Composition dependent band offsets of ZnO and its ternary alloys

0.38 MB·

by  Haitao Yin

Compositiondependent band offsets ofZnO and itsternary alloys
Haitao Yin,1,2 Junli Chen,1 Yin Wang,2,3,∗ Jian Wang,2,3 and Hong Guo4,2
1KeyLaboratory forPhotonic and ElectronicBandgap Materials of Ministryof Education,
School ofPhysicsandElectronicEngineering, HarbinNormal University, Harbin 150025, China
2Department of Physics and the Center of Theoretical and Computational Physics,
The University of Hong Kong, Pokfulam Road, Hong Kong SAR, China
3TheUniversityofHongKongShenzhenInstituteofResearchandInnovation,Shenzhen, Guangdong518057,China
4Center for the Physics of Materials and Department of Physics,
7 McGill University, Montreal, Quebec H3A 2T8, Canada
1 (Dated:January26,2017)
0
2 We report the calculated fundamental band gaps of wurtzite ternary alloys Zn1−xMxO (M=Mg, Cd) and
the band offsets of the ZnO/Zn1−xMxO heterojunctions, these II-VI materials are important for electronics
n
andoptoelectronics. Ourcalculationisbasedondensityfunctionaltheorywithinthelinearmuffin-tinorbital
a
(LMTO)approachwherethemodifiedBecke-Johnson(MBJ)semi-localexchangeisusedtoaccuratelyproduce
J
thebandgaps,andthecoherentpotentialapproximation(CPA)isappliedtodealwithconfigurationalaveragefor
5
theternaryalloys. ThecombinedLMTO-MBJ-CPAapproachallowsonetosimultaneouslydetermineboththe
2
conductionbandandvalencebandoffsetsoftheheterojunctions.ThecalculatedbandgapdataoftheZnOalloys
] scaleasEg =3.35+2.33xandEg =3.36−2.33x+1.77x2forZn1−xMgxOandZn1−xCdxO,respectively,
i wherexbeingtheimpurityconcentration. Thesescalingaswellasthecompositiondependentbandoffsetsare
c
quantitativelycompared totheavailableexperimental data. Thecapabilityof predictingtheband parameters
s
andbandalignmentsofZnOanditsternaryalloyswiththeLMTO-CPA-MBJapproachindicatethepromising
-
l applicationofthismethodinthedesignofemergingelectronicsandoptoelectronics.
r
t
m
. NOTE:ThisarticlehasbeenacceptedbyScientificReports centrationofsomeimpurityspecieB.Ifxissmallasisusually
t
a inarevisedform(www.nature.com/articles/srep41567). thecase,e.g.x=0.1%,onehastocalculateasystemcontain-
m ingatleast1000hostatomsA in orderto accommodatejust
- asingleimpurityatomB. Namelythetotalnumberofatoms
d are very large when x is small which makes first principles
I. INTRODUCTION
n
analysisextremelydifficulttodo. Second,onehastoperform
o
aconfigurationalaverageofthecalculatedresultsbecausethe
c Zincoxide(ZnO),havingadirectbandgap(E )of3.37eV
[ g dopedimpurityatomsarerandomlydistributedinthesample,
and a high excitonbindingenergyof about60 meV at room andtheconfigurationalaverageismuchmorecomputationally
1 temperature,1,2 is a promising material for electronics and
costly. Finally,itiswell-knownthatthefirstprinciplesmeth-
v optoelectronics, including applications to solar cells,3 light- odsofdensityfunctionaltheory(DFT)11,12withlocal-density
7 emitting diodes4,5 and ultraviolet micro-lasers.6 In electron- approximation(LDA)13–15 and generalizedgradientapproxi-
4
1 icstechnology,semiconductorheterojunctionsarecommonly mation (GGA)16,17 underestimate the band gaps of semicon-
used to provide potential barriers and/or quantum wells that
7 ductorsand, without a correctband gap, the accuracyof the
0 tune and control carrier transport. Composition controlling calculated band offset could be questionable. For these rea-
1. techniques are the most widely applied method to alter the sons,thecompositiondependentbandoffsetsofmanyimpor-
bandgap ofmaterials in the so called “bandengineering”to
0 tantsemiconductors,includingthatofZnOanditsternaryal-
build heterojunctions. For ZnO, composition doping of Mg
7 loys,havenotbeentheoreticallyinvestigatedtoasatisfactory
1 andCdatomshasbeenusedtoincreaseordecreasetheband level.
