Summary of the PhD thesis Theoretical and mathematical physics: Electronic transport in semiconductor nanostructure based on polar material AlGaN/GaN and Penta-Graphene nanoribbon

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The purposes of research "Electronic transport in semiconductor nanostructure based on polar material AlGaN/GaN and Penta-Graphene nanoribbon" studying on electronic transport phenomena in semiconductor nano structures such as AlGaN/GaN and penta-graphene nanoribbon.

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Nội dung Text: Summary of the PhD thesis Theoretical and mathematical physics: Electronic transport in semiconductor nanostructure based on polar material AlGaN/GaN and Penta-Graphene nanoribbon

MINISTRY OF EDUCATION VIETNAM ACADEMY OF AND TRAINING SCIENCE AND TECHNOLOGY GRADUATE UNIVERSITY SCIENCE AND TECHNOLOGY ……..….***………… PHAM THI BICH THAO ELECTRONIC TRANSPORT IN SEMICONDUCTOR NANOSTRUCTURE BASED ON POLAR MATERIAL AlGaN/GaN AND PENTA-GRAPHENE NANORIBBON Speciality: Theoretical and mathematical physics Code: 9 44 01 03 SUMMARY OF THE PHD THESIS Ha Noi – 2020
This thesis was completed at Graduate University of Science and Technology, Vietnam Academy of Science and Technology Supervisors: Assoc. Prof. Dr. Nguyen Thanh Tien Prof. Dr. Doan Nhat Quang Referee 1: Assoc. Prof. Dr. Dinh Van Trung Referee 2: Prof. Dr. Dao Tien Khoa Referee 3: Dr. Pham Ngoc Dong This dissertation will be defended in front of the evaluating assembly at academy level, place of defending: meeting room, Graduate University of Science and Technology, Vietnam Academy of Science and Technology. This thesis can be studied at: - The Library of Graduate University of Science and Technology - The Vietnam National Library
1 Introduction Nowadays, semiconductor technology is one of the most important fields in the development of science and technology. Semiconductor technology is a foundation of the information society that has been motivating human so- ciety forward by changing in production, living, communication and even in human. In semiconductor technology, semiconductor materials play a cru- cial role. The first transistor was invented in 1947 based on germanium (Ge) semiconductor with a band gap at room temperature of 0.66 eV. The first integrated circuit was born in 1958 and the bulk integrated circuit appeared in 1961 using germanium and silicon (Si) with a band gap of 1.12 eV. Since 1965, silicon has become the main material for semiconductor integrated circuits. Today, most semiconductor, integrated circuit or photovoltaic in- dustries are still based on silicon. Silicon and germanium are often referred as the first generation of semi- conductors. The second generation including gallium arsenide (GaAs, band gap at room temperature is 1.42 eV) and indium phosphide (InP, band gap at room temperature 1.35 eV) was introduced in the 1970s. The second gen- eration is primarily used in high-speed devices, microwave devices and inte- grated circuits. Besides the larger band gap, the electron mobility of GaAs is more than six times larger than that of silicon. In addition, the saturation velocity of GaAs is higher, i.e. two times larger than that of silicon. There- fore, devices based on GaAs are suitable for high-frequency operations. In addition, field-effect transistors based on GaAs also have advantages such as low noise, high performance, ect. However, GaAs has lower thermal con- ductivity and disruptive potential than semiconductors like GaN and SiC have, resulting in capacity limitations. At the end of the twentieth century, the third generation of semicon- ductors (wide band gap) such as gallium nitride (GaN, band gap at room temperature: 3.45 eV) and silicon carbide (SiC, band gap in room temper- ature: 3.25 eV for 4H-SiC) show remarkable features, which have attracted a lot of attention. III nitride semiconductors including GaN, InN, AlN and their heterostructures can be widely applied to electronic and optoelectronic
