4 FIG. 3: Strain induced between two stacked hexagonal (real space) lattices in the case where one lattice (black / dashed line) is (a) stretched but not rotated and (b) stretched and rotated with respect to the undistorted lattice (blue / solid line). (c) displays equal strain for both lattices. For rotated lattices under relative strain, the angu- lar change in the lattice vectors a GLYPH<11> ect the interference pattern di GLYPH<11> er- ently for each direction. As a result, the moir´e pattern observed in STM will appear stretched or distorted. Significantly higher strain is required for similarly sized distortions if the lattices are equally strained along the same direction. At the low moir´e angles ( GLYPH<18> ), the lattice vectors (when equally strained) will obtain similarly sized an- gular distortions ’$_{i}$ which will nullify each other. cosines, Lattice vectors ( a$_{GLYPH<6>}$ ) which are not along the direction of strain will undergo an additional rotation, ’$_{GLYPH<6>}$ . In the most general case of arbitrary changes in length ( GLYPH<15>$_{i}$ ) and arbitrary rotations ( ’$_{i}$ ), this becomes: where i = 0 ; GLYPH<6> , and the distortion in the angular orientation of the moir´e is given by In order to approximate the values of GLYPH<15>$_{i}$ for a uniform uniaxial stress, we make use of Poisson’s ratio GLYPH<23> and the strain matrix " by, GLYPH<15>$_{x}$ = "; GLYPH<15>$_{y}$ = GLYPH<0> GLYPH<23>" . 29 The strain matrix is: where the axes have been chosen to diagonalize the strain ma- trix. For strain in the arm chair direction, we do not need to rotate the lattice vectors a$_{i}$ with respect to " . In this case the lattice is stressed along a$_{0}$ . We can relate the strain matrix to the values GLYPH<15>$_{i}$ by applying the original definintion, Thus, under a small strain approximation ( " GLYPH<28> 1) we obtain We use the values from equations (10) and (11), along with GLYPH<23> = 0 : 5 to plot the overall distortion in the wavelength of the interference pattern. If we put these values into (6), we can compare the length distortion ( D$_{+}$ = D$_{GLYPH<0>}$ ) of the resulting moir´e pattern. Figure 4 clearly shows the expected behavior. As the rotation angle GLYPH<18> between the lattices increases, the e GLYPH<11> ect of the distortion is quickly diminished; however, for a large moir´e pattern the rotation angle is small, leaving the pattern easily distorted by the relative strain. The reason is that small angular di GLYPH<11> erences become comparable to the rotation angle ( ’ GLYPH<25> GLYPH<18> ), and lattice distortions cannot be neglected. Addi- tionally, in this regime small changes in the lattice constant a contribute greatly to the moir´e rotation angle. The shape of the moir´e hexagon su GLYPH<11> ers additional skewing in GLYPH<30>$_{i}$ due to the angular distortion of the reciprocal lattice vectors. Obtaining an exact measure of the strain is di GLYPH<14> cult because it can be hard to determine the direction of the applied stress. FIG. 4: Curves display the length distortion between two vectors of the moir´e cell as a function of the moir´e rotation angle GLYPH<18> . The amount of distortion for a small strain rapidly increases with decreas- ing moir´e rotation angles (i.e. when the rotation angle is comparable to the angular distortion, GLYPH<18> GLYPH<24> ’ ). " = 1 + " 0 0 1 GLYPH<0> GLYPH<23>" ! ; (8) j a 0 $_{i}$j = (1 + GLYPH<15>$_{i}$ ) j a$_{i}$ j = j " a$_{i}$ j : (9) GLYPH<15>$_{0}$ = "; GLYPH<15>$_{GLYPH<6>}$ GLYPH<25> 1 4 (1 GLYPH<0> 3 GLYPH<23> ) ": (10) ’$_{0}$ = 0 ; ’$_{GLYPH<6>}$ GLYPH<25> GLYPH<6> p 3 " GLYPH<12> GLYPH<12> GLYPH<12> 1 + GLYPH<23> 4 + " (1 GLYPH<0> 3 GLYPH<23> )$^{GLYPH<12>}$ GLYPH<12> GLYPH<12> (11) D$_{0}$ = 2 GLYPH<25> j k$_{0}$ GLYPH<0> k 0 $_{0}$j = a (1 + GLYPH<15>$_{0}$ ) 2$^{q}$(1 + GLYPH<15>$_{0}$ ) sin 2 ( GLYPH<18>= 2) + GLYPH<15> 2 $_{0}$=4 ; (5) D$_{i}$ = 2 GLYPH<25> j k$_{i}$ GLYPH<0> k 0 $_{i}$j = a (1 + GLYPH<15>$_{i}$ ) 2$^{q}$(1 + GLYPH<15>$_{i}$ ) sin 2 [( GLYPH<18> + ’$_{i}$ ) = 2] + GLYPH<15> 2 i = 4 : (6) GLYPH<30>$_{i}$ = arcsin$^{0}$ B B B B @ (1 + GLYPH<15>$_{i}$ ) sin ( GLYPH<18> + ’$_{i}$ ) 2$^{q}$(1 + GLYPH<15>$_{i}$ ) sin 2 [( GLYPH<18> + ’$_{i}$ ) = 2] + GLYPH<15> 2 i = 4$^{1}$ C C C C A (7)