Articleshttps://doi.org/10.1038/s41565-017-0022-x

Strain distributions and their influence on electronic structures of WSe2–MoS2 laterally strained heterojunctionsChendong Zhang   1,2*, Ming-Yang Li3,4, Jerry Tersoff5, Yimo Han6, Yushan Su1,7, Lain-Jong Li   3, David A. Muller   6,8 and Chih-Kang Shih1*

1Department of Physics, University of Texas at Austin, Austin, TX, USA. 2School of Physics and Technology, Wuhan University, Wuhan, China. 3Physical Sciences and Engineering Division, King Abdullah University of Science and Technology, Thuwal, Saudi Arabia. 4Research Center for Applied Sciences, Academia Sinica, Taipei, Taiwan. 5IBM Research Division, T. J. Watson Research Center, Yorktown Heights, NY, USA. 6School of Applied and Engineering Physics, Cornell University, Ithaca, NY, USA. 7School of the Gifted Young, University of Science and Technology of China, Hefei, Anhui, China. 8Kavli Institute at Cornell for Nanoscale Science, Cornell University, Ithaca, NY, USA. *e-mail: cdzhang@whu.edu.cn; shih@physics.utexas.edu

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1

Strain distributions and their influences on electronic structures of

WSe2-MoS2 laterally strained heterojunctions

Chendong Zhang, Ming-Yang Li, Jerry Tersoff, Yimo Han, Yushan Su，Lain-Jong Li,

David A. Muller, and Chih-Kang Shih

Supplementary Section 1: Thickness measurements on isolated ML MoS2 and WSe2

flakes

Supplementary Section 2: Determination of strain tensor from moiré pattern

Supplementary Section 3: Detailed analysis of the moiré patterns formed between

TMDs and HOPG

Supplementary Section 4: Statistics of the density of misfit dislocation

Supplementary Section 5: Measurements of the quasiparticle band gaps of MoS2

Supplementary Section 6: Decay constant spectroscopy of the valence band edge

Supplementary Section 7: 1D interface state imaged by a Kappa mapping

Supplementary Section 8: Conduction band bending in WSe2 region

2

Supplementary Section S1: Thickness measurements on isolated ML MoS2 and

WSe2 flakes

Figure S1 a and c are the STM images of isolated monolayer MoS2 and WSe2,

respectively. b and d are the height profiles taken along the dashed lines in a and c,

respectively. Scanning parameters for a and c are 3.0 V, 10 pA.

Figure S1 a and c are the STM images of isolated monolayer MoS2 and WSe2 grown on HOPG,

respectively. The sample biases for Figs. S1a and S1c are the same as the one used in Fig. 1b. Fig.

S1b and S1d show the height profiles along the paths marked in Fig. S1a and S1c. The apparent

height difference of ~ 1 Å is consistent with the value measured on the WSe2-MoS2 lateral

heterojunctions.

3

Supplementary information S2: Determination of strain tensor from moiré

pattern

i) The general principle behind the formation of moiré patterns has been well established (ref.

S1-S4). The periodicity of a moiré pattern depends on the lattice constants of both layers (L1 and

L2) and the rotation angle between themIn the case of a moiré pattern created by the

superposition of two 2D lattices of the same symmetry with slightly different lattice constants, its

periodicity follows the equation:

1 2

2 2

1 2 1 22 cos

L L

L L L L

(S1)

Similarly, the rotation angle of moiré pattern with respect to L1 lattice can be given by:

1

2 2

1 2 1 2

sinsin

2 cos

L

L L L L

(S2)

These expressions also hold for the case of square lattices and rectangular lattices with same

aspect ratio. In the case of rotationally aligned case ( = 0), the moiré pattern periodicity is

simplified further as

1 2

1 2| |

L L

L L

or 2L

(S3)

with the relative mismatch 1 2

1

| |L L

L

4

This expression tells us that there exists an amplification factor between moiré periodicity and

the atomic lattice is inversely proportional to the relative mismatch, AL = 1/.

At a finite but very small rotational angle, i.e. << we can make the approximation that

2cos 1 / 2 . In addition, with 1 2

1

| |L L

L

, we make another approximation that

221 2

1 2

( )L L

L L

. With these two approximations, one can express the moiré pattern periodicity and

the rotation angle as

2

2

2(1 )

2

L

(S4)

Figure S2 Simulations of moiré patterns formed by two lattices with 4% mismatch. a-c

are for three types of lattices: a, square lattices, b, rectangular lattices and c, hexagonal

lattices. The ratio between the constants of X(blue) and Y(red) lattices is 1:0.96 in all

three cases. The middle and right columns show the simulations when rotational angle

between X and Y are 0° and 1°, respectively.