: gap,respectively.7–10
v
It could be quite tedious and difficult to experimentally During the past decades, considerable theoretical efforts
i
X measurethebandgapsofsemiconductoralloysandbandoff- have been devoted to solve the above band gap and config-
r setsoftheirheterojunctionsovertheentiredopingconcentra- urationalaverageissues. AmongthemthecombinedDFTap-
a tion,thereforefirstprinciplestheoreticalcalculationsandpre- proachofcoherentpotentialapproximation(CPA)18andmod-
dictionsareofgreatfundamentalinterestaswellaspractical ified Becke-Johnson (MBJ) semilocal exchange19 under the
relevance. Inparticular,asnanoelectronicdevicesarereach- linear muffin-tin orbital (LMTO) scheme,20,21 has been suc-
ingthesub-10nmscale,atomisticfirstprinciplespredictionof cessfulincalculatingthebandinformationofmanysemicon-
bandparametersof semiconductormaterialsand heterojunc- ductors, their alloy and heterojunctions. So far, the LMTO-
tions is becoming very important in order to design and/or CPA-MBJ approach has been applied to predict the band
selectnewmaterialsthathavedesiredproperties. However,it gapsofallgroupIII-Vsemiconductors,22 thealloyInGaN,23
hasbeenaseriouschallengeforexistingfirstprinciplesmeth- AlGaAs,24 the group IV semiconductor alloys SiGeSn,25 as
odsto accuratelypredictbandinformationof semiconductor wellasthe bandoffsetofgroupIII-Vsemiconductorhetero-
alloys and heterojunctions for several reasons. First of all, junctionsGaAs/AlGaAs.24 Quantitativecomparisonbetween
let’s consider calculating the band offset of a heterojunction the calculated results and measured data were very satisfac-
A1−xBx/AwhereAisasemiconductorandxthedopingcon- tory. However, a more challenging and important semicon-
2
initioSimulationPackage(VASP).26,27AsshowninFig.1(a),
theagreementisexcellent-suggestingtheappropriatenessof
our ASA scheme. Note that the calculated LDA band gap,
though consistent with that reported in the literature,28,29 is
underestimatedwhichisawell-knownissueofLDA.
HavingobtainedthecorrectbandstructureofpureZnOby
LMTO at the LDA level, we further determine the c-values
of the MBJ potential(See MethodsSection) in orderto pro-
duce the experimental E , 3.37 eV, of pure ZnO. Fig. 1 (b)
g
showstheMBJresult,andtheopeningoftheE ascampared
g
toourLDAresultaswellasthepreviouslyreportedLDAE
g
of 0.74 eV28,29 and GGA E of 0.80 eV,30 is evident. Till
g
now,wehavedecidedalltheparametersfortheLMTO-CPA-
MBJapproachtoproducetheexperimentalbandgapofpure
ZnO,includingtheLMTOrelatedASAparametersforsphere
FIG. 1. (Color online) (a) The calculated LDA band structure of
positions and radii and the MBJ related parameter c-values
pureZnO.Blacksolidlinewasobtainedbytheplanewaveelectronic
for increasing the band gap of semiconductors. More com-
package VASP, and red circles by NANODSIM. (b) The calculated
plicatedASAschemeandmorec-valuesfordifferentspheres
MBJ band structure of pure ZnO crystal, giving the experimental
mayproducemuchmoreaccuratebandparameters,wewould
band gap of 3.37 eV. (c) Thecalculated CPAband structure of the
Zn0.85Mg0.15O alloy, the band gap isincreased to 3.7 eV. (d) The notethatourchoiceisagoodcompromisebetweensimplicity
calculatedCPAbandstructureoftheZn0.85Cd0.15Oalloy,theband and reasonable accuracy. Though a few parameters need to
gapisdecreasedto3.07eVcomparedtothatofpureZnO. be pre-determinedin our first principlecalculations, it is ac-
ceptablesincewearedealingaverychallengingproblemand
these parameters can be easily obtained. In the further cal-
ductorfamily-wurtzite groupII-VIsemiconductors,hasnot culations of the band gap of semiconductor alloys and band
beeninvestigatedbytheLMTO-CPA-MBJapproachyet. offset of semiconductor heterojunctions, we will fix all the
In this work, we have calculated the band gap of wurtzite parametersobtainedinthepureZnOcalculations.
