Supplemental Material for A structural distortion induced magneto-elastic locking in Sr 2 IrO 4 revealed through nonlinear optical harmonic generation
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1 Supplemental Material for A structural distortion induced magneto-elastic locking in Sr 2 IrO 4 revealed through nonlinear optical harmonic generation D. H. Torchinsky, 1, 2 H. Chu, 1, 3 L. Zhao, 1, 2 N. B. Perkins, 4 Y. Sizyuk, 4 T. Qi, 5 G. Cao, 5 and D. Hsieh 1, 2 1) Institute for Quantum Information and Matter, California Institute of Technology, Pasadena, CA 91125, USA 2) Department of Physics, California Institute of Technology, Pasadena, CA 91125, USA 3) Department of Applied Physics, California Institute of Technology, Pasadena, CA 91125, USA 4) School of Physics and Astronomy, University of Minnesota, Minneapolis, MN 55116, USA 5) Center for Advanced Materials, Department of Physics and Astronomy, University of Kentucky, Lexington, Kentucky 456, USA 1
2 CONTENTS I. Mathematical Expressions, Symmetry Adapted Tensors and Fits to Selected Point Groups for Lowest Allowed Order of Second Harmonic Generation 4 A. Comparison between 4/m (C 4h ), 4/mmm (D 4h ) and m (C s ) point groups 5 B. Monoclinic Point Groups (C 2 ) 6 2. m (C s ) /m (C 2h ) 9 C. Orthorhombic Point Groups (D 2 ) mm2 (C 2v ) mmm (D 2h ) 17 D. Tetragonal Point Groups (C 4 ) (S 4 ) (D 4 ) mm (C 4v ) m (D 2d ) /mmm (D 4h ) /m (C 4h ) 31 II. Mathematical Expressions, Symmetry Adapted Tensors and Fits to 4/mmm (D 4h ) and 4/m (C 4h ) Point Groups for Third Harmonic Generation 35 A. 4/mmm (D 4h ) 36 B. 4/m (C 4h ) 39 III. Visual Depiction of Loss of c and d Glide Planes Through Staggered Tetragonal Distortion 42 IV. Experimental Details 45 2
3 V. Explanation of the Apparent Four-Fold Rotational Symmetry of the THG-RA data below T N 45 3
4 I. MATHEMATICAL EXPRESSIONS, SYMMETRY ADAPTED TENSORS AND FITS TO SELECTED POINT GROUPS FOR LOWEST ALLOWED ORDER OF SECOND HARMONIC GENERATION Below, we provide fits to selected crystallographic point groups among monoclinic symmetry, orthorhombic symmetry and tetragonal symmetry. The lowest order allowed symmetry adapted tensor and signal are given for all point groups: for non-centrosymmetric media, the bulk electric dipole response dominates; for centrosymmetric media the bulk magnetic dipolar, bulk electric quadrupolar and surface electric dipolar responses are