This is the submitted version of the article: Zollner K., Cummings A.W., Roche S., Fabian J.. Graphene on two-dimensional hexagonal BN, AlN, and GaN: Electronic, spin-orbit, and spin relaxation properties. Physical Review B, (2021). 103. 075129: - . 10.1103/PhysRevB.103.075129. Available at: https://dx.doi.org/10.1103/PhysRevB.103.075129 Graphene on two-dimensional hexagonal BN, AlN, and GaN: Electronic, spin-orbit, and spin relaxation properties Klaus Zollner,1, ∗ Aron W. Cummings,2 Stephan Roche,2, 3 and Jaroslav Fabian1 1 Institute for Theoretical Physics, University of Regensburg, 93040 Regensburg, Germany 2 Catalan Institute of Nanoscience and Nanotechnology (ICN2), CSIC and The Barcelona Institute of Science and Technology, Campus UAB, Bellaterra, 08193 Barcelona, Spain 3 ICREA - Institució Catalana de Recerca i Estudis Avançats, 08010 Barcelona, Spain We investigate the electronic band structure of graphene on a series of two-dimensional hexagonal nitride insulators hXN, X = B, Al, and Ga, with first principles calculations. A symmetry-based model Hamiltonian is employed to extract orbital parameters and spin-orbit coupling (SOC) from the low-energy Dirac bands of the proximitized graphene. While commensurate hBN induces a staggered potential of about 10 meV into the Dirac band structure, less lattice-matched hAlN and hGaN disrupt the Dirac point much less, giving a staggered gap below 100 µeV. Proximitized intrinsic SOC surprisingly does not increase much above the pristine graphene value of 12 µeV; it stays in the window of (1-16) µeV, depending strongly on stacking. However, Rashba SOC increases sharply when increasing the atomic number of the boron group, with calculated maximal values of 8, 15, and 65 µeV for B, Al, and Ga-based nitrides, respectively. The individual Rashba couplings also depend strongly on stacking, vanishing in symmetrically-sandwiched structures, and can be tuned by a transverse electric field. The extracted spin-orbit parameters were used as input for spin transport simulations based on Chebyshev expansions of the time-evolution and spin operators, yielding interesting predictions for the electron spin relaxation. Spin lifetime magnitudes and anisotropies depend strongly on the specific (hXN)/graphene/hXN system, and can be efficiently tuned by an applied external electric field as well as the carrier density in the graphene layer. A particularly interesting case for experiments is graphene/hGaN, in which the giant Rashba coupling is predicted to induce spin lifetimes of 1-10 ns, short enough to dominate over other mechanisms, and lead to the same spin relaxation anisotropy as that observed in conventional semiconductor heterostructures: 50%, meaning that out-of-plane spins relax twice as fast as in-plane spins. Keywords: spintronics, graphene, heterostructures, proximity spin-orbit coupling I. INTRODUCTION and theoretical35–37 evidence of graphite-like hAlN, which can be thinned down to the monolayer limit. Hexagonal boron nitride (hBN) has become one of the Graphene/hAlN heterostructures have already been con- most important insulator materials for electronics and sidered theoretically49 and just recently have been spintronics. In the two-dimensional (2D) few-layer limit grown50 . it is often used as a tunnel barrier to overcome the con- Similarly, GaN is an important semiconductor material ductivity mismatch in spin injection devices, or as an en- for technological applications, especially in the wurtzite capsulation material to protect other 2D materials from form with a direct band gap in the ultraviolet range51,52 . degradation. The current generation of graphene-based Monolayer hGaN is therefore a potentially important ma- spintronic devices also relies on the excellent properties of terial for ultra-compacted electronics and optics53 . Inter- hBN as a substrate material, leading to outstanding spin estingly, graphene also seems to play a major role for the and charge transport properties1–17 , which are highly im- growth of hGaN54 . Graphene on hGaN was considered portant for the realization of spin logic devices8,18–33 . from first-principles55 as a tool to control the Schottky However, there are other layered hexagonal nitride semi- barrier and contact type by strain engineering. conductors/insulators that may prove useful for elec- There are already several fields now in which hAlN tronic and spintronic applications. In particular, first- and hGaN show promising results. For example, principles calculations have demonstrated the chemical TMDC/hAlN or blue phosphorene/hGaN heterostruc- stability of hexagonal aluminum nitride (hAlN) and gal- tures are predicted to be important visible-light-driven lium nitride (hGaN)34–40 , as well as revealing their me- photocatalysts for water splitting56,57 . Excitonic effects chanical and optical properties. have also been studied in hAlN and hGaN38,58 , with Bulk AlN is an important material41 . It is used absorption spectra dominated by strongly bound exci- as a high-quality substrate for the growth of topolog- tons. Both hAlN and hGaN are gaining interest for opto- ical insulators42,43 and transition-metal dichalcogenides electronic applications53 due to their two-dimensional na- (TMDCs)44,45 . To grow strain-free AlN on a sapphire ture. Stacks of hAlN and hGaN59 also show novel elec- substrate, graphene can be used as a buffer layer46 . The tronic and optical properties. two-dimensional counterpart to bulk AlN is also becom- From the theoretical perspective, the properties of ing more important. There is strong experimental47,48 hAlN and hGaN are not yet well studied. Mono- 2 layer hAlN has an indirect band gap ranging between ters, we then perform spin transport simulations to gain 2.9–5.8 eV49,60 , which is strain dependent34,61 . The band insight into spin relaxation in (hXN)/graphene/hXN het- gap also depends on the number of hAlN layers34 . Mono- erostructures. In all studied systems, we find large spin layer hGaN has an indirect band gap ranging between lifetimes in the nanosecond range and above, reaching 2.1–5.5 eV, which is also tunable by strain52,53,60,62 . up to a few seconds in encapsulated structures. Depend- G0 W0 -calculations provide band gaps of hAlN (5.8 eV60 ) ing on the heterostructure, strong electron/hole asym- and hGaN (4.55 eV52 ) comparable to the theoretical and metry and giant anisotropy in the spin lifetimes can be experimental values for hBN (6.0 eV63 ). It then seems observed, which are experimentally testable fingerprints natural to consider hGaN and hAlN as alternatives to of our results. For example, the weak intrinsic combined hBN in transport experiments, even though few- and with the strong Rashba SOC in graphene/hGaN results monolayer materials are difficult to synthesize at the mo- in spin lifetimes between 1 and 10 ns and a nearly con- ment. stant spin relaxation anisotropy of 50% for all carrier Layered hBN is a perfect substrate for graphene spin- densities. Adding a hBN capping layer, meanwhile, in- tronics, allowing for giant mobilities in