Signatures of dephasing by mirror-symmetry breaking in weak-antilocalization magnetoresistance across the topological transition in Pb 1−x Sn x Se
Alexander Kazakov, Wojciech Brzezicki, Timo Hyart, Bartłomiej Turowski, Jakub Polaczyński, Zbigniew Adamus, Marta Aleszkiewicz, Tomasz Wojciechowski, Jaroslaw Z. Domagala, Ondrej Caha, Andrei Varykhalov, Gunther Springholz, Tomasz Wojtowicz, Valentine V. Volobuev, Tomasz Dietl
DDephasing by mirror-symmetry breaking and resulting magnetoresistance across thetopological transition in Pb − x Sn x Se Alexander Kazakov ID , ∗ Wojciech Brzezicki ID ,
1, 2
Timo Hyart ID ,
1, 3, † Bart(cid:32)lomiej Turowski, Jakub Polaczy´nski, Zbigniew Adamus, Marta Aleszkiewicz, Tomasz Wojciechowski ID , Jaroslaw Z. Domagala, Ondˇrej Caha, Andrei Varykhalov, Gunther Springholz, Tomasz Wojtowicz ID , Valentine V. Volobuev ID ,
1, 8, ‡ and Tomasz Dietl ID
1, 9, § International Research Centre MagTop, Institute of Physics,Polish Academy of Sciences, Aleja Lotnikow 32/46, PL-02668 Warsaw, Poland M. Smoluchowski Institute of Physics, Jagiellonian University,prof. S. (cid:32)Lojasiewicza 11, PL-30348 Krak´ow, Poland Department of Applied Physics, Aalto University, FI-00076 Aalto, Espoo, Finland Institute of Physics, Polish Academy of Sciences,Aleja Lotnikow 32/46, PL-02668 Warsaw, Poland Department of Condensed Matter Physics, Faculty of Science,Masaryk University, Kotl´aˇrsk´a 2, 611 37 Brno, Czech Republic Helmholtz-Zentrum Berlin f¨ur Materialien und Energie,Albert-Einstein Strasse 15, 12489 Berlin, Germany Institut f¨ur Halbleiter- und Festk¨orperphysik, Johannes Kepler University, Altenbergerstrasse 69, A-4040 Linz, Austria National Technical University ”KhPI”, Kyrpychova Str. 2, 61002 Kharkiv, Ukraine WPI Advanced Institute for Materials Research, Tohoku University,2-1-1 Katahira, Aoba-ku, Sendai 980-8577, Japan
Many conductors, including recently studied Dirac materials, show saturation of coherence lengthon decreasing temperature. This surprising phenomenon is assigned to external noise, residualmagnetic impurities or two-level systems specific to non-crystalline solids. Here, by consideringSnTe-class of compounds as an example, we show theoretically that breaking of mirror symmetrydeteriorates Berry’s phase quantization, leading to additional dephasing in weak antilocalizationmagnetoresistance (WAL-MR). Our experimental studies of WAL-MR corroborate these theoreticalexpectations in Pb − x Sn x Se thin film with Sn contents x corresponding to both topological crys-talline insulator and topologically trivial phases. In particular, we find the shortening of the phasecoherence length in samples with intentionally broken mirror symmetry. Our results indicate thatthe classification of quantum transport phenomena into universality classes should encompass, inaddition to time-reversal and spin-rotation invariances, spatial symmetries in specific systems. Introduction
One of the most powerful characterizations of quantumsystems is in terms of ten universality classes that cor-respond to different ways fermionic Hamiltonians trans-form under time-reversal T , particle-hole and chiral sym-metry operations [1–3]. This generic approach, immuneto space symmetry details, allows describing specifici-ties of transport and topological phenomena in a broadrange of normal and superconducting materials [1–4]. Inthe sector of normal conductors, this classification leadsto three major experimentally realized cases dependingon the presence (+) or the absence (–) of T and thespin-rotation invariance S . The instances in the presenceof time-reversal symmetry are referred to as orthogonal(++) and symplectic (+–) class, whereas the universalityclass in the absence of time-reversal symmetry is knownas the unitary class [1, 5]. The unitary class is sometimes ∗ Electronic address: [email protected] † Electronic address: [email protected] ‡ Electronic address: [email protected] § Electronic address: [email protected] divided into subclasses depending on the existence of thespin-rotation symmetry and spin-polarization in systemswhere the effects of carrier interactions are relevant [6, 7].It becomes, however, increasingly clear that this pic-ture is not complete. For instance, weak antilocaliza-tion (WAL) magnetoresistance (MR) described by theHikami-Larkin-Nagaoka (HLN) formula [5] is expectedfor the symplectic class in the limit of strong spin-orbitscattering, for example, due to: (i) spin-momentum lock-ing of carriers encircling 2D gapless Dirac cones at sur-faces of 3D topological materials [8–11]; (ii) a large pre-cession frequency in the interfacial Rashba field com-pared to the inverse momentum relaxation time [12]; anda strong Elliott-Yafet mechanism due to sizable mixingof spin states in the carrier wave functions [13]. Sur-prisingly, however, robust WAL MR is also observed forgraphene [14]. It has actually been found that becauseof isospin-momentum locking, the carriers encircling 2Dgapless Dirac cones acquire the Berry phase ϕ = π , whicheliminates backscattering and, thus, results in MR thatmimics the symplectic case, even though the spin-orbitinteraction is negligible. However, in graphene also thespace group symmetries are important because both theintervalley scattering and trigonal warping of the cones, a r X i v : . [ c ond - m a t . m e s - h a ll ] N ov which makes momentum p nonequivalent to − p , sup-press the WAL effect [14, 15]. Similarly, the theoreti-cal discovery of topological indices associated with crys-tal point group symmetries [16] showed that the tenfoldclassification has to be much extended to incorporate arather abundant family of topological crystalline insu-lators (TCIs) and superconductors [17]. In particular,it was predicted theoretically [18, 19] and confirmed ex-perimentally [20–24] that the mirror symmetry M canprotect the presence of gapless topological Dirac coneson (001) and (111) surfaces of cubic SnTe-type semicon-ductors with the inverted band structure.Here, by taking thin films of cubic SnTe-type semi-conductors as an example, we show—combining analyticand numerical approaches developed recently [25]—thatthe Berry phase for carriers encircling the Fermi loops isquantized to π in these systems once inversion symmetryis broken by, for instance, due to the differences in topand bottom surfaces. This quantization is independent ofthe Fermi level position in respect to the bulk bandgap,and occurs for the band arrangement corresponding totopologically nontrivial and trivial phases accessible inPb − x Sn x Se materials with x > x c and x < x c , re-spectively, where x c = 0 .
16 [26]. Our detailed analysisdemonstrates that in the studied (111) case, this quan-tization is associated with crossings of quantum well 2Dsubbands, protected by T and M symmetries, and ap-pearing for both inverted ( x > x c ) and normal ( x < x c )band arrangements. This finding, supported by a directconductance determination, indicates that WAL MR inthese layers might be affected not only by doping withmagnetic impurities, as found previously in a range ofmaterials [27–30], but also by breaking M .In order to test these theoretical predictions, we havegrown a series of epitaxial (111) Pb − x Sn x Se thin filmswith the Sn content 0 ≤ x ≤ . in-situ part ofthe epilayers by amorphous Se. Angle-resolved photoe-mission spectroscopy (ARPES) confirms that our sam-ples cover both sides of the topological phase transition.Nevertheless, but in agreement with our theoretical ex-pectations, we observe for any x robust WAL MR, welldescribed by the HLN formula in the limit of strong spin-orbit scattering and with the prefactor α = − / ϕ = π and inter-subbandscattering time shorter than the phase coherence time τ φ . This insight elucidates why robust WAL MR haspreviously been observed not only for topological SnTeepilayers [30–33] or Pb . Sn . Se quantum wells [34], butalso for non-topological PbTe/(Pb,Eu)Te quantum wells[35]. Importantly, our data reveal striking differences be-tween WAL MR in uncapped films compared to samplesin which the amorphous Se surface layer intentionallybreaks the mirror symmetry. This breaking of M , rele-vant as electrons penetrate the cap, has two consequencesrevealed here experimentally and explained theoretically:(i) τ φ ( T ) determined from WAL MR saturates below 4 Kin the Se covered samples but not in samples withouta Se cap; (ii) the conductance decay with the in-plane magnetic field faster in films covered by Se. Our re-sults, together with those for graphene [14, 15], lead to arather striking conclusion that specific space symmetriescan account for an apparent low-temperature saturationof τ φ ( T ) determined from WAL MR studies. This find-ing suggests that hidden spatial-symmetry properties, to-gether with decoherence specific to amorphous solids [36],might have often been responsible for a hitherto myste-rious low-temperature saturation of τ φ ( T ) observed inmany systems [37], including recently studied Dirac ma-terials [34, 38–41]. Theoretical results
Mirror and time-reversal symmetry protectedquantization of the Berry phases.