5

2

2

sinsin (1 )

2

(S5)

In the case that << , these two expressions are further simplified to 2 sinand sin

L

.

Namely, the moiré pattern periodicity and rotation angle are both amplified by the same factor of

~ 1/. Depending on the value of /, an additional correction factor of /2 may be needed.

Shown in Fig. S2 are simple simulations of moiré patterns for lattices with 4% mismatch for three

different kinds of lattices (square, rectangular and hexagonal) at = 0 and = 1°, respectively. In

all three cases, one obtains an amplification AL = 24 for = 0 and AL = 22 and 𝜑 = 24° for = 1°.

Even without the correction factor, the approximation of AL = 1/ for both periodicity and

rotational angle is precise to about 10% accuracy. With the correction factor of /2, the accuracy

improves to about 1%. Note that, if the rectangular lattices (𝒂 ⃗⃗ ⃗ = 𝑎�̂� and 𝒃 ⃗⃗ ⃗ = 𝑏�̂�) have different

aspect ratios a/b, equations S3 and S4 should also hold for each direction separately, just like the

anisotropic strain in our case. As we discussed below, the rotational angle amplification expression,

can actually be extended to the case of small shear angle distortion.

ii) Rectangular lattices with a small shear angle

We next consider the effect of a small shear strain on the moiré pattern, in particular for the

rectangular lattices. Fig. S3 shows the simulation using the same X and Y lattices as the ones in

Fig. S2b. When the Y lattice is sheared by 1 degree along a direction, it becomes a parallelogram

but not a rectangle. The resulting moiré pattern as shown in Fig. S3b is also a parallelogram but

with a much larger shear angle of about 24 degrees. Note that if we keep the primitive vector

unchanged, namely the a and b are still perpendicular, the periodicities in Y lattice along the a and

b directions remain the same as the case before shear. Therefore, the a and b as labeled for the

sheared moiré pattern should also be the same with the case before shear.

The accurate formula for the amplification of shear angle can be derived as following. As

discussed in ref. S1, the derivations of equations (S1) and (S2) are based on a one-dimensional

model, i.e. the overlap of two sets of parallel lines with different interspaces. Here, the moiré

pattern formed by rectangular lattices can be considered as the combination of two of such 1D

model which consist of the lines along perpendicular directions, i.e. line sets (a) and (b) in Fig.

S3a.

6

Therefore, the shear of lattice Y in Fig. S3a is actually equivalent to the situation that only the

line set (b) rotates by an angle of . The shear angle of moiré pattern should follow the equation

Sas

1

2 2

1 2 1 2

sinsin

2 cos

a

a d a d

,

and 2 1/2 1 2

2 2

1 2 1 2

| | coscos (1 sin )

2 cos

a a

a d a d

Figure S3 Simulation of the moiré pattern with a shear strain in Y lattice. a is a schematic showing

the lattices used for simulation. The rectangular lattices are identical with the ones used in Fig. S1b

except that the Y lattice is now sheared by 1˚ along the a direction. The sheared rectangular lattices

can be considered as the combination of two line sets (a) and (b). Therefore, the shear of Y is equivalent

to the rotation of line set (b). The resulting moiré pattern is shown in b. The black parallelogram

represents the supercell which is also but with a much larger shear angle of about 24 degrees.

7

Here, 2 2cosd a is the interspace of the line set (b) after the rotation. From the above

equations, one can get:

1 2

1

| |tantan , with a

a

a a

a

(S6)

Note that the amplification of shear angle follows the amplification along the a direction, 1/a.

This result comes naturally since the real lattice shear will lead to a small displacement along a.

iii) In the studies of the strained MoS2 lying on strain-free WSe2, we choose a centered

rectangular lattice with lattice vectors 𝒂 and 𝒃 to represent the hexagonal lattice of ML-TMD,

where 𝒂 is along the zigzag direction in parallel to the interface and the 𝒃 is along the

perpendicular arm-chair direction. As demonstrated in Fig. S2c, the strain-free moiré pattern can

also be treated as a centered rectangular lattice with the periodicities a and b.

Figure S4 The simulation with a gradient for both tensile and compressive strain is shown in a. It

resembles the experimental image (Fig. S3b) well. The dashed lines in a and b, which are drawn in

order to guide the eye, represent the gradual decay of the tensile strain along a direction. c is a

schematic of the supercell marked by red dashed line in b. All geometric parameters are labelled as

shown. The 2D tensor matrix for the center spot (red) can be determined with these parameters.