considered equally. Fits were only performed to responses for which all four geometries (SS, PS, SP, and PP) are allowed or if the corresponding point group is discussed in the main text. In the following, θ refers to the incidence angle of the incoming beam relative to sample normal, ψ is the angle of sample rotation relative to ψ = which is taken to be the a axis, and k = (k x,, k z ) is the wavevector of the incident light for ψ =. We begin with a side-by-side comparison of the 4/m (C 4h ), 4/mmm (D 4h ) and m (C s ) point groups. Expressions for the individual fitting functions can be found in the respective sections below. 4
5 A. Comparison between 4/m (C 4h ), 4/mmm (D 4h ) and m (C s ) point groups FIG. 1. (a)-(d) Fits to 4/m (C 4h ) bulk electric quadrupole. (e)-(h) Fits to 4/mmm (D 4h ) bulk electric quadrupole. (i)-(l) Fits to m (C s ) bulk electric dipole. 5
6 B. Monoclinic Point Groups 1. 2 (C 2 ) χ (2) tensor for 2 (C 2 ) (C 2 symmetry in xy-plane) bulk electric dipole (non-centrosymmetric): χ ED ijk = yxz yyz yxz yyz zxx zxy zxy zyy zzz I SS (2ω) = I P S (2ω) sin 2 (2θ) [ 2χ yxz cos 2 (ψ) 2χ sin 2 (ψ) (χ χ yyz ) sin(2ψ) ] 2 I SP (2ω) sin 2 (θ) [ χ zyy cos 2 (ψ) + χ zxx sin 2 (ψ) 2χ zxy cos(ψ) sin(ψ) ] 2 I P P (2ω) sin 2 (2θ) [ cos(θ) [ (χ + χ yxz ) sin(2ψ)/2 χ cos 2 (ψ) + χ yyz sin 2 (ψ) ]] 2 + sin 2 (θ) [ sin 2 (θ)χ zzz + cos 2 (θ)(χ zxx cos 2 (ψ) + χ zyy sin 2 (ψ) + 2χ zxy cos(ψ) sin(ψ)) ] 2 6
7 2. m (C s ) χ (2) tensor for m (C s ) (xy-plane mirror symmetry) bulk electric dipole (non-centrosymmetric): χ ED ijk = xxx xxy xxy xyy xzz yxx yxy yxy yyy yzz zxz zyz zzx zzy I SS (2ω) [χ xxx sin 3 (ψ) χ yyy cos 3 (ψ) (2χ xxy χ yxx ) cos(ψ) sin 2 (ψ) +(2χ yxy + χ xyy ) cos 2 (ψ) sin(ψ)] 2 I P S (2ω) [sin 2 (θ)(χ yzz cos(ψ) χ xzz sin(ψ))+cos 2 (θ)(χ yxx cos(ψ) 3 +(2χ yxy χ xxx ) cos 2 (ψ) sin(ψ)+ (χ yyy 2χ xxy ) cos(ψ) sin(ψ) 2 ) χ xyy sin 3 (ψ)] 2 I SP (2ω) cos 2 (θ)[χ xyy cos 3 (ψ) + χ yxx sin 3 (ψ) + (χ xxx 2χ yxy ) cos(ψ) sin 2 (ψ) + (χ yyy 2χ xxy ) cos(ψ) 2 sin(ψ)] 2 I P P (2ω) cos 2 (θ)[sin 2 (θ)(χ xzz cos(ψ)+χ yzz sin(ψ))+cos 2 (θ)(χ xxx cos 3 (ψ)+χ yyy sin 3 (ψ)+ (2χ xxy +χ yxx ) cos 2 (ψ) sin(ψ)+(2χ yxy +χ xyy ) cos(ψ) sin 2 (ψ))] 2 +sin 2 (2θ) sin 2 (θ)(χ zxz cos(ψ)+ χ zyz sin(ψ)) 2 7
8 FIG. 2. Fits to m (C s ) bulk electric dipole. 8
