Table Of ContentTunability of the k-space Location of the Dirac Cones in the Topological Crystalline
Insulator Pb Sn Te
1−x x
Y. Tanaka,1 T. Sato,1 K. Nakayama,1 S. Souma,2 T. Takahashi,1,2
3 3 3 3
Zhi Ren, M. Novak, Kouji Segawa, and Yoichi Ando
1Department of Physics, Tohoku University, Sendai 980-8578, Japan
2WPI Research Center, Advanced Institute for Materials Research,
Tohoku University, Sendai 980-8577, Japan and
3Institute of Scientific and Industrial Research, Osaka University, Ibaraki, Osaka 567-0047, Japan
3 (Dated: December 11, 2013)
1
Wehaveperformedsystematicangle-resolvedphotoemissionspectroscopyofthetopologicalcrys-
0
2 talline insulator (TCI) Pb1−xSnxTe to elucidate the evolution of its electronic states across the
topological phase transition. As previously reported, the band structure of SnTe (x = 1.0) mea-
n
suredonthe(001)surfacepossessesapairofDirac-conesurfacestateslocatedsymmetricallyacross
a theX¯ point in the (110) mirror plane. Upon approaching thetopological phase transition into the
J
trivialphaseat xc ≈0.25, we discovered that Diraccones gradually movetoward theX¯ point with
7 its spectral weight gradually reduced with decreasing x. In samples with x ≤ 0.2, the Dirac-cone
surfacestateiscompletelygone,confirmingtheoccurrenceofthetopologicalphasetransition. Also,
]
l the evolution of the valence band feature is found to be consistent with the bulk band inversion
al takingplaceat xc. Thetunability ofthelocation of theDiracconesin theBrillouin zonewould be
h usefulforapplicationsrequiringFermi-surfacematchingwithothermaterials,suchasspininjection.
-
s PACSnumbers: 73.20.-r,71.20.-b,75.70.Tj,79.60.-i
e
m
. Topological insulators (TIs) materialize a new quan- served in the TCI phase, the ARPES measurements for
t
a tumstateofmatterwhereanunusualmetallic statepro- isostructualPbTe[10]andPb0.8Sn0.2Te[11]revealedthe
m
tected by time-reversal symmetry (TRS) appears at the absence of any surface states, which strongly suggests
- edgeorthesurfaceofabandinsulator[1,2]. Thediscov- a trivial-to-nontrivial topological quantum phase tran-
d
n ery of TIs triggered the search for new types of topolog- sition (QPT) in the solid-solution system Pb1−xSnxTe
o ical materials protected by other symmetries [3–7], and [10, 11]. This is in line with the established band di-
c a recent theory predicted the existence of “topological agram of this compound which shows a band inversion
[ crystalline insulators” (TCIs) in which metallic surface somewhere near x ∼ 0.3 [14, 15] that is supported by
1 statesareprotectedbypoint-groupsymmetryofthecrys- band-structure calculations [9, 16]. However, it is still
v tal structure [8]; in particular, mirror symmetry leads unclearhowexactlythesurfaceandbulkelectronicstates
7 to the topological invariant called mirror Chern number evolveasafunctionofSncompositionx,whichwouldbe
7
[9]. Such a TCI phase has been experimentally verified useful for establishing practical understanding of TCIs.
1
1 by angle-resolved photoemission spectroscopy (ARPES) Infact,before Pb1−xSnxTe became relevantinthe topo-
. experiments for narrow-gap IV-VI semiconductors SnTe logicalcontext,therewerenoARPESstudies ofthe evo-
1
0 [10], Pb0.6Sn0.4Te [11], and Pb0.77Sn0.23Se [12], all hav- lution of its electronic structure in the mixed composi-
3 ing the same crystal structure shown in Fig. 1(a). In tions despite the continued interest in this system in the
1 those materials, the topological surface states measured semiconductor community [15, 16].
: onthe(001)surfaceconsistofDiracconeslocatedatmo- In this Letter, we report comprehensive ARPES stud-
v
i mentaslightlyawayfromthetime-reversal-invariantmo- ies of Pb1−xSnxTe with various Sn compositions (x =
X mentum(TRIM)X¯ pointinthe(110)mirrorplaneofthe 0.0, 0.2, 0.3, 0.5, 0.6, 0.8, and 1.0). The double Dirac-
r crystal [Fig. 1(b)]; since there are four X¯ points on the cone surface state persists in the whole TCI phase of
a
boundary of the first surface Brillouin zone (BZ), there x ≥ 0.3, but surprisingly, the k position of the Dirac
are a total of four Dirac cones within it [10–12]. This point (called Dirac momentum) was found to systemat-
is distinct from the three-dimensional (3D) TIs whose ically move within the mirror plane toward the X¯ point
surface states are characterized by an odd number of with decreasing x. This means that one can tune the
Dirac cones [13], although the Dirac-cone surface states Dirac momentum, albeit in a limited range,by changing
in TIs and TCIs are both produced by bulk band inver- the composition in this system, which is new to a Dirac
sion caused by a strong spin-orbit coupling. Note that, system. Inaddition,wewereabletotracethebulk-band
due to the periodicity of the BZ, one always sees a pair dispersion in the 3D k space, and found that the spec-
ofDiracconesacrossX¯ inTCIs,whichleadstoacharac- tralsignatureisconsistentwiththebandinversionatthe
teristic“doubleDirac-cone”structureinvolvingaLifshitz QPT.
