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Antiferromagnetism of two-dimensional electronic gas on light-irradiated SrTiO3 and at LaAlO3/SrTiO3 interfaces
This content has been downloaded from IOPscience. Please scroll down to see the full text. 2015 J. Phys.: Condens. Matter 27 252001 (http://iopscience.iop.org/0953-8984/27/25/252001) View the table of contents for this issue, or go to the journal homepage for more
Download details: IP Address: 128.226.37.5 This content was downloaded on 02/06/2015 at 14:20
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Journal of Physics: Condensed Matter J. Phys.: Condens. Matter 27 (2015) 252001 (5pp)
doi:10.1088/0953-8984/27/25/252001
Fast Track Communication
Antiferromagnetism of two-dimensional electronic gas on light-irradiated SrTiO3 and at LaAlO3/SrTiO3 interfaces L P Gor’kov NHMFL, Florida State University, 1800 East Paul Dirac Drive, Tallahassee, FL 32310, USA L.D. Landau Institute for Theoretical Physics of the RAS, Chernogolovka 142432, Russia E-mail: [email protected] Received 16 February 2015 Accepted for publication 1 April 2015 Published 28 May 2015 Abstract
To gain an insight into the origin of tunable two-dimensional (2D) electronic liquid at the interfaces of transition-metal oxides, we address properties of a conducting layer on the light-irradiated surfaces of SrTiO3 ; the energy spectrum of the latter is known and consists of the titanium dxz /dyz and dxy bands. Recently, Santander-Syro et al (2014 Nature Mater. 13 1085) revealed that the dxy bands actually comprise two chiral branches with the Kramers degeneracy at the zone center lifted in the absence of a magnetic moment. We suggest that interacting electrons on the irradiated SrTiO3 go over into a magnetic phase as the result of one of the instabilities of the 2D Fermi liquid with exchange interactions, and point out the concrete antiferromagnetic order parameter. Large energy scales of the order of Fermi energy ∼0.1 eV inherent in this mechanism warrant stability of the magnetic ground state against ever-present effects of disorder. Arguments are given that electrons at the irradiated SrTiO3 surface and at the LaAlO3 /SrTiO3 interfaces undergo a kind of first-order transformation into one and the same phase of the 2D electronic Fermi liquid with reduced magnetic symmetry. Keywords: transition-metals oxides, interface, two-dimensional electron gas, insulator–metal transition, itinerant antiferromagnetism (Some figures may appear in colour only in the online journal)
including, in many instances coexisting 2D ferromagnetism [3–5] and 2D superconductivity [6–11]. A decade-long work using various advanced experimental techniques in combination with the theoretical calculations allowed the reconstruction of the structure of the quantum well at the intersection of two such wide-band insulators reasonably well. However, no major breakthrough at the level of microscopic physics was achieved regarding the very origin of the in-layer electronic gas. In particular, it concerns the nature of the insulator-to-metal transition. The latter bears the threshold character and occurs when thickness d of the LaAlO3 layer on top of SrTiO3 surpasses the value d = dcr of 4 unit cells (u.c.) [12].
1. Introduction
The groundbreaking discovery by Ohtomo and Hwang [1] of a metallic electronic layer at interfaces between the two oxides LaAlO3 (LAO) and SrTiO3 (STO) has led to the new exciting field of oxide electronics. A tunable two-dimensional gas of electrons (2DEG) with the surface density of charge in the range ns ∼ 1013 –1015 cm2 and a mobility as high as few 103 cm2 V−1 s −1 presents a perfectly new conducting liquid, thus opening high prospects for technologic advance [2]. Further developments have led to disclosure at the LAO/STO interfaces of a variety of remarkable phenomena of great interest for fundamental condensed matter physics in general, 0953-8984/15/252001+05$33.00
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© 2015 IOP Publishing Ltd Printed in the UK
J. Phys.: Condens. Matter 27 (2015) 252001
The situation is changing following the discovery of a metallic layer on the illuminated TiO2 -terminated surface of bare SrTiO3 ; pure crystalline SrTiO3 being an insulator with a broad gap ≈3.5 eV, electrons are doped into a quantum well at the bending conduction band by irradiating surface in the ultraviolet light [13–15]. The spectrum of 2DEG on the irradiated surface is comprised of the two groups of bands formed of the titanium 3d-levels: the two light dxy bands and pair of the heavier dxz /dyz ones. The four Fermi surfaces seen by ARPES are two concentric rings originating from the dxy bands and two dxz /dyz ellipsoids elongated along the kx and ky directions [14, 15]. Recent spin and angle resolved photoemission (SARPES) experiments [16] revealed that these two light dxy bands in reality are a pair of spin-polarized branches with spins winding in the opposite directions, as in the case of a surface band in the presence of the Rashba spin–orbit interaction [17]. Surprisingly, the two branches seem to be split at the 2D Brillouin zone center with the Kramers degeneracy at the momentum p = 0 lifted. That is, the time-reversal invariance is broken without a visible magnetic moment. The broken time-reversal symmetry in the dxy band [16] signifying a hidden magnetism is most intriguing. Below we propose the interpretation in terms of the Fermi liquid effects. We argue that the exchange interactions between electrons in the surface layer make the latter unstable towards spontaneous transition into a magnetic phase. Gaining energy, 2DEG in the new phase becomes stable with respect to disorder introduced by the oxygen vacancies and other irregularities at the surface. Our symmetry analysis leads to the antiferromagnetic order parameter in the form of a vector perpendicular to the plane and depending on the azimuth angle. Consistent with the anisotropy of out-of-plane component of the spins polarization, this order parameter results in minor changes in shape of the two concentric dxy Fermi surfaces. We show that these changes are compatible with the accuracy of data [16, 18]. The outstanding question then is whether the metallic energy spectrum observed on the irradiated surface of SrTiO3 may have any relation to the spectrum of 2DEG in heterostructures. The main experimental facts do not contradict the idea that electrons supplied by overlaying LAO layers onto the Ti 3d levels of SrTiO3 at the metal-to-insulator transition form the same 2DEG as on illuminated surfaces of bare SrTiO3 , keeping the topology of the dxy and dxz /dyz Fermi surfaces intact.
