Design of mid-IR and THz quantum cascade laser cavities with complete TM photonic bandgap

0.56 MB·English

by  Michael Bahriz

Designofmid-IR andTHzquantum cascadelasercavitieswithcomplete TMphotonic bandgap
MichaelBahriz,1 OrionCrisafulli,2 VirginieMoreau,1 RaffaeleColombelli,1, andOskarPainter2,†
∗
1Institutd’Electronique Fondamentale, Bat. 220, Universite Paris-Sud, 91405 Orsay, France
2Thomas J. Watson, Sr., Laboratory of Applied Physics,
California Institute of Technology, Pasadena, California 91125
(Dated:February2,2008)
Wepresentthedesignofmid-infraredandTHzquantumcascadelasercavitiesformedfromplanarphotonic
7 crystalswithacompletein-planephotonicbandgap. Thedesignisbasedonahoneycomblattice,andachieves
0 afull in-plane photonic gap for transverse-magnetic polarized light whilepreserving aconnected patternfor
0 efficientelectrical injection. Candidatedefects modes for lasingareidentified. Thislatticeisthen usedasa
2 modelsystemtodemonstrateanovel effect: under certainconditions -thataretypicallysatisfiedintheTHz
range-acompletephotonicgapcanbeobtainedbythesolepatterningofthetopmetalcontact.Thispossibility
n
greatlyreduces therequired fabrication complexity and avoidspotential damage of thesemiconductor active
a
region.
J

1
PACSnumbers:
1

]
s I. INTRODUCTION
c
i
t
p Quantumcascade(QC)lasersaresemiconductorlasersourcesbasedonintersubband(ISB)transitionsinmultiplequantum
o wellsystems1. Theiremissionwavelengthcanbetunedacrossthemid-infrared2,3 (mid-IR,5µm<l <24µm)andTHz4 (65
. µm<l <200µm)rangesoftheelectromagneticspectrum. Inthemid-IR,wheretheyarebecomingthesemiconductorsource
s
c ofchoicethankstotheirgoodperformance,thepotentialapplicationsarechemicalsensing,spectroscopyandfree-spaceoptical
i communications. In the THz rangethe mostpromisingapplicationis the imagingofconcealedobjects. Biologicalmaterials,
s
y semiconductorchippackagingandclothingareallTHz-transparent(whiletheyareopaqueatshorterwavelengths),makingTHz
h imagingusefulforsecurityandmedicalapplications.
p
MostoftheactivityintheQClaserfieldhasconcentratedonedge-emittinglasersduetotheintrinsictransverse-magnetic(TM)
[
polarizationofISBtransitions,andcorrespondingdifficultyinimplementingvertical-cavitysurfaceemittinglasers(VCSELs).
1 Surfaceemissionhasbeenobtained,rather,byintegratingsecond-ordergratingsonedgeemittingdevices5,6 orbyreplacingthe
v standard Fabry-Perotcavity with a photonic crystal (PC) resonator7,8. The latter solution, the application of photonic crystal
4 technologytoQClasers9,10,isparticularlyappealingbecauseoftheflexibilitythatitallowsthedesigner.Two-dimensional(2D)
4
photoniccrystalscanbeusedtocreatelocalizedmicrocavitylasersourcesthatcanbebuiltastwo-dimensionalarraysonasingle
1
chip11,12 orforlarge-area,high-power,single-modesurfaceemittinglasersources13.
1
Ofparticularimportanceindeterminingthepropertiesofanyphotoniccrystalstructureistheeffectiverefractiveindexcontrast
0
7 attainable. Thehighertheindexcontrast,thestrongertheopticaldispersionofthephotonicbandsandthegreatertheabilityto
0 localize,diffract,andreflectlightwithinthephotoniclattice.Ahighindexcontrastisthereforecrucialfordeviceminiaturization.
/ Inplanarwaveguidedevices, thepresenceofa fullopticalbandgapin two-dimensionsisbeneficial(althoughnotcompletely
s
c necessary) in forming ultra-low-volume laser cavities with minimal optical loss. As is discussed in Ref.14, for 2D photonic
i crystals,TMopticalbandgapsarefavoredinalatticeofisolatedhigh-e regions.Unfortunately,thisconfigurationisincompatible
s
y withanelectricalinjectiondeviceduetoitsnon-connectednatureandanalternateapproachisrequired.
h Inthispaperwestudytheuseofaconnectedhoneycomblatticeforcreating2DphotoniccrystalQClaserstructures.Webegin
p
inSectionIIwithareviewofthepropertiesofthis2Dlatticeviaaplanewaveexpansion(PWE)analysis. Weshowthatafull
:
v 2DopticalbandgapforTMpolarizationcanbeobtainedinthisconnectedlattice, andstudythelocalizedresonantmodesthat
i formaroundasimplepointdefectinthelattice. InSectionsIIIandIVfullthree-dimensional(3D)finite-differencetime-domain
X
(FDTD)simulationsareusedtoanalyzethepropertiesofthehoneycomblatticeintwodifferentverticalwaveguidestructures.