We consider thesymmetry-enriched berryology of the (111) multilayersystem in the SnTe material class [18] (see Sec. S1 inSupplementary Information) for more details). By dop-ing the system we obtain Fermi loops around the high-symmetry points Γ and M i ( i = 1 , ,
3) (see Fig. 1).The type of doping (electron or hole doping) is not im-portant for our theoretical considerations because both Г or M point φ = π � fully symmetricbroken time-reversalbroken (110) mirrorand time-reversal ≤φ <2 π abc ⊗ a ×a a ‒ a a +a ≤φ <2 π � m M or M point Г M M M k x ~ k y ~ Г M M M k x ~ k y ~ Г or M point φ = π �� Г or M point FIG. 1:
Dependence of Fermi loops and Berry phases ϕ on the presence of time-reversal T and (110) mirror M symmetries for doped Pb − x Sn x Se thin films. a , If T and M are obeyed the Berry phases of the Fermi loops insubbands of a two-dimensional slab are quantized to ϕ = π ,protecting subband crossings at high-symmetry points Γ and M i (black dot). This also holds when only M is broken. b ,If T is broken (e.g. by a Zeeman field shown by arrows) but M is obeyed the Berry phases are quantized to π only forFermi loops going around a subbands’ crossing at the Γ and M points lying in the mirror plane. c , The Berry phasesare arbitrary when both T and M are simultaneously broken(arrows and asymmetric surface layer). Гθ m θ Гθ + π θ - ba MT FIG. 2:
Action of time-reversal T and mirror M sym-metries on states belonging to the Fermi loop around Γ point . Points along the loop are parametrized by the an-gle θ in respect to the mirror line (dashed). a , Time-reversalmaps state | ψ ( n ) θ (cid:105) onto | ψ ( n ) θ + π (cid:105) . b , Mirror symmetry mapsstate | ψ ( n ) θ (cid:105) onto | ψ ( n ) − θ (cid:105) . types of doping yield similar Fermi loops and our results follow from generic symmetry arguments. In our calcu-lations, all 2D subbands are non-degenerate along theFermi loops because of the spin-orbit interaction and theinversion asymmetry due to the presence of inequivalentsurfaces. We show that both time-reversal T and (110)mirror M symmetries lead to quantization of the Berryphases (see Fig. 2). Due to three-fold rotational symme-try, there exists also two other mirror symmetries whichwould lead to equivalent considerations.In the presence of time-reversal symmetry obeying T = −
1, we obtain that all the Fermi loops have quan-tized Berry phase ϕ = π (see Fig. 1a). To prove this weconsider the eigenstates | ψ ( n ) θ (cid:105) , θ ∈ [ − π, π ), belongingto the n th energy band forming a Fermi loop around Γ(or M ) point (see Fig. 2). Since θ is periodic variable | ψ ( n ) θ (cid:105) can acquire a Berry phase by a parallel shift alongthe loop. The gauge-invariant form of the Berry phase isgiven by, ϕ n = arg (cid:104)(cid:68) ψ ( n ) − π (cid:12)(cid:12)(cid:12) ψ ( n ) − π + δθ (cid:69)(cid:68) ψ ( n ) − π + δθ (cid:12)(cid:12)(cid:12) ψ ( n ) − π +2 δθ (cid:69) . . . (cid:68) ψ ( n ) π − δθ (cid:12)(cid:12)(cid:12) ψ ( n ) π − δθ (cid:69)(cid:68) ψ ( n ) π − δθ (cid:12)(cid:12)(cid:12) ψ ( n ) − π (cid:69)(cid:105) . (1)where δθ is an infinitesimal step in angle θ . In thepresence of a time-reversal symmetry we obtain the states | ψ ( n ) θ (cid:105) for − π ≤ θ < ≤ θ < π states as | ψ ( n ) θ + π (cid:105) = T | ψ ( n ) θ (cid:105) .By this construction we find that most of the phases inEq. (1) cancel and we are left with (see Sec. S2) ϕ n = arg (cid:20) − (cid:12)(cid:12)(cid:12)(cid:68) ψ ( n ) − δθ (cid:12)(cid:12)(cid:12) ψ ( n )0 (cid:69)(cid:12)(cid:12)(cid:12) (cid:21) = π. (2)Hence, in the presence of time-reversal symmetry sat-isfying T = − π .Equivalently we can say that the Berry phases are equal π because each Fermi loop encircles a crossing (i.e. Diracpoint) of the subbands protected by Kramers degener-acy at high-symmetry points of the Brillouin zone (BZ).We point out that for time-reversal symmetry obeying T = 1 (no spin-orbit interaction) the Berry phaseswould be quantized to 0. Thus, our symmetry analysis ofthe Berry phases reproduces the well-known result thatmaterials with strong (weak) spin-orbit coupling supportWAL (WL) due to belonging to the symplectic (orthog-onal) universality classes [1, 5]. This result holds bothin the topologically trivial and non-trivial regimes, andtherefore we expect WAL independently of the Sn con-tent in Pb − x Sn x Se thin films.It turns out that the presence of crystalline mirror sym-metry M can lead to the quantization of Berry phaseseven if time-reversal symmetry T is broken, e.g., by anon-zero Zeeman field, as shown in Fig. 1b. Namely, inthe presence of mirror symmetry we obtain the states | ψ ( n ) θ (cid:105) for 0 ≤ θ ≤ π by diagonalizing the Hamiltonianand we define − π ≤ θ < | ψ ( n ) − θ (cid:105) = M| ψ ( n ) θ (cid:105) (see Fig. 2). From Eq. (1) we find that most of the phasescancel due to unitarity of M and we get (see Sec. S2) ϕ n = arg (cid:104)(cid:68) ψ ( n ) π (cid:12)(cid:12)(cid:12) M (cid:12)(cid:12)(cid:12) ψ ( n ) π (cid:69)(cid:68) ψ ( n )0 (cid:12)(cid:12)(cid:12) M (cid:12)(cid:12)(cid:12) ψ ( n )0 (cid:69)(cid:105) , (3)where | ψ ( n )0 (cid:105) and | ψ ( n ) π (cid:105) are eigenstates of M witheigenvalues ±
1. Thus the product under arg functionis either +1 or −
1, and hence the Berry phases for allmirror-symmetric Fermi loops are quantized to 0 or ππ (Fig. 1b). Again, we find that ϕ = π if the Fermiloop encloses a Dirac point i.e. a crossing of subbands.For weakly broken T the crossings stay inside Fermiloops within the mirror plane. However, the Berry phasechanges to 0 when they move outside at a topologicalphase transition for strongly broken T (see Sec. S3).Finally, we find that the Berry phases are arbitrarywhen both T and M are simultaneously broken (Fig. 1c).We emphasize that this symmetry analysis is completelygeneric in the sense that the microscopic details of thebreaking of the time-reversal and mirror symmetries arenot important, but we have also confirmed these findingsby explicitly calculating the Berry phases in the presenceof specific perturbations breaking of T and M indepen-dently of each other (see Sec. S3).This theoretical analysis leads to two important predic-tions that can be directly tested experimentally. First, weobtain a similar behaviour of the Berry phases for bothtopologically non-trivial and trivial materials, becausethe symmetries of the system rather than topologicalsurface states are important. Therefore, we expect thatsimilar WAL-like behaviour is observed in Pb − x Sn x Sealloys independently of the Sn content. Secondly, in re-alistic condensed matter systems, both T and M sym-metries are always weakly broken. Therefore, the de-viations of the Berry phases from the quantized valuescan be increased by intentionally breaking the T and M symmetries more strongly until there is no systematicdestructive interference of the backscattering paths thatcould lead to WAL. The destruction of the WAL effectby intentionally breaking the time-reversal symmetry hasbeen experimentally demonstrated, but we predict thatthis could be achieved also by breaking the mirror sym-metry provided that time-reversal symmetry is alreadyweakly broken. The crystalline mirror symmetry can bebroken in a controllable way by covering the surface of thesample with a suitable material. Amorphous solids haveshort range order in the sense that the distances betweenneighboring atoms are similar to those in the crystal, butthe translational symmetry is absent, so that there is nolong-range order and all point group symmetries are vio-lated in crystallographic sense (the symmetry operationwill not result in the same structure). Therefore, the im-portance of the crystalline mirror symmetry on the WALeffect can be tested, for instance, by proximitizing thesample with an amorphous semiconductor.We have confirmed both of these predictions exper-imentally. However, before discussing our experimentalfindings, we will next calculate the quantum correction tothe conductivity coming from the Cooperon propagator.By a rigorous treatment we obtain a generalized HLN for-mula that takes into account the length scale related tothe breaking of the mirror symmetry and time-reversalsymmetries. We show that in the case of perpendicu-lar magnetic field this symmetry-breaking leads to thesaturation of the effective dephasing length at low tem-perature and to the slower decrease of the conductivityas function of the field. For the in-plane field we get theopposite trend in the regime where the energy scale re-lated to the field is smaller than the energy scale relatedto the mirror symmetry breaking. Our theoretical resultsare in excellent agreement with the experimental findingsdiscussed in the following sections. Quantum correction to the conductivity.
In thevicinity of the band crossings appearing at the high-symmetry points we derive a low-energy 2D Hamiltonianfor a single pair of subbands in a form of H (cid:126)k,(cid:126)σ = (cid:126) m e (cid:126)k + α so ( σ x k y − σ y k x ) + gσ z . (4)Here m e is the effective mass of the electron, α so is aneffective spin-orbit-like coupling that arises from break-ing of the inversion symmetry (always present due to thesurface in these samples), g is the mass term induced bythe breaking of the mirror symmetry and weak break-ing of the time-reversal symmetry, and σ is an effective pseudospin variable which describes entangled spin andorbital degrees of freedom.The quantum correction to the conductivity can bewritten as [42]∆ σ xx = − e π (cid:126) D e L (cid:88) (cid:126)Q (cid:88) α,β = ± C αββα ( (cid:126)Q ) (5)where D e = v F l/ l = v F τ is the elastic mean-free path, v F is the Fermi velocity, τ is the elastic scattering time, L is the area of the sample,and α, β = ± are the pseudospin indices of the Cooperonpropagator C . For weak disorder ( τ E F / (cid:126) (cid:29) E F is the Fermi energy) the Cooperon propagator canbe approximated as C ( (cid:126)Q ) = τ (cid:18) − (cid:90) d Ω2 π − iτ Σ / (cid:126) (cid:19) − , (6)where Σ( (cid:126)Q ) = H (cid:126)Q − (cid:126)k,(cid:126)σ (cid:48) − H (cid:126)k,(cid:126)σ (7)is a 4 × (cid:126)σ (cid:48) and (cid:126)σ . The integral in Eq. (6) is over allangles of velocity (cid:126)v = (cid:126) (cid:126)k/m e on the Fermi surface. Bytaking lowest order terms in (cid:126)Q and α so , we can write thequantum correction to the conductivity in the absence ofthe magnetic field as∆ σ xx = e π (cid:126) L (cid:88) (cid:126)Q Tr (cid:34) Γ (cid:18) D e τ φ + H c (cid:19) − (cid:35) , (8)where H c is a non-Hermitian Cooperon Hamiltonian H c = (cid:126)Q + 2 Q so (cid:126)Q · ˆ a(cid:126)S + Q (cid:0) S x + S y (cid:1) − i g (cid:126) D e ( σ (cid:48) z − σ z ) , (9) Q so = 2 m e α so / (cid:126) and τ φ is the dephasing time. Singletand triplet interference is encoded in matrix Γ havingthree − σσ (cid:48) triplet sector and +1 inthe singlet sector. The perpendicular magnetic field canbe introduced in the Cooperon Hamiltonian by minimalsubstitution (cid:126)Q → (cid:126)Q + 2 e (cid:126)A/ (cid:126) , and the summation in thiscase should be taken over the Landau levels (see Sec. S4for details).The parameter g = 0 unless both time-reversal andmirror symmetry are simultaneously broken (see Sec. S4for details). We can assume that the time-reversal sym-metry is always weakly broken by the same amount dueto the intrinsic mechanisms, but the breaking of the mir-ror symmetry is tunable and depends on how the sam-ple is covered. Thus, in uncovered samples we assumethat g ≈ g becomes significantly larger andwe assume that it is given by g = 1 .