9 3. 2/m (C 2h ) χ (2) tensor for 2/m (C 2h ) bulk electric quadrupole (centrosymmetric): χ EQ ijkl = xxxx xyxx xyxx xyxy xzxz xxyx xxyy xxyy xyyy xzyz z z xzzx xzzy yxxx yyxx yyxx yyxy yzxz yxyx yyyx yyyx yyyy yzyz yxzz yyzz yzzx yzzy zzxx zzxy zzxx zzxy zzyx zzyy zzyx zzyy zxzx zxzy zyzx zyzy zzzz I SS (2ω) sin 2 (θ)[χ yyxy cos 4 (ψ) χ xxyx sin 4 (ψ)+(2χ xyxx χ xyyy +χ yxxx 2χ yyyx ) cos 2 (ψ) sin 2 (ψ)+ (2χ xxyy + χ yxyx χ xxxx ) cos(ψ) sin 3 (ψ) (2χ yyxx χ yyyy + χ xyxy ) cos(ψ) 3 sin(ψ)] 2 I P S (2ω) sin 2 (θ)[cos 2 (θ)(χ yxxx cos 4 (ψ) χ xyyy sin 4 (ψ) + (χ yyxy χ xxyx 2χ xyxx + 2χ yyyx ) cos 2 (ψ) sin 2 (ψ)+(2χ yyxx χ xxxx +χ yxyx ) cos 3 (ψ) sin(ψ)+(χ yyyy χ xyxy 2χ xxyy ) cos(ψ) sin 3 (ψ)+ (χ z + χ yxzz + χ xzzy + χ yzzx ) cos 2 (ψ) + (χ yyzz χ z χ xzzx + χ yzzy ) sin(2ψ)/2 (χ z + χ xzzy )) sin 2 (θ)((χ xzyz + χ yzxz ) cos 2 (ψ) + (χ yzyz χ xzxz ) sin(2ψ)/2 χ xzyz )] 2 I SP (2ω) sin 2 (2θ)[(χ xyxy cos 4 (ψ)+χ yxyx sin 4 (ψ)+(χ xxxx 2χ xxyy 2χ yyxx +χ yyyy ) cos 2 (ψ) sin 2 (ψ)+ (χ xyyy 2χ xyxx + χ yyxy ) cos 3 (ψ) sin(ψ) + (yxxx 2yyyx + xxyx) cos(ψ) sin 3 (ψ)) 2 + χ 2 zyzy] I P P (2ω) sin 2 (2θ)[(cos 2 (θ)(χ z + χ xzzx χ xxxx cos 4 (ψ) χ yyyy sin 4 (ψ) + (χ z + χ xzzy + χ yxzz + χ yzzx ) sin(2ψ)/2 (χ z + χ xzzx χ yyzz χ yzzy + (χ xyxy + 2χ xxyy + χ yxyx + 2χ yyxx ) cos 2 (ψ)) sin 2 (ψ) (χ xxyx + 2χ xyxx + χ yxxx ) cos 3 (ψ) sin(ψ) (χ xyyy + χ yyxy + 2χ yyyx ) cos(ψ) sin 3 (ψ)) sin 2 (θ)(χ xzxz +(χ xzyz +χ yzxz ) sin(2ψ)/2 (χ xzxz χ yzyz ) sin 2 (ψ))) 2 + (cos 2 (θ)(χ zyzy + (χ zxzy + χ zyzx ) sin(2ψ)/2 + (χ zxzx χ zyzy ) cos 2 (ψ)) sin 2 (θ)(2χ zzyy χ zzzz + (χ zzxy + χ zzyx ) sin(2ψ) + 2(χ zzxx χ zzyy ) cos 2 (ψ))) 2 ] 9
10 FIG. 3. Fits to 2/m (C 2h ) bulk electric quadrupole. 1
11 χ (2) tensor for 2/m (C 2h ) surface electric dipole (centrosymmetric): χ ED ijk = yxz yyz yxz yyz zxx zxy zxy zyy zzz I SS (2ω) = I P S (2ω) sin 2 (2θ) [ 2χ yxz cos 2 (ψ) 2χ sin 2 (ψ) (χ χ yyz ) sin(2ψ) ] 2 I SP (2ω) sin 2 (θ) [ χ zyy cos 2 (ψ) + χ zxx sin 2 (ψ) 2χ zxy cos(ψ) sin(ψ) ] 2 I P P (2ω) sin 2 (2θ) [ cos(θ) [ (χ + χ yxz ) sin(2ψ)/2 χ cos 2 (ψ) + χ yyz sin 2 (ψ) ]] 2 + sin 2 (θ) [ sin 2 (θ)χ zzz + cos 2 (θ)(χ zxx cos 2 (ψ) + χ zyy sin 2 (ψ) + 2χ zxy cos(ψ) sin(ψ)) ] 2 11
12 χ (2) tensor for 2/m (C 2h ) bulk magnetic dipole (centrosymmetric): χ MD ijk = yxz yyz yxz yyz zxx zxy zxy zyy zzz I SS (2ω) [ k x (χ zxx sin 2 (ψ) + χ zyy cos 2 (ψ) 2χ zxy cos(ψ) sin(ψ)) ] 2 I P S (2ω) [k x (sin 2 (θ)χ zzz + cos 2 (θ)(χ zxx cos 2 (ψ) + χ zyy sin 2 (ψ) + 2χ zxy cos(ψ) sin(ψ))) + k z cos(θ) sin(θ)((χ + χ yxz ) sin(2ψ) 2(χ cos 2 (ψ) + χ yyz sin 2 (ψ)))] 2 I SP (2ω) = I P P (2ω) = [ k sin(2θ)(2χ yxz cos 2 (ψ) 2χ sin 2 (ψ) (χ χ yyz ) sin(2ψ)) ] 2 12
