supplementary information - nature · dislocations can be seen as dark spots in the topograph (an...

Strain engineering Dirac surface states in heteroepitaxial topological

crystalline insulator thin films

Ilija Zeljkovic, Daniel Walkup, Badih A. Assaf, Kane L. Scipioni, R. Sankar, Fangcheng Chou and Vidya

Madhavan

Supplementary Note 1. STM topographs of SnTe/PbSe(001) heteroepitaxial thin films as a function of thickness

Due to substrate-film lattice mismatch, edge dislocations (crystal defects involving missing half planes of atoms) are

created at the interface during the film growth. The Burgers vector of these dislocations is parallel to the interface

(see Ref. 1 for more details on the heteroepitaxial film growth). STM topographs of our SnTe/PbSe(001)

heteropetixial thin films as a function of film thickness are shown in Supplementary Figure 1.

Supplementary Figure 1. STM topograph film thickness dependence. STM topographs of SnTe/PbSe(001) thin films as a function of thickness for: (a) <1, (b) 2.5, (c) 20 and (d) 40 monolayers (MLs). Each dark line in a topograph corresponds to an edge dislocation formed at the film-substrate interface. Density of these dark lines increases as a function of thickness as expected from previous studies on similar heteroepitaxial thin films 1. Red arrow in (c) denotes an example of a single termination point of an edge dislocation.

Each dark line in the topograph corresponds to an edge dislocation formed at the film-substrate interface, and has

to be terminated at a free crystal interface. This can be seen in our STM topographs as the ending point of dark

lines (Supplementary Figure 2). This termination point also denotes the end (or the start) of the missing half plane

of atoms. In the field-of-view used for QPI and strain analysis in the main text, we observe no such terminating

points (based on Ref. 1 we know they have to be outside of our field-of-view). In turn, this allows us to investigate

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stoichiometric films, and exclude the effects of grain boundaries on the topological surface states which was

problematic in previous studies 2.

Supplementary Figure 2. Termination points of edge dislocations. (a) STM topograph of ~40 ML thin film. Edge dislocations can be seen as dark spots in the topograph (an example of one such dislocation is enclosed by the dashed square). (b) Fourier-filtered section of the topograph in (a) enclosed by the dashed square. Small cross in (b) denotes the start of the edge dislocation.

Supplementary Note 2. Large area STM topographs of ~40 ML SnTe/PbSe(001) thin film

To investigate the large scale morphology of our SnTe thin films, we acquire large STM topographs which

demonstrate that these thin films grow uniformly across microscopic regions of the sample, even across the steps

present in the PbSe substrate (Supplementary Figure 3). Different regions of the sample shown in Supplementary

Figure 3 are separated by tens of micrometers.

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Supplementary Figure 3. Large area STM topographs. (a-d) STM topographs of microscopically large areas of ~40 ML SnTe thin film separated by tens of micrometers.

Supplementary Note 3. QPI experimental data, CEC anisotropy and determination of the Dirac point

Complete experimental data set of dI/dV FTs used to extract SS dispersion in the main text is shown in

Supplementary Figure 4. In order to compare the anisotropy of the constant energy contours (CECs) along Γ-X and

X-M directions in SnTe and Pb0.63Sn0.37Se, we look at the Q1 QPI signature. We pick two representative energies in

each compound such that the position of the QPI peak in the FT image is the same along the X-M direction (blue

arrow in Supplementary Figure 5). Then we compare the positions of the QPI peaks along Γ-X direction, which show

a pronounced difference (yellow arrows in Supplementary Figure 5). Since the QPI signature represents a simple

autocorrelation of the CECs, we can conclude that CECs in SnTe in the energy range below the Lifshitz transition are

significantly more anisotropic than those in Pb0.63Sn0.37Se. Assuming comparable Fermi velocities between the two

compounds, we can further conclude that the two Dirac points symmetric around X are closer together in

momentum space in Pb0.63Sn0.37Se than those in SnTe.





Supplementary Figure 4. Complete QPI data set. FTs of the dI/dV images used to extract the dispersions in the main text.

Supplementary Figure 5. Constant energy contour anisotropy comparison between different TCI materials. QPI peak comparison between SnTe (right) and Pb0.63Sn0.37Se (left). Blue arrow denotes the position of the QPI peak along X-M direction, while the yellow arrows represent the peak position along Γ-X direction. It can be seen that although the peak position is nearly identical along X-M, it differs significantly along Γ-X direction.

Finally, we comment on the extraction of the Dirac point momentum space position. If we take the minimum of the

dI/dV as the approximate energy of the Dirac point (which is also consistent with ARPES 3,4 and our QPI

measurements), from the dispersion shown in Fig. 2(f) of the main text, we can approximate the magnitude of the

Q2 peak to be ~0.91Å-1 at the Dirac point energy. Since within the energy regime close to the Dirac point, Q2 wave

vector arises from scattering between disconnected Fermi pockets close to X point, using simple geometry we can

approximate the momentum space position of each Dirac point to be ~0.06Å-1 away from X, consistent with the

theoretically expected value of 0.05 Å -1 (Ref. 5).





