Velocity-field characteristics of MgxZn1−xO/ZnO heterostructures

In this work, electron transport in MgxZn1−xO/ZnO heterostructures at room temperature is simulated by the ensemble Monte Carlo (EMC) method. Electron scattering mechanisms including acoustic deformation potential, piezoelectric acoustic phonon, polar optical phonon (POP), interface roughness (IFR), dislocation, electron escape (ESC) and capture (CPR) by optical phonons, and random alloy are considered in EMC. The electron drift velocity in MgxZn1−xO/ZnO heterostructures is calculated for various Mg mole fractions x (0.1–0.3) at electric fields up to 25 kV/cm. We find that no obvious velocity saturation occurs in the range of the electric field considered. The results show that ESC scattering is one of the main physical mechanisms limiting the drift velocity. On the other hand, the competition between IFR and intersubband POP scattering is found to play an important role in the change in electron drift velocity with the increasing Mg mole fractions.


Introduction
The third-generation wide-bandgap semiconductor ZnO has shown great potential in optoelectronic device applications such as high-power electronic devices and UV light emitters [1]. ZnO can be alloyed with MgO to form Mg x Zn 1−x O, in which the bandgap is dependent on the Mg mole fraction x. The conduction band discontinuity formed at the MgZnO/ ZnO heterointerface due to the difference in energy bandgap between MgZnO and ZnO can be 1.0 eV [2]. Therefore, the MgZnO/ZnO heterostructure system is an effective candidate to confine carriers in the ZnO layer [3]. It is this high conduction band offset energy together with the strong polarization fields existing in MgZnO/ZnO heterostructures that induces a high sheet carrier concentration. Recently, high electron mobility exceeding 1 × 10 6 cm 2 /Vs was reported in MgZnO/ZnO heterostructures [4].
The power and speed performance of MgZnO/ZnO highelectron-mobility transistors (HEMT) could be improved if HEMT channels had higher electron drift velocity. On the other hand, the characteristics of HEMTs is not only determined by conductance of the transistor channel in a weak electric field, but also depend on its variation with increasing field strength. Therefore, the field dependence of the electron drift velocity is very important to device analysis and design [5][6][7][8][9].
The electron drift velocity in heterostructures is limited by a combination of various scattering mechanisms. Begum et al. [10] investigated the role of acoustic phonon (AC) and polar optical phonon (POP) scattering in the electron mobility at MgZnO/ZnO heterojunction. Recently, Zan and Ban [11] theoretically analyzed the electron mobility limited by interface optical (IF) and confined optical (CO) phonon modes in a symmetric MgZnO/ZnO quantum well (QW) structure. Thongnum et al. [12] reported the role of interface roughness (IFR) in electron transport in MgZnO/ZnO, finding that the scattering potential in the in-plane direction can affect electron mobility. Cohen et al. [13] found that increasing the Mg mole fraction in Zn 1-x Mg x O films can decrease the conductivity, mobility, and electron density, which was explained by a combination of increasing electron effective mass and alloy disorder scattering. Sang et al. [14] calculated the low-temperature mobility of MgZnO/ZnO heterostructures, and found that dislocation scattering was dominant in the low-electron concentration region. Wang et al. [15] analyzed electron transport in MgZnO/ZnO heterostructures considering polarization roughness scattering (PRS) and IFR scattering. Li et al. [16] numerically calculated the temperature dependence of the electron mobility of ZnMgO/ZnO heterostructures. To our knowledge, a comprehensive investigation of the roles of all various scattering in electron transport in MgZnO/ZnO QWs is still lacking.
ZnO can hold a dislocation density up to 4.94 × 10 9 cm −2 in MgZnO/ZnO heterostructures [17]. Li et al. [16] pointed out that polar optical phonon (POP) scattering is usually the dominant scattering mechanism in MgZnO/ZnO material systems at room temperature. Compared with other wide bandgap materials, ZnO has a remarkably high electromechanical coupling coefficient (EMCC) [18,19], which results in a relatively high rate of piezoelectric acoustic phonon (PAP) scattering. Khalil et al. [20] studied electron capture (CPR) and escape (ESC) due to the polar optical phonon scattering in QWs. When CPR occurs, electrons can be captured from bulk states (3D) into confined states (2D). On the contrary, in ESC, electrons escape from the 2D to the 3D state. The obtained rates of capture and escape scattering can be near 10 12 s −1 in the low-energy range. Up to now, the characteristics of electron drift velocity with an increasing electric field are still not clear, especially at high temperature. In this work, we use an ensemble Monte Carlo (EMC) method considering various electron scattering mechanisms [including ESC, CPR, random alloy (RAS), IFR, POP, PAP, and acoustic deformation potential (ADP)] to simulate the electron transport, and analyze the roles of different electron scattering in determining the velocity-field characteristics of Mg x Zn 1−x O/ZnO heterostructures at room temperature.

