Electronic Structure and Defect Modeling of

III-V(Sb)Superlattices with hybrid DFT

Tristan Garwood

April 26, 2013

1 Introduction

Infrared detectors are widely used in both military and civilian applicationsand have been researched extensively in the last two decades. Such applicationsinclude detection and tracking of hot objects at a long range, passive night vi-sion, finding people in collapsed buildings, astronomy, and bio-medical imaging.The IR portion spans a wavelength range from 1.0 to around 300 µm and isloosely further classified into near-IR (1-5 µm), mid-wave IR (5-12 µm), long-wave IR (12 -25 µm), and far IR (25-300 µm). Transitions between vibrationalquantum states typically occur in the infrared. Thus, in applications that desirethe observation and identification of chemical species using point detection orstand-off detection, such as pollution monitoring, gas leak detection, gas sensing,and spectroscopy, one needs to work in these wavelength bands. The spectrumof black-body radiation has a peak of around 10µm at room temperature. Thismakes detectors that have a range between 8-12µm very helpful to identify heatradiation from a target at around 300K.

Earlier military applications withstood the expense of the less stable andmore convenient HgCdTe alloy. The extremely small change of lattice constantwith composition makes it possible to grow high quality layers and heterostruc-ture. However, there are multiple problems associated with HsCdTe IR detec-tors such as large tunneling dark currents caused by the narrow band gap (lessthan 0.1 eV). InAs/GaSb type-II strained layer superlattices (SLS) have at-tracted considerable research interest for use as IR photodetectors due to theirsuppressed Auger recombination relative to bulk mercury-cadmium-telluride(MCT) materials which leads to improved temperature limits of spectral de-tectivities[18]. Superior performance of InAs/GaSb SLS has been theoreticallyexpected, but is yet to be observed experimentally. It is believed that this is aresult of Shockley-Read-Hall (SRH) recombination which causes increased levelsof dark current.

The eventual goal of my research project for the REU program at CHTM isto apply Sandia Labs’ Socorro code to InAs/GaSb SLS structures and obtainphysically accurate results for ideal structures and to then incorporate defects in

1

Figure 1: HRTEM image of an InAs/GaSb SLS grown at CHTM

the SLS. The results of these calculations will then be compared to experimentaldata to see if we can get a better idea of what kind of defects are causing thehigh levels of dark current associated with our InAs/GaSb detectors.

1.1 InAs/GaSb type-II strained layer superlattices

A superlattice is a periodic structure of layers of two or more materials, thelayers of which tend to be on the order of several nanometers. The materialsused to grow a superlattice are typically semiconductors with different band-gaps. This technique creates a set of selection rules which affects the conditionsfor charges to flow through the structure. Due to the periodic structure ofsuperlattice, it has periodic potential. To simplify the physical picture of SL,we can consider a set of quantum wells with finite barrier height U. The currentstandard for classifying superlattices is to group them by their confinementenergy schemes. These confinement schemes are usually labeled type I and typeII. In type I superlattices, the electrons and holes are both confined within thesame layers. Type II superlattices are composed of spatially indirect band gapsemiconductors which confines the electrons and holes in different layers.

The Type II InAs/GaSb superlattice was first introduced by the Nobel lau-reate L. Esaki in the 1970s [9] and then proposed for infrared detection ap-plications by Smith and Mailhiot in 1987. Since then, the Type II Sb-basedsuperlattice has evolved drastically with many variants for different purposes.It can be grown uniformly on comparatively large GaSb wafers with precisecontrol of the SLS structures and has a higher electron effective mass renderinglower tunneling currents. In a InAs/GaSb SLS the valence band of GaSb isabove the conduction band for InAs which creates minibands in the superlat-tice region by the coupling of electron wavefunctions. Infrared detection will bedone by the intersubband transitions between two minibands. The mechanismfor SRH recombination in InAs/GaSb SLS is unknown, however it appears toinvolve transitions to energy levels in the middle of the band-gap and gallium

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Figure 2: Illustration of a InAs/GaSb SLS

is believed to be a limiting factor as gallium-free SLS structures exhibit longercarrier lifetimes. It has been suspected that the SRH mechanism is due to de-fects in the crystal lattice of the SLS structures, but it is not known what typeof defect could be causing shorter carrier lifetimes.