r
SectionIIIdealswithmid-IRPCQClasers. Therealactive-regionandwaveguidestructureofamid-IRsurface-plasmon(SP)
a
QClaserisconsidered. Thehigh-indexcontrastinthiswaveguidestructureisobtainedbyairholesthatpenetratedeeplyinto
the semiconductor layers. Section IV focuses on THz PC QC laser waveguide structures. In the metal-metal waveguides of
THz lasers we find that the effective index contrast, and therefore the photonic bandstructure, is strongly dependent on the
waveguidethickness37. Inparticular,belowacertaincriticalwaveguidethickness,afullphotonicgapcanbeinducedbythesole
patterningofthemetallayers. Thisnoveleffectcouldbeusefulforthe developmentof a varietyofTHz lasers, includingPC
surface-emittinglasers15,duetothesimplefabricationrequirements.
2
II. TWO-DIMENSIONALANALYSIS:PLANEWAVEEXPANSION
A. Theplanewaveexpansionmethod
The calculationsof this section are based on a planewaveexpansion(PWE) method. Itis a frequency-domainmethodthat
allowsextractionoftheBlochfieldsandfrequenciesbydirectdiagonalization.ThecalculationswereperformedusingtheMIT
PhotonicBands(MPB)simulationtool16.WiththePWEmethoditispossibletocomputethebandstructuresandelectromagnetic
modesofperfectlyperiodicdielectricstructures. CalculationofdefectmodeswillbetackledinsectionIICusingasuper-cell
approach.
B. PhotoniccrystalstructureandTMgap
The2Dphotoniccrystalconsideredinthisworkisthatofahoneycomblattice,consistingofahexagonalBravaislatticewith
abasisoftwoairholesperunitcell(seeFig. 1(a)). Theholeradiusisr andthelatticeconstantisa. Thereciprocallattice is
alsohexagonal,withG ,X andJthehighsymmetrypointsofthelattice(seeFig. 1(b)).Inthe2Dsimulationsofthissectionthe
relativedielectricconstantofthedielectricbackgroundistakenase =11.22(n=3.35),correspondingtotheindexofrefraction
ofmostIII-Vsemiconductormaterialsatfrequenciesbelowtheenergygap.
FIG. 1: Structure of the honeycomb lattice. (a) Two-dimensional honeycomb latticewith two ”dielectric atoms” (A and B) per unit cell.
a1=(a,0)anda2=(a/2,√3a/2)aretheprincipallatticevectors(a= a1 = a2),ristheholeradius.(b)Two-dimensionalreciprocalspace
forthehoneycomblattice. ThereducedBrillouinzoneisshadedinblu|e. |Rec|ipr|ocallatticevectorsaresuperpositionsofG1=(0,4p /√3a)
andG2=(2p /a,2p /√3a).
FIG.2: Photonicbandcalculationsfora2Dhoneycomblatticeofairholesinadielectriclattice(e =11.22). (a)Photonicbandstructurefor
TMpolarizationwithr/a=0.234.(b)Completebandgap(shadedblueregion)asafunctionofr/a.AfullTMgapexistsonlyforr/a>0.18
3
Fig. 2(a)showsthephotonicbandstructureforalatticewithr/a=0.234.Forthisr/aratioafullTMgapispresent,centered
arounda/l 0.225,withawidthapproximately12%ofthecentralgapfrequency. Theexistenceofagap,however,depends
≈
stronglyonr/a. InFig.2(b)weplotthegap-mapforthislattice,whichclearlyshowsthatonlyaboveacriticalvalueofcircular
hole size (r/a 0.18)can a TM photonicgap be obtained. This is a commonbehaviorof lattices of air holes; the largerthe
holes,theclose≥rthelatticeistoasystemofisolatedhigh-e regions,andthelargertheTMphotonicbandgap9. Theadvantageof
thehoneycomblatticeisthattheTMgapopenswhentheholesizeandporosityofthelatticearestillreasonablysmall,allowing
formoreefficientelectricalinjection.
C. 2Ddefectdesign:supercellmethod
Inthissectionwestudy,in2D,thelocalizedresonancesthatformaroundapoint-likedefectofthehoneycomblattice. The
defectthatweconsiderisobtainedbyremovingahexagonofsixholes,asshowninFig. 3(a). Thedefectmodesarecalculated
with a supercell method using the same PWE solver. The presence of the defect cavity breaks the periodicity of the lattice,
requiringthecreationofasupercelloverwhichthestructureisassumedperiodic. Thissupercellcontainsthedefectcavityand
istiledinspace. TheWigner-Seitzcellofthereciprocallatticeofthesupercellwillbecorrespondinglysmallerthanthatofthe
underlyingphotoniclattice,resultinginafoldingofthehoneycomblatticephotonicbands(seeFig. 3(b)). Thelocalizeddefect
statesofthesupercellappearasbandswithalmostnodispersion,lyinginsidethephotonicbandgapofthehoneycomblattice.