13 C. Orthorhombic Point Groups (D 2 ) χ (2) tensor for 222 (D 2 ) bulk electric dipole (non-centrosymmetric): χ ED ijk = yxz yxz zxy zxy I SS (2ω) = I P S (2ω) [ sin(2θ)(χ yxz cos 2 (ψ) χ sin 2 (ψ)) ] 2 I SP (2ω) [sin(θ)χ zxy sin(2ψ)] 2 I P P (2ω) cos 2 (θ) sin 2 (2θ)((χ + χ yxz ) 2 + χ 2 zxy) sin 2 (2ψ) 13
14 FIG. 4. Fits to 222 (D 2 ) bulk electric dipole. 14
15 2. mm2 (C 2v ) χ (2) tensor for mm2 (C 2v ) bulk electric dipole (non-centrosymmetric): χ ED ijk = yyz yyz zxx zyy zzz I SS (2ω) = I P S (2ω) [cos(θ) sin(θ)(χ χ yyz ) sin(2ψ)] 2 I SP (2ω) [ sin(θ)(χ zxx sin 2 (ψ) + χ zyy cos 2 (ψ)) ] 2 I P P (2ω) [ cos(θ) sin(2θ)(χ cos 2 (ψ) + χ yyz sin 2 (ψ)) ] 2 + [ cos 2 (θ) sin(θ)(χ zxx cos 2 (ψ) + χ zyy sin 2 (ψ)) + sin 3 (θ)χ zzz ] 2 15
16 FIG. 5. Fits to mm2 (C 2v ) bulk electric dipole. 16
17 3. mmm (D 2h ) χ (2) tensor for mmm (D 2h ) bulk electric quadrupole (centrosymmetric): χ EQ ijkl = xxxx xyxy xzxz xxyy xxyy z xzzx yyxx yyxx yxyx yyyy yzyz yyzz yzzy zzxx zzxx zzyy zzyy zxzx zyzy zzzz I SS (2ω) sin 2 (θ) sin 2 (2ψ)[(χ yyyy 2χ yyxx χ xyxy ) cos 2 (ψ)+(2χ xxyy +χ yxyx χ xxxx ) sin 2 (ψ)] 2 I P S (2ω) sin 2 (θ) sin 2 (2ψ)[sin 2 (θ)(χ xzxz χ yzyz )+cos 2 (θ)((χ yyzz +χ yzzy χ z χ xzzx )+ (χ xxxx χ yxyx 2χ yyxx ) cos 2 (ψ) + (2χ xxyy + χ xyxy χ yyyy ) sin 2 (ψ))] 2 χ 2 zyzy] I SP (2ω) sin 2 (2θ)[(χ yxyx sin 4 (ψ)+χ xyxy cos 4 (ψ)+(χ xxxx 2χ xxyy 2χ yyxx +χ yyyy ) sin 2 (2ψ)/4) 2 + I P P (2ω) sin 2 (2θ)[cos 2 (θ)(χ yyyy sin 4 (ψ) χ xxxx cos 4 (ψ) + (χ z + χ xzzx (2χ xxyy + 2χ yyxx + χ xyxy + χ yxyx ) sin 2 (ψ)) cos 2 (ψ) + (χ yyzz + χ yzzy ) sin 2 (ψ)) sin 2 (θ)(χ xzxz cos 2 (ψ) + χ yzyz sin 2 (ψ))] 2 + sin 2 (2θ)[cos 2 (θ)(χ zxzx cos 2 (ψ)+χ zyzy sin 2 (ψ))+sin(θ)(χ zzzz 2χ zzxx cos 2 (ψ) 2χ zzyy sin 2 (ψ))] 2 17
18 FIG. 6. Fits to mmm (D 2h ) bulk electric quadrupole. 18
19 χ (2) tensor for mmm (D 2h ) surface electric dipole (centrosymmetric): χ ED ijk = yyz yyz zxx zyy zzz I SS (2ω) = I P S (2ω) [cos(θ) sin(θ)(χ χ yyz ) sin(2ψ)] 2 I SP (2ω) [ sin(θ)(χ zxx sin 2 (ψ) + χ zyy cos 2 (ψ)) ] 2 I P P (2ω) [ cos(θ) sin(2θ)(χ cos 2 (ψ) + χ yyz sin 2 (ψ)) ] 2 + [ cos 2 (θ) sin(θ)(χ zxx cos 2 (ψ) + χ zyy sin 2 (ψ)) + sin 3 (θ)χ zzz ] 2 19