Every solid has defects in its structure. In the case of crystals there are char-acteristic defects (because of periodicity) such as: point defects, like vacanciesand interstitials; one-dimensional defects, like dislocations; and two-dimensionaldefects, like grain boundaries, or interfaces. Trap assisted recombination occurswhen an electron falls into a ”trap”, an energy level within the band-gap causedby the presence of a foreign atom or a structural defect. Once the trap is filledit can not accept another electron. The electron occupying the trap energy canin a second step fall into an empty state in the valence band, thereby complet-ing the recombination process and possibly producing dark current. In physicsand in electronic engineering, dark current is the relatively small electric cur-rent that flows through photosensitive devices such as a photo-multiplier tube,photo-diode, or charge-coupled device even when no photons are entering thedevice.

Accurate defect level calculations are needed to determine whether theseaccount for the SRH mechanism or whether there could be some other type ofdefect such as anti-site bonding, interstitial atoms, etc. Reasonably accurateab initio electronic structure calculations have been performed using densityfunctional theory (DFT) with hybrid functionals [2]. Calculation of defects insemiconductor superlattices necessitates high-performance, parallel computa-tion as a unit cell size of a few hundred atoms may be needed to sufficientlylocalize the defect states.

1.2 Density Functional Theory

For the present purposes, we define the modern electronic structure prob-lem as finding the ground-state energy of non-relativistic electrons for arbitrarypositions of nuclei within the Born-Oppenheimer approximation [3]. If we can

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perform many of these calculations in a reasonable time-frame, properties of ar-bitrary materials can be predicted. Many traditional approaches to solving thisdifficult many-body problem begin with the HartreeFock (HF) approximation,in which Ψ is approximated by a single Slater determinant of orbitals (single-particle wave-functions)[15] and the energy is minimized [16]. Other approachesuse reduced descriptions, such as the density matrix or Greens function, whichleads to an infinite set of coupled equations that must somehow be truncated.Density functional theory is a successful approach to finding solutions to theSchrodinger equation in practical settings, and is rapidly becoming a standardtool for materials modeling problems in physics, chemistry, and materials sci-ence.

The situation we are interested in, where multiple electrons are interactingwith multiple nuclei, is a very complicated system to model. With each electronin the system, 3 dimensions are added to the problem, so if we are interestedin a cluster of 100 Pt atoms, the full wave function requires more than 23,000dimensions. The fundamental tenet of density functional theory is that anyproperty of a system of many interacting particles can be viewed as a functionalof the ground state density n0(r); that is, one scalar function of position n0(r),in principle, determines all the information in the many-body wave-functionsfor the ground state and all excited states[5]. The modern formulation of DFToriginated from a paper written by P. Hohenberg and W. Kohn in 1964 [7].The paper shows that the electron density can be thought of as a ”basic vari-able” used to determine all properties of the system; i.e. all the properties ofthe system can be thought of as functionals of the ground state density. Thisformulation can be used for any system of interacting particles in an externalpotential Vext(r). In our case we are interested in the problem of electrons andfixed nuclei, where the Hamiltonian can be written as:

H =h2

2me∗∑i=0

∆i + Vext +1

2∗∑i6=j

1

|ri − rj |(1)

(the nuclei-nuclei interaction can be added later). DFT is based on two theo-rems first proved by Hohenberg and Kohn [7].

Theorem 1: For any system of interacting particles in an external potentialVext(r), the potential Vext(r) is determined uniquely, except for a constant, bythe ground state particle density n0(r) [5].

Once we have Vext(r), the Hamiltonian is then determined, except for a con-stant energy. If we know the Hamiltonian then we know the many-body wave-functions for all states, therefore all properties are determined by the electrondensity n0(r).

Theorem 2: A universal functional for the energy E[n] in terms of the den-sity n(r) can be defined, valid for any external potential Vext(r).

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For any particular Vext(r), the exact ground state energy of the system is theglobal minimum value of this functional, and the density n(r) that minimizesthe functional is the exact ground state density n0(r) [5]. We can then use thisground state density to determine the Hamiltonian and all the properties of thesystem. Thus it has been established that a functional can be defined for anydensity (subject to certain conditions), and that by minimizing this functionalone would find the exact density and energy of the true interacting many-bodysystem.

