Electrical Control over Phonon Polarization in Strained Graphene
Jens Sonntag, Sven Reichardt, Bernd Beschoten, Christoph Stampfer
EElectrical Control over Phonon Polarization in Strained Graphene
J. Sonntag,
1, 2, ∗ S. Reichardt, B. Beschoten, and C. Stampfer
1, 2 JARA-FIT and 2nd Institute of Physics, RWTH Aachen University, 52074 Aachen, Germany Peter Gr¨unberg Institute (PGI-9), Forschungszentrum J¨ulich, 52425 J¨ulich, Germany Department of Physics and Materials Science, University of Luxembourg, L-1511 Luxembourg, Luxembourg (Dated: March 3, 2021)We explore the tunability of the phonon polarization in suspended uniaxially strained grapheneby magneto-phonon resonances. The uniaxial strain lifts the degeneracy of the LO and TO phonons,yielding two cross-linearly polarized phonon modes and a splitting of the Raman G peak. We utilizethe strong electron-phonon coupling in graphene and the off-resonant coupling to a magneto-phononresonance to induce a gate-tunable circular phonon dichroism. This, together with the strain-inducedsplitting of the G peak, allows us to controllably tune the two linearly polarized G mode phononsinto circular phonon modes. We are able to achieve a circular phonon polarization of up to 40 %purely by electrostatic fields and can reverse its sign by tuning from electron to hole doping. Thisprovides unprecedented electrostatic control over the angular momentum of phonons, which pavesthe way toward phononic applications.
Phonons – collective excitations of lattice vibrations –play a fundamental role in solid state physics and materi-als science. They affect a wide variety of material prop-erties and phenomena relevant for electronics, thermaltransport, and optics [1, 2] and are pivotal for quantumeffects such as superconductivity. Of particular interestare material systems in which phonons carry additionaldegrees of freedom, such as chirality [3–6], polarization,or angular momentum [7, 8]. They are crucial for novelphonon-driven phenomena such as the phonon Hall ef-fect [9, 10], the phonon ac Stark [11] and Edelstein ef-fects [12], or the phonon Zeeman effect [13] and can evendrive electronic phase transitions [14–16] and topologicalstates [16, 17]. A control over phonons and their de-grees of freedom is thus a key goal on the pathway topotential phononic applications [1, 2], such as phonon-based quantum information devices [18]. However, a mi-croscopic control over phonons is very challenging dueto their spin-less nature and the large inertia of the nu-clei, which makes it hard to control them directly byelectromagnetic fields. In systems with strong electron-phonon interaction, however, a control over phonons canbe achieved by manipulating the electrons. In this re-gard, graphene is a prime candidate as it features strongelectron-phonon interaction and allows for excellent ex-ternal control over its electronic system. In particular,the degenerate longitudinal (LO) and transverse optical(TO) phonons at the Γ point of the first Brillouin zoneare strongly coupled to the electronic system, which en-ables the tuning of their frequencies and lifetimes via anelectrostatic gate [19]. The electron-phonon coupling canfurther be significantly modified by a large external mag-netic field, in which the electronic system condenses intodiscrete Landau levels, giving rise to so-called magneto-phonon resonances (MPRs) [20–34].Here we show that the control over the electron-phononcoupling in graphene allows to tune the movement of thenuclei from a linear oscillatory motion along or perpen- dicular to the bonds to a circular motion about theirequilibrium positions. We thus induce a circular polar-ization of the phonon modes, which yields control overthe phonon angular momentum. We achieve this purelyvia external static electric and magnetic fields in com-bination with uniaxial strain ( ∼ . − mode, which is linearly polarizedin the direction of the strain, and a harder G + mode,which is linearly polarized perpendicular to the