Faraday rotation due to surface states in the topological insulator (Bi$_{1-x}$Sb$_{x}$)$_{2}$Te$_{3}$

5.7 MB·

by  Yinming Shao

Faraday rotation due to surface states in the topological insulator (Bi Sb ) Te
1−x x 2 3
Y. M. Shao1,2,∗, K. W. Post,2, J. S. Wu2, S. Dai2, A. J. Frenzel2, A. R. Richardella3, J. S. Lee3,
N. Samarth3, M. M. Fogler2, A. V. Balatsky4,5, D. E. Kharzeev6,7 and D. N. Basov1,2
1Department of Physics, Columbia University, New York, New York 10027, United States
2Physics Department, University of California-San Diego, La Jolla, California 92093, United States
3Department of Physics, The Pennsylvania State University,
University Park, Pennsylvania 16802, United States
4Nordita, KTH Royal Institute of Technology and Stockholm University,
Roslagstullsbacken 23, SE-106 91 Stockholm, Sweden
5Institute for Materials Science, Los Alamos National Laboratory, Los Alamos, New Mexico 87545, United States
7 6Department of Physics and Astronomy, Stony Brook University,
1 Stony Brook, New York 11794-3800, United States
0 7Department of Physics and RIKEN-BNL Research Center,
2 Brookhaven National Laboratory, Upton, New York 11973, United States

n
a Using magneto-infrared spectroscopy, we have explored the charge dynamics of (Bi,Sb)2Te3 thin
J filmsonInPsubstrates. Fromthemagneto-transmissiondataweextractedthreedistinctcyclotron
0 resonance (CR) energies that are all apparent in the broad band Faraday rotation (FR) spectra.
1 This comprehensive FR-CR data set has allowed us to isolate the response of the bulk states from
the intrinsic surface states associated with both the top and bottom surfaces of the film. The FR

] datauncoveredthatelectron-andhole-typeDiracfermionsresideonoppositesurfacesofourfilms,
ll which paves the way for observing many exotic quantum phenomena in topological insulators.
a
h
-
Three-dimensional (3D) topological insulators (TIs) tional insights come from the magnetic field dependence
s
e are insulating in the bulk but host gapless, topologi- oftheLLtransitionsenergy(∆E ),satisfyingthedipole
n
m
cally protected surface states (TSSs) [1, 2]. The Dirac- selectionrule[19,20]: ∆|n|=±1. Forinstance, inasys-
. like TSSs are robust against disorder, as they are pro- tem with parabolic bands, including bulk bands in TIs,
t
a tected by topological properties of the bulk electronic LLtransitionsscalelinearlywithmagneticfieldaseB/m,
m
wave functions [3]. The linear band dispersion of TSSs wheremisthebandmass(classicalCR).Drasticallydif-
- [4–6] has been identified by angle resolved photoemis- ferentLLenergiesareanticipatedandobservedforDirac
d
sion spectroscopy (ARPES), whereas scanning tunneling fermions in TSSs [19–21]:
n √
o spectroscopy (STS) [7, 8] has verified the distinctive B
(cid:112)
c dispersing Landau levels (LLs) of the TSS. With the ad- E =E ±v 2e(cid:126)|n|B (1)
±n DP F
[
vent of the new generation of thin films of 3D-TIs, it be-

1 came possible to minimize bulk conduction, yielding an where EDP is the Dirac Point energy, n is the LL in-
v electrodynamic response that is dominated by the TSS dex, v is the Fermi velocity and B is the magnetic
F
3 [9–11]. However,whensurfacesensitiveprobes(ARPES, field. Therefore, TSS/bulk carriers can be further dis-
0
4 STS)areappliedtothesenewTIfilms,theyareonlyable tinguished by their d√ispersion of LL transition energies
tocharacterizethetop,vacuum-facingsurfaceofthesam- with magnetic field ( B/linear in B).
2
0 ple. Whiletransportmeasurementsprobeallconducting Recently, we used terahertz/far-infrared transmission
. channels in TI thin films, the decomposition of the re- spectroscopytostudythedynamicresponseofBSTthin
1
0 sponse to each individual TSS/bulk layer is not trivial. films grown on InP (111)A substrates (see Supporting
7 Information Sec. I for more details). We found an ex-
1 HerewereportonFaradayrotation(FR)andcyclotron ceptionallysmallDrudeweightthatlieswithintheupper
:
v resonances (CR) magneto-transmission spectroscopy of limit imposed by the conductivity sum rule for TSS [9].
i (Bi Sb ) Te (BST) thin films. The combination of Although the sum rule criterion is instructive, this anal-
X 1−x x 2 3
these techniques yields a comprehensive characterization ysis does not allow one to discriminate between the con-
r
of the TSS dynamics both at the top and bottom sur- tributions due to the surface and the bulk. In this work,
a
faces, complementary to transport and ARPES data. wedisentanglethesecontributionsbasedontheirunique,
[5, 6, 12–14]. We stress that FR, the rotation of light dynamic responses in a magnetic field. The broadband
polarizationafterpassingthroughamediuminmagnetic FR-CRmeasurementsenableamulti-componentanalysis
field, is particularly informative in distinguishing mul- ofthedataanduncovermarkeddistinctionsbetweentop
tiple conductive channels in complex materials [15, 16]. andbottomTSSsinourfilms. Thetwosurfaceshavedif-
Whereas both CR and FR experiments are sensitive to ferentFermienergiesandhostDiracfermionsofopposite
transitions between the LLs, only the FR allows one to sign, which is required for an observation of a number of
unambiguouslydiscriminatetheresponseofelectronsand exoticeffects, includingtopologicalexcitoncondensation
holesviathesignoftheFaradayangleθ [17,18]. Addi- in TIs [6, 22].
F
2

