Electrical generation and detection of spin waves in a quantum Hall ferromagnet
Di. S. Wei Toeno van der Sar Seung Hwan Lee Kenji Watanabe Takashi Taniguchi Bertrand I. Halperin Amir Yacoby
cond-mat.mes-hall

	
	
	
	
	John A. Paulson School of Engineering and Applied Sciences, Harvard University, Cambridge, Massachusetts 02138
	
	
	Department of Physics, Harvard University, Cambridge, Massachusetts 02138
	
	
	Department of Physics, Harvard University, Cambridge, Massachusetts 02138
		
	
	Advanced Materials Laboratory, National Institute for Materials Science, Tsukuba, Ibaraki 305-0044, Japan
	
	
	Advanced Materials Laboratory, National Institute for Materials Science, Tsukuba, Ibaraki 305-0044, Japan
		
	
	Department of Physics, Harvard University, Cambridge, Massachusetts 02138
		
	
	John A. Paulson School of Engineering and Applied Sciences, Harvard University, Cambridge, Massachusetts 02138
	Department of Physics, Harvard University, Cambridge, Massachusetts 02138

	
	
	  
	 	
	 	Spin waves are collective excitations of magnetic systems. An attractive setting for studying long-lived spin-wave physics is the quantum Hall (QH) ferromagnet, which forms spontaneously in clean two-dimensional electron systems at low temperature and in a perpendicular magnetic field. We used out-of-equilibrium occupation of QH edge channels in graphene to excite and detect spin waves in magnetically ordered QH states. Our experiments provide direct evidence for long distance spin wave propagation through different ferromagnetic phases in the N=0 Landau level, as well as across the insulating canted antiferromagnetic phase. Our results will enable experimental investigation of the fundamental magnetic properties of these exotic two-dimensional electron systems.
	 
 
	  		
	  	Electrical generation and detection of spin waves in a quantum Hall ferromagnet
    Amir Yacoby
    January 19, 2018
===============================================================================

Quantum Hall (QH) ferromagnetism arises from the interaction of electrons in massively degenerate, quantized energy levels known as Landau levels (LLs) <cit.>. When disorder is low enough for Coulomb interactions to manifest, the electrons in partially filled LLs spin-polarize spontaneously to minimize their exchange energy, with the single-particle Zeeman effect dictating their polarization axis <cit.>. In graphene, these phenomena give rise to ferromagnetic phases when the N=0 LL is at quarter- and three-quarter-filling <cit.>. Such QH ferromagnets have an insulating topological bulk and spin-polarized edge states. Furthermore, a canted antiferromagnetic (CAF) state is believed to emerge at one-half filling, with a canting angle determined by the competing valley anisotropy and Zeeman energy <cit.>. Spin waves, also known as magnons, are the lowest energy excitation in both the QH ferromagnet and CAF <cit.>, and could provide crucial information about these topologically non trivial magnetic states. 
	
	In our experimental setup, we generate magnons by creating an imbalance of chemical potential between two edge states of opposite spin that run along the boundary of a QH magnet. If this imbalance is smaller than the energy required for generating magnons in the QH magnet (and there are no thermal magnons already present in the system), scattering between these two edge states is forbidden because the change in angular momentum of a scattered electron cannot be absorbed by the system. Indeed, previous measurements have shown that oppositely spin-polarized edge channels do not equilibrate as long as the imbalance is small <cit.>. However, we find edge channel equilibration commences when the imbalance exceeds the minimum energy required for exciting magnons in the QH ferromagnet. Because the magnetization of the QH ferromagnet is extremely dilute, there are negligible demagnetizing fields and the minimum energy to excite magnons is given by the Zeeman energy E_Z=gμ_B B <cit.>, where g is the electron g-factor, μ_B is the Bohr magneton, and B is the external magnetic field. Although magnon generation does not directly affect the conductance of the system, the reverse process of magnon absorption by far-away edge states does, allowing us to detect the propagation of magnons electrically, in close analogy to the conventional detection of magnons in insulators via the inverse spin Hall effect <cit.>.
	
