Spin Transport in Hydrogenated Graphene
David Soriano, Dinh Van Tuan, Simon M.-M. Dubois, Martin Gmitra, Aron W. Cummings, Denis Kochan, Frank Ortmann, Jean-Christophe Charlier, Jaroslav Fabian, Stephan Roche
aa r X i v : . [ c ond - m a t . m e s - h a ll ] A p r Spin Transport in Hydrogenated Graphene
David Soriano , Dinh Van Tuan, , , Simon M.-M. Dubois , Martin Gmitra , Aron W. Cummings ,Denis Kochan , Frank Ortmann, , , Jean-Christophe Charlier , Jaroslav Fabian and Stephan Roche, , ICN2 - Institut Catal`a de Nanoci`encia i Nanotecnologia, Campus UAB, 08193 Bellaterra (Barcelona), Spain Department of Physics, Universitat Aut´onoma de Barcelona, Campus UAB, 08193 Bellaterra, Spain Universite catholique de Louvain (UCL), Institute of Condensed Matter and Nanosciences (IMCN),Chemin des ´etoiles 8, 1348 Louvain-la-Neuve, Belgium Institute for Theoretical Physics, University of Regensburg, 93040 Regensburg, Germany Institute for Materials Science and Max Bergmann Center of Biomaterials,Technische Universit¨at Dresden, 01062 Dresden, Germany Dresden Center for Computational Materials Science,Technische Universit¨at Dresden, 01062 Dresden, Germany ICREA, Instituci´o Catalana de Recerca i Estudis Avanc¸ats, 08070 Barcelona, Spain (Dated: April 8, 2015)In this review we discuss the multifaceted problem of spin transport in hydrogenated graphene from a the-oretical perspective. The current experimental findings suggest that hydrogenation can either increase or de-crease spin lifetimes, which calls for clarification. We first discuss the spin-orbit coupling induced by local σ − π re-hybridization and sp C-H defect formation together with the formation of a local magnetic moment.First-principles calculations of hydrogenated graphene unravel the strong interplay of spin-orbit and exchangecouplings. The concept of magnetic scattering resonances, recently introduced [1] is revisited by describing thelocal magnetism through the self-consistent Hubbard model in the mean field approximation in the dilute limit,while spin relaxation lengths and transport times are computed using an efficient real space order N wavepacketpropagation method. Typical spin lifetimes on the order of 1 nanosecond are obtained for 1 ppm of hydrogenimpurities (corresponding to transport time about 50 ps), and the scaling of spin lifetimes with impurity densityis described by the Elliott-Yafet mechanism. This reinforces the statement that magnetism is the origin of thesubstantial spin polarization loss in the ultraclean graphene limit.
PACS numbers: 72.80.Vp, 73.63.-b, 73.22.Pr, 72.15.Lh, 61.48.Gh
Introduction .-Remarkable electronic and transport prop-erties of graphene (see for instance [2]) can be further tai-lored towards higher functionality through the use of chem-ical functionalization [3], irradiation (defect formation) [4]or structural patterning (such as creating a nanomesh super-lattice) [5]. Among the wealth of possible modifications,hydrogenation, ozonization or fluorination have demon-strated the tunability of graphene from a weakly disorderedsemimetal to a wide band-gap or Anderson insulator depend-ing on the nature and density of impurity atoms, varyingtypically from . to few percent [6–8]. Together withRaman spectroscopy which provides an estimate of defectdensity, analysis of the low-temperature conductance be-haviour and weak localization regime enable evaluation ofthe main transport length scales (such as the mean free pathand localization length) [8, 9]. Surprisingly, strongly hydro-genated graphene, with anomalously large Ioffe-Regel ratioof /k F ℓ e ∼ still exhibits quantum Hall effect featuresin the high magnetic field regime [10].Hydrogen defects are particularly interesting since, to-gether with a resonant scattering state created locally inspace, the breaking of the sublattice symmetry entails theformation of a zero-energy mode and a local magnetic mo-ment on the order of 1 Bohr magneton for an isolated hydro-gen adatom [11]. Meanwhile, the coupling between different induced magnetic moments is either ferromagnetic or anti-ferromagnetic, depending on whether the H-adatoms corre-spond to the same or to different sublattices of the graphenelattice, respectively. Theoretical calculations have reportedspecific magnetoresistance signals for specific long rangemagnetic ordering situations [12], which could be realizedby substrate-induced chemical reactivity as proposed in Ref.[13].Graphene spintronics has attracted a lot of attention sincethe pioneering demonstration that spin could be efficientlyinjected and propagated over long distances at room tem-perature [14]. This has opened an opportunity for the de-velopment of lateral spintronics [15–21], that would benefitfrom the unique features of graphene, such as a high mobil-ity and remarkable electronic and transport features like rel-ativistic energy dispersion, Klein tunneling phenomenon,...[23]. In Table 1, we compile the typical values obtained forcharge and spin transport in a variety of graphene devices,from graphene supported on silicon oxide or boron nitrideto suspended graphene. One observes that the variation ofmobility varies by up to two orders of magnitude, whereasspin lifetime seems difficult to relate to the quality of thematerial.Understanding spin transport in hydrogenated graphene is TABLE I: Charge/spin transport parameters in graphene on various substratesSubstrate µ (cm /V s ) D c ( m s − ) D s ( m s − ) τ s ( ps ) λ sf ( µm ) SiO [14] × × − × − ∼
100 1 . − SiO [15] − × − . × − ∼
500 2 . Suspended[20] × . − . . − . ∼
150 4 . SiC [17] × × − × − . − SiC [18] × − − − > hBN [19] × .