( a ) ( b )
1 T 2 T
A 4 0
1 .2 3 T 4 T
a 5 T 6 T
7 T 8 T
3 0
b
c
C
B )
) d ( c )
(0 ra
B) / T1 .1 2 0 (cid:1) (mF
(
T
1 0
1 .0
0
0 1 0 2 0 3 0 4 0 6 0 8 0 0 1 0 2 0 3 0 4 0 6 0 8 0
E n e r g y ( m e V ) E n e r g y ( m e V )
FIG. 1. Panel (a): magneto-transmission data normalized by the spectrum at zero field of the BST thin film. Panel (b):
Faraday rotation (FR) data. Data at different fields are offsetted by (a) 0.02 and (b) 4 mrad for clarity. Opaque regions due
to InP absorption are greyed-out. Black dotted lines are Drude-Lorentz model fits for 8 T data and black arrows indicate the
positions of infrared active phonons (∼6 and 8 meV). Panel (c) is a schematic showing the FR definition and measurement
setup.
Generally, multiple types of carriers (electrons/holes presence of an external magnetic field. Similar multi-
inTSS/bulkbands)cancoexistinTIsandallcontribute componentmodelshavebeenusedtosuccessfullydecode
tothemagneto-optical(MO)response. Accordingtothe the complex MO spectrum for multilayer graphene [17]
Drude-Lorentz (DL) model, the total conductivity is the and Bi Se thin films [10].
2 3
sum of oscillators: In the far-infrared region under the thin-film approxi-
N mation,themeasuredbroadbandCRandFRspectraare
σ (ω,B)=(cid:88)Dj γj −iω directly related to real part of σ (ω,B) and σ (ω,B),
xx π ω2 −(ω+iγ )2 xx xy
j=1 c,j j (2) respectively. Forsimplicity, σxx andσxy willbeassumed
−(cid:88)N iωωp2,jε0 −iωε (ε −1) ttoionbseinthtehereraelstpaorftthofetphaepecro.rresponding complex func-
ω2 −ω2−iωΓ 0 ∞
j=1 o,j j Magneto-transmissionspectranormalizedbythespec-
trumatzerofieldT(ω,B)/T(ω,0)areshowninFig. 1(a).
σxy(ω,B)=(cid:88)N Dπj ω2 −(−ωω+c,jiωγ )2 (3) Ttifiheedminostthperosmpeicnternutmfieolbdt-ianindeudceadtfeBat=ure8sTca:nabmeairdkeend-
j=1 cj j increase (∼10%, labeled as A) in the transmission is ob-
The first term in (2) and (3) stands for the Drude re- servedatlowfrequency,followedbyadip(B)andaweak
sponseoffreecarriersinmagneticfield. Eachoscillatoris downturn (C) at higher frequencies. The gross features
characterized by its own Drude weight (D = πne2/m), of the experimental data are accounted for within the
j
cyclotron frequency ω =qB/m and scattering rate γ . Drude model of the cyclotron resonance (Eq. (2)). Both
c,j j
The second term in (2) represents Lorentzian oscillators A and B features can be identified in the data at smaller
centeredatω ,withplasmafrequencyω andscatter- fields. As the magnetic field increases (from 2 T to 8 T),
o,j p,j
ingrateΓ . TheLorentziantermsdescribethecontribu- the positions of these features increase: an observation
j
tions of phonons and are absent in the Hall conductivity consistent with the multi-component CR interpretation
σ . The high frequency dielectric constant ε accounts of the observed features. The black arrows in Fig. 1
xy ∞
forabsorptionsabovethemeasuredfrequencyrange,and indicate the position of two phonon modes (∼6 and 8
is associated with interband transitions. In Eq. (2), meV)inBST[9]. Thephononfrequenciesdonotexhibit
eachoftheDrudeoscillatorscandecribeaCR/LLtransi- frequency shifts with magnetic field.
tionintheclassical/quantumregime. Themodelimplies Faraday rotation spectra, shown in Fig. 1(b), of-
that the Drude absorption peak is shifted from zero fre- fer complementary insights into the magneto-optical re-
quency at B = 0 to finite cyclotron frequency ω in the sponse. In addition to the low-frequency phonons (ar-
c
3