direc-tion of the strain (see Figure 1c,d) [35, 37]. The strain-induced splitting allows us to track the Raman signal ofeach mode separately and, more importantly, to manip-ulate the vibrational pattern of the nuclei from a linearto a circular motion via tuning the electronic system andthus induce a circular phonon polarization. We achievethe latter by applying an out-of-plane magnetic field B ,which forces the electrons on cyclotron orbits and quan-tizes the electronic density of states into discrete Landaulevels (LLs). In a simple picture, this forces the nucleiinto a circular motion due to their coupling to the elec-trons. More precisely, the phonons couple to circularlypolarized electronic transitions between the LLs. At thecharge neutrality point, the electron-hole symmetry im-plies that neither helicity, i.e., circular polarization, is a r X i v : . [ c ond - m a t . m e s - h a ll ] M a r ε strained graphene -1 Raman shift (cm ) (g) - G + G R a m an i n t en s i t y ( a . u . ) G strain (b) (c)(d) (e)(a) + G - G x B-fielddoping (f) unstrained + - σ Gate
LOTO ŷ + σ + - G - σ + σ + + G ε FIG. 1. (a,b) Illustration of the vibrational pattern of the degenerate LO and TO phonon modes that give rise to the G peakin pristine graphene. (c,d) The application of a uniaxial strain ε leads to a splitting of the degenerate phonon modes intotwo branches G ± that are linearly polarized along and perpendicular to the direction of strain. (e,f) The application of astatic magnetic field B in combination with a finite charge carrier density n induces a net circular polarization of the phononmodes. The resulting vibrational patterns can be visualized as linear combinations of circular motions of the nuclei abouttheir equilibrium positions with different weights for left- and right-handed movements, as indicated by the different levels ofopaqueness. (g) Raman spectrum of graphene featuring the G peak before (bottom) and after (top) the current annealing step.The purple line represents a fit to the two-subpeak structure with a sum of two Lorentzians (red and blue dotted lines). Upperleft inset: Schematic illustration of graphene (purple) suspended from two gold contacts (yellow). The current annealing stepinduces uniaxial strain as illustrated by the two insets on the right. preferred as electrons and holes rotate in opposite direc-tions. As a result, the phonons remain linearly polar-ized. However, when tuning the charge carrier density n with a gate voltage, we break this symmetry. As aconsequence, both phonon branches become increasinglycircularly polarized with opposite helicity, i.e., the nucleiwill increasingly adopt a circular motion about their equi-librium positions (see Figure 1e,f). The resulting phononmodes can be thought of as a linear superposition of thetwo helicities σ ± with unequal weights, as indicated bythe different levels of opaqueness in Figure 1e,f.Our experiments are based on a suspended graphenefield effect device, see schematic in the inset of Figure 1g.It features a high charge carrier mobility and a lowcharge carrier density inhomogeneity, which are both cru-cial to observe magneto-phonon resonances. A current-annealing step is employed to both induce uniaxial strainand to effectively clean the graphene sheet [38]. Thecleaning results in a sharp peak in device resistance at thecharge neutrality point with a charge carrier inhomogene-ity of n ∗ ≈ . × cm − (see Figure S1 in SupportingInformation) and allows the observation of the quantumHall effect for B ≈ . µ ≈
100 000 cm / (Vs). The presence of uniaxialstrain after annealing is clearly visible in the splitting ofthe Raman G peak into two separate peaks, see Figure 1g.To quantify the peak splitting and the induced strain, wefit the G peak with a sum of two Lorentzians which sharethe same spectral width Γ G . The two subpeaks are cen-tered around a mean frequency of ¯ ω G ≈ . − and split by ∆ ω G ≈ . − . From ∆ ω G we es-timate the amount of uniaxial strain in our sample as ε ≈ .
