Electronic transport in semiconductor nanostructure based on polar material algan/gan and penta - Graphene nanoribbon

l 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- 8covery, 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. 91.3 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 L d and N I , spacer L s . 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: ζ (z) = { Aκ1/2exp (κz/2) z < 0, Bk1/2 (kz + c) exp (−kz/2) z > 0. (2.1) In equation (2.1), 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  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 = Bk1/2c, Aκ3/2 / 2 = Bk3/2 (1− c/2) , A2 +B2 ( c2 + 2c+ 2 ) = 1. (2.2) 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 + V tot (z) , (2.3) T is the kinetic energy and V tot (z) is the overall confining potential: V tot (z) = V b (z) + Vσ (z) + VH (z) , (2.4) where V b (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: V b (z) = V0θ (−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: Vσ (z) = 2pi εa eσ |z| . (2.6) 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 dz2 V H (z) = 4pie2 εa [N I (z)− n (z)] . (2.7) In which, N I (z) and n (z) are the density of donors along the growth direction and the one of electrons. Sample is modulation-doped: N I (z) = { N I −z d ≤ z ≤ −z s , 0 z < −z d , z > −z s . (2.8) where, z s = L s v  z d = L s + L d , L s and L d as the thicknesses of the spacer and doping layers, respectively. The bulk density of electrons along the z-axis is determined by n (z) = n s |ζ (z)|2 , (2.9) with n s as their sheet density. For heterostructure, the donors and the 2DEG is neutral, so its electric field is vanishing z = ±∞: ∂V H ∂z (±∞) = 0. (2.10) 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: ∂V H ∂z (−∞) = 0 v  V H (−∞) = E I , (2.11) with E I as the binding energy of an ionized donor. As a result, the Hartree potential may be represented in the form: V H (z) = V I (z) + V s (z) . (2.12) The potential due to remote donors V I (z) V I (z) = E I + 4pie2n I εa  0 z < −z d , (z + zd) 2 2Ld −z d ≤ z ≤ −z s , z + (zs + zd) 2 −z s < z. (2.13) The potential due to 2DEG V s (z) V s (z) = −4pie 2n s εa { f (z) z < 0, g (z) + z + f (0)− g (0) z > 0. (2.14) with the auxiliary functions f (z) v  g (z): f (z) = A2 κ eκz, g (z) = B2 k e−kz [ k2z2 + 2k (c+ 2) z + c2 + 4c+ 6 ] . (2.15) 2.3 Total energy per electron in the lowest subband E0 (k, κ) = 〈T 〉+ 〈Vb〉+ 〈Vσ〉+ 〈VI〉+ 〈Vs〉 /2 (2.16) The average kinetic energy 〈T 〉 = − h¯ 8mz [ A2κ2 +B2k2 ( c2 − 2c− 2)] , (2.17) 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 〈Vb〉 = V0A2. (2.18) 〈Vσ〉 = 2pieσ εa [ A2 κ + B2 k ( c2 + 4c+ 6 )] . (2.19) 〈V I 〉 = E I + 4pie2n I εa { d+ s 2κ + A2 κ (d− s) [ χ2 (d)− χ2 (s)− dχ1 (d) + sχ1 (s) + d 2 2 [χ0 (d)− 1] − s 2 2 [χ0 (s)− 1] ] + B2 k ( c2 + 4c+ 6 )} . (2.20) 〈V s 〉 = −4pie 2n s εa [ A2 κ − A 4 2κ (2.21) + B2 k ( c2 + 4c+ 6 )− B4 4k ( 2c4 + 12c3 + 34c2 + 50c+ 33 )] . 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 ′ (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 function in an AlGaN/GaN HS for 2DEG density: n s = 1012, 5× 1012, 1013 cm−2, 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, 5×1018 and 1019 cm−3, labeled a, b, and c, respectively. (b), Wave function in an AlGaN/GaN HS for spacer thickness: L s = 0  A, 70  A v  150  A, labeled a, 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: µ = eτ m∗ (3.1) 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 τtot = 1 τ AD + 1 τ CR (3.2) 18 At rather high 2DEG densities (n s > 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: 1 τ = 1 2pih¯E F 2k F∫ 0 dq q2 (4k2 F − q2) 〈 |U (q)|2 〉 ε2 (q) (3.3) q: the momentum transfer vector by a scattering, q = |q| = 2k F sin (ϑ/2), with ϑ as the scattering angle. The Fermi wave number is fixed by the 2DEG density: k F = √ 2pin s . Fermi energy: E F = h¯2k2 F / 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: ε (q) = 1 + q s q F s (q/k) [1−G (q/k)] , q ≤ 2k F (3.4) where q s = 2m∗e2ε a h¯2 is the inverse 2D Thomas-Fermi screening length, ε a 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 