2 devices. The band gap of these structures range from the near infrared (0.7 eV, InN) to the far ultraviolet (6.2 eV, AlN). In particular, compared with conventional III-V and II-IV semiconductors, spontaneous and piezoelectric polarization in GaN and AlN with wuztzite structure is about ten times larger. Therefore, GaN, AlGaN / GaN, InGaN / GaN, ect can be applied to high electronic mobility transistor and heterojunction field effect transistor. Currently, semiconductor heterostructure is widely used in many fields due to its great advantages. Specifically, in the telecommunication field with semiconductor transistors, satellite television, warning systems, ect; Energy field with solar cell, light-emitting diode, information storage device, ect; Medical field with water filtration system, data processing system, ect. Many studies show that the electric and optical properties of heterostructure semiconductor vary significantly compared to those of the bulk semiconduc- tor and by the external field. Moreover, the heterostructure semiconductor also has superior intrinsic properties. One of the properties is that the po- larization depends on the direction of the material and the structure of the material, especially the low-dimensional structures. Therefore, the het- erostructure semiconductor has been an attractive topic in modern material research in recent decades. Strong polarization effect exists in many materials such as GaN, ZnO, MgO, InN, ect. Although there have been a number of studies on the ef- fects of polarization on the electric properties of these structures, these constructions have not been fully and systematically studied. In particular, the heterostructure semiconductor with polarizing confinement effect needs to be studied extensively. In addition, the relationship between confinement effects and electronic transport characteristics should be studied in more detail. Along with the heterostructures, the allotropes of carbon are currently at- tracting much attention. Until the mid-twentieth century, the two most com- mon allotropes of carbon in nature were diamonds and graphite. Graphite is a material consisting of two layers of carbon (2D) atoms arranged in a hexagonal lattice. Because only 2 of the 3 orbitals p make bonds, there is a unpaired orbit, pz , which can be used in electron transport. There- fore, graphite is a good conductor. The next carbon allotrope, Buckminster Fullerenes, was discovered in 1984 by Richard Smalley et. al. Next, the single-walled carbon nanotubes were discovered by Iijima in 1990. Intensive study of graphene, a two-dimensional (2D) material consist- ing of carbon atoms in the hexagonal network, began in 2004. The most notable characteristic of graphene is the thinnest crystal with extremely hardness, outstanding elasticity and thermal conductivity. With a two- dimensional structure and a large surface area of about 2675 m2 /g, graphene exhibits some unique physical properties. Unlike three-dimensional crystals, all atoms in graphene are surface atoms, i.e. they can participate in vari-
3 ous chemical reactions and interactions. This creates great prospects for changing the properties of graphene. Over the years, many theoretical and empirical studies for graphene have been carried out. Specifically, the syn- thesis of single-layer, multi-layer graphene or graphene nanoribbon on the metal substrate were performed. The electronic, chemical, magnetic and electrochemical properties of graphene have also been considered. Although graphene has excellent physical and chemical properties, graphene is a gapless material, which makes it difficult to apply graphene in field effect transistors and other electronic devices. In 2015, a new allotrope of carbon, penta-graphene is predicted by Zhang et. al. Penta-graphene (PG) exhibits mechanical and kinetic stability even when the temperature reaches 1000 K. In addition, penta-graphene has a direct band gap about 3.25 eV, which is higher than that of other allotropes of carbon. Penta-graphene exhibits many unique electric, thermal and opti- cal properties. Studies on penta-graphene also show that hydrogenation can increase the thermal conductivity of PG. PG doping Si, B, N Ge and Sn can reduce the band gap of PG, while doping transition metal can increase or decrease the band gap and simultaneously greatly enhances the absorption of hydrogen. With a large surface area ratio and band gap, PG is beneficial for adsorption of gas molecules. Recently, due to the superior physical and chemical properties, PG has been studied through theoretical calculations and showed that they have great potential for application in the field of nanoelectronics, nano mechanics and catalysts. From penta-graphene, one can form four types of penta-graphene nanorib- bon (PGNR) with