8

Due to the lattice mismatch at the interface, there are anisotropic strains in the MoS2 region, i.e.

tensile strain along a and compressive strain along b. Moreover, these anisotropic strains have a

spatial gradient which will further induce shear distortions. Fig. S4a is a simulating moiré pattern

where aa is an exponential decay with a decay length of 100 nm and bb = 0.3aa (to simplify the

simulation). It resembles the experimental image (Fig. S4b) reasonably well. As labeled by red

dashed lines in Fig. S3b, the centered rectangular lattice seen in the case of uniform strain now

becomes a centered trapezoid. Shown in Fig. S4c is a schematic of the centered trapezoid unit cell

of moiré pattern. From its geometry parameters (as labeled in Fig. S4c), one can determine the

averaged second-order strain tensors aa ab

ba bb

at its center point.

Note that the zig-zag lines of the moiré spots always remain parallel with the WSe2-MoS2

interface. Therefore, the periodicities of moiré pattern along a and b directions (a and b) should

simply follow the equation (S3) with no need for the rotation angle. The strain tensors along a and

b directions can be written as:

1W aa

Mo W a

a

a a

, 1W b

b

Mo W b

b

b b

(S7)

, , and W W Mo Moa b a b are the lattice parameters of strain-free ML- WSe2 and MoS2. A positive

value of the tensors means a tensile strain while the negative one means a compressive strain.

The ab = ba is essentially the tangent of the shear angle in MoS2 atomic lattices. With the

equation (S6), ab can be determined based on the moiré lattice shear angle and the atomic lattice

mismatch along a direction, namely tanW Mo

ab ba

W

a a

a

. For a centered trapezoid

supercell, one can take the average of 1 and 2 to approximately represent the ab at the center

point. Similarly, the average of a1 and a2 is used to deduce the a at the center point. Using the

above method, we can determine all the second-order strain tensors at each moiré spot.

Supplementary Section 3: Detailed analysis of the moiré patterns formed

between TMDs and HOPG

9

Due to a larger difference in lattice constants between graphite (2.46 Å) and TMDs (~ 3.2 Å), the

moiré pattern formed by TMD on graphite provides a magnification factor of only ~ 3.5. Although

the direct visualization of strain distribution is difficult, the relative small magnification factor

provides a finer spatial resolution which can help us to take a glance at the strain near the interface.

In Fig. S5, labeled near the interface are two length marks (black dashed line V and the red dashed

line) spanning across five lattice spacings along the zigzag direction. Within the experimental

uncertainty, the lattice constant along the zigzag direction across the interface is unchanged,

consistent with a seamless interface reported previously using TEM. At ~ 7.5 nm away from the

interface in MoS2, however, the same length mark (white) failed to span the same number of lattice

spacings. Instead, a 7 ± 2 % longer length mark is needed, which corresponds to a reduction of 2.2

± 0.5% in the atomic lattice along the zigzag direction. In other words, it implies that a 3.8% tensile

strain along the zigzag direction at the interface quickly decays to ~ 1.6% in about 7.5 nm.

Similar measurements can be taken on the WSe2 region. In Fig. S5, we show five black length

marks (labeled from I to V) each spanning over five moiré lattice spacings at difference distances

from interface in the WSe2 region. In contrast to the obvious change in MoS2, all five length marks

have the same length within the experimental uncertainty indicating that the WSe2 lattice has

negligible distortion.

Figure S5 Zoomed-in image of the straight interface segment as labelled in Fig. 1c. The moiré

pattern with a period of ~ 1 nm formed with the underlying graphite can be seen. All length marks

are labelled along the zigzag interface with the same length. The red one is near the interface at

the MoS2 side. The black ones (I to V) locate at the difference distance from the interface at the

WSe2 side.

10

Supplementary Section 4: Statistics of the density of misfit dislocation

Here, we show the statistical results of the density of misfit dislocation based on large field-

of-view atomic resolution ADF-STEM images and STM images. We applied geometric phase

analysis (GPA) (ref. S5) to the atomic resolution ADF-STEM images to easily locate misfit

dislocations, which are shown as the dipoles in the strain field map. By measuring the distance

between neighboring dislocations, we achieved the histogram as shown in Fig. S6b. Although,

the counting here includes two types of dislocation (0˚ and 60˚ as shown in Fig. 2d), our STEM

observations show that the overwhelming majority have the burger’s vector being parallel to the

interface (0˚), only a few are at 60 ˚ to the interface.