graphene/hBN troduces a sizeable orbital band gap in the Dirac spec- structures6 , ultralong spin lifetimes1,4,11,14,64 , for ex- trum, resulting in giant out-of-plane spin lifetimes and tracting spin-orbit gaps in bilayer graphene65 , or for re- anisotropies near the charge neutrality point. vealing surprisingly strong spin-orbit anisotropy in bi- layer graphene66,67 . On the other hand, hBN induces only weak SOC in graphene5 , limiting its potential for II. COMPUTATIONAL DETAILS AND investigating proximity effects. Would hAlN and hGAN GEOMETRY provide the same protection for the electron spins in graphene as hBN? What is the expected spin relaxation In Fig. 1 we show the supercell geometries of the if graphene is stacked with the two alternative insula- graphene/hBN, graphene/hAlN, and graphene/hGaN tors? In the absence of experimental studies, theoretical heterostructures. The lattice constant of graphene68 is answers to these questions are particularly important. a = 2.46 Å, the one of hBN69 is a = 2.504 Å, the Here, we use first principles calculations to study one of hAlN36 is a = 3.12 Å, and the one of hGaN36 graphene on monolayers of hBN, hAlN, and hGaN. We is a = 3.25 Å, which we slightly adapt (straining the lat- investigate the proximity-induced SOC in graphene by tices by less than 2%) for the periodic first-principles cal- fitting a low energy model Hamiltonian to the Dirac culations. For the graphene/hBN structure, we consider bands. In particular, we find that the lattice-matched 4 × 4 (5 × 5) supercells of graphene and hBN, both with hBN induces a strong sublattice asymmetry, resulting in a lattice constant of a = 2.4820 Å, and we use the most an orbital gap on the order of 20 meV, while there is energetically favorable stacking for these commensurate an averaging effect for the sublattice asymmetry in the lattices5 . In the case of graphene on hGaN, we place a less lattice-matched hAlN and hGaN substrates, result- 4 × 4 graphene unit cell, with a compressed lattice con- ing in orbital gaps below 200 µeV. The intrinsic SOC stant of a = 2.4486 Å, on a 3 × 3 hexagonal hGaN unit parameters stay below 15 µeV for all hXN substrates, cell, with a stretched lattice constant of a = 3.2649 Å 36 . while the Rashba SOC increases from a few to tens of For graphene on hAlN, we place a 5 × 5 graphene unit µeV with ascending atomic number of the boron group. cell, with a stretched lattice constant of a = 2.4816 Å, By varying the interlayer distance between graphene and on a 4 × 4 hexagonal hAlN unit cell, with a compressed a given hXN substrate, we not only find the energeti- lattice constant of a = 3.102 Å 35,36,47 . cally most favorable geometry, but also show that the or- First-principles calculations are performed with a full bital gap as well as proximity-induced SOC parameters potential linearized augmented plane wave (FLAPW) (especially Rashba and pseudospin-inversion asymmetry code based on density functional theory (DFT), as im- (PIA) ones) are highly tunable by the van der Waals gap plemented in WIEN2k70 . Exchange-correlation effects between the layers (more than 100% by tuning the inter- are treated with the generalized-gradient approximation layer distance by only 10%). In addition, we show that (GGA)71 , including a dispersion correction72 . an external transverse electric field can be used to tune For graphene/hGaN, we use a k-point grid of 18 × 18 × the SOC of the proximitized graphene. While the in- 1, and the values of the Muffin-tin radii are RC = 1.33 trinsic couplings can be tuned between 5 to 20 µeV, the for C atoms, RGa = 1.90 for Ga atoms, and RN = Rashba and PIA couplings can reach up to a few hundred 1.64 for N atoms, with the plane wave cutoff parame- µeV, within our considered field limits of ±3 V/nm. By ter RKMAX = 4.7. For graphene/hAlN, we use a k- encapsulating graphene between two hXN layers, the in- point grid of 12 × 12 × 1, and the Muffin-tin radii are duced SOC results from an interplay of both layers (hBN RC = 1.35 for C atoms, RAl = 1.72 for Al atoms, and induces a sizeable orbital gap, while additionally, e. g., RN = 1.64 for N atoms, with the plane wave cutoff pa- hGaN provides a large Rashba and PIA SOC) allowing rameter RKMAX = 4.2. We consider only the stack- for the customization of spin transport in graphene via ing configurations shown in Fig. 1. For these two het- layer engineering. erostructures we vary the distance between the graphene Combining the model Hamiltonian and fitted parame- and hXN layers to find the lowest energy separation. 3 FIG. 1. (Color online) Top view of the supercell geometry of (a) graphene on hBN (4 × 4), (b) graphene on hAlN, and (c) graphene on hGaN. Different colors correspond to different atom types. (a) 4 (b) 4 (c) 4 grp grp grp hBN hAlN hGaN 3 3 3 2 2 2 E − EF [eV] E − EF [eV] E − EF [eV] 1 1 1 εCB 2 CB ε 1 0 0 0 εVB 2 εVB 1 −1 −1 −1 −2 −2 −2 −3 −3 −3 Γ M K Γ Γ M K Γ Γ M K Γ FIG. 2. (Color online) Calculated band structures of graphene on (a) hBN, (b) hAlN, and (c) hGaN. Bands corresponding to different layers (graphene, hBN, hAlN, hGaN) are plotted in different colors (black, green, purple, red). The inset in (a) shows a sketch of the low energy dispersion close to the K point. Due to the presence of the substrate, graphene’s low energy bands CB/VB are split into four states ε1/2 , with a band gap. When we then consider graphene encapsulated in hAlN and hBN/graphene/hAlN sandwich structures. In these or hGaN, we use these interlayer distances. For the en- cases, we take the geometries for graphene/hAlN and capsulated structures, a second hAlN (hGaN) layer is graphene/hGaN with their energetically most favorable added on top of the graphene/hAlN (graphene/hGaN) interlayer distances and place another hBN layer above stack, and we use an AA0 stacking of the two hAlN graphene at a distance of d = 3.48 Å. The in-plane lat- (hGaN) layers with graphene sandwiched between. In tice constant of hBN is adapted to that of graphene. For order to avoid interactions between periodic images of the sandwich structure including hGaN (hAlN), we use our slab geometries, we add a vacuum of at least 24 Å in a k-point grid of 18 × 18 × 1 (12 × 12 × 1). The Muffin- the z direction for all structures we consider. tin radii are RC = 1.33 (1.35) for C atoms, RB = 1.26 Graphene/hBN heterostructures were already exten- (1.28) for B atoms, RN = 1.39 (1.41) for N atoms, and sively discussed in Ref. 5. For purposes of comparison, RGa = 1.90 (RAl = 1.72) for Ga (Al) atoms, with the we consider here 4 × 4 (5 × 5) graphene/hBN supercells, plane wave cutoff parameter RKMAX = 4.7 (4.2). using a k-point grid of 18 × 18 × 1 (12 × 12 × 1) and an interlayer distance of d = 3.48 Å. The Muffin-tin radii are RC = 1.35 for C atoms, RB = 1.28 for B atoms, III. MODEL HAMILTONIAN and RN = 1.41 for N atoms, with the plane wave cutoff parameter RKMAX = 4.7 (4.2). The band structure of proximitized graphene can We also consider asymmetric hBN/graphene/hGaN be modeled by symmetry-derived Hamiltonians73 . For 4 graphene in heterostructues with C3v