05 0 .
05 200 4 . hBN/G/hBN [21] . × . − . .
05 3000 12 a challenging and important issue. The underlying goal is touse a low enough level of hydrogenation to preserve a siz-able transport signal while also inducing local or long rangemagnetic ordering (ideally a ferromagnetic ground state),and evaluate how spin diffusion is affected by interactionbetween itinerant spins and local magnetic moments. Ulti-mately, one could envision spin devices such as spin valvesthat do not use ferromagnetic materials to inject and detectspins, but rather utilize the varying signal response of mag-netized graphene to manipulate the spin degree of freedomand engineer logic functions, a long-standing quest of spin-tronics [24].From that perspective, the first experimental result show-ing some interaction between spin diffusion and magneticmoments produced by H adatoms was reported in 2012 byMcCreary and coworkers [25]. Low-temperature spin trans-port measurements on graphene spin valves (T=15K) wereshown to exhibit a dip in the non-local spin signal as afunction of the external magnetic field. This effect wastunable with hydrogen density, and was related to spin re-laxation spin relaxation by exchange coupling with para-magnetic moments. Spin lifetimes were estimated to be τ s ∼ − ps in absence of hydrogen, and were slightlyenhanced upon increasing the H-adatom density up to n H ∼ . ), and accompanied by a reduction of charge mobilityby one order of magnitude to µ ∼ / Vs .A subsequent experiment also reported a surprising in-crease of spin lifetime with hydrogen density, with valuesenhanced by a factor of two after hydrogenation of the sam-ples for n H ∼ . [26]. The upscaling of τ s with hy-drogen density suggests a Dyakonov-Perel relaxation mech-anism [27]. In these experiments, the effect of magneticmoments is proposed to be strong enough to counteract theexpected increase of spin-orbit coupling, which would beexpected to produce shorter spin lifetimes with increasingH-adatom density.In 2013, J. Balakrishnan and coworkers [28] reported ex-perimental evidence for a room temperature spin Hall ef-fect (SHE) in weakly hydrogenated graphene, with non-local spin signals up to (orders of magnitude largerthan in metals). The non-local SHE revealed by Larmorspin-precession measurements was assigned to a colossalenhancement of the spin-orbit interaction induced by H- adatoms for density in the range n H ∼ . − . ,mobilities µ ∼ . − / Vs ) and spin lifetimeon the order of τ s ∼ ps. The spin-orbit interactionwas estimated to be about 2.5 meV, one order of magni-tude higher than ab initio calculations [29]. The observa-tion of the large SHE-signal was exclusively assigned to anenhancement of spin-orbit coupling by H-adatoms, whichlimits spin lifetimes. It is however not clear whether the pre-viously observed contribution of magnetic moments in spintransport [25, 26] remains marginal or not for the forma-tion and strength of the SHE signal, and which mechanism(Dyakonov-Perel [27] or Elliott-Yafet [34]) can explain thevariety of conflicting experimental data.In this context, it is of prime importance to clarify the spe-cific impact of H-adatoms on spin relaxation in graphene,and to quantify the relative contribution of spin-orbit cou-pling and magnetic effects. A first fundamental advance inthis direction has been made by Kochan, Gmitra and Fabian[1] who introduced a new relaxation mechanism in hydro-genated graphene driven by resonant scattering by magneticimpurities. By neglecting the spin-orbit coupling effect, spinlifetimes of τ s ∼ ps were estimated for 1 ppm Hydrogenat room temperature and for large contribution of electron-hole puddles. LOCAL MAGNETIC MOMENTS AND SPIN-ORBITCOUPLING FROM FIRST-PRINCIPLES
Here we remind the main ingredients and predictions ofthe model introduced in Ref.[1], and present new resultsobtained with a more generalized microscopic approach tostudy spin dynamics and relaxation effects induced by mag-netic moments. In the latter model, we describe the localmoment formation using the self-consistent Hubbard modelon hydrogenated graphene [12]. This model is shown tocorrectly reproduce ab initio results for spin splittings ob-tained for small supercells, whereas it provides, by a scalinganalysis, a more extended spreading of local magnetic mo-ments around the H impurity when compared with the an-alytic model used in [1]. Next, the Hubbard tight-bindingparameters are implemented into a real space wavepacketpropagation method which gives direct access to spin dy-namics and spin relaxation effects [30].