FIG.3: (a)Illustrationofthe2Dhoneycomblatticedefectcavitysupercell(backgrounddielectricmaterialshownasgrey,airholesshownas
white).Theprinciplelatticevectorsarea1,a2,where a1 = a2 =a.Thedefectregionconsistsofremovalofthecentralhexagonofairholes
| | | |
(centralhexagonshownasadashedline). (b)FoldedTMphotonicbandstructurefora20-periodsupercellofthedefectcavityshownin(a)
withr/a=0.24.Theblue-linescorrespondtothefoldedbandstructureofthehoneycomblattice.Theredlinesarethedefectfrequencylevels.
Localizeddefectmodes:(c)hexapoleand(d-e)dipole-likemodes.
APWEcalculatedbandstructureofthesupercelldefectstructurewithr/a=0.24isshowninFig. 3(b).Inthesecalculations,
asupercellconsistingof20periodsofthehoneycomblatticewasusedinordertoobtainwelllocalizedresonantmodes. Three
defect modes lie within the complete band gap: two degenerate dipole-like modes and a hexapole-like mode (Figs. 3(c-e)).
Thesearethedefectmodesthatarepredictedforthe honeycomblattice throughsimplesymmetryarguments11. Ofthedefect
modespresentinthebandgap,thedipole-likemodesaremoreconcentratedinthecenterofthedefectregion,thusgivingthem
themostoverlapwiththeQCgainmaterial.Assuch,thesearethecavitymodeswewillfocusoninthe3Dsimulationsdescribed
below.
Ourchoiceofholeradius(r/a=0.24)intheabovecavitydesignwasmadebaseduponatrade-offbetweentheextentofthe
photonicbandgapand cavitymodelocalization, with thatof the electricalandthermalresistance incurredin a semiconductor
realizationofsuchastructure.Asalreadymentioned,thereducedconnectivityofaphotoniclatticemakeselectricalinjectionof
adefectcavitymorechallenging.Electricalcurrentinsuchstructuresistypicallyinjectedfromtheedgeofthephotoniccrystal
verticallythroughthedeviceactiveregion. Injectionintothedefectregionofthecavityisaresultoflateralcurrentspreading
intothecenterofthephotoniccrystal. Theincreasedlateralresistanceofhighlyporousphotoniclatticessignificantlyreduces
theinjectionefficiencyofthelaser,resultinginaddedheatingandreducedgain. Otherinjectiongeometriesarepossible,such
astheuseofsurface-plasmonlasercavitiesinwhichthetopmetalcontactextendsovertheentirephotoniccrystalandprovides
notonlyverticalwaveguidingbutalsoelectricalinjection. Thisgeometrywasusedinthedemonstrationofthefirstelectrically
injected 2D photonic crystal microcavity laser7. Nonetheless, high-aspect ratio 2D photonic crystal lattices of high porosity
poseagreaterchallengetofabricateduetothereducedcriticaldimensionsize. Itshouldbenotedthatfurtheroptimizationof
the honeycomblattice can be obtained through use of air holes of a modified geometry. Lattices formed from air holes with
a truncated circular cross-section were found to have largerphotonicbandgapsfor a givencritical dimension than that of the
standardlattice. Inwhatfollows,however,wefocusonthehoneycomblatticewithcircularairholes.
4
III. 3D-FDTDANALYSISOFMID-IRDEVICES
ThePWEmethodallowstherapidsolutionofstructureeigenmodesinthefrequencydomain.However,forthe2Danalysisde-
scribedaboveonecanonlyapproximatethelaseractiveregionwithaneffectiveindexofrefraction.Theobjectiveofthissection
istoverifythedesigndevelopedwiththe2DmodelinSectionII,andtoapplyittoarealisticmid-IRQClaserstructure.Anac-
curaterepresentationofthestructurewaveguideintheepitaxialdirectionwillbetakenintoaccountwithina3Dfinite-difference
time-domain(FDTD) approach. It is well known that the material system of choice for mid-IR QC lasers is InGaAs/AlInAs
latticematchedtoInP2,3. Inaddition,waveguidesbasedonsurface-plasmons17havebeenshowntobeadvantageousforPCQC
lasersinthemid-IR18. Thisisthereforethemodelsystemthatwewilluseforthe3Dsimulations.
A. Mid-IRsurface-plasmonwaveguidesforQClasers
SemiconductordiodelasersandconventionalQClasersrelyonopticalwaveguideswhereahigher-index-coreissandwiched
between thick cladding layers of lower refractive index, thus confining the light inside the active region stack. The small
refractiveindexdifferencebetweentheactivecoreandthewaveguidecladdingsresultsinastandardmid-IRQClaserwhichis
typically6to9µmthick.