group II-VI semiconductor alloys Zn1−xMxO (where M = Band gaps of ZnO ternary alloys. Having correctly de-
MgorCd)andthebandoffsetofZnO/Zn1−xMxO usingthe terminedthebandgapofpureZnOcrystal,exactlythesame
LMTO-CPA-MBJ approach. The calculated band gaps of ASAschemeandtheMBJsemi-localexchangepotentialwere
Zn1−xMgxO and both conduction and valence band offsets usedtocalculatetheelectronicstructuresofZnOalloys. Note
ofZnO/Zn1−xMgxO heterojunctionexhibita linearfunction that when ZnO crystal is doped with (substitutional) impu-
of Mgconcentration,while a nonlinearbehaviorofthe band rity atoms at random sites, translational symmetry is bro-
gapsofZnCdOandbandoffsetsofZnO/ZnCdOispredicted. ken and momentum k is no-longer a good quantum num-
The calculated band gaps and band offset of heterojunctions ber such that Bloch’s theorem no longer exists. In our CPA
agree well with the corresponding experimental measure- calculations,18,31 the spirit of the approachis to constructan
ments. Our calculatedband gapdata of ZnO alloysare well effectivemediumbycompletingconfigurationalaverageover
fit to the followingscaling expressions: Eg = 3.35+2.33x therandomdisorderthatrestoresthetranslationalinvariance,
for alloy Zn1−xMgxO, and Eg = 3.36 − 2.33x + 1.77x2 assuchonecanagainspeakofa“bandstructure”withbands
foralloyZn1−xCdxO.Togetherwith thepreviouslyreported thatarebroadenedbyimpurityscattering.
results on group III-V semiconductors22–24 and group IV Fig.1(c,d)plotsthecalculatedCPAbandstructureofZnO
semiconductors,25wecanconfidentlyexpectthatthebandpa- alloywith15%MgOand15%CdOcomposition,respectively.
rametersandbandalignmentsofmostconventionalsemicon- FromFig.1(c),weobtainE =3.70eVthatis0.33eVhigher
g
ductorscanbewellpredictedfromfirstprinciplesbyLMTO- than that of pure ZnO. This is very close to the experimen-
CPA-MBJapproach. tallymeasuredvalueofEg = 3.68eVforZn0.85Mg0.15O.10
ForZn0.85Cd0.15O,thecalculatedbandgapisEg =3.07eV,
namely,0.3eVsmallerthanthatofpureZnO.
