Classification of the symmetry of photoelectron dichroism broken by light
Y. Ishida, D. Chung, J. Kwon, Y. S. Kim, S. Soltani, Y. Kobayashi, A. J. Merriam, L. Yu, C. Kim
aa r X i v : . [ c ond - m a t . s t r- e l ] J u l Classification of the symmetry of photoelectron dichroism broken by light
Y. Ishida,
1, 2, ∗ D. Chung, J. Kwon,
1, 4
Y. S. Kim,
1, 4
S. Soltani,
1, 4, 5, † Y. Kobayashi, A. J. Merriam, L. Yu, and C. Kim
1, 4 Center for Correlated Electron Systems, Institute for Basic Science, Seoul 08826, Republic of Korea ISSP, The University of Tokyo, Kashiwa-no-ha, Kashiwa, Chiba 277-8561, Japan College of Liberal Studies, Seoul National University, Seoul 08826, Republic of Korea Department of Physics and Astronomy, Seoul National University, Seoul 08826, Republic of Korea Institute of Physics and Applied Physics, Yonsei University, Seoul 03722, Republic of Korea Lumeras LLC, 207 McPherson Street, Santa Cruz, California 95060, USA Beijing National Laboratory for Condensed Matter Physics, IOP, CAS, Beijing 100190, China (Dated: July 7, 2020)We investigate how the direction of polarized light can affect the dichroism pattern seen in angle-resolved photoemission spectroscopy. To this end, we prepared a sample composed of highly-orientedBi(111) micro-crystals that macroscopically has infinite rotational and mirror symmetry of the pointgroup C ∞ v and examined whether the dichroism pattern retains the C ∞ v symmetry under thestationary configuration of the light and sample. The direction of the light was imprinted in thepattern. Thereby, we apply group theory and classify the pattern with the configuration of lighttaken into account. We complete the classification by discussing the cases when the out-of-planecomponent of the polarization can be neglected, when the incidence angle is either 0 ◦ or 90 ◦ , whenthe polarization is either elliptic or linear, and also when the sample is a crystal. PACS numbers:
I. INTRODUCTION
The optical response of matter can depend on the po-larization of the light. Circular dichroism (CD) is the dif-ference in the response when the polarization is switchedfrom left-circular to right. Similar to the relationshipbetween left and right hands, the left and right circularpolarizations (LCP and RCP) are exchanged by a mir-ror operation. Therefore, the existence of CD can bejudged through evaluating the handedness [1], or chiral-ity of the experimental setup: Compare the LCP setupand the mirror image of the RCP setup; if the latter canbe superimposed on the former, CD does not exist. Someexamples of the evaluation after Ref. [1] are shown inFig. 1, in which we included the cases when the samplesare magnetized and when the polarizations are elliptic.In angle-resolved photoemission spectroscopy(ARPES), light illuminates a crystal from a certaindirection, and the emitted photoelectrons are analyzedin energy and emission angle. The intensity distributionof the photoelectrons depends on the incident polar-ization, from which the electronic-state informationmay be disclosed. Recently, it was articulated thatthe CD pattern in the distribution could be associatedwith angular-momentum textures [2–12], Berry curva-tures [13, 14], and degree of surface localization of thewavefunctions [15, 16]. CD in ARPES [17–20] has alsobeen utilized in a variety of ways such as to explore the ∗ [email protected] † Current address: MAX IV Laboratory, Lund University, PO Box118, SE-221 00 Lund, Sweden m mm m Finite dichroism Finite dichroism Left hand (a) (b) (c)
LCP RCP
LL R
No dichroismSymmetric handSymmetric hand
LL R
No dichroism
LEP REP
LɛLɛ Rɛ xzy
No dichroism ¹ i L = i R i R ¹ i L = i R i R = i L = i R i R = i L ɛ = R ɛ i R ɛ ii L == i R i R Symmetric& magnetized
LL R
Symmetric& magnetized
L R i L ɛ i L ɛ i R ɛ i R m i R ɛ i R MirrorMirror L m FIG. 1: Handedness and dichroism. (a, b) Setups to measurethe current i L/Lε and i R/Rε when the sample is illuminatedwith LCP/LEP (a) and RCP/REP (b), respectively. (c) Themirror image of (b). When the images in (a) and (c) areidentical, i L/Lε = i R/Rε . Light is directed along the mirrorplane ( y = 0) of the symmetric sample; otherwise, dichroismis allowed to exist in all the setups. m is a pseudovector andis invariant with the reflection about y = 0 when m k y -axis. symmetry breaking in cuprate superconductors [21, 22],to highlight the symmetry-reduced surface states out ofbulk states [23], and to unravel the orbital character ofheavy fermions [24].While being fruitful, the diverse utility and interpreta-tions of CD ARPES are also under scrutiny [25–38]. Ithas been debated which of the symmetry breaking, timereversal or mirror reflection, is sensed in the dichroic sig-nals of the cupartes [27]; some studies showed that thesign of CD flips when the incident photon energy ( hν ) isvaried [28, 32–35]; the CD pattern can also evolve as theincidence angle is increased [35]. Surprisingly, even theform of the light-electron interaction responsible for pho-toemission varies among the studies: The starting formis of the dipole interaction [39, 40], while some studiesinclude the terms for relativistic correction [2] and/orsurface photoemission [26, 30]. Thus, it is still under dis-cussion to what extent the magnitude and pattern of thedichroic signals are reflecting the electronic-state proper-ties. A solid basis for understanding CD ARPES is calledfor. C ∞v (a)