82 %, where we used the previously measuredGr¨uneisen parameters[35] ∂ω G + /∂ε ≈ − . − /%and ∂ω G − /∂ε ≈ − . − /%. This significant amountof uniaxial strain induced by the current annealing step isin line with a recent report by Jung et al. [37]. Note thatwe neglect biaxial strain, as it should be negligible in ourdevice geometry and would result in the same shift forboth peaks, which is irrelevant for the main experimentalfindings, i.e., the phonon polarization analysis.We now turn to the control over the linearly polar-ized G mode phonons via the manipulation of the elec-tronic system. To this end, we first focus on magneto-phonon resonances, i.e., the coupling of the phonons tothe circularly polarized LL transitions, by measuring the B field dependence of the linearly polarized modes atthe charge neutrality point ( E F = 0). The measured Ra-man spectra for magnetic fields ranging from 0 to 7 . ≈ . + and G − peaks and the resonant coupling to LL tran-sitions at B ≈ B ≈ reso-nance [20, 26, 27, 29]. It occurs when the phonon energy, E G = ¯ h ¯ ω G ( B = 0 , n = 0), matches the energy differ-ence between two LLs with energies E j = ± v F √ e ¯ hB · j ,where v F is the Fermi velocity and j is the LL index.The most prominent T j -MPRs involve LL transitions − j → j + 1 (T + j ) and − ( j + 1) → j (T − j ) with an orbital + T - T + σ - σ E F E DOS (b)(a) - R a m an s h i ft ( c m ) B (T)
Counts (a.u.) + G - GT E E E -1 E -2 E E - Γ ( c m ) G B (T) T L (c) T T E G L E ( m e V ) DOS + σ - σ L + T - T FIG. 2. (a) Raman intensity around the strain-split G peak of graphene as a function of B field at n ≈ − and G + peak). (b) Illustration of the electronic transitions of the dominating MPRs and the density of states. Transitionswith ∆ j = ( − )1 only couple to right (left) circularly polarized phonons σ +( − ) , color coded as blue (red). (c) Lower panel:Interband LL transition energies T n (solid lines) and L (grey, dotted lines) under the assumption of v T ≈ . × m/sand v L ≈ . × m/s. The closely spaced dashed red and dotted blue lines indicate the energies of the G − and G + modephonons at B = 0 T, respectively. The energy of the most relevant T resonance is also shown in (a) as dashed line. Upperpanel: B field-dependent width Γ G of the two subpeaks G − and G + . The black line represents a fit based on Dyson’s equation. angular momentum of ± h [26, 27]. Importantly, an-gular momentum conservation implies that these transi-tions selectively couple to the respective circular phononmodes σ ± , as the latter also carry angular momentum ± h , due to the circular motion of the nuclei. The cor-responding transition energies T j and the resonance con-dition are then given by T j = E j +1 − E − j = v T j √ e ¯ hB ( (cid:112) j + 1+ (cid:112) j ) = E G . (1)The lower panel in Figure 2c visualizes this resonancecondition. Note that we replaced v F by the effectiveFermi velocity v T j , which can generally be a functionof n , B , and the LL index j due to many-body andexcitonic effects [29, 39, 40]. To highlight the couplingof the two phonon modes to the LL transitions, we ex-tract their frequency shifts ∆ ω G ± = ω G ± − ω ± relative totheir respective frequencies at B = 0 T as well as theirshared width Γ G = Γ G + = Γ G − by fitting the sum oftwo Lorentzians. Most noticeably, the coupling leads toa decrease of the phonon lifetime at the resonance due tothe excitation of electron-hole pairs, which results in theincreased width Γ G of both modes at the T resonance,see upper panel in Figure 2c. We also observe a minorcontribution of the L ( − →
1) transition at B ≈ . B T = 2 .
73 T, at whichthe T -MPRs occurs, corresponds to an effective Fermivelocity of v T ≈ . × m/s, as seen from Equation 1.Although strain has been predicted to change v F [41, 42],we find that this value of v F is in excellent agreement toprevious results on the Fermi velocity renormalization inthe presence of LLs in unstrained graphene [29].As shown in Figure 3a, we observe identical magneto- phonon resonances of the two cross-linearly polarized G − and G + phonon modes at n = 0. Note that the split-ting of ∆ ω G = 14 . − is larger than the axis rangein Figures 3a-c. This indicates that the strain-inducedsplitting dominates over the individual phonon frequencyshift from the MPR. This can be intuitively understoodsince at zero doping, the electron-hole symmetry is notbroken and the T ± LL transitions remain degenerate andcouple with equal strength to the phonons [34]. As aresult, there is no net circular phonon polarization at n = 