Self-consistent calculations of 2DEG in the conduction band
In the effective mass approximation, the energy levels and the corresponding wave functions for electrons in the heterostucture are obtained by self-consistently solving the Schrodinger equation, and Poisson equation, using the finite element method [21], where z is the space coordinate and parallels the growth, m * is the space-dependent electron effective mass, is the dielectric constant, e is the free-electron charge, ℏ is the reduced Planck constant, E i and i are energy eigenvalue and envelope wave function of the ith subband, respectively, V(z) is the total potential energy, which is defined by where e (z) is the Hartree potential due to the electrostatic interaction, ΔE c (z) is the conduction band-edge offset in the heterostructure. P(z) in Eq. (2) is the total polarization induced by polarization effect, N + d is the ionized donor concentration. N + d (z) and electron density n(z) are given by the formulas as follows, respectively, where N d and E d are the donor density and donor energy level, respectively, k B is the Boltzmann constant, T is the absolute temperature, E f is the Fermi level. The temperaturedependent bandgap of ZnO is calculated from [22], The non-centrosymmetric piezoelectric würtzite (WZ) crystal structure MgZnO considered in this work is a direct bandgap material, with the energy bandgap located at the Γ-point Brillouin zone center. The bandgap Eg of Mg x Zn 1−x O is described by the following formula [23] where x is the Mg molar fraction, b(x) is the bowing parameter, b(x) = + x , E g Z n O and E g M g O are bandgaps of ZnO and MgO, respectively. = 0.6 , = 3.4.
The real MgZnO/ZnO heterostructures contain a high density of dislocations in the epitaxial layers of ZnO, which form because of a lattice-matched substrate for growth. The dislocation density can be calculated [24], 2 , which is the magnitude of the Burgers vector [25]. a MgZnO is the lattice constant of MgZnO, which is calculated from the corresponding values of MgO and ZnO for the given Mg mole fraction x using a linear relation. is the strain, which can be estimated by [26], where r Zn = 0.6 Å, r Mg = 0.57 Å, r o = 1.38 Å are Pauli covalent radii of Zn and O atoms, x is the Mg mole fraction.

Electron scattering and ensemble Monte Carlo method
Electron transport in MgZnO/ZnO heterostructures is assumed to be influenced by acoustic phonon (AC), polar optical phonon (POP), interface roughness (IFR), dislocation (DIS), and random alloy (RAS) scattering. The corresponding probabilities for intersubband and intrasubband scattering have been discussed in detail in [27,28]. Electron escape (ESC) and capture (CPR) by optical phonons are also included in this study, since different Mg mole fractions may give rise to potential well depths. The ESC and CPR scattering are actually caused by the interaction of electron and polar optical phonon; therefore, the corresponding formulae are similar to that of intersubband POP scattering rate. The corresponding scattering rates are calculated by the method in [20], where LO is the longitudinal optical phonon energy, n q is the phonon occupation number, Q( ) = k ∕∕ − k 2d , Q( , ) = k 2d − k ∕∕ , k ∕∕ and k 2d are the parallel components of wave vector in 2D and 3D states, respectively. k z is the component of the bulk wave vector in the z-direction. q s = √ e 2 N∕ s k B T , is the screening factor. f k x,n (Q) is expressed as follows where k z ,n (z) = * k z n (z). In EMC simulation, if ESC happens to a simulation electron, then the wave vector (K 3d ) of the electron in 3D state is obtained by the energy/momentum conservation and the wave vector (K 2d ) of the electron in 2D. EMC deals with CPR in a similar way: calculate K 2d based on K 3d and energy/momentum conservation. Then, the electron drift velocity is calculated according to the electron wave vectors.