1 .1 5 broadband spectra, yielding both negative and positive
S S 1 , A ( a ) ( b ) FRs.
S S 2 , B
We used the DL model (Eqs. (2),(3)) to fit the CR
1 .1 0 B u lk , C 1 0
0) C o m b in e d and FR spectra at each measured magnetic field. The
) / T(1 .0 5 rad)0 a c mthirneiemduimffermenotdeDlrruedqeuoirsecdilltaotofirst.bAotphladuastibaleseitntceornptraeitnas-
B = 8T1 .0 0 A B C (cid:1) (mF b ttiioonnochfathnenseelstharseseocoisactielldatwoirtshisthinetbeurmlks, otofptharneedcboontdtuomc-
T( -1 0 surfaces of our films. In Figs. 2(a), 2(b) we plot each
S S 1 , a
0 .9 5 of these three contributions along with the experimental
-1) ( c ) SB Su l k2,, bc ( d ) data at 8 T (red solid line).
cm 2 0 C o m b in e d ThecoexistenceofthreemobilecarriersinBSTispar-
-1(cid:1) 2 0 ticularly evident from the decomposition of the fit for
(cid:2)T) - (0T) (xx --42 000 )-1-1cm24 00 00 BB == 80 TT -1 -1(cid:2)(cid:1)(cm)xy0 a b c θwScF)pelealc(tFditfieiecgssac.tlrliytb2o,(etbdth)we)b.oytwtdThoihsefteeminagtucrutlotrsiep-ssc-tofwyempiatptehuocrnneaeesrngratoietfFrivsRtehwerseliotsshpppeoedscntis(ffreaae[ra1ean7nr]det.
(8x-6 0 (Wxx 00 1 0 2 0 -2 0 Drude weight and scattering rate. We depict these con-
(cid:2)x s E n e rg y (m e V ) tributions with the solid and dashed grey lines, respec-
0 1 0 2 0 3 0 0 1 0 2 0 3 0 tively. The hump feature with positive slope (b) origi-
E n e r g y ( m e V ) E n e r g y ( m e V ) nates from n-type carriers and is indicated with the grey
dash-dotted line. In the limit of small Faraday rotation
FIG. 2. Experimental and modeling results for (a) transmis- (θ (cid:28) 1 rad) [28], the total rotation angle equals the
F
sionand(b)FRspectraat8Tusingmulti-componentDrude-
sum of rotations due to the three terms a, b and c. This
Lorentz model. In each panel, the red solid line is the data
net rotation (black dotted line) reproduced all features
and black dotted line is the fit. Dashed grey lines represent
of our data (red line). The very same parameter set also
contributionsfromthebulk. Solidanddash-dottedgreylines
representcontributionsfromthetwosurfacesstates. Inpanel capturesthebehaviorofmagneto-transmissionspectrain
(c) we plot the difference between the diagonal conductivity Fig. 2(a). All fitting parameters for different magnetic
(σxx) spectra obtained at 8 T and at 0 T. The inset displays fields are summarized in Fig. 3 (see fits for all magnetic
σxxat8Tand0T,whicharedominatedbyphonons(6and8 fieldsinSupportingInformationFig. S1). Westressthat
meV)ofBST.Inpanel(d)theHallconductivity(σ )isplot-
xy the same model describes the entire data of CR and FR
ted. In (b) and (d) the signal from SS 2 (n-type) is opposite
spectra at all different fields. This consistent description
in sign compared to SS 1 and bulk (p-type).
of the entire data set attests to the validity of the multi-
component analysis.
The extracted diagonal conductivity σ at B = 0
xx
rows), one can readily observe three inflection points (a, T and 8 T are plotted in the inset of Fig. 2(c). As
b andc)below30meV.Eachinflectionpointcorresponds mentioned before, two phonons dominate the low en-
to a peak in σxx associated with an underlying CR/LL ergy conductivity and do not change with the magnetic