symmetry, the ef- k [10−4/Å] (c) 1 (a) 1 0 1 0.5 < sx > fective low energy Hamiltonian is 0 0.5 Model −0.5 H = H0 + H∆ + HI + HR + HPIA , (1) DFT −1 M K E − EF [meV] ← → Γ H0 = ~vF (τ kx σx − ky σy ) ⊗ s0 , (2) 0 (d) 1 H∆ = ∆σz ⊗ s0 , (3) 0.5 < sy > HI = τ (λA B I σ+ + λI σ− ) ⊗ sz , (4) 0 −0.5 HR = −λR (τ σx ⊗ sy + σy ⊗ sx ), (5) −0.5 −1 HPIA = a(λA B M ← K → Γ M ← K → Γ PIA σ+ − λPIA σ− ) ⊗ (kx sy − ky sx ). (6) (b) 40 (e) 1 splitting [µ eV] Here vF is the Fermi velocity and the in-plane wave 30 0.5 < sz > vector components kx and ky are measured from ±K, 20 0 10 val −0.5 corresponding to the valley index τ = ±1. The Pauli cond spin matrices are si , acting on spin space (↑, ↓), and 0 −1 M ← K → Γ M ← K → Γ σi are pseudospin matrices, acting on sublattice space (CA , CB ), with i = {0, x, y, z} and σ± = 12 (σz ± σ0 ). FIG. 3. (Color online) Calculated low energy band properties The lattice constant of graphene is a and the staggered of graphene on hAlN in the vicinity of the K point with an in- sublattice potential gap is ∆. The parameters λA I and terlayer distance of 3.48 Å. (a) First-principles band structure λB I describe the sublattice-resolved intrinsic SOC, λR (symbols) with a fit to the model Hamiltonian (solid line). (b) stands for the Rashba SOC, and λA B PIA and λPIA are for The splitting of conduction band ∆ECB (blue) and valence the sublattice-resolved pseudospin-inversion asymmetry band ∆EVB (red) close to the K point and calculated model (PIA) SOC. The basis states are |ΨA , ↑i, |ΨA , ↓i, |ΨB , ↑i, results. (c)-(e) The spin expectation values of the bands εVB 2 CB/VB and εCB 1 and comparison to the model results. and |ΨB , ↓i, resulting in four eigenvalues ε1/2 . k [10−4/Å] (c) 1 (a) 2 1 0 1 2 0.5 < sx > IV. FIRST-PRINCIPLES AND FIT RESULTS 0 1 Model −0.5 DFT −1 In Fig. 2, we show the calculated band structures of M K E − EF [meV] ← → Γ graphene on hBN, hAlN, and hGaN. We find that the 0 (d) 1 Dirac bands are always preserved and are located inside 0.5 the band gap of the substrate. The low energy bands of < sy > 0 CB/VB graphene are split into four states ε1/2 with a band −1 −0.5 −1 gap, as shown in the inset in Fig. 2(a), which is caused M ← K → Γ M ← K → Γ by the pseudospin symmetry breaking and the proximity- (b) induced SOC. (e) 1 splitting [µ eV] 120 0.5 90 < sz > 0 60 val −0.5 A. Low energy bands 30 cond −1 M ← K → Γ M ← K → Γ In Fig. 3 we show the low energy band properties of FIG. 4. (Color online) Calculated low energy band properties the graphene/hAlN stack. We find perfect agreement of of graphene on hGaN in the vicinity of the K point with an in- the band structure, band splittings and spin expectation terlayer distance of 3.48 Å. (a) First-principles band structure values with the first-principles data, in the vicinity of (symbols) with a fit to the model Hamiltonian (solid line). (b) the K point. In Fig. 3(a) we show a zoom to the low The splitting of conduction band ∆ECB (blue) and valence energy bands with a fit to our low energy model Hamil- band ∆EVB (red) close to the K point and calculated model tonian. The splittings, shown in Fig. 3(b), are around results. (c)-(e) The spin expectation values of the bands εVB 2 30 µeV near the K point, and are of the same order as and εCB 1 and comparison to the model results. for graphene on hBN5 . The sx and sy spin expectation values show a clear signature of Rashba SOC, while the sz expectation value sharply decays away from the K point, The fit parameters are summarized in Table I. We find see Fig. 3(c-e). In Fig. 4 we show the low energy band that hGaN induces larger band splittings in graphene properties of the graphene/hGaN stack. The overall re- than hAlN, as reflected in the 4-times larger Rashba pa- sults are very similar, except that the band splittings are rameter. The origin of this sizeable Rashba coupling is much larger, around 120 µeV near the K point. The the deformation of C pz orbitals along the z direction74 model is also in perfect agreement with the calculated (see Appendix), being much more pronounced in the bands. hGaN case compared to the other substrates. In Ta- 5 TABLE I. Summary of the fit parameters of Hamiltonian H, for graphene/hBN, graphene/hGaN, and graphene/hAlN systems with interlayer distances of 3.48 Å. We have artificially turned off SOC on N, Al, Ga, or C atoms, to resolve the contributions of individual atoms to the band splittings. The fit parameters are the Fermi velocity vF , gap parameter ∆, Rashba SOC parameter λR , intrinsic SOC parameters λA B A B I and λI , and PIA SOC parameters λPIA and λPIA for sublattices A and B. System SOC on vF [105 m/s] ∆ [µeV] λR [µeV] λA I [µeV] λB I [µeV] λA PIA [µeV] λB PIA [µeV] grp/hGaN C, Ga, N 8.328 94.9 -60.3 -12.5 -9.2 -117.5 -117.5 grp/hGaNa C, Ga, N 8.330 48.3 -60.4 -12.5 -9.2 -117.1 -117.1 grp/hGaN Ga, N 8.329 96.3 -64.9 -1.7 1.6 -111.1 -111.1 grp/hGaN C 8.329 94.9 4.6 -10.8 -10.8 0 0 hGaN/grp/hGaN C, Ga, N 8.272 27.6 0.1 -10.6 -10.6 0 0 grp/hAlN C, Al, N 8.172 87.1 14.7 -4.6 -15.6 70.3 70.3 grp/hAlNa C, Al, N 8.170 4.5 15.2 -4.9 -16.2 67.5 67.5 grp/hAlN Al, N 8.172 89.1 9.5 5.4 -5.6 81.0 81.0 grp/hAlN C 8.172 87.5 5.1 -10.0 -10.0 0 0 hAlN/grp/hAlN C, Al, N 8.077 10.0 0 3.2 3.2 0 0 grp/hBN (4 × 4) C, B, N 8.206 13.3b 7.9 -8.8 -5.8 25.6 12.1 grp/hBN (4 × 4)a C, B, N 8.206 13.4b 7.9 -8.8 -5.8 25.9 12.2 grp/hBN (5 × 5) C, B, N 8.246 11.3b 6.8 8.1 5.4 23.9 12.7 hGaN/grp/hBN C, B, Ga, N 8.301 12.0b -69.7 -9.7 -6.7 -123.3 -116.1 hAlN/grp/hBN C, B, Al, N 8.151 10.8b 7.0 -2.4 -10.9 3.6 3.6 a The rippled structure from the relaxation of internal atomic positions. b For the graphene/hBN structures, the sublattice-symmetry breaking is much larger (see Ref. 5) and the parameter ∆ is given in meV. ble I we also summarize fit results for the graphene/hBN of the systems are minimized for interlayer distances stacks (4 × 4 and 5 × 5 supercells). Both give similar re- of 3.48 Å, in agreement with literature showing weak sults, as already found in Ref. 5, which further validates vdW bonding62 . Additionally, we find that the Rashba our DFT calculations. In contrast to the graphene/hAlN and PIA parameters decrease with increasing distance, and graphene/hGaN stacks, the staggered potential gap while the intrinsic SOC parameters converge towards the parameter ∆ is much larger in the graphene/hBN case. known value of about 10 µeV for pristine graphene74 . In the case of hBN, the individual graphene sublattices The different Fermi velocities for the two systems can (A and B) are perfectly aligned above the boron and hol- be attributed to the different graphene lattice constants low positions of hBN, see Fig. 1(a), leading to a strong used in the heterostructure calculations, which affects the sublattice-symmetry breaking and a large staggered po- magnitude of the nearest-neighbor hopping integral. tential gap. In contrast, for hGaN and hAlN the sub- lattice atoms of graphene are nearly arbitrarily aligned above the substrates, see Fig. 1(b,c), and very little sub- C. Encapsulated Structures lattice asymmetry arises. We additionally