The dislocation (DIS) plays important role in electron transport and the resulting mobility. The effect of dislocation on electron transport can be analyzed by introduction of dislocation scattering. The corresponding scattering rate is obtained as [29] (10) where k F = √ 2 n s is the Fermi wave vector, which depends on the 2DEG carrier concentration n s . N dis is the dislocation density. q TF = 2∕a * B is the 2D Thomas-Fermi wave vector, a * B is the effective Bohr radius in the material. L = ef ∕c 0 is the line charge density, where c 0 is the lattice constant in the (0001) direction of ZnO.
In the nonparabolic model, the dispersion relation within a specific valley is given by where the energy E and wavenumber k are given with respect to the center of the valley, is nonparabolicity constant.

Results and discussion
The doping level in both MgZnO and ZnO layers is set to be 1 × 10 14 cm −3 , to keep consistent with the level for unintentional doping. With the self-consistent solution of the Schrödinger and Poisson equations described in Sect. 2, the band diagram, eigenvalues, and subband wave functions are calculated. The adopted material parameters of MgO and ZnO are listed in Table 1, and the respective material properties of alloys are obtained based on Vegard's law. Table 2 presents the main material parameters for the calculations of scattering rates. Figure 1 shows the conduction band diagram and the corresponding first three lowest wave functions of Mg 0.1 Zn 0.9 O/ZnO at temperature 300 K. The triangular potential well in the conduction band resulted from the polarization-induced electric field. In this case, the wave functions are confined in the narrow z-direction range from 30 to 45 nm, which becomes the 2D electron channel.
An external electric field formed by the drain voltage is applied in this channel.
From Eqs. (8) and (9), we can obtain the dependence of dislocation density on Mg mole fraction x. Referring to the experimental data of dislocation density in [17], 1.8 × 10 9 , 3.3 × 10 9 , 5.0 × 10 9 , 7.0 × 10 9 cm −2 , read from Fig. 2. Γ and Λ used to calculated IFR scattering rate are assumed as 6.0 nm and 0.6 nm, respectively, which refer to the values used in [34]. The calculated results are demonstrated in Fig. 3. It is found that in the electric field range of 0-25 kV/cm, the MgZnO/ZnO QW with higher Mg mole fraction gives rise to lower electron drift velocity. This result is due to the increase of scattering rates that are closely dependent on Mg mole fraction, here including RAS, DIS and IFR. Figure 4 presents the rates of IFR and DIS scattering used in Fig. 3. We can see that IFR and DIS scattering rates both increase as Mg mole fractions increase. At the same time, except the intrasubband POP scattering, the IFR scattering is the dominant one in this case. These two reasons account for the result in Fig. 3,  As noted in Sect. 1, MgZnO/ZnO systems can produce high EMCC compared with other material systems. Therefore, we further consider adding the PAP scattering into EMC, and repeat the simulations carried out for Fig. 3 to determine the effect of PAP scattering on electron transport. The EMCC ( K 2 = 0.145 ) is obtained from [18]. The calculated electron drift velocities as a function of electric field are drawn in Fig. 5a. We can see that, compared with the results in Fig. 3, the electron drift velocity in Fig. 5a obviously decreases owing to the additional PAP scattering. Figure 5b explicitly demonstrates the difference in electron drift velocity due to the added PAP scattering. The dashed lines are for the cases with considering the PAP scattering. We find that the difference for x = 0.1 is obvious, and that for x = 0.3 is very small, nearly negligible. The reason behind this is that the IFR scattering is strengthened considerably with the increasing Mg mole fraction, as illustrated in Fig. 4a, so that the role of PAP is weakened. Here, we can conclude that, at least for the low Mg mole fractions, the PAP scattering should be considered an important factor that influences electron transport in MgZnO/ZnO QWs.
We find that in an electric field range of 0-25 kV/cm, all electron velocities increase monotonically. Although electron velocity does not display an apparent saturation, it apparently deviates from a linear change. Sasa et al. [35] studied the velocity-field characteristics obtained from the experimental I-V data. According to their results, the velocity field maintains a near-linear relationship with the electric field in the range of 0-22 kV/cm. Wang [36] carried out Monte Carlo simulation of electron velocity with an electric field in the range from 0 to 200 kV/cm. In [36], the results showed that drift velocity also increases linearly when the electric field is lower than 30 kV/cm. This result is also consistent with the result in Fig. 5. On the other hand, the dependence of election velocity as a function of the applied electric field with different Mg mole fractions in Fig. 5 agrees with the reported results in [36].