The analysis of STM images in Fig. S6c is based on the assumption that each adsorbate at the

interface (the bright spots in Fig. 1b) corresponds to a misfit dislocation. Note that, in STM work,

we are dealing with smaller sample size that the WSe2 core ranges between 250 – 300 nm with the

rim ranging from 90 – 120 nm. For STEM sample, WSe2 is usually ~ 1um, and the MoS2 rim is >

300 nm. Moreover, the STEM sample is grown on sapphire substrate, which requires slightly

different synthesis parameters with the ones on HOPG substrate. These might cause the minor

difference in the dislocation densities as shown in Fig. S6b and S6c.

Figure S6 a, the overlay of ADF-STEM image and its strain map indicating the WSe2/MoS2

interface and the periodic dislocations along the interface. b, c, Histograms of the spacing

between neighboring dislocations from STEM and STM images, respectively.

11

Supplementary Section 5: Measurements of the quasiparticle band gaps of

MoS2

Figure S7 a and b are the color rending of (Z/V)I and decay constant mapping for the

valence band of MoS2. The total length is 120 nm with a step size of 3.5 nm.

A constant I spectroscopy mapping for the VB of MoS2 is taken on a long length scale. Fig. S7a

and S7b are the color rending of (Z/V)I and the decay constant mapping for the valence band

of MoS2, respectively. The location of the interface is marked by the black arrows. The strain

induced states and point are labelled as shown. It is interesting that except for the bending near

the interface, the state at the pointalso shows a slight alteration on a long length scale with a ~

0.05 eV shift over 100 nm.

12

Figure S8 a, STM image showing the spatial locations where the (Z/V)I spectra given in b are

carried out. c, Separated kappa measurements at the same region with their distance from the

interface labeled.

A constant I spectroscopy mapping for the CB of MoS2 on a long length scale is shown in Fig.

S8b. An STM image showing the spatial locations where the (Z/V)I spectra is given in Fig. S8a.

The corresponding kappa spectra are shown in Fig. S8c with their distance to the interface labelled.

In our previous measurements on isolated ML-MoS2, there is two prominent peaks located at +0.3

and +0.5 eV in the (I/V)I spectroscopy, and the CBM that is located at around +0.3 eV

corresponds to a peak in kappa (K point) while the second lowest one, which is located at around

+0.5 eV, can be determined as the Q point (ref. 34). However, only one peak around +0.45 ± 0.04

eV is observed on the MoS2 region of the lateral HJ. As shown in Fig. S8b and S8c, this single

peak corresponds to a rise of kappa and a slight shift towards the Fermi level when moving far

away from the interface. This behavior is similar to that of the states at the point in the valence

13

band. We tentatively think the conduction band undergoes influence from strain as well, but the

CBM is located at the corner of the Brillouin zone all the time in our measurements.

Figure S9 a and b show the criteria to determine the energy locations of band edges. b, The energy locations of

the VBM and CBM of MoS2 as functions of the distance from the interface. The discrete experimental results are

plotted by solid markers. The experimental errors for energy locations of band edges in c are ± 0.03 eV (see ref.

34).

As we discussed recently in ref. 34, the band edge should be determined at the mid-point of the

transition from the graphite to the TMD states. We use spectrum 15 in Fig. 4d and spectrum 17 in

Fig. 4c as examples to illustrate this criteria (see Fig. S9a and b). Based on the spectroscopic

measurements as shown in Fig. S7 and S8, the energy location of the CBM and VBM of MoS2 as

a function of the distance from the interface can be plotted in Fig. S9c. Therefore, the spatial

dependent quasiparticle gap can be extracted like shown in Fig. 3g.

14

Supplementary Section 6: Decay constant spectroscopy of the valence band

edge

Figure S10 a, Colour coded rendering of mapping of the valence band across the interface. An

STM image shown above displays the path line which the spectroscopy was taken along. The total

length is about 25 nm with a step size of 1.3 nm. The pink dashed line in a represents the estimated

energy location of the K point in strain-free MoS2. b, the selective subsets of the spectra. The

stabilized bias and set-point current: -2.3 V, 8 pA.