F s (q) depends on the electron dis- tribution confined along the growth direction and is determined as follows: F s (t) = A4a t+ a + 2A2B2a 2 + 2c (t+ 1) + c2 (t+ 1) 2 (t+ a) (t+ 1) 3 + B4 2 (t+ 1) 3 [ 2 ( c4 + 4c3 + 8c2 + 8c+ 4 ) + 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: G (t) = t 2 (t2 + t2 F ) 1/2 , (3.7) 19 where, tF = kF /k. Scattering by a Gaussian random field is specified by its autocorrelation function in wave vector space, 〈 |U (q)|2 〉 . In which, U (q) is a 2D Fourier transform, is described as follows: U (q) = +∞∫ −∞ dz |ζ (z)|2 U (q, z). (3.8) The autocorrelation function for alloy disorder scattering〈 |U AD (q)|2 〉 = x (1− x)u2 al Ω0fAD, (3.9) with f AD as the form factor for alloy disorder scattering: f AD = A4κ 2 [ e−2κLa − e−2κLb] . (3.10) In equation 3.9 , x is the Al content in the barrier, L b is the barrier thickness, u al is the alloy potential and Ω0 is the volume occupied by one atom. The alloy potential is an adjustable parameter for fitting to experi- mental data and is often assumed to be close to the conduction deviation between the two layers forming the alloy: u al = ∆E c . The atomic volume Ω0 is calculated by the volume of alloy unit cells Ωc, for hexagonal wurtzite crystals: Ω0 = Ωc/4, ð ¥y Ωc = (√ 3 / 2 ) a2 (x) c (x), a (x) and c (x) is the lattice constants of the alloy. The autocorrelation function for combined roughness scattering Combined roughness scattering is the sum of barrier roughness scattering and polarization roughness scattering. The autocorrelation function for this scattering mechanism is given by:〈 |U CR (q)|2 〉 = |F CR (t)|2 〈 |∆ q |2 〉 , (3.11) with F CR (t) = F BR + F PR (t) . (3.12) Where, F BR is form factor due to barrier roughness scattering and F PR (t) is form factor due to polarization roughness scattering. The form factor due to barrier roughness scattering F nloc BR = 〈V ′σ〉+ 〈V ′I 〉+ 〈V ′s 〉 (3.13) 20 with V ′ = ∂V (z)/∂z. The average interface polarization charges potential: 〈V ′σ (z)〉 = 4pie2 ε a σ 2e ( 1− 2A2) . (3.14) The average ionized donors potential: 〈V ′ I 〉 = 4pie 2n I ε a { 1−A2 − A 2 d− s [χ1 (d)− χ1 (s)− dχ0 (d) + sχ0 (s)] } (3.15) The average electrons potential n s : 〈V ′ s (z)〉 = −4pie 2n s εa [ 1−A2 + A 4 2 − B 4 2 ( c4 + 4c3 + 8c2 + 8c+ 4 )] (3.16) The form factor due to polarization roughness scattering The scattering potential of polarized charges in wave vector space is shown as follows: UPR,± (q, r) = ±2pieσ εar± ∆ q e∓qz. (3.17) "+" for z > 0 with r+ = rε and "-" for z < 0 with r− = 1. In modulation-doped polar heterojunction, the electron is described by a FangHoward wave function modified by Ando. Therefore, the polarization roughness potential is determined by: UPR (q) = FPR (t) ∆q (3.18) The form factor due to polarization roughness scattering is given by: F PR (t) = 2pieσ ε a { B2 rε [ 2 (t+ 1) 3 + 2c (t+ 1) 2 + c2 t+ 1 ] −A2 a t+ a } . (3.19) with t = q/k v  a = κ/k. 3.2 Numerical results and discussion 3.2.1 Partial mobility due to alloy disorder scattering and combined roughness scattering 21 Figure 3.1: Partial mobilities limited by alloy disorder (AD), combined rough- ness (CR), and overall (Tot) scatterings vs. polarization charge density σ/e in AlGaN/GaN modulation-doped heterojunction. The inset shows the wave func- tion in an AlGaN/GaN HJ with polarization-charge densities σ/e = 5×1012, 1013, 5× 1013 cm−2, labeled a, b, and c. As obtained results, it can be clearly seen that:: - In infinite barrier model, AD scattering is ignored because 2DEG is separated from alloy disorder. The mobility due to the combined roughness scattering decreases as the polarization charge density increases because the two-dimensional electron gas moves near the interface and the distribution peak increases. In this model, the role of modulation doping is not important because they move away from the electron distribution region. - In contrast, in the finite barrier model, both alloy disorder scattering and combined roughness scattering significantly affect electronic mobility. Specifically, the mobility due