FIG.4: Normalizedintensityprofileofthefundamentalopticalmodecomputedforaquantumcascadelaserdielectricwaveguide(a)anda
surface-plasmonwaveguide(b)designedforawavelengthof8µm. Theshadedgreenareasindicatethestackofactiveregionsandinjectors.
Theverticaldashedlineindicatesthepositionofthedevicesurface.Inthedielectriccasethethicknessoftheepitaxiallygrownmaterialis 6
≈
µm. Theoriginoftheabscissaisattheair-semiconductorinterface. Inthesurface-plasmoncasethethicknessisinstead 3µm. Theoptical
confinementfactorisindicatedbyG . ≈
Maxwell’s equations, however, also allow for another type of optical waveguiding based upon surface-plasmons19. TM
polarized electromagnetic guided modes exist at the interface of two dielectrics with opposite sign of the real part of their
dielectric constants. Negative dielectric constants are typical of metals below the plasma frequency. Thus, guided surface-
plasmonmodesatametal-semiconductorinterfaceareausefulwaveguidingsolutionforQClasers20. Thisisduetotheintrinsic
TM polarization, i.e. normal to the layers, of intersubband transitions. SP waveguides need a smaller thickness of grown
material,whileyieldingevenlargeropticalconfinementfactorsG (seeFig. 4).
Thesurface-plasmondampingalongthepropagationdirectioncanbeapproximatedwiththefollowingformula21:
a 4p nm·n3d, (1)
≈ l · k3
m
where k (n ) is the imaginary (real) part of the metal index of refraction, n is the real part of the semiconductor index of
m m d
refraction,andl isthewavelength. The1/k3l dependenceofthepropagationlossesshowsthatSPwaveguidesareespecially
m
appealing at long wavelengths. In particular the 1/k3 factor, which pushes the field out of the metal region and reduces the
m
losses,becomesverysmallatlongwavelengths.AsimpleDrudemodelshowsthattheimaginarypartoftheindexofrefraction,
foragenericmetal,increasesdramaticallywhenmovingfromshort(l =1-3µm)tolong(l =100-200µm)wavelengths.Recent
advanceshavealsoshownthatlow-lossmetallicwaveguidescanbeimplementedatmid-IRwavelengths17.Wethereforeemploy
thiswaveguidestructureasamodelsystemforthe3DnumericalsimulationsofourQClaserstructures.Thecorrespondinglayer
sequenceisdisplayedinTableI,togetherwiththecorrespondingindecesofrefractionthatareusedforthesimulations.
5
TABLEI:Layerstructureforthemid-IRSPQClaser17modeledinthe3DFDTDsimulations.Nominaloperatingwavelengthisl 8µm.
≈
Material Doping(cm 3) Thickness(µm) n Function
−
air n.a. 4 1 topcladding
Au n.a. 0.3 perfectconductor metalcontact
InGaAs 1017/1018 0.05 3.3/3.47 contactlayers
InGaAs-AlInAsMQW 1.7x1016 2.6 3.374 activeregion
structure
InGaAs 5x1016 0.5 3.475 Buffer/lowercladding
layer
InP 1017 3 3.077 substrate
WhiletheverticalconfinementisprovidedbytheSPwaveguide,thein-planeopticalconfinementisinducedbythephotonic
crystal. Ahighindexcontrastcanbeobtainedbypenetrationoftheairholesthroughthetopmetallayeranddeepintothesemi-
conductorwaveguidestructure. Intuitively,reducedscatteringlossesareobtainedforaholedepthwhichoverlapsasignificant
fraction of the guided mode energy22. In Fig. 5(a) we show a schematic of the vertical cross-section of the photonic crystal
SP waveguide structure, and in Fig. 5(b) we plot the vertical mode profile of a localized defect mode (see Fig. 6(a)) of the
honeycomblattice of air holeswith a holedepthof4.7µmin the semiconductorheterostructure. Atanoperatingwavelength
of8µmthisresultsinaverticalmodeoverlap(energyoverlap)withtheairholesofalmost90%. Itisclearfromthisexample
whysurface-plasmonQClasersareideallysuitedtoPCtechnology;theirreducedthicknessallowsforasignificantlyshallower
etchofthesemiconductormaterialincomparisontoconventionallaserwaveguidestructuresinwhichanetchdepthof10µmor
morewouldberequiredinthemid-IR.
FIG. 5: (a) Surface-plasmon optical mode profile superimposed on a sketch of the vertical section of a mid-IR photonic crystal QC laser
withairholesextending4.7µmdeepintothesemiconductor. (b)3D-FDTDcalculatedelectricfieldintensitycross-sectionofadefectmode
(top-viewshowninFig. 6(a))ofthedeeplypatternedhoneycombphotoniclattice. Theverticalmodeprofileshowsasimilarbehaviorasthe
1D surface-plasmon simulationof (a), whereas theconfinement inthe in-plane direction isprovided by distributedBragg reflectionof the
photoniclattice.