In order to comprehensively investigate the physical mechanism, we intentionally exclude ESC and CPR scattering mechanisms from EMC and recalculate the electron drift velocity under the same conditions used in Fig. 3. The corresponding results are presented in Fig. 6. When ESC scattering occurs, the electron escapes from the 2D state to the 3D state, from which the electron can fly away from the channel. In this case, the electron is considered not to be accelerated by the externally applied electric field, since the electric field only exists in the very narrow channel. We can see that the electron drift velocities in Fig. 6 are much higher than those in Fig. 3. Ardaravicius et al. [6] measured and simulated the electron drift velocity in AlGaN/GaN using EMC under different uniform electric fields and at room temperature. EMC simulation showed that the calculated velocities were markedly larger than the measured ones. As the authors pointed out, neglecting electron escape is one of the reasons, which is consistent with the result in Fig. 6. This comparison suggests that ESC and CPR scattering are important for the analysis of the velocity-field relationship in heterostructures.
In all calculations above, the non-parabolicity of the conduction band has been considered. The non-parabolicity of ZnO is = 0.3eV −1 [37]. In order to determine the effect of non-parabolicity on electron drift velocity, we recalculated the electron drift velocities with the non-parabolicity neglected. The corresponding results are drawn with the red line in Fig. 7. We find that the difference in electron drift velocity between these two cases is remarkable, nearly 22% at 25 kV/cm.
Then we let Γ and Λ be 3.0 nm and 0.28 nm, respectively, that is, changing the interface roughness and keeping the other parameters the same as those used in Fig. 5a. The obtained electron drift velocities as a function of electric field are shown in Fig. 8. Compared with the results in Fig. 5a, the obvious difference in Fig. 8 is that the electron drift velocities with x = 0.3 are not the smallest, and   Fig. 3, the IFR scattering rate here obviously decreases (Fig. 9a), meaning that IFR is not dominant over other scattering mechanisms. In this case, we find that the intersubband POP scattering is dominant except for the intrasubband POP scattering. Figure 9b presents the intersubband POP emission scattering from subband 0 to subband 3. We consider that the results in Fig. 8 are due to the competition of IFR and intersubband POP scattering. For example, below 0.2 eV, the intersubband POP emission scattering is depressed. Naturally, in this energy range, the electrons are mainly controlled by IFR scattering. At the same time, we see from Fig. 9b that in the energy range over 0.2 eV, the rates of intersubband POP emission scattering for Mg mole fraction x = 0.3 are lower than those for other x, which is thought to explain why the electron drift velocities for x = 0.3 are higher than those for x = 0.2 and 0.25 when the electric field is greater then 10 kV/cm (see Fig. 8).
Finally, for Mg x Zn 1−x O/ZnO QWs with Mg mole fraction x in the range of 0.1-0.3, DIS scattering is not found to play a significant role in electron transport, which is consistent with the result found for Ga x In 1−x N/GaN systems [38].

Summary and conclusion
In this study, we have investigated the transport properties of Mg x Zn 1−x O/ZnO heterostructures using an ensemble Monte Carlo (EMC) method. Electron drift velocities under various transverse electric fields (≤ 25 kV/cm) are calculated at temperature T = 300 K . No obvious saturation is found for the electron drift velocity in the range of electric fields considered. ESC scattering is one of the main scattering mechanisms limiting electron drift velocity. The effect of Mg mole fraction on electron drift velocity is analyzed. It is found that the higher Mg mole fraction generally gives rise to lower electron drift velocity, due to the increase of IFR, RAS and DIS scattering rates. On the other hand, in the case that IFR scattering is sufficiently low, electron drift velocity can increase with the increase of Mg mole fractions. This result is found due to the decreasing intersubband POP scattering rate with the increase of Mg mole fractions. Nonparabolicity of conduction band has a remarkable influence on electron drift velocity. The work presented here could help in the design of future devices, such as ZnO-based high electron mobility transistors (HEMT). Data availability The datasets generated during and/or analysed during the current study are not publicly available, but are available from the corresponding author on reasonable request.