Most explicitly, the Fermi liquid effects show up in the non-vanishing spectral weight seen by ARPES at higher binding energies [14]. Because of interactions, a coherent quasi-particles spectrum gives way below the Fermi surface to spectrum of incoherent excitations and, in fact, in figure 1(a) [14] and figures 1(e) and (h) [16] (i.e., at the Dirac point and below) ARPES spectra are significantly smeared by incoherent contributions. Besides, the very existence of a highly conducting metallic layer in spite of inevitable disorder caused by the generated oxygen vacancies [14, 15] speaks in favor of interactions between electrons prevailing over the tendency to weak localization. 3. Energy spectrum of 2D magnetic phases
Following [16], consider the Hamiltonian: Hˆ = p2 /2m∗ + α(px σˆ y − py σˆ x ) + σˆ z S.
(1)
Here m∗ is the effective mass for a parabolic band; the vector p lies in the plane. The second term is the Rashba spin– orbit interaction [17]; σˆ x,y,z are the Pauli spin matrices, S is the perpendicular-to-plane component of some Zeeman field. (Dimensionless parameter κ = m∗ α/pF is a measure of strength of the Rashba term in equation (1).) The spectrum consists of the two branches λ = (±): Eλ (p) = p2 /2m∗ + λ α 2 p2 + S 2 . (2) As in [16], one finds that the chirality has the opposite signs on the two (λ = ±) branches. For the z-component of the electronic spin sz it follows: sz = λS/2 α 2 p2 + S 2 . (3) Besides that ferromagnetism has never been confirmed in SrTiO3 , non-zero magnetic moment would inevitably lead to formation of the ferromagnetic domains; local contributions from the non-zero moments in SARPES experiments are then expected to be averaged to zero [16]. According to [16], equation (2) reproduces the experimental spectrum reasonably well. We find that equation (3) is actually in contradiction with certain experimental results, the significance of which, as it seems, went unnoticed there. In fact, in figures 3(c)–(f ) [16] the z-component, sz has a different sign for each of the two pairs (1,4) and (2,3) shown in figure 3(a) [16], i.e., on the opposite side of one and the same branch; in equation (3) the sign of sz depends only on choice of the branch. Such behavior as in figure 3 [16] is not compatible with a constant Zeeman vector S in equation (1). Below for spectrum of the dxy band we propose the Hamiltonian also in the form of equation (2), but a constant vector S is now substituted with the vector S¯E (p) depending on the azimuth angle ϕ as:
2. Evidence of electron–electron correlations
Many transition-metals oxides with incompletely filled 3d-shells possessing narrow electronic bands display the features typical for the Fermi liquid. In particular, to that category belong the so-called high Tc cuprates. For the materials in hand, strong exchange correlations between electrons seem to play a role in the stability of 2DEG on the illuminated SrTiO3 [16]. Large band splitting 2S ≈ 90 meV corroborates this idea. Electron–electron correlations were proven to contribute at the analysis of the STM data [19].
S¯E (p) = |S| sin ϕ. 2
(4)
J. Phys.: Condens. Matter 27 (2015) 252001
Interactions are absorbed into the so-called f -function p )eˆ + (σˆ · σˆ )ξ(p; p ). The second fˆ(p; p |σ ; σ ) = ϕ(p; term stands for the spin–spin interactions in the exchange approximation; ξ(p; p ) can be expanded over the same irreducible representations as in equation (6): ξ(p, p ) = Zl χ l,t (p)χ l,t (p ), (9)
4. Stability of 2DEG
Phase transitions that may occur in the electronic Fermi liquid can be listed invoking the so-called Pomeranchuk instabilities [19, 20]. We search for the order parameters that violate the time-reversal invariance; only phases with the unchanged lattice periodicity are considered below. In the Hamiltonian for the exchange interactions in the system ˆ Hexc = I (p, p )([aˆ α+ (p) σˆ αβ aˆ β (p − k)]
t,l
To probe stability of the system, let σˆ z Sl (p) a small term having Such the form of equation (7) be added to a ‘bare’ ε0,αβ (p). perturbation causes in return a change εˆ (p) ⇒ εˆ (p) + σˆ z S¯l (p) The relation between in energy of the excitation εαβ (p). S¯l (p) follows from the linearized equation (8): Sl (p)and ¯ = σˆ z Sl (p) + ( σ · σ γ δ )ξ(p; p )[δn(p )]δγ σˆ z Sl (p)
αβλδ
·[aˆ γ+ (p )σˆ λδ aˆ δ (p
d2 p d2 p d2 k + k)])
(5)
assume a certain crystalline symmetry for the interaction I (p, p ); at the Fermi surface I (p, p ) can be expanded over the normalized functions χ l,t (p) belonging to the all irreducible representations (l) of a given group (the group C4v in our case): Il χ l,t (p)χ l,t (p ). (6) I (p, p ) =
× d2 p /(2π )2 .
(10)
(Here [δn(p )]δγ means the difference [δn(p )]δγ = n[ε(p) + δγ − n[ε0 (p)]; n[ε(p)] is the Fermi function.) In σz S¯l (p)] fact, in the right-hand side of equation (10) [δn(p )]δγ can Leaving at the moment only the linear be expanded in S¯l (p). term: in S¯l (p) (∂n0 (p )/dε)(p dp /π ), = Sl (p) + S¯l (p)Z l S¯l (p)
t,l
(t numerates all basis functions if a representation is degenerate.) The general form for the magnetic order parameter is a certain vector Sl (p): = ηtl,i χ l,t (p). (7) Sli (p)
λ
(11) one finds:
t
(A specific vector Sl (p) in (7) originates from one of the so-called ‘anomalous’ averages of the particle operators PF aˆ α+ (p) σˆ αβ aˆ β (p) in equation (5): Sl (p) ∝ < 0 + ∗ aˆ α (p) σˆ αφ aˆ β (p) > (m dp/2π) integrated over |p| below the Fermi momentum and multiplied by one of the interaction constants Il in (6).) We consider only the vectors Sz (p) that are perpendicular to the plane (an in-plane vector S cannot split the Dirac point) and non-identical representations χ l,t(p) in (7). If averaged over the in-plane angle, Sz (p) ⇒ χ ;l,t (p) dϕ ≡ 0; as distinguished from the ferromagnetic vector at the identical representation, such ‘itinerant antiferromagnetic’ order parameter does not imply domains formation.
= Sl (p)/ 1 + Zl S¯l (p)
νλ (εF ) ≡ Sl (p)/(1 + Z¯ l ).