75 meV. Note thatthe gap opened by g is still too small to be observed - - - - - - - - - - � T � � � ��� � � � � � � ��� � � � xx � e � ℏ � a cb d M obeyedfull formula M brokenfull formula - - - - - - - - - - � T � M obeyedHLN with � φ eff M brokenHLN with � φ eff l � ��� ( � m ) ��� e ff T � K � M obeyed M broken e xx � � ��� � � � � � � ��� � � � xx � e � ℏ � xx M obeyed M broken - - - ∥ /B ∥ � � ��� � � ∥ � � � � ��� � � � xx � e � ℏ � xx f FIG. 3: a - d , ∆ σ xx as a function B in the fully symmetric and symmetry-broken cases for temperatures T = 1 . , , , , ,
25 K. a , b , Numerically calculated quantum correction to the conductivity. c , d , ∆ σ xx calculated from the modified HLN formula (10)employing the effective dephasing time τ eff φ [Eq. (11)]. e , Effective dephasing length l eff φ = (cid:113) D e τ eff φ as function of temperaturefor fully symmetric and symmetry-broken cases. f , ∆ σ xx as a function of the in-plane field B (cid:107) in the fully symmetric andsymmetry-broken cases. by ARPES, but the WAL measurement is a very sen-sitive probe of the symmetry-breaking field, so that al-ready such small value of g can dramatically show up inthe transport experiments: The resulting dependenciesof ∆ σ xx on B , in the fully symmetric and symmetry-broken cases, are shown in Figs. 3a and 3b (lines labelled T = 1 . v F = 7 . × m/s, l = v F τ = 20 nm and l φ ( T = 1 . (cid:112) D e τ φ ( T = 1 . µ m. (In gen-eral the phase-coherence length depends on temperature,as discussed below, and therefore we fix here the phase-coherence length at temperature T = 1 . Q so = 1 . × m − , so thatthe pseudospin precession length is l so = 2 π/Q so ≈
60 nm(see Sec. S4 for details).The breaking of mirror and time-reversal symmetriesby the symmetry-breaking field g leads to an openingof a small energy gap at the band crossings and tothe destruction of the quantization of the Berry phase(see Sec. S4 for details). Due to the latter reason, thesymmetry-breaking field randomizes the phases of thebackscattering paths destroying their systematic destruc-tive interference that was caused by the quantization ofthe Berry phase to ϕ = π . Therefore, in the presence ofthe symmetry-breaking field g there is a new length scalewhich limits the increase of the WAL effect with lower-ing temperature (see Fig. 3). Although, the symmetry-breaking field is a quantum-coherent effect the qualitativepicture discussed above suggests that it has a similar ef- fect as phase breaking phenomena described by l φ . Wehave confirmed this expectation by demonstrating (seeFig. 3) that the conductivity calculated from the full ex-pression can be reproduced by a modified HLN formulagiven by∆ σ xx ( B ) − ∆ σ xx (0) = − e π (cid:126) (cid:20) ψ (cid:32)
12 + 14 τ eff φ B (cid:126) eD e (cid:33) − log (cid:32) τ eff φ B (cid:126) eD e (cid:33) (cid:21) , (10)where τ eff φ is an effective dephasing time τ eff φ ( T ) = 1 D e (cid:16) l φ ( T ) + l g (cid:17) (11)and l g is the new length scale related with the symme-try breaking field g l g = 2 π (cid:115) D e (cid:126) √ g . (12)Thus, if the conductivity is measured as a functionof perpendicular magnetic field, the symmetry-breakingfield shows up as an effective dephasing length as l eff φ = (cid:113) D e τ eff φ , which saturates at low-temperatures to l g . Ad-ditionally, to describe the full temperature dependenceobserved experimentally we assume that dephasing timedepends on temperature as l φ ( T ) = l φ ( T ) T /T . Thefull temperature-dependence of the conductivity from T = 1 . T = 25 K obtained this way is shown inFigs. 3a and 3b both in the absence and presence of thesymmetry-breaking field. The corresponding tempera-ture dependencies of the effective dephasing lengths l eff φ are shown in Fig. 3e. In the absence of the symmetry-breaking field l eff φ continues to increase at low tempera-tures, whereas in the presence of the symmetry-breakingfield l eff φ saturates to l g at low temperatures. Althoughthe symmetry-breaking field g is small, it results in a de-crease of l eff φ by almost an order of magnitude at T =1 . B (cid:107) (breaking both mirror and time-reversal sym-metries) is that it increases the symmetry-breaking field g so that we can write the low-energy Hamiltonian (4)with g → (cid:0) g + g ∗ µ B (cid:12)(cid:12) B (cid:107) (cid:12)(cid:12)(cid:1) σ z . The important scale of thein-plane magnetic field is therefore B (cid:107) = g/g ∗ µ B , wherethe symmetry-breaking field due to the in-plane field be-comes equal to the symmetry breaking field due to thecovering of the sample. Then, in the limit of zero perpen-dicular magnetic field and T = T we obtain the resultshown in Fig. 3f for ∆ σ xx . For | B (cid:107) | < B (cid:107) the conduc-tivity decreases faster in the symmetry-broken case as afunction of the in-plane field.We point out that in this analysis the effect of the in-plane field can also include orbital effects, because alsothe orbital effects of the in-plane magnetic field breakmirror and time-reversal symmetries, and therefore leadeffectively to an increase of g in the low-energy theory.Although the explicit dependence of the energy gap on | B (cid:107) | can be more complicated the result that the conduc-tivity decreases faster in the symmetry-broken case seemsto be relatively robust if the applied in-plane field is rea-sonably small. (For this result the assumption that thegap increases linearly with | B (cid:107) | is not necessary.) This re-sult is also consistent with analysis based on Berry phasessince the increase of the gap leads to larger deviation ofthe Berry phases from the quantized value. Experimental results
We test the theory on (111)Pb − x Sn x Se 50-nm thickfilms deposited by MBE on BaF substrates (see Meth-ods and Supplementary Information for details of filmgrowth, characterization, processing, and experimentalmethodology). Two series of films have been grown tostudy the influence of both topological transition andbreak of the M symmetry on the WAL phenomena: thefirst series consists of bare epilayers A-E, while the sam-ples in the second series (epilayers F-J) are covered by a100 nm-thick amorphous and insulating Se cap [43]. Ineach series, Sn content is varied to drive part of the filmsthrough the topological transition at low temperatures.To ensure that the topological transition indeed takes place, the band structure at the surface of G and D epi-layers with trivial and non-trivial compositions, respec-tively, have been characterized by ARPES, as shown inFig. 4. Good agreement of the gaps determined here withthe semi-empirical Grisar formula [44] and the ARPESdata for bulk samples [20] demonstrates that residualstrains detected by XRD have a minor effect on the bandstructure. At the same time, the Rashba splitting ofbands, appreciable in IV-VI semiconductor epilayers un-der certain growth conditions [45], is not significant.Despite that ARPES confirms the expected n -typecharacter of Pb − x Sn x Se, a positive sign of the Hall coef-ficient is observed, pointing out to a relatively large con-tribution from holes at the interface to BaF , as foundearlier for PbTe/BaF epilayers [49]. High-field parabolicpositive MR for the field perpendicular to the film plane,shown in Fig. 5, is consistent with a multichannel charac-ter of charge transport (valleys, 2D subbands, n -type and p -type layers), whereas a linear component in the high-est field suggests and an admixture of the Hall resistancecaused by lateral inhomogeneities [50].Interestingly and crucially for this work, we find the ex-istence of low-field temperature-dependent positive MRin all epilayers regardless of their composition. Exceptfor PbSe, this MR dominates only in the diffusive regime, l (cid:28) l B , where l is the mean free path and l B is the mag-netic length. According to the theory developed here, weassign this MR to the Berry phase quantization broughtabout by symmetries rather than by a non-trivial char-acter of the topological phase. In particular, the mirrorsymmetry leads to WAL even if time-reversal symme-try is slightly broken. Within this scenario, and by not-ing that we expect the phase coherence length l φ to begreater than the film thickness d = 50 nm, the MR is de-scribed by the HLN theory in the limit l φ (cid:29) l so , where l so is the spin diffusion length limited be spin-orbit inter-actions, corresponding to the HLN prefactor α = − / l g → ∞ . In our case, as shownin Fig. 5d, MR for all epilayers can be fitted by the one-channel formula, treating l φ ( T ), as the only fitting pa-rameter. This means that l φ is longer than length scalescharacterizing scattering between subbands and valleys(including surface ones in the topological case) as wellas between n and p -type layers. Alternatively, and moreprobably, because of short length scales characterizingthe p -type region, the corresponding WAL or WL MR isshifted to a high field region, so the low field features aresolely due to electrons residing closer to the outer surface.It is important recalling that if only one of the parallellayers shows MR, the one channel formula remains valid[9]. As shown in Supplementary Information, by fittingthe data to the full HLN formula we find l so ≈ l , whichsubstantiates our conjecture that in our case WAL MRstems from the Berry phase quantization, and not froma sequence of spin rotations in a varying spin-orbit fieldresulting in l so (cid:29) l .In Methods, we discuss precautions undertaken in or-der to eliminate Joule heating of carriers. Furthermore, T (K) FIG. 4:
ARPES results . a-d , Dispersion and correspond-ing 2nd derivative of ARPES data taken at 12 K with photonenergy of 18 eV in the vicinity of the Γ point. a,c , Resultstrivial Pb . Sn . Se (band gap of 84 meV) and b,d , topo-logical Pb . Sn . Se epilayers (gapless states with Dirac dis-persion, as observed previously [24, 43, 46]). Sample surfacesare n -type. c , 2nd derivative plots confirm the presence ofgaped precursor surface states in the trivial phase [47, 48]. e ,Surface band gaps measured at several temperatures (points)are well described, above the topological phase transition, bythe semi-empirical Grisar formula [44] for the bulk band gap(solid lines) thus proving a negligible effect of strains on theband structure of the studied epilayers. unintentional magnetic doping cannot be responsible forthe difference between capped and uncapped samples be-cause both types of epilayers have been grown in thesame MBE chamber. Relatively large Hall bar dimen-sions (10-100 µ m) exclude finite size effects. Further-more, as shown in Supplementary Information, fitting ofto the full HLN expression, i.e., containing l so and themean free path l explicitly, confirms that l so ≈ l (cid:28) l φ (Fig. 5e), which rules out a cross-over from WAL to WL. Therefore, we assume that the dependence of the WALmagnitude on temperature and the perpendicular mag-netic field shown in Fig. 5d) is solely determined by pro-cesses deviating the Berry phase from π , and control-ling the magnitude of l φ . In the samples without Se lay-ers, l φ ( T ) follows the power law T − p/ , with p rangingfrom 1.4 to 2.6, which corresponds to electron-phonon de-phasing mechanism, without any tendency to saturationdown to 1.5 K (Fig. 5e). By contrast, in the Se coveredepilayers l φ ( T ) tends to saturate at temperatures below ≈ l φ ( T ) in Se covered epilayers with( A + A T p ) − / , which results in the similar values for p , ranging from 1.8 to 2.8 with l φ (1 . K ) ≈ − µ mand 150 −
400 nm in uncovered and Se covered samples,respectively.Actually, these striking findings provide a strong sup-port to the theory proposed here (c.f. Figs. 3 a - e andFigs. 5 d , e ). At high temperature, WAL is still governedby the thermally suppressed l φ even if time-reversal sym-metry is weakly broken. In bare epilayers, WAL is pro-tected by the quantized Berry phase ϕ = π due to the M , thus l φ continues to increase with cooling down. InSe capped epilayers there is a different situation: longinterference paths, which are relevant for large values of l φ , do not contribute to WAL, since scattering betweenstates at the Fermi level, allowed by mirror symmetrybreaking, randomizes the wave function phase φ and av-erage it to zero. Thus, there is a new length scale l g given in Eq. 12, which limits an increase of WAL MRwith lowering the temperature, similarly to the effect ofspin-disorder scattering and Zeeman splitting consideredpreviously [5, 51, 52].The theory developed here shows explicitly that theapplication of the in-plane Zeeman field (cid:126)h = g ∗ µ B (cid:126)B/ ϕ = π and larger change of con-ductivity, if reflection symmetry is violated, as shown inFig. 3 f . In order to test this prediction, we have carriedout MR measurements for the magnetic field parallel tothe film plane, as in this configuration the role of time-symmetry breaking by the vector potential is reduced[53–55], making the Zeeman effect more important, par-ticularly considering a relatively large magnitude of theelectron Land´e factor, g ∗ = 35 ± τ φ in weak magnetic fields,we present in Fig. 6 MR as a function of B/B φ , where B φ = (cid:126) / el φ . As seen, MR is systematically stronger insamples covered by the Se cap, in agreement with theo-retical expectations. Discussion
In summary, our theoretical results demonstrate, takingthin films of SnTe-class of materials as an example, thatthe inversion asymmetry and mirror symmetry, ratherthan the topological phase, are essential for the Berryphase quantization to π and, hence, to the appearance Pb Sn SePb Sn Se/Se -200 D s xx ( m S ) -0.2 -0.1 0.0 0.1 0.2-40-200 B (T) a PbSe/SePb Sn Se/SePb Sn SePb Sn Se uncovered l f T -1.00 l f ( m m ) T (K) l f T -1.31 l f T -1.00 l f T -1.17 Se coveredl SO
30 nm l SO
120 nm Pb Sn Se/Se -8 -4 0 4 8204208212216 r xx ( W / s q . ) B (T) -80080160 r yx ( W ) -0.2 0.0 0.2205.5206.0 Pb Sn Se -8 -4 0 4 8122123124125126 r xx ( W / s q . ) B (T) -1000100200 r yx ( W ) -0.2 0.0 0.2122.2122.4 b c d e Photo of Pb Sn Se Pb Sn Se -0.2 -0.1 0.0 0.1 0.2-40-30-20-100 D s xx [ m S ] B [T] Pb Sn Se/Se -0.2 -0.1 0.0 0.1 0.2-40-30-20-100 D s xx [ m S ] B [T] Pb Sn Se / Se -0.2 -0.1 0.0 0.1 0.2-40-35-30-25-20-15-10-50 D s xx [ m S ] B [T] 𝐵 ∥ FIG. 5:
Determination of the phase coherence length l φ ( T ) from MR data in the magnetic field perpendicularto the epilayer plane . a , Photo of a processed Hall-bar (epilayer B, x Sn = 0 . substrate (scale bar is 200 µ m). b,c , Longitudinal ρ xx (black symbols) and Hall (redsymbols) ρ yx resistivities in high (main figures) and weak fields (insets) in epilayers that are uncovered ( b ) and covered by Se( c ). ( d ) Evolution of the WAL-like low-field MR with increasing temperature in uncovered (upper panel) and Se covered (lowerpanel) epilayers. Experimental points (empty squares) are fitted to the one-channel HLN expression in the strong spin-orbitapproximation (solid lines) treating l φ ( T ) as an adjustable parameter. e , Determined values of l φ ( T ) in epilayers uncoveredbe Se increase down to 1.5 K (black and red), while in Se covered epilayers l φ ( T ) saturates at lower temperatures (blue andmagenta). Typical values of l so ( T ) obtained from the full HLN expression (see Supplementary Information). of the robust WAL MR. This implies the existence ofthe hitherto overlooked length scale in quantum coher-ence phenomena. This new length is associated with thecrystal symmetry breaking rather than with a topologi-cal phase transition or with the violation of time reversalor spin rotation symmetries considered so far. This offersnew prospects in controlling carrier quantum transportby system architectures.These theoretical expectations have been verified byMR studies on a series of Pb − x Sn x Se epilayers, whichhave revealed the existence of WAL MR on the both sidesof topological phase transitions as well pointed out tostriking differences in the dependencies of WAL MR ontemperature and the magnetic field for samples with themirror symmetry maintained compared to the films inwhich mirror symmetry is intentionally broken, as theelectron penetration length into the thick amorphous Selayer is longer than into a native surface oxide in un- capped films.It is certainly appropriate to analyze carefully othermechanisms that could elucidate a strong influence ofamorphous Se overlayers on WAL MR. In principle, areduction of l φ ( T ) in Se covered samples might be ex-plained by additional decoherence due to the presence oftwo-level systems in amorphous solids. However, the cor-responding theory [36] suggests that this would not leadto full saturation of l φ ( T ) at low temperatures. Further-more, according to formulae describing WAL MR in theparallel configuration [53–55], a stronger WAL MR forthe in-plane magnetic field might result from effectivelygreater thickness of samples covered by Se.A fascinating question then arises to what extent sur-prising saturations of τ φ ( T ) at low temperature observedsince decades in many systems [37], and most recently inDirac materials [34, 38–41], have been caused by the hid-den spatial symmetry breaking, the mechanisms brought Δ σ xx ( μ S ) B || / B φ x Sn = 0.16 x Sn = 0.15 Δ σ xx ( μ S ) B || / B φ x Sn = 0.30 x Sn = 0.29 Δ σ xx ( μ S ) B || / B φ x Sn = 0.10x Sn = 0.11 x Sn = 0.06 Δ σ xx ( μ S ) B || / B φ x Sn = 0.03 uncovered Se covered
FIG. 6:
Effect of mirror-symmetry breaking on MRfor magnetic field applied parallel to the film plane .Comparison of MR measured at 4.2 K in the field parallel tothe film plane (for which the Zeeman field gives a substantialcontribution) for uncovered (full symbols, solid lines) and Se-covered (open symbols, dashed lines) samples with similar x Sn (lines are guide for the eye); see Supplementary Informationfor fitting results obtained employing various formulae for MRin parallel fields; B φ = (cid:126) / el φ . into light by our work. Methods
Sample growth.