different boundary forms. Investigation of the electronic properties of four types of PGNRs shows that structures exhibit semicon- ductor or metal properties. A number of works have focused on the elec- tronic properties of pure PGNR structures, passivation by different atoms or magnetic properties of PGNR forms. From the above analysis, the effect of polarization on electronic trans- port in low-dimensional systems, in particular 2D systems, need to be stud- ied in detail. In addition, the electronic transport of novel material like graphene, namely penta-graphene nanoribbon, should be investigated more fully. Therefore, problems such as polar heterostructure, electronic proper- ties in low-dimensional systems have to consider the influence of polarized charge, structural properties and electronic transport properties in penta- graphene nanoribbon will be presented in this thesis. The purposes of research Studying on electronic transport phenomena in semiconductor nano struc- tures such as AlGaN/GaN and penta-graphene nanoribbon. The ob jects of research Electronic properties and electronic transport phenomena in AlGaN/GaN
4 and penta-graphene nanoribbon. The contents of research - Overview of AlGaN/GaN and penta-graphene materials. - Electronic confinement phenomenon in semiconductor nanostructures. - Electronic transport phenomenon in semiconductor nanostructures Al- GaN/GaN and penta-graphene nanoribbon. The methods of research - Using the variational method to determine the electronic properties and deriving analytical expressions related to the mobility specific to the electronic transport phenomenon in the AlGaN/GaN system. - Using numerical methods, programming by Mathematica to identify the variational parameters and illustrate the physical quantities graphically. - Using the density functional theory and the non-equilibrium Green's function to investigate electronic characteristics (band structure, density of state, ...) and electronic transport properties (I(V), T(E), ...) in penta- graphene nanoribbon material system. - Using Origin software to process data. - Comparing with several experimental results. The structure of thesis The thesis is presented as follows: Introduction: An overview of the thesis. Content Chapter 1: Overview of research materials. Chapter 2: Electronic distribution in AlGaN / GaN structure. Chapter 3: Electronic transport phenomenon in AlGaN / GaN structure. Chapter 4: Phenomenon of electronic transport in doped penta - graphene nanoribbon. Conclusion: Summarizing the contributions of the thesis and stating prospects for further research. (Chapter 2 is presented according to the published content 1, chapter 3 is presented according to the published content 2, chapter 4 is presented according to the published content 3).
5 Chapter 1 Overview of research materials 1.1 Heterostructure AlGaN/GaN 1.1.1 Heterostructure When the lattice mismatch between between the substrate and the layers or between the layer and the layer is suitable, an ideal crystal was made by two different material. These structures are called heterostructures. The discontinuity of the material arising in the heterojunction leads to a change in some important electrical and optical properties, such as carrier confine- ment (due to the discontinuity of the conduction band or the valence band) or radiation confinement (due to band gab discontinuity). The heterostruc- ture is made from a thin semiconductor layer (about 100 nm), sandwiched between two other semiconductor layers creating a potential well in the con- duction or valence band and is often referred to as a quantum well. (QW) (eg AlGaAs/GaAs/AlGaAs). Heterostructures are largely based on semi- conductor alloys. Semiconductor alloys can consist of two elements, three elements or four elements (for example, GaAs, InAs, InP, GaAs, GaP, Al- GaAs, InGaAsP, ...). 1.1.2 Polar heterostructure Polarization is an important characteristic of group III nitride semicon- ductors. Wurtzite group III nitride structures do not have inversion symme- try along the c axis. The strong ionicity of the metal - nitrogen bond results in a strong polarization along the crystall direction [0001]. The polarizing effect occurs in group III nitride structures when the strain is zero, so this polarization is called spontaneous polarization. The non-ideal of the lattice will affect the spontaneous polarization intensity.