(12)
λ
Z¯ l in (12) is Zl multiplied by the summary density of states ν(εF ). (On every Fermi surface: p dp/π ⇒ (pm∗ /π(p + λαm∗ ) dε ≡ νλ (ε) dε; λ = ± stands for the solutions ε± (p) = p2 /2m∗ ± αp; pF,λ are the two Fermi momenta.) Factor 1/(1 + Z¯ l ) defines the ‘magnetic’ susceptibility of pole at the system in response to the perturbation σˆ z Sl (p).The Z¯ l = −1 would signify instability of the system with respect to the spontaneous transition into a phase with one of the vectors in (7) as the order parameter. Sl (p) 4.2. Symmetry arguments
Vector S¯l (p) substituted, equations (2) and (3) acquire dependence on the azimuth angle. Besides the identical representation x 2 + y 2 , group C4v has two one-dimensional representations B1 : xy ∝ sin 2ϕ and B2 : x 2 − y 2 ∝ cos 2ϕ, and one two-dimensional representation E : x, y ⇒ cos ϕ, sin ϕ. B1 and B2 are both even at ϕ ⇒ ϕ ± π. That is, figures 3(c)–(f ) [16] for sz seem to be consistent only with the choice of the E-representation. Of the two components (cos ϕ, sin ϕ) one must choose one; in correspondence with = |S| sin ϕ in figures 3(c)–(f ) [16] we suggested S¯E (p) equation (4). Substitution of (4) into (2) gives for the energy spectrum: Eλ (p) = p2 /2m∗ + λ α 2 p2 + S 2 sin2 ϕ. (13)
4.1. Derivation of the stability criterion
It is useful to present here a simple derivation of the criterion [19]. The basic notion of the Landau Fermi liquid is that the energy of the whole system is a functional of the occupation numbers of the interacting Fermi particles constituting it. Correspondingly, the energy εαβ (p) of an elementary excitation is made up of a ‘bare’ energy ε0,αβ (p) and of the second contribution accounting for the action on the part of all particles disturbed by emergence of the excitation εαβ (p): = ε0,αβ (p) + fαβ;γ δ (p; p )[δn(p )]δγ d2 p /(2π )2 . εαβ (p) (8) 3
J. Phys.: Condens. Matter 27 (2015) 252001
(a)
(b)
Figure 1. The upper branch of the energy spectrum (13) drawn in a close vicinity of the Brillouin zone center E+ (p) ≈
α 2 p2 + S 2 sin2 ϕ. (a) Projections of contours of the constant energy at different E > 0. (b) A three-dimensional view. The singular character of the energy spectrum at p = 0 becomes smoothed in the presence of the Fermi liquid effects.
Near the zone center one has Eλ (p) ≈ λ α 2 p2 + S 2 sin2 ϕ. Let E > 0. Projections of the contours of constant energy E encircle figures that are oblate along the y-axis, as shown near in figure 1(a). A 3D view of the spectrum E+ (p) the zone center is drawn in figure 1(b). The Fermi liquid interactions will smear the singular behavior of the spectrum at p = 0. E+ (p)
is S¯E (p) = η1 cos ϕ + η2 sin ϕ. To general form of S¯E (p) determine the ratio between parameters (η1 , η2 ), one needs, as in the Landau theory of the second order phase transitions, a functional F (η1 , η2 ) to present the two equations for (η1 , η2 ) in the equivalent form as equations of the equilibrium δF (η1 ; η2 )/δηt = 0. Simple but tedious calculations give F (η1 , η2 ) ∝ {(1 + Z¯ E ) (η12 + η22 ) + (A/4εF2 )[(η12 + η22 )2 − (1/2)(η14 + η24 )]}. (For definition of A, see below). Minimum in F (η1 , η2 ) (if A > 0) is reached at the non-zero either (η1 ), or (η2 ). In conjunction with data in figures 3(c)–(f ) [16] we choose η1 = 0. With the term quadratic in the temperature one obtains: (14) |1 + Z¯ E | = (A/8)[(S¯E2 + TC2 )/εF2 ]. 2 2 2 In (14) A = −2 /ν(εF ); 2 = εF ± (∂ νi (ε)/∂ε ) is the sum of the second derivatives of density of states on the two branches, εF = pF2 /2m∗ . In a three-dimensional isotropic system ∂ 2 ν(ε)/∂ε 2 = −(1/4ε2 ) and one may infer the second-order character of the phase transition at the Pomeranchuk instability. The two-dimensional density of states is a constant; the second derivative equals zero identically even for the energy spectrum with the Rashba term Eλ (p) = p2 /2m∗ + λαp. A small cubic term in the spin–orbit interaction αp ⇒ α0 p(1 + γp2 /pF2 ) produces a non-zero 2 ∝ −α0 γ . The sign is not theoretically determined. At 2 < 0 equation (14) describes the second-order phase transition. At 2 > 0 the transition would be of the first order and higher terms in the expansion become necessary. Although doping is not the thermodynamic process, by analogy to the insulator–metal transition in LAO/STO interfaces we infer that in (14) A < 0 and the would-be transition is of the first order. The notable feature of a phase transition driven by the Pomeranchuk instability is that the only characteristic energy scale inherent in the mechanism is the Fermi energy εF = pF2 /2m∗ . For materials in hand one infers that in the magnetic phase 2S¯E ∼ εF ∼ 0.1 eV, as is indeed found in [16]. With such high TC ∼ 1000 K in experiments [13–16] we are always dealing with the magnetic state at much lower temperatures.
Shape of the dxy Fermi surfaces. Equation (13) implies the two-fold axial symmetry also for the Fermi surfaces of the two dxy bands. To estimate the magnitude of deviations from the ring-like shape of two concentric Fermi surfaces in the isotropic spectrum equation (2) with S = const, rewrite the 4.2.1.
2 (ϕ) − pF2 = λ (2m∗ αpF,λ )2 + (2m∗ S)2 sin2 ϕ difference pF,λ 2 in the following dimensionless form: pF,λ (ϕ)/pF2 − 2 1 = λ (2αm∗ /pF )2 + (S/εF )2 sin ϕ;pF,λ (ϕ) are the Fermi momenta on the two branches λ = (±); difference between pF,λ (ϕ) and pF under the square root is neglected. Taking for the purpose of illustration all parameters m∗ = 0.65me , α = 5 × 10−11 eV m, kF,+ 0.18 A−1 , kF,− 0.12 A−1 ; kF = (kF,+ + kF,− )/2 0.15 A−1 , and S ∼ 40 meV from [16] and substituting numbers, we find |kF,λ (π/2) − kF,λ (0)|/2 ≈ 0.05kF ∼ 0.0075 A−1 . That is, deviations from the shape of a ring are small and choice of the order parameter equation (4) would not contradict experimental data [16].