0. In this regime, we can describe the MPRs bycalculating the phonon self-energy and solving Dyson’sequation [20, 26, 27] (see Supporting Information for amore detailed discussion). A combined fit of the solutionof Dyson’s equation to both ω G − and Γ G is shown asblack lines in Figure 2d and Figure 3a. Evidently, the fitdescribes ω G + just as well.In the following, we show that a net circular phononpolarization can be induced by electrostatic gating, i.e.,by tuning the system with a finite charge carrier density n . After briefly discussing the influence of n on T , wewill particularly focus on the gate-tunable off-resonantcoupling of the phonons to the T ± transition, which oc-curs at large magnetic fields of around B ≈
20 T. Here,the finite n breaks the symmetry between T ± , which re-sults in a circular phonon dichroism [32–34] and therebya net circular phonon polarization.Figures 3b,c show that the T -MPR gradually vanishesfor higher charge carrier densities for both G − and G + .This is a consequence of the Pauli blocking of the re-spective transition (see Figure 2b). An electron densityof n ≈ . × cm − corresponds to a filling factor ν = nh/ ( eB ) ≈
10 at B ≈ − → + T + - T + T + - T - Δ ω ( c m ) B (T) (a) - Δ ω ( c m ) - Δ ω ( c m ) B (T)
B (T) (b) (c)
12 -2 xn = 0.7 10 cm
12 -2 xn = 0.3 10 cm -2 n = 0.0 cm T E B E E
B ≈ 20 T ν = 0+ T + - T ν = 2 ν = 6 ν =1.7 ν =3.9 ν =0 (d) (f)(e) E ± G E ± G E ± G B B ≈ 20 T B B ≈ 20 T + G - G + G - G + G - GT L FIG. 3. (a) Change in phonon frequency ∆ ω as a function of B field for both the G − (red) and the G + mode (blue) at n ≈ . − . The black line represents a fit based on Dyson’s equation. (b)and (c) show the same measurement as (a), but at intermediate and high n , respectively. (d-f) Illustration of the effect of thefilling factor on the T -MPR. The colored lines are E ± G = ¯ hω ± G and the dashed black line is T . At ν = 2 (e) the T − transitionis completely blocked while the T +0 increases in strength. For comparison, the ν = 0 case shown in (d) is reproduced in (e) ascolored dotted lines. (f) At ν = 6 all relevant LLs are completely filled and no transitions are possible. Note that ν is assumedto be constant for all B for clarity. The area shaded in grey illustrates the measurement range shown in panels (a-c). and the − → ω G at higher magneticfields even at high | n | , which we attribute to the tail ofthe T resonance at B ≈
20 T. Note that the T res-onance becomes unblocked as the effective filling factordecreases with B , see also purple dotted lines dotted linesin Figure 3d-f. Most strikingly, we observe that the evo-lution of the two phonon modes ∆ ω G − and ∆ ω G + splitfor large magnetic fields. This is most pronounced at anintermediate electron density of n ≈ . × cm − (seeFigure 3b). The splitting can be attributed to the chargecarrier-dependent change of the coupling strengths of theT ± transitions, which leads to a net circular phonon po-larization. For increasing n , the coupling to the T +0 tran-sition increases due to the increasing number of statespartaking (see Fermi level shift for electron doping inFigure 2b). This leads to an enhanced anti-crossing forthe σ + phonon branch, which is illustrated for a con-stant filling factor of ν = 2 in Figure 3d-f. Comparedto the situation for filling factor ν = 0, the enhancedanti-crossing of σ + with T +0 leads to an increase in σ + frequency for B fields below the resonance and conse-quently to an increased σ + circular polarization of thehigher frequency peak G + . This holds up to a densitywhere the second LL is completely filled and the cou-pling strength decreases again, resulting in a reductionof ω G + for ν >
2. In contrast, the T − transition is sup-pressed by filling the zeroth LL (Figure 2b), resulting in aweaker anti-crossing for σ − (see Figure 3e), i.e., a mono-tonic decrease in ω G − and an increased σ − polarizationof G − . Indeed, as seen in Figure 4a, both the G − and G + peaks show a qualitatively different dependence on n at B = 7 . ω G − monotonically decreases in fre-quency with increasing | n | , ω G + first increases for small | n | before declining at larger densities. As indicated bythe grey dotted lines in Figure 4a, the maxima in ω G + are located precisely at the complete filling of the zerothLL ( ν = 2) at n ≈ . × cm − .Most importantly, we can exploit the asymmetry be-tween ω G ± ( n ) to directly calculate the degree of circularpolarization of the phonon modes purely from the mea-sured frequencies and widths. We first note that purelycircularly