20 χ (2) tensor for mmm (D 2h ) bulk magnetic dipole (centrosymmetric): χ MD ijk = yxz yxz zxy zxy I SS (2ω) [k x χ zxy sin(2ψ)] 2 I P S (2ω) (k x cos 2 (θ)χ zxy k z cos(θ) sin(θ)(χ + χ yxz )) 2 sin 2 (2ψ) I SP (2ω) = I P P (2ω) [ k sin(2θ)(χ yxz cos 2 (ψ) χ sin 2 (ψ)) ] 2 2
21 D. Tetragonal Point Groups 1. 4 (C 4 ) χ (2) tensor for 4 (C 4 ) bulk electric dipole (non-centrosymmetric): χ ED ijk = zxx zxy zxy zxx zzz I SS (2ω) = I P S (2ω) sin 2 (2θ)χ 2 I SP (2ω) sin 2 (θ)χ 2 zxx I P P (2ω) [cos(θ) sin(2θ)χ ] 2 + [ cos(θ) sin(2θ)χ zxx /2 + sin 3 (θ)χ zzz ] 2 21
22 2. 4 (S 4 ) χ (2) tensor for 4 (S 4 ) bulk electric dipole (non-centrosymmetric): χ ED ijk = zxx zxy zxy zxx I SS (2ω) = I P S (2ω) sin 2 (2θ)(χ cos(2ψ) χ sin(2ψ)) 2 I SP (2ω) sin 2 (θ)(χ zxx cos(2ψ) + χ zxy sin(2ψ)) 2 I P P (2ω) cos 2 (θ) sin 2 (2θ)[(χ cos(2ψ)+χ sin(2ψ)) 2 +(χ zxx cos(2ψ)+χ zxy sin(2ψ)/2) 2 ] 22
23 (D 4 ) χ (2) tensor for 422 (D 4 ) bulk electric dipole (non-centrosymmetric): χ ED ijk = zxy zxy I SS (2ω) = I P S (2ω) sin 2 (2θ)χ 2 I SP (2ω) = I P P (2ω) = 23
24 FIG. 7. Fits to 422 (D 4 ) bulk electric dipole. 24
25 4. 4mm (C 4v ) χ (2) tensor for 4mm (C 4v ) bulk electric dipole (non-centrosymmetric): χ ED ijk = zxx zxx zzz I SS (2ω) = I P S (2ω) = I SP (2ω) sin 2 (θ)χ 2 zxx I P P (2ω) [cos(θ) sin(2θ)χ ] 2 + sin 2 (θ) [ cos 2 (θ)χ zxx + sin 2 (θ)χ zzz ] 2 25
26 5. 42m (D 2d ) χ (2) tensor for 42m (D 2d ) bulk electric dipole (non-centrosymmetric): χ ED ijk = zxy zxy I SS (2ω) = I P S (2ω) [sin(2θ)χ cos(2ψ)] 2 I SP (2ω) [sin(θ)χ zxy sin(2ψ)] 2 I P P (2ω) cos 2 (θ) sin 2 (2θ)(χ 2 + χ 2 zxy/4) sin 2 (2ψ) 26
27 6. 4/mmm (D 4h ) χ (2) tensor for 4/mmm (D 4h ) bulk electric quadrupole (centrosymmetric): χ EQ ijkl = xxxx xyxy xzxz xxyy xxyy z xzzx xxyy xxyy xyxy xxxx xzxz z xzzx zzxx zzxx zzxx zzxx zxzx zxzx zzzz I SS (2ω) [sin(θ)(χ xyxy χ xxxx + 2χ xxyy ) sin(4ψ)/4] 2 I P S (2ω) [cos 2 (θ) sin(θ)(χ xyxy χ xxxx + 2χ xxyy ) sin(4ψ)/4] 2 I SP (2ω) cos 2 (θ) sin 2 (θ)[χ 2 zxzx + (χ xyxy + (χ xxxx χ xyxy 2χ xxyy ) sin 2 (2ψ)/2) 2 ] I P P (2ω) [cos(θ)(sin 3 (θ)χ xzxz +cos 2 (θ) sin(θ)(χ xxxx 1/2(χ xxxx +χ xyxy +2χ xxyy ) sin 2 (2ψ)) cos 2 (θ) sin(θ)(χ z + χ xzzx ))] 2 + [sin(θ)(cos 3 (θ)χ zxzx 2 cos(θ) sin 2 (θ)(χ zzxx χ zzzz ))] 2 27
28 FIG. 8. Fits to 4/mmm (D 4h ) bulk electric quadrupole. 28
29 χ (2) tensor for 4/mmm (D 4h ) surface electric dipole (centrosymmetric): χ ED ijk = zxx zxx zzz I SS (2ω) = I P S (2ω) = I SP (2ω) sin 2 (θ)χ 2 zxx I P P (2ω) [cos(θ) sin(2θ)χ ] 2 + sin 2 (θ)[cos 2 (θ)χ zxx + sin 2 (θ)χ zzz ] 2 29