2D micro-crystals ana l yz e r T i m e - o f - f li gh t I ( , ) q x q y - q y + q x qq q x q q q q q + q q + q qqq + q y - q x P r i n c i pa l a x i s C ∞v ¯ (b) h A R o r A L o r I L I R L R ē y ē z ē x FIG. 2: CD ARPES setup in which only light breaks thesymmetry of C ∞ v . (a) Sample and slit-less photoelectron an-alyzer set in the C ∞ v configuration. The sample is composedof Bi(111) micro-crystals randomly oriented on HOPG. (b)LCP and RCP incident on the sample-and-analyzer setup.The symmetry is lowered from C ∞ v because of the incidence. In the present study, we focus on the configuration oflight that can affect the dichroism pattern in the pho-toelectron distribution. Significant though it may be, itis not trivial to distinguish the effect of the orientationof light from the others related to the electronic statesand light-electron interaction responsible for photoemis-sion. To this end, we performed CD ARPES on a highlysymmetrical setup, in which the entity of the sample andelectron analyzer has infinite rotational and mirror sym-metry characterized by the point group C ∞ v [Fig. 2(a)].We let only the incidence of light to break the C ∞ v sym-metry of the entire experimental setup [Fig. 2(b)]. In thisway, the effects related to the crystal symmetry can beaveraged out and the effect of the orientation of light canbe singled out. Namely, we conducted an experiment toclarify whether the light-induced reduction of the symme-try is reflected in the pattern. After the clarification, wepresent a systematic classification of the pattern aidedby the group theory [41]. The classification takes intoaccount the configuration of light (incidence angle andellipticity) and includes the cases when the crystallinesample has some mirror planes.The paper is structured as follows: After the presentIntroduction (Section I), we describe the setup to attainthe C ∞ v symmetry of the sample-and-detector entity in Section II; then, we present the CD ARPES results inSection III, in which we show that the light-induced re-duction of the symmetry is imprinted in the CD pattern;thereby in Section IV, we present a systematic classifica-tion of the CD pattern, in which we investigate the casesfor a variety of incidence angles, ellipticity, and also whenthe sample is a crystal; the summary and remarks aremade in Section V; in Appendix A, we provide the irre-ducible representations for C ∞ v , which is relevant whenthe light is in normal incidence to the sample surface. II. THE C ∞ v CONFIGURATION
The experimental setup is unique in that the entity ofthe sample and electron analyzer has the symmetry ofC ∞ v during the data acquisition. The key to the high-symmetry setup is twofold: (1) We prepared a samplethat is effectively C ∞ v ; (2) we used a slit-less photoelec-tron analyzer [42–47].The sample with C ∞ v symmetry was prepared by evap-orating bismuth (Bi) of ∼ ∼ − Torr atroom temperature and then heated to 370 K, as describedelsewhere [48]. HOPG is composed of stacked layers ofgraphite micro-crystals oriented randomly in plane, andBi grown thereon forms into micro-crystals with the (111)face oriented normal to the surface [48–51]; see the il-lustration of the sample in Fig. 2(a). Macroscopically,Bi/HOPG is invariant with respect to whatever rotationabout the z -axis along the surface normal and whatevermirror reflection about the plane containing the z -axis,and thus its symmetry can be characterized by the pointgroup C ∞ v .Photoelectrons were collected by using a slit-less ana-lyzer, namely, the angle-resolved time-of-flight (ARToF)analyzer of Scienta-Omicron. In contrast to the analyz-ers that can accept photoelectrons emitted into a zero-dimensional hole or one-dimensional slit, the slit-less typecan collect photoelectrons emitted into a two-dimensionalsolid-angular cone. This enabled us to acquire the pho-toelectron distribution in the cone without rotating thesample, or with the configuration of the light and samplefixed in space. We set the sample surface normal alongthe principal axis of the ARToF analyzer as shown inFig. 2(a), and hence, the symmetry of the entity of thesample and analyzer can be characterized with C ∞ v .For the polarized light source, we adopted the 10.8-eV laser-based source [52] commercialized by Lumeras.The 10.8-eV harmonics was generated in a xenon-filledtube through a four-wave mixing (4 Ω × Ω = 9 Ω ) atthe repetition rate of 1 MHz, and the polarization of the10.8-eV output was controlled by varying the polariza-tion of the fundamental laser Ω [53]. We set the basis ofthe Cartesian coordinate { ¯ e x , ¯ e y , ¯ e z } so that the planeof incidence is y = 0, and the incidence angle was set to η = 50 ◦ ; see Fig. 2(b). Because the 10.8-eV beam passedthrough a Brewster prism after its generation [52], the SR1BS SR4SS1,2SS3,4 SR5SR6SSR4SSR6SR2,3
FIG. 3: CD ARPES of Bi(111)/HOPG. (a and b) Cuts of the total intensity I L + I R at θ x = 0 ◦ ( k x = 0) (a) and at θ y = 0 ◦ ( k y = 0) (b). The left panel is the distribution in energy ( E − E F ) and emission angle; the middle and right panels arerespectively the plots in momentum ( k ) space and its second derivative along the energy. (c) Constant-energy cuts of I L + I R and I L − I R . (d and e) CD seen in the cuts at θ x (d) and θ y (e). The cuts shown in (a, b, d and e) and (c) are the average ofthe intensity distributed within θ x,y ± . ◦ , and ( E − E F ) ± ¯ e y component of the polarization was slightly lost andbecame elliptic. The effect of the ellipticity will be dis-cussed in Section IV D. The 10.8-eV ARToF system wasoperated according to Ref. [45]. The direction of the pho-toelectron is described with the polar-angular notation( θ x , θ y ) [54], where θ x = 0 ◦ ( θ y = 0 ◦ ) is for the emissioninto x = 0 ( y = 0) plane. The temperature of the samplewas maintained at ∼