Thin Pb − x Sn x Se films have beendeposited on freshly cleaved (111) BaF substrates bymolecular beam epitaxy (MBE) in the PREVAC 190growth chamber with the base pressure below 10 − mbar,and equipped with elemental Pb, Sn and Se sources. Theflux ratio is controlled by a beam flux monitor placed inthe substrate position. Typical selenium to metal flux ra-tio was of the order of 3:2. The structural quality of thefilm surface is monitored in-situ by the reflection high-energy electron diffraction (RHEED). The film growthrate determined by pronounced RHEED oscillations is inthe range 0.1-0.3 nm/s. Growth conditions, namely theratio of beam fluxes and the growth temperature havebeen thoroughly optimized to obtain high quality thinfilms of a thickness of 50 nm with Sn content varying from0 to 0.40. The amorphous Se cap layer was confirmed tobe insulating in a separately checked Se/BaF structure.XRD structural analysis of Se covered samples revealedno additional strains compared to the bare epilayers. Wehave also checked that the mobilities in Se covered filmsare typically higher than in the absence of the cover (seeSupplementary Information for details), indicating that the Se cover does not lead to any kind of reduction of thesample quality. The Se cap also serves as a protectionagainst contamination for ARPES measurements. Sample characterization
The XRD measurementswere performed by PANalytical X’Pert Pro MRD diffrac-tometer with a 1.6 kW x-ray tube (vertical line focus)with CuK α radiation ( λ = 1 . × Ge (220) monochromator and for high resolution mea-surements a channel-cut Ge(220) analyzer. AFM imageswere obtained in tapping mode using Veeco NanoscopeIIIa microscope. Additional morphological and compo-sition characterization was accomplished by field emis-sion scanning electron microscopy (FE-SEM) with Neon40-Auriga Carl Zeiss microscope equipped with energydispersive x-ray spectroscopy (EDX) system QUAN-TAX 400 Bruker. The electronic band structure of thefilms was verified by angle-resolved photoemission spec-troscopy (ARPES) at UE112 PGM-2a-1 beamline ofBESSY II (Berlin) in the photon energy range 15-90 eVand horizontal light polarization using a six-axes auto-mated cryomanipulator and a Scienta R8000 electronspectrometer. Typical energy and angular resolutionswere better than 20 meV and 0.5 ◦ , respectively. Sample patterning and magnetotransport mea-surements
Grown epilayers were further processed formagnetoresistance (MR) measurements to the form ofHall bars by e-beam lithography and Br wet etching withthe long arm along a (cid:104) (cid:105) direction revealed by cleavageof BaF . The in-plane magnetic field was oriented alongthe current in the tilted field experiments. Resistivitymeasurements were performed in an 8 T/1.5 K cryostat,using a standard lock-in technique at 20 – 30 Hz withthe excitation current from 1 µ A down to 10 nA at thelowest temperature. We checked that lowering of currentdown to 1 nA does not affect positive low-field magne-toresistance, meaning that the saturation of the phasebreaking length at low temperatures in samples coveredby Se cannot be explained by Joule heating.
Data availability
Experimental results are available as source data withthe paper. All other data that support the plots withinthis paper and other findings of this study (includingthose in Supplementary Information) are available fromthe corresponding authors upon reasonable request.
Code availability
The codes supporting theory are available from W.B.and T.H. upon request. [1] Beenakker, C. W. J. Random-matrix theory of quantumtransport.
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The International Center for Interfacing Magnetismand Superconductivity with Topological Matter Mag-Top is supported by the Foundation for Polish Sciencethrough the IRA Programme co-financed by EU withinSG OP (Grant No. MAB/2017/1). We acknowledge theHelmholtz-Zentrum Berlin for provision of synchrotronradiation beamtime at UE112 PGM-2a-1 of BESSY IIunder the EU CALIPSO Grant number 312284. W.B.also acknowledges support by Narodowe Centrum Nauk(NCN, National Science Centre, Poland) Project No.2019/34/E/ST3/00404. G.S. also acknowledges supportby Austrian Science Funds, Projects No. P30960-N27and I3938-N27. Competing interests
The authors declare no competing interests.
Additional information
Author contributions
A.K. and W.B. contributed equally to this work.W.B and T.H. developed the theory with input fromT.D. The samples were grown and characterized byx-ray by V.V.V. with the assistance of B.T. and J.J.D,respectively. A.K. carried out processing and magne-totransport measurements with the assistance of J.P.and Z.A., respectively. V.V.V., O.C. and G.S., withthe help of A.V., performed ARPES measurements.AFM data were collected by M.A. and EDX by T.W.The manuscript was written by A.K., W.B., T.H.,V.V.V. and T.D. All authors discussed the results andcommented on the manuscript. T.D. and T. Wojtowiczsupervised the project.
Supplementary information is available for thispaper at https://doi.org/xxx.
Correspondence and requests for materials shouldbe addressed to A.K, T.H., V.V. or T.D.1
Supplementary Information
Dephasing by mirror-symmetry breaking and resulting magnetoresistance acrossthe topological transition in Pb − x Sn x Se S1. THE MULTILAYER HAMILTONIAN AND THE SYMMETRIES
Our starting point is the tight-binding Hamiltonian for SnTe-material class [1], H = m (cid:88) j ( − j (cid:88) r ,α ˆ c † jα ( r ) · ˆ c jα ( r ) + (cid:88) j,j (cid:48) t jj (cid:48) (cid:88) (cid:104) r , r (cid:48) (cid:105) ,α ˆ c † jα ( r ) · ˆ d rr (cid:48) ˆ d rr (cid:48) · ˆ c j (cid:48) α ( r (cid:48) ) − (cid:88) j iλ (cid:88) r ,α,β ˆ c † jα ( r ) × ˆ c jβ ( r ) · ˆ σ α,β , (S1)where ˆ c jα ( r ) are vectors of fermionic operators corresponding to p x -, p y - and p z -orbitals and the indices denote thesublattice j ∈ { , } [(Sn,Pb)/(Te,Se) atoms], spin α and lattice site r . Here ˆ σ α,β is a vector of Pauli matrices, ˆ d rr (cid:48) are unit vectors pointing from r to r (cid:48) and the next-nearest-neighbour hoppings satisfy t = − t .Defining the unit cell as two atoms at positions (0 , ,
0) and (0 , ,
1) (taking one interatomic distance as a length unit)and the lattice translation vectors as (cid:126)a = (1 , , (cid:126)a = (0 , ,
1) and (cid:126)a = (0 , ,
2) we find that the three-dimensionalbulk Hamiltonian can be represented in momentum space as [2], H ( (cid:126)k ) = m ⊗ ⊗ τ z + t (cid:88) α = x,y,z ⊗ (cid:0) − L α (cid:1) ⊗ h (1) α ( (cid:126)k ) + t (cid:88) α (cid:54) = β ⊗ (cid:16) − ( L α + ε αβ L β ) (cid:17) ⊗ h (2) αβ ( (cid:126)k )+ (cid:88) α = x,y,z λσ α ⊗ L α ⊗ , (S2)where (cid:126)k = ( k , k , k ), ε αβ is a Levi-Civita symbol, L α = − iε αβγ are the 3 × L = 1 matricesand spin-orbit coupling is given by λ . The mass difference between the two sites of the unit cell is encoded in apseudospin τ z Pauli matrix and matrices h (1) α ( (cid:126)k ) and h (2) αβ ( (cid:126)k ) describe hopping between nearest neighbors, h (1) x = [cos k + cos( k − k )] τ x + [ − sin k + sin( k − k )] τ y ,h (1) y = [cos k + cos( k − k )] τ x + [ − sin k + sin( k − k )] τ y ,h (1) z = [1 + cos k ] τ x − sin k τ y , and next-nearest neighbors h (2) xy = 2 cos( k + k − k ) τ z , h (2) yx = 2 cos( k − k ) τ z ,h (2) xz = 2 cos k τ z , h (2) zx = 2 cos( k − k ) τ z ,h (2) yz = 2 cos k τ z , h (2) zy = 2 cos( k − k ) τ z . The multilayer system composed of N L (111) layers can be obtained from H ( (cid:126)k ) by replacing quasimomenta k by areal-space hopping matrix structure, H (1 , , ( k , k ) = H in H out H † out H in H out H † out H in . . . 00 0 . . . . . . H out H † out H in , (S3)where diagonal blocks are given by H in ( k , k ) = m ⊗ ⊗ τ z + t (cid:88) α = x,y,z ⊗ (cid:0) − L α (cid:1) ⊗ h (1) α,in ( k , k )+ t (cid:88) α (cid:54) = β ⊗ (cid:104) − ( L α + ε αβ L β ) (cid:105) ⊗ h (2) αβ,in ( k , k ) + (cid:88) α = x,y,z λσ α ⊗ L α ⊗ , (S4)2and off-diagonal ones by H out ( k , k ) = t (cid:88) α = x,y,z ⊗ (cid:0) − L α (cid:1) ⊗ h (1) α,out ( k , k ) + t (cid:88) α (cid:54) = β ⊗ (cid:104) − ( L α + ε αβ L β ) (cid:105) ⊗ h (2) αβ,out ( k , k ) . (S5)The matrices describing hopping are now given by h (1) x,in = cos k τ x − sin k τ y , h (1) x,out = e − ik τ x + i e − ik τ y ,h (1) y,in = cos k τ x − sin k τ y , h (1) y,out = e − ik τ x + i e − ik τ y ,h (1) z,in = τ x , h (1) z,out = τ x + i τ y , for the nearest neighbors and for the next-nearest neighbors the only non-vanishing matrices are h (2) yx,in = 2 cos( k − k ) τ z , h (2) xy,out = e − i ( k − k ) τ z ,h (2) xz,in = 2 cos k τ z , h (2) zx,out = e − ik τ z ,h (2) yz,in = 2 cos k τ z , h (2) zy,out = e − ik τ z . (S6)Additionally, we may add a surface potential term to the Hamiltonian H (1 , , ( k , k ) in a form of V surf = V s diag N (0 , . . . , , , , , , ⊗ ⊗ ⊗ , (S7)where diag N ( d , . . . , d N ) means a diagonal matrix with entries given by d , . . . , d N and V s is the height of thepotential.The important symmetries of the model are mirror reflection symmetry with respect to the (110) plane, MH (1 , , ( k , k ) M − = H (1 , , ( k , k ) , M = √ N ⊗ ( σ x − σ y ) ⊗ (cid:2) ( L x − L y ) − (cid:3) ⊗ (S8)and the time-reversal symmetry T H (1 , , ( k , k ) T − = H (1 , , ( − k , − k ) , T = i K N ⊗ σ y ⊗ ⊗ . (S9)Finally, the orthogonal surface quasimomenta k ˜ x and k ˜ y , used in Figs. 1, S1 and S2, are defined as k ˜ x = k , k ˜ y = 1 √ k − k ) . (S10) S2. MIRROR AND TIME-REVERSAL SYMMETRY PROTECTED QUANTIZATION OF THE BERRYPHASES FOR NON-DEGENERATE BANDS