The problem that we are faced with is approximating these functionals sinceno one has provided a method of constructing the exact functionals. The mainreason DFT is so widely used today is because of the approach proposed byKohn and Sham in 1965 [8]. The idea of the Kohn-Sham approach is to replacethe original many-body problem obeying the Hamiltonian by an auxiliary inde-pendent particle problem that can be solved more easily. This approach leadsto independent-particle equations for the non-interacting system with all thedifficult many-body terms incorporated into an exchange-correlation functionalof the density. By solving these equations one can determine the ground stateproperties of the original system with the accuracy only limited by the approx-imations in the exchange-correlation functional. I won’t go into much detail asfar as the derivation, but essentially the Kohn-Sham approach is to rewrite theHohenberg-Kohn expression for the ground state energy functional in the form:EKS = Ts[n] +

∫dr ∗ Vext(r)n(r) + EHartree[n] + EII + Exc[n] where Vext is

the external potential due to the nuclei and any external fields and EII is theinteraction between nuclei [5]. Vext, EHartree and EII are all well defined so wejust need to find Exc[n] (Ts is given explicitly as a functional of the orbitals).

If the universal functional Exc[n] were known, then the exact ground stateenergy and density of the many-body electron problem could be found by solv-ing the Kohn-Sham equations. As long as we can make a good approximationof Exc[n] we then have a useful approach to calculating ground state electronicproperties of a many-body system. The remarkable successes of the local den-sity approximation (LDA) and the generalized-gradient approximation (GGA)functionals within the Kohn-Sham approach have led to widespread interest indensity functional theory as the most promising approach for accurate, practicalmethods in the theory of materials.

1.2.1 The Local Density Approximation

The true form of the exchange-correlation functional, whose existence is guar-anteed by the Hohenberg-Kohn theorem, is simply not known. There is however,one case where this functional can be derived exactly: the homogeneous elec-tron gas. In this situation, the electron density is a constant at all points inspace (i.e. n(r) = C). This approximation only uses the local density to definethe approximate exchange-correlation functional, so it is called the local den-sity approximation (LDA) [6]. The LDA may seem limited for use in materialsscience since variations in n(r) are what define chemical bonds and properties,but the homogeneous electron gas does provide a practical way to actually use

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Figure 3: Illustration of how inaccurate LDA can be for band gap calculations[19]

a nearly 1-eV discrepancy remains. The standard deviationof this set is about 0.35 eV compared with the original de-viation in LDA band gaps of greater than 1 eV.

Proceeding further, we examine the consequences includ-ing of the second-order correction. The second-order term inEq. ~26!, however, includes the interaction~g! betweenneighboring bonds. As stated previously, this interaction is aconstant for the near-neighbor bonds but nevertheless mustbe realized by appeal to some physical property in theabinitio calculation. Here we shall associate it with the band-width of the upper valence-bandp-type states because thesestates produce the prototypical bonding orbitals in the tetra-hedrally bonded semiconductors, and their near-neighbor in-teractions are then directly related to the tight-binding band-width. We also recognize that a number of semiconductorsrequire inclusion of local-field effects in their dielectric de-scriptions to obtain accurateGW corrections. These tend tobe the wide-band semiconductors C, BN, etc., so, on thisbasis,g is reasonably expected to be related to the band-width. Incorporation of this effect at the level of a lineardependence then introduces a single additional parameterleading to a subsequent correction of the form

FIG. 1. Theoretical bands gaps calculated within the LDA of thesemiconductors listed in Table I. Circles, diamonds, and squaresrepresent group IV, group III-V, and group II-VI semiconductors,respectively.

TABLE I. Material parameters utilized in the modelGW correction along with resulting gaps.^r1/3&c

represents the expectation value ofr1/3(r ) for the lowest conduction-band state at the Brillouin-zone center,e` is the static-dielectric constant, and BW is the measured LDA bandwidth of the upperp-like valencebands taken from theab initio results. Numbers in parentheses indicate the spin-orbit correction~one-third ofwhich estimates the band-gap decrease applied to the model band gaps!. The notion 3C, 2H, and WZ indicatecubic, hexagonal, and wurtzite for those semiconductors that have a number of recognized structures~e.g.,GaN has also been grown in the zinc-blende phase!.