80 K with liquid nitrogen and thevacuum level of the analyzer chamber was 2 × − Torr.
III. RESULTS
The left panels of Figs. 3(a) and 3(b) show the cuts ofthe total photoelectron distribution I L + I R at θ x = 0 ◦ and θ y = 0 ◦ , respectively. The middle and right pan-els are, respectively, the distribution mapped on energy-momentum space and its second derivative along theenergy; the latter is the distribution of the negative-curvature strength that highlights the bands. The bandsseen in Figs. 3(a) and 3(b) appear identical. This sup-ports that Bi(111)/HOPG was successfully formed tohave the rotational symmetry about the surface nor-mal. Based on the literature [55, 56], we can assignthe bands to surface state (SS), surface resonance (SR),strongly hybridizing surface resonance (SSR), and bulkstate (BS), as indicated in Fig. 3(b). For example, thesurface Rashba bands along ¯ Γ - ¯ M (SR2, 3) and ¯ Γ - ¯ K (SR4) are simultaneously observed in the cuts. In thepanels of Fig. 3(c), we show the cuts of I L + I R atsome selected energies. The circular contours seen inthe constant-energy cuts further support that the sam- ple was formed into the C ∞ v symmetry. For the cuts of I L ± I R at a variety of energies and angles, see Supple-mental Material movie file [57].In Fig. 3(c), we also displayed the constant-energy cutsof the CD distribution I L − I R . First of all, CD is finite; inother words, CD is not vanished even when the sample-and-analyzer is configured to have the C ∞ v symmetry.Patterns in the CD distribution can be characterizedby nodes, or where the CD disappears. In the CD pat-terns of the constant-energy cuts shown in Fig. 3(c), thereis always a horizontal node at θ y = 0 ◦ and the patternappears as a reflection of itself with a sign flip with re-spect to the node. The magnitude of CD does not strictlyobey the anti-symmetry argument partly because the ef-ficiency of the multi-channel-plate detector as well as thetransmission of ARToF can be inhomogeneous over thetwo-dimensional detection plane. Nevertheless, the locusof the node is indifferent to the inhomogeneity of the two-dimensional detection, and therefore, the existence of thehorizontal node at θ y = 0 ◦ is solid.Figures 3(d) and 3(e) respectively show the cuts ofthe CD distribution I L − I R at some selected θ x and θ y .The signal of CD is substantial in all the cuts exceptfor that at θ y = 0 ◦ where the intensity is relatively, orvanishingly, small; this cut corresponds to the horizontalnode. The cuts at constant θ x [Fig. 3(d)] are virtuallyanti-symmetric with respect to the node at θ y = 0 ◦ . Onthe other hand, the cuts at θ y away from 0 ◦ [Fig. 3(e)]exhibit a variety of nodes that winds in the image.Summarizing the results, the prepared Bi(111)/HOPGsample successfully formed to have the C ∞ v symmetry, asjudged from the patterns of I L + I R . The CD distributionof the photoelectrons emitted from the C ∞ v -symmetricBi(111)/HOPG was not null but finite, and the patternof the CD distribution was anti-symmetric with respectto the node on the y = 0 plane set by the direction of thelight. IV. DISCUSSIONA. Classifying CD with a point group
Taking the opportunity that the setup is highly sym-metric, let us apply group theory and interpret the re-sults. It will be shown that the CD distribution I L − I R and total distribution I L + I R can be related to, if notidentified to, the base for representing the symmetry.Hereafter, the incidence angle η is neither 0 ◦ nor 90 ◦ unless described otherwise. ÅÅ =Y L I L A L j +( ) ¤ 2 Y L Y R = Å Å Y R I R A R j (a) LCP RCP ÊÊ (c) (b)(d) Ê s y Ê s y A R Ê Ê s y -( ) ¤ 2 Y L Y R ÊÊ s y A L I L j I R j j A L A R ( )+ ¤ 2 A L A R ( ) ¤ 2 - Æ Symmetric ( A ′) Anti-symmetric ( A ′) ( )+ ¤ 2 I L I R ( )- ¤ 2 I L I R ē y ē z ē x FIG. 4: Symmetry operations on the experimental setup. (a)Setup for LCP, Ψ L . (b) Mirror reflection of Ψ L , which is iden-tical to the setup for RCP. (c and d) Symmetric (c) and anti-symmetric component (d) extracted from the two illustrations Ψ L and Ψ R . Note, the two no more correspond to any real-istic setups but are mathematical entities; the horizontally-polarized-light setup (c) would not result in the distribution( I L + I R ) /
2, and vertically-polarized light incident on nullsample-and-analyzer (d) is not a realistic experimental setup.