	 

	To demonstrate spin wave propagation, we begin with a dual-gated monolayer graphene device (device 1) where the central region can be tuned to a different filling factor than the adjacent regions (Fig. 1A). Connecting the two leads is a chiral edge state that carries spin-polarized electrons aligned with the magnetic field, which we call spin-up. We tune the central region to a three-quarters-filled LL (ν = 1), whereas the outer regions are tuned to a non-magnetic fully filled LL (ν = 2). We apply a source-drain voltage V_dc to induce a difference in chemical potential μ  = -e V_dc between the edge channels emerging from the two contacts, where e is the electron charge.  Once |μ| ≥ E_Z, an electron traveling in a high-energy (“hot”), spin-down edge state can relax into a low-energy (“cold”), spin-up edge state by emitting a magnon into the ferromagnetic bulk (Fig. 1, B and C). Because equilibration must occur close to the ferromagnetic bulk in order to launch magnons, the edge states must equilibrate over short length scales at localized “hot spots” where the hot and cold edges meet. This makes graphene an ideal platform to observe this phenomenon, where edge state equilibration can occur over length scales < 1 μm <cit.> (See <cit.> for further discussion). Because only spin-down angular momentum can be propagated into the spin-up bulk, magnon generation occurs at the location denoted by an encircled minus sign when μ≥ E_Z (Fig. 1B) and at the location denoted by an encircled plus sign when μ≤ -E_Z (Fig. 1C).  These magnons propagate through the insulating QH ferromagnet and can be absorbed by the reverse process between other edge channels (Fig. 1B-C), which causes a deviation in the conductance from a well-quantized ν = 1 QH state. 


	When we measure the conductance of the graphene device (Fig. 1E, atomic force microscopy image in fig. S3) as a function of V_dc, we find that the ν = 1 QH ferromagnet remains precisely quantized at the expected value of e^2/h, and then changes once the applied bias reaches the Zeeman threshold (V_dc= ± V_EZ= ∓ E_Z/e), as expected from our model (Fig. 1F). Interestingly, we find that thanks to contact doping <cit.> we can tune the entire device to ν = 1 and find the same phenomenon of conductance deviation at the Zeeman threshold (fig. S4).
	
	
	 

	



By tilting the external magnetic field with respect to the sample-plane normal axis, we verify that the change in conductance occurs when the applied chemical potential exceeds the bare Zeeman energy E_Z = g μ_B B_T (g=2), which is given by the total field B_T (Fig. 1G – sample is tuned entirely to ν=1). In contrast, previous transport studies of spin and valley excitations in graphene and GaAs have only found excitations related to the exchange energy gap <cit.>, which depends on the component of the field perpendicular to the sample plane (B_⊥). Our tilted-field measurements therefore corroborate our magnon-based interpretation of the observed change in sample conductance. All subsequent experiments described in this work are done at perpendicular field.

The conductance change at E_Z can either be positive or negative, depending on the number of magnons absorbed at each contact. To examine this, we use different sets of leads in the same device (Fig. 2A, device 2) to perform two-terminal conductance measurements. We start with leads L_2 and L_1 in Fig. 2B. We label the amount of redistributed chemical potential at each of the absorption sites ε_i, with i indexing the absorption site (note that ε_i = 0 for -E_Z < μ < +E_Z), where ε_i is proportional to the number of magnons absorbed at site i. Absorption at ε_1 and ε_2 have opposite effects on the conductance, as magnon absorption transfers chemical potential from the outer edge to the inner edge. Therefore, for μ≥ E_Z, magnon absorption at ε_1 decreases the particle current (I_P = - I/e where I is the charge current) whereas magnon absorption at ε_2 increases I_P (Fig. 2B). For μ≤ -E_Z, the hot and cold reservoirs are reversed, and we now consider the change to the negative particle current -I_P. Although ε_1 still decreases the particle current, I_P is now negative, and so ε_1 actually increases the magnitude of the particle current (|-I_P|); similarly, for μ≤ -E_Z, ε_2 decreases |-I_P| (Fig. 2C). We can quantify this using current conservation to formulate the differential conductance as a function of ε_i and μ:


    dI/dV = dI_P/dμ 
    = 1/R_Q( 1+dε_2/dμ-dε_1/dμ)



where R_Q =h/e^2 is the resistance quantum, V = V_ac + V_dc, and we have neglected contact resistance (see <cit.> for a derivation of Eq. 1, which takes contact resistance into account). We find that the conductance decreases at negative bias and increases at positive bias (Fig. 2D) – indicating that ε_1 > ε_2 for both positive and negative bias. This implies that more magnons are absorbed at ε_1 than at ε_2. Because our contacts have all been fabricated identically, we conclude this is because ε_1 is closer to magnon generation than ε_2 (for both positive and negative bias, see Fig. 2, A and B). Using different sets of contacts and top gates (Fig. 2, E to H) we can change the relative distances of ε_i to the locations of magnon generation. We confirm that for each configuration, the conductance values after E_Z correspond to a greater number of magnons absorbed at the site closer to magnon generation. 

