Electronic structure of semiconductor nanostructures: A modified localization landscape theory
D. Chaudhuri, J. C. Kelleher, M. R. O'Brien, E. P. O'Reilly, S. Schulz
aa r X i v : . [ c ond - m a t . m e s - h a ll ] D ec Electronic structure of semiconductor nanostructures: A modified localizationlandscape theory
D. Chaudhuri, ∗ J. C. Kelleher, M. R. O’Brien,
1, 2
E P. O’Reilly,
1, 2 and S. Schulz Tyndall National Institute, University College Cork, T12 R5CP, Ireland Department of Physics, University College Cork, Cork, T12 YN60, Ireland (Dated: January 1, 2020)In this paper we present a modified localization landscape theory to calculate localized/confinedelectron and hole states and the corresponding energy eigenvalues without solving a (large)eigenvalue problem. We motivate and demonstrate the benefit of solving ˆ H u = 1 in the modifiedlocalization landscape theory in comparison to ˆ Hu = 1, solved in the localization landscape theory.We detail the advantages by fully analytic considerations before targeting the numerical calculationof electron and hole states and energies in III-N heterostructures. We further discuss how the solutionof ˆ H u = 1 is used to extract an effective potential W that is comparable to the effective potentialobtained from ˆ Hu = 1, ensuring that it can for instance be used to introduce quantum correctionsto drift-diffusion transport calculations. Overall, we show that the proposed modified localizationlandscape theory keeps all the benefits of the recently introduced localization landscape theory butfurther improves factors such as convergence of the calculated energies and the robustness of themethod against the chosen integration region for u to obtain the corresponding energies. We findthat this becomes especially important for here studied c -plane InGaN/GaN quantum wells withhigher In contents. All these features make the proposed approach very attractive for calculation oflocalized states in highly disordered systems, where partitioning the systems into different subregionscan be difficult. I. INTRODUCTION
Over the past two decades the calculation of theelectronic structure of semiconductor nanostructuressuch as quantum wells (QWs) and quantum dots (QDs)has attracted enormous attention.
This stems onthe one hand from understanding and tailoring theirfundamental electronic and optical properties. Onthe other hand, insight gained into the fundamentalproperties are also key for optimizing or designing deviceswith new or improved characteristics and capabilities.Energy efficient light emitting diodes (LEDs) areamongst such devices.
However, from an atomisticstandpoint, to model the single-particle states of QDs,multi-QW (MQWs) or even full LED structures, the(time-independent) Schr¨odinger equation (SE) has tobe solved for systems that can easily contain up toseveral million atoms.
Given the large number ofatoms, standard density functional theory cannot beapplied and empirical models have been widely used.
Even when employing these more empirical models, ingeneral, large eigenvalue problems have to be solved,which can numerically still be demanding. The numericaleffort is even further amplified when calculations haveto be performed self-consistently, as for instance whendescribing transport properties of LED structures. Recently, and originally used to describe Andersonlocalization in disordered systems, a new approach hasbeen introduced in the literature, which circumventssolving a large eigenvalue problem to obtain (groundstate) wave functions and energies of, for instance, aQW. This approach is the so-called localization landscapetheory (LLT).
Here, instead of solving the time-independent SE, ˆ Hψ = Eψ , and thus a (large) eigenvalue problem, the idea is to solveˆ Hu = 1 . (1)The benefit of this approach is that only a set oflinear equations needs to be evaluated, which reducesthe computational load significantly, while giving resultsin very good agreement with the solution of thetime-independent SE. A detailed analysis of thecomputational benefit of LLT can be found in Ref. 15,where “standard” self-consistent SE-Poisson calculationsfor transport properties in InGaN/GaN-based LEDs arecompared to the results of a model that utilizes drift-diffusion in combination with LLT. A speed up by a factorof order 50 has been reported in Ref. 15 by the use of theLLT based framework.However, LLT is not only attractive from a numericalpoint of view, it allows also to predict and capturephysics that may be missing in, for instance, semi-classical approaches. An example that was mentionedalready above and will be discussed in more detail belowis that it allows to establish quantum corrections todrift diffusion models. Furthermore, LLT can be usedto describe Urbach tail energies observed in absorptionspectra of InGaN/GaN QW systems. Recently it hasalso been applied to study localized vibrational modesin enzymes. Finally, a recent development is also toapply it to the Dirac equation for studying properties ofgraphene or topological insulators. Taking all this together, LLT has several attractiveadvantages and can give good agreement with the directsolution of the SE. However care must be taken whencalculating energies and eigenfunctions from u . Asdescribed in Ref. 18, the zero of energy (referenceenergy) has to be carefully chosen to obtain goodagreement between energies E calculated via LLT andSE. Additionally, the region over which u is integratedto obtain E has to be selected carefully, as alsodemonstrated in Ref. 18 for a single c -plane GaN/AlGaNQW. From this, complications may arise in highlydisordered systems, such as InGaN wells with localvariations in In content, where the system has to bepartitioned into “appropriate” regions to obtain energiesand wave functions that match closely the resultsobtained by solving the SE.Keeping all this in mind, here we describe a modifiedLLT (MLLT), which keeps the benefits of the LLT, buthas several advantages as we will discuss and demonstratebelow. Our starting point for the MLLT is:ˆ H u = 1 . (2)Obviously the MLLT keeps the advantage of the LLTthat 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/fasterconvergence of the ground state energy with respectto the SE results. This is especially true for higherIn contents. Also, we will demonstrate, by solvingthe SE, LLT and MLLT numerically for electron andhole ground state energies in c -plane InGaN/GaN QWsthat the results of the MLLT are less sensitive tothe choice of the region over which u is integrated toobtain these energies. Finally, we will discuss how toextract an effective potential W from MLLT that reflectsand possesses similar features as the effective potentialobtained from LLT. This is important, given that W isfor instance used in drift-diffusion studies of InGaN/GaNQW-based LEDs, to account for quantum correctionsin the transport calculation frame. All this makes theMLLT approach very attractive for studying for instanceAnderson localization or carrier transport in III-N basedLEDs where partitioning of the potential landscape inthese highly disordered systems might be difficult. Wenote that the MLLT approach was discussed briefly inRef. 16 but no detailed study has yet been presentedcomparing the two approaches.The manuscript is organized as follows. In Sec. II webriefly summarize aspects of the theoretical backgroundof the LLT which helps us to motivate the idea underlyingthe MLLT. In Sec. III we apply LLT and MLLT to aparticle-in-a-box problem, since this allows us to flesh outfundamental aspects of the LLT and MLLT approach. Tofurther investigate fundamental aspects and differencesof LLT and MLLT, in an Appendix we briefly investigatethe solution of an infinite triangular well. This analysisreveals that LLT diverges for this problem while MLLTconverges, but to a ground state energy that is noticeablydifferent from the SE solution. To apply LLT, MLLTalong with the SE to systems with a triangular but finitepotential profile, we study c -plane In x Ga − x N/GaNsingle QWs. To do so, we first introduce basic propertiesof III-N heterostructures in Sec. IV. In Sec. V the results from LLT and MLLT for c -plane In x Ga − x N/GaN singleQWs are presented and compared to the solutions fromthe SE. We conclude and summarize our work in Sec. VI.