Let us regard the illustration of the LCP experiment[Fig. 4(a)] as a function Ψ L in xyz space. The illustra-tion consists of the followings: the sample-and-analyzer ϕ ( x, y, z ); the polarized photon field A L ( x, y, z ; η ) = A ( a + e − iωt + c. c. ) in the scalar gauge [58], where a ± =(cos η ¯ e x ± i ¯ e y + sin η ¯ e z ) / I L ( x, y, z ), which is a functional of the polarizedphoton field. Ψ L can be formulated as the direct summa-tion of light, sample-and-analyzer, and the distribution: Ψ L = A L ⊕ ϕ ⊕ I L . (1) TABLE I: Character table for C σ .C σ Basis ˆ E ˆ σ y A Ψ L + Ψ R A Ψ L − Ψ R Having regarded the illustration as a function, we cannow apply operators on it and obtain new illustrations.Here, we apply two operators that consist the point groupC σ = { ˆ E, ˆ σ y } . By applying the mirror operator ˆ σ y on Ψ L , a new illustration Ψ R is constructed: Ψ R ≡ ˆ σ y Ψ L = ˆ σ y A L ⊕ ˆ σ y ϕ ⊕ ˆ σ y I L = A R ⊕ ϕ ⊕ I R ;also see, how the illustration Ψ R of Fig. 4(b) is con-structed from Ψ L of Fig. 4(a). Ψ R happens to be iden-tical to the illustration of the CD ARPES setup withthe incidence of RCP, and therefore, the mirror imageof I L is nothing but the distribution obtained with RCP: I R = ˆ σ y I L . Note, ˆ σ y A L = A − e − iωt + c. c. = A R becauseˆ σ y { ¯ e x , ¯ e y , ¯ e z } = { ¯ e x , − ¯ e y , ¯ e z } , and ˆ σ y ϕ = ϕ [59].The two illustrations, or the two functions Ψ L and Ψ R ,are exchanged when operated on by ˆ σ y , while they re-main themselves when operated on by the identity ˆ E ;see Figs. 4(a) and 4(b). Thus, the set { Ψ L , Ψ R } formsa two-dimensional basis for representing the point groupC σ : ˆ E { Ψ L , Ψ R } = { Ψ L , Ψ R } (cid:20) (cid:21) , (2)ˆ σ y { Ψ L , Ψ R } = { Ψ L , Ψ R } (cid:20) (cid:21) . (3)The matrix representation displayed in eqs. (2) and(3) is reducible. It is { Ψ L + Ψ R } and { Ψ L − Ψ R } thatrespectively form the one-dimensional irreducible repre-sentations A and A of C σ ; see, Table I. In terms of theillustration, ( Ψ L + Ψ R ) / Ψ L − Ψ R ) / Ψ L and Ψ R ; see Figs. 4(c) and 4(d). I L + I R and I L − I R are the subset of, or part of theillustration of, { Ψ L + Ψ R } and { Ψ L − Ψ R } , respectively.To summarize, I L + I R and I L − I R are respectivelyidentified to the subset of the bases for the A and A representations of C σ . If there is no confusion about I L ± I R being the subset of { Ψ L ± Ψ R } , it may be restatedas follows: I L + I R and I L − I R respectively have A and A symmetry of C σ .If the photoelectron distribution is irrelevant to thesymmetry reduction due to light, then either the illus-tration of the light is formally omitted from Fig. 4 or theincidence angle η is set to zero, and { Ψ L ± Ψ R } becomesthe base for the irreducible representation D ± of C ∞ v ;see Appendix A. As a result, I L − I R , which is the subsetof { Ψ L − Ψ R } , becomes a null function.Now, the question asked in Introduction (Section I) canbe reformatted as follows: I L − I R is related to the base TABLE II: Character table for C .C Basis ˆ E ˆ C ˆ σ y ˆ σ x A Ψ ′ L + Ψ ′ R , Ψ L + Ψ R ( η = 90 ◦ ) 1 1 1 1 A Ψ ′ L − Ψ ′ R B B Ψ L − Ψ R ( η = 90 ◦ ) 1 -1 -1 1 for a point-group representation, but of which group,C ∞ v or C σ ? Given the results shown in Fig. 3, the an-swer is C σ , because I L − I R is not null but anti-symmetricwith respect to the horizontal node. B. The C case In the previous Section IV A, we showed that I L − I R acquires a horizontal node that reflects the symmetry ofC σ even though the sample-and-detector had the C ∞ v symmetry. In this section, we show that there are twocases when I L − I R retains a higher symmetry of C . Wealso derive some implications from the fact that I L − I R of Bi/HOPG did not exhibit the pattern of C .The first case can occur when the z component of thephoton field ( A z = A · ¯ e z ) is neglected from the light-electron interaction. As shown in Figs. 5(a) and 5(b),the A z -omitted photon field A ′ L,R = A L,R − ( A L,R · ¯ e z ) ¯ e z orbits on an oval elongated along y -axis in the xy plane.The two functions Ψ ′ L,R = A ′ L,R ⊕ ϕ ⊕ I L,R are exchanged(remain themselves) when operated on with ˆ σ y and ˆ σ x ( ˆ E and ˆ C ); in other words, the set { Ψ ′ L , Ψ ′ R } forms a two-dimensional base that represents the point group C = { ˆ E, ˆ C , ˆ σ y , ˆ σ x } . Here, ˆ C and ˆ σ x are the operators for thetwo-fold rotation about the principal ( z ) axis and mirrorreflection with respect to x = 0, respectively. It caneasily be verified that { Ψ ′ L ± Ψ ′ R } forms the base for theirreducible representation of C (Table II). Particularly, I L − I R becomes the subset of the base Ψ ′ L − Ψ ′ R for the A representation of C . Thus, I L − I R acquires thevertical node in addition to the horizontal node.Under what condition can A z be neglected from thelight-electron interaction responsible for photoemission?When the dipole approximation is valid so that the sur-face photoelectric term ∇· A can be neglected, the matrixelement of the interaction reads h f | A · ˆ p | i i = A x h f | ˆ p x | i i + A y h f | ˆ p y | i i + A z h f | ˆ p z | i i , (4)where | i i and | f i are the initial state and photoelectronfinal state, respectively. If |h f | ˆ p x,y | i i| ≫ |h f | ˆ p z | i i| ∼ , (5)then the third term in eq. (4) and hence A z can be ne-glected and the vertical node may emerge.Condition (5) can be fulfilled when the wavefunc-tion of the initial state ψ i = h xyz | i i is localized bulk vacuum Ĉ | á f | p | i ñ | ~ z Ĉ | á f | p | i ñ | ≠ z z ~ λ dB ~ a Surface state | i ñ Photoelectron state| f ñ (c) I L (b) ÅÅ =Y L ¢ ¢ I L A L j ¢ A L = = (a) j A L I R j A R Ê s y Ê s y Ê s x Ê s x ,,, Ê Ĉ LCP - component A z RCP - component A z , Ê Ĉ I L I R = Å Å Y R I R A R j ¢ ¢ A R ¢ j j ¢ A L A R ¢ A R I L A L j I L j A L j A R (e) ÅÅ =Y L I L A L j = = (d) I R LCP ( ) h = 90° I R = Å Å Y R I R A R j Ê s x , Ê , Ê s x , Ê , Ê s y , Ĉ Ê s y , Ĉ j h = ° h = ° RCP ( ) h = 90° ē y ē z ē x ē y ē z ē x FIG. 5: The C case. (a and b) Illustrations for LCP (a) andRCP (b) experiments when the A z component can be omittedfrom the light-electron interaction. (c) Spatial profile of theinitial and final states. When the initial state is localized onthe surface within a length scale a shorter than the de Brogliewavelength of the photoelectron state, then |h f | ˆ p z | i i| ∼ ◦ . in the surface region within a length scale a shorterthan the de Broglie wavelength λ dB of the final-state wave function ψ f = h xyz | f i , as shown inFig. 5(c) [15]. Then, R dz ψ ∗ f ( x, y, z ) ∂ z ψ i ( x, y, z ) ∼ ψ ∗ f ( x, y, R a − a dz ∂ z ψ i ( x, y, z ) = 0, and hence, condi-tion (5) can hold and the vertical node may emerge.When | i i is a two-component spinor, the condition is readfor both the up- and down-spin components.In the CD patterns for the surface-related states shownin Fig. 3(c), there is no apparent vertical node at θ x = 0 ◦ .This implies that condition (5) is not fulfilled, or thatthe surface states observed in the ARPES image are notmuch localized in the surface region compared to λ dB . λ dB can be estimated from the kinetic energy of the pho-toelectron ε k : λ dB ∼ h/ √ mε k . Here, h and m are thePlanck constant and electron mass, respectively. By us-ing the relationship ε k = hν − w + ( E − E F ) and settingthe work function w ∼ hν = 10 . λ dB is es-timated to be at most 5 ˚A for the photoelectrons directlygenerated from the initial states at E − E F ≥ -0.7 eV.According to the theoretical calculations [60, 61], theBi(111) surface-state wavefunctions could penetrate intobulk for more than 5 bismuth bilayers, or &
20 ˚A [62]. Theabsence of the vertical node thus supports these estima-tions. The deep penetration of the surface states can beattributed to their interaction with the bulk states [63].Alternatively, if λ dB can be sufficiently elongated by low-ering hν , then the vertical node may emerge. A vertical-node-like pattern occurs around E − E F = − .