5. Transition into the magnetic phase
It is instructive to consider the thermodynamic transition into a new phase near the stability threshold of a non-magnetic Fermi liquid at |Z¯ l + 1| 1. To derive equations for the parameters of the new phase at temperatures below the critical in equation (11) and expand temperature TC , omit ‘bare’ Sl (p) multiply [δn(p )]δγ up to terms of the third order in S¯l (p); and integrate along the both sides in turns by functions χ l,t (p) Fermi surface. Such common procedure leads to the equations for the whole set of parameters ηt in equation (7). For the two-dimensional representation E : x, y ⇒ cos ϕ, sin ϕ the 4
J. Phys.: Condens. Matter 27 (2015) 252001
6. Comparison with 2DEG at the LAO/STO interface
Acknowledgments
To start with, one needs to recall that the metal–insulator transition at LAO/STO interfaces bears the extremely abrupt character. Thus, at the 4 u.c. thick LaAlO3 layer the conductance jumps by five orders in magnitude from being below the measurable limit [12]. At the same thickness the scanning tunneling spectroscopy (STM) experiments reveal non-zero density of states in the conducting layer [21]. Four unit cells of LAO are critical both for ferromagnetism [22, 23] and for 2D superconductivity [7, 24]. The metal–insulator transition seems to coincide as well with a structural change occurring at 4 u.c. described in [18] as a polar Ti–O-buckling. Taken together, the observations of such a sharp threshold draw a pattern of the first order-type transition into the distinct new phase. 2DEG on the buried LAO/STO interfaces was studied by the spectroscopic means in [11, 18]. Mapped in the k-space [11], the band states at the interface with LAO layers exceeding the critical thickness 4 u.c. reveal the same topology of the Fermi surfaces, as in [13]. Another instructive fact is this. While the process of doping in the LAO/STO interfaces generally goes with increasing thickness of the LAO layer, both the itinerant and localized electrons in all cases occupy the same Ti 3d-levels on the side of SrTiO3 [11, 25, 26]. Taken together, the above experimental facts give strong support to the idea that electrons at the irradiated SrTiO3 surface and at the LAO/STO interfaces undergo a kind of first-order transformation into one and the same phase of twodimensional electronic Fermi liquid with the reduced magnetic symmetry.
The author thanks L Levitov for drawing his attention to experiments [16], E Rashba for many stimulating discussions and G Teitel’baum for the help with figure 1. The work was supported by the NHMFL through NSF Grant No. DMR1157490, the State of Florida and the US Department of Energy. References [1] Ohtomo A and Hwang H Y 2004 Nature 427 423 [2] Thiel S, Hammer G, Schmeh A, Schneider C W and Mannhart J 2006 Science 313 1942 [3] Lee J-S, Xie Y W, Sato H K, Bell C, Hikita Y, Hwang H Y and Kao C-C 2013 Nat. Mater. 12 703 [4] Bi F, Huang M, Bark C-W, Ryu S, Eom C-B, Irvin P and Levy J 2013 arXiv:1307.5557v2 [5] Joshua A, Ruhman J, Pecker S, Altman E and Ilani S 2012 Nat. Commun. 3 1129 [6] Reyren N et al 2007 Science 313 1196 [7] Gariglio S, Reyren N, Caviglia A D and Triscone J-M 2009 J. Phys.: Condens. Matter 21 164213 [8] Dikin D A, Mehta M, Bark C W, Folkman C M, Eom C B and Chandrasekhar V 2011 Phys. Rev. Lett. 107 056802 [9] Li L, Richter C, Mannhart J and Ashoori R C 2011 Nat. Phys. 7 762 [10] Bert J A, Kalisky B, Bell C, Kim M, Hikita Y, Hwang H Y and Moler K A 2011 Nat. Phys. 7 767 [11] Berner G et al 2013 Phys. Rev. Lett. 110 247601 [12] Mannhart J, Blank D H A, Hwang H Y, Millis A J and Triscone J-M 2008 MRS Bull. 33 1027 [13] Santander-Syro A F et al 2011 Nature 469 189 [14] Meevasana W, King P D C, He R H, Mol S-K, Hashimoto M, Tamai A, Songsiriritthigul P, Baumberger F and Shen Z-X 2011 Nat. Mater. 10 114 [15] Plumb C et al 2014 Phys. Rev. Lett. 113 086801 [16] Santander-Syro A F, Fortuna F, Bareille C, R¨odel T C, Landolt G, Plumb N C, Dil J H and Radovic M 2014 Nat. Mater. 13 1085 [17] Bychkov Yu and Rashba E I 1984 JETP Lett. 39 78 [18] Plumb N C et al 2013 arXiv:1304.5948v1 [19] Pomeranchuk I Ya 1959 J. Exp. Theor. Phys. 8 361 [20] Gor’kov L P and Sokol A 1992 Phys. Rev. Lett. 69 2586 [21] Breitschaft M et al 2010 Phys. Rev. B 81 153414 [22] Kalisky B, Bert J A, Klopfer B B, Bell C, Sato H K, Hosoda M, Hikita Y, Hwang H Y and Moler K A 2012 Nat. Commun. 3 922 [23] Salman Z et al 2012 Phys. Rev. Lett. 109 257207 [24] Reyren N et al 2007 Science 317 1196 [25] Berner G et al 2010 Phys. Rev. B 82 241405 [26] Salluzzo M et al 2009 Phys. Rev. Lett. 102 166804 [27] Cancellieri C, Reinle-Schmitt M L, Kobayashi M, Strocov V N and Willmott P R 2013 arXiv:1307.6943v1
7. Conclusions
In summary, magnetism of the conducting layer was inferred from the fact of lifted Kramers degeneracy in [16]. We proposed the Pomeranchuk-type instability as the mechanism for formation of the magnetic phase and pointed out the concrete symmetry of the antiferromagnetic order parameter. We argued that a large energy scale of the order of the Fermi energy inherent in this mechanism protects the ground state of 2DEG against ever-present random effects of disorder. The theoretical concepts put forward in the above prompt for further elaboration on the part of experiment. In particular, we suggest analysis of the perpendicular-to-plane spin polarization component. It would be equally interesting if ARPES were able to discern the two-fold symmetry in the vicinity of the Dirac point at the center of the Brillouin zone.