^r1/3&c e` BW ~eV! Egapda ~eV! Egap

st ~eV!a Egapmodel ~eV! Egap

expt ~eV!

C 0.519 5.7 13 4.09 6.56 5.48b 5.48Si 0.321 12.1 7 0.45 1.17b 1.17b 1.17Ge 0.347 15.9 8 0.00 0.57 0.58~0.18! 0.74Sn 0.399 24.1 7 20.30 20.12 20.08 ~0.80! 0.10SiC ~3C! 0.434 6.5 11 1.30 3.11 2.59~0.00! 2.39SiC ~2H! 0.402 6.6 9 2.05 3.68 3.37~0.00! 3.30BN 0.456 4.5 10 4.34 7.09 5.93~0.00! 6.10a

BP 0.391 10.8 11 1.31 2.33 2.18~0.00! 2.05AlN 0.421 4.8 7 4.18 6.55 6.01~0.00! 6.28AlP 0.327 7.5 6 1.44 2.62 2.55~0.00! 2.51AlAs 0.313 8.2 6 1.29 2.22 2.18~0.33! 2.23AlSb 0.376 10.2 6 1.16 1.94 1.93~0.67! 1.67GaN ~WZ! 0.429 5.6 8 2.45 4.54 4.07~0.00! 3.42GaP 0.344 9.1 7 1.50 2.52 2.44~0.00! 2.35GaAs 0.331 10.6 7 0.42 1.16 1.17~0.34! 1.52GaSb 0.384 14.4 6 0.00 0.51 0.52~0.70! 0.81InN 0.608 9.3 5 0.00 1.77 1.79~0.00! 1.89InP 0.397 9.6 6 0.43 1.55 1.54~0.00! 1.42InAs 0.399 12.3 6 0.00 0.76 0.78~0.38! 0.42ZnO 0.4469 3.7 5 0.60 3.87 3.23~0.00! 3.44ZnS 0.3977 5.1 5 1.64 3.75 3.64~0.00! 3.78ZnSe 0.3867 5.4 5 0.82 2.63 2.46~0.40! 2.82CdS 0.3624 5.5 4 0.77 2.62 2.63~0.00! 2.58CdSe 0.3533 6.2 4 0.24 1.65 1.63~0.42! 1.83

aExperimental gaps for BN range from 6.0 to 6.5 eV. A recent result from Ref. 37 is used here for compari-son.

bThe semiconductor was used as a reference for the model parameters.

PRB 58 15 553CORRECTIONS TO DENSITY-FUNCTIONAL THEORY . . .

the Kohn-Sham equations. The LDA is also often surprisingly accurate and forsystems with slowly varying charge densities generally gives very good results.However, it is a well known problem that the LDA leads to a substantial un-derestimate of calculated band gaps in semiconductors and insulators (fig.3).Thus, the LDA method isn’t accurate enough for our purposes.

1.2.2 The Generalized Gradient Approximation

The best known class of functionals after LDA uses information about thelocal gradient in the electron density as well as the local electron density. Thisapproach is called the generalized gradient approximation (GGA). There are alarge number of different GGA functionals because there many ways to form thegradient of the electron density. Two of the most widely used GGA functionalsare the Perdew-Wang functional (PW91)[10] and the Perdew-Burke-Ernzerhoffunctional (PBE) [2]. The local density approximation can be considered tobe the zeroth order approximation to the semi-classical expansion of the den-sity matrix in terms of the density and its derivatives. A natural progressionbeyond the LDA is thus to the gradient expansion approximation (GEA) inwhich rst order gradient terms in the expansion are included. In the GGA afunctional form is adopted which ensures the normalization condition and that

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the exchange hole is negative definite. This leads to an energy functional thatdepends on both the density and its gradient but retains the analytic proper-ties of the exchange correlation hole inherent in the LDA. The GGA improvessignificantly on the LDAs description of the binding energy of molecules.

