Magnonics in collinear magnetic insulating systems
MMagnonics in collinear magnetic insulating systems
B. Flebus Department of Physics, Boston College, 140 Commonwealth Avenue Chestnut Hill, MA 02467 (Dated: 18 February 2021)
In the last decades, collinear magnetic insulating systems have emerged as promising energy-saving information carri-ers. Their elementary collective spin excitations, i.e., magnons, can propagate for long distances bypassing the Jouleheating effects that arise from electron scattering in metal-based devices. This tutorial article provides an introductionto theoretical and experimental advances in the study of magnonics in collinear magnetic insulating systems. We startby outlining the quantum theory of spin waves in ferromagnetic and antiferromagnetic systems and we discuss theirquantum statistics. We review the phenomenology of spin and heat transport of the coupled coherent and incoherentspin dynamics and the interplay between magnetic excitations and lattice degrees of freedom. Finally, we introducethe reader to the key ingredients of two experimental probes of magnetization dynamics, spin transport and NV-centerrelaxometry setups, and discuss experimental findings relevant to the outlined theory.
I. INTRODUCTION
The possibility of using the electron’s spin as a new de-gree of freedom for transmitting information has received sig-nificant attention in the last decades [1,2]. Of particular in-terest for novel device applications are magnetic insulatingsystems [3,4]: solids in which electrons are “frozen” in theiratomic positions while a pure spin current, i.e., a flow of spinangular momentum, can be transmitted through a wave-likecollective motion of the electrons’ spins. Transmitting sig-nals via spin flows in insulators has the potential to circum-vent the heating and energy loss that is generated when theelectrons collide, e.g., with impurities, as they move withina crystal lattice. The discovery of the spin Hall effect (SHE)and the inverse spin Hall effect (ISHE) [5–11], which converta charge current into a spin current and vice versa , has furtheradvanced the field by allowing electrical injection and detec-tion of spin currents in heterostructures comprised of mag-netically ordered insulators interfaced with a normal metal.More recently, Uchida and co-authors showed that a pure spinflow can be generated in a magnetic insulator not only viaelectrical injection, but by heat gradients as well via the spinSeebeck effect (SSE) [12], suggesting the possibility of con-verting waste heat into spin signals and, thus, opening up in-triguing prospects for “greener” information technology [13].Experimental observations of electrically- and thermally-generated spin transport have confirmed the potential ofmagnetic insulating systems as long-range information carri-ers [14–17].There is a plethora of magnetic insulating systems,which display ground-state spin configurations ranging fromcollinear to non-collinear depending on the intrinsic spin-spininteractions. Non-collinear configurations stem from spincouplings that can endow the system with a non-trivial topo-logical band structure and engender exotic spin phases andtextures. However, for the purpose of spin current transmis-sion, these systems are not ideal platforms due to the lack ofspin conservation; thus, the magnetic materials currently de-ployed in spin and heat transport setups provide spins that aremainly collinear.In this tutorial we aim at introducing the readers to the keyproperties of collective spin excitations in collinear ferromag- netic and antiferromagnetic insulating systems. In section I,we outline the fundamental ingredients of the theory of spinwaves and we review the coupled equations describing thecoherent (long-wavelength) magnetization dynamics and thespin and heat transport driven by the thermal magnon cloud.Finally, we briefly discuss the coupling between magnetic andlattice degrees of freedom.In section II, we introduce spin transport and NV relaxom-etry setups as probes of magnetization dynamics and we dis-cuss recent experimental results making use of the theoreticaltools outlined in section I.
II. COLLINEAR MAGNETIC INSULATORS
In this section, we introduce the fundamental principles ofspin-wave theory taking as examples monoatomic ferromag-netic and two-sublattice antiferromagnetic collinear insula-tors. We discuss the interaction between their coherent andincoherent spin dynamics, and their coupled spin and heattransport properties. Finally, we review the coupling betweenmagnetic and lattice degrees of freedom. A. U (1)-symmetric ferromagnetic systems We consider a U (1)-symmetric ferromagnetic insulatingsystem. At each i th site of the monoatomic lattice, we candefine a spin S i = S n i , where S is the classical spin and theunit vector n i the spin orientation. At a temperature T farbelow the magnetic ordering (Curie) temperature T c , we canmap spin fluctuations around the ground state onto second-quantized operators via the Holstein Primakoff (HP) transfor-mation [18]. For a ground state with spins uniformly orientedalong the − z -direction, the HP transformation reads, in themacrospin limit, asˆ S iz = ˆ a † i ˆ a i − S , ˆ S i − = ˆ S ix + i ˆ S iy (cid:39) √ S ˆ a i , (1)where ˆ a † i ( ˆ a i ) is the canonical bosonic second-quantized op-erator that creates (annihilates) a magnon carrying spin angu-lar momentum ¯ h . In the limit of small excitation amplitude, a r X i v : . [ c ond - m a t . m e s - h a ll ] F e b i.e., for T (cid:28) T c , the magnon-magnon interaction correctionsare small [19]. Thus, any term higher than quadratic in themagnon creation and annihilation operators can be omittedfrom the magnetic Hamiltonian H m , i.e., H m = ∑ k ¯ h ω k ˆ a † k ˆ a k + E . (2)Here E is the ground state classical energy, ω k the spin-wavedispersion, andˆ a k = √ N ∑ i e i k · r i ˆ a i , ˆ a † k = √ N ∑ i e − i k · r i ˆ a † i , (3)where N is the number of lattice sites.The Hamiltonian H m possesses U (1) symmetry, i.e., [ ˆ S iz , H m ] =
0, i.e., it is invariant for rotations in spin spacearound the z -axis. Owing to this symmetry, Noether’s theo-rem guarantees that the z -component of the spin, S iz = (cid:104) ˆ S iz (cid:105) ,and, consequently, the total magnon number, ˜ N = ∑ k (cid:104) ˆ a † k ˆ a k (cid:105) ,are conserved quantities, where (cid:104) ... (cid:105) stands for the equilibrium(thermal) average. Such conservation law is, however, an ap-proximation: in reality, magnon-magnon and magnon-latticeinteractions that invalidate spin conservation are present inany insulating system. Even in the long-wavelength and low-temperature limit, one must include a dimensionless Gilbertdamping parameter α , with α (cid:28)
1, in the Landau-Lifshitz-Gilbert equation [20,21] in order to account for the experi-mentally observed broadening of the ferromagnetic resonance(FMR), i.e.,˙ n ( r ) = − γ n ( r ) × H ( r ) − α n ( r ) × ˙ n ( r ) . (4)Here, γ (>0) is the gyromagnetic ratio, H ( r ) is the effective(Landau-Lifshitz) field acting on the order parameter n ( r ) ,with n ( r ) = n i in the continuum limit. Equation (4) can besolved in the linear regime, i.e., by neglecting terms quadraticin spin fluctuations, i.e., n i n j for i , j = x , y . By performing aHP and Fourier transformation (1), one can find a direct corre-spondence between the linearized classical dynamics (4) andthe second-quantized Hamiltonian via the Heisenberg relation˙ˆ a k = i ¯ h [ H m , ˆ a k ] . (5)According to Eqs. (4) and (5), however, non-Hermitian termsviolating magnon number conservation will appear in theHamiltonian H m [22]. Until recently, such terms have beenneglected in the quantum theory of spin waves; a discussionof such non-Hermicity will be addressed elsewhere.Here we focus on magnetic systems with low damping,for which spin nonconserving terms can be (approximately)neglected. Namely, we discuss the limit in which the en-ergy scales of dissipative processes are very small comparedto the exchange interactions that control the thermalizationof the magnon distribution function. It has been extensivelydebated whereas the elementary quanta of spin