Electronic structure of semiconductor nanostructures: A modified localization landscape theory

Over the past two decades the calculation of the electronic structure of semiconductor nanostructures such as quantum wells (QWs) and quantum dots (QDs) has attracted enormous attention. Dierks and Czycholl (1998); Stier et al. (1999); Fry et al. (2000); Baer et al. (2004); Christmas et al. (2005); Winkelnkemper et al. (2006); Schliwa et al. (2007); Marquardt et al. (2008); Singh and Bester (2009); Schulz et al. (2015) This stems on the one hand from understanding and tailoring their fundamental electronic and optical properties. On the other hand, insight gained into the fundamental properties are also key for optimizing or designing devices with new or improved characteristics and capabilities. Energy efficient light emitting diodes (LEDs) are amongst such devices. Andreev et al. (2012); Piccardo et al. (2017); Piprek (2017) However, from an atomistic standpoint, to model the single-particle states of QDs, multi-QW (MQWs) or even full LED structures, the (time-independent) Schrödinger equation (SE) has to be solved for systems that can easily contain up to several million atoms. Bester et al. (2003); Schulz et al. (2015) Given the large number of atoms, standard density functional theory cannot be applied and empirical models have been widely used. Schliwa et al. (2007); Marquardt et al. (2008); Singh and Bester (2009); Schulz et al. (2015) Even when employing these more empirical models, in general, large eigenvalue problems have to be solved, which can numerically still be demanding. The numerical effort is even further amplified when calculations have to be performed self-consistently, as for instance when describing transport properties of LED structures. Li et al. (2017)

Recently, and originally used to describe Anderson localization in disordered systems, a new approach has been introduced in the literature, which circumvents solving a large eigenvalue problem to obtain (ground state) wave functions and energies of, for instance, a QW. This approach is the so-called localization landscape theory (LLT). Filoche and Mayboroda (2012); Arnold et al. (2016); Filoche et al. (2017) Here, instead of solving the time-independent SE, $\hat{H}\psi=E\psi$, and thus a (large) eigenvalue problem, the idea is to solve

The benefit of this approach is that only a set of linear equations needs to be evaluated, which reduces the computational load significantly, while giving results in very good agreement with the solution of the time-independent SE. Filoche et al. (2017) A detailed analysis of the computational benefit of LLT can be found in Ref. Li et al., 2017, where “standard” self-consistent SE-Poisson calculations for transport properties in InGaN/GaN-based LEDs are compared to the results of a model that utilizes drift-diffusion in combination with LLT. A speed up by a factor of order 50 has been reported in Ref. Li et al., 2017 by the use of the LLT based framework.

However, LLT is not only attractive from a numerical point of view, it allows also to predict and capture physics that may be missing in, for instance, semi-classical approaches. An example that was mentioned already above and will be discussed in more detail below is that it allows to establish quantum corrections to drift diffusion models. Li et al. (2017) Furthermore, LLT can be used to describe Urbach tail energies observed in absorption spectra of InGaN/GaN QW systems. Piccardo et al. (2017) Recently it has also been applied to study localized vibrational modes in enzymes. Chalopin et al. (2019) Finally, a recent development is also to apply it to the Dirac equation for studying properties of graphene or topological insulators. Lemut et al. (2019)

Taking all this together, LLT has several attractive advantages and can give good agreement with the direct solution of the SE. However care must be taken when calculating energies and eigenfunctions from $u$. As described in Ref. Filoche et al., 2017, the zero of energy (reference energy) has to be carefully chosen to obtain good agreement between energies $E$ calculated via LLT and SE. Additionally, the region over which $u$ is integrated to obtain $E$ has to be selected carefully, as also demonstrated in Ref. Filoche et al., 2017 for a single $c$-plane GaN/AlGaN QW. From this, complications may arise in highly disordered systems, such as InGaN wells with local variations in In content, where the system has to be partitioned into “appropriate” regions to obtain energies and wave functions that match closely the results obtained by solving the SE.