23 eV[Fig. 3(c)], but this is in the bulk-band region [see, theright panel of Fig. 3(b)], and therefore, the node around-0.23 eV is understood as an accidental one, or beyondthe classification scheme presented herein.The argument for neglecting A z through condition (5)is similar to that adopted when explaining the verti-cal node that occurred in the CD patterns for interca-lated [15] and aged [16] topological insulators. In thosestudies, hν was as low as 7 eV and the confinement ofthe surface states could be enhanced by the intercala-tion [64, 65] and aging [66]. However, the samples had tobe rotated step by step during the data acquisition be-cause a slit-type analyzer was used instead of a slit-lessanalyzer. Therefore, the group theoretical argument wasnot rigorously applicable in those studies [15, 16].The second case can occur when the incidence angleis η = 90 ◦ , namely when the circularly polarized pho-ton field rotates on the yz plane. The illustration Ψ L [Fig. 5(d)] operated on by either ˆ σ y or ˆ C overlaps tothe RCP illustration Ψ R [Fig. 5(e)], provided that thechange in the direction of the light can be neglected, andthat is valid when the long-wave-length approximationholds. Then, { Ψ L + Ψ R } ( { Ψ L − Ψ R } ) with η = 90 ◦ formsthe basis for the irreducible representation A ( B ) ofC ; see Table II. The near-grazing-incidence configura-tion η ∼ ◦ can be achieved, for example, in the so-called Takata setup [67], wherein the electron-lens axisof the analyzer is placed perpendicular to the incidenthard-X-ray beam that can be circularly polarized [68].The Takata setup was the key to attain the through-put [69] high enough for conducting ARPES even in thehard-X-ray regime [70]. C. The case for crystalline samples
So far, we have investigated CD ARPES when thesample-and-detector ϕ has C ∞ v symmetry. The re- sults evidenced that the symmetry of the experimentalsetup including the light is imprinted in the CD pat-tern. Thereby, we classified the CD pattern by using thegroup theory. We identified three types characterized bythe point groups: C σ (Section IV A), C (Section IV B),and C ∞ v (Appendix A).Here, we extend the argument to the case when thesample is a crystal so that the symmetry of ϕ is lowerthan C ∞ v . Four types will be identified as describedbelow.First, when the crystal has a mirror plane and that ismatched to the plane of incidence ( y = 0), the argumentsfor C σ presented in Section IV A can readily be applied; I L − I R will have A symmetry of C σ and acquires thehorizontal node. Second, when the crystal has anothermirror plane at x = 0 besides that at y = 0, the argu-ments for C presented in Section IV B become appli-cable; when A z can be neglected from the light-electroninteraction (when η = 90 ◦ ), I L − I R will have A ( B )symmetry of C . Third, when the crystal has a mirrorplane matched to x = 0 but not at y = 0, the argu-ments for C ′ = { ˆ E, ˆ σ x } can be applied; when A z canbe neglected from the light-electron interaction, I L − I R will have A symmetry of C ′ . Finally, when the sampledoes not have a mirror plane at y = 0, the two setupsdescribed by Ψ L and Ψ R cannot be converted to one otherby any geometrical symmetry operations, and each of Ψ L and Ψ R is at most the base for the A representation ofthe most primitive point group C = { ˆ E } .To summarize, when the sample is a single crystal, weidentify four types in the CD pattern characterized bythe point groups C σ , C , C ′ σ and C . D. Elliptical dichroism
The group theoretical classification of I L − I R owes tothe fact that RCP is the reflection of LCP with respectto the incidence plane y = 0. In other words, the pair { A L , A R } being invariant with ˆ σ y , or having the mirrorsymmetry in short, was the essential ingredient for thearguments to hold. Thus, as illustrated in the bottomrow of Fig. 1, there is no need for the pair to be composedof LCP and RCP. As we shall explicate below, most ofthe arguments can be retained even when the pair is ofleft- and right-elliptical polarizations (LEP and REP) aslong as the pair has the mirror symmetry with respect to y = 0.To begin with, we set the pair of LEP and REP asfollows. We first regard a particular polarized light di-rected along ¯ e Z as LEP, which can be described as thesuperposition of two orthogonal transverse waves as A Lε = A [cos ξ cos( ωt + δ )¯ e X + sin ξ sin( ωt )¯ e Y ]= A + ε e − iωt + c. c., and then regard its mirror reflection with respect to theincidence plane y = Y = 0 as REP: A Rε ≡ ˆ σ y A Lε = A − ε e − iωt + c. c. Here, A cos ξ ( A sin ξ ) is the amplitude of the transversewave polarized along ¯ e X (¯ e Y ), δ sets the phase differencebetween the two waves, and A ± ε = A [ e − iδ cos ξ (cos η ¯ e x +sin η ¯ e z ) ± i sin ξ ¯ e y ] /
2. Some pairs of LEP and REP areshown in Fig. 6, in which the cases for linear and circu-lar polarizations are included. For the moment, we ex-clude the special cases for the linear polarizations along¯ e X ( A X ) and ¯ e Y ( A Y ), which can respectively be ob-tained by setting ( δ, ξ ) to (90 ◦ , ◦ ) and (90 ◦ , ◦ ); see,Section IV E. By definition, { A Lε , A Rε } becomes sym-metric with respect to ˆ σ y , and the arguments for C σ and C related to Figs. 4, 5(d) and 5(e) are retained.Thus, the horizontal node seen in the CD pattern ofBi(111)/HOPG (Fig. 3) ensures that the polarized pairhad the mirror symmetry with respect to y = 0. A Rε A Lε (0°, 45°) = ( δ , ξ ) (0°, 60°) (45°, 60°) (90°, 60°) LEPREP ωtωt ωtωtωtωt ē ( )= ē Y y ē X ( cos η )= + ē x sin η ē z Ê Y σ Ê X σ, Ê Y σ Ê X σ, FIG. 6: The mirror-symmetric pair of the polarized light.REP (bottom) is constructed by reflecting LEP (top) about Y = 0. From left to right, the polarization changes fromcircular to linear as the parameter set ( δ, ξ ) is varied. Note,LEP and REP are also symmetric with respect to X = 