transition at high binding energies [8, 10]. High-quality single crystals of the two end-member
In contrast to the double Dirac-cone signature ob- samples SnTe (x = 1.0) and PbTe (x = 0.0) were grown
2
FIG. 2: (Color online): (a)-(c) Photon-energy dependenceof
near-EF ARPES intensity map around the X¯ point along a
FIG.1: (Coloronline): (a)CrystalstructureofPb1−xSnxTe. cutacrossthekpointwherethebulkVBislocatedclosestto
(b) Bulk fcc BZ and corresponding tetragonal (001) surface EF at each hν, shown for x = 0.2, 0.5, and 0.8, respectively.
BZ. The (110) mirror plane is indicated by the green shaded (d)-(f)Comparison of theenergy distribution curves(EDCs)
aproeian.t (focr) Mxa=pp1i.n0g, 0o.f5A, 0R.P3,EaSndint0e.n0siptlyotattedEFasaarofuunndcttiohne Xo¯f aatndky0.=8,0reesxptercatcivteedly.from theintensity plots for x = 0.2, 0.5,
in-plane wave vector (kx and ky) measured with the He Iα
line(hν =21.218 eV)atT =30K.Theintensityisobtained
by integrating the spectra within ±10 meV of EF. (d) Cor-
responding near-EF ARPES intensity along the Γ¯X¯ cut. (e) and the Photon Factory, respectively. The energy and
Band dispersions derived from the second derivatives of the angular resolutions were set at 10-30 meV and 0.2◦, re-
momentum distribution curves (MDCs) along the Γ¯X¯ cut. spectively. Samples were cleaved in situ along the (001)
The k location of the Dirac point (the Λ¯ point) is indicated crystal plane in an ultrahigh vacuum of 1×10−10 Torr.
bywhitearrowsin(e). (f)Comparisonofthebanddispersions
All the measured samples present shiny mirror-like sur-
extracted from the peak position of the second derivativesof
theMDCs for x = 1.0, 0.5, and 0.3. face after cleaving, indicative of their high quality. The
Fermilevel(EF)ofthe sampleswasreferencedtothatof
a gold film evaporated onto the sample holder.
by the modified Bridgeman method [10], and the solid- Figure 1(c) displays the Fermi-surface (FS) mapping
solution samples (x = 0.2, 0.3, 0.5, 0.6, and 0.8) were aroundthe X¯ pointofthe surfaceBZmeasuredwiththe
prepared by a vapor transport method. High-purity ele- He Iα resonance line (hν = 21.218 eV) for Pb1−xSnxTe
ments of Sn (99.99%), Pb (99.998%), and Te (99.999%) with four representative Sn compositions x including
areused for the growths. The actualSncontentx in the both end members SnTe (x = 1.0) and PbTe (x = 0.0).
grown crystals was measured by the inductively-coupled At x = 1.0, one can immediately notice a dumbbell-
plasma atomic emission spectroscopy and was found to shaped intensity pattern elongated along the Γ¯X¯ direc-
be alwaysconsistentwith the nominalx (whichisshown tion (k direction) with its intensity maxima located
x
in this paper) within ±0.02. All the Pb1−xSnxTe single awayfromtheX¯ point. Thisfeaturearisesfromthedou-
crystals are single domain and show good crystallinity ble Dirac-cone surface state [10] whose Dirac points are
in x-ray Laue analyses. ARPES measurements were located at each intensity maxima (defined the Λ¯ point),
performed with the MBS-A1 and VG-Scienta SES2002 as one can see from the near-EF band dispersion along
electronanalyzers with a high-intensity helium (He) dis- the Γ¯X¯ cut in Fig. 1(d) showing the band maxima on
chargelampat TohokuUniversityand alsowith tunable both sides of the X¯ point. Such an “M”-shaped band
synchrotronlightsatthe beamlineBL28AatthePhoton dispersion is characteristic of the surface state since it
Factory (KEK). To excite photoelectrons, we used the is absent in the bulk-band calculations [9, 11, 17–19].