B. Defectcavitydesign
Thecavity investigatedin thissectionis the sameas in SectionIIC. Itis obtainedbyremovinga fullhexagonofair holes
fromthePClattice. AsdepictedinFig. 5,we“etch”4.7-µmdeepholesintothesemiconductorlaserstructure,withther/aratio
oftheairholessetto0.21. ThepropertiesofthedifferentlaserstructurelayersusedinthesimulationaregiveninTableI. Full
3D-FDTDsimulations11,12,22 wereperformedwith aneffectivegridresolutionof100nm, correspondingtoroughly20points
perwavelengthinthehigh-indexsemiconductormaterialatanoperatingwavelengthof8µm. Theuseofperfectlyconducting
boundariesforthetopmetalcontactlayerisasmallapproximationat8µmduetothemuchhigherplasmafrequencyoftheAu
metalcontactsusedinpractice.
A simulationof aphotoniccrystalcavitywith n =7 periodsinthe xˆ-directionandn =4 periodsin theyˆ-directionyields
x y
thethreelocalizeddefectmodesshowninFig. 6(a-c). Thesemodesarethesamemodesasfoundinthe2DanalysisofFig. 3.
Comparisonof the 2D and 3D mode propertiesare summarizedin Table II. The normalizedfrequenciesof the defectmodes
lie within the TM-like bandgap of the 3D honeycomb lattice waveguide structure, and are centered around a/l 0.2. At a
∼
wavelength of 8 µm this correspondsto a lattice constant of approximatelya=1.6 µm, a hole radius of r=0.34 µm, and a
minimumdimensiongivenbythegapbetweennearestneighborholesofa(1/√3 2r/a)=0.25µm.
−
Effectivequalityfactorsassociatedwithin-plane(Q ),topsidevertical(Q),andbottomsidevertical(Q )radiationlossesare
t b
alsocalculatedforthe3DFDTDsimulations. ThetopksideverticaleffectiveQ-factorforallofthedefectmodesisestimatedat
6
TABLEII:Comparisonbetween2D(PWE)and3D(FDTD)simulateddefectmodes.
Parameter x-dipolemode y-dipolemode hexapolemode
PWEa/l 0.2115 0.2115 0.2162
FDTDa/l 0.196 0.197 0.205
Q 326 288 169
Qkt 1.2 107 9 106 1.1 107
× × ×
Q 92 93 114
b
>106, dueto thefactthatthe cavitymodeslie predominantlybelowthe lightconeofthe topair-cladding23. Q onthe other
b
hand, can be increased only slightly fromthat reportedin Table II throughdeeper etchingof the air holes, limited mainly by
thehigh-indexofthesemiconductorsubstrateandthelocalizednatureofthecavity. DuetotheTM-likepolarizationin-plane
bandgap provided by the honeycomblattice, an increase in the number of PC mirror periods to a value above 10 effectively
eliminatesin-planeradiationloss,andtheQ-factorofthecavitymodesislimitedbybothradiationintothesubstrateaswellas
materialabsorptionlossduetoohmicheatinginthemetalsurface-plasmonlayer(forwavelengthsinthemid-IR,andaAumetal
surface-plasmonlayer,materialabsorptionlimitstheQ-factortoapproximately1038).
FIG.6:In-planemodeprofiles(Ez)forthe(a)x-dipole,(b)y-dipole,and(c)hexapoledefectmodesofthesurface-plasmonverticalwaveguide
structurewithan“etched”hexagonaldefectcavityinthehoneycomblattice.
IV. ANALYSISOFTHZDEVICES
OurFDTDanalysisofmid-IRPCQClasersexaminedstructuresinwhichthehoneycomblatticewaspatternedthroughthe
QC semiconductoractive region. This is necessary in order to induce a strong index contrast. However, a different solution
is possible if the optical mode is highly confined in the vertical direction, as is the case for THz QC lasers with metal-metal
waveguidestructures24,25. Inthiscase,andundercertainconstraintsrelatedtothestructurethickness,thesolepatterningofthe
topmetalcontactiscapableofinducingacompletephotonicbandgap. Thesemetal-insulator-metal(MIM)structures26,aswe
willrefertothemhere(seeFigs. 7and8),willbethefocusofourtimedomainmodelinginthefollowingsections. Theappeal
ofthisdesignliesinthefactthatonlythetopmetallayerneedstobepatterned27,28,whichbothsimplifiesthefabricationprocess
andimprovestheefficiencyofcarrierdiffusionintheactiveregion.
