature; all of these contributions to the low- eld electron drift mobility exhibit the same trend in terms of their dependence on the crystal temperature. The 524.3. GaN bulk results 100 200 300 400 500 600 700 800 900 1000 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 x 104 Temperature (K) Mobility (c m2  / Vs ) GaN Figure 4.4: The acoustic deformation potential contribution to the low- eld electron drift mobility as a function of the crystal temperature for the case of GaN. 534.3. GaN bulk results 100 200 300 400 500 600 700 800 900 1000 102 103 104 105 106 Temperature (K) Mobility (c m2  / Vs ) GaNPolar optical Acoustic Piezoelectric Combined n=1017 cm−3 Figure 4.5: The polar optical phonon, piezoelectric, and acoustic deforma- tion potential contributions to the low- eld electron drift mobility, and the combined low- eld electron drift mobility, plotted as functions of the crystal temperature for the case of GaN. For the purposes of this analysis, the ion- ized impurity contribution is neglected, i.e., it is set to in nity. The electron concentration, n, is set to 1017 cm 3 for all cases. 544.4. AlN bulk results dependence of the combined low- eld electron drift mobility on the crystal temperature, for various electron concentration selections, is presented in Figure 4.6, the ionized impurity contribution to the low- eld electron drift mobility being neglected, as with Figure 4.5. With ionized impurity scattering introduced, the combined low- eld elec- tron drift mobility exhibits a very di erent dependence on the crystal tem- perature for the case of GaN, as is seen in Figure 4.7. For the purposes of this analysis, the bulk ionized impurity concentration, NT , is set to 1017 cm 3. As the contribution to the low- eld electron drift mobility associated with ionized impurity scattering monotonically increases with the crystal tem- perature, and the other contributions to the low- eld electron drift mobility exhibit a contrary functional dependence, it should come as no surprise that at low crystal temperatures the low- eld electron drift mobility exhibits a dependence that is dominated by ionized impurity scattering while at high crystal temperatures, a dependence that is dominated by the polar optical phonon, piezoelectric, and acoustic deformation potential scattering mecha- nisms occurs. In Figure 4.8, the combined low- eld electron drift mobility is depicted for various electron and ionized impurity concentrations as a func- tion of the crystal temperature. For all cases, the electron concentration, n, is set to the bulk ionized impurity concentration, NT . 4.4 AlN bulk results The ionized impurity contribution to the low- eld electron drift mo- bility for the case of AlN is shown as a function of the crystal temperature 554.4. AlN bulk results 100 200 300 400 500 600 700 800 900 1000 0 1000 2000 3000 4000 5000 6000 7000 8000 9000  n=1016 cm−3 Temperature (K) Mobility (c m2  / Vs ) GaN  n=1017 cm−3  n=1018 cm−3 Figure 4.6: The combined low- eld electron drift mobility as a function of the crystal temperature with the polar optical phonon, piezoelectric, and acoustic deformation potential scattering mechanisms taken into account for various electron concentrations for the case of GaN. For all cases, the ionized impurity contribution is neglected, i.e., it is set to in nity. 564.4. AlN bulk results 100 200 300 400 500 600 700 800 900 1000 102 103 104 105 106 107 108 Temperature (K) Mobility (c m2 /Vs ) n=NT=10 17 cm−3 GaN Polar optical Piezoelectric Acoustic Impurity Total Figure 4.7: The ionized impurity, polar optical phonon, piezoelectric, and acoustic deformation potential contributions to the low- eld electron drift mobility as a function of the crystal temperature for the case of GaN. The bulk ionized impurity concentration, NT , is set to 1017 cm 3. The electron concentration, n, is set to the bulk ionized impurity concentration, NT . 574.4. AlN bulk results 100 200 300 400 500 600 700 800 900 1000 103 104 n=NT=10 16 cm−3  n=NT=10 17 cm−3  n=NT=10 18 cm−3 Temperature (K) Mobility (c m2 /Vs ) GaN Figure 4.8: The combined low- eld electron drift mobility as a function of the crystal temperature for the case of GaN. The contributions to the low-  eld electron drift mobility considered include those related to the ionized impurity, polar optical phonon, piezoelectric, and acoustic deformation po- tential scattering mechanisms. For all cases, the electron concentration, n, is set to the bulk ionized impurity concentration, NT . 584.4. AlN bulk results in Figure 4.9. Three cases are considered for the purposes of this partic- ular analysis; (1) the case of NT set to 1016 cm 3, (2) the case of NT set to 1017 cm 3, and (3) the case of NT set to 1018 cm 3, NT denoting the bulk ionized impurity concentration. In keeping with the general principle of bulk charge neutrality, the electron concentration, n, is set to the bulk ionized impurity concentration, NT , for all cases. It is seen that this contri- bution to the low- eld electron drift mobility monotonically increases with the crystal temperature for all cases, as was observed for the case of GaN; recall Figure 4.1. This monotonic dependence occurs as the thermal veloc- ity increases monotonically with the crystal temperature, and therefore, the interaction time with the ionized impurities is reduced with greater crystal temperatures. For a  xed crystal temperature, the low- eld electron drift mobility is seen to monotonically increase with a reduction in the ionized impurity concentration. This dependence arises as a consequence of the in- creased ionized impurity scattering that occurs at higher ionized impurity concentrations. In Figure 4.10, the contribution to the low- eld electron drift mobility associated with polar optical phonon scattering is plotted as a function of the crystal temperature for the case of AlN. It is seen that this contribu- tion to the low- eld electron drift mobility diminishes with increases in the crystal temperature, as was observed for the case of GaN; recall Figure 4.2. Greater crystal temperatures correspond to greater numbers of polar optical phonons. This results in the observed decrease in the low- eld electron drift mobility corresponding to increases in the crystal temperature. In Figure 4.11, the contribution to the low- eld electron drift mobility 594.4. AlN bulk results 100 200 300 400 500 600 700 800 900 1000 0 0.5 1 1.5 2 2.5 3 x 104 Temperature (K) Mobility (c m2  / Vs )  n=NT=10 16 cm−3  n=NT=10 17 cm−3  n=NT=10 18 cm−3 AlN Figure 4.9: The ionized impurity contribution to the low- eld electron drift mobility as a function of the crystal temperature for the case AlN. For all cases, the electron concentration, n, is set to the bulk ionized impurity concentration, NT . 