First-principles calculations of local magnetism and spin-orbitcoupling effects in hydrogenated graphene
When a hydrogen atom is chemisorbed, graphene locallyundergoes a structural deformation, breaking the sp sym-metry in favor of an sp -like hybridization. After ab ini-tio structural optimization, the hydrogenated carbon atom isfound to be slightly displaced out of the plane ( ∼ . ˚ A ),forming a C-H bond of length . ˚ A . As measured experi-mentally, the σ − π rehybridization induced by the hydrogenadatom comes with the formation of a local magnetic mo-ment of the order of 1 µ B . Recently, spin-unpolarized first-principles calculations suggested that the local sp distortionand its resulting pseudospin inversion asymmetry induce agiant enhancement of the spin-orbit coupling (SOC) [29]. Inthis section, spin-polarized first-principles calculations areused to investigate the interplay between the local magneticmoment and the enhanced spin-orbit interaction induced byhydrogen adatoms. Using unconstrained spinors to representthe one-particle wave-functions, the respective contributionsto the energy bands splitting of both SOC and electronic ex-change are computed. The spin textures associated with thelow energy electronic spectrum are also investigated.Hydrogenated graphene in the dilute limit is representedby a large supercell (5x5x1) containing a single hydrogendefect, leading to a hydrogen concentration of ∼ . Theout-of-plane dimension of the cell ensures a distance > ˚ A between neighboring graphene planes in order to avoid inter-action between periodically repeated images. The one parti-cle Hamiltonian is computed within the framework of non-collinear spin-polarized density functional theory as imple-mented in the Vienna Ab initio Simulation Package (VASP)[31]. The projector augmented wave method is used to ex-pand the one-particle wave-functions up to an energy cut-off of eV [32]. The eigenstates of the self-consistentHamiltonian are populated according to an electronic tem-perature of 300K. Electronic exchange and correlation aretreated within the generalized gradient approximation bymeans of the PBE functional [33]. Integrals over the Bril-louin zone are performed using a 6x6x1 Monkhorst-Packgrid of k-points. The geometry is fully optimized until re-maining atomic forces and stresses are lower than . eV/ ˚ A and . eV/ ˚ A respectively. Various magnetization axes, ei-ther parallel or perpendicular to the graphene plane, havebeen considered. The different configurations are energeti-cally equivalent, i.e. the computed total energies vary by lessthan 1 µ eV. In what follows, we focus on the most symmet-rical and experimentally relevant configuration where themagnetization axis is chosen as the direction perpendicularto the graphene plane.The computed low-energy band structure is depicted in FIG. 1: (a,b)
Ab initio band structures of hydrogenated graphenein the dilute limit ( ∼ ) including spin-orbit coupling calcu-lated within both (a) spin-polarized and (b) spin-unpolarized frame-works. The color scale is associated with the in-plane componentof the spin expectation values. Hence, the blue and red extrema ofthe color scale correspond to eigenstates whose spin is parallel andperpendicular to the graphene plane. Black dots are determined bysymmetry. (c,d) In plane projection of the spin texture of the high-est valence band in both the spin-polarized (c) and -unpolarized (d)frameworks. The arrows indicate the in-plane projections of thespin directions (rescaled for eye convenience). The color scale in-dicates the actual values of the in-plane projections. Red and bluearrows respectively correspond to spin momenta that are paralleland perpendicular to the graphene plane. Fig.1 (a). The hydrogen impurity opens an energy gapof . eV and produces two nearly-flat bands around theFermi energy, accounting for the computed total magneticmoment of . µ B . In this configuration, the spin degen-eracy is broken and the energy splitting of the bands origi-nates from both electronic exchange and spin-orbit interac-tion. For the sake of comparison, the low energy spectrumcomputed within a spin-unpolarized framework is also re-ported in Fig.1 (b). We note that spin-unpolarized meansthat the density matrix is constrained to be diagonal withsame values for both spin-directions. In this case, the energysplitting of the bands only originates from the spin orbit in-teraction. At the scale of Fig.1(b), the splitting is not visibleto the eye.The computed energy splittings corresponding to the low-energy bands are reported in Fig.2 (a,c). The energy splittingcomputed within the spin unpolarized framework [see Fig.2(c)] is in perfect agreement with previous estimates [29]. Inorder to clarify the impact of the spin-orbit interaction, theSOC-induced splitting ( Λ SOC ) here defined as the differencebetween the energy splittings computed with and without thespin-orbit interaction is also reported in Fig.2 (b). Λ SOC isshown to be at least two orders of magnitude smaller than theenergy splitting associated with the stabilization of the lo-cal