0. The mirror symmetric pair of LEP and REP about the y = Y = 0 plane thus defined automatically fulfills themirror symmetry about the X = 0 plane because thefollowing relationship holds: A Rε = − ˆ σ X A Lε . Hence,the A z -omitted pair { A ′ Lε , A ′ Rε } also becomes symmetricwith respect to ˆ σ x , where A ′ Lε,Rε = A Lε,Rε − ( A Lε,Rε · ¯ e z )¯ e z . Thus, the arguments for C related to Figs. 5(a)and 5(b) are also retained.With the change of the pair from circular to ellipticalpolarizations, the only case that is modified in the classi-fication is when the light is incident in the surface normal; η = 0 ◦ (Appendix A). At η = 0 ◦ , XY Z and xyz coor-dinates are matched and the light-sample-detector entitybecomes mirror symmetric with respect to both x = 0and y = 0 planes. Thus, the elliptic dichroism pattern I Lε − I Rε at η = 0 ◦ will have the A symmetry of C and acquire nodes along both x = 0 and y = 0. E. Linear vertical/horizontal polarization
The linear vertical A Y and linear horizontal A X polar-izations were excluded from the elliptic dichroism argu-ments presented in the previous Section IV D. Both A X and A Y remain themselves when operated on by ˆ σ Y aswell as by ˆ σ X . Thus, Ψ X,Y ≡ A X,Y ⊕ ϕ ⊕ I X,Y becomesa one-dimensional base for the most primitive represen-tation (all indices in the character table are one) of thecorresponding symmetry group of the experimental setupincluding the configuration of the light. As a result, thephotoelectron distribution I X,Y itself will have the primi-tive symmetry with the reminder that I X,Y is understoodas the subset of the base Ψ X,Y .Specifically, when the sample-and-detector has theC ∞ v symmetry, I X,Y will be mirror symmetric with re-spect to y = 0 ( A symmetry of C σ ) [Fig. 4(a)]; if theconditions for the C symmetry hold, then I X,Y alsowill be mirror symmetric with respect to x = 0 ( A sym-metry of C ) [Fig. 5(a)]; when η = 90 ◦ [Fig. 5(d)] andthe condition holds so that the direction of the light canbe ignored, I Y will be symmetric with respect to both y = 0 and x = 0 ( A symmetry of C ), whereas I X willbe isotropic ( D +0 symmetry of C ∞ v ) because A X hasthe infinite rotational and mirror symmetry of C ∞ v ; andwhen η = 0 ◦ [Fig. 7(a)], I X,Y will have the A symmetryof C .The classification for the cases when the sample is acrystal can also be obtained systematically. For example,when the crystal has a mirror plane at y = 0, then I X,Y will have a mirror-symmetric distribution with respectto y = 0 ( A symmetry of C σ ). Note, the classificationscheme presented herein is different from the symmetryarguments for the selection rule that is often used to iden-tify the orbital character of the bands; for example, seeRef. [71]. Those arguments apply to the probability am-plitude h f | A · ˆ p | i i for the photoelectrons emitted into themirror plane of the crystal, and whether the amplitude iszero or not depends on whether A · ˆ p | i i can be regardedas even (allowed) or odd (forbidden) with respect to themirror operation that keeps the crystal invariant. V. SUMMARY AND REMARKS
The motivation of the present study was to clarifyhow the direction of light can affect the dichroism seenin ARPES. To this end, we prepared a C ∞ v -symmetricsample, illuminated the sample with polarized light deliv-ered from a laser-based source, recorded the distributionof the photoelectrons emitted from the sample by us-ing a slit-less ARToF analyzer, and investigated whetherthe dichroism pattern seen in the distribution retainedthe C ∞ v symmetry or not. The dichroism pattern wasreduced from the C ∞ v symmetry and exhibited a nodealong the plane of incidence set by the direction of thelight. Thereby, we applied group theory and systemati-cally classified the dichroism pattern with the directionof the light taken into account.The group-theoretical classification of the dichroismpattern described in the present study does not dependon the microscopic mechanism of the light-electron in-teraction, but with two exceptions: The long-wavelengthapproximation ∇· A = 0 was applied before condition (5)and also when justifying the arguments in the experimen-tal configuration of η = 90 ◦ . ∇ · A can be non-negligiblearound the surface region when the dielectric responseof the dipolar surface region ( ∼ ∼
10 - 30 eV [39, 72, 73], andcan profoundly modify the matrix element for the very-surface-localized states [30, 74]. Alternatively, the finger-print of the surface photoelectric effect may manifest asthe invalidation of the classification presented herein.We also remark that time-reversal operation ˆ T is notused in the classification. In fact, the event of a photo-electron emission cannot be symmetric with respect to ˆ T .The photoelectron final state | f i has a time-reversal part-ner ˆ T | f i ; | f i and ˆ T | f i are the inverse LEED and LEEDstates, respectively, where LEED stands for low-energyelectron diffraction. The two are degenerate in energybut are orthogonal to each other: h f | ˆ T | f i = 0; see Sec-tion 29 of Ref. [75]. Thus, even when the hamiltonianof the system and its N -body initial state may possessˆ T -symmetry, the photo-excited ( N − i , whereasthe response detected in ARPES is the distribution inspace. Thus, the scheme presented herein can be appliedto classify the patterns in the distribution outputted fromparticle-in particle-out experimental setups, in which theincoming particle field is polarized. Acknowledgments
This work was conducted under the ISSP-CCES Col-laborative Program and was supported by the Institutefor Basic Science in Republic of Korea (IBS-R009-Y2and IBS-R009-G2) and by JSPS KAKENHI (17K18749,19K22140 and 19KK0350). Y.I. acknowledges the finan-cial support by the University of Tokyo for the sabbati-cal stay at Seoul National University. S.S. acknowledgessupport from the Yonsei University BK21 program.