Keeping all this in mind, here we describe a modified LLT (MLLT), which keeps the benefits of the LLT, but has several advantages as we will discuss and demonstrate below. Our starting point for the MLLT is:

Obviously the MLLT keeps the advantage of the LLT that instead of solving a (large) eigenvalue problem, one is left with evaluating a system of linear equations. Additionally, we will show that when compared to LLT, the MLLT provides in general a better description/faster convergence of the ground state energy with respect to the SE results. This is especially true for higher In contents. Also, we will demonstrate, by solving the SE, LLT and MLLT numerically for electron and hole ground state energies in $c$-plane InGaN/GaN QWs that the results of the MLLT are less sensitive to the choice of the region over which $u$ is integrated to obtain these energies. Finally, we will discuss how to extract an effective potential $W$ from MLLT that reflects and possesses similar features as the effective potential obtained from LLT. This is important, given that $W$ is for instance used in drift-diffusion studies of InGaN/GaN QW-based LEDs, to account for quantum corrections in the transport calculation frame. Li et al. (2017) All this makes the MLLT approach very attractive for studying for instance Anderson localization or carrier transport in III-N based LEDs where partitioning of the potential landscape in these highly disordered systems might be difficult. We note that the MLLT approach was discussed briefly in Ref. Filoche and Mayboroda, 2012 but no detailed study has yet been presented comparing the two approaches.

The manuscript is organized as follows. In Sec. II we briefly summarize aspects of the theoretical background of the LLT which helps us to motivate the idea underlying the MLLT. In Sec. III we apply LLT and MLLT to a particle-in-a-box problem, since this allows us to flesh out fundamental aspects of the LLT and MLLT approach. To further investigate fundamental aspects and differences of LLT and MLLT, in an Appendix we briefly investigate the solution of an infinite triangular well. This analysis reveals that LLT diverges for this problem while MLLT converges, but to a ground state energy that is noticeably different from the SE solution. To apply LLT, MLLT along with the SE to systems with a triangular but finite potential profile, we study $c$-plane In_xGa_1-xN/GaN single QWs. To do so, we first introduce basic properties of III-N heterostructures in Sec. IV. In Sec. V the results from LLT and MLLT for $c$-plane In_xGa_1-xN/GaN single QWs are presented and compared to the solutions from the SE. We conclude and summarize our work in Sec. VI.

In this section we present the theoretical background of our studies. As already discussed in the introduction, the “standard” approach to calculate the electronic states and energies of semiconductor heterostructures is based on solving the time-independent SE:

Here, $\hat{H}$ is the Hamilton operator, $\psi_{i}$ the wave function of state $i$ and $E_{i}$ the corresponding energy eigenvalue. To calculate $E_{i}$ and $\psi_{i}$ for a system described by $\hat{H}$, Eq. (3) can be treated as an eigenvalue problem. To do so, the Hamiltonian matrix, corresponding to the Hamilton operator $\hat{H}$ in Eq. (3), has to be constructed. The exact form of this matrix depends on the choice of the underlying electronic structure theory, Schulz and O’Reilly (2017) which for semiconductor heterostructures usually ranges from empirical pseudo-potential methods (EPM), Singh and Bester (2009) to empirical tight-binding models (ETBM), Caro et al. (2013) over to $\mathbf{k}\cdot\mathbf{p}$ Stier et al. (1999) or single-band effective mass approximations (EMA) Wojs et al. (1996). The dimension of the Hamiltonian matrix depends on several factors; in an atomistic framework, such as EPM or ETBM, for instance on the number of atoms in the system. Taking a Stranski-Krastanov grown QD as an example, where both the dot and also the barrier material region have to be taken into account in the theoretical modeling of its electronic structure, several millions of atoms have to be considered. Bester (2009) As a consequence, one is left with a large scale eigenvalue problem. Even though efficient numerical routines are available, calculating the eigenstates and energies is still demanding. The numerical burden further increases if self-consistent calculations for optical properties, such as self-consistent Hartree or Hartree-Fock calculations, are required. Kindel et al. (2010); Patra and Schulz (2017)

To circumvent solving large eigenvalue problems, but at the same time to gain insight into wave functions and corresponding energies of a quantum system, the LLT was introduced in 2012, especially focusing on Anderson localization in highly disordered systems. Filoche and Mayboroda (2012) Recently this approach gained strong interest for calculating the electronic structure of nitride-based QW systems. Filoche et al. (2017); Piccardo et al. (2017); Li et al. (2017) Instead of evaluating the SE, Eq. (3), LLT targets solving Eq. (1), where $\hat{H}$ is again the Hamiltonian operator of the system under consideration. As shown by Filoche et al., Filoche et al. (2017) and as we will briefly outline below, the function $u$ can be used to calculate the ground state energy and wave functions of the system described by $\hat{H}$. This summary of the LLT allows us also to motivate the MLLT.