II. LOCALIZATION LANDSCAPE THEORY:THEORETICAL BACKGROUND
In this section we present the theoretical backgroundof our studies. As already discussed in the introduction,the “standard” approach to calculate the electronic statesand energies of semiconductor heterostructures is basedon solving the time-independent SE:ˆ Hψ i = E i ψ i . (3)Here, ˆ H is the Hamilton operator, ψ i the wave functionof state i and E i the corresponding energy eigenvalue.To calculate E i and ψ i for a system described byˆ H , Eq. (3) can be treated as an eigenvalue problem.To do so, the Hamiltonian matrix, corresponding tothe Hamilton operator ˆ H in Eq. (3), has to beconstructed. The exact form of this matrix depends onthe choice of the underlying electronic structure theory, which for semiconductor heterostructures usually rangesfrom empirical pseudo-potential methods (EPM), toempirical tight-binding models (ETBM), over to k · p or single-band effective mass approximations (EMA) .The dimension of the Hamiltonian matrix depends onseveral factors; in an atomistic framework, such asEPM or ETBM, for instance on the number of atomsin the system. Taking a Stranski-Krastanov grownQD as an example, where both the dot and also thebarrier material region have to be taken into accountin the theoretical modeling of its electronic structure,several millions of atoms have to be considered. Asa consequence, one is left with a large scale eigenvalueproblem. Even though efficient numerical routines areavailable, calculating the eigenstates and energies is stilldemanding. The numerical burden further increases ifself-consistent calculations for optical properties, such asself-consistent Hartree or Hartree-Fock calculations, arerequired.
To circumvent solving large eigenvalue problems, butat the same time to gain insight into wave functions andcorresponding energies of a quantum system, the LLTwas introduced in 2012, especially focusing on Andersonlocalization in highly disordered systems. Recentlythis approach gained strong interest for calculating theelectronic structure of nitride-based QW systems.
Instead of evaluating the SE, Eq. (3), LLT targets solvingEq. (1), where ˆ H is again the Hamiltonian operator ofthe system under consideration. As shown by Filoche etal. , and as we will briefly outline below, the function u can be used to calculate the ground state energy and wavefunctions of the system described by ˆ H . This summaryof the LLT allows us also to motivate the MLLT.As discussed in Ref. 18, the function/state u can beexpressed in the basis formed by the eigenfunctions ψ i ofˆ H : | u i = X i α i | ψ i i (4)with α i = h u | ψ i i = Z Z Z u ( r ) ψ i ( r ) d r . (5)Due to the self-adjointness of ˆ H , α i can be obtained via α i = h u | ψ i i = 1 E i h u | ˆ Hψ i i = 1 E i h | ψ i i . (6)From Eq. (6) one can see that contributions fromenergetically higher lying states to u , Eq. (4), dependon the factor 1 /E i . Therefore, if the energy separationbetween state i and i + 1 is small, for example betweenthe ground state ( i = 1) and the first excited state( i = 2), several states may contribute significantly tothe expansion in Eq. (4). This is obviously undesirablewhen u should approximate for instance the ground statewave function ψ obtained from the SE.Furthermore, assuming as an example a QW system,so that electron and hole wave functions are localized in asubregion of the full well-barrier system, the energeticallylowest states contributing to u in Eq. (4) are basicallythe fundamental, local quantum states in this subregion.In many cases, for example when looking at radiativerecombination of carriers, these are the states one isinterested in. Therefore, in each localization subregionΩ m , u can be estimated from | u i ≃ h | ψ m i E m | ψ m i = α | ψ m i , (7)where | ψ m i is the local fundamental state in thesubregion Ω m . Following Ref. 18, | ψ m i can be assumedto be proportional to u in subregion Ω m : | ψ m i ≈ | u i|| u || . (8)Finally, using Eq. (8) one can approximate thefundamental/ground state energy in subregion Ω m from: E m = h ψ m | ˆ H | ψ m i ≈ h u | ˆ H | u i|| u || = h u | i|| u || = RRR Ω m u ( r ) d r RRR Ω m u ( r ) d r . (9)Thus, from this equation it is clear that the function u ( r ) = h r | u i provides a direct estimate of the (groundstate) energy.However, u ( r ) is not only connected to the ground stateenergy and wave function, it also defines an “effectiveconfining potential”, which is given by W = 1 /u . One can show that W is related to the exponential decayof localized states away from their (main) localization subregion. This decay of the wave function is thenconnected to tunneling effects, as shown for instanceby the Wenzel-Kramers-Brillouin (WKB) approximation.Therefore, the effective confining potential W hasattracted interest for drift-diffusion calculations, giventhat W then introduces quantum corrections into thesesemi-classical transport models. Taking all this together, several points can beconcluded from the above. First, to obtain u and forinstance E m , the system has to be partitioned intosubregions Ω m so that Eq. (8) is a good approximation.Secondly, the reference energy or zero of energy shouldbe chosen so that the expansion of | u i , Eqs. (4) and (6),respectively, is dominated by the expansion coefficient α . In other words contributions from energeticallyhigher 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 forinstance semiconductor QDs, very often the so-calledfolded spectrum method (FSM) is applied to turn aninterior eigenvalue problem into finding the lowest energyeigenvalue. More precisely, in the FSM, instead ofsolving the eigenvalue problem ˆ Hψ = Eψ , one evaluates( ˆ H − ǫ ) ψ = ˜ Eψ . Here, is the unit operator and ǫ isthe so-called reference energy around which the spectrumis folded. In case of ǫ = 0, ˜ E = E . Working with ˆ H forthe LLT, thus resulting in MLLT, has now the followingadvantages for the expansion of u in terms of | ψ i i . UsingEq. (4) the expansion coefficients α i are given by: α MLLT i = h u | ψ i i = h u | ˆ H ψ i i E i = h ˆ H u | ψ i i E i = h | ψ i i E i . (10)Therefore: | u i = X i h | ψ i i E i | ψ i i . (11)As one can see from this equation, the contributionsfrom higher lying energy states come in with 1 /E i instead of 1 /E i as in the “standard” LLT. Therefore,the here proposed MLLT should lead to an even betterapproximation of the fundamental wave function in asubregion and therefore a better approximation of thecorresponding energy.While this clearly shows the benefit of using MLLT incalculating wave functions and energies, the question ishow to obtain the effective potential W from MLLT?Here, care must be taken since u itself has now thedimension inverse energy squared. To obtain W MLLT from MLLT one can define W MLLT = ( E l · u MLLT ) − ,where E l is for example the ground state energyof the systems under consideration. However, aswe see from Eq. (11), several different energies maycontribute to the expansion of u . Another option isfor instance to define the effective potential W MLLT via W MLLT = ( √ u MLLT ) − . Given the importance of theeffective potential W for describing localized states andalso tunneling effects, it is therefore important to analyzethe effective potential in more detail and compare it to W LLT obtained from“standard” LLT.To highlight and demonstrate the benefits of the MLLTfurther for wave functions and energies, but also to gaininsight into W MLLT , we first study a simple particle-in-a-box problem with infinitely high barriers in the nextsection. This calculation can be done fully analyticallyand offers therefore a very transparent test case for thetwo methods and to compare the results directly with theresults from solving the SE.