transition[10,17]. Allthreeinflectionpointshardenwith field. Their frequencies (ω ), oscillator strengths (ω2)
o p
increasing magnetic field. and widths (Γ) are therefore kept constant at their zero
The highest measured FR angle in the low-frequency field values (listed in Table S1 in Supporting Informa-
limit of our data (∼15 mrad) is an order of magnitude tion) during the fitting at all magnetic fields. The elec-
smaller compared to previous work on Bi Se [10]. We tronic background is modified by the field. Specifically,
2 3
remark that the quantized FR is expected to occur in thespectralweight(SW),definedasSW=(cid:82)Ωcσ dω(Ω :
0 xx c
multiples of α ∼ 7 mrad both in graphene and TIs in cutoff frequency), shifts from low energy (∼3 meV) to
the limit of ω (cid:28) ω ,E [23, 24], where E is the energy higherenergy(∼15meV)intheB =8Tspectrum. This
c g g
gapassociatedwiththebulkresponse. Theexperimental SW shift is more evident in the field-induced changes of
observation of quantized FR has been reported recently σ (Fig. 2(c)). Accordingtothef-sumrule[29],theto-
xx
in different TI systems at THz frequencies [25–27]. In talSWshouldbeconservedandindependentofmagnetic
this work, the condition ω (cid:28) ω is not strictly fulfilled field,givenanappropriateΩ . Indeed,fromFig. 2(c)we
c c
sincethelowestexperimentalfrequency(∼3meV)isstill findthatthemagneticfieldinducedaccumulationofSW
very close to ω (∼8 meV) at 8 T due to the broadness around (cid:126)ω is qualitatively compensated by loss of SW
c c
of the resonance. Therefore, the observed small rotation at low energies.
is unlikely to be related to quantized FR. Below we will Phonon absorptions also affect the shape of FR spec-
demonstrate that the small rotation is the result of an tra at low frequencies, shown in Fig. 2(b). On the other
interplay of multiple transitions, clearly resolved in our hand, according to Eq. (3), the Hall conductivity (σ )
xy
4
is expected to only contain electronic contributions. In- law is the hallmark of LL spacings for massless Dirac
deed,theσ spectra(redlineinFig. 2(d))extractedvia fermions and therefore we assign these two terms to the
xy
thin-film approximation are free of phonon structures. responseoftheTSSsinourfilms. Notably,thisbehavior
The extracted total σ spectra show an excellent agree- can only be observed provided the Fermi energy is close
xy
ment with the three-layer DL model calculation (black to the Dirac point when only the lowest few LLs are oc-
dotted line), justifying the use of thin-film approxima- cupied[18,30]. WethusconcludethattheFermienergies
tions (see Supporting Information Sec. I). We now turn associatedwithtwosurfacesinBSTareveryclosetothe
Dirac point and we will report estimates of E below.
F
1 5 In contrast, the broader CR-like transition (Fig. 3(a))
2 0 (a ) 12 E F (b ) showsconventionallinearfielddependence. Weattribute
eV) 1 5 c eV) 1 0 0 b tthhiesinlastetteorfmFiogd.e3t(oa).buTlkhepc-ytyclpoetrcoanrrmiearsss,eaxstrpaicctteudrefdroimn
(cid:1)(m (cid:1)c 1 0 (cid:1)(m| (cid:1)c 5 a tmhaesssloopfe0.i0s80.m055dmedeu,cwedhicfrhomis sqmuaamlletrutmhaonsctihlleateiffonecstifvoer
5 E F | - 01 E F a p-type BST esample [31]. The discrepancy could be
0 0 due to difference in Fermi energy (BST valence band are