This change to the conductance is not a consequence of QH breakdown. Conductance deviations after the Zeeman threshold that depend on the sign of V_dc are not explained by any current breakdown theories <cit.>. Additionally, we find that the threshold voltage bias does not depend on the lead configuration (Fig. 2), the size of the  ν = 1 region (fig. S4), or the density of the ν = 1 region (fig. S6) – which is all inconsistent with trivial QH breakdown, but consistent with our magnon model. In total, we have measured this  ν = 1 conductance deviation occurring at the Zeeman energy for eight devices of widely varying geometries (figs. S3, S4, and S11).  

	 

Thus far we have established that we are able to generate and absorb magnons at current carrying contacts. If these chargeless excitations propagate through the insulating bulk, we also expect to see signatures of magnon propagation and absorption via non-local voltage measurements (dV_NL/dV—referred to as nonlocal signal S_NL), away from the source-drain current. To measure S_NL we use L_3 and L_2 in device 2 as source-drain contacts, and use contacts L_4 and L_5 as voltage probes (Fig. 3A). These contacts are separated from the source-drain contacts by a top gate (TG2) which we tune between ν_TG2 = -2 and ν_TG2 = 2, where all other regions are tuned to ν=1. The conductance between L_3 and L_2 drops at V_EZ in accordance with our model (Fig. 3B), whereas magnon generation is largely unaffected by TG2 (fig. S7, A). At ν_TG2 = 1 we measure a change in S_NL at ± V_EZ due to the relative absorption at each magnon absorption site (ε_i).

The sign of S_NL indicates that there is more magnon absorption at sites closer to where magnon generation occurs. Through current conservation (<cit.>) we find that the measured differential voltage (unitless) is: 


    d V_NL/dV = ( dε_4/dμ - dε_5/dμ )


The site labeled by ε_4 is closer to magnon generation than ε_5 for both negative and positive bias, so |dε_4| > |dε_5|. However, the differential change in voltage (dε_i/dμ) is negative for V_dc≥ V_EZ and positive for V_dc≤ -V_EZ, corresponding to an overall negative value for S_NL at V_dc≥ V_EZ and a positive value at V_dc≤ -V_EZ (Fig. 3C). 

The device geometry used for our non-local measurements allows us to tune TG2 away from  ν_TG2 = 1, and thereby examine magnon transmission through different filling factors. We make two surprising observations. We observe that when ν_TG2 = -1 the signal s_NL is almost identical signal to when ν_TG2 = 1 (Fig. 3C and fig. S8). This signal arises in the absence of any charge leakage across the ν_TG2 = -1 region (fig. S9), so that changes in S_NL can be attributed to magnon transport through the ν_TG2 = -1 ferromagnet. This suggest that there is neither spin nor valley mismatch between the ferromagnetic states on either side of the boundary. We therefor propose an ordering of the LLs that does not require a spin or valley flip for magnons to travel across the interface between ν_BG = 1 and ν_TG2 = -1 (Fig. 3D; see <cit.> for a theoretical discussion.)