III. LOCALIZATION LANDSCAPE THEORYAND MODIFIED LOCALIZATION LANDSCAPETHEORY: APPLICATION TO A SQUARE WELLWITH INFINITELY HIGH BARRIERS
In this section, we apply both LLT and MLLTto the simple particle-in-a-box problem with infinitelyhigh barriers, since here fully analytic solutions can bederived. The benefit of this is twofold: (i) it sheds lightonto general features of the LLT and (ii) it demonstratesthe advantages of the proposed MLLT. We compare theresults obtained from LLT and MLLT with those fromthe 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 for0 < z < L , the SE in this region reads: − ~ m d dz ψ n ( z ) = E n ψ n ( z ) . (12)For z ≤ z ≥ L the potential energy is infinitelylarge. Due to the boundary conditions ψ n (0) = 0 and ψ n ( L ) = 0, the eigenvalues E n and the normalizedeigenstates ψ n ( z ) are given by: E n = n π ~ mL (13)and ψ n ( z ) = r L sin (cid:16) nπzL (cid:17) . (14)The ground state energy eigenvalue E = π ~ / mL will now serve as a reference for our calculations usingLLT and MLLT, respectively. A. LLT solution
Following Eq. (4), u ( z ) can be expressed as a linearcombination of the eigenfunctions ψ n ( z ), Eq. (14), whichform a complete basis set for the Hilbert space: u = X n α n ψ n . (15) Now, exploiting the LLT equation ˆ Hu = 1 and usingEq. (14), from Eq. (15) we obtain:ˆ Hu = X n E n α n ψ n = 1 ⇒ X n α n E n r L sin (cid:16) nπzL (cid:17) = 1 . (16)Here, we recall that the constant function 1 can berepresented by π ∞ X n odd n sin (cid:16) nπzL (cid:17) = 1 . (17)Thus, combining Eq. (16) and (17), the coefficients α n are zero for even n and for the odd values of n they read: α n = 2 √ LE πn . (18)Using this expression for α n , Eq. (18), u , Eq. (15), istherefore given by: u ( z ) = ∞ X n odd √ LE n π ψ n ( z )= 2 √ LE π ∞ X m =1 ψ m − ( z )(2 m − = λ (cid:18) ψ ( z ) + 127 ψ ( z ) + 1125 ψ ( z ) + ... (cid:19) , (19)where λ = √ LE π . From this equation it followsthat the series expansion of u converges as 1 /n withsignificantly lesser contributions from the higher orderterms. Furthermore, only every second basis state of theinfinite square well eigenstates contributes. Thus, for thisproblem, the LLT gives a very good approximation of theground state wave function ψ ( z ), but, since for instance ψ ( z ) is missing in the expansion, the first excited statecannot be described by u ( z ) in general. However, wehighlight here that when applying LLT to disorderedsystems where several minima/subregions Ω m can bedefined, LLT can be applied to the different Ω m andone can find the fundamental state for each subregion.While locally this is the ground state, globally thesestates will be excited states. In addition, an analysisbased on Weyl’s Law has shown that the LLT can givea very good estimate of the integrated density of statesover a significant energy range, despite that it cannotbe used to estimate individual higher state energies ina given local minimum. Turning back to our problemhere, u gives a very good description of the fundamentalstate in the subregion Ω m = [0 , L ]. It is important toremember that the 1 /n convergence resulted directlyfrom ˆ Hu = 1. So the MLLT approach, utilizing ˆ H u = 1should lead to an even faster convergence of the seriesexpansion of u in terms of the eigenstates ψ n ( z ) in the E ne r g y ( e V ) W LLT W MLLT W MLLT z (¯ )
Infinite Square Well
FIG. 1. (Color online) Comparison of the effective potentialsfor a square well with infinitely high potential barriers. Theinfinite square well potential is given by the (black) dasheddotted line. The effective confining potential calculatedvia LLT is given by the red solid line. Effective confiningpotentials obtained from MLLT via two different approaches(see main text) are given by the (blue) dashed and (green)dotted line. subregion Ω m = [0 , L ]. Before discussing this in moredetail we turn and calculate the ground state energy ofthe one-dimensional (1-D) infinite square well potentialproblem within LLT.Using Eq. (9), and keeping in mind h ψ n | ψ m i = δ n,m ,the energy E m , LLT is given by: E m , LLT = h u | H | u i|| u || = h u | i|| u || = 2 λ √ Lπλ P ∞ m =1 1(2 m − P ∞ m =1 1(2 m − = 2 √ Lλπ π . (20)Substituting the value of λ = √ LE π into Eq. (20) one isleft with E m ≈ . · E . (21)Thus, the ground state or fundamental energy E m in the subregion Ω m = [0 , L ] is in excellent agreementwith the result obtained directly from the SE; E m is just over 1% larger than the ground state energyeigenvalue E . However, as we will discuss below andin an appendix, it is not guaranteed that always such agood agreement is achieved and that LLT might even failfor certain confinement potentials.Having discussed energy eigenvalues, we turn nowto consider the effective potential W LLT resulting fromthe LLT. This is given by W LLT = u − and shown inFig. 1 by the red solid line along with the potential ofa square well (black dashed dotted line) of width 50 ˚A and with infinitely high barriers. Figure 1 showsthat W LLT softens the potential near the boundaries.As we will discuss further below, this effect is alsoseen in a well with finite barriers, where it providesthe above discussed quantum corrections to transportsimulation. Therefore, it is important that the MLLTcaptures these pertinent aspects as well, to be of use forsuch simulations. In the following section we discuss theMLLT for a square well with infinitely high barriers.