604.4. AlN bulk results 100 200 300 400 500 600 700 800 900 1000 101 102 103 104 105 106 Temperature (K) Mobility (c m2  / Vs ) AlN Figure 4.10: The polar optical phonon contribution to the low- eld electron drift mobility as a function of the crystal temperature for the case of AlN. 614.4. AlN bulk results 100 200 300 400 500 600 700 800 900 1000 0 200 400 600 800 1000 1200 n=1016 cm−3 n=1017 cm−3 n=1018 cm−3  Temperature (K) Mobility (c m2  / Vs ) AlN Figure 4.11: The piezoelectric contribution to the low- eld electron drift mobility as a function of the crystal temperature for various selections of the electron concentration for the case of AlN. 624.4. AlN bulk results associated with piezoelectric scattering is plotted as a function of the crystal temperature for various selections of the electron concentration for the case of AlN. As with the polar optical phonon case, it is seen that this contri- bution to the low- eld electron drift mobility monotonically decreases with increases in the crystal temperature, as was observed for the case of GaN; recall Figure 4.3. For a  xed crystal temperature, it is seen that the contri- bution to the low- eld electron drift mobility increases monotonically with increases in the electron concentration. This trend occurs as increases in the electron concentration enhance the screening, thereby diminishing the strength of the piezoelectric scattering that occurs. In Figure 4.12, the contribution to the low- eld electron drift mobility associated with acoustic phonon deformation potential scattering is plotted as a function of the crystal temperature for the case of AlN. It is seen that this contribution to the low- eld electron drift mobility monotonically de- creases with increases in the crystal temperature, as was observed for the case of GaN; recall Figure 4.4. Greater crystal temperatures correspond to a larger number of acoustic deformation potential phonons, i.e., lattice po- sition  uctuations of greater intensity. This results in the observed decrease in the low- eld electron drift mobility. In Figure 4.13, the contributions to the low- eld electron drift mobility associated with the polar optical phonon, the piezoelectric, and the acoustic deformation potential scattering mechanisms are plotted as a function of the crystal temperature for the case of AlN. Using Eq. (3.10), the resul- tant combined low- eld electron drift mobility is also plotted as function of the crystal temperature. For the purposes of this analysis, ionized impurity 634.4. AlN bulk results 100 200 300 400 500 600 700 800 900 1000 0 1000 2000 3000 4000 5000 6000 Temperature (K) Mobility ( c m2  / Vs ) AlN Figure 4.12: The acoustic deformation potential contribution to the low-  eld electron drift mobility as a function of the crystal temperature for the case of AlN. 644.4. AlN bulk results 100 200 300 400 500 600 700 800 900 1000 102 103 104 105 106 Temperature (K) Mobility (c m2  / Vs ) n=1017 cm−3Polar optical Acoustic Piezoelectric Combined AlN Figure 4.13: The polar optical phonon, piezoelectric, and acoustic defor- mation potential contributions to the low- eld electron drift mobility, and the combined low- eld electron drift mobility, plotted as functions of the crystal temperature for the case of AlN. For the purposes of this analysis, the ionized impurity contribution is neglected, i.e., it is set to in nity. The electron concentration, n, is set to 1017 cm 3 for all cases. 654.4. AlN bulk results scattering is neglected, i.e., its contribution to the low- eld electron drift mobility is set to in nity. The electron concentration, n, is set to 1017 cm 3 for all cases. It is seen that the combined low- eld electron drift mobility monotonically decreases with increases in the crystal temperature, as was observed for the case of GaN; recall Figure 4.5; all of these contributions to the low- eld electron drift mobility exhibit the same trend in terms of their dependence on the crystal temperature. The dependence of the com- bined low- eld electron drift mobility on the crystal temperature, for various electron concentration selections, is presented in Figure 4.14, the ionized im- purity contribution to the low- eld electron drift mobility being neglected, as with Figure 4.13. With ionized impurity scattering introduced, the combined low- eld elec- tron drift mobility exhibits a very di erent dependence on the crystal tem- perature for the case of AlN, as is seen in Figure 4.15. For the purposes of this analysis, the bulk ionized impurity concentration, NT , is set to 1017 cm 3. As the contribution to the low- eld electron drift mobility as- sociated with ionized impurity scattering monotonically increases with the crystal temperature, and the other contributions to the low- eld electron drift mobility exhibit a contrary functional dependence, it should come as no surprise that at low crystal temperatures the low- eld electron drift mo- bility exhibits a dependence that is dominated by ionized impurity scatter- ing while at high crystal temperatures, a dependence that is dominated by the polar optical phonon, piezoelectric, and acoustic deformation potential scattering mechanisms occurs, as was observed for the case of GaN; recall Figure 4.7. In Figure 4.16, the combined low- eld electron drift mobility is 664.4. AlN bulk results 100 200 300 400 500 600 700 800 900 1000 0 200 400 600 800 1000 1200 n=1016 cm−3 n=1017 cm−3 n=1018 cm−3  AlN Temperature (K) Mobility (c m2  / Vs ) Figure 4.14: The combined low- eld electron drift mobility as a function of the crystal temperature with the polar optical phonon, piezoelectric, and acoustic deformation potential scattering mechanisms taken into account for various electron concentrations for the case of AlN. For all cases, the ionized impurity contribution is neglected, i.e., it is set to in nity. 674.4. AlN bulk results 100 200 300 400 500 600 700 800 900 1000 102 103 104 105 106 n=NT=10 17 cm−3 AlN Temperature (K) Mobility (c m2  / Vs ) Impurity Acoustic Polar optical Total Piezoelectric Figure 4.15: The ionized impurity, polar optical phonon, piezoelectric, and acoustic deformation potential contributions to the low- eld electron drift mobility as a function of the crystal temperature for the case of AlN. The bulk ionized impurity concentration, NT , is set to 1017 cm 3. The electron concentration, n, is set to the bulk ionized impurity concentration, NT . 