www.sciencemag.org/cgi/content/full/science.1239451/DC1
Supplementary Material for
Observation of Dirac Node Formation and Mass Acquisition in a
Topological Crystalline Insulator
Yoshinori Okada, Maksym Serbyn, Hsin Lin, Daniel Walkup, Wenwen Zhou, Chetan
Dhital, Madhab Neupane, Suyang Xu, Yung Jui Wang, R. Sankar, Fangcheng Chou,
Arun Bansil, M. Zahid Hasan, Stephen D. Wilson, Liang Fu, Vidya Madhavan
Published 29 August 2013 on Science Express
DOI: 10.1126/science.1239451
This PDF file includes:
Materials and Methods
SupplementaryText
Figs. S1 to S4
Reference (24)
2
I.
MATERIALS AND METHODS
A.
Sample growth
The growth of Pb1−x Snx Se (nominal x =0.5) crystals was carried out by self-selecting vapor growth (SSVG) involved
two stages. In the first stage, we grew bulk Pb1−x Snx Se crystals by a conventional melt growth. Stoichiometric
amounts of a mixture of Pb, Sn and Se powders with purity of 99.999% were put into a quartz-glass ampoule in an
argon-filled glove box. The ampoule was sealed after being evacuated to about 3 × 10−4 Torr. The sealed ampoule
was slowly heated to 850 ◦ C, holding for 24 hours. Then, the furnace was allowed to slowly cool down to 600 ◦ C at a
cooling rate of 5 ◦ C/min, followed by shutting off the furnace. The obtained Pb1−x Snx Se crystals were subsequently
ground into powder as the source material for the SSVG growth. In the second stage, 10g of Pb1−x Snx Se powder
were carefully placed in the quartz tube and sealed. The sealed ampoule was placed inside a three zone horizontal
furnace with a large temperature gradient. The SSVG process was carried out by keeping the hot zone, the middle
zone and the cold zone at 830 ◦ C, 775 ◦ C and 725 ◦ C, respectively for about 4 to 5 days. Mirror like cubic crystals
were obtained in the middle zone after the growth completion. The nominal concentration was estimated by the
Pb/Se mixture weight ratio before the growth.
B.
Chemical analysis
In this study, after STM/STS measurements, we carefully analyzed the chemical composition of the measured samples. The stoichiometry and actual doping level of Sn in Pb1−x Snx Se sample was determined via EDS measurements
in a JEOL 6340f scanning electron microscope using varying magnification and with spectra collected along multiple
spots in the same sample. All samples were found to be homogeneous, and the measured Sn concentration, x, revealed
an average value of xEDS =0.3404.
The samples were further characterized using x-ray diffraction measurements on a Bruker D2 Phaser diffractometer
at room temperature. The crystallinity of samples used for STM measurements were checked through measuring a
sharp mosaic of (h, 0, 0) single crystal peaks, and then the samples were ground into powder to refine the lattice
constants and verify the absence of impurity phases. The powder pattern was refined using the known cubic space
group Fm-3m (225) with the lattice constants of a=b=c=6.085 Å. As a further check, this lattice constant value was
used to determine the Sn-concentration via the relation: x=50.636-8.2648 × a [24].
The diffraction determined Sn content, x, yields a value xXRD =0.339, which agrees within resolution with the value
obtained from our EDS measurement (xEDS =0.3404).
C.
Typical topographic images
Samples were cleaved at liquid nitrogen temperature in ultra high vacuum (UHV). All the STM/STS measurements
in this study were carried out at ∼5K. Although most of the sample surface typically showed step edges and terraces
(Fig. S1), flat regions could be found on extensive search. We constrained our measurements to flat enough areas of
at least a few thousand angstroms. This is important to avoid LLs modulation from potential created by step edges
because of LL wave expansion effects, as discussed previously [15].
D.
Angle Resolved Photoemission Spectroscopy (ARPES)
Single crystal samples obtained from same batch from which sample for STM/STS measurement was performed,
were characterized by Angle-resolved photoemission spectroscopy (ARPES). ARPES measurements were performed
on PGM beamline at the Synchrotron Radiation Centre (SRC), Madison, Wisconsin. Samples were cleaved in situ
at 20 K with chamber pressure better than 5 x 10−11 Torr at SRC, resulting in shiny surfaces. Typical energy and
momentum resolution was better than 20 meV and 1% of surface Brillouin zone, respectively. (Fig. S2) shows ARPES
dispersion map measured along Γ̄-X̄-Γ̄ momentum direction using photon energy of 18 eV at temperature of 100K.