Supplementary Note 4. Determination of local strain and Fourier transform masking

procedure

To determine the strain at the nanoscale, we start with an atomically resolved STM topograph 𝑇𝑇(𝐫𝐫), and apply the

Lawler-Fujita drift correction algorithm with small drift-correction length scale of 2-3 lattice constants 6. This

algorithm applies a transformation 𝐫𝐫 → 𝐫𝐫 + 𝐮𝐮(𝐫𝐫) to the topograph, such that the transformed topograph 𝑇𝑇′(𝐫𝐫)

contains a perfectly periodic atomic lattice.

First, let us examine what effects contribute to the displacement field 𝐮𝐮(𝐫𝐫). In addition to any local strain, 𝐮𝐮(𝐫𝐫) also

includes the effects of piezo nonlinearity, thermal drift, and hysteresis. There may also be small terms linear in 𝐫𝐫,

since the reciprocal lattice vectors (which the Lawler-Fujita algorithm requires as input) are rounded to the nearest

pixel. The above effects are slowly varying and can usually be expanded as a power series in 𝐫𝐫. To remove them, we

first subtract a 3rd-order polynomial sheet fit in x and y from each component of 𝐮𝐮(𝐫𝐫).

Importantly, such normalized displacement field 𝐮𝐮(𝐫𝐫) can be viewed as the displacement vector in elasticity theory 7, whose derivatives give the strain tensor:

𝑢𝑢𝑖𝑖𝑖𝑖 = 12 (𝜕𝜕𝑢𝑢𝑖𝑖

𝜕𝜕𝑥𝑥𝑖𝑖+ 𝜕𝜕𝑢𝑢𝑖𝑖

𝜕𝜕𝑥𝑥𝑖𝑖)

The relative change in the two-dimensional volume element is given by Tr(𝑢𝑢𝑖𝑖𝑖𝑖) = 𝛁𝛁 ∙ 𝐮𝐮. In two dimensions, the

average change in the element of length is 12 𝛁𝛁 ∙ 𝐮𝐮. The “strain maps” shown in Fig. 3(b) and Supplementary Figure

7(d-f) represent the spatially-varying value of - 12 𝛁𝛁 ∙ 𝐮𝐮. The positive values in a strain map denote tensile strain

(lattice constant increases), while the negative values denote the compressive strain (lattice constant decreases).

Supplementary Figure 6. Strain map components. Two dimensional maps of (a) -uxx, (b) -uyy, and (c) -1/2(uxx + uyy) over the identical area of the sample used for analysis in the main text.





To demonstrate that the strain measured is independent of any STM setup conditions, we apply the previously

described algorithm to STM topographs acquired over the identical area of the sample at different bias

(Supplementary Figure 7(a,b)) and obtain nearly identical resulting strain maps (Supplementary Figure 7(d,e)).

Supplementary Figure 7. Bias independence of STM topographs and strain maps. STM topographs of identical 130 nm area acquired at (a) -100 mV and (b) +100 mV. (c) Line cuts through both topographs show that the amplitude of the checkerboard pattern remains the same at the two different bias voltages. (d,e) Strain maps calculated from the topographs in (a,b). (f) Line cuts from points A to B denoted in (d,e), which show that the measured strain pattern is independent of bias polarity.

Next, in order to determine the QPI dispersion as a function of strain, we utilize the following procedure. First, we

derive a normalized reference layer F(r) by taking a topograph (Supplementary Figure 8(a)) and isolating the

checkerboard structural pattern by: (1) smoothing out the atomic corrugations, and (2) Fourier-filtering out the

long-wavelength components larger than 21 nm. The resulting image (Supplementary Figure 8(e)) is then used to

separate all the pixels in the map into three different `masks’ Mi according to the rule:

𝑀𝑀1(𝐫𝐫) = {1, if 𝐹𝐹(𝐫𝐫) is in the bottom 13𝑟𝑟𝑟𝑟 of the values of 𝐹𝐹

0, otherwise

M2 and M3 were defined by selecting the middle and the top 1/3 of the values in F(r), respectively.

Then, we multiply the differential conductance image dI/dV with each one of the masks Mi to create strain-

separated dI/dV maps Gi (Supplementary Figure 8(b-d)). To reduce high wave vector noise generated by sharp

edges, each Mi was convolved with a Gaussian of decay length ~10 Å before multiplication. We also subtract out

the average value of Gi to avoid introducing step functions at the transition from 1 to 0. The final Fourier transforms

of the dI/dV images in Supplementary Figure 8(b-d) are shown in Supplementary Figure 8(f-h).





Supplementary Figure 8. FT masking procedure. (a) Drift-corrected STM topograph acquired at +100 mV. (e) Masking function 𝐹𝐹(𝐫𝐫). (b-d) Strain-separated dI/dV maps Gi for compressive strain, no strain, and tensile strain, respectively. Four-fold symmetrized FTs of images in (b-d) are shown in (f-h).

Supplementary References

1. Springholz & Wiesauer, Phys. Rev. Lett. 88, 015507 (2001).

2. Liu et al., Nat. Phys. 10, 294–299 (2014).

3. Tanaka, Y. et al. Nat. Phys. 8, 800–803 (2012).

4. Xu, S.-Y. et al. Nat. Commun. 3, 1192 (2012).

5. Liu, Duan & Fu, Phys. Rev. B 88, 241303 (2013).

6. Lawler et al., Nature 466, 347–351 (2010).

7. See e.g. Landau and Lifshitz, Theory of Elasticity (1960).





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