6 Figure 1.1: Atomic arrangement on Ga and N surface of GaN crystal. Arrows indicate spontaneous polarization. The additional polarization in group III nitride structures due to strain is called piezoelectric polarization Pz . The calculations for interface bound charge and two-dimensional electronic gas (2DEG) depend on both piezo- electric polarization Pz and spontaneous polarity Psp . The value of Psp in a material is constant, while Pz is a function of strain and can be determined from the following expression: a − a0 C13 Pz = 2 e31 − e33 (1.1) a0 C33 with a0 is the lattice constant when the system in not affected by stress or compression, a is the lattice constant when the system is affected by stress or compression, e31 and e33 as piezoelectric coefficients. C13 and C33 as elastic constants. 1.1.3 The effect of polarized charge on electronic trans- port in AlGaN/GaN polar heterojunction In AlGaN/GaN polar heterostructure, 2DEG density is very high, can reach n2d ≈ 2 × 1013 cm−2 and moblity at room temperature µ ≈ 1500 cm2 /V.s. The 2DEG sheet carrier density can be modulated by varying the AlGaN barrier thickness as well as the Al content. When a strong polariza- tion occurs (∼ 1 MVcm−1 ), the electrons are electrostaticly repelled close to AlGaN/GaN interface and the center of wave function is brought close to the heterojunction. This results in an increase in alloy disorder scattering
7 and surface roughness scattering. These scattering processes predominate at low temperatures, and even at room temperature when 2DEG is high density. In polar heterostructures, a new scattering mechanism does not exist in non-polar semiconductors and weak polarized semiconductors called dipole scattering. This is due to the alloy disorder, resulting in the dipole moments in each unit cell being non-periodic with the lattice. The charge distribution in the direction of [0001] of the polar heterostruc- turs can be determined by solving the Schr odinger equation and the Poisson equation in the approximate effective mass. 1.2 Graphene and penta-graphene 1.2.1 Graphene Graphene is a single graphite layer with high carrier mobility, in the range of 20005000 cm2 /Vs. Therefore, graphene is applicable to field-effect transistors (FET) operating at high frequencies. Graphene also exhibits high optical transparency of up to 97.7% which should be a potential candidate for solar cell, storing three-dimensional data. In addition, graphene also has high thermal conductivity, high Young's modulus and large surface area. However, graphene is a gapless structure, so it is limited for applications in the field of optoelectronics. As a result, different methods have been implemented to open the band gap of graphene such as doping, changing edge, applied field, ... 1.2.2 Graphene nanoribbon Graphene nanoribbon (GNR) is a one-dimensional structure formed by cutting graphene in different crystal directions. Based on the edge of graphene nanoribbon, there are two types of graphene nanoribbon: zigzag graphene nanoribbons (ZGNR) and armchair graphene nanoribbons (AGNR). In gen- eral, the properties of GNR are sensitive to many factors, such as doping, defects, edge changes, adsorption and external electric fields. This offers many opportunities to tune and expand GNR applications. Among the pro- posed methods, doping is one of the most frequently ways to adjust the properties of GNR. 1.2.3 Penta-graphene Penta-graphene can be exfoliated from T12-carbon, has an intrinsic bandgap about 3.25 eV and contains both sp2 and sp3 hybridization. Since its dis-
8 covery, penta-graphene has attracted much attention by a number of pre- eminent properties to become a potential candidate in the field of opto- electronics. Many studies on the structural properties of penta-graphene have been conducted. Studied results of penta-graphene doped by B, N and Si showed that band gap of penta-graphene area depends not only on the doped element but also the doped position. In addition, the bond lengths and the bond angles of penta-graphene structure is also affected by doping, passivation, ... Figure 1.2: Penta-graphene nanoribbons. 1.2.4 Penta-graphene nanoribbon So far, 4 types of penta-graphene nanoribbon (PGNR) have been created: zigzag PGNR (ZZPGNR), armchair PGNR (AAPGNR), zigzag-armchair PGNR (ZAPGNR) and sawtooth PGNR (SSPGNR) (Figure 1.2). According to the research results of Yuan et al., SSPGNR is the most durable structure when considering the same structural width in four types of PGNR. The analysis of band structure also show that the ZZPGNR, AAPGNR, ZAPGNR structures show the metallic properties, while SSPGNR is a semiconductor.
9 1.3 AlGaN/GaN-based high mobility electronic tran- sistors and graphene based field-effect transistor High mobility electronic transistors (HEMT) are basically heterojunctions formed by semiconductors having a different band-gap. GaN-based HEMTs have the same structure as regular GaAs-based HEMTs. However, in AlGaN/GaN HEMT does not require doping. Instead, elec- trons due to spontaneous polarization appear in the wurtzite structure GaN. The accumulation of free carriers results in high carrier concentrations at the interface leading to 2DEG channels. Two-dimensional electron gas is a function of the barrier, AlGaN layer thickness and positive charge at the interface. Inheriting the traditional bipolar transistors research, the production and research of graphene field effect transistors (GFET) have been implemented. Thanks to the superior properties of graphene, GFET can be effectively applied in a range of various technologies. Conclusions In this chapter, AlGaN/GaN polar heterostructure, penta-graphene nanorib- bon and their properties are presented. The analysis showed that AlGaN/GaN polar heterostructure and penta-graphene nanoribbon are suitable materials for optoelectronic devices. However, to put into practical application, they need to be investigated in detail about the electronic distribution, mobility, electronic properties, I(V) curve, ... These properties are governed by scat- tering processes. An important new scattering process to be investigated is scattering due to polarized charge. Moreover, the effect of polarized charge on the electronic transport properties of the AlGaN/GaN material system should also be considered. For penta-graphene nanoribbon material systems, the effects of doping, passivation, applied field, ... on structural and trans- port properties should be investigated in more detail. The posed problems will be systematically studied in subsequent chapters for specific systems with analytic calculations and detailed numerical calculations for electronic distribution, mobility, and band structure, density of state, transmission spectrum and I(V) curve.