684.4. AlN bulk results 100 200 300 400 500 600 700 800 900 1000 0 100 200 300 400 500 600 n=NT=10 16 cm−3 n=NT=10 17 cm−3 n=NT=10 18 cm−3 AlN Temperature (K) Mobility (c m2  / Vs ) Figure 4.16: The combined low- eld electron drift mobility as a function of the crystal temperature for the case of AlN. The contributions to the low-  eld electron drift mobility considered include those related to the ionized impurity, polar optical phonon, piezoelectric, and acoustic deformation po- tential scattering mechanisms. For all cases, the electron concentration, n, is set to the bulk ionized impurity concentration, NT . 694.5. InN bulk results depicted for various electron and ionized impurity concentrations as a func- tion of the crystal temperature. For all cases, the electron concentration, n, is set to the bulk ionized impurity concentration, NT . 4.5 InN bulk results The ionized impurity contribution to the low- eld electron drift mo- bility for the case of InN is shown as a function of the crystal temperature in Figure 4.17. Three cases are considered for the purposes of this particu- lar analysis; (1) the case of NT set to 1016 cm 3, (2) the case of NT set to 1017 cm 3, and (3) the case of NT set to 1018 cm 3, NT denoting the bulk ionized impurity concentration. In keeping with the general principle of bulk charge neutrality, the electron concentration, n, is set to the bulk ionized impurity concentration, NT , for all cases. It is seen that this contribution to the low- eld electron drift mobility monotonically increases with the crystal temperature for all cases, as was observed for the cases of GaN and AlN; re- call Figures 4.1 and 4.9. This monotonic dependence occurs as the thermal velocity increases monotonically with the crystal temperature, and there- fore, the interaction time with the ionized impurities is reduced with greater crystal temperatures. For a  xed crystal temperature, the low- eld electron drift mobility is seen to monotonically increase with a reduction in the ion- ized impurity concentration. This dependence arises as a consequence of the increased ionized impurity scattering that occurs at higher ionized impurity concentrations. In Figure 4.18, the contribution to the low- eld electron drift mobility 704.5. InN bulk results 100 200 300 400 500 600 700 800 900 1000 0 0.5 1 1.5 2 2.5 3 x 105  n=NT=1 x 10 17 cm−3  n=NT=1 x 10 18 cm−3  n=NT=1 x 10 16 cm−3 InN Temperature (K) Mobility (c m2  / Vs ) Figure 4.17: The ionized impurity contribution to the low- eld electron drift mobility as a function of the crystal temperature for the case InN. For all cases, the electron concentration, n, is set to the bulk ionized impurity concentration, NT . 714.5. InN bulk results 100 200 300 400 500 600 700 800 900 1000 101 102 103 104 105 106 Temperature (K) Mobility ( c m2  / Vs ) InN Figure 4.18: The polar optical phonon contribution to the low- eld electron drift mobility as a function of the crystal temperature for the case of InN. 724.5. InN bulk results associated with polar optical phonon scattering is plotted as a function of the crystal temperature for the case of InN. It is seen that this contribu- tion to the low- eld electron drift mobility diminishes with increases in the crystal temperature, as was observed for the cases of GaN and AlN; recall Figures 4.2 and 4.10. Greater crystal temperatures correspond to greater numbers of polar optical phonons. This results in the observed decrease in the low- eld electron drift mobility corresponding to increases in the crystal temperature. In Figure 4.19, the contribution to the low- eld electron drift mobility associated with piezoelectric scattering is plotted as a function of the crystal temperature for various selections of the electron concentration for the case of InN. In contrast to the cases of GaN and AlN, however, a distinctive non-monotonic dependence of the low- eld electron drift mobility on the crystal temperature is observed for the case of the electron concentration, n, set to 1018 cm 3. In contrast, a monotonic dependence on the crystal temperature is observed for the cases of n set to 1016 cm 3 and 1017 cm 3, as was found for the cases of GaN and AlN; recall Figures 4.3 and 4.11. This probably arises as a consequence of the fact that the Fermi energy level is in the conduction band for this particular free electron concentration selection, i.e., the doping is degenerate. For the other cases, however, it is seen that this contribution to the low- eld electron drift mobility monotonically decreases with increases in the crystal temperature. In Figure 4.20, the contribution to the low- eld electron drift mobility associated with acoustic phonon deformation potential scattering is plotted as a function of the crystal temperature for the case of InN. It is seen that this 734.5. InN bulk results 100 200 300 400 500 600 700 800 900 1000 0 1 2 3 4 5 6 x 105 Temperature (K) Mobility (c m2  / Vs ) InN n=1018 cm−3 n=1017 cm−3 n=1016 cm−3 Figure 4.19: The piezoelectric contribution to the low- eld electron drift mobility as a function of the crystal temperature for various selections of the electron concentration for the case of InN. 744.5. InN bulk results 100 200 300 400 500 600 700 800 900 1000 0 0.5 1 1.5 2 2.5 3 3.5 4 x 106 Temperature (K) Mobility (c m2  / Vs ) InN Figure 4.20: The acoustic deformation potential contribution to the low-  eld electron drift mobility as a function of the crystal temperature for the case of InN. 754.5. InN bulk results contribution to the low- eld electron drift mobility monotonically decreases with increases in the crystal temperature, as was observed for the cases of GaN and AlN; recall Figures 4.4 and 4.12. Greater crystal temperatures correspond to a larger number of acoustic deformation potential phonons, i.e., lattice position  uctuations of greater intensity. This results in the observed decrease in the low- eld electron drift mobility. In Figure 4.21, the contributions to the low- eld electron drift mobility associated with the polar optical phonon, the piezoelectric, and the acoustic deformation potential scattering mechanisms are plotted as a function of the crystal temperature for the case of InN. Using Eq. (3.10), the resultant combined low- eld electron drift mobility is also plotted as function of the crystal temperature. For the purposes of this analysis, ionized impurity scattering is neglected, i.e., its contribution to the low- eld electron drift mobility is set to in nity. The electron concentration, n, is set to 1017 cm 3 for all cases. It is seen that the combined low- eld electron drift mobility monotonically decreases with increases in the crystal temperature, as was observed for the cases of GaN and AlN; recall Figures 4.5 and 4.13; all of these contributions to the low- eld electron drift mobility exhibit the same trend in terms of their dependence on the crystal temperature. The dependence of the combined low- eld electron drift mobility on the crystal temperature, for various electron concentration selections, is presented in Figure 4.22, the ionized impurity contribution to the low- eld electron drift mobility being neglected, as with Figure 4.21. With ionized impurity scattering introduced, the combined low- eld elec- tron drift mobility exhibits a very di erent dependence on the crystal tem- 764.5. InN bulk results 100 200 300 400 500 600 700 800 900 1000 103 104 105 106 107 108 Temperature (K) Mobility ( c m2  / Vs ) InNPolar optical Acoustic Piezoelectric Combined n=1017 cm−3 Figure 4.21: The polar optical phonon, piezoelectric, and acoustic defor- mation potential contributions to the low- eld electron drift mobility, and the combined low- eld electron drift mobility, plotted as functions of the crystal temperature for the case of InN. For the purposes of this analysis, the ionized impurity contribution is neglected, i.e., it is set to in nity. The electron concentration, n, is set to 1017 cm 3 for all cases. 774.5. InN bulk results 100 200 300 400 500 600 700 800 900 1000 0 1 2 3 4 5 6 x 105 n=1016 cm−3 n=1018 cm−3 n=1017 cm−3  Temperature (K) Mobility (c m2  / Vs ) InN Figure 4.22: The combined low- eld electron drift mobility as a function of the crystal temperature with the polar optical phonon, piezoelectric, and acoustic deformation potential scattering mechanisms taken into account for various electron concentrations for the case of InN. For all cases, the ionized impurity contribution is neglected, i.e., it is set to in nity. 784.6. ZnO bulk results perature for the case of InN, as is seen in Figure 4.23. For the purposes of this analysis, the bulk ionized impurity concentration, NT , is set to 1017 cm 3. As the contribution to the low- eld electron drift mobility associated with ionized impurity scattering monotonically increases with the crystal tem- perature, and the other contributions to the low- eld electron drift mobility exhibit a contrary functional dependence, it should come as no surprise that at low crystal temperatures the low- eld electron drift mobility exhibits a dependence that is dominated by ionized impurity scattering while at high crystal temperatures, a dependence that is dominated by the polar optical phonon, piezoelectric, and acoustic deformation potential scattering mech- anisms occurs, as was observed for the cases of GaN and AlN; recall the Figures 4.7 and 4.15. In Figure 4.24, the combined low- eld electron drift mobility is depicted for various electron and ionized impurity concentra- tions as a function of the crystal temperature. For all cases, the electron concentration, n, is set to the bulk ionized impurity concentration, NT . 4.6 ZnO bulk results The ionized impurity contribution to the low- eld electron drift mo- bility for the case of ZnO is shown as a function of the crystal temperature in Figure 4.25. Three cases are considered for the purposes of this par- ticular analysis; (1) the case of NT set to 1016 cm 3, (2) the case of NT set to 1017 cm 3, and (3) the case of NT set to 1018 cm 3, NT denoting the bulk ionized impurity concentration. In keeping with general principle of bulk charge neutrality, the electron concentration, n, is set to the bulk 794.6. ZnO bulk results 100 200 300 400 500 600 700 800 900 1000 102 103 104 105 106 107 108 Temperature (K) Mobility (c m2 /Vs ) n=NT=10 17 cm−3 InN Polar optical Acoustic Piezoelectric Impurity Total Figure 4.23: The ionized impurity, polar optical phonon, piezoelectric, and acoustic deformation potential contributions to the low- eld electron drift mobility as a function of the crystal temperature for the case of InN. The bulk ionized impurity concentration, NT , is set to 1017 cm 3. The electron concentration, n, is set to the bulk ionized impurity concentration, NT . 804.6. ZnO bulk results 100 200 300 400 500 600 700 800 900 1000 0 1 2 3 4 5 6 7 8 9 x 104 n=NT=10 16 cm−3 n=NT=10 17 cm−3 n=NT=10 18 cm−3 InN Temperature (K) Mobility (c m2 /Vs ) Figure 4.24: The combined low- eld electron drift mobility as a function of the crystal temperature for the case of InN. The contributions to the low-  eld electron drift mobility considered include those related to the ionized impurity, polar optical phonon, piezoelectric, and acoustic deformation po- tential scattering mechanisms. For all cases, the electron concentration, n, is set to the bulk ionized impurity concentration, NT . 814.6. ZnO bulk results 100 200 300 400 500 600 700 800 900 1000 0 2 4 6 8 10 12 14 x 104  n=NT=10 16 cm−3  n=NT=10 17 cm−3  n=NT=10 18 cm−3 Temperature (K) Mobility (c m2 /Vs ) ZnO Figure 4.25: The ionized impurity contribution to the low- eld electron drift mobility as a function of the crystal temperature for the case ZnO. For all cases, the electron concentration, n, is set to the bulk ionized impurity concentration, NT . 824.6. ZnO bulk results ionized impurity concentration, NT , for all cases. It is seen that this contri- bution to the low- eld electron drift mobility monotonically increases with the crystal temperature for all cases, as was observed for the cases of GaN, AlN, and InN; recall Figures 4.1, 4.9, and 4.17. This monotonic dependence occurs as the thermal velocity increases monotonically with the crystal tem- perature, and therefore, the interaction time with the ionized impurities is reduced with greater crystal temperatures. For a  xed crystal temperature, the low- eld electron drift mobility is seen to monotonically increase with a reduction in the ionized impurity concentration. This dependence arises as a consequence of the increased ionized impurity scattering that occurs at higher ionized impurity concentrations. In Figure 4.26, the contribution to the low- eld electron drift mobility associated with polar optical phonon scattering is plotted as a function of the crystal temperature for the case of ZnO. It is seen that this contribution to the low- eld electron drift mobility diminishes with increases in the crystal temperature, as was observed for the cases of GaN, AlN, and InN; recall Figures 4.2, 4.10, and 4.18. Greater crystal temperatures correspond to greater numbers of polar optical phonons. This results in the observed decrease in the low- eld electron drift mobility corresponding to increases in the crystal temperature. In Figure 4.27, the contribution to the low- eld electron drift mobility associated with piezoelectric scattering is plotted as a function of the crystal temperature for various selections of the electron concentration for the case of ZnO. As with the polar optical phonon case, it is seen that this contri- bution to the electron drift mobility monotonically decreases with increases 834.6. ZnO bulk results 100 200 300 400 500 600 700 800 900 1000 101 102 103 104 105 106 Temperature (K) Mobility (c m2 /Vs ) ZnO Figure 4.26: The polar optical phonon contribution to the low- eld electron drift mobility as a function of the crystal temperature for the case of ZnO. 844.6. ZnO bulk results 100 200 300 400 500 600 700 800 900 1000 0 2000 4000 6000 8000 10000 12000 Temperature (K) Mobility (c m2 /Vs )  ZnOn=10 17 cm−3 n=1016 cm−3 n=1018 cm−3 Figure 4.27: The piezoelectric contribution to the low- eld electron drift mobility as a function of the crystal temperature for various selections of the electron concentration for the case of ZnO. 854.6. ZnO bulk results in the crystal temperature, as was observed for the cases of GaN, AlN, and InN for low electron concentrations; recall Figures 4.3, 4.11, and 4.15. For a  xed crystal temperature, it is seen that the contribution to the low- eld electron drift mobility increases monotonically with increases in the electron concentration. This trend occurs as increases in the electron concentration enhance the screening, thereby diminishing the strength of the piezoelectric scattering that occurs. In Figure 4.28, the contribution to the low- eld electron drift mobility associated with acoustic phonon deformation potential scattering is plotted as a function of the crystal temperature for the case of ZnO. It is seen that this contribution to the low- eld electron drift mobility monotonically decreases with increases in the crystal temperature, as was observed for the cases of GaN, AlN, and InN; recall Figures 4.4, 4.12, and 4.20. Greater crystal temperatures correspond to a larger number of acoustic deformation potential phonons, i.e., lattice position  uctuations of greater intensity. This results in the observed decrease in the low- eld electron drift mobility. In Figure 4.29, the contributions to the low- eld electron drift mobility associated with the polar optical phonon, the piezoelectric, and the acoustic deformation potential scattering mechanisms are plotted as a function of the crystal temperature for the case of ZnO. Using Eq. (3.10), the resultant combined low- eld electron drift mobility is plotted as function of the crystal temperature. For the purposes of this analysis, ionized impurity scattering is neglected, i.e., its contribution to the low- eld electron drift mobility is set to in nity. The electron concentration, n, is set to 1017 cm 3 for all cases. It is seen that the combined low- eld electron drift mobility monotonically 864.6. ZnO bulk results 100 200 300 400 500 600 700 800 900 1000 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 x 105 Temperature (K) Mobility (c m2 /Vs ) ZnO Figure 4.28: The acoustic deformation potential contribution to the low-  eld electron drift mobility as a function of the crystal temperature for the case of ZnO. 874.6. ZnO bulk results 100 200 300 400 500 600 700 800 900 1000 102 103 104 105 106 Temperature (K) Mobility (c m2 /Vs ) ZnOPolar optical Acoustic Piezoelectric Combined n=1017 cm−3 Figure 4.29: The polar optical phonon, piezoelectric, and acoustic defor- mation potential contributions to the low- eld electron drift mobility, and the combined low- eld electron drift mobility, plotted as functions of the crystal temperature for the case of ZnO. For the purposes of this analysis, the ionized impurity contribution is neglected, i.e., it is set to in nity. The electron concentration, n, is set to 1017 cm 3 for all cases. 