The Dirac-like surface states are observed to be present along the mirror line (Γ̄-X̄-Γ̄) direction. One Dirac cone lies
inside the first surface Brillouin zone whereas the other one is located outside. No other states are found along any
other momentum directions on the Fermi level. The chemical potential of our sample is found to be about 50 meV
from the Dirac point and the Dirac point is located on the Γ̄-X̄ line at 0.02 Å−1 on the near side of X̄. Our crystals are
n-type, which allows observing the both upper and lower Dirac cones. The observation of metallic Dirac-like surface
3
states along mirror symmetry direction establishes that the sample used in this study is in topological crystalline
insulator phase. It is important to note that the ARPES intensity is observed to be suppressed in the vicinity of
Dirac point. The suppression of intensity can be possibly understood by considering the strong scattering in the
disordered alloy system, crystalline distortion (not detected in the bulk) or ARPES matrix element effects. Basic
trend is consistent with the schematic drawing shown in (Fig. 1D) and experimentally obtained LLs (Fig. 3).
II.
A.
THEORETICAL MODEL
k · p model and rhombohedral distortion
Let us briefly review the basics of k · p model for the surface states of TCI which was recently derived in [6]. The
starting point is the bulk k · p description of the band structure near L points in the Brillouin zone [2].To study the
(001) surface, the projection has to be done. Under this projection, two out of four inequivalent L points are mapped
onto X̄1 point in the two-dimensional Brillouin zone (see Fig. S3). Remaining two points are mapped onto X̄2 point.
Naive projection gives two degenerate species of Dirac fermions in vicinity of X̄1 point,
0
HX̄
(k) = (v1 k1 sy − v2 k2 sx ) ⊗ I,
1
(S1)
with momentum k now being measured from the X̄1 point, and k1,2 ||Γ̄X̄2,1 . The matrices ⃗s act in the space of
Kramers-degenerate states, whereas matrix I is an identity matrix in the flavor space, indicating that we have two
identical copies of Dirac fermions.
In reality, the effective Hamiltonian (S1) has to be supplemented by all possible hybridization terms, which are not
prohibited by symmetry. For this, one has to find all symmetries which leave X̄1 point invariant and their action on the
effective surface Hamiltonian. In total there are three spatial symmetries, present in the system. These are two-fold
rotation around surface normal, C2 , and mirror symmetry with respect to reflection of k1 and k2 axes, M1 : k1 → −k1
and M2 : k2 → −k2 . In addition, there is time-reversal symmetry, denoted as Θ. Physically, mirror symmetry M1
acts only on the electron spin, and leaves L1,2 point intact. The mirror M2 and rotation of π interchanges L1,2 . Using
notations of (S1) the action of symmetries may be summarized as [6]:
C2 : −iτx sz ,
M1 : −isx ,
M2 : −iτx sy ,
Θ : isy K,
(S2)
where ⃗τ Pauli matrices act on the flavor space and time-reversal symmetry includes complex conjugation, K. Note,
that this is the action of symmetries in vicinity of X̄1 point. For the X̄2 the action of symmetries will look different,
but we will not need it here.
Symmetries (S2) restrict the set of possible terms which may hybridize two flavors of Dirac fermions. To zeroth
order in momentum, the only allowed perturbations are τx and τy sx . This gives us effective Hamiltonian for the case
when no additional symmetries are broken to be [6]
HX̄1 (k) = (v1 k1 sy − v2 k2 sx ) + mτx + δsx τy ,
(S3)
with m and δ controlling the strength of these hybridization terms. The band structure resulting from this Hamiltonian
can be easily visualized in three steps. First, if we have only the first term in (S3), it would give us two degenerate
Dirac cones. The mτx term splits this degeneracy, by shifting the origin of Dirac cones to ±m in energy. Nevertheless,
two cones at zero energy are degenerate on the ring. The effect of the last term in (S3) is to lift this degeneracy
everywhere except for the points with k1 = 0. Thus resulting band structure has two emergent Dirac cones on the
axis k1 = 0, and Van-Hove singularities located on the axis k2 = 0, see Fig. S3.