waves, i.e.,magnons, can be described as quasi-particles whose den-sity is (approximately) conserved in collinear magnetic sys-tems with low damping. The assumption of magnon density (quasi-) conservation carries important consequences. Froma statistical standpoint, it implies the existence of a well-defined magnon chemical potential, and thus of magnon Bose-Einstein condensation, achievable even at relatively high tem-peratures [23]. Magnon density conservation has also beenpredicted to underlie a spin superfluid phase that shares manysimilarities with supercurrent of electric charge in supercon-ductors and the mass superflow in helium [24–26].Recent experiments have shown that the injection of spincurrent into a collinear magnet can drive the magnon gas intoa quasi-equilibrium state described by a Bose–Einstein statis-tics with non-zero chemical potential [27–29], suggesting thatthe inclusion of a chemical potential is necessary to properlycapture the experimental features of incoherent magnon trans-port. Thus, we will describe thermal magnons as a (quasi-equilibrium) thermalized Bose-Einstein ensemble with a well-defined chemical potential µ , i.e., n BE (cid:18) ¯ h ω k − µ k B T (cid:19) , (6)where n BE ( x ) = ( e x − ) − is the Bose-Einstein distributionfunction and k B the Boltzmann constant. The equation gov-erning the diffusive dynamics of thermal magnons density ˜ n can be written as˙˜ n ( r ) + ∇ · j s ( r ) = − g n µ µ ( r ) . (7)Here, g n µ parametrizes the spin relaxation rate and j s ( r ) = − σ ∇ µ ( r ) the spin current, where σ is the magnon spin con-ductivity. Thermal and long-wavelength magnons interact,e.g., via exchange coupling or single-ion anisotropies. Theform of such interaction between coherent (4) and incoher-ent (7) spin dynamics can be derived phenomenologically,using the symmetry and reciprocity principles discussed inRef. [30]. For a U (1)-symmetric system, it suffices to sup-plement Eqs. (4) and (7) with the following terms [30]¯ h ˙ n ( r ) = − η n ( r ) × [ ¯ h ˙ n ( r ) − µ ( r ) z × n ( r )] , (8)˙˜ n ( r ) = − η ˜ s z · n ( r ) × (cid:20) ˙ n ( r ) − µ ( r ) ¯ h n ( r ) × z (cid:21) , (9)where η (cid:28) n and the thermal magnon cloud. Here, wehave introduced the reduced spin density ˜ s = S / V − ˜ n , with V being the system volume. Equations (8) and (9) showthat the magnon chemical potential can be tuned by drivingthe coherent spin dynamics. The later, in turn, can pumpspin angular momentum into the incoherent thermal magnoncloud [30]. Using this mechanism, Du and coauthors mea-sured the magnon chemical potential via single spin magne-tometry based on nitrogen-vacancies [29], as we will describein detail later. B. U (1)-symmetric antiferromagnetic systems Due to their lack of a net magnetic moment, antiferromag-nets can not be controlled via an external magnetic field withthe same ease as ferromagnetic systems. However, the ab-sence of production of stray fields, which limit the packingdensity of ferromagnetic elements, and their fast spin dy-namics ( ∼ THz) make them desirable for the developmentof spintronic devices [31,32]. To illustrate the fundamentalproperties of antiferromagnetic magnons, here we focus ona U (1)-symmetric antiferromagnetic insulator with a ground-state staggered magnetic order. The neighbouring spins arealigned in opposite directions: this magnetic arrangement canbe conveniently described in terms of a unit cell with two mag-netic sublattices, A and B . For each i th unit cell, we can de-fine the spin S A ( B ) i = S n A ( B ) i , where the unit vector n A ( B ) i rep-resents the spin orientation of the magnetic sublattice A ( B ) .We consider sublattices with the same magnitude of magneti-zation; if the sublattices have different magnetizations, i.e., | S A | (cid:54) = | S B | , the system is ferrimagnetic and in several re-spects behaves like a ferromagnet, e.g., yttrium iron garnet(YIG) [33].We set the orientations of the order parameters as n A ( B ) i (cid:107)± ˆ z . This assumption is valid only below the critical fieldat which a spin-flop transition occurs, i.e., when the two-sublattice spins rotate suddenly to a direction perpendicular tothe easy-magnetization direction, as shown in Fig. 1(a). Be-low the spin-flop transition and far below the magnetic order-ing (Néel) temperature T N , the HP transformation reads, in themacrospin limit, asˆ S Aiz = S − ˆ a † i ˆ a i , ˆ S A + = √ S ˆ a i , ˆ S Biz = ˆ b † i ˆ b i − S , ˆ S B + = √ S ˆ b † i , (10)where ˆ a † i ( ˆ a i ) and ˆ b † (ˆ b ) are the creation (annihilation) opera-tors for spin deviations on the sublattice A and B , respectively.The sublattice spins are coupled by, e.g., nearest-neighbor ex-change interactions; thus, terms of the form ˆ a k ˆ b − k ( ˆ a † k ˆ b † − k )appear in the Hamiltonian upon HP and Fourier transforma-tion. The magnetic Hamiltonian can be brought in a diagonalform, i.e., H m = ∑ k (cid:104) ¯ h ω k , α ˆ α † k ˆ α k + ¯ h ω k , β ˆ β † k ˆ β k (cid:105) , (11)via a Bogoliubov-de-Gennes transformation [35], i.e.,ˆ α k = u k ˆ a k − v k ˆ b † − k , ˆ β k = u k ˆ b k − v k ˆ a † − k , (12)where ˆ α † k ( ˆ β † k ) is the creation operator for a magnon modewith dispersion ω k , α ( β ) . Setting u k − v k = z -component of the spinoperator (in units of ¯ h ) in terms of the Bogoliubov quasi-particles (12) as [34]ˆ S z = ∑ i ( S Aiz + S Biz ) = ∑ k ¯ h (cid:16) − ˆ α † k ˆ α k + ˆ β † k ˆ β k (cid:17) , (13)we can identify bosonic operator ˆ α † k ( ˆ β † k ) as the operatorcreating a magnon with spin angular momentum ∓ ¯ h . U (1)symmetry implies the conservation of the z -component of the FIG. 1. (a) Spin configurations in collinear antiferromagnets sub-jected to a magnetic field H oriented along the magnet symmetryaxis. Below the spin-flop transition, the order parameters n A and n B are antiparallel. Increasing the magnetic field induces a spin-floptransition, leading to a canted spin structure. (b) Magnon eigen-modes, labelled as α and β . The antiferromagnetic mode α ( β ),which carries ± ¯ h spin angular momentum, can be visualized as acombination of both up and down spins, precessing counterclock-wise (clockwise) at the frequency ω α ( β ) . The two antiferromagneticmodes are associated with an equal and opposite chemical potential µ . total spin S z = (cid:104) ˆ S z (cid:105) (13). Generally, the magnon chemicalpotential accounts for how much the free energy of the sys-tem in thermal equilibrium will change by adding a quanta ofspin angular momentum. In an antiferromagnetic system, netspin injection is achieved by creating an imbalance betweenthe population of the two magnon species (13); thus, thereis a single chemical potential associated with both magnonicspecies [36]. Assuming (approximate) spin conservation, wecan assign a chemical potential ∓ µ to the Bose-Einstein dis-tribution function n BE , α ( β ) of the antiferromagnetic eigen-mode α ( β ) [36], i.e., n BE , α ( β ) (cid:18) ¯ h ω α ( β ) k ± µ k B T (cid:19) . (14)Thermal fluctuations engender a finite thermal magnondensity of both antiferromagnetic modes, i.e., ˜ n α and ˜ n β ; sincethe contributions of the two modes to the spin current have op-posite signs, it is convenient to rewrite the spin diffusion equa-tion, in the continuum limit, in terms of the net density of ther-mal magnons carrying angular momentum ¯ h , i.e., ˜ n = ˜ n β − ˜ n α ,as ˙˜ n ( r ) + ∇ · j s ( r ) = − g µ µ ( r ) , (15)which is formally analogous to Eq. (7). Their classical cou-pled dynamics can be modeled by the phenomenologicalLandau-Lifshitz equation [37]˙ n A ( B ) = − γ × (cid:2) H A ( B ) − H c n B ( A ) (cid:3) , (16)where H A ( B ) is the effective (Landau-Lifshitz) field acting onthe order parameter n A ( B ) and H c is the, e.g., exchange, fieldcoupling the spin dynamics of the two sublattices.The linearized dynamics of the long-wavelength magneticorder parameters (16) can be diagonalized in terms of two nor-mal modes n α and n β [37]. In the mode α both up and downspins precess clockwise with frequency ω α k , while in the β mode the spins undergo a counterclockwise precession withfrequency ω β k , as depicted in Fig. 1(b). The correspondingLandau-Lifshitz-Gilbert equation reads as [37,38]˙ n α ( β ) = − γ n α ( β ) × H − α α , β n α ( β ) × ˙ n α ( β ) , (17)where we have included a mode-selective Gilbert damping pa-rameter α α ( β ) [39]. In terms of the normal modes, the inter-play between coherent and incoherent spin dynamics can beaccounted for by supplementing Eqs. (15) and (17) with [36]¯ h ˙ n α = − η α n α × ( ¯ h ˙ n α − µ z × n α ) , (18)¯ h ˙ n β = − η β n β × ( ¯ h ˙ n β − µ z × n β ) , (19)˙˜ n = − η α ˜ s α z · n α × ( ˙ n α − µ ¯ h n α × z ) − η β ˜ s β z · n β × ( ˙ n β − µ ¯ h n β × z ) , (20)with ˜ s α ( β ) = s − ˜ n α ( β ) . Here, η α ( β ) (cid:28) n α ( β ) and the ther-mal magnon cloud. C. Coupled spin and heat transport