He Iα resonance line (hν = 21.218 eV) and the circu- The M-shaped dispersion and the dumbbell-shaped FS
larly polarized lights of 75-100 eV at Tohoku University are also observed for x = 0.5, which signifies that the
3
FIG. 3: (Color online): (a), (b) x dependence of near-EF ARPES intensity and corresponding EDCs, respectively, measured
along the ky cut across the bulk VB top that can be accessed with hν = 78.5 eV. (c) Comparison of the EDCs at ky = 0
[thickcurvesin(b)]. Black andredarrows denotethefeatures stemmingfrom thebulkVBandthesurface state,respectively.
(d) Plots of the ∆EVB value (black circles) and the intensity at EF (red circles) as a function of x. The intensity at EF is
normalized to that at theVB top.
system still belongs to the TCI phase, although the in- determined the bulk-band dispersion along the k (wave
z
tensitydistributionaroundEF appearstobeweakerand vector perpendicular to the surface) direction. The rep-
broader than that for x = 1.0. At x = 0.3, however, resentative band dispersions obtained near the Λ¯ point
the observed ARPES intensity looks different, and nei- along the k direction (perpendicular to the Γ¯X¯ direc-
y
ther the dumbbell-shaped FS nor the M-shaped disper- tion) for selected x values (0.2, 0.5, and 0.8) are shown
sion are clearly resolved. This is because the relative in Figs. 2(a)-(c). At x = 0.2, there exists a highly dis-
intensity of the bulk band gains strength and the sur- persive band showing a holelike dispersion whose top is
face band is becoming obscured. Nevertheless, we are dependentonthephotonenergy;itislocatedatthebind-
still able to trace the surface-band dispersion by taking ing energy (EB) of 0.22 eV at hν = 92 eV, but it gradu-
second derivatives of the intensity along the kx direction allymovestowardEF (seewhitearrows)withdecreasing
and, as shown in Fig. 1(e), the M-shaped dispersion is hν. This trend can also be seen in Fig. 2(d) where the
still discernible. This suggests that x = 0.3 is in the energy distribution curves (EDCs) for k = 0 measured
y
vicinity of the QPT but is still on the TCI side. In con- withvariousphotonenergiesfromhν =78.5to92eVare
trast, the spectralfeature for x = 0.0looks substantially overlaid,whichclearlypresentsa peakshift. We found a
different. The near-EF intensity is strongly suppressed similartrendforx=0.5[Fig. 2(b)],althoughtheenergy
and no Dirac-cone surface band is observed, reflecting location of the band is quantitatively different. Similar
the topologically trivial (ordinary) nature of PbTe [10]. to the case of x = 0.2, the prominent band at x = 0.5
The most important feature found in Fig. 1 is the is attributed to the bulk valence band (VB) because it
evolution of the surface state in the TCI phase. In par- shows a finite kz dispersion [see also Fig. 2(e) for the
ticular, as one can see in Fig. 1(e), the k position of the peak shift in EDCs]. On the other hand, the kz disper-
Dirac point in the BZ (i.e., the Λ¯ point which we also sion is not so obvious for x = 0.8 [Figs. 2(c) and (f)],
call Dirac momentum) systematically moves toward the likely due to the stronger surface-bandintensity and rel-
X¯ pointwithdecreasingx(seewhitearrows). Acompar- ative suppression of the bulk-band weight, as we discuss
isonoftheextractedbanddispersionsshowninFig. 1(f) in more detail later.
further reveals that the slope of the Dirac cones (Fermi We have previously elucidated that hν ∼78.5 eV hits
velocity) is essentially unchanged despite the systematic to the rightk valueto probe the L pointofthe bulk BZ
z
shift of the Dirac momentum. This peculiar evolutionof projectedtothe X¯ pointofthe surfaceBZand,hence,is
the double Dirac-cone structure in the TCI phase is the maximally suited to measure the top of the bulk VB in
main finding of this work. the3Dkspace[10](notethatthebulkFSisexpectedto
Since a topologicalphasetransitionmustbe accompa- be locatedatthe L point in this system[18–20]). There-
niedby abandgapclosing,itis usefulto lookforits sig- fore,to elucidate the evolutionofthe bulk VB asa func-
nature in the ARPES data. We have performed photon- tion of x, we have measured all the samples with hν =
energy dependent ARPES measurements for each x and 78.5eValongthekcuttocrossthebulkVBtop. Figures
4
interband scatterings promoted by the proximity of the
bulk band.
Based on the present ARPES experiments, one may
draw a schematic picture of the band evolution in
Pb1−xSnxTe as shown in Fig. 4. The band inversion
and the QPT takes place at x ≈ 0.25 which separates
c
the TCI and trivial phases. The double Dirac-cone sur-
face state is realizedonly in the TCI phase,in which the
separation between the two Dirac cones, and hence the
Dirac momentum, is dependent on x.