1.2.3 Hybrid Functionals

The hybrid approach to constructing density functional approximations wasintroduced by Axel Becke in 1993 [11]. Hybridization with Hartree-Fock (exact)exchange provides a simple scheme for improving many molecular properties,such as atomization energies, bond lengths and vibration frequencies, which tendto be poorly described with simple ”ab initio” functionals [12]. Hybrid DFTfunctionals mix results from different functionals to get the Exchange-correlationenergy. The results are combined with a weighted sum, with the coefficients are

Figure 4: Comparison of calculated band-gap (eV) vs. experimental using hy-brid functionals [20]

N other electrons in the system, when it should interact with N-1 electrons only (so called the self interaction problem [9]). In the DFT in the LDA approximation all occupied bands are pushed up in energy by this interaction which has for a consequence an on-site/diagonal Coulomb/Exchange repulsion. This leads to too small energy band gap in DFT-LDA. On the other hand, in Hartree-Fock (HF) theory the Coulomb and exchange interactions cancels exactly, i.e. Jii = Kii, so there is no self interaction. HF theory overestimates band gaps due to the lack of dynamical screening. It appears that energy gaps predicted with hybrid functionals with ~20% HF and ~80% DFT exchange, exactly the proportions optimised for use in the calculation of ground state energetics in the B3LYP functional, are also in good agreement with experimentally measured ones.

0 1 2 3 4 5 6 70

1

2

3

4

5

6

7

(a)

B3LYP PBE0

1 InSb 2 InAs 3 InN (wz) 4 GaSb 5 GaAs 6 InP 7 AlSb 8 AlAs 9 GaP10 AlP11 GaN (wz)12 AlN (wz)

12

111098

76

5

43

21

Ene

rgy

gap

(theo

r.) (e

V)

Energy gap (exp.) (eV)

5 6 7

5

6

7

(b)

B3LYP PBE0 GGA

1 GaP 2 AlP 3 GaAs 4 AlAs 5 InP 6 InAs 7 GaSb 8 AlSb 9 InSb

9

876

5

43

21

La

ttice

con

stan

t (th

eory

) (A

)

Lattice constant (exp) (A)

FIGURE 1. A comparison of experimental energy gaps (a) and experimental lattice constants (b), with those predicted using DFT with B3LYP, PBE0, and GGA functionals. The energy gaps are computed at the experimental lattice constant. It has been shown previously, that the B3LYP functional, can provide a remarkable agreement with measured energy gaps, for most of the III-V semiconductor binaries [10]. In figure 1(a) the calculated energy gaps of 9 representative ZB binaries and 3 group III�–N WZ materials are shown for the B3LYP and PBE0 hybrid functionals. The B3LYP functional provides slightly better agreement with the experiment then PBE0 when the calculation is performed assuming experimental lattice constants. In general the PBE0 functional results in an overestimate of the band gap. In figure 1(b) the optimized lattice constants of the nine ZB binaries is presented for calculations using the B3LYP and PBE0 functionals. The lattice constants predicted using the PBE0 functional are closer to the experimental values than the B3LYP results. The B3LYP calculations systematically overestimate values of the lattice constant by 1-2% and are in somewhat worse agreement with experiment than those predicted from the GGA functional.

The trends in the band gap and lattice constant can be understood from the proportion of Fock exchange retained in the functional, which is

25% in PBE0 and 20% in B3LYP. The larger exchange component opens a larger band gap as it compensates for electronic self interaction. Increasing the exchange also tends to decrease the lattice constant; in the limit of a pure HF calculation the lattice constant would be significantly underestimated in a covalently bonded system. One therefore expects the use of PBE0 to lead to higher band gaps and shorter bond lengths than B3LYP in covalent systems. For wurtzite structures, with generally very small spin orbit splitting, it is observed here that there is remarkable agreement between ab-initio predicted and experimental energy gaps and optimized lattice constants for both B3LYP and PBE0 approaches. The overall agreement is competitive with that achieved by more sophisticated methods [11,12] and with empirical theories tailored for this purpose [13].

CONCLUSIONS

In conclusion, the hybrid exchange methodology provides an efficient and robust basis for large scale calculations of III�–V semiconductors reliably predicting both the ground state energetics and the electronic structure.