calculated the low energy band struc- In the symmetric encapsulated graphene het- tures when SOC is artificially turned off in the nitrogen, erostructures, namely hAlN/graphene/hAlN and gallium, aluminum or carbon atoms. The fit parameters hGaN/graphene/hGaN, we find that, compared to the for these situations are also given in Table I. From these non-encapsulated cases, Rashba SOC is strongly sup- fits we can resolve the atomic contributions to the SOC pressed because inversion symmetry is nearly restored, parameters induced solely by the substrate. When SOC as shown in Table I. Thus, the encapsulated geometries of the substrate is neglected, we recover the intrinsic SOC should in principle lead to larger spin lifetimes, as is of pristine graphene74 , and strongly reduce the Rashba the case for graphene/hBN structures5 . In the case and PIA contributions to the band splittings. of the hBN/graphene/hGaN and hBN/graphene/hAlN structures, the fit parameters are also summarized in Table I and result from an interplay between the B. Distance Study different top and bottom encapsulation layers. The hBN layer significantly enhances the orbital gap parameter In Fig. 5, we show the evolution of the fit param- ∆ due to commensurability with the graphene lattice eters for graphene/hAlN and graphene/hGaN as we and the resulting sublattice-symmetry breaking. In the vary the interlayer distance, similar to Ref.5 for the case of hBN/graphene/hGaN, the SOC parameters are graphene/hBN stack. We find that the total energy similar to the non-encapsulated graphene/hGaN case, 6 (a)(a) (e)(e) (a) (e) EVBE − EF [meV] 1010 0 600600 5 5 −5 ] Etot [meV] I [µ eV] I [µeV] I [µeV] 400400 hAlN hAlN 0 0 −0.25 −10 hGaN hGaN −5−5 hGaN tot [ 200200 −15 λA hAlN λA λA −10−10 0 0 −15−15 −0.5 3.13.1 3.23.2 3.33.3 3.43.4 3.53.5 3.13.1 3.23.2 3.33.3 3.43.4 3.53.5 −2 0 2 −2 0 2 distance distance [Å][Å] distance distance [Å][Å] electric field [V/nm] electric field [V/nm] (b) (f) (b)(b) (f)(f) 8.4 8.48.4 −5−5 −5 vF/105 [m/s] vF/10 [m/s] vF/10 [m/s] I [µ eV] −10−10 8.3 λBIB [µeV] λI [µeV] 8.28.2 −15−15 8.2 −10 8 8 5 −20−20 8.1 −15 λB 7.87.8 −25−25 8 7.67.6 −30−30 3.13.1 3.23.2 3.33.3 3.43.4 3.53.5 3.13.1 3.23.2 3.33.3 3.43.4 3.53.5 −2 0 2 −2 0 2 electric field [V/nm] electric field [V/nm] distance distance [Å][Å] distance distance [Å][Å] (c) (g) c)(c) 0.50.5 (g)(g) 400400 110 PIA [µ eV] 100 ∆ [µeV] PIA [µeV] PIA [µeV] 0.40.4 200200 100 ∆ [meV] ∆ [meV] 0.30.3 0 0 0 90 0.20.2 −100 −200 −200 λA 0.10.1 80 −200 λA λA −400 −400 0 0 −2 0 2 −2 0 2 3.13.1 3.23.2 3.33.3 3.43.4 3.53.5 3.13.1 3.23.2 3.33.3 3.43.4 3.53.5 electric field [V/nm] electric field [V/nm] distance distance [Å][Å] distance distance [Å][Å] (d) (h) d)(d) (h)(h)400400 25 PIA [µ eV] 0 100 λR [µ eV] 5050 [µeV] λBPIA [µeV] 200200 R [µ ] λR [µeV] 0 0 −25 0 −50−50 0 0 −50 −100 −100 −100 −75 λB −200 −200 λBPIA −150 −150 −200 −200 −200 −400 −400 −2 0 2 −2 0 2 3.13.1 3.23.2 3.33.3 3.43.4 3.53.5 3.13.1 3.23.2 3.33.3 3.43.4 3.53.5 electric field [V/nm] electric field [V/nm] distance distance [Å][Å] distance distance [Å][Å] FIG. 6. (Color online) Fit parameters as a function FIG. 5. (Color online) Fit parameters as a function of distance of the transverse electric field for graphene/hAlN and between graphene and hAlN or hGaN. (a) Total energy, (b) graphene/hGaN at fixed interlayer distances of 3.48 Å. Fermi velocity vF (c) gap parameter ∆, (d) Rashba SOC pa- (a) The valence band edge with respect to the Fermi level, rameter λR , (e) intrinsic SOC parameter λA I for sublattice A, (b) the Fermi velocity vF , (c) gap parameter ∆, (d) Rashba (f) intrinsic SOC parameter λB I for sublattice B, (g) PIA SOC SOC parameter λR , (e) intrinsic SOC parameter λA I for sub- parameter λA PIA for sublattice A, and (h) PIA SOC parameter lattice A, (f) intrinsic SOC parameter λB I for sublattice B, λBPIA for sublattice B. (g) PIA SOC parameter λA PIA for sublattice A, and (h) PIA SOC parameter λB PIA for sublattice B. as if the hBN layer would have no effect. In particular, the Rashba and PIA SOC parameters have roughly the intrinsic SOC parameter for sublattice A (B) decreases same values. In contrast, for the hBN/graphene/hAlN (increases) in magnitude, while both PIA parameters can structure, the formerly large PIA parameters from the be drastically enhanced by a factor of 2 when the field non-encapsulated graphene/hAlN case are now strongly is tuned from negative to positive values. In the case suppressed, while the intrinsic and Rashba SOC are of of hAlN as the substrate, the electric field tunability is similar magnitude. somewhat similar, but not as pronounced as in the case of hGaN. However, the PIA couplings show also a strong tunability ranging from 40 to 170 µeV within our electric D. Electric Field Effects field limits. Gating can be used to tune the SOC, especially the Rashba SOC, in the Dirac bands of graphene5,74 . Tun- E. Rippling Effects ing the SOC can have a significant impact on spin trans- port and relaxation properties. Consequently, gating So far, we only considered flat monolayers stacked on is a potentially efficient control knob in experimental top of each other, with fixed interlayer distances. In ex- studies. Here we apply a transverse electric field, mod- periment, the layers are typically not completely flat and eled by a sawtooth potential, across graphene/hAlN and rippling occurs due to imperfections and impurities. How graphene/hGaN heterostructures for a fixed interlayer does rippling affect the proximity-induced SOC? The distance of 3.48 Å. The results are summarized in Fig. 6. starting point are the graphene/hXN heterostructures Indeed, most of the parameters can be tuned by the with the energetically most favorable interlayer distance field. Especially in the case of hGaN, the tunability of of 3.48 Å, to which we apply the force relaxation75–77 the Rashba, intrinsic, and PIA SOC parameters is giant implemented in the WIEN2k code. Thereby, we deter- within our considered field limits. Tuning the field from mine the equilibrium positions of all atoms by minimiz- −3 to 3 V/nm, we find that the Rashba SOC can be ing the forces on the nuclei, while respecting the C3v - enhanced in magnitude from about 40 to 90 µeV. The symmetry of the heterostructures. When all forces are 7 below 0.5 mRy/bohr, we continue to calculate the low to an exponential decay, si (E, t) = exp(−t/τs,i (E)). We energy bands and apply the model Hamiltonian fit rou- also calculate the momentum relaxation time τp (E) = 2D(E)/vF2 , where D(E) = max 12 dt  d tine. The fit results are summarized in Table I. ∆X 2 (E, t) is the t After relaxation of a given graphene/hXN heterostruc- semiclassical diffusion coefficient. ture, we also calculate the mean values, z̄, and the standard deviations, ∆z, of the z coordinates of the To induce charge scattering and thus spin relaxation atoms belonging to the individual monolayers. From the in our simulations, we include a random distribution mean values, we calculate the average interlayer distance, of Gaussian-shaped electrostatic impurities which are