Table Of ContentTopologicalinsulators forhighperformance terahertz to infrared applications
Xiao Zhang,1 Jing Wang,2 and Shou-Cheng Zhang3
1Department of Electrical Engineering, Stanford University, Stanford, CA 94305, USA
2Department of Physics, Tsinghua University, Beijing 100084, China
3Department of Physics, McCullough Building, Stanford University, Stanford, CA94305-4045
(Dated:January20,2011)
TopologicalinsulatorsintheBi Se familyhaveanenergygapinthebulkandagaplesssurfacestateconsist-
2 3
ingofasingleDiraccone. Lowfrequencyopticalabsorptionduetothesurfacestateisuniversallydetermined
1 bythefinestructureconstant. Whenthethicknessofthesethreedimensionaltopologicalinsulatorsisreduced,
1 theybecomequasi-twodimensionalinsulatorswithenhancedabsorbance. Thetwodimensionalinsulatorscan
0 betopologicallytrivialornon-trivialdependingonthethickness,andwepredictthattheopticalabsorptionis
2 largerfortopologicalnon-trivialcasecomparedwiththetrivialcase. Sincethethreedimensionaltopological
n insulatorsurfacestateisintrinsicallygapless,weproposeitspotentialapplicationinwidebandwidth,highper-
a formancephoto-detection coveringabroadspectrumrangingfromterahertztoinfrared. Theperformanceof
J photodetectioncanbedramaticallyenhancedwhenthethicknessisreducedtoseveralquintuplelayers,witha
8 widelytunablebandgapdependingonthethickness.
1
PACSnumbers:78.68.+m78.20.Bh,78.20.Ci,78.56.-a
]
i
c
Topological insulators (TIs) are a new state of quantum ductionsurfacebandsdependsolelyonthespin-statesincon-
s
- matterwithaninsulatingbulkgapandgaplessedgeorsurface trast to the conventional semiconductors. When the thick-
l
r states interesting for condensedmatter physics, materialsci- ness of such thin film is less than 6 QLs, a gap is opened
mt enceandelectricalengineering1–8. Thetwo-dimensional(2D) up for the surface states around the Γ point9,14,15,20, and the
TI, with quantumspin Hall (QSH) effect has been predicted resulting 2D insulator can either be topologically trivial, or
.
t and observed in HgTe/CdTe quantum well2,3. Recently, 3D non-trivial, depending the thickness of the film15. We show
a
m TIsuchasBi2Se3 andBi2Te3 weretheoreticallypredictedto here that the opticalabsorbance near the band edge is either
have bulk energy gap as large as 0.3eV, and gapless surface smallerorlargerthanπα,dependingonitisconventionalin-
-
d statesconsistingofasingleDiraccone6,7. Theangle-resolved sulator or 2D QSH insulator. This suggests an optical mean
n photoemission spectroscopy (ARPES) observed such linear to measureitstopologicalnatureotherthantransport3. With
o Dirac spectrum dispersing from the Γ point in both of these suchstrongabsorbance,thethinfilmBi Se maybeapromis-
2 3
c materials7,8. Bi Se and Bi Te are stoichiometric rhombo- ing candidate for high performancephotodetectorin the ter-
[ 2 3 2 3
hedral crystals with layered structure consisting of stacked ahertz (THz) to infrared frequency range. The gapless 3D
1 quintuple layers (QLs), with relatively weak Van der waals TI can cover this full spectrum just by a single device with
v couplingbetweenQLs(eachQLisabout1nmthick). There- highsignal-to-noiseratio(SNR)comparabletomultiplestruc-
3
fore high quality thin films have been successfully grown tures using the conventional photodetecting material - bulk
8
on silicon and silicon carbidesubstrates via molecular beam Hg Cd Te21–23 with different fraction x. The SNR can be
5 1 x x
epitaxy9–11, layer by layer, which enables further scientific furth−er enhanced to be about 15 times better with reduced
3
. study and applications integratable with today’s electronics. thicknessof Bi2Se3 less than 6 QLs. Graphene has been re-
1 ThesematerialshavealsobeengrownbyAu-catalyzedvapor- cently demonstrated as a THz photodetector18. With almost
0
liquid-solid12andcatalyst-freevapor-solidchemicalvaporde- the same high absorbance17, TI photodetectorproposedhere
1
1 positionmethods13 on silicon, silicondioxideandsiliconni- hasmanyadvantages.Theprominentoneisatunablesurface
: tridesubstrates.Thesurfacestatesofsuchthinfilmhavebeen bandgapwhichiseasilyachievedbyreducingthethickness,
v
predicted6,14,15andobservedtoopenagapwhentheyarethin- whichisnecessaryinmanyoptoelectronicsapplications24.