REFERENCES

1. S. Kurth, J. P. Perdew, P. Blaha, Int. J. Quantum. Chem 75, 889 (1999)

2. C.M. Zicovich-Wilson, F. Pascale, C. Roetti, V.R. Saunders, R. Orlando, R. Dovesi, J. Comput. Chem.25, 1873 (2004).

3. B. Montanari, B. Civalleri, C.M. Zicovich-Wilson, R. Dovesi, Int. J. Quantum Chem. 106, 1703 (2006).

4. R. Dovesi et al, CRYSTAL06 User's Manual, University of Torino, Torino, 2006.

5. C.L. Bailey, A. Wander, S. Mukhopadhyay, B.G.Searle and N.M. Harrison, Phys. Chem. Chem. Phys. 10, 2918 (2008)

6. L. Pisani, B. Montanari and N. M. Harrison, New J. Phys. 10, 033002 (2008)

7. L. Ge, B. Montanari, J. H. Jefferson, D. G. Pettifor, N. M. Harrison, G.A.D. Briggs, Phys. Rev. B 77, 235416 (2008)

8. J. Heyd, J.E. Peralta, G.E. Scuseria, R.L. Martin, J. Chem. Lett. 123, 174101 (2005).

9. J. P. Perdew and A. Zunger, Phys. Rev. B 23, 5048 (1981).

10. S. Tomic, B. Montanari, and N.M. Harrison, Physica E 40, 2125 (2008).

11. S. Kummel and L. Kronik, Rev. Mod. Phys. 80, 3 (2008).

12. P. Rinke et al., Phys. Rev. B 77, 075202 (2008). 13. J. R. Chelikowsky and M. L. Cohen, Phys. Rev. B 14,

556 (1976)

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to be determined by reference to a system for which the exact result is known.Hybrid functionals of this type are now very widely used for chemistry andphysics applications. Computed binding energies, geometries and frequenciesare systematically more reliable than the best GGA functionals. One exampleof a successful hybrid functional is the Becke, three-parameter, Lee-Yang-Parrfunctional (B3LYP).

EB3LY Pxc = ELDA

xc +a0(EHFx −ELDA

x )+ax(EGGAx −ELDA

x )+ac(EGGAc −ELDA

c )(2)

Unfortunately, although the results obtained with these functionals are usu-ally sufficiently accurate for most applications, there is no systematic way ofimproving them. Hence in the current DFT approach it is not possible to esti-mate the error of the calculations without comparing them to other methods orexperiments.

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1.2.4 Pseudopotentials

We have simplified our quantum problem so that we just need to solve a single-particle equation (Kohn-Sham approach), but we need to solve for enough statesfor every electron. The core electrons spend all their time near the nucleus.They repel the outer electrons, so the outer electrons feel a weaker potentialfrom the nucleus, but otherwise they dont affect the chemical properties etc.Provided we reproduce this screening effect, we can ignore these core electronsaltogether! We consider each atoms nucleus and core electrons as an ion (LDA),and produce a pseudopotential that has the same effect on the outer electrons.Not only have pseudopotentials reduced the cut-off energy we need, they havealso let us concentrate on the valence electrons, reducing the number of stateswe need from our Schrdinger equation.

Figure 5: Comparison of a wavefunction in the Coulomb potential of the nu-cleus (blue) to the one in the pseudopotential (red). The real and the pseudowavefunction and potentials match above a certain cutoff radius rc.

2 Activities So Far

When I applied for the REU program at CHTM, I was hoping to get anidea of what grad school would be like and at the same time start learningabout some of the problems that modern science is currently taking on. Mostof my time working at CHTM so far has been spent building up backgroundknowledge in the hopes that I could actually start understanding what theseproblems even are. Looking back on my educational career so far, I can’t thinkof any time in my life that I have learned so much. The research project I havebeen working on with Dr. Greg Von Winckel is, in his words ”not appropriate

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for an undergraduate”, but I feel that I have flourished from being thrown inthe deep end. Along with building up my knowledge on materials science, Ihave been evaluating various DFT calculation packages.

2.1 DFT software

There is a wide spectrum of software packages to choose from that can performab initio DFT calculations. A quick internet search reveals a few dozen piecesof software for solving these systems. Each one comes with its own learningcurves, availability of documentation, and quirks. In an effort to narrow downthe software that we thought would suit our needs we set a benchmark that thegiven software package should be able to make band gap calculations for III-Vsemiconductors withing a reasonable deviation (20 percent) of the experimentalvalues.