As discussed in Ref. Filoche et al., 2017, the function/state $u$ can be expressed in the basis formed by the eigenfunctions $\psi_{i}$ of $\hat{H}$:

with

Due to the self-adjointness of $\hat{H}$, $\alpha_{i}$ can be obtained via

From Eq. (6) one can see that contributions from energetically higher lying states to $u$, Eq. (4), depend on the factor $1/E_{i}$. Therefore, if the energy separation between state $i$ and $i+1$ is small, for example between the ground state ($i=1$) and the first excited state ($i=2$), several states may contribute significantly to the expansion in Eq. (4). This is obviously undesirable when $u$ should approximate for instance the ground state wave function $\psi_{1}$ obtained from the SE.

Furthermore, assuming as an example a QW system, so that electron and hole wave functions are localized in a subregion of the full well-barrier system, the energetically lowest states contributing to $u$ in Eq. (4) are basically the fundamental, local quantum states in this subregion. In many cases, for example when looking at radiative recombination of carriers, these are the states one is interested in. Therefore, in each localization subregion $\Omega_{m}$, $u$ can be estimated from Filoche et al. (2017)

where $|\psi^{m}_{1}\rangle$ is the local fundamental state in the subregion $\Omega_{m}$. Following Ref. Filoche et al., 2017, $|\psi^{m}_{1}\rangle$ can be assumed to be proportional to $u$ in subregion $\Omega_{m}$:

Finally, using Eq. (8) one can approximate the fundamental/ground state energy in subregion $\Omega_{m}$ from:

Thus, from this equation it is clear that the function $u(\mathbb{r})=\langle r|u\rangle$ provides a direct estimate of the (ground state) energy.

However, $u(\mathbb{r})$ is not only connected to the ground state energy and wave function, it also defines an “effective confining potential”, which is given by $W=1/u$. Filoche and Mayboroda (2012); Filoche et al. (2017) One can show that $W$ is related to the exponential decay of localized states away from their (main) localization subregion. This decay of the wave function is then connected to tunneling effects, as shown for instance by the Wenzel-Kramers-Brillouin (WKB) approximation. Therefore, the effective confining potential $W$ has attracted interest for drift-diffusion calculations, given that $W$ then introduces quantum corrections into these semi-classical transport models. Li et al. (2017)

Taking all this together, several points can be concluded from the above. First, to obtain $u$ and for instance $E^{m}_{1}$, the system has to be partitioned into subregions $\Omega_{m}$ so that Eq. (8) is a good approximation. Secondly, the reference energy or zero of energy should be chosen so that the expansion of $|u\rangle$, Eqs. (4) and (6), respectively, is dominated by the expansion coefficient $\alpha_{1}$. In other words contributions from energetically higher lying states to $u$ are then of secondary importance.

The last aspect motivates the modified LLT (MLLT), Eq. (2), and is triggered by two factors. First, when calculating eigenvalues and eigenfunctions of for instance semiconductor QDs, very often the so-called folded spectrum method (FSM) is applied to turn an interior eigenvalue problem into finding the lowest energy eigenvalue. Wang and Zunger (1994) More precisely, in the FSM, instead of solving the eigenvalue problem $\hat{H}\psi=E\psi$, one evaluates $(\hat{H}-\epsilon\mathds{1})^{2}\psi=\tilde{E}\psi$. Here, $\mathds{1}$ is the unit operator and $\epsilon$ is the so-called reference energy around which the spectrum is folded. In case of $\epsilon=0$, $\tilde{E}=E^{2}$. Working with $\hat{H}^{2}$ for the LLT, thus resulting in MLLT, has now the following advantages for the expansion of $u$ in terms of $|\psi_{i}\rangle$. Using Eq. (4) the expansion coefficients $\alpha_{i}$ are given by:

Therefore:

As one can see from this equation, the contributions from higher lying energy states come in with $1/E^{2}_{i}$ instead of $1/E_{i}$ as in the “standard” LLT. Therefore, the here proposed MLLT should lead to an even better approximation of the fundamental wave function in a subregion and therefore a better approximation of the corresponding energy.