II. RESULTSANDDISCUSSIONS WehavecalculatedE ofwurtziteZnOalloysversusimpu-
g
rityconcentrationxuptox=40%,forZn1−xMxOwhereM
BandstructureofpuresemiconductorZnO.Tocarryout =MgorCd,asshowninFig.2.Beyondx=40%,itisknown
DFT calculations within the LMTO scheme, atomic sphere experimentally7,8 thatZnOalloytendstoformacubicphase
approximation (ASA) should be employed for space fill- (whichisnotthefocusofthiswork).ForZn1−xMgxOalloys,
ing with placing atom and vacancy spheres at appropriate ourcalculatedE increaseslinearlywiththeMgOconcentra-
g
locations.20,21DifferentASAscheme,i.e.,differentspherepo- tion as E = 3.35+2.33x (the solid squaresline in Fig.2).
g
sitionsandradii,willaffectthecalculatedelectronicstructure. Thisscalingisconsistentwiththe experimentalobservations
To verifyour ASA scheme of the LMTO method, we calcu- of E = 3.4+2.3x in Ref. 32, and E = 3.37+2.51x in
g g
lated the band structure of pure ZnO crystal at the level of Ref. 33. According to our calculations, E of the wurtzite
g
local LDA to compare with that obtained by the Vienna Ab Zn1−xMgxOalloycanbetunedto4.29eVatx = 40%. Our
3

4.5
Zn MgO
1-x x CBO
0.6
Zn1-xCdxO V) VBO
)4.0 e
V (
e Eg=3.35+2.33x ts0.4 2
( e VBO=2.17x-1.70x
s
g3.5 f
E f
o
2 0.2
Eg=3.36-2.33x+1.77x d
n 2
3.0 a CBO=-0.07x-0.07x
b
0.0
0.0 0.1 0.2 0.3 0.4 0.0 0.1 0.2 0.3 0.4
concentration x concentration x
FIG. 2. Solid squares and dots are calculated band gap Eg of the FIG. 4. Solid squares and dots present the CBO and VBO of the
Zn1−xMgxOandZn1−xCdxOternaryalloys, respectively. Thein- ZnO/Zn1−xCdxOheterojunctionsforx<40%,respectively.
creaseofMg(Cd)compositionenlarges(narrows) theband gapof
thesewurtzitealloyswiththeMg(Cd)compositionfrom0to40%.
istopredictthebandoffsetsfortheZnO/Zn1−xMxOhetero-
junctions. Since ourmethodcan accuratelypredictthe band
gaps of pure ZnO and its ternary alloys, both the conduc-
tion band offset (CBO) and the valence band offset (VBO)
0.6 CBO can be calculated in one shot. In comparison, for those cal-
V) VBO culations where Eg was not correctly obtained (such as us-
e 0.4 ingLDA/GGA functional),the conductionbandoffsetcould
( CBO=1.57x
s not be predicted unless one amends E after the valence
et 0.2 band offset is calculated. Fig. 3 plots tghe calculated band
s
f
of 0.0 offsets of ZnO/Zn1−xMgxO heterojunctions for x < 40%.
d VBO=-0.79x The band offsets linearly scale as VBO= −0.79x eV and
n-0.2 CBO= +1.57xeV. Here theminussign inVBO meansthat
a
b thevalencebandmaximumoftheZnMgOalloyislowerthan
-0.4 thatofthepureZnO;whiletheplussigninCBOindicatesthe
conductionbandminimumoftheZnMgOishigherthanthat
0.0 0.1 0.2 0.3 0.4
ofthepureZnO.Fromthefittedcurves(blacklines),theband
concentration x offset ratio (VBO:CBO) is about 2. This is in the range of
FIG. 3. Solid squares and dots present the CBO and VBO of the the experimentallyreportedvalues: 3:2 to 7:3 as reportedin
ZnO/Zn1−xMgxOheterojunctionsforx<40%,respectively. Ref.39,and1.5to2asreportedinRef.9. AtMgconcentra-
tionof15%,ourcalculatedCBO(VBO)is0.22eV(0.11eV),
which agreeswith the experimentallymeasured data of 0.16
fitted linear slope agrees outstandingly with previous calcu- eV(0.09eV).9
lated value of 2.03 by LDA,34 though LDA gives underesti- Fig. 4 shows the calculated band offsets of
matedbandgaps. Moreover,if quadraticfitting isappliedto ZnO/Zn1−xCdxOforx < 40%,andanonlinearrelationship
our calculated data, the coefficient of the x2 term gives the withtheCdconcentrationisobtained:VBO=2.17x−1.70x2
bowingparameter of 0.18 eV, which is close to the previous eVandCBO=−0.07x−0.07x2eV.Thenegativecoefficient
calculated bowing parameter of 0.44.35 For the Zn1−xCdxO of x for CBO indicates that the conduction band minimum
alloys, we found that E changes with the Cd concentration of the ZnCdO is lower than that of the pure ZnO, while the
g
asE = 3.36−2.33x+1.77x2whichisnonlinear(thedots positive coefficient of x for VBO shows a higher valence
g
linelineinFig.2),andtheEg ofthewurtziteZn1−xCdxOal- band maximum of the ZnCdO. In particular, our calculated
loy can be narrowed to 2.71 eV when x = 40%. Here, the VBOvaluesof0.20eVatx=10%and0.104eVatx=5%
coefficientofthex2 termgivesthebowingparameterof1.77 agree very well with the experimentalvalues of 0.203eV at
eV,whichagreesextremelywellwiththeexperimentalvalue x=9.6%40and0.17±0.03eVatx=5%41.