A. MIMstructures:waveguidesforTHzQClasers
Metal-metalwaveguides26 areusuallyemployedforQClasersintheTHzrange24,25 becausetheycanprovidealmostunity
opticalconfinementfactors,andsimultaneously,relativelylowwaveguidelosses. Theactivelasercoreissandwichedbetween
twometallayers(typicallyTi/AuorGe/Au/Ni/Au)whichactassurface-plasmonlayers(Fig. 7(a)). Thetwosurface-plasmon
modesof both the top andbottommetal-coreinterfacesbecomecoupledand formtwo guidedmodes, one of evenparityand
one of odd parity (assuminga nearly symmetricmetal-core-metalstructure, and where the vector parityis determinedby the
symmetryofthemagneticfield).Thedispersiondiagramofadouble-metalwaveguidewithactivelayercorethicknessofL =1
a
µmisdisplayedinFig. 7(e). Forsub-wavelengthcorethicknessestheoddparitysurfacemodeiscut-off,whiletheevenparity
7
modeexistsallthe way downto zerofrequency. The electric andmagneticfieldprofilesofthe evenparitysurfacemodeata
wavelengthofl =100µmareshowninFig. 7(b-d),showingthenear-unityconfinementfactorofthefieldsinthecoreregion.
0
FIG.7: (a)Schematicshowingthecross-sectionofaMIMwaveguide(metallayersareassumedsemi-infinite). (b)Ez normalelectricfield
component,(c)Exlongitudinalfieldcomponent,and(c)Hy magneticfieldplotoftheevenparityguidedsurfacemodeatl =100µmforan
activeregioncorethicknessofLa=1µm. (e)Dispersiondiagramofthedouble-metal(Au)waveguidestructure(La=1µm). Thecoreis
modeledwithaconstantrefractiveindexofna=3.59,whilethemetallayersaremodeledusingaDrude-LorentzmodelforAu(background
dielectricconstante b=9.54,plasmonfrequencyw p=1.35 1016rad/s,andrelaxationtime8 10−15s).
× ×
As has been noted by other authors29, a reduction in the dielectric core region thickness of MIM structures results in an
increaseinthein-planewavevectoroftheevenparityguidedmodeforaparticularfrequency. Thisisaresultofthe“pushing”
oftheelectromagneticenergyintothemetalcladdingregionswithdecreasedcorethickness. Ofimporttothecurrentworkis
theabilitytocreatealargeeffectiveindexcontrastthroughpatterningonlyofthemetallayers. Insuchapatternedstructurethe
guidedmodesees two effectiveindecesas it propagates,the largeeffectiveindexof the guidedsurface-plasmonmodewhere
the metal is left intact, and a lower effective index, more delocalized mode of the dielectric core and air-cladding where the
metalisremoved.Thatthispictureisinfactvalid,andcanbeusedtogreateffect,isdemonstratedbelowwhereweanalyzethe
bandstructureandlocalizedresonancesofdouble-metalwaveguideswithonlyasurface-patterninginthetopmetalcontact.
B. PatternedMIMwaveguidebandstructureanalysis
FIG.8: Schematicof thehoneycomb latticeandMIM structureusedinFDTDmodeling. (a) Topview of thelatticewithacentral defect
consistingoftheremovingofthecentralhexagonofairholes. (b)Cross-sectionofthesimulatedstructure. Onlythetopmetallayer(perfect
conductor) is patterned. The displayed layer thicknesses are for an operating wavelength of l 0=100 µm and a normalized frequency of
a/l 0=0.17withinthebandgapofthehoneycomblattice.
ThemodelMIMstructurethatwe willconsiderhere, shownschematicallyinFig. 8(b), consistsofthe followingsequence
oflayers(frombottomtotop): abottomunpatternedmetalcontact,aQCactiveregiondielectriccore(modeledwithauniform
8
indexofn =3.59forsimplicity30),atoppatternedmetalcontact,andaregionofairabovethestructure. Themetallayersare
a
assumedtobeperfectconductors;lossduetoabsorptioninthemetalregionsisdiscussedinsectionIVD.Murabsorbingbound-
aryconditions31areusedtomodelradiationlossoutofthestructure.ForeachoftheTHzFDTDsimulationsdescribedbelow,a
gridresolutionof58pointsperlatticeconstantaofthehoneycomblatticewasused. Atanominaloperatingwavelengthof100
µm,andfornormalizedfrequenciesa/l 0.17withinthebandgapofthehoneycomblattice(seebelow),thelatticeconstant
0
∼
isa=17µm. Atthiswavelengthandnormalizedfrequency(hereinafterthenominaloperatingconditions),thecorresponding
spatialresolutionis0.3µm(roughly95pointsperwavelengthinthedielectriccorematerial),andtheMIMlayerthicknessesare
asshowninFig. 8(b).
FIG.9:FDTDcalculatedbandstructureofapatternedMIMwaveguidewithanormalizeddielectriccorethicknessofLa=La/a=0.176anda
normalizedairholeradiusofr/a=0.25inthetopmetalcontact.Metallayersaresimulatedasperfectconductingboundaries.Atanoperating
wavelength of l 0 =100 µm within the photonic bandgap (a/l 0 =0.17), the corresponding physical sizes of the patterned double-metal
waveguideare a, r, La = 17, 4.25, 3 µm.