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Antiferromagnetism of two-dimensional electronic gas on light-irradiated SrTiO3 and at LaAlO3/SrTiO3 interfaces
This content has been downloaded from IOPscience. Please scroll down to see the full text. 2015 J. Phys.: Condens. Matter 27 252001 (http://iopscience.iop.org/0953-8984/27/25/252001) View the table of contents for this issue, or go to the journal homepage for more
Download details: IP Address: 128.226.37.5 This content was downloaded on 02/06/2015 at 14:20
Please note that terms and conditions apply.
Journal of Physics: Condensed Matter J. Phys.: Condens. Matter 27 (2015) 252001 (5pp)
doi:10.1088/0953-8984/27/25/252001
Fast Track Communication
Antiferromagnetism of two-dimensional electronic gas on light-irradiated SrTiO3 and at LaAlO3/SrTiO3 interfaces L P Gor’kov NHMFL, Florida State University, 1800 East Paul Dirac Drive, Tallahassee, FL 32310, USA L.D. Landau Institute for Theoretical Physics of the RAS, Chernogolovka 142432, Russia E-mail: [email protected] Received 16 February 2015 Accepted for publication 1 April 2015 Published 28 May 2015 Abstract
To gain an insight into the origin of tunable two-dimensional (2D) electronic liquid at the interfaces of transition-metal oxides, we address properties of a conducting layer on the light-irradiated surfaces of SrTiO3 ; the energy spectrum of the latter is known and consists of the titanium dxz /dyz and dxy bands. Recently, Santander-Syro et al (2014 Nature Mater. 13 1085) revealed that the dxy bands actually comprise two chiral branches with the Kramers degeneracy at the zone center lifted in the absence of a magnetic moment. We suggest that interacting electrons on the irradiated SrTiO3 go over into a magnetic phase as the result of one of the instabilities of the 2D Fermi liquid with exchange interactions, and point out the concrete antiferromagnetic order parameter. Large energy scales of the order of Fermi energy ∼0.1 eV inherent in this mechanism warrant stability of the magnetic ground state against ever-present effects of disorder. Arguments are given that electrons at the irradiated SrTiO3 surface and at the LaAlO3 /SrTiO3 interfaces undergo a kind of first-order transformation into one and the same phase of the 2D electronic Fermi liquid with reduced magnetic symmetry. Keywords: transition-metals oxides, interface, two-dimensional electron gas, insulator–metal transition, itinerant antiferromagnetism (Some figures may appear in colour only in the online journal)
including, in many instances coexisting 2D ferromagnetism [3–5] and 2D superconductivity [6–11]. A decade-long work using various advanced experimental techniques in combination with the theoretical calculations allowed the reconstruction of the structure of the quantum well at the intersection of two such wide-band insulators reasonably well. However, no major breakthrough at the level of microscopic physics was achieved regarding the very origin of the in-layer electronic gas. In particular, it concerns the nature of the insulator-to-metal transition. The latter bears the threshold character and occurs when thickness d of the LaAlO3 layer on top of SrTiO3 surpasses the value d = dcr of 4 unit cells (u.c.) [12].
1. Introduction
The groundbreaking discovery by Ohtomo and Hwang [1] of a metallic electronic layer at interfaces between the two oxides LaAlO3 (LAO) and SrTiO3 (STO) has led to the new exciting field of oxide electronics. A tunable two-dimensional gas of electrons (2DEG) with the surface density of charge in the range ns ∼ 1013 –1015 cm2 and a mobility as high as few 103 cm2 V−1 s −1 presents a perfectly new conducting liquid, thus opening high prospects for technologic advance [2]. Further developments have led to disclosure at the LAO/STO interfaces of a variety of remarkable phenomena of great interest for fundamental condensed matter physics in general, 0953-8984/15/252001+05$33.00
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J. Phys.: Condens. Matter 27 (2015) 252001
The situation is changing following the discovery of a metallic layer on the illuminated TiO2 -terminated surface of bare SrTiO3 ; pure crystalline SrTiO3 being an insulator with a broad gap ≈3.5 eV, electrons are doped into a quantum well at the bending conduction band by irradiating surface in the ultraviolet light [13–15]. The spectrum of 2DEG on the irradiated surface is comprised of the two groups of bands formed of the titanium 3d-levels: the two light dxy bands and pair of the heavier dxz /dyz ones. The four Fermi surfaces seen by ARPES are two concentric rings originating from the dxy bands and two dxz /dyz ellipsoids elongated along the kx and ky directions [14, 15]. Recent spin and angle resolved photoemission (SARPES) experiments [16] revealed that these two light dxy bands in reality are a pair of spin-polarized branches with spins winding in the opposite directions, as in the case of a surface band in the presence of the Rashba spin–orbit interaction [17]. Surprisingly, the two branches seem to be split at the 2D Brillouin zone center with the Kramers degeneracy at the momentum p = 0 lifted. That is, the time-reversal invariance is broken without a visible magnetic moment. The broken time-reversal symmetry in the dxy band [16] signifying a hidden magnetism is most intriguing. Below we propose the interpretation in terms of the Fermi liquid effects. We argue that the exchange interactions between electrons in the surface layer make the latter unstable towards spontaneous transition into a magnetic phase. Gaining energy, 2DEG in the new phase becomes stable with respect to disorder introduced by the oxygen vacancies and other irregularities at the surface. Our symmetry analysis leads to the antiferromagnetic order parameter in the form of a vector perpendicular to the plane and depending on the azimuth angle. Consistent with the anisotropy of out-of-plane component of the spins polarization, this order parameter results in minor changes in shape of the two concentric dxy Fermi surfaces. We show that these changes are compatible with the accuracy of data [16, 18]. The outstanding question then is whether the metallic energy spectrum observed on the irradiated surface of SrTiO3 may have any relation to the spectrum of 2DEG in heterostructures. The main experimental facts do not contradict the idea that electrons supplied by overlaying LAO layers onto the Ti 3d levels of SrTiO3 at the metal-to-insulator transition form the same 2DEG as on illuminated surfaces of bare SrTiO3 , keeping the topology of the dxy and dxz /dyz Fermi surfaces intact.