884.6. ZnO bulk results decreases with increases in the crystal temperature, as was observed for the cases of GaN, AlN, and InN; recall Figures 4.5, 4.13, and 4.21; all of these contributions to the low- eld electron drift mobility exhibit the same trend in terms of their dependence on the crystal temperature. The dependence of the combined low- eld electron drift mobility on the crystal temperature, for various electron concentration selections, is presented in Figure 4.30, the ionized impurity contribution to the low- eld electron drift mobility being neglected, as with Figure 4.29. With ionized impurity scattering introduced, the combined low- eld electron drift mobility exhibits a very di erent dependence on the crys- tal temperature for the case of ZnO, as is seen in Figure 4.31. For the purposes of this analysis, the bulk ionized impurity concentration, NT , is set to 1017 cm 3. As the contribution to the low- eld electron drift mo- bility associated with ionized impurity scattering monotonically increases with the crystal temperature, and the other contributions to the low- eld electron drift mobility exhibit a contrary functional dependence, it should come as no surprise that at low crystal temperatures the low- eld electron drift mobility exhibits a dependence that is dominated by ionized impurity scattering while at high crystal temperatures, a dependence that is domi- nated by the polar optical phonon, piezoelectric, and acoustic deformation potential scattering mechanisms occurs, as was observed for the cases of GaN, AlN, and InN; recall the Figures 4.7, 4.15, and 4.23. In Figure 4.32, the combined low- eld electron drift mobility is depicted for various electron and ionized impurity concentrations as a function of the crystal tempera- ture. For all cases, the electron concentration, n, is set to the bulk ionized 894.6. ZnO bulk results 100 200 300 400 500 600 700 800 900 1000 0 0.5 1 1.5 2 2.5 3 x 104 Temperature (K) Mobility (c m2 /Vs ) ZnO n=1018 cm−3 n=1017 cm−3 n=1016 cm−3 Figure 4.30: The combined low- eld electron drift mobility plotted as a function of the crystal temperature with the polar optical phonon, piezo- electric, and acoustic deformation potential scattering mechanisms taken into account for various electron concentrations for the case of ZnO. For all cases, the ionized impurity contribution is neglected, i.e., it is set to in nity. 904.6. ZnO bulk results 100 200 300 400 500 600 700 800 900 1000 102 103 104 105 106 Temperature (K) Mobility (c m2 /Vs ) n=NT=10 17 cm−3 ZnO Polar optical Acoustic Piezoelectric Impurity Total Figure 4.31: The ionized impurity, polar optical phonon, piezoelectric, and acoustic deformation potential contributions to the low- eld electron drift mobility as a function of the crystal temperature for the case of ZnO. The bulk ionized impurity concentration, NT , is set to 1017 cm 3. The electron concentration, n, is set to the bulk ionized impurity concentration, NT . 914.6. ZnO bulk results 100 200 300 400 500 600 700 800 900 1000 0 1000 2000 3000 4000 5000 6000 n=NT=10 16 cm−3 n=NT=10 17 cm−3 n=NT=10 18 cm−3 ZnO Temperature (K) Mobility (c m2 /Vs ) Figure 4.32: The combined low- eld electron drift mobility as a function of the crystal temperature for the case of ZnO. The contributions to the low- eld electron drift mobility considered include those related to the ion- ized impurity, polar optical phonon, piezoelectric, and acoustic deformation potential scattering mechanisms. For all cases, the electron concentration, n, is set to the bulk ionized impurity concentration, NT . 924.7. A comparison of the bulk results impurity concentration, NT . 4.7 A comparison of the bulk results In Figure 4.33, the dependence of the low- eld electron drift mobility on the crystal temperature is depicted for all the semiconductor materials that were considered in the analysis, i.e., GaN, AlN, InN, and ZnO. For all cases, the ionized impurity, polar optical phonon, piezoelectric, and acous- tic deformation potential scattering mechanisms are included, Eq. (3.10) allowing for the determination of the combined low- eld electron drift mo- bility. It is noted that InN exhibits the highest combined low- eld electron drift mobility when compared with the other materials. This undoubtedly arises as a consequence its light conduction band e ective mass; recall Fig- ure 2.6 and Table 4.1. In contrast, AlN exhibits the lowest low- eld electron drift mobility, this being due to its heavy conduction band e ective mass; recall Table 4.1. ZnO and GaN exhibit a low- eld electron drift mobility de- pendence on the crystal temperature that coincides for temperatures below 150 K, but separates for temperatures in excess of 150 K. The greater high temperature low- eld electron drift mobility associated with GaN is related to the fact that its polar optical phonon energy is larger than that associated with ZnO; from Table 4.1, it is seen that the polar optical phonon energy, i.e., associated with GaN and ZnO are 91.2 and 72 meV, respectively. This implies that more thermal energy will be required to produce polar optical phonons in GaN than in ZnO. Thus, at a given temperature, the population of polar optical phonons will be greater for the case of ZnO than for GaN. 934.7. A comparison of the bulk results 100 200 300 400 500 600 700 800 900 1000 102 103 104 105  n=NT=1 x 10 17 cm−3 Temperature (K) Mobility (c m2  / Vs ) AlN GaN InN ZnO Figure 4.33: The combined low- eld electron drift mobility dependence on the crystal temperature for the cases of GaN, AlN, InN, and ZnO. For cases, the bulk ionized impurity is set to 1017 cm 3. For all cases, the electron concentration, n, is set to the bulk ionized impurity concentration, NT . 944.8. Two-dimensional electron gas As a result, GaN exhibits a higher low- eld electron drift mobility; polar optical phonons play a very small role at low temperatures, thus, there is very little di erence in the low temperature results for these materials. 