magnetic moment by the electronic exchange (Fig.2 (a)).This highlights the prevailing role of electronic exchange inruling the low-energy spectrum in the presence of local mag-netic moments. The comparison of Fig.2 (b) and (c) furtheremphasizes the interplay between spin-orbit and exchangecouplings.The reduction of SOC-induced spin splitting due to ex-change coupling is explained by considering a simple ef-fective Hamiltonian. Consider electrons with momentum k along x and energy ε k , experiencing Rashba SOC α whichis in general momentum dependent. If the electrons also feelan exchange coupling ∆ with a magnetic moment orientedalong z , as is our case, the Hamiltonian is H = ε k + ∆ σ z + α ( k ) σ y . (1)The energy splitting of the eigenvalues of this operator is p ∆ + α . (2)In the absence of exchange, the splitting is linear in α , sinceRashba SOC splits two degenerate levels. In the presenceof exchange, and for realistic ∆ ≫ α , as in our case, theRashba contribution to the splitting is only second order in α ( α ∆ ). This rough estimate gives for SOC of 100 µ eV and ∆ of 0.1 eV, a Rashba coupling contribution to the splittingof approximately 0.1 µ eV, significantly below what is ob-served in Fig.2 (b). It is then likely that the difference in thecalculated exchange coupling with and without SOC is dueto the Rashba dependence of the exchange itself. This couldbe linear with α , with a numerical factor. The understand-ing of this dependence in terms of simple phenomenologicalmodels is still elusive and is currently the object of furtherinvestigation.The spin texture of the valence band is illustrated in Fig.1(c). When the electronic density matrix is constrained to bespin-unpolarized, the frustration of the electronic exchange”artificially” leads to a highly non-collinear spin texture en-tirely ruled by SOC (see Fig.1 (d)). On the contrary, in theunconstrained framework, the stabilization of the magneticmoment dominates the energetics of the low energy spec-trum and the spin texture is found to be nearly collinear withthe spin-orbit interaction only accounting for ∼ . of thedeviation from collinearity (Fig.1 (c)). While the local rehy-bridization of graphene induces an enhancement of the spin-orbit interaction, the magnetic properties of graphene as wellas its spin texture are quasi-exclusively ruled by the elec-tronic exchange in the presence of hydrogen-induced local Γ M K
Γ Γ
M K
Γ Γ
M K ΓΓ M K
Γ Γ
M K
Γ Γ
M K ΓΓ M K
Γ Γ
M K
Γ Γ
M K
Γ Γ
M K Γ E n e r g y ( e V ) E n e r g y ( e V ) E n e r g y ( e V ) E n e r g y ( μ e V ) E n e r g y ( μ e V ) E n e r g y ( μ e V ) E n e r g y ( μ e V ) E n e r g y ( μ e V ) E n e r g y ( μ e V ) Total splitting SOC induced ( Λ SOC ) Total splitting (spin polarized) (spin polarized) (spin unpolarized) (a) (b) (c)
FIG. 2: Energy splittings of the conduction (red curves), impu-rity (blue curves) and valence (green curves) bands along high-symmetry lines in reciprocal space. (a) Total energy splitting withinthe spin -polarized framework. (b) Contribution of SOC to the en-ergy splitting within the spin polarized framework (i.e. differencebetween the energy splittings computed respectively with and with-out accounting for the contribution of the spin-orbit interaction toHamiltonian). (c) Total energy splitting within the spin-unpolarizedframework (i.e. only arising from SOC as the electronic density isconstrained to be spin-up polarized). magnetic moments. Hence, the contribution of the spin-orbitinteraction to the Hamiltonian is neglected in the followingdiscussion on the spin lifetimes in hydrogenated graphene.
ONE-ORBITAL MEAN-FIELD HUBBARDAPPROXIMATION
The electronic structure of graphene is modeled by a nearest-neighbor tight-binding model with a single p z -orbital persite. When a H atom is adsorbed on top of a carbon (C)atom, the sp -symmetry is locally broken, and the electronfrom the C p z orbital is removed from the π bands to forma σ bond with the H atom (Fig.3 (a)). Figure 3(b) shows a5x5 supercell where the absence of a p z -orbital in the cen-ter of the green region (yellow site) represents hydrogen ad-sorption. To remove the p z orbital, we use a sufficientlylarge on-site potential V ∞ ≈ eV. To properly describemagnetism in hydrogenated graphene, we introduce on-siteCoulomb repulsion between electrons with opposite spins bymeans of the Hubbard model in its mean-field approximation H = − γ X h i , j i ,σ c † i ,σ c j ,σ + U X i ( n i , ↑ h n i , ↓ i + n i , ↓ h n i , ↑ i ) , (3)where t is the first-neighbors hopping term, c † i ,σ ( c j ,σ ) is thecreation (annihilation) operator in the lattice site i ( j ) withspin σ , U is the on-site Coulomb repulsion, and h n i , ↓ i , h n i , ↑ i are the converged expectation values of the occupation num-bers for spin-down and spin-up electrons, respectively. Theratio U/γ = 1 has been chosen to accurately reproducefirst-principles calculations. FIG. 3: (a) Schematic representation of the sp hybridizationbreakdown upon hydrogen adsorption, equivalent to the creationof a single vacancy. (b) Single-vacancy 5x5 supercell used