B. MLLT
Having solved the particle-in-a-box problem withinthe SE and LLT, we target it now within MLLT byemploying: ˆ H u = 1 . (22)Using Eq. (4), in the case of the MLLT one is left withˆ H u = X n E n α n ψ n = 1 . Following the steps outlined above for the LLT, theexpansions coefficients α n , again taking only odd n values, are given by α n = 2 √ LE πn . (23)Comparing the above equation with Eq. (18), we findhere already that α n scales as 1 /n instead of 1 /n . Withthis u reads: u ( z ) = 2 √ LE π ∞ X n odd n ψ n ( z )= λ ′ ∞ X m =1 ψ m − ( z )(2 m − = λ ′ (cid:18) ψ ( z ) + 1243 ψ ( z ) + 13125 ψ ( z ) + ... (cid:19) , (24)with λ ′ = 2 √ L/E π . When comparing this resultwith the expansion of u in the LLT frame, Eq. (19), itis evident that the MLLT yields an even faster/betterconvergence/approximation of u ( z ) with respect to theground state/fundamental state ψ . Thus, within theMLLT approach the approximation ψ m ≈ u/ || u || ,Eq. (8), should be even better justified.The square of the energy eigenvalue ( E m ) isgiven by: ( E m ) = h u | H | u i|| u || == 2 √ Lπλ ′ P ∞ m =1 1(2 m − P ∞ m =1 1(2 m − (25) ≈ . · E . Therefore, E m ≈ . · E yields an evenbetter approximation of the true ground state energy,when compared to the LLT result discussed above( E m ≈ . · E ). Again, the reason for thisimprovement can be traced back to the series expansionof u where the expansion coefficients α n decrease rapidlyin magnitude with increasing n .Having seen the improved ground state energyconvergence in MLLT, we now turn our attentionto the calculation of the effective confining potential W MLLT within MLLT. In the previous section wehave already discussed two approaches to obtain W MLLT from u MLLT , namely f W MLLT = ( E l · u MLLT ) − or W MLLT = ( √ u MLLT ) − . From Eq. (24), it is clear that f W MLLT = ( E l · u ) − with E l = E will give an effectivepotential f W MLLT that will be in excellent agreementwith W LLT , given that λ ′ = 2 √ L/ ( E π ). This isconfirmed in Fig. 1, where f W MLLT (green dashed line)matches almost perfectly W LLT , thus keeping the featureof softening the potential at the infinitely high barriers.Also the second approach, f W MLLT = ( √ u MLLT ) − givesa reasonable description of the potential, however, witha less pronounced softening near the barrier.Overall, we see for the infinite square well potentialthat f W MLLT = ( E · u ) − gives an effective potential thatmatches closely W LLT , reflecting that each expansioncoefficient α n , cf. Eq. (23), only depends on the groundstate energy E . However, as indicated already in Sec. II,Eq. (10), this might not be the case for other potentials.We discuss this further briefly in the appendix, wherewe apply the MLLT to a triangular shaped well with infinitely high barriers. For such a potential we find thatLLT does not converge to give a finite estimate of theground state energy E ; MLLT does converge but to anenergy that is noticeably different from the solution ofthe SE. Given that both LLT and MLLT have difficultiesin dealing with a triangular shaped potential with infinitebarriers, we investigate a triangular shaped well with finite barriers in the following. Such a system is relevantfor studying electronic and optical properties of III-N-based QW systems, as we describe in the next section. IV. BACKGROUND ON NITRIDE-BASEDHETEROSTRUCTURES AND InGaN QUANTUMWELL
III-N materials, such as InN, GaN and AlN haveattracted considerable interest for optoelectronic devices,since their alloys are in principle able to cover emissionwavelengths from infrared to deep ultra-violet. InGaNheterostructures, such as QWs, are of particular interestfor emission in the visible spectral range. Whencompared to other III-V materials, such as InAs orGaAs, III-N materials preferentially crystallize in thewurtzite crystal phase while InAs and GaAs crystallizein the zinc blende phase. The wurtzite crystal structure,
TABLE I. Band gap E g , lattice constants a , c , spontaneouspolarization P sp , piezoelectric coefficients e ij , elasticconstants C ij and effective electron m e and hole m h mass for wurtzite InN and GaN. The hole mass has beendetermined from the equations given in Ref. 33, using the A i -parameters from Ref. 32.Parameters GaN InN E g (eV) 3.44 0.64 a (˚A) 3.189 3.545 c (˚A) 5.185 5.703 P sp (C/ m ) -0.034 -0.042 e (C/ m ) -0.45 -0.52 e (C/ m ) 0.83 0.92 C (GPa) 106 92 C (GPa) 398 224 m e ( m ) 0.209 0.068 m h ( m ) 1.876 1.811 due to its lack of inversion symmetry, allows for astrain-induced piezoelectric polarization vector field butalso a spontaneous polarization vector field, which iseven present in the absence of any strain effects. Discontinuities in the polarization vector fields leadto very strong electrostatic built-in fields (MV/cm) inInGaN/GaN QW systems grown along the wurtzite c -axis, which is the standard growth direction forthese systems. Often these systems, especially whendealing with transport properties of InGaN/GaN MQW-based LED structures, are treated as 1-D systems inwhich the conduction and valence band profiles aremodified by the presence of the intrinsic electrostaticbuilt-in potentials.
It should be noted that this isa simplified description of these systems; more recentlyit has been shown that the alloy microstructure of InGaNQWs significantly affects the electronic structure, so thatlocal potential fluctuations play an important role.