Spin and heat transport in collinear magnetic insulatorshave been extensively investigated both theoretically and ex-perimentally in a variety of setups. Here, for simplicity, wepresent the theory of coupled spin and heat transport by con-sidering a normal metal | magnetic insulator | normal metal het-erostructure in a 1D geometry, shown in Fig. 2. The metallicreservoirs act as thermal baths, set at two different tempera-tures, i.e., T l and T r . The temperature bias generates a lin-ear temperature gradient ∇ T across the sample. The normalmetals have strong spin–orbit coupling: if a charge currentflows through them, a spin accumulation is generated at themetal | magnetic insulator interface and vice versa .The bulk spin, j s , and heat, j q , currents carried by magnonsin the magnetic insulating system can be written as (cid:18) j s j q (cid:19) = − (cid:18) σ ζρ κ (cid:19) (cid:18) ∇ µ ∇ T (cid:19) , (21)where the tensors σ , κ , ζ , and ρ ( = T ζ by the Onsager-Kelvinrelation [40, 41]) are, respectively, the spin and (magnetic)heat conductivities, and the spin Seebeck and Peltier coeffi-cients. The hydrodynamics equations for spin and heat trans-port can be easily constructed within the Boltzmann transporttheory [42], i.e.,˙˜ n + ∇ · j s = − g n µ µ − g nT ( T − T p ) , ˙ u + ∇ · j q = − g u µ µ − g uT ( T − T p ) . (22)Here, T ( T p ) is the magnon (phonon) temperature, u the en-ergy density of the thermal cloud, while g n ( u ) µ and g n ( u ) T FIG. 2. Metal|magnetic insulator|metal heterostructure. Viathe spin Hall effect, a charge current I sent through the metalalong the y -direction generates a spin accumulation µ l = µ l ˆ z at themetal | magnetic insulator interface, which injects a z -polarized spincurrent j s into the magnetic insulator. Due to the Joule heating as-sociated with the charge current, a temperature gradient ∇ T is setacross the sample. The injected spin current and the temperaturegradient trigger the dynamics of the magnetic order parameter n andof the thermal magnon cloud density ˜ n . The coherent and incoherentspin dynamics at the magnetic insulator | metal interface pump a spincurrent j s at x = L , leading to a spin accumulation µ r = µ r ˆ z that canbe converted into a measurable voltage in the metal via the inversespin Hall effect. parametrize, respectively, the relaxation of magnons via in-elastic magnon-magnon and magnon-phonon interactions andthe magnon thermalization to the phonon temperature. Therelations (21) and (22), complemented by Eqs. (4,8,9) and byEqs. (17,18,19,20) for, respectively, a ferromagnetic and anantiferromagnetic system, describe the coupled bulk spin andheat transport of the coherent and incoherent spin dynamics.The transport equations must be determined consistentlywith the boundary conditions for spin and heat transport atthe interfaces. For a ferromagnet, we take the order param-eter to be aligned along the z -direction, i.e., n (cid:107) − ˆ z . For atwo-sublattice antiferromagnet, it is convenient to introducethe Néel unit vector, i.e., n = ( n A − n B ) /
2. In this geom-etry, the relevant nonequilibrium spin accumulation at themetal | magnetic insulator interface at x = ( L ) is polarizedalong z -direction [43,44], i.e., µ l ( r ) = µ l ( r ) ˆ z . The boundaryconditions read as j s | x = ( L ) = G l ( r ) (cid:0) µ l ( r ) − µ (cid:1) | x = ( L ) + S l ( r ) (cid:0) T l ( r ) − T (cid:1) | x = ( L ) , j q | x = ( L ) = K l ( r ) (cid:0) T l ( r ) − T (cid:1) | x = ( L ) + Π l ( r ) (cid:0) µ l ( r ) − µ (cid:1) | x = ( L ) . (23)Here, the coefficients G l ( r ) , K l ( r ) , S l ( r ) , and Π l ( r ) (= T S l ( r ) byOnsager reciprocity) are the magnon spin and thermal con-ductances and spin Seebeck and Peltier coefficients, respec-tively, at the metal | magnetic insulator interface at x = ( L ) .Furthermore, the coherent magnetization dynamics pumps aspin current into an adjacent conductors [25,43–46], i.e., j s | x = ( L ) = − g l ( r ) ˆ z · ( n × ˙ n ) | x = ( L ) , (24)where g l ( r ) is the spin pumping efficiency of themetal | magnetic insulator interface at x = ( L ) , relatedto the magnon spin conductance G l ( r ) by Onsager reci-procity [44, 45]. The bulk and interfacial transport coef-ficients appearing, respectively, in Eqs. (21) and (23) canbe obtained via algebraic manipulation and integration ofthe magnon Boltzmann equation over momentum. Theirestimate depends on the details of the transport regime underinvestigation; for a detailed discussion of some examples, werefer the reader to Refs. [27] and [30].It is worth remarking that the outlined phenomenology canbe easily extended to a setup comprised of an insulating non-magnetic substrate | magnetic insulator | normal metal, com-monly used, e.g., in longitudinal SSE measurements. At thesubstrate|magnetic insulator interface, the spin flow is blockedas there are no spin carriers in the substrate. Nevertheless, heatstill can be transmitted via inelastic spin-preserving scatteringprocesses between magnons and phonons. The correspond-ing boundary conditions at the substrate | insulator interface at x = j s | x = = , j q | x = = K l ( T l − T ) | x = . (25) D. Magnon-phonon coupling
Coupling between collective magnetic and elastic excita-tions, i.e., magnons and phonons, is ubiquitous in magneticinsulating systems. It relies on relativistic effects such asdipole-dipole interactions and spin-orbit coupling, as well ason the dependence of the spin-spin exchange interactions onthe phonon coordinates, and it provides a pathway for ther-malization and dissipation of the magnetization dynamics.While the exact form of the phonon-driven magnon thermal-ization and dissipation rates can be computed from micro-scopic models [47], often they are accounted for by introduc-ing phenomenological parameters, see, e.g., Eq. (22).However, magnon thermalization and dissipation are notthe only relevant mechanisms emerging from the magne-toelastic coupling. When the magnetoelastic coupling termthat is quadratic in magnon and phonon operators does notvanish, magnons and phonons can hybridize to form quasi-particles that are an admixture thereof, dubbed as magnon-polarons [48, 49]. The Hamiltonian of a coupled magnon-phonon system reads generally as H = H m + H ph + H mec , (26)where H m ( ph ) is the magnon (phonon) quadratic Hamilto-nian and H mec the long-wavelength magnetoelastic couplingHamiltonian, whose explicit form depends on the symmetriesof the underlying lattice. The Hamiltonian (26) can be broughtinto diagonal form via a Bogoliubov-de-Gennes transforma-tion [35], which maps magnon and phonon operators intocomposite quasi-particles, i.e., magnon-polarons. In the ab-sence of coupling, transversal acoustic (TA) and longitudinalacoustic (LA) phonon bands are degenerate with the spin-wave dispersion at certain energies, as depicted in Fig. 3(a).When the coupling is sufficiently strong, the mutual interac-tion between the magnons and phonons lifts the degeneracy,leading to an anticrossing around which the magnon-polaron FIG. 3. (a) The dispersions of uncoupled magnons and transverseacoustic (TA) phonons are degenerate at certain wavevectors. (b)In the presence of magnon-phonon coupling, the system can be di-agonalized in terms of magnon-polarons, i.e., hybridized magnon-phonon quasi-particles. The magnetoelastic coupling lifts the de-generacy between the magnonic and phononic bands and the cross-ing points become anticrossing points. At the