30 χ (2) tensor for 4/mmm (D 4h ) bulk magnetic dipole (centrosymmetric): χ MD ijk = zxy zxy I SS (2ω) = I P S (2ω) = I SP (2ω) = I P P (2ω) k 2 sin 2 (2θ)χ 2 3
31 7. 4/m (C 4h ) χ (2) tensor for 4/m (C 4h ) bulk electric quadrupole (centrosymmetric): χ EQ ijkl = xxxx xyxx xyxx xyxy xzxz xxyx xxyy xxyy xyyy xzyz z z xzzx xzzy xyyy xxyy xxyy xxyx xzyz xyxy xyxx xyxx xxxx xzxz z z xzzy xzzx zzxx zzxy zzxx zzxy zzxy zzxx zzxy zzxx zxzx zxzy zxzy zxzx zzzz I SS (2ω) sin 2 (θ)[χ xxyx +(2χ xxyy +χ xyxy χ xxxx ) sin(4ψ)/4 (2χ xyxx +χ xxyx χ xyyy ) sin 2 (2ψ)/2] 2 I P S (2ω) sin 2 (θ)[cos 2 (θ)(χ xyyy χ z χ xzzy + (χ xxxx 2χ xxyy χ xyxy ) sin(4ψ)/4 + (χ xxyx + 2χ xyxx χ xyyy ) sin 2 (2ψ)/2) + sin 2 (θ)χ xzyz ] 2 I SP (2ω) sin 2 (2θ)[((χ xxyx +2χ xyxx χ xyyy ) sin(4ψ)/4+(2χ xxyy +χ xyxy χ xxxx ) sin 2 (2ψ)/2 χ xyxy ) 2 + χ 2 zxzx] I P P (2ω) cos 2 (θ) sin 2 (θ)[cos 2 (θ)((χ xxxx χ z χ xzzx )+(χ xxyx +2χ xyxx χ xyyy ) sin(4ψ)/4+ (2χ xxyy χ xxxx + χ xyxy ) sin 2 (2ψ)/2) + sin 2 (θ)χ xzxz ] 2 + sin 2 (2θ)[cos 2 (θ)χ zxzx + sin 2 (θ)(χ zzzz 2χ zzxx )] 2 31
32 FIG. 9. Fits to 4/m (C 4h ) bulk electric quadrupole. 32
33 χ (2) tensor for 4/m (C 4h ) surface electric dipole (centrosymmetric): χ ED ijk = zxx zxy zxy zxx zzz I SS (2ω) = I P S (2ω) sin 2 (2θ)χ 2 I SP (2ω) sin 2 (θ)χ 2 zxx I P P (2ω) [cos(θ) sin(2θ)χ ] 2 + sin 2 (θ)[cos 2 (θ)χ zxx + sin 2 (θ)χ zzz ] 2 33
34 χ (2) tensor for 4/m (C 4h ) bulk magnetic dipole (centrosymmetric): χ MD ijk = zxx zxy zxy zxx zzz I SS (2ω) k 2 xχ 2 zxx I P S (2ω) [k z sin(2θ)χ xzz k x (χ zxx cos 2 (θ) + χ zzz sin 2 (θ))] 2 I SP (2ω) = I P P (2ω) k 2 sin 2 (2θ)χ 2 34
35 II. MATHEMATICAL EXPRESSIONS, SYMMETRY ADAPTED TENSORS AND FITS TO 4/mmm (D 4h ) AND 4/m (C 4h ) POINT GROUPS FOR THIRD HARMONIC GENERATION Below, we give the symmetry-adapted bulk electric dipole tensors, detailed mathematical expressions and fits for third harmonic generation. Only the 4/mmm (D 4h ) and 4/m (C 4h ) point groups are considered as the others have been definitively ruled out by second harmonic generation measurements. Two separate temperatures are presented in the fits below. 35
36 A. 4/mmm (D 4h ) χ (3) tensor for 4/mmm (D 4h ) bulk electric dipole: χ ED ijkl = xxxx xxyy z xxyy xxyy z z xxyy xxyy xxyy xxxx z z z zzxx zzxx zzxx zzxx zzxx zzxx zzzz I SS (3ω) [χ xxxx + (3χ xxyy χ xxxx ) sin 2 (2ψ)/2] 2 I P S (3ω) [cos 3 (θ)(3χ xxyy χ xxxx ) sin(4ψ)/4] 2 I SP (3ω) [cos(θ)(3χ xxyy χ xxxx ) sin(4ψ)/4] 2 I P P (3ω) cos 4 (θ)[cos 2 (θ)(χ xxxx + (3χ xxyy χ xxxx ) sin 2 (2ψ)/2) + 3 sin(θ) 2 χ z ] 2 + sin 4 (θ)[3 cos 2 (θ)χ zzxx + sin 2 (θ)χ zzzz ] 2 36