inthe mean-field Hubbard calculations with periodic boundary con-ditions. The p z orbital at the center of the vacancy (yellow siteinside the green region) is removed by adding a large on-site po-tential ( V ∞ ) to simulate hydrogen adsorption. Hydrogen defects in graphene may induce sublattice sym-metry breaking leading to the appearance of zero-energymodes in the density of states (DOS), as predicted by Inui et al. [40], which are mainly localized around the impu-rities [35, 36]. These zero-energy states become spin-polarized upon switching on the Coulomb repulsion, lead-ing to semi-local S = 1 / magnetic moments with astaggered spin density primarily localized on one sublat-tice [11, 37, 38]. For a finite concentration of defects, thelong range ordering of magnetic moments is dictated bythe type of sublattice functionalization, being co-polarizedor ferromagnetic (FM) for the same sublattice and counter-polarized or antiferromagnetic (AFM) otherwise. The totalspin S of the macroscopic ground state is given by the ex-cess of magnetic moments on one chosen sublattice [39],although S = 0 is the most likely value on simple statisticalgrounds (equal H occupation of both sublattices). Here weconsider the dilute limit so that long range magnetic statesare neglected and we get local paramagnetic (PM) impuri-ties.In order to reach a pure paramagnetic state, we have stud-ied the evolution of the local magnetic moments m i =1 / n i, ↑ − n i, ↓ ) with the varying supercell size (Fig.4). Inparticular, Fig.4 (left panel) shows the values of the magneticmoment only at nearest neighbor sites of the defect m nn . FIG. 4: Evolution of magnetic moments on first neighbor atom forincreasing distance between impurities ( d H − H ). The value con-verges apparently for d H − H ≥ . ˚A where long range mag-netic interactions start to vanish precluding the formation of FM orAFM ordering and leading to isolated paramagnetic defects. Onthe right side of the figure, we show the magnetic moments m i corresponding to each lattice site for different supercell sizes. As the supercell size is increased and the defects move awayfrom each other, m nn decreases quickly and converge for asupercell size 14x14, which corresponds to . % of ad-sorbed H and d H − H = 33 . ˚A. This decay is related to theoverlap between wavefunctions corresponding to neighbor-ing defects. For strongly overlapping states (large concen-tration of H), long-range interactions induce a FM state be-tween magnetic moments located in neighboring cells whichincreases the values of m i . When the overlap is very small(diluted limit), long-range interactions have no influence onthe local magnetism leading to almost constant m i valuesaround the defect (PM state).Figure 5(a) shows the band structure of a single hydro-genated 28x28 graphene supercell, corresponding to . of adsorbed H, calculated using the mean-field Hubbardmodel (Eq. 3). The splitting ( ∆ s ) of the mid-gap stateformed during H adsorption is plotted in Fig. 5(b) for dif-ferent supercell sizes at different k -points. At the K point ∆ s decays as ∝ d − . H − H , while at the Γ and M-points ∆ s ∝ d − H − H . These results confirm the existence of very smallsplittings at tiny concentrations of H (of the order of 1ppm).In order to compare the values of the energy splitting ofspin polarized bands obtained using the mean-field Hubbardmodel with published DFT results [1], we have plotted ∆ s along M-K- Γ path in the inset of Fig. 5(b). Bottom, mid-dle and top panels correspond to the valence, mid-gap andconduction bands respectively. Although DFT results for a5x5 supercell (dashed lines) show a slightly higher splittingof the spin polarized bands with respect to our results, thismethodology nicely capture the main physics of magneticresonances induced by H adatoms in graphene. FIG. 5: (a) Band structure corresponding to . of H adsorbed(28x28 supercell). The spin degeneracy is broken at the Fermi en-ergy ( E = 0 ) due to the formation of local magnetic fields aroundimpurities. (b) Evolution of ∆ s at three symmetry points K, Γ and M, for increasing values of the distance between impurities( d H − H ). The inset shows the evolution of ∆ s along the M-K- Γ path in the hexagonal Brillouin zone of the valence (bottom), impu-rity (center) and conduction (top) bands. Dashed lines correspondto DFT results. SPIN DYNAMICS AND RELAXATION
We now investigate the spin dynamics and relaxation phe-nomena in hydrogenated graphene by comparing two com-plementary theoretical approaches and contrasting our re-sults with state-of-the-art experimental data.
Single impurity limit