4.8 Two-dimensional electron gas At the interface of a wide energy gap/narrow energy gap semicon- ductor discontinuity, i.e., a heterojunction, a two-dimensional electron gas can accumulate; recall Figure 3.4. The electron concentration within the resultant quantum well can exceed the bulk ionized impurity concentration, electrons from the neighbouring regions pouring into such a well. Thus, in order to study how the low- eld electron drift mobility within the two- dimensional electron gas is in uenced by the presence of such a well, with the bulk ionized impurity concentration, NT , held  xed at 1017 cm 3, the depen- dence of the low- eld electron drift mobility on the electron concentration, n, is examined, increases in the electron concentration beyond the bulk ion- ized impurity concentration being expected. This dependence is determined for GaN, AlN, InN, and ZnO, as is shown in Figures 4.34, 4.35, 4.36, and 4.37. For all cases, the ionized impurity, polar optical phonon, piezoelectric, and acoustic deformation potential scattering mechanisms are considered, Eq. (3.10) being used in order to evaluate the resultant combined low- eld electron drift mobility. The observed enhancements in the combined elec- tron drift mobility within the two-dimensional electron gas beyond the bulk results is quite signi cant in many of the cases. The exact reasons for these functional dependencies lies beyond the scope of this examination. 954.8. Two-dimensional electron gas 100 200 300 400 500 600 700 800 900 1000 102 103 104 Temperature (K) Mobility (c m2 /Vs ) NT=10 17 cm−3 GaN n=2 x 1018 cm−3 n=1 x 1018 cm−3 n=5 x 1017 cm−3 n=2  x 1017 cm−3 n=1017 cm−3 Figure 4.34: The combined low- eld electron drift mobility as a function of the crystal temperature for the case of GaN for various electron concentra- tions. The bulk ionized impurity concentration, NT , is set to 1017cm 3 for all cases. 964.8. Two-dimensional electron gas 100 200 300 400 500 600 700 800 900 1000 102 103 NT=10 17 cm−3 Temperature (K) Mobility (c m2 /Vs ) AlN n=1018 cm−3 n=2 x 1018 cm−3 n=1017 cm−3 n=5 x 1017 cm−3 n=2 x 1017 cm−3 Figure 4.35: The combined low- eld electron drift mobility as a function of the crystal temperature for the case of AlN for various electron concentra- tions. The bulk ionized impurity concentration, NT , is set to 1017cm 3 for all cases. 974.8. Two-dimensional electron gas 100 200 300 400 500 600 700 800 900 1000 0 1 2 3 4 5 6 x 104 Temperature (K) Mobility (c m2 /Vs ) NT=10 17 cm−3 InN n=1018 cm−3 n=5 x 1017 cm−3 n=1017 cm−3 n=2  x 1017 cm−3 n=2 x 1018 cm−3 Figure 4.36: The combined low- eld electron drift mobility as a function of the crystal temperature for the case of InN for various electron concentra- tions. The bulk ionized impurity concentration, NT , is set to 1017cm 3 for all cases. 984.8. Two-dimensional electron gas 100 200 300 400 500 600 700 800 900 1000 102 103 104 Temperature (K) Mobility (c m2 /Vs ) NT=10 17 cm−3 ZnO n=1 x 1018 cm−3 n=5 x 1017 cm−3 n=2  x 1017 cm−3 n=2 x 1018 cm−3 n=1017 cm−3 Figure 4.37: The combined low- eld electron drift mobility as a function of the crystal temperature for the case of ZnO for various electron concentra- tions. The bulk ionized impurity concentration, NT , is set to 1017cm 3 for all cases. 99Chapter 5 Conclusions In conclusion, the dependence of the low- eld electron drift mobility associated with a number of wide energy gap semiconductors on the crystal temperature was determined. The materials considered in this analysis in- cluded the wide energy gap semiconductors GaN, AlN, and ZnO. InN was also considered as it is a material that can be alloyed with GaN to produce a wide energy gap semiconductor alloy. The bulk results were determined assuming that the electron concentration, n, is equal to the bulk ionized im- purity concentration, NT , for all the cases. It was found that the low- eld electron drift mobility associated with InN is greater than those associated with GaN and ZnO, which in turn are greater than that exhibited by AlN. This is due to the light e ective mass associated with the electrons in InN and the much heavier e ective mass associated with the electrons in AlN; the e ective mass of the electrons within GaN and ZnO is between these extreme cases. The two-dimensional electron gas results were obtained by examining the sensitivity of the results to variations in the electron concen- tration, n, with the bulk ionized impurity concentration, NT , set to a  xed value, the variations in the electron concentration beyond the bulk ionized donor concentration arising as a result electrons from the neighbourhood 100Chapter 5. Conclusions of a two-dimensional quantum well leading to an increase in the electron concentration beyond the bulk ionized impurity concentration, NT . These electrons within the resultant two-dimensional electron gas will act to screen the ionized impurities, and thus, lead to dramatic enhancements in the low-  eld electron drift mobility beyond the bulk results. In this thesis, a number of original contributions have been made. The assembly of material parameters corresponding to the wide energy gap semi- conductors GaN, AlN, and ZnO represents a useful contribution to the liter- ature, as does that corresponding to the narrow energy gap semiconductor InN. The dependence of the low- eld electron drift mobility on the crystal temperature, something that had heretofore only been done for the case of GaN by Shur et al: [18], also represents a useful addition to the literature. The identi cation of the individual contributions to this mobility, something that had only been done for the case of GaN by Shur et al: [18], also rep- resents an original contribution to the literature. Finally, an examination of the low- eld electron drift mobility within a two-dimensional electron gas con guration is an novel contribution that points to fertile avenues for further investigation. There are a number of issues that could be further explored in the near term future. These topics for further investigation include the application of the present low- eld electron drift mobility results to electron device mod- eling, the further development of expressions for the various contributions to the low- eld electron drift mobility, and a more proper analysis of the distribution of charge, and its e ect on the low- eld electron drift mobility, associated with the two-dimensional electron gas that is present at the inter- 101Chapter 5. Conclusions face of a wide energy gap/narrow energy gap semiconductor discontinuity, such as that present at the AlGaN/GaN interface. Each of these topics is brie y addressed below. The low- eld electron drift mobility results that have been developed within this thesis may be applied for the purposes of electron device model- ing. The transport of electrons within an electron