37 FIG. 1. Fits to 4/mmm (D 4h ) bulk electric dipole third harmonic generation at 295 K 37
38 FIG. 11. Fits to 4/mmm (D 4h ) bulk electric dipole third harmonic generation at 18 K 38
39 B. 4/m (C 4h ) χ (3) tensor for 4/m (C 4h ) bulk electric dipole: χ ED ijkl = xxxx xxxy xxxy xxyy z xxxy xxyy xxyy xyyy z z z z z xyyy xxyy xxyy xxxy z xxyy xxxy xxxy xxxx z z z z z zzxx zzxx zzxx zzxx zzxx zzxx zzzz I SS (3ω) [χ xxxx + (3χ xxxy χ xyyy ) sin(4ψ)/4 + (3χ xxyy χ xxxx ) sin 2 (2ψ)/2] 2 I P S (3ω) cos 2 (θ)[cos 2 (θ)(χ xyyy +(3χ xxyy χ xxxx ) sin(4ψ)/4+(3χ xxxy χ xyyy ) sin 2 (2ψ)/2)+ 3 sin(θ) 2 χ z ] 2 I SP (3ω) cos 2 (θ)[χ xyyy + (3χ xxyy χ xxxx ) sin(4ψ)/4 + (3χ xxxy χ xyyy ) sin 2 (2ψ)/2] 2 I P P (3ω) cos 4 (θ)[cos 2 (θ)(χ xxxx +(3χ xxxy χ xyyy ) sin(4ψ)/4+(3χ xxyy χ xxxx ) sin 2 (2ψ)/2)+ 3 sin 2 (θ)χ z ] + sin 4 (θ)[3 cos 2 (θ)χ zzxx + sin 2 (θ)χ zzzz ] 2 39
40 FIG. 12. Fits to 4/m (C 4h ) bulk electric dipole third harmonic generation at 295 K 4
41 FIG. 13. Fits to 4/m (C 4h ) bulk electric dipole third harmonic generation at 18 K 41
42 III. VISUAL DEPICTION OF LOSS OF C AND D GLIDE PLANES THROUGH STAGGERED TETRAGONAL DISTORTION In Figs. 14 and 15, we depict the effects of the c and d glide operations on the Sr2IrO4 unit cell using the notations of Figs. 4a and 4b of the main text. Each glide operation involves first applying a reflection operator followed by a translation operator. Both glide operations leave the lattice invariant in the absence of the proposed staggered tetragonal distortion 1 = 2. However with the staggered distortion 1 2, both glide planes are lost and the crystal symmetry is reduced from I4 1 /acd to I4 1 /a. 42
43 σ bc +c/2 z=7/8 z=5/8 z=3/8 z=1/8 A B C A FIG. 14. Loss of the c glide plane. The first operation is a reflection about the bc plane (operator σ bc ) followed by a translation of the unit cell by c/2 in the second step. In the case that the tetragonal distortions on the two sublattices are equivalent, the original unit cell would be recovered. However, in the presence of a staggered tetragonal distortion, the c glide symmetry is lost. 43