First, we consider the magnetic scattering problem in thesingle impurity limit, which can be simplified by consider-ing only the spin-flip processes stemming from the exchangecoupling on the resonant scatterer (H atom) site. This wasdone in details in Ref. [1] on the basis of the Hamiltonian ˆ H = ˆ H + ˆ H S which involves the usual p z -orbital Hamil-tonian for the pristine graphene ˆ H = − γ P h ij i c + i c j together with the term ˆ H S describing graphene-hydrogenchemisorption including the interaction of electron spin ˆ s h and impurity moment ˆ S (exchange coupling) ˆ H S = X σ ǫ h h † σ h σ + T h h + σ c C H ,σ + c + C H ,σ h σ i − J ˆ s h · ˆ S (4)Here h + σ ( h σ ) and c + m,σ ( c m,σ ) are fermionic operators creat-ing (annihilating) electron with spin σ on the hydrogen andgraphene carbon site m , respectively. The orbital chemisorp-tion parameters entering the Hamiltonian ˆ H S are extracted from the ab-initio calculation in Refs. [1, 42]—the hydro-gen on-site energy ǫ h = 0 . eV and graphene-hydrogenhybridization T = 7 . eV. The exchange term − J ˆ s h · ˆ S de-scribes an effective interaction of the local magnetic moment ˆ S induced on the hydrogen site with an itinerant electronspin ˆ s h when this hops into the hydrogen | h i ≡ h + | i -levelthat hybridizes with the graphene host. Both, ˆ S and ˆ s h arevectors of spin -operators (in our definition without con-ventional ¯ h factor) and J is a constant with the dimensionof energy. For the exchange we take J = − . sincethis value is consistent with a more detailed parametrizationof the magnetic impurity model as discussed in Ref. [1]. Infact the precise value and the sign of J are not important aslong as | J | is greater that the orbital resonance width (for thecase of hydrogen the resonance width is meV). In the inde-pendent electron-impurity picture (we do not discuss Kondophysics), the exchange term − J ˆ s h · ˆ S can be diagonalizedintroducing the singlet ( ℓ = 0 ) and triplet ( ℓ = 1 ) compositespin states | ℓ, m ℓ i ( m ℓ = − ℓ, . . . , ℓ ). Transforming the ini-tial Hamiltonian ˆ H S , Eq. (4), to the new spin basis we arriveat ˆ H S = h + h ⊗ X ℓ,m ℓ (cid:2) ǫ h − J (4 ℓ − (cid:3) | ℓ, m ℓ ih ℓ, m ℓ | + T (cid:2) h + c C H + c + C H h (cid:3) ⊗ X ℓ,m ℓ | ℓ, m ℓ ih ℓ, m ℓ | . (5)Down-folding the hydrogen | h i -state by means of theL¨owdin transformation we get for each spin component ℓ, m ℓ an independent effective delta-function problem ˆ H eff S ( E ) = X ℓ,m ℓ V ℓ ( E ) c + C H c C H ⊗ | ℓ, m ℓ ih ℓ, m ℓ | , (6)located on the hydrogenated carbon site C H with the energydependent coupling V ℓ ( E ) = T E − ǫ h + J (4 ℓ − . (7)From here it is straightforward to compute T-matrix ele-ments for Bloch states | κ i and | κ ′ i of the unperturbedgraphene (to shorthand the notation κ comprises band in-dex and crystal momentum). Assuming the Bloch states arenormalized to graphene unit cell the result is as follows T κ ,ℓ,m ℓ | κ ′ ,ℓ ′ ,m ℓ ′ ( E ) = δ ℓℓ ′ δ m ℓ m ℓ ′ V ℓ ( E )1 − V ℓ ( E ) G ( E ) , (8)where V ℓ ( E ) is given by Eq. (7) and G ( E ) is Green’s func-tion per site and spin for the unperturbed graphene, i.e., G ( E ) ≃ EW ln (cid:12)(cid:12)(cid:12) E W − E (cid:12)(cid:12)(cid:12) − iπ | E | W Θ( W − | E | ) . (9)The above Green’s function is valid near the graphene chargeneutrality point in the energy window from -1 eV to 1 eV,where the linearized bandwidth W = p √ πγ ≃ eV.For practical reasons we need relaxation rates /τ σσ ′ ( E ) that characterize spin-conserving ( σ = σ ′ ) and spin-flipping( σ = σ ′ ) processes in graphene at the given Fermi energy E in presence of magnetic active impurities. For that we takeinto account all scattering processes | κ , σ, Σ i → | κ ′ , σ ′ , Σ ′ i at the given energy E ≡ E ( κ ) with the requested incom-ing and outgoing electron (hole) spins σ and σ ′ and allowedimpurity spins Σ and Σ ′ (a charge carrier in graphene flipsits spin only if the magnetic moment does the same to con-serve the total angular momentum). To get /τ σσ ′ ( E ) westart with the transition rate W κ σ Σ | κ ′ σ ′ Σ ′ and trace out the Σ -spin degrees of freedom corresponding to magnetic mo-ment ˆ S induced on the hydrogen site, then we integrate overall outgoing momenta | κ ′ i and finally we average the resultover all incoming states | κ i at the given Fermi energy E .This can be done because the angular dependence of the T-matrix is trivial, it does not depend on the angle of κ neitherof κ ′ . Assuming the distribution of magnetic moments isdilute and they are in average spin-unpolarized (e.g. by in-teraction with phonons) the result is as follows τ σσ ′ ( E ) = η π ¯ h ν ( E ) f σσ ′ h V ( E )1 − V ( E ) G ( E ) , V ( E )1 − V ( E ) G ( E ) i . (10)Here η is the concentration of hydrogen impurities per car-bon atom, ν ( E ) = − π Im G ( E ) is the graphene densityof states per atom and spin, V ℓ ( E ) for ℓ = 0 , is given byEq. (7) and function f σσ ′ [ x, y ] is defined by f σσ ′ [ x, y ] = 12 δ σσ ′ (cid:12)(cid:12) x (cid:12)(cid:12) + 18 (cid:12)(cid:12) x + ( σ · σ ′ ) y (cid:12)(cid:12) . (11)The function f σσ ′ originates from the decomposition of theactual spin states | σ, Σ i and | σ ′ , Σ ′ i with respect to thesinglet-triplet basis | ℓ, m ℓ i after the tracing out the spinstates of the induced magnetic moment. The symbol σ · σ ′ entering its definition equals 1 (-1) for the parallel (antipar-allel) spin alignments and (cid:12)(cid:12) (cid:12)(cid:12) stands for the absolute value.Knowing the partial rates /τ σσ ′ ( E ) we can define the spin-relaxation rate /τ s ( E ) = 1 /τ ↓↑ ( E ) + 1 /τ ↑↓ ( E ) and momentum-relaxation rate /τ ( E ) = 1 /τ ↑↑ ( E )+ 1 /τ ↑↓ ( E ) at zero temperature as functions of Fermi energy. FromEq. (10) we immediately see that both /τ s and /τ shouldgo to zero at the charge neutrality point since there are nostates that can participate in scattering, ν (0) = 0 . Secondly,the denominators − V ( E ) G ( E ) and − V ( E ) G ( E ) that enter the function f σσ ′ can minimize at certain energies E and E and this would manifests as two sharp peaks (sin-glet and triplet one) in the spin and momentum relaxationrates. Both mentioned features are clearly seen in Fig. 7.To account for finite temperature effects one should ther-mally broaden /τ s implementing rate equations for thegraphene charge-carriers that obey Fermi-Dirac statistics.The concise formula for the spin-relaxation rate at finite tem- perature T then becomes τ s ( E, T ) = P k (cid:2) − ∂ ǫ k F( ǫ k , E, T ) (cid:3) /τ s ( ǫ k ) P k (cid:2) − ∂ ǫ k F( ǫ k , E, T ) (cid:3) , (12)where F( ǫ k , E, T ) is the equilibrium Fermi-Dirac distri-bution at temperature T and Fermi energy E . Experi-ments show that charged impurities and electron-hole pud-dles within the sample still further affect system’s Fermi en-ergy. To account for all that effects one usually convolves /τ s ( E, T ) by the Gaussian kernel with standard deviation σ b , τ s ( E, T ) = 12 √ πσ b ∞ Z −∞ d ǫ τ s ( ǫ, T ) exp (cid:18) − ( E − ǫ ) σ b (cid:19) . (13)Depending on the sample quality the typical value of σ b tobe taken for best fits is about 70-110 meV (see discussionnext section). Tight-binding Model of hydrogenated graphene
In our second approach to spin relaxation, we study thespin dynamics using the following tight-binding Hamilto-nian, ˆ H S r = ǫ h X m h + m h m + T X h mi i h + m c i (14) + X i J i c + i σ z c i which includes the self-consistent Hubbard terms describ-ing isolated magnetic moments (dilute limit), and a ran-dom distribution of H-adatoms in the ppm range. The longrange nature of magnetic moment induced by hydrogen isinvolved in this Hamiltonian by considering up to ninth-nearest-neighbor exchange coupling term J i . Spin-orbitcoupling is found to yield negligible corrections to the re-sults, so it is neglected in the following. The spin dynamicsare investigated by computing the time-dependence of thespin polarization defined by [30] S ( E, t ) = Tr h δ ( E − ˆ H ) ˆs ( t ) i Tr h δ ( E − ˆ H ) i where ˆs ( t ) = e i ˆ Ht ¯ h ˆs e − i ˆ Ht ¯ h . Using random phase states toperform the trace efficiently, we get S ( E, t ) = 12Ω ρ ( E ) h ψ ( t ) | δ ( E − ˆ H ) ˆs + ˆs δ ( E − ˆ H ) | ψ ( t ) i and the initial wavepacket can be prepared in an arbitraryspin polarization as | Ψ RP i = 1 √ N N X i =1 (cid:18) cos (cid:0) θ i (cid:1) e iΦ i sin (cid:0) θ i (cid:1)(cid:19) e iπα i | i i In what follows, the wavepackets are prepared (at t = 0 )with an in-plane spin polarization and their time-dependentpropagation and S x ( E, t ) are evaluated applying the evolu-tion operator (Schr¨odinger equation) to the wavepackets . Inabsence of spin-orbit interaction, the spin dynamics is influ-enced by the existence of local magnetism, with H-relatedmagnetic moments pointing out-of-plane.Fig.6 shows the time-evolution of the spin polarization atthree different energies, namely the Dirac point, an energyclose to the expected resonance and some high energy value(see inset). The curves first exhibit an sudden drop (espe-cially for the Dirac point) followed by an exponential decay,which dictates the values of the spin relaxation times τ s us-ing S x ( t ) = S x ( t ) e − t/τ s (fitting the numerical results from t = 75 ps). The initial fast decay of S x ( t ) is understoodas follows. We study the propagation of wavepackets which(for computational efficiency) are at time t = 0 in a randomphase state, that is extended through the whole system. Forsmooth disorder, this method well captures the main trans-port length scales as described in Ref.[2]. Here however,the hydrogen adatoms (or vacancies) produce strong mag-netic moments localized around the impurity. This intro-duces some transient decay of the initial spin polarizationwhich is not representative of the long time exponential be-havior. Spin relaxation times in hydrogenated graphene
In Fig.7, the inverse spin relaxation times obtained forthe two models are superimposed. Within the single im-purity approximation, two sharp magnetic resonances areclearly formed (for singlet and triplet states), at which /τ s is maximum at zero temperature (green dashed lines). Thisapproximation of τ s however diverges close to the Diracpoint, which disagrees with most low-temperature exper-imental data. Room temperature spin-relaxation experi-mental data can however be reproduced using thermal andcharge-puddles broadenings. A choice of 0.36 ppm of H-impurities together with a finite temperature broadening(300 K) and charge density fluctuations of 110 meV arenecessary (blue dashed line) to reproduce experimental data(black filled circles) obtained for high-quality graphene spin-valve devices. [26].The