i
X nerthan6QLs9. Upto now,fewstudyonopticalproperties
r of the topological surface states has been reported. On the
a otherhand,thesingleDiracconeontheBi Se surfacecanbe I. EFFECTIVEMODELFORTHINFILMTI
2 3
imaginedas1/4of graphene16, it is straightforwardto study
theopticalpropertiesandrelevantapplicationsofTIinanal- 3DTIlikeBi Se ischaracterizedbyasurfacestateconsist-
2 3
ogytotheoptoelectronicsapplicationsofgraphene17–19. ingofasingleDiraccone,wherethespinpointsperpendicular
tothemomentum6
Starting from an effective k p Hamiltonian and ignoring
·
intra-unitcelldipolecontributiontoabsorbance,weshowthat = A k σx k σy , (1)
surf 2 y x
thelowenergyopticalabsorbanceofthinfilmBi Se thicker H −
2 3 (cid:16) (cid:17)
than 6 QLs is a universalquantityπα/2, which does notde- with A = ~v , v is the Fermi velocity and σx,y are Pauli
2 F F
pend on the photon energy or chemical composition of the matricesdescribingthespin. Whenthethicknessofafilmis
material, where α is the fine structure constant. This origi- reduced,overlappingbetweenthesurfacestatewavefunctions
natesfromDirac natureofthe 2D topologicalsurfacestates. from the upperand lower surfaces of the film becomesnon-
Furthermore,the opticaltransitionsfromthe valence to con- negligible,andhybridizationbetweenthemwillinduceagap
2
atDiracpointtoavoidcrossingofbands20. Furthermore,the
(a)
2D energygap is predictedto oscillate between the ordinary
insulatinggapandQSHgapasafunctionofthickness15. The
e
effective model of the quasi-2D system can be derived from
thek pHamiltonianfor3DTI25, photon (cid:127)
·
(k) A k 0 A k h
1 z 2
MA k (k) A k 0− k
H03D = A210k+z −AM20k+ −MA2(1kk−)z −−MA1(kkz)+ǫ0(k), (2) TI
with k = k ik , ǫ (k) = C + D (k2 + k2) + D k2,
± x ± y 0 0 2 x y 1 z
(k) = M + B (k2 +k2)+ B k2, and the four basis of the
3MD Hamilto0nian2arexdenyoted a1s zP1+, , P2 , , P1+, , (b)
| z ↑i | −z ↑i | z ↓i
|pPa2ri−zt,y↓ainwdith (th)efsourpseprsincriuppt (±doswtann)d.inWghfeonr ethveenTIanfidlmodids e 3,4
↑ ↓
thin, k is no longer a good quantum number, by replacing
z
photon
k i∂ andwiththein-planemomentumk goodquantum (cid:127)
nzum→b−ers,zthen3Dmodelcanbeexpressedas k 3D(k , i∂ ) k
3D(k = 0, i∂ ) + δ 3D, where 3D(kH0= 0k, −i∂z) i≡s h 1,2
bHlo0ck dkiagonal−andz can bHe0solved exaHctl0y, thken 2D th−inzfilm
modelcan be obtainedby projecting 3D on the eigenstates TI
H0
of 3D(k =0, i∂ )(denotedas 1, , 2, , 1, , 2, ),
H0 k − z | ↑i | ↑i | ↓i | ↓i Hybridization
˜(k ) 0 0 A˜ k
2
H02D =MA˜200k+k −MA˜˜20k(k+k) MA˜˜20(kk−k) −M˜00(k−k) +ǫ˜0(kk), (3) FpsuaIrGirfa.gc1ee:nsO.erpattiiocanldaubrsinogrpptihoontoonfasubsrofarpcteiosntaftoers.(Ta)he3Dpaartnidcle(ba)n2dDhoTleI
it is nothing but the model of Bernevig, Hughes and Zhang
(BHZ) for the 2D QSH insulator2,15,20 with ǫ˜ (k ) = C˜ + thethevalencetoconductionsurfacebands,
0 0
D˜ (k2+k2), ˜(k ) = M˜ +B˜ (k2+k2),andC˜ , Dk˜ , M˜ , B˜
an2d Ax˜2 arye pMaramketers w0hich2arexthicykness dep0end2ent.0The2 H =H0+H1
iesnedroguybdleisdpeegrseinoenraisteǫa±n(kd)g=ivǫe˜0s(ka)g±apqoAf˜222kMk2˜+Ma˜t(Γkkp)o2,inwt.hiAchs = 4 ǫk,ia†k,iak,i+ ~eA2A(t)· 4 a†k,iDij(k)ak,j, (4)
hasbeenexperimentallydemonstratedin|Re0f.| 9, a finite sur- Xk Xi=1 iX,j=1
facegapisinducedwhentheTIfilmofBi Se islessthan6
2 3 wherea arethebandoperatorswithenergydispersionǫ =
QLs. k,i 1,2
ǫ and ǫ = ǫ , respectively. A(t) is the vector potential.