0 .1 5 (c ) 0 .1 0 L L 1 2 (d ) non-parabolic [31]) and/or stain in our thin films. Note
L L -1 0 that there is a peak structure in γ (Fig. 3(e)) around 4
)0 .1 0 ) T, where the corresponding (cid:126)ωc ∼8 meV coincides with
eV eV 0 .0 5 one of the phonons of BST. It is possible that the peak
(m0 .0 5 (m arises from the scattering between bulk CR and phonon
D B u lk C R D modes, as suggested in a recent terahertz MO study on
T o ta l D W Bi Se [10].
0 .0 0 0 .0 0 2 3
(e ) 1 5 (f) It is worth emphasising that the transition energies of
the two SS transitions (a and b) are fairly close to each
1 5
other, especially at low fields. We can nevertheless dif-
) )1 0
eV 1 0 eV ferentiate these two contributions because of the hump
(cid:1) (m 5 (cid:1) (m 5 sthtreuctwtuoreopinpotshiteeFpRolasrpizeacttiroan, rwohtaicthioniss cnheaarralyctcearinscteiclinogf
each other. Thus, the FR spectra allow us to unambigu-
0 0 ously differentiate and assign these two responses to two
0 1 2 3 4 5 6 7 8 9 0 1 2 3 4 5 6 7 8 9
B ( T ) B ( T ) different surfaces.
The evolution of Drude weight and scattering rate of
FIG. 3. Parameters obtained from the Drude-Lorentz fit of the two SS transitions with field are intriguing as well
magneto-transmissionandFaradayrotationspectra(Fig. 1).
(Figs. 3(d), 3(f)). Both Drude weights increase with
(a) Transition energies of the CR-like peak for bulk carriers;
increasing magnetic field, in accord with the theoretical
dashedlineisalinearfittothedata. (b)IntrabandLLtran-
predictionforDiracfermions[21]. Incontrast,theDrude
sition energies for SS 1 (red circle) and SS 2 (blue diamond).
Solid lines are model curves (Eq. (1)) of LL and LL weight for semiclasical CR is usually field-independent
1→2 −1→0
with v = 2.49×105 m/s and 0.9×105 m/s, respectively. and equals the zero-field Drude weight [30]. The scatter-
F
Drude weight of the bulk and TSS carriers are displayed in ing rates of the two SS transitions decrease with B field
(c) and (d), respectively. The decreasing Drude weight of and show an overall 1/B dependence, which is indica-
the bulk CR is unexpected for typical CR transitions. The
tive of charged impurity scattering [32, 33]. The scatter-
sumofallthreeDrudeweightsisalsodisplayedin(c)asgrey
ing rates of the TSS carriers are also much smaller than
triangles. (e) and (f) are scattering rates for bulk and TSS
their bulk counterpart, indicating much higher mobility
carriers.
associated with the surface states.
to the analysis of the fitting parameters extracted from The requirement of E lying very close to the Dirac
√ F
the three-layer DL description of magneto-optics data. point ( B dispersion) constrains the assignment of the
Each layer is parameterized by its own cyclotron fre- observedSSLandauleveltransitionstotwopossibilities:
quency ((cid:126)ω ), Drude weight (D) and scattering rate (γ). LL andLL oralternativelyLL andLL
c 0→1 1→2 −1→0 −2→−1
All these contributions are additive in the MO response forhole-liketransitions. HigherintrabandLLtransitions
in the thin-film limit [10, 28]. (|n| (cid:62) 2) are excluded for two reasons: (i) At |n| ∼ 2,
In Fig. 3 we plot the field dependence of (cid:126)ω , D and these transitions are expected to produce discontinuous
c
γ extracted from simultaneous fitting of the CR and FR jumps in transition energy that are not observed in our
spectraatdifferentmagneticfields. Thisanalysisreveals data; (ii) At |n| (cid:29) 2, corresponding to high Fermi en-
√
the B-dependence of the transition energy for the two ergy, the evolution of the transition energy with the B
√
lower energy transitions, shown in Fig. 3(b). The B fieldresemblesthelinearscaling(quasi-classicalCR)[34],
5
√
whereas our data show the B behavior, requiring low
Fermi energy. Fitting the two SS transitions with linear
dependence will result in finite intercept in energy, con-
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F
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82, 075401 (2010).
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[40] and topological exciton condensation [22]. [34] A. M. Witowski et al., Phys. Rev. B 82, 165305 (2010).
Acknowledgements - This works is supported by DOE [35] M. Eschbach et al., Nat. Commun. , 8816 (2015).
Grant NO. DE-FG02-00ER45799. Sample growth and [36] D. E. Kharzeev, Ann. Phys. 325, 205 (2010).
characterization at Penn State was supported by ONR [37] X.-L. Qi, T. L. Hughes, and S.-C. Zhang, Phys. Rev. B
78, 195424 (2008).
(Grant No. N00014-15-1-2370) and ARO-MURI (Grant
[38] M.Kargarian,M.Randeria, andN.Trivedi,Sci.Rep.5,
No. W911NF-12-1-0461). D.K. is supported by DOE
12683 (2015).
Grants No. DE-FG-88ER40388 and DE-SC0012704.
[39] A. Akrap et al., Phys. Rev. Lett. 117, 136401 (2016).
D.N.B.istheMooreFoundationInvestigator,EPIQSIni- [40] T. Morimoto, A. Furusaki, and N. Nagaosa, Phys. Rev.
tiative Grant GBMF4533. B 92, 085113 (2015).