In addition, we unexpectedly find that S_NL is suppressed at ± V_EZ when ν_TG2 = 0. For non-magnetic regions such as ν_TG2=2, it is expected that magnons will be blocked from passing through, as experimentally confirmed in Fig. 3C (the non-local signal occurring at the transition between ν = 1 and ν =2 is explained in Fig. S7E). However, ν = 0 is purportedly a canted antiferromagnet which is theoretically capable of hosting even zero-energy magnons <cit.>. It appears that the probability for an incident magnon to be transmitted across the junction between the ν = 0 and ν = 1 regions is very small for energies close to E_Z. This may be caused by, in part, the mismatch in propagation velocities in the two phases, or a barrier due to the complex nature of the interface region. Close to the boundary with a ν = 1 phase, the ground state of the ν = 0 phase may not have canted spins but may instead be in an aligned antiferromagnet state, where spins are parallel to the magnetic field on one sublattice and antiparallel on the other. Eventually, far from the boundary, we may expect the local spin arrangement to rotate into the CAF orientation (Fig. 4B). In the transition region, the minimum magnon energy will be larger than E_Z due to effects of the valley-dependent interaction terms <cit.>, which were initially responsible for the antiferromagnet arrangement to be favored over the ferromagnetic arrangement. In order to cross from the ν = 1 region to the CAF region, a magnon with energy close to E_Z would have to tunnel through the barrier region, and we would expect the transmission rate to be low. If the magnons have enough energy to overcome this barrier, they should be able to more easily enter the CAF region. Fig. 4C shows that we can experimentally exceed this barrier, where we see non-local signals at higher |V_dc| with signs in agreement with our magnon model. The onset of this magnon signal is unaffected by any charge transport across the ν_TG2 = 0 region (fig. S10). Closely examining the signal at ν_TG2 = 0, we see signals commencing at ± V_EZ which we attribute to tunneling events across this ν_BG = 1/ν_TG2 = 0 barrier (Fig. 4D).  

Note that all non-local signals (occurring at ν_TG2 =-1, 0, and 1) appear only in a finite band of V_dc. This suppression of the differential voltage signal indicates that either magnon generation is suppressed, or alternatively, that the differently-spaced contacts begin to see identical amounts of magnon absorption once the system has reached a certain magnon density threshold. We further speculate that this cut-off could be related to the magnon bandwidth, but leave this to a future investigation. 

The experiments presented here introduce a method of using magnons to probe the SU(4) spin and valley anisotropies of graphene QH systems, whichc can be used to probe highly correlated states such as the fractional QH regime <cit.>, or the quantum-spin Hall phase of monolayer graphene <cit.>. Owing to the theoretical prediction for spin superfluidity in the CAF state <cit.>, this study paves the way for exploring and realizing dissipationless spin waves in a Bose-Einstein condensate (BEC) of magnons. Such condensates should result in a coherent precession of the spin in the QH magnet, which may be probed through emitted microwave radiation. Furthermore, coherent spin waves associated with a BEC may be able to propagate long distances with negligible dissipation, which could be tested by careful length dependence measurements. 


	

§ METHODS
 
	
	

 §.§ Sample Fabrication
 All devices consist of graphene encapsulated by two layers of hexagonal boron nitride (hBN) on doped Si chips with a 285 nm layer of SiO_2 that acts as a dielectric for the Si back gate. Graphene is mechanically exfoliated from bulk graphite obtained from NGS Naturgraphit GmbH using 1009R tape from Ultron Systems and subsequently encapsulated in hexagonal boron nitride (hBN) using a dry transfer process <cit.>. Before the first metal deposition step, we annealed the devices in vacuum at 500^∘C to improve device quality. We then created top gates using electron-beam lithography and thermal evaporation of Cr/Au. We etched the devices into the desired geometry by reactive ion etching in O_2/CHF_3 using a PMMA/HSQ bilayer of resist (patterned by electron-beam lithography) as the etch mask. To fabricate edge-contacts to the graphene we etched through the entire hBN/graphene stack. We then created edge contacts by thermally evaporating Cr/Au while rotating the sample using a tilted rotation stage. 
	

 §.§ Measurement
 Our measurements were performed in a Leiden dry dilution refrigerator with a base temperature of 20 mK. Measurements of differential conductance were performed using a lock-in amplifier with an a.c. excitation voltage of 50 μV at 17.77 Hz. All measurements of differential conductance were corrected for contact/line resistances, which were independently determined by lining up the robust ν = 2 QH conductance plateau with 2e^2/h. 
	
	naturemag
	
	
	

§ ACKNOWLEDGEMENTS

	