Most explicitly, the Fermi liquid effects show up in the non-vanishing spectral weight seen by ARPES at higher binding energies [14]. Because of interactions, a coherent quasi-particles spectrum gives way below the Fermi surface to spectrum of incoherent excitations and, in fact, in figure 1(a) [14] and figures 1(e) and (h) [16] (i.e., at the Dirac point and below) ARPES spectra are significantly smeared by incoherent contributions. Besides, the very existence of a highly conducting metallic layer in spite of inevitable disorder caused by the generated oxygen vacancies [14, 15] speaks in favor of interactions between electrons prevailing over the tendency to weak localization. 3. Energy spectrum of 2D magnetic phases
Following [16], consider the Hamiltonian: Hˆ = p2 /2m∗ + α(px σˆ y − py σˆ x ) + σˆ z S.
(1)
Here m∗ is the effective mass for a parabolic band; the vector p lies in the plane. The second term is the Rashba spin– orbit interaction [17]; σˆ x,y,z are the Pauli spin matrices, S is the perpendicular-to-plane component of some Zeeman field. (Dimensionless parameter κ = m∗ α/pF is a measure of strength of the Rashba term in equation (1).) The spectrum consists of the two branches λ = (±): Eλ (p) = p2 /2m∗ + λ α 2 p2 + S 2 . (2) As in [16], one finds that the chirality has the opposite signs on the two (λ = ±) branches. For the z-component of the electronic spin sz it follows: sz = λS/2 α 2 p2 + S 2 . (3) Besides that ferromagnetism has never been confirmed in SrTiO3 , non-zero magnetic moment would inevitably lead to formation of the ferromagnetic domains; local contributions from the non-zero moments in SARPES experiments are then expected to be averaged to zero [16]. According to [16], equation (2) reproduces the experimental spectrum reasonably well. We find that equation (3) is actually in contradiction with certain experimental results, the significance of which, as it seems, went unnoticed there. In fact, in figures 3(c)–(f ) [16] the z-component, sz has a different sign for each of the two pairs (1,4) and (2,3) shown in figure 3(a) [16], i.e., on the opposite side of one and the same branch; in equation (3) the sign of sz depends only on choice of the branch. Such behavior as in figure 3 [16] is not compatible with a constant Zeeman vector S in equation (1). Below for spectrum of the dxy band we propose the Hamiltonian also in the form of equation (2), but a constant vector S is now substituted with the vector S¯E (p) depending on the azimuth angle ϕ as:
2. Evidence of electron–electron correlations
Many transition-metals oxides with incompletely filled 3d-shells possessing narrow electronic bands display the features typical for the Fermi liquid. In particular, to that category belong the so-called high Tc cuprates. For the materials in hand, strong exchange correlations between electrons seem to play a role in the stability of 2DEG on the illuminated SrTiO3 [16]. Large band splitting 2S ≈ 90 meV corroborates this idea. Electron–electron correlations were proven to contribute at the analysis of the STM data [19].
S¯E (p) = |S| sin ϕ. 2
(4)
J. Phys.: Condens. Matter 27 (2015) 252001
Interactions are absorbed into the so-called f -function p )eˆ + (σˆ · σˆ )ξ(p; p ). The second fˆ(p; p |σ ; σ ) = ϕ(p; term stands for the spin–spin interactions in the exchange approximation; ξ(p; p ) can be expanded over the same irreducible representations as in equation (6): ξ(p, p ) = Zl χ l,t (p)χ l,t (p ), (9)
4. Stability of 2DEG
Phase transitions that may occur in the electronic Fermi liquid can be listed invoking the so-called Pomeranchuk instabilities [19, 20]. We search for the order parameters that violate the time-reversal invariance; only phases with the unchanged lattice periodicity are considered below. In the Hamiltonian for the exchange interactions in the system ˆ Hexc = I (p, p )([aˆ α+ (p) σˆ αβ aˆ β (p − k)]
t,l
To probe stability of the system, let σˆ z Sl (p) a small term having Such the form of equation (7) be added to a ‘bare’ ε0,αβ (p). perturbation causes in return a change εˆ (p) ⇒ εˆ (p) + σˆ z S¯l (p) The relation between in energy of the excitation εαβ (p). S¯l (p) follows from the linearized equation (8): Sl (p)and ¯ = σˆ z Sl (p) + ( σ · σ γ δ )ξ(p; p )[δn(p )]δγ σˆ z Sl (p)
αβλδ
·[aˆ γ+ (p )σˆ λδ aˆ δ (p
d2 p d2 p d2 k + k)])
(5)
assume a certain crystalline symmetry for the interaction I (p, p ); at the Fermi surface I (p, p ) can be expanded over the normalized functions χ l,t (p) belonging to the all irreducible representations (l) of a given group (the group C4v in our case): Il χ l,t (p)χ l,t (p ). (6) I (p, p ) =
× d2 p /(2π )2 .
(10)
(Here [δn(p )]δγ means the difference [δn(p )]δγ = n[ε(p) + δγ − n[ε0 (p)]; n[ε(p)] is the Fermi function.) In σz S¯l (p)] fact, in the right-hand side of equation (10) [δn(p )]δγ can Leaving at the moment only the linear be expanded in S¯l (p). term: in S¯l (p) (∂n0 (p )/dε)(p dp /π ), = Sl (p) + S¯l (p)Z l S¯l (p)
t,l
(t numerates all basis functions if a representation is degenerate.) The general form for the magnetic order parameter is a certain vector Sl (p): = ηtl,i χ l,t (p). (7) Sli (p)
λ
(11) one finds:
t
(A specific vector Sl (p) in (7) originates from one of the so-called ‘anomalous’ averages of the particle operators PF aˆ α+ (p) σˆ αβ aˆ β (p) in equation (5): Sl (p) ∝ < 0 + ∗ aˆ α (p) σˆ αφ aˆ β (p) > (m dp/2π) integrated over |p| below the Fermi momentum and multiplied by one of the interaction constants Il in (6).) We consider only the vectors Sz (p) that are perpendicular to the plane (an in-plane vector S cannot split the Dirac point) and non-identical representations χ l,t(p) in (7). If averaged over the in-plane angle, Sz (p) ⇒ χ ;l,t (p) dϕ ≡ 0; as distinguished from the ferromagnetic vector at the identical representation, such ‘itinerant antiferromagnetic’ order parameter does not imply domains formation.
= Sl (p)/ 1 + Zl S¯l (p)
νλ (εF ) ≡ Sl (p)/(1 + Z¯ l ).