However, for the analysis here, a simplified 1-D modelis a good starting point for comparing LLT and MLLTand to highlight the benefits of the MLLT in terms ofconvergence and “robustness” of the solution against, forinstance, the choice of the sub-region Ω m over which u is being evaluated to obtain the ground state energy inthe given region. Since the methodology of the MLLTis the same as that of the LLT, MLLT can directlybe applied to a landscape with energy fluctuations dueto alloy fluctuations. However, as discussed alreadyabove, further consideration must then be given as tohow best to calculate W within MLLT, which we willdo below. To flesh out the benefits of the MLLT, wefocus on the often used 1-D description of the electronicstructure of c -plane In x Ga − x N/GaN QWs with differentIn contents x . As mentioned above, due to the underlyingwurtzite crystal structure and growth along the c -axis, c -plane InGaN/GaN QW systems exhibit very strongelectrostatic built-in fields. This electrostatic field arisesfrom discontinuities in spontaneous and piezoelectricpolarization vector fields. The corresponding total built-in potential φ QW , assuming that the wurtzite c -axis isparallel to the z -axis of the coordinate system, can beexpressed as: φ QW ( z ) = φ QWsp ( z ) + φ QWpz ( z )= ( ( P Wsp − P Bsp ) + P Wpz ǫ ǫ Wr ) ( | z | − | z − h | ) . (26)Here, h is the height/width of the QW with well barrierinterfaces at z =0 and z = h . The dielectric constant ofthe QW material is denoted by ǫ Wr and P Wsp ( P Bsp ) is thespontaneous polarization in the well (barrier). Assumingthat the barrier material is strain-free, a strain fieldis only present in the InGaN QW, since InGaN has alarger lattice constant than GaN. Thus one is left witha piezoelectric polarization component in the well P Wpz ,which in the 1-D case can be written as: P Wpz = 2 ǫ e W + ǫ e W . (27)Here, e Wij and ǫ ij are the (well) piezoelectric coefficientsand the strain tensor components. The strain tensorcomponents are given by ǫ = ( − C W /C W ) ǫ and ǫ = ( a B − a W ) /a W ; a W ( a B ) is the in-plane latticeconstant of the well (barrier) material and C Wij are theelastic constants of the well material. The materialparameters used in this study are summarized in Tab. I.When calculating the electrostatic built-in potential ofIn x Ga − x N/GaN QWs as a function of the In content x ,a linear interpolation of the involved material parametersis applied. We neglect contributions from second-orderpiezoelectric effects. Using Eq. (26) and the materialparameters from Tab. I, the resulting built-in potentialis similar to that of a capacitor. Since we are interested in a general comparisonbetween LLT and MLLT results, we calculatethe electronic structure of the above discussedIn x Ga − x N/GaN QW systems in the framework ofa single-band effective mass approximation for electronsand holes. The confining potential for electrons andholes is then given by the conduction band (CB) andvalence band (VB) edge alignment between GaN andInGaN. In Eq. (28) below we assume that the VB edgeof bulk GaN (no built-in field) denotes the zero of energyin our system; the GaN CB edge, in the absence of thebuilt-in field, is at the band gap energy E GaN g of bulkGaN. The In x Ga − x N CB edge, E InGaNCB , and the VBedge, E InGaNVB , are calculated as a function of the Incontent x as follows: E InGaNCB = x ( E InN g + ∆ E VB ) + (1 − x ) E GaN g − b CB x (1 − x ) ,E InGaNVB = x ∆ E VB − b VB x (1 − x ) . (28)Here, ∆ E VB is the natural VB offset between pureInN and GaN, which has been taken from HSE-DFTcalculations. The (composition dependent) CB and VB edge bowing parameters are denoted by b CB and b VB . 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 andholes are spatially separated along the growth direction( c -axis/ z -axis). This situation is also know as thequantum confined Stark effect (QCSE). Building on this potential profile we use a single-band effective mass approximation to construct theHamiltonian matrix of this system. Here, we use differenteffective masses for electrons and holes, with the valuesgiven in Table I. A linear interpolation between theeffective masses of InN and GaN has been applied toobtain the corresponding masses for InGaN. However,differences in the effective mass inside and outside thewell are not considered. Given that we are interested inground state energies and more generally in a comparisonbetween results obtained from the SE, LLT and MLLT,applying a constant effective mass should be sufficientfor our purposes here. To numerically solve ˆ Hψ = Eψ ,ˆ Hu = 1 and ˆ H u = 1 we use the finite difference methodand assume a well of width of L w = 35 ˚A with a barrierwidth of L b = 100 ˚A on each side of the well; thediscretization step size is ∆ = 0 .
05 ˚A.
V. RESULTS FOR c -PLANE In x Ga − x N/GaNQUANTUM WELLS
In this section we compare and discuss ground stateenergies, wave functions and the effective potential W of c -plane In x Ga − x N/GaN QWs obtained by solving SE,LLT and MLLT in the numerical framework discussedabove. Special attention is paid to the impact of theIn content x on the results, given that with increasingIn content the piezoelectric contribution to the built-inpotential, 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 veryhigh In contents ( x > c -plane wells. Suchan analysis will help us to compare the impact of strongasymmetries in the potential landscape on the results ofthe LLT and MLLT, respectively. Strong fluctuations inthe potential landscape may occur locally in InGaN QWswith higher In contents (e.g. 25% In) due to random alloyfluctuations.Several aspects of the following analysis are to benoted. Firstly, the solution of the SE represents thereference/benchmark for the results of LLT and MLLT.Secondly, since we are using a single-band effective massapproximation in the framework of a finite differencemethod, we treat electrons and holes separately. In doingso, especially for LLT and MLLT, care must be takenwhen defining the zero of energy. As discussed in Sec. II,the expansion coefficients α n for constructing u from theeigenstates of the system are inversely proportional to thecorresponding state energies. Ideally, the zero of energy E CB W eLLT W eMLLT E ne r g y ( e V ) a) E VB W hLLT W hMLLT b) E ne r g y ( e V ) z (nm) FIG. 2. (Color online) a) Conduction E CB (black solid line)and b) valence band edge E VB (black solid line) in a c -planeInGaN/GaN QW with 25 % In. The effective potentialscalculated from LLT, W LLT , and MLLT, W MLLT , are given bythe green dashed lines and the red dotted lines, respectively.More details on the calculation of W MLLT are given in themain text. should be chosen close to the “true” ground state energyof the system in a given subregion Ω m . By doing so,the expansion coefficient α is then large as comparedto higher order terms that have lesser contributions. Asa consequence u is then a very good approximation ofthe ground state wave function, resulting also in a goodestimate of the corresponding energy. Here, we alwayschoose the zero of energy as the minimum energy in theband edge profile of the confining potential for electronsand holes. An illustration of this situation is displayedin Fig. 2.Finally, following Eq. (9), when calculating the groundstate energy from u , the subspace region Ω m over which u is integrated has to be chosen. To illustrate theimpact of Ω m on the results, three different subregionshave been considered for electrons and holes. Forelectrons these are labeled as Ω em, , Ω em, and Ω em, . Thefirst electron subregion, Ω em, , corresponds to the entiresimulation cell (-10 nm to 13.5 nm). The subregionΩ em, considers slightly more than the QW region, i.e.