{ } { }
Thebanddiagramforsurface-patternedMIMwaveguidestructureswith variousdielectriccore thicknesseswere computed
to confirm the existence of a complete in-plane bandgap, and to examine the dependence of the bandgap frequency width
on structure thickness. The simulation volume consists of a single unit cell of the honeycomb lattice with Bloch periodic
boundaryconditionsappliedintheplaneofperiodicityandMurboundaryconditionsinthetopandbottomdirections(normal
to the metaland semiconductorlayers). Figure9 showsthe banddiagramfor a structurewith a normalizedcorethicknessof
L =L /a=0.176 and a normalized air hole radius of r/a=0.25 in the top metal layer. The correspondingphysical core
a a
thickness is L =3 µm under nominal operating conditions (l =100 µm, a/l =0.17). In the band diagram, truly guided
a 0 0
modesliebelowtheairlight-line,whileleakymodeslieaboveit(inthiscaselightcanonlyleakintothetopverticaldirection
where patterning of the metal has been applied). The bottom-mostband is a surface wave, lying just below the light-line of
theactiveregioncore. Itisakintotheevenparityguidedsurface-plasmonmodeofthesymmetric(unpatterned)double-metal
waveguidestudiedabove. Thehigher-lyingfrequencybandsconstitutezone-foldedversionsofthissurfacewave. Anin-plane
photonicbandgapexistsfor guidedmodesof thisstructurebetween normalizedfrequenciesa/l =0.165 0.18. Due to the
0
−
extremethinnessofthedielectriccore,thehigher-orderverticalmodesoftheMIMwaveguidearenotpresentinthisdiagram,
butlieatmuchhigherfrequencies(a/l >0.75). Asaresult,thephotonicbandgapisatruefullin-planebandgapforguided
0
resonancesof the MIM structure, not just for a single polarizationor mode symmetry. Note, the modesthat lie along the air
light-lineinthediagramareradiationmodes,predominantlylocalizedintheairregionsabovetheMIMstructure.
InFig. 10we plotthe dispersionof thebandsdefiningthe(lowest)photonicbandgapbetweenthe highsymmetryX andJ
pointsofthehoneycomblatticeforvaryingdielectriccorethicknesses. Thebandgapisseentoshrinkwithincreasingdielectric
corethickness,closingforanormalizedactiveregionthicknessgreaterthanL =0.294(nominalphysicalthicknessL =5µm).
a a
Modefieldplots(see Fig. 11) attheX andJ pointsindicatethatthehighandlow frequencymodesdefiningthe bandgapare
ofmixedtype: thelow-frequency“dielectric”or“valence”bandmodeispredominantlya surfacewave attheinterfaceofthe
toppatternedmetalsurface,whereasthehigh-frequency“air”or”conduction”bandmodeispredominantlyasurfacewaveof
thebottomunpatternedmetalinterface. Additionally,thehigh-frequencygapmodesitslargelybeneaththeunpatternedregions
ofthetopmetalboundaryandthelow-frequencymoderesidesbelowthepatternedareaswith moreenergyresidinginthe air
cladding. Thephotonicbandgapinthiscaseisaresultofdifferencesinthesetwosurfacewaves. Thissimplepicture,although
correct,betraysitscomplexityfortworeasons: (i)perfectmetalconductorboundarieswereusedinthesimulations,whichdo
notsupportsurfacewaveswhencontinuousandflat,and(ii)thepatternedairholesaremuchsmallerthanthewavelengthinair
9
FIG. 10: (a) Valence and conduction band dispersion between the X and J points for varying active region core thickness (La =
0.088( ),0.176( ),0.294( ),0.412((cid:3)) ; nominal physical thicknesses La= 1.5( ),3( ),5( ),7((cid:3)) µm). The symbols correspond to
{ ◦ × ⋄ } { ◦ × ⋄ }
datapointsfromFDTDsimulationsandthelinesareguidestotheeye.Thenormalizedholeradiusforallsimulationswasfixedatr/a=0.25.
Notethatthecompletebandgapshrinksasdevicethicknessincreases,andisclosedforLa>0.294(La>5µm).
oftheguidedmode. Thatasurfacewavecanindeedexistattheinterfaceofapatterned perfectconductinglayerwaspointed
outinRef.32, whereitwasshownthatitis preciselythecut-offfrequencyoftheair holesinthe perfectconductorwhichsets
the effectiveplasmafrequencyof the layer. The thinnerthe dielectriccorelayer, the largerthe fractionof energythatresides
at the top patterned metal interface, and a greater degree of mode delocalization that occurs into the air-cladding where the
topcontactisremoved,botheffectswhichincreasetheeffectiveindexcontrastofthetwosurfacewavesandthusthephotonic
bandgap. ItshouldbenotedthattheuseinourmodelofperfectconductingboundariesasopposedtoaAumetalcontactlayer,
for instance, doesnotaffectthe photonicbandstructureshownin Fig. 9 and Fig. 10, as the plasma frequencyof Au is much
greaterthantheTHz frequenciesunderconsideration. Furtherdiscussionandanalysisof photonicbandgapstructuresformed
fromsurfacepatternedmetalandperfectlyconductinglayerswillbepresentedinafuturestudy33. Inpractice,laserstructures
withthicknessesmuchbelowthatofseveralmicronsmaybedifficulttofabricateandincurinsurmountableopticallosses(see
belowfor discussion), butthe trend shownin Fig. 10 clearly indicatesthat suchstructureswouldsupportcompletebandgaps
thatarequitesubstantial,approaching15%ofthegapcenterfrequencyforL =1.5µmatanoperatingwavelengthofl =100
a 0
µm.