(12)
λ
Z¯ l in (12) is Zl multiplied by the summary density of states ν(εF ). (On every Fermi surface: p dp/π ⇒ (pm∗ /π(p + λαm∗ ) dε ≡ νλ (ε) dε; λ = ± stands for the solutions ε± (p) = p2 /2m∗ ± αp; pF,λ are the two Fermi momenta.) Factor 1/(1 + Z¯ l ) defines the ‘magnetic’ susceptibility of pole at the system in response to the perturbation σˆ z Sl (p).The Z¯ l = −1 would signify instability of the system with respect to the spontaneous transition into a phase with one of the vectors in (7) as the order parameter. Sl (p) 4.2. Symmetry arguments
Vector S¯l (p) substituted, equations (2) and (3) acquire dependence on the azimuth angle. Besides the identical representation x 2 + y 2 , group C4v has two one-dimensional representations B1 : xy ∝ sin 2ϕ and B2 : x 2 − y 2 ∝ cos 2ϕ, and one two-dimensional representation E : x, y ⇒ cos ϕ, sin ϕ. B1 and B2 are both even at ϕ ⇒ ϕ ± π. That is, figures 3(c)–(f ) [16] for sz seem to be consistent only with the choice of the E-representation. Of the two components (cos ϕ, sin ϕ) one must choose one; in correspondence with = |S| sin ϕ in figures 3(c)–(f ) [16] we suggested S¯E (p) equation (4). Substitution of (4) into (2) gives for the energy spectrum: Eλ (p) = p2 /2m∗ + λ α 2 p2 + S 2 sin2 ϕ. (13)
4.1. Derivation of the stability criterion
It is useful to present here a simple derivation of the criterion [19]. The basic notion of the Landau Fermi liquid is that the energy of the whole system is a functional of the occupation numbers of the interacting Fermi particles constituting it. Correspondingly, the energy εαβ (p) of an elementary excitation is made up of a ‘bare’ energy ε0,αβ (p) and of the second contribution accounting for the action on the part of all particles disturbed by emergence of the excitation εαβ (p): = ε0,αβ (p) + fαβ;γ δ (p; p )[δn(p )]δγ d2 p /(2π )2 . εαβ (p) (8) 3
J. Phys.: Condens. Matter 27 (2015) 252001
(a)
(b)
Figure 1. The upper branch of the energy spectrum (13) drawn in a close vicinity of the Brillouin zone center E+ (p) ≈
α 2 p2 + S 2 sin2 ϕ. (a) Projections of contours of the constant energy at different E > 0. (b) A three-dimensional view. The singular character of the energy spectrum at p = 0 becomes smoothed in the presence of the Fermi liquid effects.
Near the zone center one has Eλ (p) ≈ λ α 2 p2 + S 2 sin2 ϕ. Let E > 0. Projections of the contours of constant energy E encircle figures that are oblate along the y-axis, as shown near in figure 1(a). A 3D view of the spectrum E+ (p) the zone center is drawn in figure 1(b). The Fermi liquid interactions will smear the singular behavior of the spectrum at p = 0. E+ (p)
is S¯E (p) = η1 cos ϕ + η2 sin ϕ. To general form of S¯E (p) determine the ratio between parameters (η1 , η2 ), one needs, as in the Landau theory of the second order phase transitions, a functional F (η1 , η2 ) to present the two equations for (η1 , η2 ) in the equivalent form as equations of the equilibrium δF (η1 ; η2 )/δηt = 0. Simple but tedious calculations give F (η1 , η2 ) ∝ {(1 + Z¯ E ) (η12 + η22 ) + (A/4εF2 )[(η12 + η22 )2 − (1/2)(η14 + η24 )]}. (For definition of A, see below). Minimum in F (η1 , η2 ) (if A > 0) is reached at the non-zero either (η1 ), or (η2 ). In conjunction with data in figures 3(c)–(f ) [16] we choose η1 = 0. With the term quadratic in the temperature one obtains: (14) |1 + Z¯ E | = (A/8)[(S¯E2 + TC2 )/εF2 ]. 2 2 2 In (14) A = −2 /ν(εF ); 2 = εF ± (∂ νi (ε)/∂ε ) is the sum of the second derivatives of density of states on the two branches, εF = pF2 /2m∗ . In a three-dimensional isotropic system ∂ 2 ν(ε)/∂ε 2 = −(1/4ε2 ) and one may infer the second-order character of the phase transition at the Pomeranchuk instability. The two-dimensional density of states is a constant; the second derivative equals zero identically even for the energy spectrum with the Rashba term Eλ (p) = p2 /2m∗ + λαp. A small cubic term in the spin–orbit interaction αp ⇒ α0 p(1 + γp2 /pF2 ) produces a non-zero 2 ∝ −α0 γ . The sign is not theoretically determined. At 2 < 0 equation (14) describes the second-order phase transition. At 2 > 0 the transition would be of the first order and higher terms in the expansion become necessary. Although doping is not the thermodynamic process, by analogy to the insulator–metal transition in LAO/STO interfaces we infer that in (14) A < 0 and the would-be transition is of the first order. The notable feature of a phase transition driven by the Pomeranchuk instability is that the only characteristic energy scale inherent in the mechanism is the Fermi energy εF = pF2 /2m∗ . For materials in hand one infers that in the magnetic phase 2S¯E ∼ εF ∼ 0.1 eV, as is indeed found in [16]. With such high TC ∼ 1000 K in experiments [13–16] we are always dealing with the magnetic state at much lower temperatures.
Shape of the dxy Fermi surfaces. Equation (13) implies the two-fold axial symmetry also for the Fermi surfaces of the two dxy bands. To estimate the magnitude of deviations from the ring-like shape of two concentric Fermi surfaces in the isotropic spectrum equation (2) with S = const, rewrite the 4.2.1.
2 (ϕ) − pF2 = λ (2m∗ αpF,λ )2 + (2m∗ S)2 sin2 ϕ difference pF,λ 2 in the following dimensionless form: pF,λ (ϕ)/pF2 − 2 1 = λ (2αm∗ /pF )2 + (S/εF )2 sin ϕ;pF,λ (ϕ) are the Fermi momenta on the two branches λ = (±); difference between pF,λ (ϕ) and pF under the square root is neglected. Taking for the purpose of illustration all parameters m∗ = 0.65me , α = 5 × 10−11 eV m, kF,+ 0.18 A−1 , kF,− 0.12 A−1 ; kF = (kF,+ + kF,− )/2 0.15 A−1 , and S ∼ 40 meV from [16] and substituting numbers, we find |kF,λ (π/2) − kF,λ (0)|/2 ≈ 0.05kF ∼ 0.0075 A−1 . That is, deviations from the shape of a ring are small and choice of the order parameter equation (4) would not contradict experimental data [16].