anticrossing points,the mixing between magnons and phonons is maximized, while atother wave-vectors the magnon-polaron bands retain a phonon- ormagnon-like character. (c) Longitudinal spin Seebeck setup com-prised of a magnetic insulating system (YIG) sandwiched between agadolinium gallium garnet (GGG) substrate and a platinum (Pt) film.A temperature gradient set across the sample generates a spin currentthat can be detected as a voltage in the Pt film [50]. (d) At the mag-netic field H TA , the magnon dispersion shifts upwards such that theTA phonon branch becomes tangential and the effects of the mag-netoelastic coupling are maximized, i.e., the magnon and phononmodes become strongly coupled over a relatively large volume inmomentum space. The voltage measured in a spin Seebeck setupdisplays a peak at the field H TA : the magnon-polaron spin transportproperties are enhanced with respect to their purely magnonic coun-terpart [50]. bands deviate from the purely phononic and magnonic ones,as shown in Fig. 3(b). Generally, strong hybridization oc-curs only in proximity of these anticrossings points and itis not likely to affect relevantly spin and heat transport co-efficients (22), whose evaluation requires integration of theBoltzmann transport equation over the full reciprocal space.However, it was recently found that, at critical magnetic fieldvalues, the magnon and phonon dispersions tangentially toucheach other and the magnon–phonon hybridization effects are LETTERS
NATURE PHYSICS
DOI: 10.1038/NPHYS3465 IM YIG
Injector Detector V α M YIG α I Injector M YIG
Detector M α a bc d − − − − − − − − V ++ −− || ( ° ) α ( ° ) α R ω R ( m Ω ) ω R ω R ( V A − ) ω d µ i µ || µ || µ µ d µ cos( ) fit α cos ( ) fit α Figure 2 | Non-local resistance as a function of angle α . a , First harmonic signal. The red line is a cos α fit through the data. b , Schematic top-view of theexperiment. A charge current I through the injector generates a spin accumulation µ i at the Pt | YIG interface. The component µ ∥ parallel to the net YIGmagnetization M YIG generates non-equilibrium magnons in the YIG, which gives rise to a cos α dependence of the injected magnon density. The magnonsthen di ff use to the detector. At the detector, a spin accumulation µ ∥ parallel to M YIG is generated. Owing to the inverse spin Hall e ff ect, µ ∥ generates acharge voltage, of which we detect the component generated by µ d . This gives rise to a cos α dependence of the detected magnon current. The total signalis a product of the e ff ects at the injector and detector, leading to the cos α dependence shown in a . c , Second harmonic signal. The red line is a cos α fitthrough the data. d , Schematic illustration of the angular dependence of the second harmonic: Joule heating at the injector excites magnons thermally,which di ff use to the detector. This process is independent of α . At the detector, the excited magnons generate a spin accumulation antiparallel to the YIGmagnetization, which is detected in the same way as for the first harmonic, giving rise to a total cos α dependence. The data shown in a and c are from adevice with an injector–detector separation distance of 200nm and a device length of 12.5 µ m, measured at a lock-in frequency f = Hz.
The first and second harmonic signals can be characterized bysimilar values of λ , indicating that thermally excited magnons arealso generated in the close vicinity of the injector. This supportsthe conclusions drawn in ref. 14—namely, for thermal magnonexcitation, the magnon signal reaches far beyond the thermalgradient generated by the applied heating. Note, however, that thesign change for the second harmonic signal (Fig. 3b inset) illustratesthatthephysicsforelectricalandthermalmagnongenerationisverydi ff erent. This is discussed further in Supplementary Section B.We verify our assumption of magnon excitation and detectionin the linear regime by performing measurements where wereversed the role of injector and detector. The results are shownin Fig. 4a (4b) for the first (second) harmonic. For the firstharmonic non-local resistance we find R ω V – I = ± and R ω I – V = ± . As we find R V – I ( B ) = R I – V ( − B ) (ref. 29), we conclude that Onsager reciprocity holds withinthe experimental uncertainty, despite the asymmetry in theinjector–detector geometry. Reciprocity does not hold for thesecond harmonic (Fig. 4b), as expected for nonlinear processes. Finally, we verify that V ω nl scales linearly with appliedcurrent (Fig. 4c). The linearity and reciprocity of the firstharmonic non-local signal demonstrate that it is due to linearprocesses only.Remarkably, the observed magnon transport is described well bythe familiar spin di ff usion model, despite the completely di ff erentcharacter of the carriers of spins in magnetic insulators (bosons)compared to metals and semiconductors (fermions). Our resultsare consistent with spin injection/detection by invasive contacts,indicating that by optimizing contact properties the signals could beenhanced further. Our observation that the YIG spin conductivityis comparable to that of metals, combined with the long magnonspin di ff usion length in YIG, provides new opportunities from atechnological point of view to enable novel magnonic devices basedon microstructured YIG. The observed similarity between electronand magnon transport begs the question how far this analogy canbe extended, and calls for the investigation of e ff ects such as themagnon Hall e ff ect and, possibly, ballistic magnon transport atlower temperatures. NATUREPHYSICS © 2015 Macmillan Publishers Limited. All rights reserved
NATURE PHYSICS
DOI: 10.1038/NPHYS3465
LETTERS
YIGGGGPt V Pt B d a c M YIG s = +1/2 s = − s = − s = +1/2 s = +1 s = +1 CreationAbsorption b I B α I + − V + − M YIG
Figure 1 | Non-local measurement geometry. a , Schematic representationof the experimental geometry. A charge current I through the left platinumstrip (the injector) generates a spin accumulation at the Pt | YIG interfacethrough the spin Hall e ff ect. Through the exchange interaction at theinterface, angular momentum is transferred to the YIG, exciting orannihilating magnons. The magnons then di ff use towards the rightplatinum strip (the detector), where they are absorbed and a spinaccumulation is generated. Through the inverse spin Hall e ff ect the spinaccumulation is converted to a charge voltage V , which is then measured. b , Schematic of the magnon creation and absorption process. A conductionelectron in the platinum scattering o ff the Pt | YIG interface transfers spinangular momentum to the YIG. This will flip its spin and create a magnon.The reciprocal process occurs for magnon absorption. c , Opticalmicroscope image of a typical device. The parallel vertical lines are theplatinum injector and detector, which are contacted by gold leads. Currentand voltage connections are indicated schematically. An external magneticfield B is applied at an angle α . The scale bar represents 20 µ m. the sample in a fixed external field. Using lock-in amplifiers, weseparate higher-order contributions in the voltage by measuringhigher harmonics, using: V = R I + R I + ··· , where R i is the i th harmonic response . Because magnon spin injection/detectionscales linearly with I , its magnitude is obtained from the firstharmonic signal. Any thermal e ff ects due to Joule heating (forwhich " T ∝ I ) will be detected in the second harmonic signal.The result of such a measurement is shown in Fig. 2a (2c) for thefirst (second) harmonic, and the observed angular dependence isexplained schematically in Fig. 2b (2d).We fabricated two series of devices with di ff erentinjector–detector separation distances d . Series A is tailoredto the short-distance regime ( d < µ m), whereas series B exploresthe long-distance regime (3 < d < µ m). For each device, anon-local measurement as shown in Fig. 2 was performed. The magnitudes of the non-local resistances were extracted forevery d , by fitting the data with R ω = R ω + R ω nl cos ( α ) (1) R ω = R ω + R ω nl cos ( α ) (2)where R ω and R ω are o ff set resistances (see Methods) and R ω nl ( R ω nl )are the magnitudes of the first (second) harmonic signal. Figure 3aandc(3bandd)showtheresultsonlinearandlogarithmicscales,forthe first (second) harmonic non-local resistance, respectively. Both R ω nl and R ω nl are normalized to device length, to compare deviceshaving di ff erent lengths.From Fig. 3 we can clearly observe two regimes, which weinterpret as follows: at large distances, signal decay is dominatedby magnon relaxation and is characterized by exponential decay.For distances shorter than the magnon spin di ff usion length weobserve di ff usive transport, and the signal follows a 1 / d behaviour(inset Fig. 3a). Both regimes are described well with a single model,using the one-dimensional spin di ff usion equation , adapted formagnon transport:d n m d x = n m λ , with λ = √ D τ (3)where n m is the non-equilibrium magnon density, λ is the magnonspin di ff usion length in YIG, D is the magnon di ff usion constantand τ isthemagnonrelaxationtime.Theone-dimensionalapproachis valid because the YIG thickness (200 nm) is much smaller thanthe injector–detector distance d , whereas the device length is muchlarger than d . We assume strong spin–magnon coupling betweenYIG and platinum, given the large spin-mixing conductance atthe Pt | YIG interface and the strong spin–orbit interaction inplatinum. We therefore impose the boundary conditions n m ( ) = n and n m ( d ) =
0, where n is the injected magnon density, which isproportional to the applied current and is determined by variousmaterial and interface parameters. These conditions imply that theinjector acts as a low-impedance magnon source, and all magnoncurrent is absorbed when it arrives at the detector. The solution toequation (3) is of the form n m ( x ) = a exp ( − x / λ ) + b exp ( x / λ ) , andfrom the boundary conditions we find for the magnon di ff usioncurrent density j m = − D ( d n m / d x ) at the detector: j m ( x = d ) = − Dn λ exp ( d / λ ) − exp ( d / λ ) (4)The non-local resistance is proportional to j m ( d )/ n , and fromequation (4) we adopt a two-parameter fitting function for the non-local resistances, capturing the distance-independent prefactors in asingle parameter C : R nl = C λ exp ( d / λ ) − exp ( d / λ ) (5)The signal decay described by equation (5) is equivalent to thatof spin signals in metallic spin valves with transparent contacts .The dashed lines shown in Fig. 3a,c are best fits to this function,where we find from the first harmonic data λ ω = ± µ m.From the second harmonic signal (Fig. 3b,d), originating frommagnons generated by heat produced in the injector strip, we find λ ω = ± µ m. For distances larger than 40 µ m, the non-localvoltage is smaller than the noise level of our set-up (approximately3 nV r.m.s. ). We compare the magnitude and sign of the signalin the short-distance measurements to a local measurement inSupplementary Section A. From this, we obtain a value for the spinconductivity of YIG, σ s ≈ × S m − , which is comparable to thatof metals. NATUREPHYSICS © 2015 Macmillan Publishers Limited. All rights reserved
NATURE PHYSICS
DOI: 10.1038/NPHYS3465
LETTERS a bc d µ m)0 10 20 30 40Distance ( µ m) 0 10 20 30 40Distance ( µ m)0 10 20 30 40Distance ( µ m)0255075100125150 0.00.20.40.60.81.01.21.40.010.101.0010.0 Series ADistance ( µ m)1 2 3 4 − − − R ( M V A − m − ) n l ω R ( M V A − m − ) n l ω R ( Ω m − ) n l ω R ( M V A − m − ) n l ω R ( Ω m − ) n l ω Series A A / d fit0 1 2 3 4 5Distance ( µ m) R ( Ω m − ) n l ω Figure 3 | Amplitude of the non-local signals as a function of injector–detector separation distance.
Every data point represents a device with a di ff erentinjector–detector separation and results from an angle-dependent measurement as shown in Fig. 2. The magnitude of the signal is extracted by fitting theangle-dependent non-local resistance to equation (1) (2) for the first (second) harmonic. The error bars represent the standard error in the fits. The signalis scaled by device length for both first and second harmonic. a , c , First harmonic data on linear and logarithmic scales, respectively. The dashed line is a fitto equation (5), resulting in λ ω = ± µ m. For d < λ , the data is described well by a A / d fit, shown in the inset of a . b , d , Second harmonic data on linearand logarithmic scales, respectively. The dashed line is again a fit to equation (5), and we find λ ω = ± µ m. The inset to b shows the short-distancebehaviour of the second harmonic signal. For very short distances, the signal changes sign. For this reason, data points with d < µ m were omitted fromthe fit in b and d . Linear fit I c ( µ A) ca − − −
60 0 60 120 180 − − −
60 0 60 120 180 0 20 40 60 80 100 − I − VV − I b − − − − I − VV − I ( ° ) α ( ° ) α V ω R ( V A − ) ω R ( m Ω ) ω V ( µ V ) n l ω Figure 4 | Demonstration of reciprocity and linearity of the non-local resistance. a , First harmonic signal as a function of angle for the I – V and V – I configurations. We extract the amplitude of the signal using a fit to equation (1), finding R ω V – I = ± m " and R ω I – V = ± m " . We thusconclude that reciprocity holds. b , Second harmonic signal as a function of angle. We extract the amplitude of the signal using a fit to equation (2), finding R ω V – I = ± V A − and R ω I – V = ± V A − . We conclude that reciprocity does not hold for the second harmonic. c , Non-local voltage as afunction of injected charge current. The data is obtained from angle-dependent measurements. A data point represents the amplitude of theangle-dependent voltage, obtained by a fit to equation (1). The error bars, representing the standard error in the fit, are also plotted but are smaller than thedata points. The red dashed line is a linear fit through the data, showing the linearity of the first harmonic signal. The data shown here is obtained from adevice with an injector–detector separation distance of 200 nm and a device length of 12.5 µ m, measured at a lock-in frequency f = Hz.
Methods
Methods and any associated references are available in the onlineversion of the paper.
Received 22 May 2015; accepted 5 August 2015;published online 14 September 2015
References
1. Johnson, M. & Silsbee, R. H. Interfacial charge–spin coupling: Injectionand detection of spin magnetization in metals.
Phys. Rev. Lett. et al.
Electrical detection of spin transport in lateralferromagnet-semiconductor devices.
Nature Phys. NATURE PHYSICS © 2015 Macmillan Publishers Limited. All rights reserved
LETTERS
NATURE PHYSICS
DOI: 10.1038/NPHYS3465 IM YIG
Injector Detector V α M YIG α I Injector M YIG
Detector M α a bc d − − − − − − − − V ++ −− || ( ° ) α ( ° ) α R ω R ( m Ω ) ω R ω R ( V A − ) ω d µ i µ || µ || µ µ d µ cos( ) fit α cos ( ) fit α Figure 2 | Non-local resistance as a function of angle α . a , First harmonic signal. The red line is a cos α fit through the data. b , Schematic top-view of theexperiment. A charge current I through the injector generates a spin accumulation µ i at the Pt | YIG interface. The component µ ∥ parallel to the net YIGmagnetization M YIG generates non-equilibrium magnons in the YIG, which gives rise to a cos α dependence of the injected magnon density. The magnonsthen di ff use to the detector. At the detector, a spin accumulation µ ∥ parallel to M YIG is generated. Owing to the inverse spin Hall e ff ect, µ ∥ generates acharge voltage, of which we detect the component generated by µ d . This gives rise to a cos α dependence of the detected magnon current. The total signalis a product of the e ff ects at the injector and detector, leading to the cos α dependence shown in a . c , Second harmonic signal. The red line is a cos α fitthrough the data. d , Schematic illustration of the angular dependence of the second harmonic: Joule heating at the injector excites magnons thermally,which di ff use to the detector. This process is independent of α . At the detector, the excited magnons generate a spin accumulation antiparallel to the YIGmagnetization, which is detected in the same way as for the first harmonic, giving rise to a total cos α dependence. The data shown in a and c are from adevice with an injector–detector separation distance of 200nm and a device length of 12.5 µ m, measured at a lock-in frequency f = Hz.