While this clearly shows the benefit of using MLLT in calculating wave functions and energies, the question is how to obtain the effective potential $W$ from MLLT? Here, care must be taken since $u$ itself has now the dimension inverse energy squared. To obtain $W_{\text{MLLT}}$ from MLLT one can define $W_{\text{MLLT}}=(E_{l}\cdot u_{\text{MLLT}})^{-1}$, where $E_{l}$ is for example the ground state energy of the systems under consideration. However, as we see from Eq. (11), several different energies may contribute to the expansion of $u$. Another option is for instance to define the effective potential $W_{\text{MLLT}}$ via $W_{\text{MLLT}}=(\sqrt{u_{\text{MLLT}}})^{-1}$. Given the importance of the effective potential $W$ for describing localized states and also tunneling effects, it is therefore important to analyze the effective potential in more detail and compare it to $W_{\text{LLT}}$ obtained from“standard” LLT.

To highlight and demonstrate the benefits of the MLLT further for wave functions and energies, but also to gain insight into $W_{\text{MLLT}}$, we first study a simple particle-in-a-box problem with infinitely high barriers in the next section. This calculation can be done fully analytically and offers therefore a very transparent test case for the two methods and to compare the results directly with the results from solving the SE.

In this section, we apply both LLT and MLLT to the simple particle-in-a-box problem with infinitely high barriers, since here fully analytic solutions can be derived. The benefit of this is twofold: (i) it sheds light onto general features of the LLT and (ii) it demonstrates the advantages of the proposed MLLT. We compare the results obtained from LLT and MLLT with those from the SE.

We start with the SE and its solution for this problem. Assuming the well boundaries to be at $z=0$ and $z=L$, and choosing the potential energy to be zero for $0<z<L$, the SE in this region reads:

For $z\leq 0$ and $z\geq L$ the potential energy is infinitely large. Due to the boundary conditions $\psi_{n}(0)=0$ and $\psi_{n}(L)=0$, the eigenvalues $E_{n}$ and the normalized eigenstates $\psi_{n}(z)$ are given by: Nolting (2017)

and

The ground state energy eigenvalue $E_{1}=\pi^{2}\hbar^{2}/2mL^{2}$ will now serve as a reference for our calculations using LLT and MLLT, respectively.

III-N materials, such as InN, GaN and AlN have attracted considerable interest for optoelectronic devices, since their alloys are in principle able to cover emission wavelengths from infrared to deep ultra-violet. Humphreys (2008) InGaN heterostructures, such as QWs, are of particular interest for emission in the visible spectral range. Humphreys (2008) When compared to other III-V materials, such as InAs or GaAs, III-N materials preferentially crystallize in the wurtzite crystal phase while InAs and GaAs crystallize in the zinc blende phase. The wurtzite crystal structure, due to its lack of inversion symmetry, allows for a strain-induced piezoelectric polarization vector field but also a spontaneous polarization vector field, which is even present in the absence of any strain effects. Bernardini et al. (1997) Discontinuities in the polarization vector fields lead to very strong electrostatic built-in fields (MV/cm) in InGaN/GaN QW systems grown along the wurtzite $c$-axis, which is the standard growth direction for these systems. Schwarz and Kneissl (2007); Patra and Schulz (2017) Often these systems, especially when dealing with transport properties of InGaN/GaN MQW-based LED structures, are treated as 1-D systems in which the conduction and valence band profiles are modified by the presence of the intrinsic electrostatic built-in potentials. Schwarz and Kneissl (2007); Caro et al. (2011) It should be noted that this is a simplified description of these systems; more recently it has been shown that the alloy microstructure of InGaN QWs significantly affects the electronic structure, so that local potential fluctuations play an important role. Watson-Parris et al. (2011); Schulz et al. (2015) However, for the analysis here, a simplified 1-D model is a good starting point for comparing LLT and MLLT and to highlight the benefits of the MLLT in terms of convergence and “robustness” of the solution against, for instance, the choice of the sub-region $\Omega_{m}$ over which $u$ is being evaluated to obtain the ground state energy in the given region. Since the methodology of the MLLT is the same as that of the LLT, MLLT can directly be applied to a landscape with energy fluctuations due to alloy fluctuations. However, as discussed already above, further consideration must then be given as to how best to calculate $W$ within MLLT, which we will do below. To flesh out the benefits of the MLLT, we focus on the often used 1-D description of the electronic structure of $c$-plane In_xGa_1-xN/GaN QWs with different In contents $x$. As mentioned above, due to the underlying wurtzite crystal structure and growth along the $c$-axis, $c$-plane InGaN/GaN QW systems exhibit very strong electrostatic built-in fields. This electrostatic field arises from discontinuities in spontaneous and piezoelectric polarization vector fields. The corresponding total built-in potential $\phi^{\textnormal{{QW}}}$, assuming that the wurtzite $c$-axis is parallel to the $z$-axis of the coordinate system, can be expressed as: Caro et al. (2011)