3,4 +
−(k)istheinterbandtransitionoperator,
D
II. OPTICALABSORBANCEANDSELECTIONRULES 0 0 eˆ 0
+
0 0 0 eˆ
bbyyCktohuepmleiAnog.detAol sHexiantmetrihnlteaolnceaialsenecto(r3fo)mg,rawagpinhtehetnitceh,efitehrleedpmAlaacisessmldeesensstcDroiibfreakdc D(k)=eˆ0− eˆ0+ 00 00− (5)
− c
spectrumleadstotheuniversalopticalabsorbance = where eˆ = eˆ ieˆ . Here we have neglected the small
graphene x y
πα17. SuchopticalpropertiesbeingdefinedbythefPundamen- wavevec±torofligh±t. Takingintoaccountthemomentumcon-
talconstantsoriginatesfromthe two-dimensionalnatureand servationk forthe initial i and final f states, onlythe ex-
|i | i
gapless electronic spectrum of graphene. This suggests the citationprocessespicturedinFig.1(a)contributetothelight
universal absorbance should also be true for the helical sur- absorption. The absorption power per unit area is given by
face states ofTI.When the thicknessof Bi Se thinfilm ex- W = Γ~ω, where Γ is the absorption events per unit time
2 3 a
ceeds6QLs,thelowenergyopticalabsorptionofitsgapless per unit area, calculated by using Fermi’s golden rule to be
surfacescanalwaysoccurwithphotonenergyrangingfrom0 Γ = (2π/~) f i 2δ(E E ~ω), and ω is the
i,f,k|h |H1|i| f − i −
to0.3eV,wherethehighenergycut-offisthebulkbandgap. optical frequency. The incident energy flux W is given by
P i
Withinthedipolemomentapproximation,wecanwritedown W = (c/4π)A(t)2ω2. Thentheabsorbanceis definedas the
i
| |
theinteractingHamiltonianduetotheopticaltransitionsfrom ratioofabsorbedlightenergyfluxdividedbytheincidentlight
3
energyflux (k 0),thetransitionoperatorisexactlythesameasinthe
→
gaplesscase,andtheabsorbanceis
W π
= a = α, (6)
gapless
P Wi 2 1
=πα . (7)
where α is the fine structure constant. The universaloptical Pgapped 1+2M˜0B˜2/A˜22
absorption does not depend on the photon energy. For the
interaction of light with relativistic helical surface states is Thus the absorbance for the gapped surface states is around
described by a single coupling constant, i.e. the fine struc- πα,abouttwicehigherthanthegaplesscase. Inparticular,for
ture constant. The Fermi velocity is only a prefactor for 2 and 3QL, the absorbanceis 1.8%and 2.05%, smaller than
both 0 and 1, accordingly, one can expect that the coef- πα, andfor4and5QL,the absorbanceis3.85%and6.22%,
H H
ficient may not change the strength of the interaction, as in- largerthanπα. ThesignofM˜ B˜ determinesthenontrivialor
0 2
deed our calculationsshow. When a TI film is suspendedin trivialtopologyofthequasi-2Dbandstructure2,15. M˜ B˜ < 0
0 2
air, α = 1/137 and thus = 1.15%. It is among materi- and > πα, the system is in the inverted regime, i.e.