In this section, we assume that all bands are non-degenerate due to the absence of the inversion symmetry. Weshow that in the presence mirror symmetry the Berry phases for all mirror-symmetric Fermi loops (the Fermi loopmaps back to itself in the mirror symmetry operation) are quantized to 0 or π . Then we show that in the presence oftime-reversal symmetry the Berry phases for all time-reversal-symmetric Fermi loops (the Fermi loop maps back toitself in the time-reversal symmetry operation) are quantized to π .Consider the eigenstates | ψ ( n ) θ (cid:105) belonging to the n th energy band forming a Fermi loop around Γ (or M ) pointparametrized by angle θ ∈ [ − π, π ). Since θ is periodic variable | ψ ( n ) θ (cid:105) can acquire a Berry phase by a parallel shiftalong the loop. The gauge-invariant form of the Berry phase is given by, ϕ n = arg (cid:104)(cid:68) ψ ( n ) − π (cid:12)(cid:12)(cid:12) ψ ( n ) − π + δθ (cid:69)(cid:68) ψ ( n ) − π + δθ (cid:12)(cid:12)(cid:12) ψ ( n ) − π +2 δθ (cid:69) . . . (cid:68) ψ ( n ) π − δθ (cid:12)(cid:12)(cid:12) ψ ( n ) π − δθ (cid:69)(cid:68) ψ ( n ) π − δθ (cid:12)(cid:12)(cid:12) ψ ( n ) − π (cid:69)(cid:105) . (S11)where δθ is an infinitesimal step in angle θ . Now we will consider the impact of time-reversal and mirror symmetrieson possible values of ϕ n .3First, we consider the mirror symmetry. We assume that for 0 ≤ θ ≤ π we obtain all | ψ ( n ) θ (cid:105) states by diagonalizingthe Hamiltonian and we define − π ≤ θ < (cid:12)(cid:12)(cid:12) ψ ( n ) − θ (cid:69) = M (cid:12)(cid:12)(cid:12) ψ ( n ) θ (cid:69) . (S12)We can decompose the Berry phase as ϕ = arg [ χ − χ + ] with χ − = (cid:68) ψ ( n ) − π (cid:12)(cid:12)(cid:12) ψ ( n ) − π + δθ (cid:69) . . . (cid:68) ψ ( n ) − δθ (cid:12)(cid:12)(cid:12) ψ ( n ) − δθ (cid:69)(cid:68) ψ ( n ) − δθ (cid:12)(cid:12)(cid:12) ψ ( n )0 (cid:69) ,χ + = (cid:68) ψ ( n )0 (cid:12)(cid:12)(cid:12) ψ ( n ) δθ (cid:69)(cid:68) ψ ( n ) δθ (cid:12)(cid:12)(cid:12) ψ ( n )2 δθ (cid:69) . . . (cid:68) ψ ( n ) π − δθ (cid:12)(cid:12)(cid:12) ψ ( n ) − π (cid:69) . (S13)Using mirror symmetry M we can relate terms of χ − and χ + as, (cid:68) ψ ( n ) − θ − δθ (cid:12)(cid:12)(cid:12) ψ ( n ) − θ (cid:69) = (cid:68) ψ ( n ) θ + δθ (cid:12)(cid:12)(cid:12) M † M (cid:12)(cid:12)(cid:12) ψ ( n ) θ (cid:69) = (cid:68) ψ ( n ) θ + δθ (cid:12)(cid:12)(cid:12) ψ ( n ) θ (cid:69) . (S14)Therefore most of the phases in ϕ cancel and we get ϕ n = arg (cid:104)(cid:68) ψ ( n ) π (cid:12)(cid:12)(cid:12) ψ ( n ) π − δθ (cid:69)(cid:68) ψ ( n ) π − δθ (cid:12)(cid:12)(cid:12) M (cid:12)(cid:12)(cid:12) ψ ( n ) π (cid:69) (cid:68) ψ ( n ) δθ (cid:12)(cid:12)(cid:12) M † (cid:12)(cid:12)(cid:12) ψ ( n )0 (cid:69)(cid:68) ψ ( n )0 (cid:12)(cid:12)(cid:12) ψ ( n ) δθ (cid:69)(cid:105) . (S15)On the other hand, we know that (cid:12)(cid:12)(cid:12) ψ ( n )0 (cid:69) and (cid:12)(cid:12)(cid:12) ψ ( n ) π (cid:69) are eigenstates of M with eigenvalues ±
1. Hence we have (cid:68) ψ ( n ) π − δθ (cid:12)(cid:12)(cid:12) M (cid:12)(cid:12)(cid:12) ψ ( n ) π (cid:69) = (cid:68) ψ ( n ) π − δθ (cid:12)(cid:12)(cid:12) ψ ( n ) π (cid:69)(cid:68) ψ ( n ) π (cid:12)(cid:12)(cid:12) M (cid:12)(cid:12)(cid:12) ψ ( n ) π (cid:69) , (cid:68) ψ ( n ) δθ (cid:12)(cid:12)(cid:12) M † (cid:12)(cid:12)(cid:12) ψ ( n )0 (cid:69) = (cid:68) ψ ( n ) δθ (cid:12)(cid:12)(cid:12) ψ ( n )0 (cid:69)(cid:68) ψ ( n )0 (cid:12)(cid:12)(cid:12) M (cid:12)(cid:12)(cid:12) ψ ( n )0 (cid:69) , (S16)and consequently ϕ n = arg (cid:104)(cid:68) ψ ( n ) π (cid:12)(cid:12)(cid:12) M (cid:12)(cid:12)(cid:12) ψ ( n ) π (cid:69)(cid:68) ψ ( n )0 (cid:12)(cid:12)(cid:12) M (cid:12)(cid:12)(cid:12) ψ ( n )0 (cid:69)(cid:105) . (S17)This proves that in the presence of the mirror symmetry M the Berry phase is quantized as ϕ n = 0 , π .Now consider a time-reversal symmetry T = U T K , where U T is unitary operator and K is complex conjugationoperator. We assume that the time-reversal symmetry satisfies T = − U T U T = −
1, where bar meanscomplex conjugate. We assume that for − π ≤ θ < | ψ ( n ) θ (cid:105) states by diagonalizing the Hamiltonian andwe define 0 ≤ θ < π states as (cid:12)(cid:12)(cid:12) ψ ( n ) θ + π (cid:69) = T (cid:12)(cid:12)(cid:12) ψ ( n ) θ (cid:69) = U T (cid:12)(cid:12)(cid:12) ψ ( n ) θ (cid:69) , (S18)Now we want to express terms in χ + by those in χ − . We have that (cid:68) ψ ( n ) θ + π (cid:12)(cid:12)(cid:12) ψ ( n ) θ + π + δθ (cid:69) = (cid:68) ψ ( n ) θ (cid:12)(cid:12)(cid:12) U T† U T (cid:12)(cid:12)(cid:12) ψ ( n ) θ + δθ (cid:69) = (cid:68) ψ ( n ) θ + δθ (cid:12)(cid:12)(cid:12) ψ ( n ) θ (cid:69) . (S19)Therefore most of the phases in ϕ cancel and we get ϕ n = arg (cid:104)(cid:68) ψ ( n ) − δθ (cid:12)(cid:12)(cid:12) ψ ( n )0 (cid:69)(cid:68) ψ ( n ) π − δθ (cid:12)(cid:12)(cid:12) ψ ( n ) − π (cid:69)(cid:105) = arg (cid:20)(cid:68) ψ ( n ) − δθ (cid:12)(cid:12)(cid:12) ψ ( n )0 (cid:69)(cid:68) ψ ( n ) − δθ (cid:12)(cid:12)(cid:12) U T† U T† (cid:12)(cid:12)(cid:12) ψ ( n )0 (cid:69)(cid:21) = arg (cid:20) − (cid:12)(cid:12)(cid:12)(cid:68) ψ ( n ) − δθ (cid:12)(cid:12)(cid:12) ψ ( n )0 (cid:69)(cid:12)(cid:12)(cid:12) (cid:21) = π. (S20)Hence, we have proved that in the presence of time-reversal symmetry satisfying T = − π . We point out that if the time-reversal symmetry satisfies T = 1 the Berry phases are quantized to 0.4 Г M M M m m M M M Г M M M Г a b c k x ~ k y ~ FIG. S1:
Fermi loops and their Berry phases (in the units of π ) in presence of the symmetry-breaking terms . a , Mirror symmetry is preserved and time-reversal symmetry is weakly broken (cid:126)h = ( − . , . , b , Mirror symmetry ispreserved and time-reversal symmetry is strongly broken (cid:126)h = ( − . , . , c , Mirror and time-reversal symmetries arestrongly broken γ = 0 . (cid:126)h = ( − . , . , π , depending on the strength of the Zeeman field. If both symmetries are broken all Berryphases are arbitrary. The other parameters in all cases are: N L = 10, m = 1 . t = 0 . t = 0 . λ = 0 . µ = − . Г M M M m M M M Г a b k x ~ k y ~ FIG. S2:
Fermi loops and their Berry phases (in the units of π ) in presence of the symmetry-breaking terms. a, Mirror symmetry is preserved and time-reversal symmetry is weakly broken (cid:126)h = ( − . , . ,
0) and b, Mirror and time-reversalsymmetries are strongly broken γ = 0 . (cid:126)h = ( − . , . , π , depending on the strength of the Zeeman field. If both symmetries arebroken all Berry phases are arbitrary. The other parameters in all cases are: N L = 10, m = 0 . t = 0 . t = 0 . λ = 0 . µ = − . S3. BERRY PHASES IN THE SYMMETRY-BROKEN CASES
In this section, we discuss how various perturbations breaking the symmetries of the model are included in thetheory, and we calculate the effect of these perturbations on the Berry phases. The mirror symmetry breaking canappear due to intentional structural distortions or unintentional inhomogeneities and time-reversal symmetry breakingperturbations can be present due to various mechanisms [3–7]. Here, our aim is not to realistically model the breakingof these symmetries in real materials but rather to demonstrate the important role of the time-reversal and mirrorsymmetries in the quantization of the Berry phase and the WAL effect.To break time-reversal symmetry we consider a Zeeman field (cid:126)h coupling to spins (cid:126)σ , H mag = N ⊗ (cid:126)h · (cid:126)σ ⊗ ⊗ . (S21)To break mirror symmetry M on one surface of the system we modify the N L -th diagonal block H in in H (1 , , ( k , k )of Eq. (S3) by setting hopping amplitude in x direction as different than in y direction, i.e., h (1) x,in → (1 + γ ) h (1) x,in , h (2) xz,in → (1 + γ ) h (2) xz,in , h (2) zx,in → (1 + γ ) h (2) zx,in h (1) y,in → (1 − γ ) h (1) y,in , h (2) yz,in → (1 − γ ) h (2) yz,in , h (2) zy,in → (1 − γ ) h (2) zy,in , - - � � � � a v - - -4 -4 � � � � � � a v � � � - � � � a v � � � � � � � a v � � � - � � � a v � � � � � � � a bc d FIG. S3:
Dependencies of deviation from the quantized value of the Berry phase (averaged over Fermi loops) on: a, strength of mirror symmetry breaking γ in presence of a weak Zeeman field (cid:126)h = (0 , . , . b,c, in-plane Zeemanfield angle θ for | (cid:126)h | = 1 in presence of weak b, γ = 0 .
008 and strong c, γ = 0 . b,c, we plot thedifference with respect to the γ = 0 case and angles θ and θ correspond to (cid:126)h ∝ − (cid:126)a and (cid:126)h ∝ (cid:126)a , respectively. d , Deviationof the Berry phase from the quantized value ϕ = π as a function of the Zeeman field oriented in the same direction as inexperiment for the case of unbroken (solid lines) and broken (dashed lines) crystalline mirror symmetry. The other parametersin all cases are: N L = 10, m = 1 . t = 0 . t = 0 . λ = 0 . µ = − .