Here, $h$ is the height/width of the QW with well barrier interfaces at $z$=0 and $z$ = $h$. The dielectric constant of the QW material is denoted by $\epsilon^{W}_{r}$ and $P_{sp}^{W}$ ($P_{sp}^{B}$) is the spontaneous polarization in the well (barrier). Assuming that the barrier material is strain-free, a strain field is only present in the InGaN QW, since InGaN has a larger lattice constant than GaN. Vurgaftman and Meyer (2003) Thus one is left with a piezoelectric polarization component in the well $P_{pz}^{W}$, which in the 1-D case can be written as: Schulz et al. (2011)

Here, $e^{W}_{ij}$ and $\epsilon_{ij}$ are the (well) piezoelectric coefficients and the strain tensor components. The strain tensor components are given by $\epsilon_{33}=(-2{C_{13}^{W}}/{C_{33}^{W}})\epsilon_{11}$ and $\epsilon_{11}=(a^{B}-a^{W})/a^{W}$; $a^{W}$ ($a^{B}$) is the in-plane lattice constant of the well (barrier) material and $C^{W}_{ij}$ are the elastic constants of the well material. The material parameters used in this study are summarized in Tab. 1. When calculating the electrostatic built-in potential of In_xGa_1-xN/GaN QWs as a function of the In content $x$, a linear interpolation of the involved material parameters is applied. We neglect contributions from second-order piezoelectric effects. Patra and Schulz (2017) Using Eq. (IV) and the material parameters from Tab. 1, the resulting built-in potential is similar to that of a capacitor. Caro et al. (2011)

Since we are interested in a general comparison between LLT and MLLT results, we calculate the electronic structure of the above discussed In_xGa_1-xN/GaN QW systems in the framework of a single-band effective mass approximation for electrons and holes. The confining potential for electrons and holes is then given by the conduction band (CB) and valence band (VB) edge alignment between GaN and InGaN. In Eq. (28) below we assume that the VB edge of bulk GaN (no built-in field) denotes the zero of energy in our system; the GaN CB edge, in the absence of the built-in field, is at the band gap energy $E^{\text{GaN}}_{g}$ of bulk GaN. The In_xGa_1-xN CB edge, $E^{\text{InGaN}}_{\textnormal{CB}}$, and the VB edge, $E^{\text{InGaN}}_{\textnormal{VB}}$, are calculated as a function of the In content $x$ as follows: Caro et al. (2013)

Here, $\Delta{E_{\textnormal{VB}}}$ is the natural VB offset between pure InN and GaN, which has been taken from HSE-DFT calculations. Moses et al. (2011) The (composition dependent) CB and VB edge bowing parameters are denoted by $b_{\textnormal{CB}}$ and $b_{\textnormal{VB}}$. Caro et al. (2013) In combination with the built-in potential from above, the band edge profile shows the well known triangular-shaped profile, leading to the situation that electrons and holes are spatially separated along the growth direction ($c$-axis/$z$-axis). This situation is also know as the quantum confined Stark effect (QCSE). Im et al. (1998)