5. Transition into the magnetic phase
It is instructive to consider the thermodynamic transition into a new phase near the stability threshold of a non-magnetic Fermi liquid at |Z¯ l + 1| 1. To derive equations for the parameters of the new phase at temperatures below the critical in equation (11) and expand temperature TC , omit ‘bare’ Sl (p) multiply [δn(p )]δγ up to terms of the third order in S¯l (p); and integrate along the both sides in turns by functions χ l,t (p) Fermi surface. Such common procedure leads to the equations for the whole set of parameters ηt in equation (7). For the two-dimensional representation E : x, y ⇒ cos ϕ, sin ϕ the 4
J. Phys.: Condens. Matter 27 (2015) 252001
6. Comparison with 2DEG at the LAO/STO interface
Acknowledgments
To start with, one needs to recall that the metal–insulator transition at LAO/STO interfaces bears the extremely abrupt character. Thus, at the 4 u.c. thick LaAlO3 layer the conductance jumps by five orders in magnitude from being below the measurable limit [12]. At the same thickness the scanning tunneling spectroscopy (STM) experiments reveal non-zero density of states in the conducting layer [21]. Four unit cells of LAO are critical both for ferromagnetism [22, 23] and for 2D superconductivity [7, 24]. The metal–insulator transition seems to coincide as well with a structural change occurring at 4 u.c. described in [18] as a polar Ti–O-buckling. Taken together, the observations of such a sharp threshold draw a pattern of the first order-type transition into the distinct new phase. 2DEG on the buried LAO/STO interfaces was studied by the spectroscopic means in [11, 18]. Mapped in the k-space [11], the band states at the interface with LAO layers exceeding the critical thickness 4 u.c. reveal the same topology of the Fermi surfaces, as in [13]. Another instructive fact is this. While the process of doping in the LAO/STO interfaces generally goes with increasing thickness of the LAO layer, both the itinerant and localized electrons in all cases occupy the same Ti 3d-levels on the side of SrTiO3 [11, 25, 26]. Taken together, the above experimental facts give strong support to the idea that electrons at the irradiated SrTiO3 surface and at the LAO/STO interfaces undergo a kind of first-order transformation into one and the same phase of twodimensional electronic Fermi liquid with the reduced magnetic symmetry.
The author thanks L Levitov for drawing his attention to experiments [16], E Rashba for many stimulating discussions and G Teitel’baum for the help with figure 1. The work was supported by the NHMFL through NSF Grant No. DMR1157490, the State of Florida and the US Department of Energy. References [1] Ohtomo A and Hwang H Y 2004 Nature 427 423 [2] Thiel S, Hammer G, Schmeh A, Schneider C W and Mannhart J 2006 Science 313 1942 [3] Lee J-S, Xie Y W, Sato H K, Bell C, Hikita Y, Hwang H Y and Kao C-C 2013 Nat. Mater. 12 703 [4] Bi F, Huang M, Bark C-W, Ryu S, Eom C-B, Irvin P and Levy J 2013 arXiv:1307.5557v2 [5] Joshua A, Ruhman J, Pecker S, Altman E and Ilani S 2012 Nat. Commun. 3 1129 [6] Reyren N et al 2007 Science 313 1196 [7] Gariglio S, Reyren N, Caviglia A D and Triscone J-M 2009 J. Phys.: Condens. Matter 21 164213 [8] Dikin D A, Mehta M, Bark C W, Folkman C M, Eom C B and Chandrasekhar V 2011 Phys. Rev. Lett. 107 056802 [9] Li L, Richter C, Mannhart J and Ashoori R C 2011 Nat. Phys. 7 762 [10] Bert J A, Kalisky B, Bell C, Kim M, Hikita Y, Hwang H Y and Moler K A 2011 Nat. Phys. 7 767 [11] Berner G et al 2013 Phys. Rev. Lett. 110 247601 [12] Mannhart J, Blank D H A, Hwang H Y, Millis A J and Triscone J-M 2008 MRS Bull. 33 1027 [13] Santander-Syro A F et al 2011 Nature 469 189 [14] Meevasana W, King P D C, He R H, Mol S-K, Hashimoto M, Tamai A, Songsiriritthigul P, Baumberger F and Shen Z-X 2011 Nat. Mater. 10 114 [15] Plumb C et al 2014 Phys. Rev. Lett. 113 086801 [16] Santander-Syro A F, Fortuna F, Bareille C, R¨odel T C, Landolt G, Plumb N C, Dil J H and Radovic M 2014 Nat. Mater. 13 1085 [17] Bychkov Yu and Rashba E I 1984 JETP Lett. 39 78 [18] Plumb N C et al 2013 arXiv:1304.5948v1 [19] Pomeranchuk I Ya 1959 J. Exp. Theor. Phys. 8 361 [20] Gor’kov L P and Sokol A 1992 Phys. Rev. Lett. 69 2586 [21] Breitschaft M et al 2010 Phys. Rev. B 81 153414 [22] Kalisky B, Bert J A, Klopfer B B, Bell C, Sato H K, Hosoda M, Hikita Y, Hwang H Y and Moler K A 2012 Nat. Commun. 3 922 [23] Salman Z et al 2012 Phys. Rev. Lett. 109 257207 [24] Reyren N et al 2007 Science 317 1196 [25] Berner G et al 2010 Phys. Rev. B 82 241405 [26] Salluzzo M et al 2009 Phys. Rev. Lett. 102 166804 [27] Cancellieri C, Reinle-Schmitt M L, Kobayashi M, Strocov V N and Willmott P R 2013 arXiv:1307.6943v1
7. Conclusions
In summary, magnetism of the conducting layer was inferred from the fact of lifted Kramers degeneracy in [16]. We proposed the Pomeranchuk-type instability as the mechanism for formation of the magnetic phase and pointed out the concrete symmetry of the antiferromagnetic order parameter. We argued that a large energy scale of the order of the Fermi energy inherent in this mechanism protects the ground state of 2DEG against ever-present random effects of disorder. The theoretical concepts put forward in the above prompt for further elaboration on the part of experiment. In particular, we suggest analysis of the perpendicular-to-plane spin polarization component. It would be equally interesting if ARPES were able to discern the two-fold symmetry in the vicinity of the Dirac point at the center of the Brillouin zone.
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