Appendix A: Representations of C ∞ v Here, we first summarize the irreducible representa-tions of the group C ∞ v after Section 4 of Ref. [41]. Then,we apply the operators of C ∞ v to an experimental setupwhich has a very-high symmetry, and show that CD un-der the symmetrical setup can be related to the base forthe representation D − of C ∞ v .The group C ∞ v consists of the rotation operationabout the principal axis of angle α , ˆ R ( α ), and the mirroroperation about any plane containing the principal axis.All elements of C ∞ v can be generated from ˆ R ( α ) and ˆ σ y ,which are related to each other by ˆ R ( α ) ˆ R ( α ′ ) = ˆ R ( α + α ′ )and ˆ R ( α )ˆ σ y = ˆ σ y ˆ R ( − α ).Let us first consider the group C ∞ = { ˆ R ( α ) } , which isan invariant subgroup of C ∞ v : C ∞ v = C ∞ ⊕ ˆ σ y C ∞ . Thebase for the irreducible representation of C ∞ can be v m ,which has the following property, ˆ R ( α ) v m = e − iλα v m ,where m is an integer. ÊR ( ) άÊR ( ) ά ÅÅ =Y L I L A L j +( ) ¤ 2 Y L Y R = Å Å Y R I R A R j Symmetric ( D ) + Anti-symmetric ( D ) - -( ) ¤ 2 Y L Y R A I R j ÊR ( ) ά Ê s y ÊR ( ) ά Ê s y ÊR ( ) άÊR ( ) ά I L j A L Ê s y ÊR ( ) ά Æ ( )+ ¤ 2 I L I R j Ê s y ÊR ( ) ά A L A R ( )+ ¤ 2 A L A R ( ) ¤ 2 - (a) LCP RCP(c) (b)(d) )- ¤ 2 ¤ 2 L I R I ē y ē z ē x FIG. 7: C ∞ v symmetry operations on the experimental setup.(a) Experimental setup Ψ L , in which LCP is in normal inci-dence to the sample. (b) The only new illustration Ψ R whichcan be constructed from Ψ L by applying the operations of thegroup C ∞ v . Ψ R is identical to the setup where RCP is in nor-mal incidence to the sample. (c and d) The illustrations thatform the bases for the irreducible representations D +0 (c) and D − (d) of C ∞ v . Now, it is easy to see that { v λ , v − λ } ( λ = 1, 2, 3, . . . )forms the base for the irreducible representations of C ∞ v :ˆ R ( α ) { v λ , v − λ } = { v λ , v − λ } (cid:20) e − iλα e iλα (cid:21) , ˆ σ y { v λ , v − λ } = { v λ , v − λ } (cid:20) (cid:21) . There are also two one-dimensional irreducible represen-tations based by v ± = v λ ± v − λ ( λ = 0):ˆ R ( α ) v ± = v ± , (A1)ˆ σ y v ± = ± v ± . (A2)To summarize, for the C ∞ v , there are two one-dimensional irreducible representations D +0 and D − anda sequence of two-dimensional irreducible representations D λ ( λ = 1, 2, 3, . . . ).Now, let us consider a highly-symmetric setup Ψ L , inwhich the LCP light is in normal incidence to the samplesurface; see Fig. 7(a). Such a setup cannot be realized inthe present ARToF system because the analyzer blocksthe light, but can be when a port is utilized to let lightthrough slit-less-type analyzers such as the display-typeanalyzers [42], momentum microscopes [44, 47] and hemi-spherical analyzers equipped with electron deflectors [54].In fact, the normal-incidence configuration was demon-strated and linear-polarization dependence was studiedfor 1 T -TaS [76] by using the display-type analyzer [19].We apply all the operators of C ∞ v to Ψ L . The onlynew illustration constructed through this procedure is Ψ R , which is displayed in Fig. 7(b), and that happens tobe identical to the setup where RCP is in normal inci-dence to the sample surface. Thus, I R that constitutethe illustration Ψ R is identified to the photoelectron dis-tribution obtained with RCP.The set { Ψ L , Ψ R } becomes a base for the representa-tion of C ∞ v , which is still reducible. 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