25 and h = 0 .
01 (all energies in eV). where γ describes the strength of mirror symmetry breaking.For Fig. 1 we have diagonalized H (1 , , ( k , k ) with N L = 10 layers taking the following parameters (all in eV, µ is the Fermi level) • Fully symmetric – Non-trivial: m = 1 . t = 0 . t = 0 . λ = 0 . µ = − . – Trivial: m = 2 . t = 0 . t = 0 . λ = 0 . V s = 0 . µ = − . • Broken time-reversal – Non-trivial: m = 1 . t = 0 . t = 0 . λ = 0 . (cid:126)h = ( − . , . ,
0) and µ = − . – Trivial: m = 2 . t = 0 . t = 0 . λ = 0 . V s = 0 . (cid:126)h = ( − . , . , µ = − . • Broken mirror and time-reversal – Non-trivial: m = 1 . t = 0 . t = 0 . λ = 0 . γ = 0 . (cid:126)h = ( − . , . ,
0) and µ = − . – Trivial: m = 2 . t = 0 . t = 0 . λ = 0 . V s = 0 . γ = 0 . (cid:126)h = ( − . , . ,
0) and µ = − . V s . Dueto this potential some of the low-energy states are localized close to the surface forming topologically trivial surfacestates in qualitative agreement with the experimental observations.In Fig. S1 we show we show Fermi loops and Berry phases in the case of broken time-reversal symmetry. Whenthe mirror symmetry is preserved the Berry phase for mirror-symmetric Fermi loops around Γ and M points arequantized as either 0 or π . In the case of weakly broken time-reversal symmetry, the Berry phases remain quantizedto π (Fig. S1a). By increasing the strength of the time-reversal symmetry breaking the Berry phases around the Γpoint become 0 (Fig. S1b). The changes of Berry phases occur at topological transitions where energies of two bands6become degenerate at a particular momentum within the Fermi loops. The Fermi loops around M remain non-trivialup to the larger value of the field and the Berry phases for Fermi loops around other high-symmetry points takenon-quantized values. In the case when all symmetries are broken (Fig. S1c), all the Berry phases are non-quantized.Fig. S3d shows the dependencies of deviation of the Berry phase from the quantized value ϕ = π as a function of theinplane Zeeman field for the cases of preserved and broken mirror symmetry. The curves follow similar trend as theone observed in the experiment, shown in Fig. 6 and calculated from the Cooperon propagator, see Fig. 3f.The Fermi loops around M points in Figs. 1 and S1 take form of the ellipses elongated in the direction perpendicularto Γ − M line. It is however possible in the present model to obtain the elongation parallel to the Γ − M line. InFig. S2 we show that effect of the time-reversal and mirror symmetry breaking perturbations on the Berry phases isthe same as before.It is important to determine how robust is the tendency of the Berry phase, in the presence of an in-plane magneticfield, to deviate stronger from the quantized value when the mirror symmetry is simultaneously broken. In Fig. S3awe show a representative dependence of the average deviation of the Berry phase | δϕ | av on γ in the presence of a smallZeeman field in the in-plane direction (cid:126)a . We notice that apart from the small interval between γ = 0 and γ = 0 . γ = 0. We also note that the curve shownin Fig. S3a gets transformed as | δϕ | av ( γ ) → | δϕ | av ( − γ ) when the Zeeman field is transformed by a mirror symmetry M : (cid:126)h ∝ (cid:126)a → (cid:126)h ∝ − (cid:126)a . Therefore, if the mirror symmetry breaking is small, the deviation can be decreased bychoosing the Zeeman field in the (cid:126)a direction and increased by the field in the − (cid:126)a direction. This is confirmed bydetermining the dependence of the deviation | δϕ | av on the direction of the in-plane Zeeman field. We parametrizethis field as (cid:126)h ∝ cos θ ( (cid:126)a + (cid:126)a ) / √ θ ( (cid:126)a + (cid:126)a ) / √ (cid:126)a i are not orthogonal) and for fixed γ we track the difference in deviation of the Berry phase with respect to γ = 0 case as function of θ . In Figs. S3b,cwe show the results for small γ = 0 .
008 and large γ = 0 .
4. We have marked the angles θ and θ that correspond tomirror-related field directions (cid:126)h ∝ − (cid:126)a and (cid:126)h ∝ (cid:126)a (in general case mirror symmetry relates (cid:126)h ( θ ) with (cid:126)h ( π − θ )). Wesee that for small γ the deviation with respect to mirror-symmetric case is larger for θ = θ and smaller for θ = θ but for large γ in both cases the deviation is larger. S4. QUANTUM CORRECTION TO THE CONDUCTIVITY
In the vicinity of the band crossings appearing at the high-symmetry points we derive a low-energy 2D Hamiltonianfor a single pair of subbands in a form of H (cid:126)k,(cid:126)σ = (cid:126) m e (cid:126)k + α so ( σ x k y − σ y k x ) + gσ z . (S22)Here m e is the effective mass of the electron, α so is an effective spin-orbit-like coupling that arises from breakingof the inversion symmetry (always present due to the surface in these samples), g is the mass term induced by thebreaking of the mirror symmetry and weak breaking of the time-reversal symmetry, and σ is an effective pseudospinvariable which describes entangled spin and orbital degrees of freedom.The quantum correction to the conductivity can be written as [8]∆ σ xx = − e π (cid:126) D e L (cid:88) (cid:126)Q (cid:88) α,β = ± C αββα ( (cid:126)Q ) (S23)where D e = v F l/ l = v F τ is the elastic mean-free path, v F is the Fermi velocity, τ isthe elastic scattering time, L is the area of the sample, and α, β = ± are the pseudospin indices of the Cooperonpropagator C . For weak disorder ( τ E F / (cid:126) (cid:29) E F is the Fermi energy) the Cooperon propagator can beapproximated as C ( (cid:126)Q ) = τ (cid:18) − (cid:90) d Ω2 π − iτ Σ / (cid:126) (cid:19) − , (S24)where Σ( (cid:126)Q ) = H (cid:126)Q − (cid:126)k,(cid:126)σ (cid:48) − H (cid:126)k,(cid:126)σ (S25)is a 4 × (cid:126)σ (cid:48) and (cid:126)σ . The integral in Eq. (S24) is over allangles of velocity (cid:126)v = (cid:126) (cid:126)km e (S26)7on the Fermi surface. To the lowest order in (cid:126)Q and α so , we getΣ = − (cid:126) (cid:126)v · (cid:126)Q − (cid:126) Q so ( S x v y − S y v x ) + g ( σ (cid:48) z − σ z ) = − (cid:126) (cid:126)v · ( (cid:126)Q + Q so ˆ a(cid:126)S ) + g ( σ (cid:48) z − σ z ) , (S27)where Q so = 2 m e α so / (cid:126) , (cid:126)S = 12 ( (cid:126)σ + (cid:126)σ (cid:48) ) , (S28)and ˆ a = (cid:18) −
11 0 (cid:19) . (S29)Thus, we can write the Cooperon as C ( (cid:126)Q ) − = 1 τ − (cid:90) d Ω2 π
11 + i τ (cid:126) (cid:104) (cid:126) (cid:126)v · (cid:16) (cid:126)Q + Q so ˆ a(cid:126)S (cid:17) − g ( σ (cid:48) z − σ z ) (cid:105) . (S30)Expanding the Cooperon to the second oder in (cid:16) (cid:126)Q + Q so ˆ a(cid:126)S (cid:17) and performing angular integral over (cid:126)v (having | (cid:126)v | = v F )we get for g/ (cid:126) (cid:28) /τ C ( (cid:126)Q ) = 1 D e (cid:16) (cid:126)Q + Q so ˆ a(cid:126)S (cid:17) − i g (cid:126) ( σ (cid:48) z − σ z ) . (S31)Thus the problem comes down to inverting a non-Hermitian Cooperon Hamiltonian in a form of: H c ≡ C − D e = (cid:126)Q + 2 Q so (cid:126)Q · ˆ a(cid:126)S + Q (cid:0) S x + S y (cid:1) − i g (cid:126) D e ( σ (cid:48) z − σ z ) . (S32)In the basis of triplet | S = 1 , m = 1 (cid:105) = | ↑↑(cid:105) , | S = 1 , m = 0 (cid:105) = ( | ↑↓(cid:105) + | ↓↑(cid:105) ) / √ | S = 1 , m = − (cid:105) = | ↓↓(cid:105) and singlet | S = 0 , m = 0 (cid:105) = ( | ↑↓(cid:105) − | ↓↑(cid:105) ) / √ H c = Q + (cid:126)Q √ Q so Q + √ Q so Q − Q + (cid:126)Q √ Q so Q + − iη √ Q so Q − Q + (cid:126)Q − iη (cid:126)Q , (S33)with Q ± = Q y ± iQ x , η = 2 g (cid:126) D e . (S34)Equation (S23) can now be written as:∆ σ xx = e π (cid:126) L (cid:88) (cid:126)Q Tr (cid:34) Γ (cid:18) D e τ φ + H c (cid:19) − (cid:35) , (S35)where we have also included the dephasing time τ φ and Γ is given byΓ = − − − . (S36)Following the standard approach (e.g., ref. [9–11]), we introduce the perpendicular magnetic field in the CooperonHamiltonian by minimal substitution (cid:126)Q → (cid:126)Q + 2 e (cid:126)A/ (cid:126) , where the vector potential can be chosen as (cid:126)A = (0 , xB, Q + → Q y + iQ x + 2 eB (cid:126) x = 2 (cid:114) eB (cid:126) √ (cid:18)(cid:114) (cid:126) eB ∂ x + (cid:114) eB (cid:126) x + Q y (cid:114) (cid:126) eB (cid:19) = 2 (cid:114) eB (cid:126) a,Q − → Q y − iQ x + 2 eB (cid:126) x = 2 (cid:114) eB (cid:126) √ (cid:18) − (cid:114) (cid:126) eB ∂ x + (cid:114) eB (cid:126) x + Q y (cid:114) (cid:126) eB (cid:19) = 2 (cid:114) eB (cid:126) a † ,(cid:126)Q → eB (cid:126) ( − ∂ ξ + ξ ) = 4 eB (cid:126) (cid:18) a † a + 12 (cid:19) ,a = 1 √ ∂ ξ + ξ ) , a † = 1 √ − ∂ ξ + ξ ) , ξ = (cid:114) eB (cid:126) x + Q y (cid:114) (cid:126) eB . (S37)Here a ( † ) are the harmonic oscillator ladder operators which obey bosonic commutation relations.The Cooperon Hamiltonian becomes: H c = 2 Q so (cid:114) eB (cid:126) ( aS + + H.c. ) + 4 eB (cid:126) (cid:18) a † a + 12 (cid:19) + Q (cid:0) P − S z (cid:1) − iηR, (S38)with matrices S + √ , S z = − , P = , R = . (S39)We can now utilize the fact that the Cooperon Hamiltonian satisfies a symmetry [ H c , M z ] = 0, where M z = a † a + S z . (S40)Namely, the spectrum of M z is composed of eigenvalues {− , , , , , , , , , , , , . . . } , and the Hamiltonian hasa block diagonal form in the eigenbasis of the M z . The first two blocks can be combined into a 4 × H (0) c Q = b b √ b − iηQ − √ b b − iηQ − b , (S41)and the other blocks are given by H ( n ) c Q = n − b √ nb √ nb n + 2) b (cid:112) n + 1) b − iηQ − (cid:112) n + 1) b n ) b − iηQ − n ) b , ( n = 1 , , . . . n max −
1) (S42)where b = eB (cid:126) Q . The maximum number of Landau levels n max is restricted by the condition that the Cooperoncyclotron radius should be larger than the elastic mean-free path. This can be written as n max = 14 τ B (cid:126) eD e . (S43)Taking also into account that the degeneracy of the Cooperon Landau levels is eBL / ( π (cid:126) ), we obtain∆ σ xx = e π (cid:126) (cid:18) Be (cid:126) (cid:19) n max (cid:88) n =0 Tr (cid:34) Γ (cid:18) D e τ φ + H ( n ) c (cid:19) − (cid:35) . (S44)Assuming that τ (cid:28) τ φ , we can take the limit n max → ∞ . In this case, we recover the Hikami-Larkin-Nagaoka (HLN)formula in the limit η = Q so = 0, i.e.,∆ σ xx ( B ) − ∆ σ xx (0) = e π (cid:126) (cid:20) ψ (cid:18)
12 + 14 τ φ B (cid:126) eD e (cid:19) − log (cid:18) τ φ B (cid:126) eD e (cid:19)(cid:21) . (S45)9 � h x /h � E / � E max FIG. S4: Gap ∆ E between two subbands that form Fermi surface around Γ points as function of mirror breaking term γ andtime reversal Zeeman field h x = − h y that preserves mirror symmetry. The maximal gap ∆ E max is defined as ∆ E at γ = 0 . h x = h with h = 0 .