10 Chapter 2 Electron distribution in AlGaN/GaN polar heterojunctions This chapter will present two-dimensional electron gas distribution in AlGaN/GaN polar modulation-doped heterojunction. Different from back- ground doping, modulation doping can help to limit scattering by ionized impurities and create confining potential for 2DEG. The research model is shown in Figure 2.1. AlGaN/GaN polar modulation-doped heterojunction is made up of two junction layers of AlGaN and GaN with polarization- charge density σP , doping thickness and donor bulk density, respectively Ld and NI , spacer Ls . In this section, the role of 2DEG and ionisation will be considered. In addition, the role of the interface polarization charges and the ionized impurities will be compared which has not been done in previous works. Figure 2.1: Modulated doped model in AlGaN/GaN structure.
11 2.1 Variational wave function for heterojunctions of finite potential barrier AlGaN/GaN polar heterojunctions of group III Nitrides will be consid- ered. At low temperature, the 2DEG is assumed to primarily occupy the lowest subband. In the realistic model of triangular QWs with a finite po- tential barrier, the electron state may be well described by a FangHoward wave function modified by Ando: Aκ1/2 exp (κz/2) z < 0, ζ (z) = 1/2 (2.1) Bk (kz + c) exp (−kz/2) z > 0. In equation (2.1), z < 0 for AlGaN; z > 0 for GaN. A and B are normal- ization parameters which are given in normalization conditions, κ v k are half the wave numbers which are determined through the boundary ζ (z) v 0 ζ (z) at z = 0. From the boundary and normalization conditions, We have a system of equations describing the relationship between A, B, c, κ and k , as follows: Aκ1/2 = Bk 1/2 c, Aκ3/2 2 = Bk 3/2 (1 − (2.2) c/2) , A2 + B 2 c2 + 2c + 2 = 1. 2.2 Confining potentials in a polar modulation-doped heterostructure The system is studied along the growth direction z ( perpendicular to the surface), is fixed by the Hamiltonian: H = T + Vtot (z) , (2.3) T is the kinetic energy and Vtot (z) is the overall confining potential: Vtot (z) = Vb (z) + Vσ (z) + VH (z) , (2.4) where Vb (z) , Vσ (z) , VH (z) are potential barrier, interface polarization charges and Hartree potential. Potential barrier with a finite height V0 at the interface plane z = 0: Vb (z) = V0 θ (−z) , (2.5)
12 where θ (z) as a unity step function. The potential barrier height is fixed by the conduction band offset between the AlGaN and GaN layers: V0 = ∆Ec (x), x as the alloy (Al) content in the AlGaN barrier. Interface polarization charges potential: Ddue to piezoelectric and spontaneous polarizations in a nitride-based strained HS there exist positive polarization charges bound on the interface. These charges create a uniform normal electric field with the potential given by: 2π Vσ (z) = eσ |z| . (2.6) εa where σ as their total density, e as electron charge and εa is the average value of the dielectric constants of AlGaN and GaN. Hartree potential is generated by the electrostatic field of the ionized bulk donor and 2DEG in heterostructure, determined by the Poisson equa- tion: d2 4πe2 dz 2 V H (z) = εa [NI (z) − n (z)] . (2.7) In which, NI (z) and n (z) are the density of donors along the growth direction and the one of electrons. Sample