2.1.1 ADF computational chemistry package

The first software package that I tried out was the Amsterdam Density Func-tional (ADF) computational chemistry package. ADF has a free trial so I wasable to try out different DFT calculations for a variety of materials. The doc-umentation for ADF is the best that I have come across so far in the realm ofDFT software, but I was still left confused about a few important details (likehow to form a superlattice). ADF also has a good GUI that is fairly intuitive(when compared to other DFT software). None of the calculations I performedfor GaAs gave the current band-gap within even 20 percent accuracy, so I endedup moving on to try other software. This is probably because ADF is meantmore for use by chemists (who would be doing calculations for complex organicmolecules for example) and not materials science.

2.1.2 Abinit

Abinit was developed using the popular Kohn-Sham formulation of DFT. It iscurrently run by Xavier Gonze and written in FORTRAN. Dozens of variablesare available for customizing the input conditions of a system. This makesAbinit fairly versatile, but also provides a hard learning-curve. Abinit has twodifferent pseudo-potentials built into it for each element. Additionally, it hasmultiple algorithms to calculate a new pseudo-potential if one is desired. OnceI was finally able to get Abinit working on my computer at work, I wasn’t ableto calculate a band-gap for GaAs that was close enough to the experimentalvalue to suit our needs.

2.1.3 Gaussian

Gaussian was originally developed by Nobel laureate John Pople in 1971 atCarnegie Mellon in FORTRAN. Initially, the program was open source andfreely distributed to any university that wanted to use it. However, more re-cently the program has become very closed-source and is sold by Gaussian Inc.

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under an extremely restrictive license. Various stipulations of the license are:not being allowed to disclose how fast the program is, work on any compet-ing program, or allow anyone that is banned from using Gaussian to use yourcopy. The official list of people banned from using Gaussian can be found atwww.bannedbygaussian.org which includes Princeton, CalTech, and the creatorof Gaussian himself (John Pople). Since it is illegal to discuss how fast theprogram is, and how accurate it is, it is hard to determine whether it is worththe price of $4500-36,000 (academic and industry prices respectively). It is es-pecially hard to justify this price when there are open-source programs that arefreely available. Since there is no trial, we selected not to use Gaussian.

2.1.4 VASP

VASP is a plane-wave code for ab-initio density-functional calculations. It at-tempts to match the accuracy of the most advanced all-electron codes by usinga projector-augmented-wave approach (PAW) [13] for describing the electron-ion interaction. In VASP, central quantities, like the one-electron orbitals, theelectronic charge density, and the local potential are expressed in plane wavebasis sets. The interactions between the electrons and ions are described usingnorm-conserving or ultra-soft pseudo-potentials, or the projector-augmented-wave method. VASP implements all of the common DFT methods includinghybrid functionals. Green’s functions methods and many-body perturbationtheory are also available in VASP. I emailed VASP and they assured me that itcould calculate reasonably accurate band-gaps for GaAs and other III-V semi-conductors. We were thinking of going with VASP, but after talking with Nor-mand Modine from Sandia Labs we decided to use Soccoro instead for variousreasons.

2.2 Collaboration with CINT

Socorro is a modular, object oriented code for performing self-consistentelectronic-structure calculations utilizing the Kohn-Sham formulation of density-functional theory and is being developed by researchers at Sandia NationalLaboratories, Vanderbilt University, and Wake Forest University under a GNUGeneral Public License. Since we were looking for software that would need tofit to a specific set of requirements, we decided that using Soccoro would be theright choice since we can work closely with Sandia to modify it to suit our needs.In early February we submitted a CINT Rapid Access proposal, which was thenaccepted. In the six weeks since the Rapid Access proposal was accepted, wehave worked with CINT Scientist Normand Modine to demonstrate that we canuse the Socorro code and hybrid functionals incorporating a suitable fractionof the exact exchange to obtain band gaps that agree well with experimentalresults for InAs and GaAs. In the case of InAs, we found that the optimizedeffective potential method implemented in Socorro did not converge reliably tothe correct ground state due to the nearly vanishing Kohn-Sham band gap ofthis system. As a result, Dr. Modine implemented the Generalized Kohn-Sham

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approach to hybrid calculations using a recently proposed compact representa-tion of the exchange-operator[14]. For the cases that we have tested, we findthat this approach converges robustly and rapidly to the ground state whilemaintaining the computational efficiency of the optimized effective potentialmethod. Based on these results, we conclude that the Socorro code will likelybe able to provide an accurate model of InAs/GaSb SLS.