Other perturbations not captured by the present model may lead to the breaking of some symmetries on the surface,
thus allowing for extra terms in the effective Hamiltonian. Let us consider the effect of mirror symmetry breaking,
assuming that the mirror symmetry M2 and rotational symmetry C2 are broken. Physically, such symmetry breaking
may arise from rhombohedral distortion along [110] direction in the real space (see (Fig. 4) in the main text). Then,
in addition to the perturbation τx , one may add term τz to the effective Hamiltonian in vicinity of X̄1 point in BZ,
′
HX̄
(k) = (v1 k1 sy − v2 k2 sx ) + mτx + δsx τy + ϵτz ,
1
(S4)
where ϵ controls the strength of the symmetry breaking. At the same time, the effect of the breaking of M2 symmetry
for the X̄2 point is equivalent to the breaking of M1 symmetry for the X̄1 point. Thus, band structure at X̄2 can be
4
deduced from (S3) by allowing for the M1 symmetry breaking. When we exclude M1 from the symmetries list (S2),
term proportional to sz τy becomes allowed, thus we have:
′′
HX̄
(k) = (v1 k1 sy − v2 k2 sx ) + mτx + δsx τy + ∆sz τy ,
1
(S5)
Note, that there is no symmetry relating ϵ and ∆ to each other, thus in principle these are two different parameters.
The effect of the perturbation, controlled by corresponding ϵ (∆) is very different in vicinity of X̄1 (X̄2 ) point. The
term ϵτz in (S4) does not open a gap in the low energy Dirac points, as they are located on the axis k2 = 0. The
gapless nature of Dirac point in such case is protected by the presence of mirror symmetry M1 , which is not broken.
In contrary, for the X̄2 point, the breaking of M2 symmetry leads to a gap opening at Dirac points, see Fig. S3
B.
Semi-classical picture
Within the semi-classical picture, the area of constant energy contour (CEC) in k-space (Sn ), at the energy corresponding to the given LL with index n at magnetic field B can be expressed as
2πe
(n + γ)B.
(S6)
~
The factor γ is zero for Dirac fermions with linear band dispersion. We plot the Sn calculated from experimentally
obtained peak positions using Eq. (S6) with γ = 0, along with CEC area theoretically calculated from the band
structure. The resulting experimental and theoretical curves are extraordinarily similar (Fig. 3D). Comparing the
fits to the dispersion confirms that our indexing for LLs and the discontinuity represents the saddle point energy in
the dispersion.
Sn =
C.
Landau levels spectrum and band structure parameters
Starting from effective Hamiltonians (S4)-(S5) for vicinity of X̄1,2 points, we use numerical procedure to obtain
Landau levels spectrum. The detailed description of the procedure can be found elsewhere. In short, operators
vy k2 ± ivx k1 are replaced by ladder operators, π, π † acting in the basis of Landau level orbitals. Their matrix elements
are given by
π|n⟩ =
~v̄ √
2n|n − 1
ℓB
(S7)
~v̄ √
2(n + 1)|n + 1
(S8)
ℓB
√
√
where the magnetic length ℓB = ~/(eB), and v̄ = v1 v2 is the geometric mean of Fermi velocities in both
directions. Using this mapping, Hamiltonian is turned into a matrix of size 4Λ × 4Λ, where Λ = 100 is the cutoff
used in the simulations for this paper. Parameters for the Landau levels structure are obtained from fitting to
experimentally observed LLs (Fig. 3).
If one simulates LLs using the Hamiltonian (S3) one can reproduce all basic features, except for the non-dispersive
∗
∗
labeled as E±
on Fig. 3 in the main text. The structure of two non-dispersive features E±
located below and above
another non-dispersive peak labelled as EDP 1 , can be naturally explained by the mirror symmetry breaking discussed
above. Qualitative effect of such symmetry breaking on the band structure is illustrated in the top panels of Fig. S4.
In principle, there are two parameters in Eqs. (S4) and (S5) controlling this symmetry breaking. Parameter ∆ controls
the magnitude of the gap for Dirac cones near X̄2 point, whereas ϵ weakly changes the band structure compared to
the case without distortion. For simplicity we assumed that ϵ = ∆, and corresponding Landau levels fan diagram is
∗
shown in the panel D of Fig. S4. The value of ∆ can be readily deduced from the distance between E±
and EDP 1
features in the data to be of order of 10 meV. Fitting of experimental data with Landau levels we find the parameters
in the effective Hamiltonian to be
√
m = 0.055 eV,
δ = 0.04 eV,
∆ = 0.011 eV.