01 eV. The other parameters are: N L = 10, m = 1 . t = 0 . t = 0 . λ = 0 . µ = − .
25 (allenergies in eV).
To obtain Q so we use the low energy expansion around Γ point of the 10-layer model in the topological phase (i.e. m = 1 . t = 0 . t = 0 . λ = − . a = 6 . × − m to get Q so = 1 . × m − .This means that the pseudospin precession length is l so = 2 π/Q so ≈
60 nm.The parameter g = 0 unless both time-reversal and mirror symmetry are simultaneously broken, see Fig. S4. We canassume that the time-reversal symmetry is always weakly broken by the same amount due to the intrinsic mechanisms,impurities and environment, but the breaking of the mirror symmetry is tunable and depends on how the sample iscovered. S5. STRUCTURAL CHARACTERIZATION
Our structural and surface morphology investigations revealed the high structural quality of obtained samples.The XRD studies show the presence of a single phase from the (111) oriented film and substrate (Fig. S5a). Thepresence of atomically sharp surface of the films, which is indicated by streaky RHEED reflections situated on Lauesemicircle (inset to Fig. S5b), is further confirmed by high-resolution XRD measurements (Fig. S5b). A large numberof well-developed Kissing fringes, the result of interference between surface and interface reflected beams, guaranteesa presence of smooth surface and interface of the films. From the relative position of the fringes, the film thicknessis precisely determined. AFM investigations revealed a monolayer flat surface as one can deduce from the AFMline profile (Fig. S5c). The contours of monolayer steps are clearly defined in the presented AFM image. Theaverage RMS roughness of 0.345 nm is also consistent with an elevation level difference of 1 monolayer (1 atomiclayer of metal + 1 atomic layer of Se). Due to relatively large lattice constant misfit between the film and substrate(∆ a/a = 1 . − BaF =18.1 10 − /K,TEC PbSe =19.4 10 − /K at room temperature [13, 14], we do not expect big changes in the value of strain upon filmcooling. In addition to residual strain determination, asymmetric RSM is used to calculate a fully relaxed latticeconstant and verify the composition of the films according to Vegard’s law [15, 16]. S6. LIST OF SAMPLES AND MAGNETOTRANSPORT CHARACTERIZATION
Upon cooling, all grown epilayers exhibit metallic behaviour (i.e. dR xx /dT > R ( T ) followpower law for most samples, except for the PbSe epilayer (Fig. S6). Carrier density and low-temperature mobilityare extracted from the slope of the Hall resistivity measured up to 2 T and zero-field resistivity. According to suchmeasurements, the PbSe epilayer has relatively low hole density, 3 × cm − , and high mobility, 3 . × cm /Vs.10
20 30 40 50 60 70 80 90 100 110 12010 Pb Sn SePb Sn SePb Sn Se (111) (222) (333) P b S n S e I n t en s i t y ( a r b . un i t s ) (444) B a F
49 50 51 52 53 5410 t = 54 nm X Sn =21.2 % (222) P b S n S e B a F I n t en s i t y ( a r b . un i t s ) Q - w (degree) Z ( n m ) X ( m m) Line profile abc d
FIG. S5:
Structural characterization of obtained samples . a , XRD spectra of three samples with different Sn contentindicating of (111) oriented single phase from films and substrate. b , High resolution XRD spectrum of Pb . Sn . Se filmaround (222) reflection showing high order of thickness fringes; the insets represent streaky RHEED pattern obtained along[110] azimuth for the same sample (left) and XRD RSM in the vicinity of symmetric (222) reflection (right). c , Typical AFM ofthe film with extracted line profile demonstrating monolayer thick roughness. d , XRD (513) asymmetric RSM with relaxationtriangle evidencing the presence of small strains in the films. Addition of Sn rapidly increases the Hall carrier density, thus all studied Pb − x Sn x Se films are p -type and their carrierdensities and mobilities are in the range of (0.5 – 2 . × cm − and (0.5 – 2) × cm /Vs, respectively. Highcarrier densities and relatively low mobilities are associated with the presence of highly disordered p -type region atthe interface with BaF [17, 18]. Table S1 contains detailed information on the studied epilayers.According to data collected in Fig. S7, magnetoresistance (MR) in all studied epilayers is similar. In particular, thereis a pronounced dip of resistance around B = 0, which is associated with the Berry phase positive magnetoresistance.In the higher field range, MR is parabolic, which smoothly changes to a linear behaviour in the strongest fields. Suchhigh-field linear MR is usually explained by semiclassical models [19–21], where it arises from spatial variations ofcarrier density. Such behaviour is well described with the semi-empirical model [20, 22, 23]: ρ xx ( B ) = ρ xx (0)1 − A + A √ µB ) . (S46)Here, µ stands for mobility. In high in-plane magnetic field ( B (cid:107) I ), most of the studied epilayers exhibited negativelongitudinal MR (Fig. 5), however, in several epilayers, longitudinal resistance increase with the field (Fig. 5). Wehave not found any correlation between the MR sign and composition of the films. Such behaviour may be associatedwith disorder; model calculations [24] showed that inhomogeneity along the growth direction can induce negativelongitudinal MR.11 TABLE S1: Parameters of the studied epilayers: tin content x ; film thickness d ; apparent hole concentration p , carrier mobility µ p and mean free path l determined from the Hall and resistivity data; the phase coherence length extracted from the HLN fit,Eq. (S47) at 1.5 K.Epilayer Se covered Sn content, % d, nm p , 10 cm − µ p , cm /Vs l , nm l φ at 1.5 K, nmA no 9.0 52 1.20 1200 34.9 1080B no 11.2 50 1.37 950 21.7 1890C no 16.6 50 1.57 490 7.9 760D no 19.5 52 2.32 540 17.7 910E no 24.0 50 1.71 930 25.1 1830F yes 0.0 44 0.03 3850 125 400G yes 6.5 47 0.49 2080 63 310H yes 13.6 53 1.26 480 5 270I yes 15.9 54 1.26 1120 18 370J yes 30.0 50 2.1 690 47 150 Supplementary Fig r xx ( W ) T (K) Pb Sn Se Pb Sn Se Pb Sn Se Pb Sn Se Pb Sn Se Model NewFunction1 (User)Equation (A0+A1*x+A2*x^2+A5*x^5+A*x^p)Plot ?$OP:F=1 ?$OP:F=2A0 71.91554 ± 0.01675 95.53978 ± 0.09237A1 0.25645 ± 3.6341E-4 0.26347 ± 0.00237A2 0.00178 ± 1.86085E-6 0.00231 ± 1.34567E-5A5 2.04019E-11 ± 4.71669E-14 -4.12095E-12 ± 4.41761E-13A 0 ± 0 0 ± 0p 1 ± 0 1 ± 0Reduced Chi-Sqr 0.26789 1.54445R-Square(COD) 0.9999 0.99942Adj. R-Square 0.9999 0.99942
PbSe/Se Pb Sn Se/Se Pb Sn Se/Se Pb Sn Se/Se Pb Sn Se/Se r D xx ( W ) T (K) a b FIG. S6:
Temperature dependence of the resistivity. a , Uncovered samples; b , samples covered by Se. S7. FITTING EXPERIMENTAL DATA WITH FULL HLN EXPRESSION
According to the theory developed here, the low-field MR is assigned to the Berry phase quantization brought aboutby the mirror and time reversal symmetries rather than by a non-trivial character of the topological phase. Withinthis scenario and by noting that the phase coherence length l φ is larger than the film thickness d = 50 nm, the MRhas been described by the HLN theory in the limit l φ (cid:29) l so, where l so is the mean free path for spin-orbit scattering[9], ∆ σ xx ( B ) = α n v (cid:88) i e π (cid:126) f (cid:32) (cid:126) c el φ,i B (cid:33) , (S47)where the prefactor α = − /
2; the magnetic field B is perpendicular to the film plane; the summation is overindependent conduction channels, f ( x ) ≡ ψ (1 / x ) − ln( x ), ψ ( x ) is the digamma function, l φ,i is the phase coherencelength in the i ’th conduction channel. In our case, as shown in Fig. 3d in the main text, MR for all epilayers canbe fitted by the one-channel formula, n v = 1 treating l φ ( T ), as the only fitting parameter. This means that l φ,i are longer than length scales characterizing scattering between subbands and valleys (including surface ones in thetopological case) as well as between n and p -type layers. Alternatively, and more probably, because of short lengthscales characterizing the p -type region, the corresponding WAL or WL MR is shifted to a high field region, so the lowfield features are solely due to electrons residing closer to the outer surface. It is important recalling that if only onelayer shows MR, Eq. (S47) with n v = 1 remains valid even in the multichannel case.The simplified HLN formula [Eq. (S47)] is assumed to be valid in the case of the topological surface state for whichthe Berry phase is quantized. Resulted presented in Fig. 5 of the main text show that this is also in the case of bulk-dominated transport (Fermi level in the conduction band) and even in the topologically trivial materials. To check12 Supplementary Fig -8 -4 0 4 875788184 Pb Sn Se r xx ( W ) B (T) B ^ B || -0.2 0.0 0.275.875.9 -8 -4 0 4 8868788 Pb Sn Se r xx ( W ) B (T) B ^ B || -0.2 0.0 0.286.086.186.2 -8 -4 0 4 8767778 Pb Sn Se r xx ( W ) B (T) B ^ B || Model NewFunction (User)Equation R0 / (1 - A + A / sqrt(1 + (mu*B)^2))Plot ?$OP:F=1 ?$OP:F=2R0 75.85014 ± 3.9331E-4 75.85014 ± 0A 0.17424 ± 4.97611E-4 -0.00932 ± 4.62488E-5mu 0.0873 ± 1.63274E-4 0.0873 ± 0Reduced Chi-Sqr 3.37833E-4 4.41308E-4R-Square(COD) 0.99947 0.76667Adj. R-Square 0.99947 0.76667 -0.2 0.0 0.275.775.8 -8 -4 0 4 8131132133134135 Pb Sn Se r xx ( W ) B (T) B ^ B || -0.2 0.0 0.2131.6131.8 -8 -4 0 4 8108109110 Pb Sn Se r xx ( W ) B (T) B ^ B || -0.2 0.0 0.2107.9108.0108.1 a b c e d -8 -4 0 4 810001500200025003000 PbSe/Se B ^ B || r xx ( W ) B (T) -0.2 0.0 0.210801100 -8 -4 0 4 8120130140150160 r xx ( W ) B (T) B ^ B || Pb Sn Se/Se -0.2 0.0 0.2121.6121.8122.0 -8 -4 0 4 8204207210213216 Pb Sn Se/Se B ^ B || r xx ( W ) B (T) -0.2 0.0 0.2 -8 -4 0 4 890919293949596 Pb Sn Se/Se B ^ B || r xx ( W ) B (T) -0.2 0.0 0.290.891.0 -8 -4 0 4 8888990 B ^ B || Pb Sn Se/Se r xx ( W ) B (T) -0.2 0.0 0.2 f g h j i Uncovered thin films
Se covered thin films
FIG. S7:
