Angle-resolved photoemission with circularly polarized light in the nodal mirror plane of underdoped Bi₂Sr₂CaCu₂O_8+δ superconductor

High-temperature superconductors have attracted much attention due to their unique properties for applications. These applications can potentially be expanded dramatically if we understand the superconducting mechanism and device ways to enhance the superconducting transition temperature. However, a prerequisite for the above tasks is to understand the nature of the normal state in cuprates–pseudogap phase–from which superconductivity emerges at low temperatures.^1–5 Among the numerous efforts that have been made in both theories and experiments, proposals to detect the possible symmetry breaking state in angle-resolved photoemission spectroscopy (ARPES) measurements using circularly polarized light have held a unique position.^6,7 An electronic order that breaks the mirror symmetry of the system would contribute to the circular dichroism in photoemission. Various orders breaking the reflection symmetry about different mirror planes could also be differentiated by the momentum dependence of the measured circular dichroism. For example, an ordered pattern of circulating currents was predicted to break the reflection symmetry about the Γ (0,0)-M (π,0) antinodal mirror plane which gives a maximum circular dichroism at the antinode (π,0).⁶ On the other hand, the antiferromagnetic order with wavevector Q = (π,π) and the “d-density wave” order would break the reflection symmetry about the Γ (0,0)-Y (π,π) nodal mirror plane.⁷ Another electronic order with increasing interest is the charge order observed in various cuprates^8–11 which has been proposed to be of a chiral nature.^12,13 Chiral symmetry breaking generally breaks all mirror symmetries of the system and would thus contribute to circular dichroism in photoemission measured in any mirror plane.

A prominent experimental realization of the proposal is the report of circular dichroism in ARPES spectra at antinode (π,0) of Bi2212 below pseudogap temperature T*.^14–19 The observed circular dichroism was first interpreted as direct evidence for the breaking of time-reversal symmetry,¹⁴ which seemed to be consistent with the theoretical proposal of a spontaneous ordered pattern of circulating currents.^6,7 This interpretation was later challenged by others considering structural supermodulations in the Bi-O layer (superstructures) which break the reflection symmetry about the Γ (0,0)-M (π,0) antinodal mirror plane (Fig. 1(a) vs. Fig. 1(b)) and give rise to the circular dichroism at antinode (π,0).^{15,17,19–21}

Compared to the Γ (0,0)-M (π,0) antinodal mirror plane (Figs. 1(a) and 1(b)), the Γ (0,0)-Y (π,π) nodal mirror plane provides an alternative platform to study possible electronic orders. As shown in Fig. 1(c), this nodal plane remains a mirror plane even with the presence of superstructures.^1,21 Therefore, the presence/absence of circular dichroism along the Γ(0,0)-Y (π,π) momentum cut in the nodal mirror plane would provide clear information about the nature of possible symmetry breaking orders in the sample.

An equally important, yet technical issue regarding circular dichroism experiments arises from the experimental geometry (geometric dichroism) which might mask the circular dichroism intrinsic to the sample.^15,16 Ideally, the geometric dichroism should be zero if the propagation vector of the light, the sample surface normal, and the final state momentum are all in a mirror plane of the sample.^14,22 However, a finite deviation from the desired perfect experimental geometry always exists and gives rise to geometric dichroism with a magnitude that depends on the deviation and the material. This leads to a natural question: how accurate the experiment needs to be performed before the intrinsic circular dichroism can be determined in this system?

In this paper, we report a study of the circular dichroism along the Γ (0,0)-Y (π,π) momentum cut in the nodal mirror plane of a Bi2212 underdoped T_c = 75 K sample (as shown in Fig. 1(c)). No circular dichroism is observed on the level of ∼2% at low temperature, which places a clear constraint on the forms of possible symmetry breaking orders that might exist in this system. Meanwhile, the geometric dichroism in this material is found to be very sensitive to any slight deviation of the sample angle from its perfect experimental geometry. An angular offset as small as 0.3°–0.5° is sufficient to induce a sizeable geometric circular dichroism.

The ARPES measurements were carried out at Beamline-9A of the Hiroshima Synchrotron Radiation Center with circularly polarized 21.2 eV photons. The system is equipped with a R4000 electron analyzer with a horizontal slit and a sample holder on a 6-axis manipulator. The experimental energy resolution was ∼12 meV, and the angular resolution was ∼0.3°. Only the center region of the detector (±10°) was selected to eliminate any edge effect of the electron analyzer. The Fermi level was referenced to that of a polycrystalline Au piece in electrical contact with the sample. Measurements were performed at 30 K with a base pressure better than 5 × 10⁻¹¹ Torr.

A perfect sample alignment is a prerequisite for the experiment. We use Fermi surface mapping over a large momentum range to align the sample.^1,5,23 With the aid of clearly resolved fine structures on the underlying Fermi surface (Fig. 2(a)), the sample was well aligned such that the propagation vector of the light, the sample surface normal, and the final state momentum are all in the Γ (0,0)-Y (π,π) nodal mirror plane. If the reflection symmetry is (not) broken by the electronic order, we should expect a finite (zero) circular dichroism along Γ (0,0)-Y (π,π) where the geometric dichroism is zero.

As presented in Fig. 2(c), when the momentum cut lies perfectly within the Γ (0,0)-Y (π,π) nodal mirror plane (Figs. 2(a) and 2(b)), the photoelectrons excited by lights with left- and right-handed circular polarizations exhibit identical intensities (I_CL = I_CR) along the entire cut, giving rise to zero circular dichroism within an error of ∼2% (Fig. 2(d)). While the intrinsic and geometric circular dichroism could happen to cancel each other and give zero dichroism at a certain momentum point when the sample is not perfectly aligned,^15,16 it is unlikely that they always have the same magnitude but with an opposite sign over the entire momentum cut. Therefore, our results suggest an absence of both the intrinsic and geometric dichroism (on the level of ∼2%) along the nodal direction of our sample.