(S9)
v̄ = v1 v2 = 2.60 eV · Å,
π † |n⟩ =
It is these parameters which were used in the panel D of Fig. S4, as well as in the Fig. 4 of the main text. Finally,
to confirm small effect of non-zero ϵ, the panel E of Fig. S4 compares Landau levels for the case without symmetry
breaking with those emerging from (S4) with ϵ = 0.011 eV. The difference is indeed very small and becomes visible
only at higher energies.
5
low
high
70 nm
low
high
30 nm
low
high
90 nm
FIG. S1: Typical topographic images of the sample Pb1−x Snx Se taken over a wide area.
FIG. S2: High-resolution ARPES dispersion of the sample used in this study. The cut shown is along mirror symmetry Γ-X-Γ
direction measured with photon energy of 18 eV at temperature 100K.
6
FIG. S3: Top (bottom) plots show band structure for the vicinity of X̄1 (X̄2 ) point. In the vicinity of X̄2 , breaking of the Mx
mirror symmetry opens up a gap between Dirac points. Whereas, gapless Dirac points near X̄1 are protected by Mx , thus the
effect of distortion on the low energy band structure is weak.
7
100
100
100
50
50
50
0
E - EDP1 (meV)
150
E - EDP1 (meV)
150
E - EDP1 (meV)
150
0
0
-50
-50
-50
-100
-100
-100
-150
-150
-150
0
2
4
6
B(T)
8
0
2
4
6
B(T)
8
0
2
4
6
B(T)
8
FIG. S4: Top panels show the cartoon of band structure without (panel A) and with mirror symmetry breaking (panel B).
Bottom panels show Landau levels fan diagrams resulting from the corresponding band structure. For very small magnetic field
the Landau levels in vicinity of EDP 1 are not well resolved due to finite cutoff Λ = 100 imposed in the numerical procedure.
Panels C and D show Landau levels without and with mirror symmetry breaking. When the mirror symmetry is broken, the gap
opens only in vicinity of X̄2 point. Although low energy Dirac cones near X̄1 point remain gapless when the mirror symmetry
is broken, the band structure is weakly modified at higher energy. This leads to hardly noticeable difference between light blue
(mirror symmetry is broken) and dark-blue (no symmetry breaking) Landau levels shown in the panel E.
References and Notes
1. L. Fu, Topological crystalline insulators. Phys. Rev. Lett. 106, 106802 (2011).
doi:10.1103/PhysRevLett.106.106802 Medline
2. T. H. Hsieh, H. Lin, J. Liu, W. Duan, A. Bansil, L. Fu, Topological crystalline insulators in the
SnTe material class. Nat.Commun. 3, 982 (2012). doi:10.1038/ncomms1969 Medline
3. Y. Tanaka, Z. Ren, T. Sato, K. Nakayama, S. Souma, T. Takahashi, K. Segawa, Y. Ando,
Experimental realization of a topological crystalline insulator in SnTe. Nat. Phys. 8, 800–
803 (2012). doi:10.1038/nphys2442
4. S. Y. Xu, C. Liu, N. Alidoust, M. Neupane, D. Qian, I. Belopolski, J. D. Denlinger, Y. J.
Wang, H. Lin, L. A. Wray, G. Landolt, B. Slomski, J. H. Dil, A. Marcinkova, E.
Morosan, Q. Gibson, R. Sankar, F. C. Chou, R. J. Cava, A. Bansil, M. Z. Hasan,
Observation of a topological crystalline insulator phase and topological phase transition
in Pb(1-x)Sn(x)Te. Nat. Commun. 3, 1192 (2012). doi:10.1038/ncomms2191 Medline
5. P. Dziawa, B. J. Kowalski, K. Dybko, R. Buczko, A. Szczerbakow, M. Szot, E. Łusakowska,
T. Balasubramanian, B. M. Wojek, M. H. Berntsen, O. Tjernberg, T. Story, Topological
crystalline insulator states in Pb(1-x)Sn(x)Se. Nat. Mater. 11, 1023–1027 (2012). Medline
6. J. Liu, W. Duan, L. Fu, Surface States of Topological Crystalline Insulators in IV-VI
Semiconductors, arXiv:1304.0430.