is modulation-doped: NI −zd ≤ z ≤ −zs , NI (z) = (2.8) 0 z < −zd , z > −zs . where, zs = Ls v zd = Ls + Ld , Ls and Ld as the thicknesses of the spacer and doping layers, respectively. The bulk density of electrons along the z -axis is determined by 2 n (z) = ns |ζ (z)| , (2.9) with ns as their sheet density. For heterostructure, the donors and the 2DEG is neutral, so its electric field is vanishing z = ±∞: ∂VH (±∞) = 0. (2.10) ∂z However, in a polar HS the 2DEG originates not only from donors, but also from polarization charges, the neutrality condition is not claimed on the donor-2DEG subsystem. Hence, the boundary condition at z = −∞
13 must be different, given as follows: ∂VH (−∞) = 0 v VH (−∞) = EI , (2.11) ∂z with EI as the binding energy of an ionized donor. As a result, the Hartree potential may be represented in the form: VH (z) = VI (z) + Vs (z) . (2.12) The potential due to remote donors VI (z) z < −zd ,   0  2 4πe nI 2 (z + zd )  −zd ≤ z ≤ −zs ,  VI (z) = EI + 2Ld (2.13) εa   z + (zs + zd )  −zs < z.  2 The potential due to 2DEG Vs (z) 4πe2 ns f (z) z < 0, Vs (z) = − (2.14) εa g (z) + z + f (0) − g (0) z > 0. with the auxiliary functions f (z) v g (z): A2 κz f (z) = e , κ B 2 −kz 2 2 k z + 2k (c + 2) z + c2 + 4c + 6 . (2.15) g (z) = e k 2.3 Total energy per electron in the lowest subband E0 (k, κ) = hT i + hVb i + hVσ i + hVI i + hVs i /2 (2.16) The average kinetic energy ¯ 2 2 h A κ + B 2 k 2 c2 − 2c − 2 , (2.17) hT i = − 8mz mz is effective mass of the GaN electron in the direction z . The average potential barrier, the average interface polarization
14 charges potential and the average Hartree potential hVb i = V0 A2 . (2.18) 2πeσ A2 B2 2 (2.19) hVσ i = + c + 4c + 6 . εa κ k 4πe2 nI d + s hVI i = EI + εa 2κ A2 d2 + χ2 (d) − χ2 (s) − dχ1 (d) + sχ1 (s) + [χ0 (d) − 1] κ (d − s) 2 s2 B2 2 (2.20) − [χ0 (s) − 1] + c + 4c + 6 . 2 k 4πe2 ns A2 A4 hVs i = − − (2.21) εa κ 2κ B2 2 B4 2c4 + 12c3 + 34c2 + 50c + 33 . + c + 4c + 6 − k 4k 2.4 Numerical results and discussion From the above results, the influence of confining source on the electron wave function in the ideal model (infinite barrier) and real model (finite barrier) is fundamentally different. In an ideal model with an infinite barrier (dashed line), the peak of the wave function is raised as the polarization- charge and ionic impurities density increase. Whereas the wave function form is almost unchanged when the spacer thickness changes. In contrast, in a real model with a finite barrier (solid line), the peak of the wave function decreases as increasing two-dimensional electron gas density, donor density and spacer thickness. The peak of the wave function is only raised when the interface polarization-charge density increases. This difference is explained as follows: σ > 0, interface polarization charge causes electronic attraction. In the infinite barrier model, the wave function cannot penetrate, so the wave function peak is raised, the local slope at the interface plane ζ 0 (z = 0) increases. In contrast, in the finite barrier model, the wave function can penetrate through the interface plane, so the peak of the wave function moves towards the barrier and the local value at the ζ(0) plane decreases. As a result, combined roughness scattering was weak.