To understand the implications of our result, we consider various electronic orders. It has been suggested that the antiferromagnetic order with Q = (π,π) and the “d-density wave” order would break the reflection symmetry about the nodal mirror plane and give rise to a finite circular dichroism in ARPES spectra.⁷ In principle, chiral symmetry breaking can also produce a non-zero circular dichroism. Nevertheless, one might ask whether the chiral symmetry breaking in cuprates can in principle be detected along the nodal direction: The pseudogap phenomena, implicated with the chiral order, are known to have a vanishing effect on the nodal electronic states near the Fermi level (E_F). Therefore, the nodal states, as the initial states of the photoemission process related to our observation, may not be chiral. Nevertheless, we note the studies on chiral molecular systems have shown that the circular dichroism in photoemission is dominated by final-state (delocalized) interactions-scattering of the outgoing photoelectrons off the chiral molecular framework.²⁴ Such a mechanism should be at work independent of the nature of chirality, whether structural or electronic. Therefore, if a chiral order exists in the material, whether it involves the nodal states, a circular dichroism associated with the latter on the mirror plane along the nodal direction should not be zero. On a quantitative level, a dichroism asymmetry factor on the order of 20% was seen in chiral molecular systems.²⁴ The absence of circular dichroism in our result does not seem to be consistent with the existence of a chiral order.

Nevertheless, a final conclusion about the existence of the aforementioned electronic orders has to be based on more theoretical efforts which aim to quantitatively determine the magnitude of the expected circular dichroism. This is especially true for the chiral order, of which limited knowledge is currently available. The expected value might be below our experimental sensitivity, different from the case of chiral molecular systems. In this regard, our study places an upper limit, ∼2%, to the circular dichroism in photoemission due to the possible existence of these electronic orders. Another possibility is that the symmetry breaking electronic orders only exist in a sample with lower doping level and/or at a particular temperature region between T_C and T*, since our measurements were performed on an underdoped 75 K sample below T_C. While more doping and temperature dependent measurements are needed to fully address this issue, we note the circular dichroism at the antinode (π,0) was observed on samples with similar doping levels and at various temperatures below T*, regardless of T_C.¹⁴ 

The absence of circular dichroism along the nodal direction also provides a unique chance to study the geometric dichroism as a function of the sample angle deviation. Fig. 3 shows the results for the momentum cuts in the vicinity of the Γ (0,0)-Y (π,π) nodal direction. The difference in angle (tilt angle θ, see Fig. 1(d) for its definition in the experimental setup) between these cuts is within 0.5°. Energy-integrated photoemission intensities for left-handed (I_CL) and right-handed (I_CR) circularly polarized lights along these cuts (Fig. 3(a)) are shown in Figs. 3(b)–3(e), respectively. A moderate geometric dichroism starts to show up at θ = 0.3° and becomes pronounced when θ reaches 0.5°. Our results suggest that the geometric dichroism in Bi2212 is very sensitive to small angle deviation from the perfect alignment.

To understand the origin of the geometric dichroism, we consider the photoemission under three-step model and sudden approximation. The photoemission intensity I is proportional to $|Mf,i|2$⁠, where $Mf,i≡⟨ϕf|H|ϕi⟩$ is the one-electron dipole matrix element describing the ejection of an electron from an initial state $|ϕi⟩$ to a final state $|ϕf⟩$⁠.¹ When the incident light is circularly polarized, the dipole operator containing the vector potential of photons with different polarizations can be written as H_CL (circular left) and H_CR (circular right). Then, the circular dichroism is denoted as D = $|⟨ϕf|HCL|ϕi⟩|2$–$|⟨ϕf|HCR|ϕi⟩|2$⁠.²² If we define an operator R as reflection of the blue plane of Fig. 1(a), then $R−1HCLR=HCR$ and $⟨ϕf|HCR|ϕi⟩$ = $⟨ϕf|R−1HCLR|ϕi⟩$⁠. When the blue plane is a mirror plane, $R|ϕi⟩=±|ϕi⟩, R|ϕf⟩=±|ϕf⟩$⁠, and thus D is zero.²² However, if there is a small angular offset and the blue plane is no longer a mirror plane, then the dichroism has a nonzero value. In this regard, materials with different electronic structure would generally have different responses in the photoemission circular dichroism to the angle deviation in experimental geometry. Theoretical calculations considering the band structure of Bi2212 are needed to understand the high sensitivity of the geometric dichroism in this system on a quantitative level.

In summary, by measuring the photoemission spectra along Γ (0,0)-Y (π,π) in the nodal mirror plane of a Bi2212 underdoped 75 K sample with circularly polarized light, we report the absence of a detectable circular dichroism on the level of ∼2% at low temperature. This result is not affected by the structural supermodulations, thus places a clear constraint on the forms of possible symmetry breaking electronic orders. Our result also reveals a high sensitivity of geometric dichroism to the slight angle deviation from the perfect experimental geometry. It highlights the importance to establish a perfect extinction of geometric dichroism before searching for any intrinsic circular dichroism in this material.

The work at Boston College was supported by a BC startup fund (J.H.), the U.S. NSF CAREER Award No. DMR-1454926 (R.-H.H.) and Graduate Research Fellowship DGE-1258923 (T.R.M.). ARPES experiments were performed at Hiroshima Synchrotron Radiation Center under Proposal No. 14-A-1. The work at Brookhaven National Laboratory was supported by the Office of Basic Energy Sciences, U.S. Department of Energy, under Contract No. DE-SC00112704.