7. I. M. Lifshitz, Sov. Phys. JETP 11, 1130 (1960).
8. R. S. Markiewicz, Phase separation near the Mott transition in La 2-x Sr x CuO 4. J. Phys.
Condens. Matter 2, 665–676 (1990). doi:10.1088/0953-8984/2/3/015
9. W. Kohn, J. M. Luttinger, New mechanism for superconductivity. Phys. Rev. Lett. 15, 524–
526 (1965). doi:10.1103/PhysRevLett.15.524
10. T. M. Rice, G. K. Scott, New mechanism for a charge-density wave instability. Phys. Rev.
Lett. 35, 120–123 (1975). doi:10.1103/PhysRevLett.35.120
11. M. Fleck, A. M. Oleś, L. Hedin, Magnetic phases near the Van Hove singularity in s-and
dband Hubbard models. Phys. Rev. B 56, 3159–3166 (1997).
doi:10.1103/PhysRevB.56.3159
12. J. González, Konh-Luttinger superconductivity in graphene. Phys. Rev. B 78, 205431 (2008).
doi:10.1103/PhysRevB.78.205431
13. R. Nandkishore, L. Levitov, A. Chubukov, Chiral superconductivity from repulsive
interactions in doped graphene. Nat. Phys. 8, 158–163 (2012). doi:10.1038/nphys2208
14. See supplementary materials on Science Online.
15. Y. Okada, W. Zhou, C. Dhital, D. Walkup, Y. Ran, Z. Wang, S. D. Wilson, V. Madhavan,
Visualizing Landau levels of Dirac electrons in a one-dimensional potential. Phys. Rev.
Lett. 109, 166407 (2012). doi:10.1103/PhysRevLett.109.166407 Medline
16. G. Li, Y. Eva, Andrei, Observation of Landau levels of Dirac fermions in graphite. Nat.
Phys. 3, 623–627 (2007). doi:10.1038/nphys653
17. T. Hanaguri, K. Igarashi, M. Kawamura, H. Takagi, T. Sasagawa, Momentum-resolved
Landau-level spectroscopy of Dirac surface state in Bi2Se3. Phys. Rev. B 82, 081305
(2010). doi:10.1103/PhysRevB.82.081305
18. G. Li, A. Luican, J. M. B. Lopes dos Santos, A. H. Castro Neto, A. Reina, J. Kong, E. Y.
Andrei, Observation of Van Hove singularities in twisted graphene layers. Nat. Phys. 6,
109–113 (2010). doi:10.1038/nphys1463
19. F. Zhang, C. L. Kane, E. J. Mele, Topological Mirror Superconductivity. arXiv:1303.4144.
20. F. D. M. Haldane, Model for a quantum Hall effect without Landau levels: Condensed-matter
realization of the “parity anomaly”. Phys. Rev. Lett. 61, 2015–2018 (1988).
doi:10.1103/PhysRevLett.61.2015 Medline
21. K. Adouby, C. Perez-Vicente, J. C. Jumas, R. Fourcade, A. Abba Toure, Structure and
temperature transformation of SnSe. Stabilization of a new cubic phase Sn4Bi2Se7. Z.
Kristallogr. 213, 343–349 (1998). doi:10.1524/zkri.1998.213.6.343
22. M. Iizumi, Y. Hamaguchi, K. F. Komatsubara, Y. Kato, Applicability of Vegard's Law to the
PbxSn1−xTe Alloy System. J. Phys. Soc. Jpn. 38, 443 (1975). doi:10.1143/JPSJ.38.443
23. M. Snykers, P. Delavignette, S. Amelinckx, The domain structure of GeTe as observed by
electron microscopy. Mater. Res. Bull. 7, 831–839 (1972). doi:10.1016/00255408(72)90133-X
24. A. Szczerbakow, H. Berger, Investigation of the composition of vapour-grown Pb1-xSnxSe
crystals ( x ≤ 0.4) by means of lattice parameter measurements. J. Cryst. Growth 139,
172–178 (1994). doi:10.1016/0022-0248(94)90042-6