The decay constant is given by 2

||

2

2b

m k , and can reflect the difference in parallel momentum

k|| of electronic states, and can be used to identify the critical points in the Brillouin zone of these

states (see ref. 34 for details). The experimental measurement of k has been introduced in the

Methods section. As the (I/V)I mapping of the valence band is discussed in Fig. 4a, shown in

Fig. S10a is the corresponding k mappingtaken along the same path line. The selected individual

kappa spectra are shown in Fig. S10b. The numbers on the spectra refer to their positions in the

15

complete set as labeled in the corresponding topography image in Fig.S10a. The strain induced

states (SIS) correspond to a rising of kappa (as labeled in spectrum 13 - 15), indicating that SIS,

which is the VBM, might still be located at the corner of the distorted Brillouin zone. An additional

peak at ~ -1.9 V has been seen in the (I/V)I spectrum 12 (Fig. 4c) which is attributed to the

interface state. The corresponding kappa measurement shows that this interface state appears as a

rising of kappa value (spectra #12 in Fig. S10b). Due to the overlapping with the dip for the

states, the maximum kappa occurs at ~ -1.91± 0.03 eV.

Supplementary Section 7: 1D interface state imaged by a Kappa mapping

Figure S11 a and b, topography image and corresponding dI/dZ (kappa)

image taken at -1.91V for a straight interface segment. The kappa image was

taken using a lock-in amplifier with a Z-modulation amplitude of 10 pm and

a modulation frequency of 914 Hz which is faster than the feedback time

constant. c, a kappa profile along the pathway labeled in b.

As shown in spectrum 12 of Fig. S10b, the interface states appear as a peak of kappa at -1.91V.

In Fig. S11b, a kappa image (dI/dZ mapping) was taken together with the regular topography (Fig.

S11a) by applying a Z modulation of 0.01nm with a lock-in amplifier (Methods). Here, the

interface states are clearly visualized as a 1D bright stripe with a width of 1nm. Fig. S11c is a

16

kappa profile along the path in Fig. S11b. The slight dropping of kappa from 4 to 6 nm in Fig.

S11c is due to the bending of the states near the interface.

Supplementary Section 8: Conduction band bending in WSe2 region

Figure S12 a, color rendering of (Z/V)I mapping for the conduction band of WSe2. The total length is

18.5 nm with a step size of 0.7nm. The location of the interface is labeled by a black arrow. The top axis

shows the spectra index. b and c, the selected individual (Z/V)I and corresponding kappa spectra. The

index is labeled for each spectrum.

The conduction band bending of WSe2 is not observed clearly in Fig. 4b due to the limited

measuring range. Here, Fig. S12a is a color rendering of (Z/V)I mapping zoomed on the

conduction band of WSe2. The selected individual (Z/V)I and corresponding kappa spectra are

shown in Fig. S12b and S12c, respectively. The numbers on the spectra refer to their positions in

the complete set as labeled on the top-axis of Fig. S12a. It is obvious here to see a conduction band

bending of a magnitude of ~ 0.1 eV. The depletion length is around 5 nm which is much shorter

than previous reported value acquired from transport measurements (ref. 14). This may be due to

the strong screening effect from the conducting substrate other than the insulating sapphire

substrate used before. In addition, a weak state near +1.35 eV which belongs to the state near the

M point (ref. 34) is observed in Fig. S12a. Note that, compared with ref. 34, the energy locations

17

for all critical points ( K, Q and M*) reported in this study are rigidly shifted by ~ -0.05 eV. This

could be the result of minor variations of the tip work function or the surrounding environment.

Note that we have reported that the CBM of an isolated ML-WSe2 flake is located at the Q point

(slightly lower than the K point) in ref. 34. As shown in Fig. S12c, this is also observed in the

WSe2 region of lateral HJs all the way up to the interface.

18

Supplementary references:

S1. Oster, G., Wasserman, M. & Zwerling, C. Theoretical Interpretation of Moiré Patterns. J. Opt. Soc.

Am. 54, 169-175(1964).

S2. Amidror, I. Moiré patterns between aperiodic layers: quantitative analysis and synthesis. J. Opt. Soc.

Am. A 20, 1900-1919 (2003).

S3. Amidror, I. The Theory of the Moiré Phenomenon. (Springer-Verlag London, 2009).

S4. Choi, J., Wehrspohn, R. B. & Gösele, U. Moiré Pattern Formation on Porous Alumina Arrays Using

Nanoimprint Lithography. Adv. Mater.15 (2003).

S5. Hytch, M.J., Snoeck, E., Kilaas, R. Quantitative measurement of displacement and strain fields from

HREM micrographs. Ultramicroscopy, 74, 131-146 (1998)