d = z̄grp − z̄hXN , between graphene and hXN, while the meant to represent the impact of charged impurity scat- standard deviations represent the amount of rippling of tering in graphene95,96 . The electrostatic P potential at the layers. The results are summarized in Table II. each atomic site i is then given by i = Vj exp(−|~ri − Overall, we find that the average interlayer distances are j barely affected and the individual monolayers are still ~rj |2 /2ξ 2 ), where ~ri is the position of each carbon atom, nearly flat with a maximum rippling below 1 pm. Com- ~rj is the position of each impurity, ξ is the width of paring the fit parameters in Table I, for the flat and rip- each impurity, and the height Vj of each impurity is pled structures, we find that mainly the staggered po- randomly√distributed in [−V, V ]. Here we use V = 2.8 tential gap ∆ is affected, while the SOC parameters stay eV, ξ = 3a, and an impurity density of 0.1%. This nearly the same. In the case of hGaN, ∆ is about twice choice of parameters leads to a charge mobility around as large in the flat structure compared to the rippled µ ≈ 1000 cm2 /Vs and intervalley scattering in the range one. For hAlN, the effect is even more drastic, since the of τiv ≈ (10−60)τp . Here we should note that this mobil- rippled structure nearly leads to a vanishing sublattice ity is quite a bit lower than what would be expected for asymmetry in graphene, while the flat structure has a graphene interfaced with these hXN materials. However, 20-times larger asymmetry, as reflected in the parameter limitations on the sample size we can simulate require ∆. us to use this value to ensure proper numerical conver- gence. Assuming the Dyakonov-Perel mechanism of spin TABLE II. The calculated average interlayer distances, relaxation (see below), the obtained spin lifetimes can be d, and the ripplings, ∆z, of the monolayers, of the scaled to expected experimental mobilities according to graphene/hXN heterostructures after relaxation of internal τs−1 ∝ µ. Finally, for each graphene/hXN heterostruc- forces. ture, we convert the continuum Hamiltonian of Eq. (1) to a real-space tight-binding form and use the first set of System d [Å] ∆zgrp [pm] ∆zhXN [pm] parameters in Table I. grp/hBN 3.47952 0.005 0.117 grp/hAlN 3.48017 0.089 0.282 In our numerical simulations we have only considered grp/hGaN 3.47997 0.042 0.093 the graphene/hGaN heterostructure, as the others have lower SOC, putting their spin lifetimes outside the range of what we can simulate numerically. Thus, to explain the features seen in our simulations and to make predic- tions for the hAlN and hBN heterostructures, we now V. SPIN RELAXATION describe a simple and transparent model of spin relax- ation in these systems. We start by assuming the only A. Model and Numerical Approach source of spin relaxation is from the SOC parameters de- fined in Eq. (1), and not from extrinsic sources such as Spin relaxation in graphene is a vibrant topic in con- ripples or magnetic impurities81,89 . There are two tra- densed matter physics64,66,67,78–93 . Therefore, we now ditional spin relaxation mechanisms driven by uniform use a combination of modeling and numerical simulations SOC, namely the Elliott-Yafet (EY) and the Dyakonov- to predict the nature and magnitude of spin relaxation Perel (DP) mechanisms97–99 . In the EY mechanism, spin in graphene/hXN heterostructures. To numerically sim- flips occurs at the point of scattering, and the spin life- ulate spin transport, we use a real-space time-dependent time is proportional to the scattering time. In graphene, approach that has been employed for the study of elec- the EY mechanism is predicted to relax in-plane spins trical and spin transport in a wide variety of disordered via the intrinsic SOC100 . In the DP mechanism, SOC- materials94 . We initialize a spin-polarized random-phase induced spin precession and dephasing occur between state in a 500×500 nm2 sample of graphene, and we scattering events, leading to an inverse scaling between evolve it in time using an efficient Chebyshev expansion spin lifetime and scattering time. of the time evolution operator. At each time step we cal- culate the spin polarization si (E, t) and the mean-square The numerical simulations show a negligible EY con- displacement ∆X 2 (E, t) of the state as a function of the tribution, primarily arising from the small value of λI Fermi energy E. The energy-dependent spin lifetime in all systems. Thus, we focus solely on the DP mech- τs,i (E) is then extracted by fitting the spin polarization anism. The spin relaxation rates in graphene are then 8 given by5,83,101 The peak in the spin lifetime around E = 0, correspond- ing to the dip in τp at the same point, is also indicative −1 τs,x = τp hωy2 i + τiv hωz2 i, of the DP mechanism. −1 τs,y = τp hωx2 i + τiv hωz2 i, (7) −1  V τs,z = τp hωx2 + ωy2 i,  6FDWWHULQJWLPH>IV@  $QLVRWURS\  iv  where τs,x/y are the lifetimes of spins pointing in the  V graphene plane, τs,z is the out-of-plane spin lifetime, τp   p is the momentum relaxation time, τiv is the intervalley         scattering time, and ωi are the components of the effec-  V (QHUJ\>H9@ (QHUJ\>H9@ tive magnetic field induced by SOC. As we will see below, s the z-component of the effective magnetic field has oppo- site sign in the ±K valleys and thus drives spin relaxation QV *UK%1 through intervalley scattering, while the in-plane compo- nents depend on the direction of electron momentum and thus are connected to τp . QV *UK$O1 The effective magnetic field can be written as (see Ap- pendix) *UK*D1 QV        ~ ωx = ±λR + akλ+ (QHUJ\>H9@    PIA + gk (k) · sin(−θ), 2 ~ ωy = ±λR + akλ+    PIA + gk (k) · cos(θ), FIG. 7. (Color online) Predicted spin lifetime in 2 graphene/hXN heterostructures. The solid (dashed) lines ~ show the out-of-plane (in-plane) spin lifetimes calculated from ωz = [λVZ + gz (k)] · τ, (8) 2 Eqs. (7)-(10). The symbols show the spin lifetime from nu- merical simulations of graphene/hGaN. The left inset shows where λ+ A B PIA = (λPIA + λPIA )/2 gives rise to a k-linear the spin lifetime anisotropy for each heterostructure, and the spin splitting similar to Rashba SOC in traditional 2D right inset shows the momentum relaxation time and inter- electron gases, λVZ = (λA B I − λI )/2 is the valley-Zeeman valley scattering time calculated from numerical simulations. SOC, and θ denotes the direction of electron momentum k in the graphene plane. The +(−) sign in ωx/y is for the To better understand these results, we now turn to valence (conduction) band, and thus a strong PIA SOC the model of Eqs. (7)-(8). We first examine the cases can induce electron-hole asymmetry in the spin relax- of graphene/hAlN and graphene/hGaN, for which all en- ation. Note that ωz is proportional to τ and thus has op- ergies in Table I are small, less than 100 µeV (for PIA posite sign in opposite valleys, as mentioned above. The SOC, the relevant energy scale is akλPIA ). If we consider second terms in the brackets, gk/z (k), represent higher- spin transport near room temperature (kB T ≈ 26 meV), order terms in the effective magnetic field. These are then the gk/z (k) terms in Eq. (8) are smeared out by the large near the graphene Dirac point (~vF k . ∆, λi ) and large thermal broadening and only the first terms in the decay as ∼1/k at higher energies. brackets are responsible for spin transport. The in-plane and out-of-plane spin relaxation rates are then given by B. Single-Sided Heterostructures 2  −1 −1 2 1 −1 τs,x = τs,y = τs,z λVZ τiv , + ~ 2 The numerical results for the graphene/hGaN het-  2   erostructure are shown as the symbols in Fig. 7. The −1 2 right inset shows the momentum relaxation time, which τs,z = ±λR + akλ+ PIA τp , (9) ~ ranges from 6 to 20 fs, and the intervalley scattering time, which ranges from 75 fs to 1 ps. The blue tri- the same as what was previously derived for angles (squares) in the main panel show the in-plane graphene/TMDC heterostructures, in which strong (out-of-plane) spin lifetime as a function of Fermi en- valley-Zeeman SOC coupled with intervalley scattering ergy. Lifetimes are between 1 and 10 ns, with a max- leads to large values of spin lifetime anisotropy83,84,101 . imum around the charge neutrality point and a slight Plugging the parameters from Table I and the values of electron-hole asymmetry. The spin lifetime anisotropy, τp and τiv from the numerical simulations into Eq. (9), defined as the ratio of out-of-plane to in-plane spin life- we obtain spin lifetimes for graphene/GaN that match time, ζ ≡ τs,z /τs,x , is shown as the blue circles in the very well with the numerical simulations, as shown by left inset. The anisotropy is nearly 1/2 over the entire the blue lines in Fig. 7. The spin lifetime anisotropy energy range, indicating that spin relaxation is driven by of ζ ≈ 1/2 also matches very well, confirming the the DP mechanism with a dominant Rashba+PIA SOC. dominance of Rashba+PIA over valley-Zeeman SOC. 9 The red lines in Fig. 7 show the expected spin lifetimes In contrast, the Rashba SOC in graphene/hAlN is for graphene/AlN. Here the Rashba SOC is somewhat somewhat weaker and has the opposite sign. The re- smaller, allowing the PIA SOC to play a more dominant sult is that the spin lifetime anisotropy is significantly role and resulting in significant electron-hole asymmetry. larger (note the log color scale for this system), and is In particular, for positive energies the PIA cancels out maximized at positive E-fields, where the magnitude of the Rashba SOC, which enhances the out-of-plane spin Rashba SOC is smallest. Although the magnitude of the lifetime. Meanwhile, the in-plane spin lifetime is sup- anisotropy is quite different, it is evident that in both pressed by an enhanced contribution from valley-Zeeman systems it is tunable via external field and doping. SOC, leading to large spin lifetime anisotropy. For neg- ative energies, PIA augments the Rashba SOC and the $QLVRWURS\ anisotropy is reduced.    *UK*D1 Now we consider the case of spin relaxation in (ILHOG>9QP@ graphene/hBN. In this system the staggered sublattice   potential ∆ is large, opening a band gap on the order of  the thermal broadening, and thus the higher-order terms in Eq. (8) cannot be ignored. These terms are compli-    cated expressions of the various parameters in the Hamil- tonian (see Appendix), but for the values given in Table   I they are well captured (to within 1%) by   ∆2 λVB + 2∆ε− ∆  *UK$O1 PIA εk (ILHOG>9QP@ gk (k) = 2 ,   ε∆ k ∆2 λVZ + 2∆λI ε∆   k gz (k) = 2 , (10)  ε∆ k −  where λVB = λR + akλ+ A B PIA , εPIA = ak(λPIA − λPIA )/2, A B   λI = (λI + λIp )/2 is the intrinsic or Kane-Mele SOC, and        ε∆k = ~vF k + (~vF k)2 + ∆2 . Equation (10) was derived (QHUJ\>H9@ for spin transport in the conduction band, but similar terms apply to the valence band by replacing λR → −λR FIG. 8. (Color online) Spin lifetime anisotropy in and ε∆ ∆ k → −εk . graphene/hGaN and graphene/hAlN heterostructures. Re- In the high-energy limit, the terms in Eq. (10) disap- sults for graphene/hGaN were calculated numerically, while pear and the spin lifetime is described by Eq. (9). How- those for graphene/hAlN were derived from Eq. (9). ever, as the Fermi energy approaches the band edges (k → 0), ωx/y → 0 and ωz → 2(λI + λVZ ). Thus, the out-of-plane spin lifetime diverges while the in-plane spin lifetime remains finite, leading to a giant spin lifetime D. Double-Sided Heterostructures anisotropy around the charge neutrality point. This be- havior is shown as the green lines in Fig. 7, and corrob- Beyond an external electric field, spin transport prop- orates previous predictions of spin lifetime anisotropy in erties may be further tuned by sandwiching graphene these systems5 . These results indicate that a large spin between different hXN layers. For example, by placing lifetime anisotropy should be visible around the charge hBN on one side of graphene and hGaN or hAlN on the neutrality point in graphene heterostructures with a large other side, one can realize a system that combines the staggered sublattice potential, even if the SOC is small. SOC properties of graphene/hGaN or graphene/hAlN, but with the large orbital gap of graphene/hBN. This can be seen by examining the final two rows of Table I. C. Electric Field Dependence Spin lifetimes for the hBN/graphene/hGaN and hBN/graphene/hAlN systems are shown in Fig. 9(a), As shown in Fig. 6 above, an external electric field with solid (dashed) lines denoting out-of-plane (in-plane) can be used to efficiently tune the various spin-orbit pa- lifetime. The spin lifetime anisotropy is shown in the in- rameters in the graphene/hGaN and graphene/hAlN het- set. As expected, each system exhibits a giant anisotropy erostructures. We plot the resulting impact on the spin around the charge neutrality point, owing to the orbital lifetime anisotropy in Fig. 8. For positive E-fields, the gap induced by the hBN layer. Away from the gap, anisotropy in graphene/hGaN remains around 1/2, as the the hBN/graphene/hGaN system behaves just like the Rashba SOC is the dominant term. For negative fields, single-sided graphene/hGaN system, with an anisotropy the Rashba SOC strength is reduced, valley-Zeeman SOC of 1/2 resulting from the dominant Rashba SOC. Mean- begins to play a role, and a modest anisotropy appears, while, the hBN/graphene/hAlN system exhibits large reaching a value of ζ ≈ 3 at high doping levels. anisotropy and strong electron-hole asymmetry, similar 10 to Fig. 7 above. Thus, these double-sided heterostruc- D  V tures appear to behave as a “superposition” of two single-  $QLVRWURS\  sided heterostructures, with hBN providing the orbital  V  gap and hGaN or hAlN providing the particular nature  of the SOC.  