 §.§ General
 We thank A. H. Macdonald, J. D. Sanchez-Yamagishi,
	S. L. Tomarken, and S. P. Harvey for helpful discussions and
	feedback, X. Liu for fabrication help, and P. Kim for providing the
	transfer setup. Funding: Supported by the Gordon and Betty
	Moore Foundation’s EPiQS Initiative through grant GBMF4531;
	the U.S. Department of Energy, Basic Energy Sciences Office,
	Division of Materials Sciences and Engineering under award
	DE-SC0001819 (D.S.W. and T.v.d.S.); NSF Graduate Research
	Fellowship grant DGE1144152 (D.S.W.); the STC Center for
	Integrated Quantum Materials, NSF grant DMR-1231319 (B.I.H.);
	and the Elemental Strategy Initiative conducted by MEXT,
	Japan, and JSPS KAKENHI grant JP15K21722 (K.W. and T.T.).
	Nanofabrication was performed at the Center for Nanoscale
	Systems at Harvard, supported in part by NSF NNIN award
	ECS-00335765. Author contributions: D.S.W., T.v.d.S., B.I.H.,
	and A.Y. conceived and designed the experiments; D.S.W.
	fabricated the devices; D.S.W. and A.Y. performed the
	experiments; D.S.W., T.v.d.S., S.H.L., B.I.H., and A.Y. analyzed
	the data and wrote the paper; and K.W. and T.T. synthesized the
	hexagonal boron nitride crystals. Competing interests: The
	authors declare no competing financial interests. Data and
	materials availability: All measured data are available in the
	supplementary materials. 
 
	
	

§ CORRESPONDENCE
 Correspondence and requests for materials should be addressed to A.Y. (email: yacoby@physics.harvard.edu).	


	
		
		

§ SUPPLEMENTARY INFORMATION

		
		
		
		Supplementary Note 1. Equilibration of edge states at ‘hot spots’

		
		One question that arises from this study is why, after decades of experimental investigation into QH ferromagnets, has this phenomenon not been observed in GaAs quantum wells? We posit that this is due to the readiness of edge states in graphene to equilibrate over small length scales due to the sharp confining potentials. Because there is a limited spatial range over which the `hot spot' magnon generation can occur adjacent to the ν = 1 ferromagnetic bulk, spin-flip induced edge equilibration must occur over short lengths. Past studies have shown that in graphene edges of the same spin are able to fully equilibrate over length scales < 1 μm <cit.>, while similar studies done in GaAs found typical lengths of around tens of microns, and sometimes up to 200 μm <cit.>. In order to experimentally verify that magnons are generated at these corner `hot spots', we have fabricated a device with gated corners showing conductance changes at E_Z in accordance with our magnon model (Fig. S11). 
		
		The difference in experimentally-determined equilibration lengths between graphene and GaAs is likely due to the sharper confining potentials in graphene, which allow for small spatial edge channel separation—increasing the likelihood of inter-channel scattering. Additionally, a smooth potential may allow for edge reconstruction <cit.>, which if present, could also affect the inter-channel scattering rate and limit magnon generation. Experiments in GaAs based systems have shown that edge recondsturction plays an important role due to the smooth confining potential <cit.>. The graphene devices investigated in this work have gate electrodes located at just a few tens of nanometers distance away, likely limiting the amount of edge reconstruction. Furthermore, the close proximity of the metal gates to the graphene may screen the electric fields that cause edge reconstruction, which is another potential difference with GaAs-based systems <cit.>.
		
		However, interestingly, we note that although an electrostatically-defined confinement potential is able to suppress the magnon signal, it does not eliminate it completely (Fig. S11). This suggests that strong edge disorder is not required for the ν = 2 and ν = 1 edge channels to equilibrate, and that it is possible that magnons could be generated in GaAs devices with a sharp electrostatic confinement potential. We note, however, that while magnon generation may be possible, magnon propagation may not be as efficient in GaAs systems due to large spin-orbit coupling <cit.> and more nuclear spins <cit.> relative to graphene, which could facilitate magnon dissipation. Such dissipative processes would be important because, as we describe in the main text, magnon generation itself does not affect the sample conductance – only when magnons are able to propagate and are absorbed in by electrons in other edge channels do we detect a change in sample conductance. 
		
		Additionally, we note that even when we do not explicitly add an extra edge state near the contacts (by gating the side regions to ν  = 2), contact doping of the graphene by the Cr/Au leads <cit.> introduces additional spin-down edge states—which also leads to magnon generation at EZ.  

		
		
		
		Supplementary Note 2. Calculating V_dc necessary to exceed V_EZ given a finite contact resistance

	
		 In a two-terminal measurement, the applied bias voltage (V_dc) drops over both the contact resistances at both the source and the drain. The d.c. current is therefore
		
		
    I_dc = V_dc/2R_C+R_Q
	
		
		where R_Q is the quantum resistance of an edge channel, and where R_C includs both the contact resistance at each lead as well as the filtering on the lines. The filtering on each line is 4.5kΩ, and the contact resistance at each lead of a typical device is about 500Ω.  
		 