44 σ d +(a-b+c)/4 z=7/8 z=5/8 z=3/8 z=1/8 A B C A FIG. 15. Loss of the d glide plane. The first operation is a diamond reflection about a (-11) plane (operator σ d ) followed by a translation of the unit cell by (a b + c)/4 in the second step. In the case that the tetragonal distortions on the two sublattices are equivalent, the original unit cell would be recovered. However, in the presence of a staggered tetragonal distortion, the diamond glide symmetry is lost. 44
45 IV. EXPERIMENTAL DETAILS Single crystals of Sr 2 IrO 4 were grown using a self-flux technique from off-stoichiometric quantities of IrO 2, SrCO 3 and SrCl 2. The ground mixtures of powders were melted at 147 C in partially capped platinum crucibles. The soaking phase of the synthesis lasted for > 2 hours and was followed by a slow cooling at 2 C/hr to reach 14 C. From this point the crucible is brought to room temperature through a rapid cooling at a rate of 1 C/hr. For our rotating scattering plane based RA experiments, we use light produced by an optical parametric amplifier pumped by a regenerative Ti:sapphire amplifier, which produces wavelength tunable laser pulses with < 1 fs duration at a 1 khz repetition rate. The beam is focused down to a 2 µm spot on the sample surface at an oblique incidence angle of 3 using a reflective objective. Less than 1 mw average incident power was used in order to avoid photoinduced sample damage. Crystals were oriented using x-ray Laue diffraction. V. EXPLANATION OF THE APPARENT FOUR-FOLD ROTATIONAL SYMMETRY OF THE THG-RA DATA BELOW T N In the electric-dipole approximation, the nonlinear polarization at the third harmonic frequency is given by P i (3ω) = χ (i) ijkl + χ(c) ijkl [ ] E j (ω)e k (ω)e l (ω), where χ (i) ijkl is a time-invariant (i-type) tensor that is allowed both above and below T N and χ (c) ijkl is a time non-invariant (c-type) tensor that is allowed only below T N. The tensor components of χ (i) ijkl and χ (c) ijkl are respectively identified by the crystallographic and magnetic point group symmetries of the system. Since both χ (i) ijkl and χ(c) ijkl contribute to the low temperature THG signal, the rotational anisotropy should be two-fold symmetric as dictated by the magnetic point group of the antiferromagnetic ordered phase. However the magnitude of χ (c) ijkl is typically much smaller than that of χ (i) ijkl, therefore the breaking of four-fold symmetry is not detected within our experimental resolution. 45
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