alswithhighestabsorbancPe,about200timeslargerthanbulk the QPSgaHppeidnsulating states; while M˜ B˜ > 0 and <
0 2 gapped
Hg1 xCdxTe of equivalent thickness, with absorption coeffi- πα, the system is just conventional insulator. ThisPsuggests
cien−t about 100cm−1 for incident infrared light21. It is also an optical way to identify the topological nature other than
just half of the value of a single layer graphene. This is be- transportmeasurement3.
causefor3DTIfilm,theupanddownsurfacesgives2gapless
Inreality,theabsorptionoccursnotjustaroundbandedge,
Diracfermions,whileforsinglelayergraphene,thetwoDirac
so from the pointof view for application, we should get the
conestogetherandtworealspinsprovideadegeneratefactor
fullsurfaceabsorptionspectrum.Thusthetransitionrateis
of4.
Astheenergygapisopenedaroundthe Diracpointin the
thin film TI like Bi2Se3 less than6 QLs, interbandrealtran- Γ=(2π/~) |hf|H1|ii|2δ(Ef −Ei−~ω) (8)
sitionsbetweentheconductionandvalencesurfacebandscan iX,f,k
beexcitedbyopticalfieldwhenthephotonenergy~ωexceeds
thesurfacebandgap2M˜ (Fig.1(b)). Aroundthebandedge Theabsorbanceis
0
| |
(ω)= Γtotal(ω)~ω = Pgapped(M˜0) 1 A˜22−(A˜22+2M˜B˜2)+ q(A˜22+2M˜B˜2)2+B˜22(~2ω2−4M˜2), (9)
Pgapped Wi r1+ B˜(22A(˜~222+ω22M˜−B4˜2M˜)22) − B˜22 ~2ω2
(M˜ ) is the band edge absorbance and the parameters tion,respectively.WecanseeclearlyinFig.2(b)thatthespin
gapped 0
P
for Bi Se are in Ref. 9. For 4 QLs Bi Se , the absorbance opticaltransitioninthethinfilmTIisverydifferentfromthe
2 3 2 3
isshowninFig.2(a). Wecanseethattheabsorbanceishigh conventionaldirect-gapsemiconductorsuch as GaAs, where
in the allowed frequencyrange, indicating a good frequency the spin wavefunctionremainsunchanged. This uniquespin
coverage when TI is used for photodetection. When the 3D opticaltransitionselectionrulesmakethethinfilmTIattrac-
TI is reduced to quasi-2D, the absorbance is enhanced only tiveforspintronicsapplicationslikefastswitching.26
near the band edge, and when the photon energy increases,
theabsorbancedropstothe3Dvalueof =πα/2.
gapless
P
The gapped surface bands in the thin film TI can be clas- III. TI-BASEDPHOTODETECTOR
sifiedintotwoclasses, withoneclassbeingthetimereversal
partneroftheother. Bands1and3areoneclass,theotheris Recently, graphenehas been experimentallydemonstrated
2and4(Fig.2(b)). AsshowninEq.(5),nearthebandedge, as a high performance THz Infrared photodetector due to
right-handed (σ+) light couples only to 1 and 3, while left- its zero band gap and high a∼bsorbance18. Similarly, the TI
handed(σ )lightcouplesonlyto2and4. Itwillcausetran- surfacemayalso beused forsuchapplication. Thefigureof
sitionbetw−eenthespin-down(up)valencestateandthespin- meritof photodetectoris characterizedby its SNR. Fig. 3(a)
up(down)conductionstate. Forconventionalsemiconductors showsatypicalphotodetectorusingtheTIthinfilm-thepho-
likeGaAsquantumwellin[001]direction,theopticalselec- toconductive resistor22. The particle and hole pairs will be
tion rulesfrom heavyhole bandsto conductionbandsare as generated through light absorption and the system becomes
σ : S / 1 (X iY ) / . Here S , X , more electric conductive. Such change in conductivity can
± | i⊗|↑ ↓i→∓√2 | i± | i ⊗|↑ ↓i | i | i
Y and / representstheorbital-andspin-partwavefunc- beeasilymeasuredinanelectroniccircuit. TheSNRinsuch
| i |↑ ↓i
4
(a) betterthanHg Cd Te(Fig.3(b)).Besidesthis,thebandgap
0.040 4QLs BiSe of such quasi-12−Dx syxstem are tunable form 5meV to 0.25eV
2 3
0.036 Graphene by varying its thickness9 (Fig. 3(b)) where the high energy
0.032 3D Bi2Se3 cut-off is 0.6eV here due to the confinementof bulk bands9.