-1.5 nm ≤ z ≤ em, , wejust consider it to be 0 nm ≤ z ≤ z = 0and extends 1 nm into the barrier region above theupper QW interface at z = 3 . em, accounts for the tilt in the band edges and thattherefore the electron wave function is expected to leakfurther into the barrier region on the + z -side of the well.For holes, the subregions one Ω hm, and two Ω hm, areidentical to the first two electron cases. Only subregionΩ hm, is different from Ω em, . For Ω hm, we have chosen-1 nm ≤ z ≤ m affects differently the results obtained fromLLT and MLLT. This insight is for instance of interestwhen treating (random) potential fluctuations, wherepartitioning the system may be difficult. Thus, a methodwhere results are less dependent on the Ω m choice is ingeneral preferred. A. Electron ground state energies, wave functionsand effective potential
In a first step we focus on the results for the electronground state energy as a function of the In content x for the above discussed c -plane In x Ga − x N/GaN QWs.The data are presented in Fig. 3, upper row, for thethree different integration regions Ω em,i . The resultsobtained by solving the SE, E e SE , are given by the blacksquares. The green circles show the results from LLT, E e LLT , while the red triangles denote MLLT data. Beforelooking at the results in detail, one can already infer fromFig. 3 that when using subregion Ω em, , cf. Fig. 3 c),a very good agreement between LLT, MLLT and SE isachieved. Clearly larger deviations are observed for LLTand MLLT with respect to the SE result when using thefull simulation cell, Ω em, , cf. Fig. 3 a).The lower row of Fig. 3 displays the deviations (in%) between LLT (MLLT) and the solution of the SEas a function of the In content x for the three differentsubregions Ω em,i . The green circles show the data forthe comparison between the SE and LLT, while thered triangles do so for the comparison between SE andMLLT. Starting with Ω em, , Fig. 3 d), we observe thatfor both LLT and MLLT the deviations increase withincreasing In content x in the well. However, LLTshows noticeably larger deviations when compared toMLLT for In contents x exceeding values of 15% ( x =0 . em, , Fig. 3 e), deviations are ingeneral strongly reduced. Nevertheless, still a noticeabledifference between LLT and MLLT is observed. Morespecifically, while LLT produces errors of above 5%, theMLLT results are a good approximation of the trueground state energy, independent of the In content x (errors below 2% are found). When restricting theintegration region further (Ω em, ), Fig. 3 f), deviationsin LLT are further reduced and only in the very high Incontent regime ( x > .
4) more pronounced deviations areobserved. The error in the MLLT result is below 1% overthe range of In composition considered. This analysisshows that the MLLT produces a better approximationof the electron ground state energy, independent ofthe In content x and chosen subregion Ω em,i . This eSE E eLLT E eMLLT E eSE E eLLT E eMLLT E eSE E eLLT E eMLLT Indium content (x)Indium content (x) E ne r g y ( e V ) E ne r g y ( e V ) D e v i a t i on ( % ) D e v i a t i on ( % ) Indium content (x) D e v i a t i on ( % ) E ne r g y ( e V ) W em,1 W em,3 W em,2 deviation of E eLLT deviation of E eMLLT deviation of E eLLT deviation of E eMLLT deviation of E eLLT deviation of E eMLLT FIG. 3. Upper row: Electron ground state energies for c -plane In x Ga − x N/GaN QWs as a function of the In content x . Theresults are shown for the three different subregions Ω em,i discussed in the text. The electron ground state energy is computedby solving the SE (black squares), LLT (green circles) and MLLT (red triangles). Lower row: Deviation of E e LLT (green circles)and E e MLLT (red triangles) with respect to the solution of the SE for the different subregions Ω em,i . makes it therefore very attractive for calculations of thefundamental state in subregion of an energy landscapewhich shows strong fluctuations so that the systemcannot be easily partitioned into different subregions.This begs the question why the energy obtained fromMLLT is more robust against changes in Ω m . To addressthis point, Fig. 4 shows the (normalized) electron groundstate wave function calculated from the SE (black solidline) and the (normalized) u functions obtained from LLT(green dashed line) and MLLT (red dotted line) using thefull simulation box Ω em, . The results are displayed forthe c -plane In . Ga . N/GaN QW. The electron groundstate wave function ψ e SE shows the expected behavior ofhaving the highest value in the QW region, with thewave function amplitude then decaying rapidly in theGaN barrier region. Turning to the result from LLT(green dashed line) first, we find that u has a maximumin the well, however, it has also a constant finite valuein the GaN barrier region, especially for z > em , a strongimpact on the obtained energy E LLT could be expectedsince contributions from u in the barrier are removedwhen reducing the subregion Ω em . This is exactly thesituation we observe in Fig. 3. More specifically changingthe subregion from Ω em, (full system) to Ω em, (mainlyQW region), the error in E LLT when compared to E SE ,reduces from 9.8% to 1.3% for the In . Ga . N/GaNQW.The situation is different in the MLLT approach. -12 -9 -6 -3 0 3 6 9 12 15 Y eSE /||Y eSE || u eLLT /||u eLLT || u eMLLT /||u eMLLT || A m p li t ude z (nm) FIG. 4. (Color online) Comparison of the (normalized)electron ground state wave functions of a c -plane InGaN/GaNQW with 25% In and a width of 3.5 nm. The wave functionsare obtained by solving the SE (solid black line), LLT (dashedgreen line) and MLLT (dotted red line). The blue dashed-dotted box and the solid magenta box indicate the subregionsΩ em, Ω em, , discussed in the text. Here, u e MLLT , at least for z < ψ e SE , with the magnitude of u e MLLT being very small, similar to ψ e SE but in contrast to u e LLT .However, for z > u e MLLT is0 hSE E hLLT E hMLLT E hSE E hLLT E hMLLT W hm,3 W hm,2 W hm,1 E hSE E hLLT E hMLLT a) b) c)f)e)d) deviation of E hLLT deviation of E hMLLT deviation of E hLLT deviation of E hMLLT D e v i a t i on ( % ) E ne r g y ( e