		The actual d.c. voltage that drops over the edge channel (V_dc') is therefore given by
		
		
    V_dc' = I_dc R_Q
	
		
		Therefore:
		
		
    V_dc = V_dc' ( 2R_C+R_Q)/R_Q
	
		
	 In our figures, we use `V_EZ' (V_EZ = -E_Z/e) to denote the bias at which -eV_dc' reaches the Zeeman energy (E_Z = g μ_B B_T). 

		
		
		
		Supplementary Note 3. Circuit analysis for two-terminal conductance measurement. 

		
		
		
		
		
		
		
		
		
		

		In Fig. S1 The (particle) current conservation equation at the source reservoir (labeled μ_1) is:
		
		
    2 μ_1/R_Q = μ_2 -ε_2/R_Q + μ_1 +ε_1/R_Q + μ -μ_1/R_C

		
		where R_Q is the resistance quantum. As described in the main text, ε_i  denotes the chemical potential redistributed between edge states at the ith contact. Additionally, although there are indeed also spin-flips occurring at the negative-bias magnon generation location when positive bias magnons are being generated (and for the reverse case), we ignore these in our analysis because they do not contribute to changes in the conductance.
		
		The equation at the drain contact is:
			
		
    μ_2 +ε_2/R_Q + μ_1 -ε_1/R_Q = 2μ_2/R_Q + μ_2/R_C

		
		Solving for μ_2 we find
		
		
    μ_2 = μ +ε_2 - ε_1/2+R_Q/R_C


		Using the chemical potential of the voltage source as μ = -eV_dc, and the charge current I = -eI_P (where I_P is defined as 1/e^2 μ_2/R_C  to normalize the units) the differential conductance measured is
		
		
    dI/dV_dc    = dI_P/dμ  = (dμ_2/R_C)/dμ

		
		
		
    dI/dV_dc = 1/2 R_C+R_Q( 1+dε_2/dμ-dε_1/dμ)

		
		This becomes Equation 1 (main text) in the absence of contact resistance (R_C =0).

		
		
	
		
		Supplementary Note 4. Circuit analysis for non-local voltage measurements.

		
		
		
		
		
		
		

  	
	In Fig. S2 the chemical potential of the edge states after magnons are absorbed are labeled as μ_i - ε_i and μ_i +ε_i where `i' denotes the contact where the chemical potential originates.  We calculate the conductance expected after the Zeeman energy has been reached. This device has 5 contacts in total. We write a current conservation equation at each contact:
		
		
    L_1: μ_2 - ε_2+μ_1 + ε_1 = 2 μ_1

		
	    
    L_2: μ_3 - ε_3+μ_2 + ε_2 = 2 μ_2 + μ_2 R_Q/R_C

		
		
    L_3: (μ - μ_3) R_Q/R_C + μ_3 + ε_3 + μ_4 - ε_4 = 2 μ_3

		
		
    L_4: μ_5 - ε_5+μ_4 + ε_4 = 2 μ_4

		
		
    L_5: μ_1 - ε_1+μ_5 + ε_5 = 2 μ_5

		

		Solving for μ_2 we find
	
		
    μ_2 = μ +ε_2 - ε_3/2+R_Q/R_C

	
		
		Therefore,
	
		
    dI/dV_dc  = dI_P/dμ = d(μ_2/R_C)/dμ = 1/2 R_C+R_Q( 1+dε_2/dμ-dε_3/dμ)

	
		
		The non-local voltage measured is:
		
    S_NL = dV_NL/dV = ( dε_4/dμ-dε_5/dμ)

		
	
		By defining S_NL as the difference between two voltage probes, any edge current which reaches the two voltage probes should not affect the measurement — although we do see some small background voltage which is explained in Fig. S7, B.
		
		A similar circuit analysis can be done for any of the configurations found in the main text or supplementary materials. 
 

		
	
	
	
	
	Supplementary Note 5. Theoretical Notes.

	
	We first note that the energy levels shown in the Figure 3D are only schematic. The actual
	Landau levels will be broadened due to electron-electron interactions and, perhaps, disorder. The curves represent more accurately the energy in the middle of the Landau level, and the ordering of the levels is more meaningful than the actual energies.
	