polarized phonons σ ± can only couple to LLtransitions of the same helicity due to angular momen-tum conservation. The only coupling between phononsof different phonon helicities thus results from the uniax-ial strain. We can therefore describe the dynamics of thetwo phonon modes by an effective Hamiltonian H eff ( n ) = ¯ h (cid:18) ˜ ω σ + ( n ) ηη ˜ ω σ − ( n ) (cid:19) , (2)written in the basis of the σ ± modes, wherein ˜ ω σ ± ( n ) = ω σ ± ( n ) − i Γ σ ± ( n ) / σ ± , which include the frequency shiftsfrom the interaction with LL transitions. The off-diagonal term η describes the coupling between the twocircular phonon modes due to strain. We emphasize thatthis Hamiltonian is exact under the restrictions of an-gular momentum selection rules and does not depend onany microscopic model. Instead, all quantities in H eff candirectly be obtained from the experimental data, as itseigenvalues ¯ h ˜ ω G ± can directly be measured in form of the + ν T ν T ν T ν T - ω + ( c m ) G - ω - ( c m ) G (a)(b)
12 -2 n (10 cm ) + P G - P G (c) ± P ( % ) G ν T j FIG. 4. (a) Measured frequencies of the G + (blue dots, leftaxis) and the G − mode (red dots, right axis) as a functionof n (at B = 7 . | ν | = 2. The lines show ω G ± as calculated via thesix-level model presented in the Supporting Information. (b)Circular phonon polarization P G + (blue) and P G − (cyan) ascalculated from the experimentally measured Raman peaksvia Equation 5. The lines correspond to the polarization ascalculated via the six-level model presented in the Support-ing Information. (c) Effective partial filling factor of the LLtransitions ¯ ν T ± j as a function of n at B = 7 . complex peak frequencies ˜ ω G ± = ω G ± − i Γ G ± /
2. Firstly,at n = 0, we have ˜ ω σ + = ˜ ω σ − due to the electron-holesymmetry, i.e., the splitting between the two measuredphonon branches ∆˜ ω G = ˜ ω G + − ˜ ω G − is purely caused bythe uniaxial strain and is equal to 2 η . Experimentally,we obtain η = ∆˜ ω G ( n =0) / . − . Secondly, ˜ ω σ ± can be obtained from the eigenvalues ¯ h ˜ ω G ± and η :˜ ω σ ± = ˜ ω G + + ˜ ω G − ± (cid:113) ∆˜ ω − η . (3)We can now determine the circular phonon polarizationby calculating the eigenvectors v G ± of H eff and projectthem on the circular basis vectors, i.e., on σ + = (1 , T and σ − = (0 , T : P G ± = | σ + · v G ± | − | σ − · v G ± | | σ + · v G ± | + | σ − · v G ± | . (4) As the eigenvectors of H eff can be calculated analytically,we are able to determine the circular phonon polariza-tion purely from the measured phonon frequencies andwidths: P G ± = ± η − | ∆˜ ω G − (cid:112) (∆˜ ω G ) − η | η + | ∆˜ ω G − (cid:112) (∆˜ ω G ) − η | . (5)The results of the conversion of the experimentally mea-sured ˜ ω G ± into circular polarization P G ± are shown inFigure 4b as blue and cyan dots, respectively. For n > P G + up to ν = 2, reach-ing circular polarization values of up to ∼
40 %, before itdecreases at larger n . The behavior of P G − is exactlyopposite and changes sign when tuning from electron( n >
0) to hole ( n <
0) doping. We emphasize thatthese n dependent circular phonon polarizations are di-rectly extracted from our measured Raman spectra anddo not dependent on any knowledge of the nature of theelectron-phonon coupling.To gain a more detailed understanding of the mi-croscopic origin of this polarization behavior, we willnow look closer into the origin of the asymmetry in ω σ ± ( n ). As discussed above, the asymmetry can betraced back to the change in coupling strength of thecircular phonon components σ ± with the respectivelypolarized T ± LL transitions. In the following we showthat the experimentally extracted phonon polarizationsin Figure 4b can be reproduced quantitatively by mod-eling the charge carrier-dependent electron-phonon cou-pling. For this, we use an effective six-level model inwhich the electron-phonon system is described by six cou-pled quantum mechanical states, i.e., in which we adopt abasis of the two circular phonon modes, the two T ± , andthe two T ± LL transitions. The details of this model arepresented in the Supporting Information together withthe full set of used parameters, which we extract from thefit shown in Figure 2d and Figure 3a at n = 0. Impor-tantly, the coupling constants g T ± j between the phononsand the LL transitions depend on the filling of the LLs as g T ± j ∝ (cid:113) ¯ ν T ± j and thus on n . Here, the effective partialfilling factor of the LL transition ¯ ν T ± j , which is shown inFigure 4c, is given by:¯ ν T + j = (1 + δ j, )( ± ¯ ν ∓ j ∓ ¯ ν ± ( j +1) ) . (6)It is a function of the partial filling factor ¯ ν j , which relatesthe filling factor ν to the fractional occupancy of the j thLL: ¯ ν j = ν + 2 − j (cid:12)(cid:12)(cid:12)(cid:12) ∈ [0 , . (7)In the context of this six-level model, we then calculateboth ω G ± and the circular polarization P G ± as a func-tion of n by diagonalizing the coupled electron-phononsystem.Evidently, we find good agreement between the six-level model and our experimental data, as shown in Fig-ure 4a and b. We want to emphasize that the relevantparameters of the six-level model are extracted from theindependent B field measurement at n = 0 (Figure 2dand Figure 3a), which makes the six-level model predic-tive for the measurements at fixed B = 7 . n . Crucially, we observe the initial increase in ω G + and P G + with | n | and the kinks at | ν | = 2. As discussedpreviously, the coupling of the circular σ +( − ) phonon tothe T +( − )0 -transition increases (decreases) with electrondoping n > ν T j inFigure 4c and their connection to the coupling constants g T ± j ∝ (cid:113) ¯ ν T ± j . This leads to an increased σ + phononfrequency for any B below the resonant magnetic fielddue to a stronger anti-crossing with the T +0 -transition,as illustrated in Figure 3e. Because the G + mode is de-fined as the higher frequency mode, its σ + polarizationwill consequently increase. For ν > ν T +0 (see Figure 4c) decreases and with it the cou-pling strength g T +0 , which results in a weakening of theanti-crossing and thus in a decreased ω G + and decreasedcircular phonon polarization. Since the G − mode will al-ways be predominantly polarized as the circular phononwith lower frequency, i.e., the circular phonon with thelower LL transition coupling strength, it shows a mono-tonic decrease in ω G − as a function of n and a predomi-nant σ − polarization. It is important to emphasize thatthe T +( − )0 -transitions swap their roles when going fromelectron ( n >
0) to hole doping ( n < + mode will be σ − polarized. This resultsin the same polarization behavior, albeit with oppositesign. Consequently, we are uniquely able to tune the cir-cular polarization of the G ± modes, reaching polarizationdegrees of up to ∼
40 %, purely by static electromagneticfields due to the interplay of the strain-induced splittingof the phonon modes and the doping-induced asymmetryin electron-phonon coupling strength. We are thus ableto control the angular momentum of the phonons, whichis directly related to the degree of polarization P G ± .In conclusion, we explored the coupling of strain-split linearly polarized phonons to circularly polarizedLandau level transitions in graphene. We have shownthat by electrostatic doping, we can selectively couplethe phonons to right- or left-handed polarized Landaulevel transitions, which imprints their circular polariza-tion onto the phonon modes. This allows us to con-trol the phonon polarization from a purely linear to acircular polarization of up to 40 %, purely by electro-static fields. This finding is verified by explicitly con-sidering the charge carrier dependent electron-phononcoupling within an effective six-level model. We expectthat this novel control of the phonon polarization and of the phonon angular momentum by static electromagneticfields will be of great interest for phononic applications[4, 5] and will contribute to the rising field of researchstriving to manipulate phonons and their novel degree offreedoms [3–8, 16]. Moreover, our observation of MPRsin strained graphene shows that MPRs can be a viabletool in the future to investigate strain gradient-inducedpseudo-magnetic fields [43]. Acknowledgments:
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Supporting Information: Electrical Control over Phonon Polarization in StrainedGraphene
METHODS
Sample fabrication:
The suspendended graphene sample is fabricated from an exfoliated graphene flake onSi/SiO . The contacts made from Cr/Au are fabricated via electron beam-lithography and a subsequent lift-off step.The flake is suspended by etching away ∼
170 nm of SiO with hydrofluoric acid. Finally, a critical point drying-stepis used to prevent the collapse of the suspended graphene flakes. The graphene flake investigated here has a width of2 µ m and the distance between the contacts is 1 . µ m . Raman measurements:
The magneto-Raman measurements are performed in a confocal, low-temperature ( T =4 . λ = 532 nm) has a power of 0.5 mW and is focusedto a spot size of ∼
500 nm. All Raman spectra are taken in the center of the graphene flake.