The first and second harmonic signals can be characterized bysimilar values of λ , indicating that thermally excited magnons arealso generated in the close vicinity of the injector. This supportsthe conclusions drawn in ref. 14—namely, for thermal magnonexcitation, the magnon signal reaches far beyond the thermalgradient generated by the applied heating. Note, however, that thesign change for the second harmonic signal (Fig. 3b inset) illustratesthatthephysicsforelectricalandthermalmagnongenerationisverydi ff erent. This is discussed further in Supplementary Section B.We verify our assumption of magnon excitation and detectionin the linear regime by performing measurements where wereversed the role of injector and detector. The results are shownin Fig. 4a (4b) for the first (second) harmonic. For the firstharmonic non-local resistance we find R ω V – I = ± and R ω I – V = ± . As we find R V – I ( B ) = R I – V ( − B ) (ref. 29), we conclude that Onsager reciprocity holds withinthe experimental uncertainty, despite the asymmetry in theinjector–detector geometry. Reciprocity does not hold for thesecond harmonic (Fig. 4b), as expected for nonlinear processes. Finally, we verify that V ω nl scales linearly with appliedcurrent (Fig. 4c). The linearity and reciprocity of the firstharmonic non-local signal demonstrate that it is due to linearprocesses only.Remarkably, the observed magnon transport is described well bythe familiar spin di ff usion model, despite the completely di ff erentcharacter of the carriers of spins in magnetic insulators (bosons)compared to metals and semiconductors (fermions). Our resultsare consistent with spin injection/detection by invasive contacts,indicating that by optimizing contact properties the signals could beenhanced further. Our observation that the YIG spin conductivityis comparable to that of metals, combined with the long magnonspin di ff usion length in YIG, provides new opportunities from atechnological point of view to enable novel magnonic devices basedon microstructured YIG. The observed similarity between electronand magnon transport begs the question how far this analogy canbe extended, and calls for the investigation of e ff ects such as themagnon Hall e ff ect and, possibly, ballistic magnon transport atlower temperatures. NATUREPHYSICS © 2015 Macmillan Publishers Limited. All rights reserved n
NATURE PHYSICS
DOI: 10.1038/NPHYS3465 IM YIG
Injector Detector V α M YIG α I Injector M YIG
Detector M α a bc d − − − − − − − − V ++ −− || ( ° ) α ( ° ) α R ω R ( m Ω ) ω R ω R ( V A − ) ω d µ i µ || µ || µ µ d µ cos( ) fit α cos ( ) fit α Figure 2 | Non-local resistance as a function of angle α . a , First harmonic signal. The red line is a cos α fit through the data. b , Schematic top-view of theexperiment. A charge current I through the injector generates a spin accumulation µ i at the Pt | YIG interface. The component µ ∥ parallel to the net YIGmagnetization M YIG generates non-equilibrium magnons in the YIG, which gives rise to a cos α dependence of the injected magnon density. The magnonsthen di ff use to the detector. At the detector, a spin accumulation µ ∥ parallel to M YIG is generated. Owing to the inverse spin Hall e ff ect, µ ∥ generates acharge voltage, of which we detect the component generated by µ d . This gives rise to a cos α dependence of the detected magnon current. The total signalis a product of the e ff ects at the injector and detector, leading to the cos α dependence shown in a . c , Second harmonic signal. The red line is a cos α fitthrough the data. d , Schematic illustration of the angular dependence of the second harmonic: Joule heating at the injector excites magnons thermally,which di ff use to the detector. This process is independent of α . At the detector, the excited magnons generate a spin accumulation antiparallel to the YIGmagnetization, which is detected in the same way as for the first harmonic, giving rise to a total cos α dependence. The data shown in a and c are from adevice with an injector–detector separation distance of 200nm and a device length of 12.5 µ m, measured at a lock-in frequency f = Hz.
The first and second harmonic signals can be characterized bysimilar values of λ , indicating that thermally excited magnons arealso generated in the close vicinity of the injector. This supportsthe conclusions drawn in ref. 14—namely, for thermal magnonexcitation, the magnon signal reaches far beyond the thermalgradient generated by the applied heating. Note, however, that thesign change for the second harmonic signal (Fig. 3b inset) illustratesthatthephysicsforelectricalandthermalmagnongenerationisverydi ff erent. This is discussed further in Supplementary Section B.We verify our assumption of magnon excitation and detectionin the linear regime by performing measurements where wereversed the role of injector and detector. The results are shownin Fig. 4a (4b) for the first (second) harmonic. For the firstharmonic non-local resistance we find R ω V – I = ± and R ω I – V = ± . As we find R V – I ( B ) = R I – V ( − B ) (ref. 29), we conclude that Onsager reciprocity holds withinthe experimental uncertainty, despite the asymmetry in theinjector–detector geometry. Reciprocity does not hold for thesecond harmonic (Fig. 4b), as expected for nonlinear processes. Finally, we verify that V ω nl scales linearly with appliedcurrent (Fig. 4c). The linearity and reciprocity of the firstharmonic non-local signal demonstrate that it is due to linearprocesses only.Remarkably, the observed magnon transport is described well bythe familiar spin di ff usion model, despite the completely di ff erentcharacter of the carriers of spins in magnetic insulators (bosons)compared to metals and semiconductors (fermions). Our resultsare consistent with spin injection/detection by invasive contacts,indicating that by optimizing contact properties the signals could beenhanced further. Our observation that the YIG spin conductivityis comparable to that of metals, combined with the long magnonspin di ff usion length in YIG, provides new opportunities from atechnological point of view to enable novel magnonic devices basedon microstructured YIG. The observed similarity between electronand magnon transport begs the question how far this analogy canbe extended, and calls for the investigation of e ff ects such as themagnon Hall e ff ect and, possibly, ballistic magnon transport atlower temperatures. NATUREPHYSICS © 2015 Macmillan Publishers Limited. All rights reserved
FIG. 4. (a) Non-local spin transport setup and experimental data [14]. Two Pt strips, separated by a distance d , are deposited on top ofa magnetic insulating film (YIG) grown on a GGG substrate. One metallic strip (injector) injects spin current into the magnetic insulatingsystem. The spin current propagates through the insulator and generates a voltage V in the other metallic strip (detector). (b,c) By tuningthe orientation of the in-plane magnetic field B , the magnetic insulator order parameter n can be oriented at a generic angle α within thesample plane, i.e., n (cid:107) cos α . A charge current in the injector leads to a spin accumulation µ l = µ l ˆ z , with µ l ∝ I , at the metal | magneticinsulator interface; the effective spin accumulation that injects non-equilibrium magnons in the insulator is parallel to the order parameter, i.e., µ eff = µ l cos α . Similarly, a spin accumulation parallel to n is generated at the detector. The charge current I yields as well a temperaturegradient ∇ T ∝ I that injects non-equilibrium magnons with no dependence on the angle α . (b) The first harmonic signal, which measureselectrically-generated spin waves, is a product of the effects at the injector and detector, leading to a cos α dependence. (c) The measuredsecond harmonic signal, which measures thermally-generated spin waves, displays a cos α dependence. (d) The measured first harmonic signalis fitted to the model in Eq. (28). maximized [50, 51]. In correspondence of these fields, spintransport properties can be enhanced or suppressed, depend-ing on the ratio between the phonon and the magnon relax-ation times. If the phonon relaxation time is longer (shorter)than the magnon relaxation time, magnon-polarons will havea longer (shorter) lifetime than the pure magnetic excitationsand thus spin transport properties will be enhanced (sup-pressed). These features have been recently observed in thefield dependence of thermally generated current in YIG in alongitudinal SSE transport setup, shown in Figs. 3(c) and (d).Similar effects have been investigated as well in antiferromag-netic systems [52–54]. III. PROBING SPIN DYNAMICS
In this section we review two experimental probes of mag-netization dynamics. We introduce the reader to spin transportsetups, by focusing on the measurements of the YIG spin dif-fusion length [14] and of thermally-generated spin currentsin the antiferromagnetic insulator MnFe [17]. Secondly, wediscuss a new minimally-invasive magnon sensing technique,i.e., NV-center relaxometry, and review the recent measure- ments of the magnon chemical potential in YIG [29] and ofthe spin diffusion length in antiferromagnetic α -Fe O [55]. A. Spin transport