V    In Figs. 7 and 9(a) we have predicted spin lifetimes (QHUJ\>H9@ K%1*UK$O1 s ranging from 1 ns up to hundreds of ns. These large QV lifetimes arise from the small SOC in these systems, but as shown in Table I, the SOC can be further reduced QV K%1*UK*D1 by encapsulating graphene between two identical hXN layers. In this case, the valley-Zeeman and PIA SOC QV disappear, and the Rashba SOC is also completely or nearly eliminated. In this situation the effective magnetic E V field becomes V K$O1*UK$O1 ~ ωx = ±λR 1 + 2λI /ε∆I   k · sin(−θ), 2 PV s ~  ωy = ±λR 1 + 2λI /ε∆I   · cos(θ), K*D1*UK*D1 $QLVRWURS\ k 2  ~ ωz = ∆ 2λI /ε∆I   · τ, (11)  V k  2    p (QHUJ\>H9@ where ε∆Ik = ~vF k + (~vF k)2 + (∆ + λI )2 . In Fig. 9(b) QV we plot the spin lifetimes for graphene encapsulated by        hGaN or hAlN. Owing to the small SOC, all spin life- (QHUJ\>H9@ times are extremely large; around one millisecond for GaN and up to several seconds for AlN. In these sys- FIG. 9. (Color online) Predicted spin lifetime in tems, ωz strongly increases at low energies, leading to a (a) hBN/graphene/hXN and (b) hXN/graphene/hXN het- sharp decrease of the in-plane spin lifetime and thus a erostructures. The solid (dashed) lines show the out-of-plane large anisotropy around the charge neutrality point for (in-plane) spin lifetimes calculated from Eqs. (7)-(11). The the GaN system. For the AlN system, the out-of-plane insets show the corresponding spin lifetime anisotropy. spin lifetime is infinite due to the absence of Rashba SOC and thus is not pictured. It should be noted that here we have used the same the cases of graphene on hAlN and hGaN, we performed disorder as before, but in these encapsulated systems τp an interlayer distance study to find the energetically most is expected to be up to 100 times higher, corresponding favorable van der Waals gap, which is about 3.48 Å. Our to a mobility around 100,000 cm2 /Vs. This would reduce calculations, combined with a low energy model Hamilto- the spin lifetimes by a factor of 100. nian, reveal that graphene’s linear dispersion is well pre- Finally, we would also like to comment on the ex- served within the insulator band gap for all considered tremely large values of spin lifetime reported in Fig. 9, heterostructures, and the fitted model SOC parameters, as well as the diverging lifetimes and anisotropies seen describing the spin splitting of the Dirac states, are below in Figs. 7 and 8. These values are reached because we 100 µeV for optimized interlayer distances. From the in- are only considering the DP mechanism of spin relax- terlayer distance study, we also find that the fitted orbital ation, which leads to large spin lifetimes when SOC be- and SOC parameters strongly depend on the distance be- comes small. However, such values will never be reached, tween graphene and the substrate. Overall, for the non- as other fundamental sources of spin relaxation – such encapsulated structures, Rashba and PIA contributions as corrugations or phonons – are expected to take over dominate the spin splitting, while they are suppressed once DP relaxation becomes sufficiently weak. Therefore, in the encapsulated cases where intrinsic SOC terms of hXN/graphene/hBN or symmetric hXN/graphene/hXN maximum 10 µeV dominate. The model parameters are heterostructures may be suitable systems for exploring then used to estimate spin lifetimes in the encapsulated these other spin relaxation mechanisms, and their role as and non-encapsulated heterostructure systems. an ultimate upper bound on spin transport in graphene. With respect to spin transport and relaxation, sev- eral interesting and surprising features appear in these graphene/hXN systems. Depending on the relative VI. SUMMARY strength of the parameters in the Hamiltonian, spin transport can be quite conventional, can exhibit strong In this work, we have considered heterostructures of electron/hole asymmetry in the lifetimes and in the graphene and the monolayer hexagonal nitride insula- spin lifetime anisotropy, or can exhibit giant anisotropy tors, hBN, hAlN, and hGaN, from first principles. In only around the charge neutrality point. This enhanced 11 anisotropy around the charge neutrality point arises from as a strong staggered sublattice potential ∆ in combina- h iT √ tion with another source of SOC. This suggests that |ΨCB , ↑i = 1 τ e−iτ θ 0 0 / 2, the graphene/hBN system, in which sublattice symme- h iT √ try breaking can open a significant band gap, is the best |ΨCB , ↓i = 0 0 1 τ e−iτ θ / 2, candidate to observe this behavior. Meanwhile, graphene h iT √ encapsulated in GaN or AlN appears to be an excellent |ΨVB , ↑i = 1 −τ e−iτ θ 0 0 / 2, candidate for exploring the upper limits of spin transport iT √ in 2D van der Waals heterostructures80–82 . h |ΨVB , ↓i = 0 0 1 −τ e−iτ θ / 2, (A3) where the CB/VB states have eigenenergy ±ε0 = ±~vF k. Writing the Hamiltonian in the basis ACKNOWLEDGMENTS {|ΨCB , ↑i , |ΨCB , ↓i , |ΨVB , ↑i , |ΨVB , ↓i} yields " # HCC HCV This work was funded by the Deutsche Forschungsge- H≡ = (A4) meinschaft (DFG, German Research Foundation) SFB HVC HVV 1277 (Project-ID 314695032), DFG SPP 1666, DFG   ε0 + τ λVZ f ∗ λCB ∆ + τ λI f ∗ λ+ CV SPP 2244, and the European Unions Horizon 2020 re-  f λCB ε0 − τ λVZ f λ− ∆ − τ λI    CV search and innovation program under Grant No.881603  ∗ − ∗ , (Graphene Flagship). ICN2 is supported by the Severo  ∆ + τ λI f λCV −ε0 + τ λVZ f λVB  Ochoa program from Spanish MINECO (grant no. SEV- f λ+ CV ∆ − τ λI f λVB −ε0 − τ λVZ 2017-0706) and funded by the CERCA Programme / Generalitat de Catalunya. where λCB = −λR + akλ+ + PIA , λVB = λR + akλPIA , and λ± − CV = εPIA ± τ λR . In the heterostructures with hBN, ∆ is much larger than the spin-orbit parameters p and the eigenenergies of H are given by E ≈ ± ε20 + ∆2 . When graphene is sandwiched by two identical hXN lay- Appendix A: Derivation of Effective Magnetic Field ers, Table I indicates that VZ, Rashba, and PIA SOC p and the eigenenergies of H are then given by disappear, E ≈ ε20 + (∆ ± λI )2 for the conduction band and the In this Appendix we derive the complete form of the opposite sign for the valence band. In Figs. 7 and 9 we effective magnetic field, ω ~ , induced by SOC. In the ba- used the eigenenergy of the upper conduction band, but sis {|ΨA , ↑i , |ΨA , ↓i , |ΨB , ↑i , |ΨB , ↓i}, the Hamiltonian the results do not change appreciably if we use the lower of Eq. (1) can be written in matrix form as conduction band instead.   