Building on this potential profile we use a single-band effective mass approximation to construct the Hamiltonian matrix of this system. Here, we use different effective masses for electrons and holes, with the values given in Table 1. A linear interpolation between the effective masses of InN and GaN has been applied to obtain the corresponding masses for InGaN. However, differences in the effective mass inside and outside the well are not considered. Given that we are interested in ground state energies and more generally in a comparison between results obtained from the SE, LLT and MLLT, applying a constant effective mass should be sufficient for our purposes here. To numerically solve $\hat{H}\psi=E\psi$, $\hat{H}u=1$ and $\hat{H}^{2}u=1$ we use the finite difference method and assume a well of width of $L_{w}=35$ Å  with a barrier width of $L_{b}=100$ Å on each side of the well; the discretization step size is $\Delta=0.05$ Å.

In this section we compare and discuss ground state energies, wave functions and the effective potential $W$ of $c$-plane In_xGa_1-xN/GaN QWs obtained by solving SE, LLT and MLLT in the numerical framework discussed above. Special attention is paid to the impact of the In content $x$ on the results, given that with increasing In content the piezoelectric contribution to the built-in potential, and thus the “tilt” in the band edge profiles, increases. More specifically, we study here In contents $x$ ranging from 5% up to 50%, even though the very high In contents ($x>25\%$) are experimentally very difficult to achieve for fully strained $c$-plane wells. Such an analysis will help us to compare the impact of strong asymmetries in the potential landscape on the results of the LLT and MLLT, respectively. Strong fluctuations in the potential landscape may occur locally in InGaN QWs with higher In contents (e.g. 25% In) due to random alloy fluctuations.

Several aspects of the following analysis are to be noted. Firstly, the solution of the SE represents the reference/benchmark for the results of LLT and MLLT. Secondly, since we are using a single-band effective mass approximation in the framework of a finite difference method, we treat electrons and holes separately. In doing so, especially for LLT and MLLT, care must be taken when defining the zero of energy. As discussed in Sec. II, the expansion coefficients $\alpha_{n}$ for constructing $u$ from the eigenstates of the system are inversely proportional to the corresponding state energies. Ideally, the zero of energy should be chosen close to the “true” ground state energy of the system in a given subregion $\Omega_{m}$. By doing so, the expansion coefficient $\alpha_{1}$ is then large as compared to higher order terms that have lesser contributions. As a consequence $u$ is then a very good approximation of the ground state wave function, resulting also in a good estimate of the corresponding energy. Here, we always choose the zero of energy as the minimum energy in the band edge profile of the confining potential for electrons and holes. An illustration of this situation is displayed in Fig. 2.

Finally, following Eq. (9), when calculating the ground state energy from $u$, the subspace region $\Omega_{m}$ over which $u$ is integrated has to be chosen. To illustrate the impact of $\Omega_{m}$ on the results, three different subregions have been considered for electrons and holes. For electrons these are labeled as $\Omega^{e}_{m,1}$, $\Omega^{e}_{m,2}$ and $\Omega^{e}_{m,3}$. The first electron subregion, $\Omega^{e}_{m,1}$, corresponds to the entire simulation cell (-10 nm to 13.5 nm). The subregion $\Omega^{e}_{m,2}$ considers slightly more than the QW region, i.e. -1.5 nm $\leq z\leq$ 5 nm. For the last subregion, $\Omega^{e}_{m,3}$, we just consider it to be 0 nm $\leq z\leq$ 4.5 nm, meaning that this region starts at the well-barrier interface at $z=0$ and extends 1 nm into the barrier region above the upper QW interface at $z=3.5$ nm. This asymmetry in $\Omega^{e}_{m,3}$ accounts for the tilt in the band edges and that therefore the electron wave function is expected to leak further into the barrier region on the $+z$-side of the well. For holes, the subregions one $\Omega^{h}_{m,1}$ and two $\Omega^{h}_{m,2}$ are identical to the first two electron cases. Only subregion $\Omega^{h}_{m,3}$ is different from $\Omega^{e}_{m,3}$. For $\Omega^{h}_{m,3}$ we have chosen -1 nm $\leq z\leq$ 3.5 nm, which reflects that the tilt in the band edges shifts electron and hole states in opposite directions.