V ) E ne r g y ( e V ) D e v i a t i on ( % ) E ne r g y ( e V ) D e v i a t i on ( % ) deviation of E hLLT deviation of E hMLLT FIG. 5. Upper row: Hole ground state energies for c -plane In x Ga − x N/GaN QWs as a function of the In content x . Theresults are shown for the three different subregions Ω hm,i discussed in the text. The hole ground state energy is computed bysolving the SE (black squares), LLT (green circles) and MLLT (red triangles). Lower row: Deviation of E h LLT (green circles)and E h MLLT (red triangles) with respect to the solution of the SE for the different subregions Ω hm,i . comparable to that of u e LLT and therefore much largerthan ψ e SE in this region. Thus, given that the magnitudeof u e MLLT is small in the region z < ψ e SE in the QW region,the analysis confirms the observation that E e MLLT is lessdependent on Ω em than E e LLT .Finally, we discuss here the effective confining potentialfor electrons obtained both within LLT, W e LLT , andMLLT, W e MLLT . In case of LLT it is obtained via W e LLT = ( u e LLT ) − , and given in Fig. 2 a) by the greendashed line for an InGaN/GaN QW with 25% In. Thepotential reveals a softening at the QW barrier interface,which, as discussed above, is an important feature forquantum corrections in drift-diffusion calculations using W e LLT for the energy landscape. The here observedpotential profile is consistent with the results reportedpreviously. To obtain the effective confining potentialfrom MLLT that reflects the behavior of W e LLT , we findthat W e MLLT = ( p u e MLLT ) − works here best, while f W e MLLT = ( u e MLLT · E ) − results in a very differenteffective potential from W e LLT (not shown). Figure 2 a)confirms that W e MLLT (red dotted line) is in very goodagreement with W e LLT (green dashed line). We note thatthis holds over the full In content x range studied here. B. Hole ground state energies, wave functions andeffective potential
Having discussed the results for the electron groundstate energies, wave functions and the effective confiningpotential, we now turn and present the results for holes,again as a function of the In content x for the abovediscussed c -plane In x Ga − x N/GaN QWs. The upper rowof Fig. 5 presents the comparison between the energiesobtained from SE, ( E h SE , black squares), LLT, ( E h LLT ,green circles) and MLLT, ( E e MLLT , red triangles). Theresults are shown for the three different subregions Ω hm,i over which u is integrated to obtain the correspondingenergy. The lower row of Fig. 5 depicts for Ω hm,i thedeviation (in %) of LLT (green circles) and MLLT (redtriangles) from the SE solution. Looking at Fig. 5 a)first, one can clearly see that when integrating over thefull simulation region (Ω hm, ), both E h LLT and E h MLLT deviate from E h SE with increasing In content x . However,deviations are larger for LLT than for MLLT. A similarbehavior was also observed for the electron ground stateenergies when the full simulation box Ω em, is considered,cf. Fig. 3. But, trends for electrons and holes are quitedifferent. For electrons, the deviation in the groundstate energies with respect to the SE solution increasewith increasing In content x , cf. Fig. 3 d). For theholes, deviations in the ground state energy also startto increase with increasing In content x but deviationssaturate at around 18% and 8% for LLT and MLLT,1 -12 -9 -6 -3 0 3 6 9 12 150.000.050.100.150.200.25 Y hSE /||Y hSE || u hLLT /||u hLLT || u hMLLT /||u hMLLT || A m p li t ude z (nm) FIG. 6. (Color online) Comparison of the (normalized)ground state hole wave functions for a c -plane InGaN/GaNQW with 25% In content and a width of 3.5 nm. The wavefunctions are obtained by solving the SE (solid black line),LLT (dashed green line) and MLLT (dotted red line), usingthe simulation box Ω hm, . The blue dashed-dotted box andthe solid magenta box represents the two integration regionsfrom -1.0 nm to 5nm (Ω hm, ) and -1.0 to 3.5 nm (Ω hm, ). respectively, when the In content exceeds 15% ( x > . hm, ), MLLT provides a better description of E h SE whencomparing errors with LLT. When adjusting/reducingthe subregion Ω hm , cf. Figs. 5 b) and c), to calculate E h LLT and E h MLLT , the agreement with E h SE clearly improves.This is in particular true for E h MLLT , as Fig. 5 e)and f) show; deviations from E h SE close to 3% or lessare observed over the full In content range x . Morespecifically, for a well with 25% In, the error in E h MLLT reduces from 8% to 1.8% (see Figs. 5 d) and f)) whenchanging from Ω hm, to Ω hm, . Looking at E h LLT forthe same situation, we observe that the deviations arereduced from 18.4% (Ω hm, ) to 8% (Ω hm, ). However,the values are still noticeably higher when compared to E MLLT . This also shows that MLLT results are robustagainst changes in the In content x , while LLT exhibitslarger deviations from the SE data, especially for higherIn contents.Following our investigations on the electron groundstate energies and wave functions, we study here alsothe hole ground state wave functions. Again we useas a test system the c -plane In . Ga . N/GaN QW.The wave functions calculated from SE (black solid line),LLT (green dashed line) and MLLT (red dotted line)are shown in Fig. 6. Before looking at the fine details,independent of the model used, the wave functions arelocalized inside the well and “decay” in the GaN barrierregion. However, how the wave functions decay in thebarrier region strongly depends on the model. Whilethe hole ground state wave function ψ h SE rapidly decays in the barrier material, this situation is only true for u h LLT and u h MLLT along the + z -direction. Even though inthe + z -direction u h LLT and u h MLLT are similar, there arealso differences. While u h MLLT is very close to 0 in thebarrier region, u h LLT is small, but has a noticeable finiteconstant value in the GaN barrier for z > z < u calculated from MLLTand LLT, given that deviations in the ground state energyincrease as integration region Ω hm,i is increased.In the last step we turn attention to the effectivepotential for holes, W h , calculated within LLT andMLLT. The results are shown in Fig. 2 b). Theconfining potential from LLT, W h LLT = ( u h LLT ) − , isgiven by the green dashed line and shows again asoftening of the potential near the well barrier interface.Also for holes we have tested calculating the effectivepotential from MLLT via W h MLLT = (cid:18)q u h MLLT (cid:19) − and f W h MLLT = (cid:0) E h · u h MLLT (cid:1) − . The conclusion that is drawnfrom this is similar to that for electrons, meaning that f W h MLLT gives a potential profile very different from W h LLT (not shown), while W h MLLT is in good agreement with W h LLT . This is confirmed by Fig. 2 b), showing thatthe confining potential obtained from MLLT (red dottedline) captures the same effects as W h LLT (green dashedline). Again this result holds over the full In contentrange investigated here.