We now reduce the full Schrödinger equation, Hψ = ∆ + τ λA εk f ∗ εA ε+ Eψ, to expressions projected onto the conduction or va- I PIA R ε− lence band only,  ε∗k −∆ − τ λB f ∗ εB   H= I R PIA  , − ∗  f εA ∆ − τ λA  PIA εR I εk  Hψ = Eψ + ∗ B ∗ B εR f εPIA εk −∆ + τ λI ⇓ (A1) " #" # " # A,B A,B HCC HCV ψCB where εk = ~vF k · τ eiτ θ , ε±R = iλR (τ ± 1), ε PIA = akλ PIA , =E ψCB iθ and f = ie . From this we want a Hamiltonian projected HVC HVV ψVB ψVB onto the conduction or valence bands with the form ⇓ HCB ψCB = EψCB 0 1 HCB/VB = HCB/VB + ~~ ω · ~s HVB ψVB = EψVB 2 where " # " # ±ε0 0 ~ ωz ωx − iωy = + . (A2)  −1  0 ±ε0 2 ωx + iωy −ωz HCB ≡ HCC + HCV (E − HVV ) HVC   −1 HVB ≡ HVV + HVC (E − HCC ) HCV 0 where HCB/VB is the graphene Hamiltonian in the ab- (A5) sence of SOC (λ = 0) and sublattice symmetry breaking (∆ = 0), ω ~ is the effective magnetic field induced by these Using the expressions for HCC etc. from Eq. (A4) above, terms, and ~s are the spin Pauli matrices. and writing HCB in the form given in Eq. (A2), we arrive When ∆ = λ = 0, the eigenstates of H can be written at the effective magnetic field given in Eq. (8), with the 12 full expressions for gk and gz in the conduction band term, as is the case for heterostructures involving hBN, given in Eq. (A6) below. The expressions for the valence the effective magnetic field is well-approximated by the band are similar, with the replacements λVB → λCB , expressions in Eq. (10). λR → −λR , and ε0 → −ε0 . When ∆ is the dominant  2  λVB ∆2 − λ2I − λ2R + ε− + 2λVZ λI ε− −   PIA PIA − ∆λR + 2 (ε0 + E) ∆εPIA − λI λR gk (k) = 2 (ε0 + E) − λ2VZ − λ2VB  2  ∆2 + λ2I − λ2R − ε− + 2λVB λI ε− −    λVZ PIA PIA + ∆λR + 2 (ε0 + E) ∆λI + λR εPIA gz (k) = 2 (A6) (ε0 + E) − λ2VZ − λ2VB Appendix B: Density of States and Valence Charge the hBN valence band edge is much closer to the Dirac Density point than the conduction band edge, see Fig. 2(a). All in all, graphene is only weakly perturbed by the hBN. In order to find out more about the origin and the Nevertheless, already such tiny substrate contributions magnitude of the proximity-induced SOC in a specific strongly affect the SOC of the Dirac bands. graphene/hXN heterostructure, we calculate the density In Fig. 11, we show the calculated DOS and valence of states (DOS) and the valence charge density in an charge density for the graphene/hAlN heterostructure. energy window of ±100 meV around the Fermi level. For From the DOS, see Fig. 11(b), we can see that mainly this purpose, we use a denser k-point grid of 42 × 42 × 1 the pz -orbitals from N and Al contribute for the valence (36 × 36 × 1) for heterostructures with hBN and hGaN Dirac bands. In contrast, for the conduction band side, (hAlN). also N px + py give a strong contribution. The calculated In Fig. 10, we show the calculated DOS and valence valence charge density, see Fig. 11(a), again supports the charge density for the graphene/hBN heterostructure. DOS picture. Similarly to the hBN case, a clear deforma- From the DOS, see Fig. 10(b), we can see that mainly tion of the C pz -orbitals along the z-axis is absent, which the pz -orbitals from B contribute near the Dirac point, is consistent with the rather small Rashba SOC. Similar while additionally N px + py -orbitals are present. The to before, we also look at the percentage atomic com- calculated valence charge density, see Fig. 10(a), clearly position of the total DOS at ±100 meV away from the supports this picture, where a hybridization to B pz - Fermi level. At 100 meV (−100 meV), the total DOS of orbitals can be directly seen. The presented results are in the graphene/hAlN structure is composed of 98.898% C agreement with recent calculations of the graphene/hBN atoms, 0.083% Al atoms, and 1.018% N atoms (97.976% system5 , from where we also know that N and B atoms C atoms, 0.164% Al atoms, and 1.860% N atoms). In give a significant contribution to the intrinsic SOC pa- contrast to the hBN case, the total N contribution is en- rameters. The Rashba SOC should depend more on the hanced by a factor of 10. The asymmetry in the total deformation of C pz -orbitals along the z-axis74 due to the hAlN DOS contribution for positive and negative ener- hXN substrate. From the charge density plot, we cannot gies is again related to the band structure, see Fig. 2(b), directly see such a deformation, hence the small Rashba where the hAlN valence band edge is only about 700 meV SOC for the hBN case. below the Dirac point. In order to gain some more insights, we have also cal- In Fig. 12, we show the calculated DOS and valence culated the percentage atomic composition of the total charge density for the graphene/hGaN heterostructure. DOS at energies of ±100 meV away from the Fermi From the DOS, see Fig. 12(b), we find that mainly level. At 100 meV (−100 meV), the total DOS of the p-orbitals from N contribute near the Dirac point, the graphene/hBN structure is composed of 99.741% C about 10-times more than for the hBN case but sim- atoms, 0.189% B atoms, and 0.069% N atoms (99.622% ilar to the hAlN case. Moreover, there is some small C atoms, 0.291% B atoms, and 0.086% N atoms). Conse- contribution from Ga pz -orbitals in the chosen energy quently, the valence band Dirac states are affected more window around the Dirac point. The calculated valence strongly by the substrate than the conduction band Dirac charge density, see Fig. 12(a), clearly supports this pic- states. This could be related to the fact that the |λB I | pa- ture, where a hybridization to N pz -orbitals can be seen, rameter, being proportional to the valence band splitting which themselves couple to Ga p orbitals. In addition, at K, deviates much stronger from the intrinsic SOC of the C atoms which sit above the Ga atoms display a bare graphene74 of about 12 µeV, compared to the |λA I | clear deformation of the pz -orbitals along the z-axis (es- parameter, being proportional to the conduction band pecially looking at the green colored isovalues). This splitting at K. In addition, one should keep in mind that deformation should be responsible for the rather large 13 Rashba SOC in the hGaN case, compared to the other substrates. At 100 meV (−100 meV), the total DOS of the graphene/hGaN structure is composed of 99.140% C atoms, 0.261% Ga atoms, and 0.599% N atoms (98.944% C atoms, 0.127% Ga atoms, and 0.929% N atoms). FIG. 10. (Color online) (a) Calculated valence charge density of the graphene/hBN heterostructure considering only states in an energy window of ±100 meV around the Fermi level. The colors correspond to isovalues between 2 × 10−5 (blue) and 5×10−8 (red) e/Å3 , while the isolines range from 1×10−3 to 1 × 10−7 e/Å3 . (b) The atom and orbital resolved DOS. The DOS of B and N atoms is multiplied by a factor of 103 for comparative reasons, while for C atoms only the d-orbital contribution is multiplied by a factor of 10. ∗ 3 klaus.zollner@physik.uni-regensburg.de Calado, V. E. et al. 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(Color online) (a) Calculated valence charge density The colors correspond to isovalues between 1 × 10−5 (blue) of the graphene/hAlN heterostructure considering only states and 1×10−7 (red) e/Å3 , while the isolines range from 1×10−3 in an energy window of ±100 meV around the Fermi level. to 1 × 10−7 e/Å3 . (b) The atom and orbital resolved DOS. The colors correspond to isovalues between 2 × 10−5 (blue) The DOS of Ga (N) atoms is multiplied by a factor of 103 and 1×10−7 (red) e/Å3 , while the isolines range from 1×10−3 (102 ) for comparative reasons, while for C atoms only the to 1 × 10−7 e/Å3 . (b) The atom and orbital resolved DOS. d-orbital contribution is multiplied by a factor of 10. The DOS of Al (N) atoms is multiplied by a factor of 103 (102 ) for comparative reasons, while for C atoms only the d-orbital contribution is multiplied by a factor of 10. nitride van der Waals heterostructures. Appl. Phys. Lett. 105, 212405 (2014). 8 Gurram, M., Omar, S. & van Wees, B. J. 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