Magnetoresistivity in a high field range . a-e , Uncovered samples; f-j , samples covered by Se. The data forthe magnetic field perpendicular and parallel to the film plane are shown by black and red symbols, respectively. Black linespresent the best fit to Eq. (S46). the validity of the simplified HLN expression, we have fitted experimental data employing the full HLN expression,∆ σ xx ( B ) = e π (cid:126) (cid:20) f (cid:18) B B (cid:19) + f (cid:18) B B (cid:19) + f (cid:18) B B (cid:19)(cid:21) , (S48)where B = B e + B SO + B s ,B = 43 B so + 23 B s + B φ ,B = 2 B s + B φ . (S49)The fields B e , B so , B s and B φ are related to the mean free path l , the spin diffusion length limited by spin disorderscattering l s and spin-orbit coupling l so , and the phase coherence length l φ according to B i = (cid:126) / el i . Results ofthe best fit of experimental data at 4.2 K are presented in Fig. S8. Since there is no magnetic doping, we assumethat B s = 0; values of the mean free path extracted from zero-field mobility are used for B e , thus there are onlytwo adjustable parameters, l φ and l so . Fitting experimental data at 4.2 K results in l so values below 60 nm, i.e.,comparable to l and several times shorter than l φ . Moreover, fitting results do not change appreciably with varying l so towards even lower values, so that using the simplified HLN expression in a strong SO coupling limit is justified.The only exception here is PbSe/Se epilayer, for which it was impossible to fit with the simplified HLN expression,as a large magnitude of the mean free path l = 125 nm makes that the condition B < B e is violated for this sample[25, 26]. It worth noting that fitting of WAL MR data at different temperatures with the full HLN expression (S48)results in l so values that are temperature independent within the accuracy of the fitting.13 Supplementary Fig -0.4 -0.2 0.0 0.2 0.4-30-20-100 Pb Sn Se D s xx ( m S ) B (T) l e = 41 nml SO = 10 nml f = 403 nm -0.4 -0.2 0.0 0.2 0.4-30-20-100 l e = 22.6 nml SO = 5.4 nml f = 452 nm Pb Sn Se D s xx ( m S ) B (T) -0.4 -0.2 0.0 0.2 0.4-30-20-100 l e = 5.6 nml SO = 32 mnl f = 380 nm Pb Sn Se D s xx ( m S ) B (T) -0.4 -0.2 0.0 0.2 0.4-20-100 l e = 23 nml SO = 17 nml f = 325 nm D s xx [ m S ] B [T] Pb Sn Se -0.4 -0.2 0.0 0.2 0.4-20-100 l e = 17 nml SO = 30 nml f = 256 nm Pb Sn Se D s xx ( m S ) B (T) -0.4 -0.2 0.0 0.2 0.4-20-15-10-50 l e = 5 nml SO = 35 nml f = 245 nm Pb Sn Se/Se s xx ( m S ) B (T) -0.4 -0.2 0.0 0.2 0.4-100 l e = 125 nml SO = 120 nml f = 366 nm s xx ( m S ) B (T) PbSe/Se -0.4 -0.2 0.0 0.2 0.4-20-100 l e = 31 nml SO = 43 nml f = 297 nm Pb Sn Se/Se s xx ( m S ) B (T) -0.4 -0.2 0.0 0.2 0.4-20-15-10-50 l e = 18 nml SO = 54 nml f = 272 nm Pb Sn Se/Se s xx ( m S ) B (T) -0.4 -0.2 0.0 0.2 0.4-10-50 Pb Sn Se/Sel e = 47 nml SO = 62 nml f = 160 nm s xx ( m S ) B (T) a b c e d f g h j i Uncovered thin films
Se covered thin films
FIG. S8: Low-field magnetoconductivity at 4.2 K fitted with the full HLN expression [Eq. (S48)]. epilayer F - l f T -1.00 epilayer G - l f T -1.22 epilayer H - l f T -1.32 epilayer I - l f T -1.14 epilayer J - l f T -0.89 l f ( n m ) T (K) l f ( n m ) T (K) epilayer A - l f T -0.68 epilayer B - l f T -1.00 epilayer C - l f T -0.76 epilayer D - l f T -0.93 epilayer E - l f T -1.31 a b FIG. S9:
Temperature dependence of the phase coherence length obtained from fit to the HLN formula . a ,Se-uncovered epilayers. b , Se-covered epilayers. No saturation is observed for uncovered epilayers down to 1.5 K. S8. COMPARISON OF COHERENCE LENGTH VALUES TO PUBLISHED DATA
Reported values for l φ for topological materials differ significantly, and depend, most likely, on particular growth andprocessing conditions. However, one can notice (see Table S2) that despite variation in particular values of l φ , datafor topological insulators (TIs) show a similar temperature dependence ≈ T − p/ (above low temperature saturation)with p ≈
1, which points to the dominance of e-e interactions in 2D [27]. Published data for TCI materials are morescarce, but there are several works [28–30], which report l φ temperature dependence with p are in the range from 1 to14 TABLE S2: Published values for l φ in topological insulatorsMaterial Thickness Temperature l φ Temperature dependenceTI materialsBi Te [34] 5-100 QL 1.5 K 100-1000 nm —Bi Te [34] 5-100 QL 1.5 K 100-1000 nm —Bi Se Te [35] 50 nm 1.9 K 320 nm ≈ T − . Bi Te [36] 5-128 QL 1.5 K 800 nm —Bi SeTe [37] 15 nm 7 K 60 nm ≈ T − . Bi . Sb . Te [38] 90 nm 4.2 K 160 nm ≈ T − . Bi Te [39] 65 nm 1 K 300 nm ≈ T − . Bi Te [40] 40 QL 2 K 1000 nm ≈ T − . Bi SeTe [41] bulk ( ∼ ≈ T − . Bi . Sb . Te . Se . [42] 130 nm 2 K 116 nm ≈ T − . Bi Se [43] 6-54 nm 2 K 40-120 nm ≈ T − . BiSbSe Te [44] 50-90 nm 2 K 86 nm ≈ T − . Bi Se Te [45] 100-120 nm 1.8 K 57 nm ≈ T − . Bi Te /Te [46] 15 nm 0.1 K 1500 nm —Bi Se [47] 30 nm 10 K 318 nm ≈ T − . Bi SeTe [48] 15 nm 2 K 120 nm ≈ T − . Sb Te [23] 8 QL 1.8 K 150 nm ≈ T − . TCI materialsSnTe [49] 30-60 nm 2 K 200-400 nm —SnTe [28] 46 nm 4 K 200 nm ≈ T − . SnTe [28] 74 nm 4 K 600 nm ≈ T − . Pb − x Sn x Se [29] 10-16 nm 2 K 250-350 nm from T − . to T − . (Pb . Sn . ) . In . [50] bulk (0.2-0.6 mm) 5 K 123 nm —Pb . Sn . Te [51] 20 nm 2 K 147 nm —SnTe [52] 10 u.c. 1.8 K 200 nm —SnTe [30] 10-100 nm 1.7 K 350-1020 nm from T − . to T − . Pb . Sn . Se [53] 55 nm 3 K 900-1700 nm from T − . to T − .
3. This suggests that the role of the e-e dephasing mechanism is reduced. Though at low temperatures the dominantmechanism of dephasing is e-e scattering [31], this source of dephasing is suppressed due to a large magnitude of thedielectric constant in the studied materials [32]. Thus, in our case electron-phonon (e-ph) interactions dominate inthe dephasing process. Indeed, e-ph interactions results in p = 2, 3, or 4; with particular value depending on thedetails of the studied system [32, 33]). Our results agree with the published data for TCI materials. Also, values of l φ rarely exceed 1 µ m at 1-2 K. Thus values of l φ in the current study, obtained in samples without Se cap at 1.5 K(1-2 µ m, see Fig. S9 and Table S1) are one of the largest among topological materials [28, 30]. With the supportof the published data, we can argue that the dephasing mechanism in IV-VI TCI compounds differs from the one inV-VI TI materials and, most likely, the role of e-e interactions is diminished. S9. FITTING MR IN PARALLEL FIELDS
As discussed in the main text and shown there in Fig. 6, we have also studied low-field MR for the magnetic fieldapplied in-plane, along the current direction. In the parallel field, the WAL effect does not disappear and exhibitsa similar magnitude as in the perpendicular field, as presented in Fig. S10. In the past, conventional WL MR in aparallel field was theoretically considered for diffusive films, l (cid:28) d , where d is the film thickness [54], later for theclean films, i.e, quantum wells, l (cid:29) d [55], and finally for the intermediate regime [56] ( l (cid:39) d ). All these theoriespredict a similar logarithmic shape of ∆ σ xx ( B (cid:107) ),∆ σ xx ( B (cid:107) ) = α e π (cid:126) ln (cid:32) β e d l φ (cid:126) B (cid:107) (cid:33) , (S50)15 Supplementary Fig l f = 463 nm a = -0.50; b = 0.30 a = -0.40; b = 0.66 Pb Sn Se D s xx ( m S ) B (T) -0.4 -0.2 0.0 0.2 0.4-30-20-100 a = -0.5l f = 463 nm b = 0.30 Pb Sn Se D s xx ( m S ) B (T) -0.4 -0.2 0.0 0.2 0.4-30-20-100 a = -0.5l f = 321 nm b = 0.07 D s xx ( m S ) B (T) Pb Sn Se Pb Sn Se D s xx ( m S ) B (T) l f = 321 nm a = -0.50; b = 0.07 a = -0.37; b = 0.47 a b c d FIG. S10:
Comparison of low-field magnetoconductance in perpendicular and parallel configurations . The data(black symbols – perpendicular field; red symbols – parallel field) have been taken at 4.2 K for samples with Sn contentcorresponding to the topologically trivial ( a,c ) and non-trivial (b,d) phases. Grey lines are best fits to black symbols employingthe HLN formula [Eq. (S47)]; orange and red lines: fitted WAL theory in the parallel field [Eq. ((S50)] with β as a singleadjustable parameter (assuming α = − /
2) and with α and β as adjustable parameters, respectively, employing l φ valuesdetermined in the perpendicular configuration (grey curves). Two-parameter fitting results in lower values of | α | < . β > /
3. A comparison of the data for samples uncovered and covered by amorphous Se is shown inFig. 6 of the main text. where the prefactor α = 1 for the WL and one channel case; the dimensionless parameter β = 1 / d (cid:28) l [54] β = (1 / d/l in the clean limit [55], and in the intermediate regime d/ l < β < / α to − / g ∗ µ B B ) in the argument of the logarithm function in Eq. (S50) [57–59], which maydominate in the small d or large g ∗ limit ( g ∗ = 35 ± α = − / β = 2 λ /d , where λ is the penetration of the surfacestate into the bulk [61, 62]. Furthermore, interfacial roughness enhances the apparent β value [63, 64].We have fitted experimental data to the conventional theory [Eq. (S50)] with α = − . l φ values obtainedfrom the fitting of the data taken in the perpendicular orientation, thus having β as the only adjustable parameter.Such a procedure leads to β : 0.48; 0.30; 0.20; 0.22 and 0.07 for epilayers A ( x Sn = 3 . x Sn = 10 . x Sn = 15 . x Sn = 24 . x Sn = 28 . d , the determined values have to be regarded as theirlower limit. Meanwhile in the DK theory [55] β = d/ l is equal to 0.077; 0.138; 0.691; 0.223 and 0.226 for epilayersA, B, C, D, and E. Within the AA theory [54], the parameter β has a fixed value 1/3. The fact that experimentalvalues of β are larger than 1/3, was reported for TI materials [65]. Theory of WAL in TIs in parallel fields [61, 62]can explain high magnitudes of β in the epilayers with composition corresponding to the inverted band structure byassuming that λ extends over almost the whole film thickness [30]. However, this approach fails to explain the similarbehaviour of MR in epilayers in the topologically trivial phase. We have also fitted MR data in the parallel field tothe formula (S50) using two adjustable parameters – α and β . Again, we have used the value of l φ determined fromfit to the MR data in the perpendicular geometry measured at the same temperature. Using this approach we haveobtained α = − . ± .
03, and higher values for β : 0.99; 0.66; 0.76; 0.35 and 0.17 for epilayers A, B, C, D, and E.16Moreover, fitting experimental data with two adjustable parameters results, not surprisingly, in a better fit than witha single one (Fig. S10).Importantly, epilayers covered by amorphous Se that breaks the mirror symmetry, behave in a similar way, but showsystematically larger β values: β = 0 .
52, 0.32, 0.79, 1.16 are obtained for epilayers G ( x Sn = 6 . x Sn = 13 . x Sn = 15 . x Sn = 30%), respectively, assuming α = − .
5. Treating α as an additional fitting parametereven larger values β = 1 .
38, 0.97, 2.11, 8.82 are found out for the same epilayers together with α = − . ± .
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