	Our ordering of levels was guided by the following observations. For a uniform graphene
	system at ν = 0, it is believed that the valley anisotropy energy is large compared to the Zeeman energy, and that the ground state is a canted antiferromagnet state <cit.>. In this half-filled N=0 Landau level, there is one electron per flux quantum on each sublattice, with spins oriented predominantly in opposite directions. In the absence of Zeeman coupling the antiferromagnetic axis could point equally well in any direction, with no difference in the energy <cit.>. In the presence of the Zeeman field, there is a small energy gain for the antiferromagnetic axis to line up in the x-y plane, perpendicular to the Zeeman field, allowing the spins on both sublattices to cant slightly in the direction of the Zeeman field. The energy gain for this is of order E_Z^2/E_A, where E_A is the valley anisotropy energy.
	
	For a general filling fraction in the range -1 < ν < 0, if one calculates the ground state energy in a restricted Hartree-Fock approximation, which assumes that only two of the possible spin-valley states are occupied by electrons, one generally finds that one valley, say the K valley, has one electron per flux quantum, while the other valley has occupancy 1 + ν < 1. For fillings very close to ν = 0, the system may remain in a canted configuration, but for |ν| exceeding a critical value, of order E_Z/E_A, it will be more favorable for the antiferromagnetic axis to align in the z-direction, so that the majority spin is fully aligned with the Zeeman field. (See, e.g. the discussion in <cit.>) Similarly, for 0 < ν < 1, we would find the antiferromagnetic spin axis to be aligned with the magnetic field, except for a small region close to ν = 0.
	
	In a situation where the electron density varies rapidly in space, the spin and valley orientationsshould be determined by the dominant exchange energy, arising from the long-range part of theCoulomb interaction, which is indifferent to the specific orientation of the occupied levels in spin-valley space, but disfavors any rapid changes or discontinuities in the occupations. In a	boundary between ν =-1 and ν = 1, we are forced to have two discontinuities in occupancy, but we can avoid any other discontinuities, if we choose to fill the levels in the order suggested in	Fig. 3D. Moreover, it is likely that in a relatively steep boundary, the canted orientation will be
	completely suppressed, and that spins will remain quantized along the z-axis. We have seen in a previous study an absence of mixing between spin states at a ν = 1/ ν = -1 interface, supporting our assumption of spin alignment in the present case <cit.>. 
	
	By contrast, when the filling fraction is ν = 0 under the center of our gate, it is likely that the system will assume the canted orientation near the center of the gate. At the same time, there should be a strip on either side of the gate, where the filling fraction is intermediate between ν = 1 and  ν = 0, where the antiferromagnetic axis is in the z-direction. An interval where the filling fraction is between  ν = 1  and  ν = -1 , with spin axes parallel to z, will act as a barrier, to a spin wave incident from a region where  ν = 1, as the energy at the bottom of the spin wave band will be raised by an amount of order the valley anisotropy energy (This should be small compared to the Coulomb exchange energy, but larger than the Zeeman energy) <cit.>. In the case where the filling under the gate is ν = -1, we would expect the barrier regions at the two sides to be relatively thin, and it is plausible that the spin waves can tunnel rather easily through the barrier region. When the filling at the gate center is ν = 0, we would expect the barriers to be much thicker, and tunneling through the barriers should be reduced accordingly.
	
	In a bulk region where the filling is very close to ν = 1, we expect that the unoccupied spin state will have its spin opposite to the magnetic field, but it will have no particular preference for either the K or K’ valley or an arbitrary linear combination of them. Different valley polarizations may be selected near the physical boundaries of the sample, but we expect that the valley orientation in the vicinity of a gate where the charge density varies rapidly should be determined by energy considerations under the gate. It should cost relatively little energy for the valley orientation to vary smoothly between the sample edges and the gate, and we would not expect spin-wave propagation to be affected by such variations.
	
	Our analysis, based on a Hartree-Fock approximation, ignores correlation effects, which can lead to fractional quantized Hall states, varying spin polarization, and transitions between different spin states in uniform graphene sample <cit.>. However, we would not expect such correlation effects to be important in the present case, where the charge density varies considerably on a sub-micron scale. 
	
		

		
		

	 
		
	


	

 	

	






		
	



	


	
	
	
	
	
		
	