PHONON SELF-ENERGY FOR LINEAR PHONON POLARIZATION
The evolution of the linearly polarized G mode phonon frequency and width with magnetic field and doping canbe obtained by solving Dyson’s equation (¯ h ˜ ω ) − (¯ h ˜ ω ) − h ˜ ω Π(˜ ω ) = 0 (S1)for the complex phonon frequency ˜ ω = ω G − i Γ G /
2. Here, ˜ ω = ω − iγ ph /
2, where ω is the phonon frequency in theabsence of a magnetic field, at zero doping, and in the adiabatic approximation and γ ph accounts for the non-electroniccontribution to the phonon decay width. Π(˜ ω ) denotes the phonon self-energy for linear phonon polarization and isthe same for both x - and y -polarization. It can be derived from the microscopic coupling of the Landau levels to thephonon within a tight binding-based model for the electron-phonon interaction [20, 26, 27] and can be written asΠ(˜ ω ) = λ T (cid:88) s = ± ∞ (cid:88) j =0 (cid:34) ¯ ν T sj T j (¯ h ˜ ω ) − ˜ T j + 2˜ T j (cid:35) . (S2)Here, ˜ T j = T j − i ¯ hγ T j / j transition energy that includes a finite decay width γ T j . The partial fillingfactors ¯ ν T sj have been defined in the main text, while λ is a dimensionless effective electron-phonon coupling constant.Finally, the experimentally observable, but in a first approximation symmetry-forbidden, L j ( − j → + j ) transitionscan be included phenomenologically via the replacementΠ(˜ ω ) → Π(˜ ω ) + λ L T ∞ (cid:88) j =0 (cid:34) ¯ ν L j L j (¯ h ˜ ω ) − ˜ L j + 2˜ L j (cid:35) , (S3)with the respective partial filling factors being given by ¯ ν L j = (¯ ν − j − ¯ ν + j ) and the complex transition energies beinggiven by ˜ L j = E + j − E − j − i ¯ hγ L j / DETAILS OF THE SIX-LEVEL MODEL
In the following, we introduce and discuss an effective six-level model to quantitatively describe the qualitativelydifferent behaviour of ω G ± . Within the model, the coupled electron-phonon system is described by six coupledquantum mechanical states. They represent the two phonon modes, the two T -, and the two T -LL excitations. Thephonons couple to the LL transitions with coupling strengths g T ± j . We choose to represent the phonon modes in thecircular basis to easily include the selection rules resulting from the conservation of angular momentum, i.e., the σ ± phonon only couples to the T ± j ( − j → j + 1) transitions. In the basis ( σ + , σ − , T +0 , T − , T +1 , T − ) the Hamiltonian isgiven by: H = ¯ hω ¯ hη g T +0 g T +1 hη ¯ hω g T − g T − g T +0 T g T − T g T +1 T g T − T − i ¯ h , (S4)where Γ = diag ( γ ph , γ ph , γ T , γ T , γ T , γ T ) is a diagonal matrix that describes the broadening of the phonons andelectronic transitions. ¯ hω = E G and T j are the bare phonon and LL transition energies, respectively. η is a parameterwhich describes the strain-induced splitting of the two modes. The modelis based on the effective Hamiltonianˆ H = (cid:88) s = ± ¯ hω ˆ b † s ˆ b s + ¯ hη (cid:16) ˆ b † + ˆ b − + ˆ b †− ˆ b + (cid:17) + (cid:88) s = ± j =0 , T j ˆ d † T sj ˆ d T sj + (cid:88) s = ± j =0 , g T sj (cid:16) ˆ b † s ˆ d T sj + ˆ d † T sj ˆ b s (cid:17) , (S5)which is written in terms of the creation and annihilation operators for the circularly polarized phonon modes, ˆ b ( † ) ± ,and for the T j =0 , LL transitions, ˆ d ( † )T ± j . The first term describes the uncoupled circularly polarized phonon modes,while the second term represents the strain-induced coupling between them. The latter can be derived by consideringa strain-induced phonon frequency splitting of 2 η in a linear basis and then rotating the Hamiltonian in a linear basisinto the circular basis. The third term in Equation S5 describes the uncoupled T ± j transitions, whereas the last termaccounts for the