inverse spin lifetimes obtained by the real space spinpropagation method are shown in Fig.7 for a H-density of 1ppm (black solid line), 2 ppm (red solid line) and 15 ppm(blue solid line). The presence of the two resonances is con-firmed by this calculation. At the resonances, the inverse S x ( t ) E DP E r1 E s* Fit. E DP Fit. E r1 Fit. E * -400 -200 0 200 400E (meV)00.0050.01 DO S ( e V - Å - ) E DP E r1 E * FIG. 6: (Main frame) time-dependent polarization for initial in-plane spin polarized wavepackets at selected energies (see inset).Numerical fits using an exponential damping are shown (dashedlines) on top of numerical simulations start from elapsed propaga-tion time t > ps. Inset: total density of state for the case of 1ppmof hydrogen on graphene. spin lifetime is greater than in the single impurity approx-imation, which is likely due to the different natures of theadatom models. In the single-impurity adatom model the themagnetic moment is strongly localized and the resonance issharp, while in the Hubbard model we have an effective de-fect model with more delocalized magnetism is real space.This broadening reduces the “residency time” of the elec-trons on the magnetic virtual bound state (as pictured in theinset of Fig.7-inset), and thus yields longer spin lifetimes.Indeed for 0.36 ppm of H-impurities, the two models differto three orders of magnitude in spin lifetime. By further con-trasting the real-space spin results to the experimental data(black dots in Fig.7), we extrapolate that a density of 10-20ppm of H-adatoms can reproduce the range of experimentalvalues of the graphene samples measured in Ref.[26].To discuss the relaxation mechanism, we compute the mo-mentum relaxation time τ p first from the sum of spin con-serving rate and spin-flip rate, which both can be calcu-lated by transforming the singlet and triplet T-matrix am-plitudes via composite spin states of electron and impu-rity [1]. The second approach uses the full Hamiltonian ˆ H S and the time dependence of the diffusion coefficient D ( E, t ) [9]. For the hydrogen coverage as low as . ,we still can observe the saturation of the diffusion coefficient D ( E, t ) −→ D max ( E ) , allowing us to extract τ p as τ p ( E ) = D max ( E )4 v ( E ) (15)where v ( E ) is the pristine graphene velocity. We use theFermi golden rule and the expected scaling of τ p ( E ) ∼ /n H , to extrapolate the values for each ppm concentration FIG. 7: (Main frame) Energy-dependence of τ s derived from theT-matrix of the single impurity model at zero temperature (greendashed lines); and at 300 K broadened by puddles with energy fluc-tuations of 110 meV (blue dashed lines). Same quantity obtainedfrom the real space spin propagation method for 1 ppm (solid blackline), 2 ppm (solid red line) and 15 ppm (solid blue line) of Hadatoms. Inset: schematic of the magnetic resonance process atthe origin of enhanced spin relaxation. (see Fig.8). The result is given in Fig.8 (green solid lines),and shows a weak energy dependence, with slight increaseclose to the Dirac point, with values in the range of 60-70 psfor 1 ppm of hydrogen impurities. In Fig.8(inset), τ p for thetwo models are reported showing some discrepancy close tothe Dirac point, as expected from the approximations made.The values match well at high enough energy.We observe that τ s ∼ τ p , while the scaling of τ s withimpurity density clearly manifests an Elliott-Yafet spin re-laxation mechanism, as predicted for such type of impuri-ties [41]. In some experiments, τ s is found to increase withhydrogen concentration, suggesting differently a Dyakonov-Perel relaxation mechanism [26], whereas others report onthe opposite trend associated to the Elliott-Yafet mechanism[28]. The origin of such inconsistency remains obscure atthat point but could stem (in particular) from the segrega-tion of H-adatoms and the alteration of fundamental effectsinduced by magnetic resonances.In conclusion, we investigated the impact of hydro-gen adatoms on charge and spin transport in transport ingraphene in the dilute limit (down to the ppm limit), forwhich no long range magnetic ordering develops. The im-portance of magnetic resonances as a new spin relaxationmechanism, as pioneered in Ref.[1] has been consolidatedand further quantified using extended models of more delo-calized magnetism, described within the Hubbard Hamilto-nian in the mean field approximation. Using efficient realspace order N wavepacket propagation methods, spin re-laxation times in the nanosecond regime were obtained for -300 -200 -100 0 100 200 300E (meV)0500100015002000 τ s , τ p ( p s ) τ s (1ppm) τ s (2ppm)10 ×τ p (1ppm) -200 -100 0 100 200E (meV)1101001000 τ p ( p s ) E DP E * E r1 FIG. 8: Main frame: Energy-dependence of spin (solid lines) andrescaled momentum (dashed line) relaxation times obtained fromthe real space wavepacket propagation method. Inset: momentumrelaxation time derived the from single impurity model [1] (solidline) and from the diffusion coefficient behavior (dashed greenline).
ACKNOWLEDGEMENTS
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