device plays an important role in determining the performance of such a device. While high- eld elec- tron device application are beyond the scope of the low- eld analysis that has been presented within this thesis, low- eld electron device applications abound. The use of the obtained low- eld electron drift mobility results to an examination of the potential of each material considered for electron de- vice applications would certainly be a worthwhile pursuit. Comparisons of the performance of various materials and device con gurations could greatly bene t the community, and would be relatively easy to implement. A second important theme that could be explored in the near term future would be a critical comparison between the low- eld electron drift mobility results obtained using the analytical expressions of Shur et al: [18] and those determined by others, such as the analyses of Nag [19], Sze [14], Chin et al: [22, 23], and Walukiewicz et al: [24]. A deeper examination of the analyses of Shur et al: [18] may allow for an assessment of the physicality of these expressions to the particular semiconductor material that are being exam- ined; in the initial analysis of Shur et al: [18], the low- eld electron drift mobility associated with GaN was the focus of attention, the applicability of these results to other semiconductor materials being worthy of further investigation. 102Chapter 5. Conclusions A  nal possible topic for investigation, that builds upon these results, is the development of a simultaneous Poisson’s and Schr odinger’s equa- tion solver that will allow for the determination of the amount of charge within the two-dimensional electron gas at the boundary of a wide energy gap/narrow energy gap semiconductor discontinuity. This would be a fun- damental contribution to the  eld that would be extremely helpful for future electron device design considerations. 103References [1] I. M. Ross, \The foundation of the silicon age," Physics Today, vol. 50, pp. 34{39, 1997. [2] R. Muller, Device Electronics for Integrated Circuits. New York: Wiley, 2003. [3] B. Streetman, Solid state electronic devices. Upper Saddle River, N.J: Pearson/Prentice Hall, 2006. [4] M. A. Green, \Intrinsic concentration, e ective densities of states, and e ective mass in silicon," Journal of Applied Physics, vol. 67, no. 6, pp. 2944 {2954, mar 1990. [5] J. B. Casady and R. W. Johnson, \Status of silicon carbide (SiC) as a wide-bandgap semiconductor for high-temperature applications: A review," Solid State Electronics, vol. 39, no. 96, pp. 1409{1422, 1996. [6] U. Hilmi, \A thermodynamic model for determining pressure and tem- perature e ects on the bandgap energies and other properties of some semiconductors," Solid State Electronics, vol. 35, no. 9, pp. 1343 { 1352. [7] D. G. Avery, D. W. Goodwin, W. D. Lawson, and T. S. Moss, \Optical 104References and photo-electrical properties of indium antimonide," Proceedings of the Physical Society. Section B, vol. 67, no. 10, p. 761, 1954. [8] S. Adachi, Properties of group-IV, III-V and II-VI semiconductors, ser. Wiley series in materials for electronic and optoelectronic applications. John Wiley & Sons, 2005. [9] H. P. Maruska and J. J. Tietjen, \The preparation and propeties of vapor-deposited single-crystal-line GaN," Applied Physics Letters, vol. 15, no. 10, pp. 327{329, 1969. [10] W. M. Yim, E. J. Stofko, P. J. Zanzucchi, J. I. Pankove, M. Ettenberg, and S. L. Gilbert, \Epitaxially grown AlN and its optical band gap," Journal of Applied Physics, vol. 44, pp. 292{296, 2009. [11]  Ozg ur, Ya, C. Liu, A. Teke, M. A. Reshchikov, S. Dougan, V. Avrutin, S. J. Cho, and H. Morkocc, \A comprehensive review of ZnO materials and devices," Journal of Applied Physics, vol. 98, no. 4, pp. 041 301+, 2005. [12] S. M. Malik, \A density of states and joint density of states analysis of hydrogenated amorphous silicon," Master’s thesis, University of Regina, 2002. [13] Y. P. Varshni, \Temperature dependence of the energy gap in semicon- ductors," Physica, vol. 34, no. 1, pp. 149{154, 1967. [14] S. M. Sze, Physics of Semiconductor Devices, 2nd ed. Wiley- Interscience, Nov. 1981. 105References [15] S. K. O’Leary, B. E. Foutz, M. S. Shur, and L. F. Eastman, \Steady- State and Transient Electron Transport Within the IIIV Nitride Semi- conductors, GaN, AlN, and InN: A Review," Journal of Materials Sci- ence: Materials in Electronics, vol. 17, pp. 87{126, 2006-02-08. [16] J. S. Blakemore, \Semiconducting and other major properties of gallium arsenide," Journal of Applied Physics, vol. 53, pp. R123{R181, 2009. [17] J. R. Chelikowsky and M. L. Cohen, \Nonlocal pseudopotential calcu- lations for the electronic structure of eleven diamond and zinc-blende semiconductors," Phys. Rev. B, vol. 14, pp. 556{582, Jul 1976. [18] M. Shur, B. Gelmont, and M. A. Khan, \Electron mobility in two- dimensional electron gas in AlGaN/GaN heterostructures and in bulk GaN," Journal of Electronic Materials, vol. 25, no. 5, pp. 777{785, 1996. [19] B. R. Nag, Electron transport in compound semiconductors. Springer- Verlag, Berlin, New York, 1980. [20] L. Shen, S. Heikman, B. Moran, R. Co e, N.-Q. Zhang, D. Buttari, I. P. Smorchkova, S. Keller, S. P. DenBaars, and U. K. Mishr, \Al- GaN/AlN/GaN High-Power Microwave HEMT," IEEE Electron Device Letters, vol. 22, pp. 457{459, 2001. [21] J. D. Albrecht, P. P. Ruden, S. Limpijumnong, W. R. L. Lambrecht, and K. F. Brennan, \High  eld electron transport properties of bulk ZnO," Journal of Applied Physics, vol. 86, no. 12, pp. 6864{6867, 1999. [22] V. W. L. Chin, T. L. Tansley, and T. Osotchan, \Electron mobilities in 106References gallium, indium, and aluminum nitrides," Journal of Applied Physics, vol. 75, no. 11, pp. 7365 {7372, jun 1994. [23] V.W.L. and Chin, \The e ect of carrier densities and compensation ratios on the electron mobility of inasxp1-x," Journal of Physics and Chemistry of Solids, vol. 53, no. 7, pp. 897 { 904, 1992. [24] W. Walukiewicz, H. E. Ruda, J. Lagowski, and H. C. Gatos, \Electron mobility in modulation-doped heterostructures," Phys. Rev. B, vol. 30, pp. 4571{4582, Oct 1984. 107

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