The aim of using these three different subregions is twofold. First, as mentioned already above, it will allow us to analyze the impact of the subregion choice on the ground state energies obtained from LLT and MLLT in comparison to the result obtained from solving the SE. Secondly, this analysis also enables us to study if the choice of $\Omega_{m}$ affects differently the results obtained from LLT and MLLT. This insight is for instance of interest when treating (random) potential fluctuations, where partitioning the system may be difficult. Thus, a method where results are less dependent on the $\Omega_{m}$ choice is in general preferred.

In this work we have proposed, motivated and analyzed, a modified localization landscape theory (MLLT). In the MLLT approach we solve $\hat{H}^{2}u=1$ instead of $\hat{H}u=1$, as in the LLT. We demonstrate the improvements resulting from using $\hat{H}^{2}u=1$ in predicting ground state energies for a particle-in-a-box (infinite square well) potential. Since this problem can be solved fully analytically in LLT and MLLT, the solution confirms that $u$ obtained from MLLT will in general give a better approximation of the true ground state wave function when compared to the result from LLT. We have also shown that this can be traced back to the energy dependence of the expansion coefficients of $u$ in terms of the particle-in-a-box eigenstates. Given that $u$ obtained from MLLT provides a very good description of the ground state wave function, it also provides an improved estimate of the ground state energy and therefore the error in this quantity is reduced in comparison to the LLT obtained value. Here, we also have provided insight into the calculation of the effective confining potential $W$ within MLLT. While this is straightforward in the case of a particle-in-a-box problem with infinitely high barriers, we highlight that care must be taken when extracting the effective confining $W$ from MLLT in general. We have discussed two strategies to obtain $W$ from MLLT that lead to results similar to those obtained from LLT, which is important when applying MLLT for instance in drift-diffusion transport calculations to account for quantum corrections.

The particle-in-a-box problem provided the ideal testbed to study the basic properties of the LLT and MLLT. However, further analysis is required to consider more realistic potentials. LLT has recently been used to evaluate the electronic structure of III-N heterostructures, where the confining potential is triangular shaped with barriers of finite height. Motivated by this, we have studied and compared ground state energies from the Schrödinger equation (SE), LLT and MLLT for $c$-plane In_xGa_xN/GaN QWs as a function of the In content $x$. Special attention was paid to the impact of the choice of the integration region of $u$ when evaluating the ground state energies. Our calculations reveal that for both electron and hole ground states, MLLT always gives a better description of the “true” ground state energy when compared to the LLT result. We also find that the subregion over which $u$ is being integrated to obtain this energy is less important for MLLT than it is for LLT. Over the composition range from 5% to 50% In in the well and when integrating over a region close to the QW, errors in the ground state energy from MLLT never exceeds 4%. While similar numbers are obtained for LLT in the lower In content range ($<$15% In) and when choosing appropriate subregions, especially for holes at higher In contents ($>$25% In), errors in the range of 5% to 10% are observed. Looking at the calculated effective potential $W$ for electrons and holes, and independent of the In content $x$, we find that using $W_{\text{MLLT}}=(\sqrt{u_{\text{MLLT}}})^{-1}$ gives in general results that match closely the effective potential $W_{\text{LLT}}$ obtained from LLT. Since $W$ plays an important role in quantum corrected drift-diffusion simulations, it is useful to see that MLLT produces an energy landscape similar to LLT, so that it can be used in such simulations.

Taking all this together the proposed MLLT keeps all the benefits of the LLT, such that only a system of linear equations has to be solved instead of a large eigenvalue problem to obtain ground state energies. At the same time the MLLT provides the following aspects: (i) “faster convergence” of the calculated ground state energies with integration region, (ii) a more “robust” behavior of the method against changes in the integration region, (iii) better agreement with results from SE, especially for higher In contents and (iv) an effective confining potential comparable to that of LLT. All these features make the MLLT method attractive for calculations of localized states in highly disordered systems, where for instance partitioning the systems into different subregions is not trivial.

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