FIG.4: (Coloronline): Schematicpictureoftheenergyband Now we discuss the implications of the present exper-
evolution around the X¯ point in Pb1−xSnxTe summarizing imental observations. The location of the Dirac cone in
thepresentARPESexperiments. CB,VB,andSSdenotethe Pb1−xSnxTe is found to be mobile in the k space, and
bulk conduction band, bulk valence band, and surface state, sucha degree of freedomis obviously a unique feature of
respectively.
the mirror-symmetry protected TCIs, in contrast to the
TIs where the Dirac point is bound to the TRIM. Intu-
itively,thedoubleDirac-conestructureinTCIsisaresult
3(a) and (b) show the resulting near-EF ARPES inten- of the hybridization of two Dirac cones, because, on the
sityandcorrespondingEDCs,respectively. Wenotethat (001) surface, two L points (which are each responsible
the features originating from deeper bands are observed for a surface Dirac cone due to the band inversion at L)
at nearly the same binding energy in all the samples, areprojectedontothesameX¯ point;therefore,thesepa-
suggesting that we can compare the positions of the VB rationofthetwoDiracconescanbetakenasameasureof
top without takinginto accountthe x dependence ofthe this “hybridization”. It is not clear at the moment what
chemical potential within the experimental uncertainty determines the strength of this “hybridization”, but the
[21]. present result seems to suggest that it has something to
Atx=1.0,weobserveahighlydispersiveholelikeband do with the size of the bulk band gap. Obviously, our
which is attributed to an admixture of bulk and surface finding is of crucial importance for establishing a con-
bandswithdominantcontributionfromthe surfacestate crete understanding of the physical mechanism to create
near EF [10, 20]. In fact, the EDCs for x = 1.0 exhibit the peculiar double Dirac-cone structure in TCIs.
a clear Fermi-edge cutoff which is a fingerprint of the From the application point of view, the x-dependence
metallic topological surface state. The Fermi edge can oftheDiracmomentummeansthatonecan“tune”itby
be clearlyrecognizeddownto x = 0.5in the EDCs atky controllingthecomposition. Thisisparticularlyusefulin
=0summarizedinFig. 3(c)forallthe xvalues,andthe applicationsrequiringFermi-surfacematchingwithother
intensity near EF at x = 0.3 still suggests its existence; materials. Forexample,toachieveanefficientspininjec-
however, at x = 0.2 the Fermi edge is obviously gone, tion, the Fermi-surface matching [22] between the topo-
which points to the topological QPT at xc ≈ 0.25. This logicalsurface state and a ferromagnetwouldbe crucial.
critical xc can also be inferred from the discontinuous The possibility to tune the location of the Dirac cone
change in the EF intensity across xc ≈ 0.25 [Fig. 3(d)]. means that one may match the Dirac cone to the Fermi
Thanks to this intensity reduction, the bulk VB be- surface of some particular ferromagnet, thereby increas-
comes more visible for smaller x, and the peak in the ing the spin injection efficiency.
EDC due to the bulk VB contribution is marked by In summary,we performedsystematic ARPESexperi-
black arrows in Fig. 3(c) (the bulk band for x = 1.0 is ments on Pb1−xSnxTe to elucidate the evolution of bulk
not distinguished because the surface contributionis too and surface electronic states as a function of Sn concen-
strong). Note that, since those data were measuredwith tration x. We found that the topological QPT between
hν ∼ 78.5 eV, the peak position in Fig. 3(c) gives the the TCI and trivial phases occurs at x ≈ 0.25 in this
c
energylocationofthe VB maximum, andintriguingly,it system and that the evolution of the energy location of
is clearly dependent on x; in particular, the VB maxi- the VB maximum is consistent with the band inversion
mumapproachesEF asxisreducedtowardxc,andthen takingplaceatthisxc. Wealsofoundthattheseparation
it moves away from EF when x passes xc. This trend between the two Dirac cones located across the X¯ point
is summarized in Fig. 3(d) where the energy difference becomes gradually smaller as the QPT is approached in
between the VB maximum and EF, denoted ∆EVB, is the TCI phase.
plotted with error bars to take into account the finite k We thank L. Fu and S. Murakami for helpful discus-
z
and lifetime broadenings. The observed trend is consis- sions, M. Nomura, T. Shoman, K. Honma, H. Kumi-
tent with the band inversion to take place at x ≈ 0.25. gashira,and K.Ono for their assistancein ARPESmea-
c
In view of this result, it is possible that the diminished surements, and T. Ueyama and K. Eto for their assis-
intensityofthesurfacebandnearx iscausedbystronger tance in crystal growth. This work was supported by
c
5
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