VI. CONCLUSIONS
In this work we have proposed, motivated andanalyzed, a modified localization landscape theory(MLLT). In the MLLT approach we solve ˆ H u = 1instead of ˆ Hu = 1, as in the LLT. We demonstratethe improvements resulting from using ˆ H u = 1 inpredicting ground state energies for a particle-in-a-box(infinite square well) potential. Since this problem can besolved fully analytically in LLT and MLLT, the solutionconfirms that u obtained from MLLT will in general givea better approximation of the true ground state wavefunction when compared to the result from LLT. We havealso shown that this can be traced back to the energydependence of the expansion coefficients of u in terms ofthe particle-in-a-box eigenstates. Given that u obtainedfrom MLLT provides a very good description of theground state wave function, it also provides an improvedestimate of the ground state energy and therefore theerror in this quantity is reduced in comparison to theLLT obtained value. Here, we also have provided insightinto the calculation of the effective confining potential W within MLLT. While this is straightforward in the case ofa particle-in-a-box problem with infinitely high barriers,we highlight that care must be taken when extracting theeffective confining W from MLLT in general. We have2discussed two strategies to obtain W from MLLT thatlead to results similar to those obtained from LLT, whichis important when applying MLLT for instance in drift-diffusion transport calculations to account for quantumcorrections.The particle-in-a-box problem provided the idealtestbed to study the basic properties of the LLTand MLLT. However, further analysis is required toconsider more realistic potentials. LLT has recentlybeen used to evaluate the electronic structure ofIII-N heterostructures, where the confining potentialis triangular shaped with barriers of finite height.Motivated by this, we have studied and compared groundstate energies from the Schr¨odinger equation (SE), LLTand MLLT for c -plane In x Ga x N/GaN QWs as a functionof the In content x . Special attention was paid to theimpact of the choice of the integration region of u whenevaluating the ground state energies. Our calculationsreveal 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 beingintegrated to obtain this energy is less important forMLLT than it is for LLT. Over the composition rangefrom 5% to 50% In in the well and when integrating over aregion close to the QW, errors in the ground state energyfrom MLLT never exceeds 4%. While similar numbersare 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. Lookingat the calculated effective potential W for electrons andholes, and independent of the In content x , we find thatusing W MLLT = ( √ u MLLT ) − gives in general resultsthat match closely the effective potential W LLT obtainedfrom LLT. Since W plays an important role in quantumcorrected drift-diffusion simulations, it is useful to seethat MLLT produces an energy landscape similar to LLT,so that it can be used in such simulations.Taking all this together the proposed MLLT keepsall the benefits of the LLT, such that only a systemof linear equations has to be solved instead of a largeeigenvalue problem to obtain ground state energies.At the same time the MLLT provides the followingaspects: (i) “faster convergence” of the calculatedground state energies with integration region, (ii) a more“robust” behavior of the method against changes in theintegration region, (iii) better agreement with resultsfrom SE, especially for higher In contents and (iv) aneffective confining potential comparable to that of LLT.All these features make the MLLT method attractivefor calculations of localized states in highly disorderedsystems, where for instance partitioning the systems intodifferent subregions is not trivial. ACKNOWLEDGMENTS
The authors acknowledge financial support fromScience Foundation Ireland under Grant Nos.17/CDA/4789, 15/IA/3082 and 12/RC/2276 P2.
Appendix: Infinite Triangular Well
Having discussed in the main text the fully analyticsolution of the particle-in-a-box problem with infinitelyhigh barriers, we study here another problem that canbe investigated fully analytically, which is a triangularwell with infinitely high barrier at z = 0; the potentialincreases from 0 at z = 0 with a slope F in the + z -direction. The aim of this study is twofold: it willillustrate (i) that in contrast to the particle-in-a-boxproblem, discussed in Sec. III, the expansion coefficientsof u can depend on multiple energies E n and (ii) thatthere are potentials where LLT and MLLT could fail togive a good approximation of (ground state) energies oreven diverge. Regarding (i), this finding is importantfor calculating the effective confining potential, showingthat it might not always be guaranteed that calculating f W MLLT = ( E · u MLLT ) − will give a good approximationof W LLT obtained from the “standard” LLT approach.For the infinite triangular potential, the SE reads: − ~ m d dz ψ n ( z ) + F zψ n ( z ) = E n ψ n ( z ) ⇔ d dz ψ n ( z ) − mF ~ (cid:18) z − E n F (cid:19) ψ n ( z ) = 0 . (A.1)Setting a = (2 mF/ ~ ) / and using γ = a ( z − E n /F ),one is left with a (cid:20) d f ( γ ) dγ − γf ( γ ) (cid:21) = 0 . (A.2)The general solution to the above differential equationcan be obtained as a linear combination of the Airyfunctions A ( z ) and B ( z ). These functions are definedas the improper Riemann integrals A ( z ) = 1 π lim c →∞ Z c cos (cid:18) t zt (cid:19) dt , (A.3) B ( z ) = 1 π lim h →∞ Z h (cid:20) exp (cid:18) t zt (cid:19) + sin (cid:18) t zt (cid:19)(cid:21) dt . (A.4)For z >
0, the function A ( z ) shows exponential decaywhereas B ( z ) diverges to infinity. Given that theconfined wave functions have to decay as z → ∞ , thefunction B ( z ) has to be discarded. Thus, the solutionsof the SE for a triangular-shaped potential with infinitelyhigh barriers, cf. Eq. (A.1), are given by: ψ n ( z ) = α n A i (cid:18) mF ~ (cid:19) / (cid:18) z − E n F (cid:19)! . (A.5)3The fact that the wave function has to go to zero at theinfinitely high barrier at z = 0 can be used to determinethe energy eigenvalues E n . To do so, the n th zero ofthe Airy function is approximated and the correspondingeigenvalue then reads: E n ≈ (cid:18) πF ~ m (cid:18) n − (cid:19)(cid:19) / . (A.6)Equipped with this solution we turn now and discussthe infinite triangular well firstly in the framework ofLLT and then of MLLT. To find a series expansion for u ,we first need an expansion for the constant function 1 interms of the eigenfunctions, over the interval [0, ∞ ): ∞ X n =1 b n ψ n = 1 . (A.7)We recall here that u = P ∞ n =1 a n ψ n , Eq. (4), so thatwhen using LLT, ˆ Hu = 1, one is left withˆ Hu = ˆ H ∞ X n =1 a n ψ n ! = ∞ X n =1 a n E n ψ n ! = 1 = ∞ X n =1 b n ψ n . (A.8)Due to the orthonormality of the wave functions, we canthus express the expansion coefficients a n as a n = b n E n . (A.9)Thus, within LLT, u can be expressed as: u LLT ( z ) = b E ψ ( z ) + b E ψ ( z ) + b E ψ ( z ) + · · · (A.10) In contrast to the infinite square-well potential, onecan show that the sum of the a n ’s does not convergefor the triangular well considered and thus the energy E ( u ) ,Eq. (9), does not converged. First, we note thatthe energies E n , Eq. (A.6), increase with increasing n as n ; numerical analysis we have undertaken indicatesthat b n decays approximately as n − α , where α ≈ . a n in LLT decays approximately as n − ( α + ) , where α + <
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