0.028 PhotodetectorsmadefromtheBi2Se3familyofTIsarepoten-
ance 0.024 tiallyeasiertomanufactureandcanovercomethelimitations
orb ofHg1 xCdxTe.
bs 0.020 −
A
0.016 I
(a) ph
0.012 V
0.008
0.08 0.12 0.16 0.20 0.24
w( eV) Light
(b)
3 4 CB -1/2 CB +1/2
l
w TI
(cid:305)+ (cid:305)- (cid:305)+ (cid:305)-
(b) 3D TI
5QL 2GDra TpIhene
1 2
HH -3/2 HH +3/2 4QL
TI surface GaAs 101
FfoopIrGtigc.ra2al:pshAeelbnesceot,iro3pnDtiroaunnledssp2ieDnctGTruIamAsusrafanandcdesTbelaIencsdtusiro.fna(cbre)uCbleaosnm.d(psa.a)rAiTshobensouorbpfasanpncidne SNR(TI)/SNR(HgCdTe) 3QL 2QL
downarrowdenotesthespinupandspindown,respectively.CBand
HHdenotestheconductionbandsandheavyholebandsinGaAs. 100
0 0.05 0.1 0.15 0.2 0.25 0.3
Photon Energy (eV)
system is defined by SNR = Iph/σIn, where Iph is the pho- FIG. 3: Photodetector device. (a) Schematics of a typical photo-
tocurrent and σIn is standard deviation of the noise current resistor.(b)TheSNRof3DTIinfullspectrum(blueline),graphene
whose major source is the thermal noise (Johnson noise) at (dashedline)and2DTIatthebandedge(redpoint)comparingwith
finite temperature T when the voltage bias V k T/q27,28. bulkHg Cd Te.
B 1 x x
I = 2qηPS/E , here a factorof 2 represents≪the electron- −
ph ph
holepair,qistheelectroncharge,ηisthequantumefficiency,
P is the incidentlight power density, S is the detecting area
and E is the incident photon energy. σ = 4k T∆f/R,
whereaphs ∆f is the bandwidth of the detecIntor, ipnverBsely pro- IV. CONCLUSIONANDOUTLOOK
portionaltoitsintegrationtimeandRistheresistance. Thus
theperformanceofaphotodetectorrelyontheintrinsicprop- In summary, we discussed in this paper the fine struc-
erties of the photodetecting material, for η is determined by ture constant defined high optical absorption of TI and its
the absorbanceandR is determinedbythe resistivityρ. The unique selection rules. Broadband and high performance
higherabsorbanceandthelargerresistivity(largerbandgap) photodetectors based on TI are proposed. They may find a
thematerialhave,thebetteritistobeusedasaphotodetector. widerangeofphotonicapplicationsincludingthermaldetec-
AdetailedcalculationoftheSNRofTIphotodetectorsispro- tion, high-speed optical communications, interconnects, ter-
videdinthesupplementaryfile29. Duetothehighabsorbance, ahertz detection, imaging, remote sensing, surveillance and
3DTIcanalsobeusedasahighperformanceTHz Infrared spectroscopy18. Inadditiontophotodetection,TImayalsobe
(0 0.3eV) photodetector, whose SNR is compa∼rable than used in future for other optoelectronicdevices, such as tera-
the∼commerciallyused bulk Hg Cd Te (Fig. 3(b)). In par- hertz laser30, waveguide31, plasmon based radiation genera-
1 x x
ticular, this full spectrum can be−covered by a single 3D TI tion and detection32 and transparentelectrode33 etc. in anal-
detector,whileforHg Cd Te,multiplestructureseachcov- ogytographene.
1 x x
eringasegmentoftheb−andwidthbytuningthefractionx(and
thenbandgap)havetobeused,tomaintainhighperformance,
which bringsmuch more complexitiesin fabrication. More- Acknowledgments
over, the resistivity ρ can be greatly enhanced when surface
opens a gap with the thickness of thin film less than several WewishtothankS.Raghu,J.Maciejko,S.B.Chung,R.D.
QLs. Indeed,theSNRcanbeimprovedtobeabout15times LiandB.F.Zhuforinsightfuldiscussions. J.W.acknowledge
5
thesupportfromNSFofChina(GrantNo.10774086),andthe 2006CB921500).
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