C. Characterizationofdefectmodes
Ultimately,onewouldliketoutilizethestrongphotonicbandgapeffectspresentinthepatternedMIMwaveguidestructures
studiedaboveto,forinstance,formlocalizedmicrocavitylaserresonators. Inordertodemonstratetheeffectivenessofthetop
metalpatterninginthisregard,wehavealsosimulatedthelocalizedresonancesthatareformedwithinandaroundpointdefects
of the honeycombmetal pattern studied in the previoussub-section. In Fig. 12(a) we show the E field plot of the localized
z
yˆ-dipole-like38modethatformsaroundapoint-defectconsistingoftheremovalofthecentral6holesinthetopmetal(seeFig.
8(a)) of a MIM waveguide with an active region thickness of L =3 µm (L =0.176). The corresponding(in-plane)spatial
a a
FouriertransformofthemodeisplottedinFig. 12(b),showingthatthemodeislocalizedatthek pointsinreciprocalspace,
X
and consistentwith a “donor”modeformedfromthe “conduction”bandedgeat the X-point11. This modeis identified as the
samedipole-likemodeofthemid-IRsurface-plasmonsimulationsofFig. 6(b)insectionIII,whereinthosestructuresadeepair
holepatterningofthesemiconductordielectricmaterialwasusedtocreatealargeindex-contrast.
For the fixedwaveguidecore thicknessof 3 µm the yˆ-dipole-likemodeof Fig. 12(a) is seen to be highly localizedin real-
space,asexpectedfromthebandgapsimulationsofFig. 10. Additionalconfirmationofthein-planelocalizationofthedefect
modedueto anin-planephotonicbandgapeffectcan beobtainedby studyingthescaling ofthe in-planeradiationlosses asa
functionofthenumberofperiodsofthePCpatterningsurroundingthecentraldefect. Thetotalqualityfactorduetoradiation
lossisdeterminedby
1 1 1
= + , (2)
Q Q Q
rad
k ⊥
10
FIG.11: High-frequency (a-c)and low-frequency(d-f) gapmode fieldplots(Ez) attheX-point ofthehoneycomb latticefor thepatterned
double-metalwaveguidestructureofFig. 10withLa=0.294(La=5µm). Thein-planefieldplotsof(a,d)correspondtoasectionthrough
thecoreoftheMIMstructurejustbelowthepatternedtopmetalcontact. Thedashedwhitelinesin(a,d)and(b,e)indicatethepositionofthe
sectionsusedinthefieldplotsof(b,e)and(c,f),respectively.
whereQ andQ quantifythein-planeandout-of-planeradiationlosses12. Theout-of-planequalityfactorisfoundtobe>105
forthesekstructu⊥res; thusthe dominantradiationlossoccursin the in-planedirection. Fig. 13 showsa plotofthe Q-factoras
a function of the number of surrounding PC periods. The radiation Q-factor increases exponentially with period number as
expectedforabandgapmode,withacavityconsistingofonly12 6.5periodsabletosustainamodewithradiationQ>103.
×
D. Effectivemodevolumeandmetalabsorptionlosses
Wehaveshownabovethatindoublemetalstructuresthefrequency-widthofthephotonicbandgapofanhoneycombphotonic
latticeincreaseswhenthestructureisthinner. Itimplies,intuitively,thatthephotoniccrystalbecomesincreasinglyeffectivein
confininglightwhenthewaveguidecorethicknessisreduced. Toquantifythemodallocalizationversuscoreregionthickness
we calculatedthe effectivemodevolumesV of theyˆ-dipole-likemodeofFig. 12for variousactiveregionthicknesses(L ).
eff a
V isdefinedas:
eff
Re E2dV
Veff= max|[e|E2], (3)
| |
wheremax[..]denotesthemaximumvalueoftheargument. TheresultsarereportedinTableIII. ThedecreaseofV withthe
eff
activecoreregionthicknessL issuper-linear,ascanbeseeninthethirdcolumnofTableIIIinwhichV /L istabulated. The
a eff a
super-lineardecreaseinV withactiveregionthinningisaclearindicationthatthephotoniccrystalformedfrompatterningof
eff
thetopmetalsurfaceprovidesmoreeffectivein-planeconfinementforthinnerwaveguidecores.
FDTD simulations to this pointhave approximatedthe metal contacts defining the MIM waveguidestructures with perfect