A non-local transport set-up is commonly used as a probeof spin transport properties of magnetic insulating systems.This setup is comprised of two metallic strips, operating, re-spectively, as spin current injector and detector. The metalsare deposited on top of a magnetic insulating sample grownon a substrate, as shown in Fig. 4(a). A charge current I mod-ulated at frequency ω is sent through the injector. The SHEconverts the charge current into a spin accumulation µ l ∝ I at the metal | magnetic insulator interface, while Joule heatinggenerates a thermal gradient ∇ T ∝ I .Spin and thermal injection trigger a spin current that mightpropagate to the magnetic insulating | detector interface. In thedetector, owing to the ISHE, the impinging spin current is con-verted into a charge current, which generates a voltage V un-der open-circuit conditions. Using lock-in amplifiers, one canseparate higher order contributions in the voltage by measur-ing higher harmonics: V = R I + R I + ... , (27)where R i is the i th harmonic response of the non-local resis-tance R = V / I [56]. The first, R , and second, R , harmonicsignals are a measure of, respectively, the electrically- andthermally-generated spin currents. An external magnetic field B , oriented at an angle α within the sample plane, sets theequilibrium direction of the magnetic order parameter n . Theeffective component of the spin accumulation, µ eff , that exertsa spin-transfer torque on the order parameter is parallel to themagnetic order, i.e., µ eff = µ l cos α . Similarly, the spin currentinjected via spin pumping into the detector is modulated by afactor cos α . Thus, the electrically-generated signal scales as R ∝ cos α , while for thermal injection one has R ∝ cos α ,as shown in Figs. 4(b) and (c).Numerous experiments have been performed using a non-local transport setup. In this tutorial, we focus on the first ex-perimental report of room-temperature long-range spin trans-port in a magnetic insulating system. Cornelissen and coau-thors [14] investigated spin transport in YIG using the setupdepicted in Fig. 4(a). In the data analysis, the spin andheat transport in the Pt | YIG | Pt heterostructure is modelled us-ing the phenomenology presented in Sec. IIC. The magnon-phonon energy relaxation length (cid:96) u = (cid:112) κ / g uT , associatedwith spin-preserving relaxation of magnon distribution to-wards the phonon temperature, is assumed to be much shorterthan the magnon relaxation (i.e., spin diffusion) length (cid:96) s ≡ (cid:112) σ / g n µ . This can be intuitively understood as spin con-serving phonon-magnon interactions can stem from Heisen-berg exchange coupling, whose strength is much larger thanthe spin-orbit interactions responsible for spin non-conservingprocesses. In this limit, by plugging Eq. (21) into Eq. (22), oneobtains, in a steady-state, the spin diffusion equation ∇ µ = µ (cid:96) s . (28)Fitting Eq. (28) to the experimental signal, shown in Fig. 4(d),leads to the estimate (cid:96) s ∼ µ m, which has been corroboratedby further experiments [15,16]. It worth noting that, few yearslater, the magnon-phonon energy relaxation length (cid:96) u in YIGwas measured to be about 250 nm [57, 58], i.e., much shorterthan the spin diffusion length, confirming the assumption thatunderlies the analysis of the experimental data in Ref. [14].An alternative technique for probing thermally-generatedspin currents is offered by a longitudinal spin Seebeck setup,comprised of a magnetic insulator sandwiched between a sub-strate and a metallic strip. A heat gradient applied acrossthe sample generates a spin current in the magnetic insulator,which is then converted into a measurable voltage in the metalvia ISHE. This setup, shown in Fig. 3(c), was used to detectthe signatures of magnon-phonon hybridization in YIG [50],and, more recently, as a probe of thermally-driven spin trans-port in the easy-axis antiferromagnetic insulator MnFe [59].Wu and coauthors revealed a dependence of the spin Seebeckvoltage on temperature that is not observable in ferromagneticsystem. The experimental signal, shown in Fig. 5, displayspeaks at temperatures much lower than the magnetic ordering heater changes approximately 10.6% from 80 to 2 K( . − . mW rms ), but this effect in comparison tothe total temperature dependence of the voltage signal issmall [16]. The temperature dependence shows a peak atlow temperatures at all magnetic fields. At low magneticfields, there appears to be a low temperature peak whoseposition increases in temperature with increasing magneticfield strength. At fields above the spin-flop transition, thispeak becomes broader and approximately matches the peakin thermal conductivity of MnF from the literature [29].Many longitudinal spin Seebeck systems have a correlationbetween the size of the spin Seebeck signal and the thermalconductivity, which is believed to be a consequence ofmagnon-phonon interaction [30,31]. In our device geom-etry, there is typically an inverse dependence on the sizeof the signal to the thermal conductivity of the thin filmsince a constant power is applied to the material insteadof a constant temperature difference Δ T [7,32,33]. Here, V ∝ Δ T ∝ ð P= κ Þ , where V is the measured voltage due tothe inverse spin Hall effect, P is the applied power, and κ isthe thermal conductivity of the film. Since our measure-ment suggests V scales with κ , there is minimally a strongerthan linear dependence of the spin Seebeck signal size on κ .This could be due to especially weak interaction betweenmagnons and phonons in this system due to higherfrequency gapped AFM magnons, leading to a largertemperature difference between nonequilibrium phononand magnon populations [34]. Both the heat capacityand thermal conductivity of MnF is dominated by phononconduction, and, therefore, the effect of the magnetic fieldon the thermal conductivity of MnF is negligible andcannot account for the spin-flop behavior in the SSE[34,35]. The lack of magnon thermal conduction is alsoevidence that the magnon-phonon relaxation times are longdue to weak interaction in MnF [35]. The inset of Fig 3shows the data from 120 kOe with the contribution at70 kOe subtracted to isolate the temperature dependence ofthe SSE in the spin-flopped phase. The data show a sharper peak at ∼ K, suggesting that the SSE in the spin-floppedphase is strongly correlated with the MnF thermalconductivity.To confirm that the origin of the jump in the spinSeebeck signal is from the spin-flop transition, measure-ments were made on a separate device fabricated simulta-neously on the same film, with the pattern oriented 90° tothe original device. In this device, spin current due to spincomponents perpendicular to the c axis are detected. Thevoltage response from the new device, performed under thesame conditions as Fig. 2(a) except with the magnetic fieldperpendicular to the c axis, is compared to the data for theold device with the magnetic field applied parallel to the c axis. The results are summarized in Fig. 4, where the jumpin the spin Seebeck signal is absent with the magnetic fieldin the ⊥ to the c -axis direction, while still present in the ∥ to c -axis case. At 80 K, above T N , both signals are roughlyequivalent. As the temperature is lowered below T N , thesignal in the ∥ c device is lower than in the ⊥ c device for H < H C , but the two signals roughly agree with each otherfor H > H C . Because the two devices are identical exceptfor the direction of the Pt bar, it is unlikely that the observedphenomena are due to proximity magnetic interactions ordiffusion of magnetic ions into the Pt layer since thisanisotropic behavior is specific to only MnF .Current theories on the origin of the spin Seebeck effectinvolve a nonequilibrium population of magnons accumulat-ing at the interface between the magnetic insulator andmetallic spin detector layer [31,36,37]. This could be dueto several mechanisms, including bulk magnon diffusion [31] FIG. 3. Temperature dependence of the spin Seebeck voltageresponse at various magnetic fields. The inset represents the140 kOe data with the 70 kOe subtracted out to judge thetemperature dependence of just the spin-flopped phase. (a) (b)(c) (d)(e) (f)
FIG. 4. (a) – (f) Spin Seebeck voltage responses on two devices,one aligned to detect spin current parallel to the c axis and onealigned to detect spin current perpendicular to the c axis. Thespin-flop transition is only present in the parallel configuration. PRL week ending4 MARCH 2016
FIG. 5. Temperature dependence of the spin Seebeck voltage re-sponse at various magnetic fields [59]. temperature. The position of the peaks varies with the inten-sity of the applied magnetic field. While a detailed model andanalysis of the data can be found in Ref. [34], here we candevelop an intuitive understanding by using the concepts out-lined in Sec. IIB. MnFe can be modeled as a two-sublatticeuniaxial antiferromagnet, with two normal magnon modesthat carry opposite spin angular momentum (13), and, thus,generate spin currents propagating in opposite directions. Thespin current density operator ˆ j s reads asˆ j s = ¯ hV (cid:104) − v k , α ˆ α † k ˆ α k + v k , β ˆ β † k ˆ β k (cid:105) , (29)where v k , α ( β ) = ∂ k ω k , α ( β ) is the group velocity of the α ( β )magnon mode. For an uniaxial antiferromagnet subjected toan external magnetic field B oriented along its symmetry axis,the dispersion can be written as ω k , α ( β ) = ω k ± γ B , (30)where ω k is the zero-field spin-wave dispersion, whose ex-plicit form depends on the specific microscopic model. Inthe absence of a magnetic field, the two magnon modes aredegenerate; thus, they have equal occupation numbers in ther-mal equilibrium. Since they also have equal group velocities,the spin current (30) vanishes. When a magnetic field is ap-plied, the magnon dispersions split according to Eq. (30). At agiven temperature, the magnon mode β has a higher occupa-tion density than the magnon mode α . The difference betweenthe occupation density of the α and β modes is proportionalto the splitting 2 γ B between their dispersions. For tempera-tures lower than the bottom of the dispersion curve of the α mode, i.e., T < ω k = , α , increasing the temperature increasesthe thermal population of the β mode and thus the net spincurrent (29). At higher temperatures, the thermal populationof the β mode start increasing, leading to a reduction of thenet current (30). SPINTRONICS