to combined roughness scattering rises with in- creasing polarization charge density sigma/e, and this scattering increases slightly as the density of NI increases. Meanwhile, the mobility due to AD scattering decreases as the polarization charge density increases and the donor density increases. This is related to the electronic distribution shown in the attached figure. As such, both alloy disorder scattering and combined roughness scattering are important. We predict that, among these two types of scattering, the 22 Figure 3.2: Partial mobilities limited by alloy disorder (AD), combined rough- ness (CR), and overall (Tot) scatterings vs. doped bulk density N I in AlGaN/GaN modulation-doped heterojunction. The inset shows the wave function in an Al- GaN/GaN heterojunction with doped bulk density N I = 1018, 5 × 1018, 1019 cm −3 . scattering that has a stronger influence on electronic mobility will depend on the characteristics of the survey system (doping, interface roughness, polarization charge density, ...). 3.2.2 Comparison between experimental data and the- oretical calculations 2DEG transport at low temperatures in the polar heterostructure is lim- ited by AD and CR scattering which are sensitive to 2DEG distribution near the interface. In this section, the influence of confining sources on the electron wave function: interface polarization charge, ionized donor and 2DEG are considered. Then, the effect of interface polarization charge and donor density on partial mobility has been investigated. Because quanti- ties are closely related, when one parameter is changed, it will affect the others. Specifically, when Al content changes leads to changes in barrier height V0 (x), density of polarized charge σ (x), 2GED density ns (x), and roughness parameters: ∆ (x) ,Λ (x). 23 Figure 3.3: Mobility limited by alloy disorder (AD), combined roughness (CR), and overall (Tot) scatterings vs Al content x in AlGaN/GaN modulation-doped heterojunction. Circles present the 77 K experimental mobility data. Figure 3.4: Mobility limited by alloy disorder (AD), combined roughness (CR), and overall (Tot) scatterings with ffinite barrier model for Al0.27Ga0.73N/GaN (left graph) and AlN/GaN (right graph). The solid and empty squares show the 20 K experimental mobility data for Al0.27Ga0.73N/GaN and AlN/GaN, respectively. 24 The results show that: - The 2DEG channel is very close to the interface, and therefore, this channel can be very sensitive to any physical processes that occur at the sample interface. - When taking into account polarization roughness scattering in total scat- tering, the compatibility between theoretical and experimental calculations for 2DEG mobility in heterojunction is determined. - With the the undoped polar AlN/GaN heterojunction, the alloy scatter- ing is insignificant due to the high barrier potential, and thus, the combined scattering becomes important. - The graph shape of mobility when the Al content is x = 0.27 and x = 1 vs 2DEG density ns is similar but the magnitude of mobility in two cases is different. Conclusions The results show that both alloy disorder scattering and combined rough- ness scattering strongly depend on Al content and the electronic distribu- tion near the interface. These two scattering mechanisms are important for 2DEG transport in modulation-doped polar heterojunction. If we consider all the roles of polarization charges (carrier supply source, a confining source, and a scattering mechanism), the finite confinings and barrier effect, the calculated results can explain experimental data of the dependence of the 2DEG mobility according to alloy content and carrier density. In addition, the calculation results also contribute to explain the influence of the AlN layer to the 2DEG mobility in AlN/GaN polar hetero- junction. Finally, from the results shown, the ideal model with the infinite barrier should only be applied to scattering mechanisms that are not sensitive to 2DEG distribution near the interface, such as ion scattering or phonon scat- tering. For scatteri