of1.75eV.36 Notethatpreviouscalculatedbowingparameter Fig. 5 intuitively show the band alignment of
wasconsiderablysmaller,1.21eVinRef. 37and1.03eV in ZnO/Zn0.85M0.15O (M=Mg, Cd) by plotting the local
Ref.38. projectedDOS42,43 alongtheheterojunctiondirection. Itcan
Band offsetsof ZnO and its ternaryalloys. Having ob- be clearly observed that the pure ZnO crystal formed type-I
tainedaccurateE for theZnO alloys, a veryimportanttask band alignment heterojunction with its Mg and Cd doped
g
4
elements locate on the same site with certain probabilities
and their average physical quantities were dealt with CPA.
ThesepreviouslyreportedresultsshowthattheLMTO-CPA-
MBJapproachcanaccuratelycalculatethebandinformation
ofgroupIII-VandgroupIVsemiconductorsandtheirbinary
andternaryalloysandheterojunctions. Inthiswork, wefur-
ther applied this approach on a more challenging system -
wurtzitegroupII-VIsemiconductors,andtheobtainedresults
are quantitatively compared with the experimental measure-
ments.
We used the effective medium CPA method underLMTO
scheme to calculate the electronic structures of semiconduc-
torsandalloys,whichdoesnotcaptureanyatomicrelaxations.
Consideringthatwe aredealinga verychallengingproblem,
which needs to solve the doping, configurational average,
band gap underestimation and computational cost issues in
onecalculation,to thisend, the combinedLMTO-CPA-MBJ
approach is a reasonably compromised method. Moreover,
formanysemiconductoralloys,theircompositionshavevery
FIG. 5. (Color online) The local projected DOS of: (a) closelatticeparameters,makingthestructuralchangenegligi-
ZnO/Zn0.85Mg0.15O and (b) ZnO/Zn0.85Cd0.15O heterojunctions. ble. Itcouldbeexpectedthatthisapproachcanbeusedinthe
ThecoolercolorindicateslowerDOS.Theblue(darker)regionin- predictionofbandparametersof wide rangeof semiconduc-
tuitively shows the band gap of the semiconductors. ZnO and its tors.
ternaryalloysZnMgOandZnCdObothformstype-Ibandalignment.