15 Figure 2.2: Wave function (a) and confining potentials (b) in an AlGaN/GaN HS for polarization-charge density: σ/e = 5 × 1012 , 1013 and 5 × 1013 cm−2 , labeled a, b, and c, respectively. Solid and dashed lines refer to the finite barrier model and the infinite barrier model. Figure 2.3: Wave−2function in an AlGaN/GaN HS for 2DEG density: n = 1012 , s 5 × 1012 , 1013cm , labeled a, b, and c, respectively. Solid and dashed lines refer to the finite barrier model and the infinite barrier model. Furthermore, the value of the wave function at z = −La near the interface is smaller, so the alloy disorder scattering also decreases. In previous studies, 2DEG transport in the heterostructure was performed
16 in an ideal model with infinite barrier, based on Fang-Howard wave func- tion. This model is mathematically simplified and is a good approximation for scattering mechanisms that are insensitive to wave functions near the interface, such as phonon scattering, ion scattering and charged dislocations scattering. Here, the scattering mechanisms are considered to be quite sensitive to the wave function near the interface. 2DEG transport in the heterostructure was investigated with a finite barrier based on the modified Fang-Howard wave function. Figure 2.4: (a), Wave function in an AlGaN/GaN HS for donor density: N = I 1018 , 5 × 1018and 1019 cm−3 , labeled a, b, and c, respectively. (b), Wave function in an AlGaN/GaN HS for spacer thickness: L = 0 A, 70 A v 150 A, labeled a, s b, and c, respectively. Solid and dashed lines refer to the finite barrier model and the infinite barrier model. Conclusions In this chapter, the electronic distribution (2DEG) in AlGaN/GaN polar heterojunction in the real model is studied. 2DEG is confined in a triangular quantum well with finite barrier and and a bent band figured by all con- finement sources. For modulation-doped structure, the effects of interface polarization charges and ionized impurities are considered. The results also show that the electronic distribution in the real model (finite barrier) and the ideal model (infinite barrier) change in opposite directions as the carrier and confining sources change. The electronic distri- bution of the two models only has the similar tendency when increasing the density of the interface polarization charges. Moreover, barrier penetration occurs the barrier height is finite.
17 Chapter 3 Electronic transport in AlGaN/GaN modulation-doped polar heterojunction In this chapter, we investigate the mobility in AlGaN/GaN modulation- doped polar heterojunction at low temperatures which are affected primarily not by ionized impurity scattering and charged dislocations scattering but by alloy disorder scattering (AD) and combined roughness scattering (CR). To do this, two-dimensional electron gas (2DEG) distribution and mobility in AlGaN/GaN modulation-doped polar heterojunction will be investigated. From obtained results, we will explain the bell shape of the 2DEG mobility dependence on the alloy content and on the 2DEG density. In addition, the proposed theory can explain the influence of the AlN layer on the 2DEG mobility in the undoped AlN/GaN heterojunction. 3.1 Analytical results At low temperature, the mobility is generally determined by: eτ µ= (3.1) m∗ The electrons of modulation-doped polar heterojunction will be governed by two scattering mechanisms: alloy disorder scattering and combined rough- ness scattering. The overall transport lifetime is determined by the mecha- nisms due to individual disorders in accordance with Matthiessen's rule 1 1 1 = + (3.2) τtot τAD τCR
18 At rather high 2DEG densities (ns > 1012 cm−2 ), the multiple scattering effects are negligibly small, and thus, we may adopt the linear transport theory as a good approximation. The inverse transport lifetimes at low temperature are then represented in terms of the ACF for each disorder as follows: Z2kF D E 2 1 1 q2 |U (q)| = dq (3.3) τ hEF 2π¯ (4kF2 − q 2 ) ε2 (q) 0 q: the momentum transfer vector by a scattering, q = |q| = 2kF sin (ϑ/2), with ϑ as the scattering angle. The Fermi wave number is fixed by the √ 2DEG density: kF = 2πns . Fermi energy: EF = h ¯ 2 kF2 2m∗ and m∗ as the effective mass of the GaN electron. The dielectric function ε (q) takes into account the screening of scattering potentials by the 2DEG. Usually, in the random phase approximation is determined as follows: qs ε (q) = 1 + Fs (q/k) [1 − G (q/k)] , q ≤ 2kF (3.4) q where qs = 2m∗ e2 εa ¯ h2 is the inverse 2D Thomas-Fermi screening length, εa is the average dielectric constant of two material layers. In a triangular well within the finite potential barrier, the electron state may be described by a FangHoward wave function modified by Ando as equation 2.1. The screening form factor Fs (q) depends on the electron dis- tribution confined along the growth direction and is determined as follows: 2 A4 a 2 + 2c (t + 1) + c2 (t + 1) Fs (t) = + 2A2 B 2 a 3 t+a (t + a) (t + 1) B4 4 3 2 + 3 2 c + 4c + 8c + 8c + 4 2 (t + 1) + t 4c4 + 12c3 + 18c2 + 18c + 9 + t2 2c4 + 4c3 + 6c2 + 6c + 3 . (3.5) with t = q/k, a = κ/k . (3.6) The local field corrections are due to the many-body exchange effect in the in-plane, given by: t G (t) = 1/2 , (3.7) 2 (t2 + t2F )