3 Plans For the Future

The goal of this work is two-fold: (1) to apply the Socorro code to InAs/GaSbSLS structures and obtain physically accurate results for an ideal structure. Inparticular, we will calculate the band offsets and electronic structure for severallayer thicknesses of interest and compare it to experimentally measured parame-ters from structures grown at CHTM, and (2) to incorporate defects in the SLS.To do so, we will introduce intrinsic point defects at various locations in the SLS,relax the defect structures, and calculate the defect formation energies as well asthe defect energy levels within the superlattice band gap. Adequately resolvingthese localized energy states due to defects is computationally challenging, asit requires that the unit cell be suffciently large that a defect not interact toostrongly with its images in adjacent super cells. This, coupled with the SLSstructure itself, necessitates a much larger unit cell than is typical in semicon-ductor band structure calculations. We will continue work with Dr. Modine toevaluate the applicability of Socorro to simulating InAs/GaSb SLS and deter-mine whether additional hybrid functionals will need to be implemented. Dr.Von Winckel will assist in the development of Socorro improve computationalefficiency as needed to handle larger unit cells.

4 References

[1] N. Gautam, M. Naydenkov, S. Myers, A.V. Barve, E. Plis, T. Rotter, L.R.Dawson, S. Krishna Appl. Phys. Lett., 98 (2011), p. 121106.[2] G. Kresse, F. Tran, Y.S. Kim, M. Marsman, and P. Blaha. Towards efficientband structure and effective mass calculations for III-V direct band-gap semi-conductors. Phys. Rev. B, 82:205212, (2012).[3] M. Born, R. Oppenheimer, Ann. Phys. 1927, 389, 457.[4] K. Burke, LO. Wagner, Int. J. Quantum Chem. 2013, 113, 96101. DOI:10.1002/qua.24259.[5] Electronic Structure: Basic Theory and Practical Methods (Vol 1) (26 April2004) by Richard M. Martin.[6] Jungwirth, P. (2010), Density Functional Theory. A Practical Introduction.By David Sholl and Janice A. Steckel. Angew. Chem. Int. Ed., 49: 485. doi:10.1002/anie.200905551.[7] P. Hohenberg and W. Kohn, ”Inhomogenous Electron Gas” phys. Rev.136:B864-871, 1964.

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[8] Kohn, Walter; Sham, Lu Jeu (1965). Physical Review 140 (4A): A1133A1138.doi:10.1103/PhysRev.140.A1133.[9] G. A. Sai-Halasz, R. Tsu, and L. Esaki, A new semiconductor superlat-tice,Appl. Phys. Lett., vol. 30, no. 12, pp. 651653, Jun. 1977.[10] J.P. Perdew, in Electronic Structure of Solids 91, edited by P. Ziesche andH. Eschrig (Akademie Verlag, Berlin, 1991), p. 11.[11] A.D. Becke (1993). J. Chem. Phys. 98 (2): 13721377. doi:10.1063/1.464304.[12] John P. Perdew, Matthias Ernzerhof and Kieron Burke (1996). J. Chem.Phys. 105 (22): 99829985. doi:10.1063/1.472933.[13] Blochl, P. E. Phys Rev B 1994, 50, 17 953.[14] I. Duchemin and F. Gygi, Computer Physics Communications, 181:855,(2010)[15] J. C. Slater, Phys. Rev. 1929, 34, 1293.[16] D. R. Hartree, W. Hartree, Proc. R. Soc. London. Ser. A Math. Phys.Sci. 1935, 150, 9.[17] Hai Xiao, Jamil Tahir-Kheli, and William A. Goddard, J. Phys. Chem.Lett. 2011, 2, 212217. DOI: 10.1021/jz101565j[18] C.H. Grein, M.E. Flatte, H. Ehrenreich, R.H. Miles, J. Appl. Phys. 74(1993) 4774;[19] Elena Plis, et al. Infrared Physics & Technology. DOI:10.1016/j.infrared.2009.09.008[20] Kurt A. Johnson, N. W. Ashcroft Phys. Rev. B 12/1998; 58(23).DOI:10.1103[21] S. Tomic, B. Montanari, N.M.Harrison Physica E, 40(2008), p.2125

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