III. SUMMARY
ternaryalloys, andthebandgapofpureZnOissmallerthan
thatofZnMgObutlargerthanthatofZnCdO. Usingastate-of-the-artatomisticapproach,wehavecalcu-
Discussions. BasedonourLMTO-CPA-MBJcalculations latedthebandgapofwurtziteZnObasedgroupII-VIternary
onZnOanditsalloys,ageneralprocedureforpredictingband alloys with Mg or Cd compositions, and the band offsets of
gapsofsemiconductoralloysandbandoffsetsofheterojunc- ZnO and its ternary alloy heterojunctions. The calculated
tions is in order. (1) Choose a proper ASA scheme and de- band gaps of the ZnO alloys scale as Eg = 3.35 + 2.33x
terminethec-valueoftheMBJsemi-localexchangetoobtain and Eg = 3.36 − 2.33x + 1.77x2 for Zn1−xMgxO and
accuratebandstructuresofthepuresemiconductor;(2)calcu- Zn1−xCdxO, respectively. Further calculations on the band
late the band information of semiconductorsalloys by CPA; offsetsshowthatthepureZnOformedtype-Ibandalignment
(3)calculatethebandoffsetofthesemiconductorheterojunc- heterojunctionwithitsMgandCddopedternaryalloys. The
tionwithcorrecteddeterminedbandgaps,fromwhichtheva- calculatedbandgapsofthealloysandbandoffsetofthehet-
lenceandconductionbandoffsetcanbepredictedinoneshot. erojunctionsquantitativelyagreewith the experimentalmea-
Sinceamajorproblemofsemiconductortechnologyistode- surements for x < 40% where the ZnO and its Mg and Cd
sign materials having desired electronic structure, the theo- doped alloys are in the wurtzite phase. The success of the
retical proceduresummarized here should be applicableto a LMTO-CPA-MBJonpredictingthebandgapsandbandoff-
widerangeofresearchissues. setsofgroupII-VIsemiconductoralloysandheterojunctions,
togetherwith its capability on calculatingand predictingthe
ThecombinedLMTO-CPA-MBJapproachisfirstlyapplied
bandgapsandbandoffsetsofgroupIV25andgroupIII-V22–24
onthebandgapcalculationofgroupIII-Vsemiconductoral-
semiconductors, make us can confidently conclude that the
loy InxGa1−xN, the results are in excellent agreement with
the measured data for the entire range of 0 ≤ x ≤ 1.23 LMTO-CPA-MBJapproachcanbewidelyusedtopredictthe
Later on, the approach is implemented in NANODSIM soft- semiconductor band parameters and band alignments from
ware package.31 Using NANODSIM Ref. 22 calculated the firstprinciplesandwillbeveryusefultothedesignofemerg-
ingelectronicsandoptoelectronics.
band structures and effective masses of all the zinc-blende
groupIII-VsemiconductorswithLMTO-MBJapproach,and
verygoodquantitativecomparisonwiththeexperimentaldata
is made; Ref. 24 for the first time calculated the band off- IV. METHODS
setofGaAs/AlxGa1−xAsheterojunctionsfortheentirerange
of the Al doping concentration 0 < x ≤ 1 using LMTO- Our calculations are based on the LMTO-CPA-MBJ self-
CPA-MBJ approach,boththe conductionband offset(CBO) consistent approach where DFT is carried out in the LMTO
andthevalencebandoffset(VBO)agreeverywellwithmany scheme with the atomic sphere approximation (ASA),20,21
experiments; Ref. 25 calculated the composition dependent as implemented in the NANODSIM software package.31 For
bandgapsofgroupIVternaryalloysSiGeSn,inwhichthree ASA, vacancy spheres were placed at appropriate locations
5
c-values for real atom spheres and vacancy spheres, respec-
TABLEI.Positionsofatomicspheresinthewurtzitestructure. VT tively. Whilethe c-valuescanbeself-consistentlycalculated
andVOdenotethevacancyspheresatthetetrahedralcenterandocta- using the electron density, in our calculations we fixed it to
hedralcenteroftheZnOprimitivecell,respectively. Theoptimized
sphereradiiof1.1481A˚,0.7654A˚,and1.3023A˚ areusedforthe bec = 1.75andc = 1.13forvacancyspheresandrealatom
realatomspheres,vacanciesatthetetrahedralcenterandvacancies spheres respectively, which gave an Eg of ZnO in excellent
attheoctahedralcenter,respectively. agreementwiththeexperimentalvalue(3.37eV).Havingob-
tained the correct band gap of ZnO, the same values of the
Site Zn O VT VO c-parameterswere used in all the subsequentcalculations of
x 0 0 0 0
thealloysandheterojunctions.