electron-phonon coupling between phonons and the LL transitions, which is diagonal in the σ ± basis.The effective, doping-dependent coupling constants g T ± j can be derived by rotating the microscopic LL electron-phonon matrix elements for linear phonon polarization into the circular basis. Alternatively, they can simply be readoff by noting that the phonon self-energy for linear phonon polarization is the average of the phonon self-energies forcircular polarization: Π( ω ) = 12 [Π + ( ω ) + Π − ( ω )] = 12 (cid:88) s = ± ∞ (cid:88) j =0 (cid:34) g sj T j (¯ hω ) − ˜ T j + g sj (cid:12)(cid:12)(cid:12) n =0 T j (cid:35) , (S6)from which we can identify g T ± j = T (cid:113) ¯ ν T ± j λ/ v F (cid:113) ¯ ν T ± j λe ¯ hB. (S7)Here, the effective partial filling factors of the LL transitions, ¯ ν T ± j , are a function of the magnetic field and aregiven by: ¯ ν T + j = (1 + δ j, )(¯ ν − j − ¯ ν j +1 ) , ¯ ν T − j = (1 + δ j, )(¯ ν − ( j +1) − ¯ ν + j ) . (S8)The partial filling factor ¯ ν j relates the filling factor ν to the fractional occupancy of the j th LL:¯ ν j = ν + 2 − j (cid:12)(cid:12)(cid:12)(cid:12) ∈ [0 , , (S9)which is defined as ¯ ν j = 0 if ν + 2 − j/ < ν j = 1 if ν + 2 − j/ > ± and T ± transitions by numerically diagonalizing Equation S4. To this end, we use the experimentalparameters extracted from the fit shown in Figure 2d and Figure 3a of the main manuscript at n = 0, which resultsin effective Fermi velocities v T ≈ . × m/s and v L ≈ . × m/s. For a more stable fit, we assumed thesame v T j = v T for all j expect for j = 0, which we set to v T = v L , since T shares the first LL with L . Theother extracted parameters are the electronic broadening of the T j -transitions γ T j ≈
392 cm − and L -transition γ L ≈
196 cm − , a phonon broadening γ ph ≈ . − , an electron-phonon coupling constant λ = 4 . × − [19, 45],and a coupling strength of the L -transition λ L ≈ . × − . Further, within this model, we assume v T = v L , γ T = γ L and use η = 8 . − , and ω = 1566 . − .To calculate the net circular phonon polarization, we first obtain the weight of the σ ± component in the G ± modeby projection on the respective component: w ( σ ± , G ± ) = | σ ± · v G ± | . We then define the circular polarization of theG ± modes as P G ± = w ( σ + , G ± ) − w ( σ − , G ± ) w ( σ + , G ± ) + w ( σ − , G ± ) = 2 w ( σ + , G ± ) − . (S10)Here, we made use of the fact that both weights need to add up to one: w ( σ + , G ± ) + w ( σ − , G ± ) = 1. Note that in theinteracting electron-phonon system, this identity does no longer hold exactly, as the phonon states can now containan admixture of electronic excitations, but it is still a very good approximation in the off-resonant electron-phononcoupling regime considered here. ADDITIONAL ELECTRICAL TRANSPORT AND RAMAN MEASUREMENTS -10 0 10234 R ( k Ω ) V (V) g s ( e / h ) -3 -2 -1
12 -2 |n| (10 cm )789101112(a) (b) holeselectronsn*
FIG. S1. (a) The resistance as a function of V g before (black) and after current annealing (red). (b) Double logarithmic plotof σ after current annealing versus charge carrier density | n | for determining the charge carrier inhomogeneity n ∗ . - ω ( c m ) D
12 -2 n (10 cm )
FIG. S2. The 2D peak position ω does not show significant variation as a function of n , indicating a negligible amount ofgate-induced strain. (a) - R a m an s h i ft ( c m ) B (T)
Counts (a.u.) (b) - R a m an s h i ft ( c m ) B (T)
Counts (a.u.) (c) - R a m an s h i ft ( c m ) B (T)
Counts (a.u.)
12 -2 xn = 0.7 10 cm
12 -2 xn = 0.3 10 cm -2 n = 0.0 cm FIG. S3. (a-c) Raman intensity around the strain-split G peak of graphene as a function of B field at n ≈ − , n =0 . × cm − , and n = 0 . × cm −2