y 0 0 0 √3/3
z 0 u c 1+u c u c/2 The experimental lattice constants47,48 of hexagonal
x 1/2 1/·2 21/2· ·0 wurtzite ZnO, a = 3.2495 A˚,47 c/a = 1.602,47 and u =
y √3/6 √3/6 √3/6 √3/3 0.382c,48 wereusedin ourcalculationsforpureZnOandall
z c/2 (1 +u) c u c/2 1+u c the alloys. The primitive cell of the wurtzite structure was
2 · · 2 · used to calculate the band structures and E of ZnO and its
g
alloys. To determine the band offset, (1120) supercell sys-
tems containing 8 unit cell of pure ZnO and 8 unit cell lay-
forspacefilling[SeeTABLEIfordetails].23Inparticular,the
ers of ZnO alloy were used to calculate the potential pro-
simulationcellisfilledbyslightlyoverlappedsphereswithin
file along the heterojunction. A 10× 10 × 10 k-mesh and
ASA,andthecellvolumeofthewholesimulatedsystemV
cell
a 6×6×3 k-mesh were usedto sample the Brillouin Zone
shouldbeequalto thetotalvolumeoftheatom andvacancy
(BZ) for the primitive cells and the heterojunctions, respec-
spheres V . With the sphere position of the wurtzite
spheres
tively. The band gap values, as well as the energy position
structure in TABLE I, we optimized the sphere radii by re-
ofthevalencebandmaximumandconductionbandminimum
producingtheelectronicbandstructuresofpureZnOcrystals.
are read fromthe CPA bandstructure. The CPA bandstruc-
Afterward,todealwiththeimpuritydopedZnOalloy,thesta-
turecanintuitivelytracetheshapeofthebandstructurewith
tistical effective medium CPA theory18 is applied which al-
broadeningdue to impurityscattering, and providemore ac-
lowsustoobtaintheconfigurationalaveragedresultswithout
curatelyandrichbandinformationforthesemiconductorand
individuallycomputingeachatomicconfiguration.
itsalloys.Thebandoffsetvalueswerecalculatedbasedonthe
We use the MBJ semi-local exchange that can accurately
electrostatistic potentialprofile of the heterojunctionand the
determinethebandgapE ofthesemiconductors.19 Follow-
g primitivecell,asmoredetailsinRef.24.
ingtheoriginalpaper,19theMBJsemi-localexchangepoten-
tialhasthefollowingform,
ACKNOWLEDGEMENTS
1 5 2t (r)
vMBJ(r)=cvBR(r)+(3c−2) σ , (1)
x,σ x,σ πr12sρσ(r) Y.W. is grateful to Drs. Yu Zhu, Lei Liu and Dongping
LiuforusefuldiscussionsregardingtheuseoftheNANODSIM
where subscript σ is spin index, ρ is the electron density software package. This work is supported by the University
σ
for spin channel σ. The quantity t is the kinetic energy Grant Council (Contract No. AoE/P-04/08) of the Govern-
σ
density and vBR(r) is the Becke-Roussel potential used in mentofHKSAR(YW,JW,HG),theOpenProjectProgramof
x,σ
Ref.44. TheaboveMBJpotentialhastwotermswhoserela- KeyLaboratoryforPhotonicandElectricBandgapMaterials
tiveweightisgivenbyaparameterc,whichdependslinearly (HY, YW), Special Program for Applied Research on Super
on the square root of the average of |∇ρ|/ρ. According to ComputationoftheNSFC-GuangdongJointFund(YW,JW),
manypreviousstudies,19,22–25,45,46thecalculatedE increases SeedFundingProgrammeforBasic ResearchofHKU(YW)
g
monotonicallywith c. For simplicity,we onlyused different andNSFC(No.11404273).
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