Control and local measurement of thespin chemical potential in amagnetic insulator
Chunhui Du, * Toeno van der Sar, * Tony X. Zhou, * Pramey Upadhyaya, Francesco Casola,
Huiliang Zhang,
Mehmet C. Onbasli,
Caroline A. Ross, Ronald L. Walsworth,
Yaroslav Tserkovnyak, Amir Yacoby † The spin chemical potential characterizes the tendency of spins to diffuse. Probing thisquantity could provide insight into materials such as magnetic insulators and spinliquids and aid optimization of spintronic devices. Here we introduce single-spinmagnetometry as a generic platform for nonperturbative, nanoscale characterization ofspin chemical potentials. We experimentally realize this platform using diamondnitrogen-vacancy centers and use it to investigate magnons in a magnetic insulator,finding that the magnon chemical potential can be controlled by driving the system ’ sferromagnetic resonance. We introduce a symmetry-based two-fluid theory describingthe underlying magnon processes, measure the local thermomagnonic torque,and illustrate the detection sensitivity using electrically controlled spin injection.Our results pave the way for nanoscale control and imaging of spin transport inmesoscopic systems. C ontrol and measurement of the chemicalpotential of a spin system can be used toexplore phenomena ranging from quan-tum phase transitions ( , ) to Bose-Einsteincondensation ( , ) and spin transport ingases and solid-state systems ( – ). In recentdecades, a large scientific effort has focused onharnessing spin transport for low-dissipationinformation processing ( , – ). In contrastto charge, spin is not a conserved quantity andnaturally decays on the nanoscale for a widerange of materials, including typical metals( , ), calling for a local detection technique.Compared to the centuries-old techniques forstudying charge transport, methods for prob-ing spin chemical potentials have only been de-veloped recently, with leading methods basedon the coupling between spin and charge trans-port ( , , , ) and inelastic light scattering( , ). Here we introduce a fundamentallydifferent approach, which uses a single sensorspin to measure the local magnetic field fluc-tuations generated by a thermal spin bath. Thisapproach is nonperturbative and provides spa- tial access to the spin chemical potential on ascale determined by the distance between thesensor spin and the system under study, openingthe door to imaging spin transport phenomena with resolutions down to the few-nanometerscale.Because the spin chemical potential is inherentlyrelated to spin fluctuations, it can be quantita-tively determined by measuring the magneticfields generated by these fluctuations. We dem-onstrate this principle using the excellent magnet-ic field sensitivity of the S = 1 electronic spinassociated with the nitrogen-vacancy (NV) centerin diamond ( , ). We measure the chemicalpotential of magnons — the elementary spin excita-tions of magnetic materials ( ) — in a 20-nm-thickfilm of the magnetic insulator yttrium-iron-garnet(YIG) on a ~100-nm-length scale (Fig. 1A). Ourmeasurements reveal that the magnon chemicalpotential can be effectively controlled by excitingthe system ’ s ferromagnetic resonance (FMR)(Fig. 1, B and C).Locally probing the weak magnetic fields gen-erated by the fluctuations of a spin system re-quires nanometer proximity of a magnetic fieldsensor to the system. We ensure such prox-imity by positioning diamond nanobeams ( )that contain individually addressable NV centersonto the YIG surface (Fig. 1, D and E). We use ascanning confocal microscope to optically lo-cate the NV centers and address their spin state( , ). A photoluminescence image (Fig. 1D)provides an overview of the system, showing anNV center (NV ) in a nanobeam that is locatedwithin a few micrometers from the gold andplatinum striplines used to excite magnons inthe YIG. RESEARCH Du et al ., Science , 195 –
198 (2017) 14 July 2017 Department of Physics, Harvard University, 17 OxfordStreet, Cambridge, MA 02138, USA. John A. PaulsonSchool of Engineering and Applied Sciences, HarvardUniversity, Cambridge, MA 02138, USA. Department ofPhysics and Astronomy, University of California, LosAngeles, 475 Portola Plaza, Los Angeles, CA 90095, USA. Harvard-Smithsonian Center for Astrophysics, 60 GardenStreet, Cambridge, MA 02138, USA. Department ofMaterials Science and Engineering, MassachusettsInstitute of Technology, 77 Massachusetts Avenue,Cambridge, MA 02139, USA. Koç University, Departmentof Electrical and Electronics Engineering, Sar ı yer, 34450Istanbul, Turkey. *These authors contributed equally to this work. † Corresponding author. Email: [email protected]
Fig. 1. Local control and measurement of the magnon chemical potential. ( A ) Sketch of anNV spin locally probing the magnetic fields generated by magnons in a 20-nm-thick YIG filmgrown on a Gd Ga O (GGG) substrate. ( B ) Sketch of the magnon dispersion and the magnondensity, which falls off as 1/energy (1/ E ), as indicated by the fading colors, at zero chemicalpotential. ( C ) Driving at the FMR increases the magnon chemical potential. The NV spin probesthe magnon density at the NV ESR frequencies w T . ( D ) Photoluminescence image showing adiamond nanobeam containing individually addressable NV sensor spins positioned on top ofthe YIG film. A 600-nm-thick Au stripline (false-colored yellow) provides MW control of the magnonchemical potential and the NV spin states. A 10-nm-thick Pt stripline (false-colored gray)provides spin injection through the spin Hall effect. ( E ) Scanning electron microscope imageof representative diamond nanobeams. on J u l y , h tt p :// sc i en c e . sc i en c e m ag . o r g / D o w n l oaded f r o m SPINTRONICS
Control and local measurement of thespin chemical potential in amagnetic insulator
Chunhui Du, * Toeno van der Sar, * Tony X. Zhou, * Pramey Upadhyaya, Francesco Casola,
Huiliang Zhang,
Mehmet C. Onbasli,
Caroline A. Ross, Ronald L. Walsworth,
Yaroslav Tserkovnyak, Amir Yacoby † The spin chemical potential characterizes the tendency of spins to diffuse. Probing thisquantity could provide insight into materials such as magnetic insulators and spinliquids and aid optimization of spintronic devices. Here we introduce single-spinmagnetometry as a generic platform for nonperturbative, nanoscale characterization ofspin chemical potentials. We experimentally realize this platform using diamondnitrogen-vacancy centers and use it to investigate magnons in a magnetic insulator,finding that the magnon chemical potential can be controlled by driving the system ’ sferromagnetic resonance. We introduce a symmetry-based two-fluid theory describingthe underlying magnon processes, measure the local thermomagnonic torque,and illustrate the detection sensitivity using electrically controlled spin injection.Our results pave the way for nanoscale control and imaging of spin transport inmesoscopic systems. C ontrol and measurement of the chemicalpotential of a spin system can be used toexplore phenomena ranging from quan-tum phase transitions ( , ) to Bose-Einsteincondensation ( , ) and spin transport ingases and solid-state systems ( – ). In recentdecades, a large scientific effort has focused onharnessing spin transport for low-dissipationinformation processing ( , – ). In contrastto charge, spin is not a conserved quantity andnaturally decays on the nanoscale for a widerange of materials, including typical metals( , ), calling for a local detection technique.Compared to the centuries-old techniques forstudying charge transport, methods for prob-ing spin chemical potentials have only been de-veloped recently, with leading methods basedon the coupling between spin and charge trans-port ( , , , ) and inelastic light scattering( , ). Here we introduce a fundamentallydifferent approach, which uses a single sensorspin to measure the local magnetic field fluc-tuations generated by a thermal spin bath. Thisapproach is nonperturbative and provides spa- tial access to the spin chemical potential on ascale determined by the distance between thesensor spin and the system under study, openingthe door to imaging spin transport phenomena with resolutions down to the few-nanometerscale.Because the spin chemical potential is inherentlyrelated to spin fluctuations, it can be quantita-tively determined by measuring the magneticfields generated by these fluctuations. We dem-onstrate this principle using the excellent magnet-ic field sensitivity of the S = 1 electronic spinassociated with the nitrogen-vacancy (NV) centerin diamond ( , ). We measure the chemicalpotential of magnons — the elementary spin excita-tions of magnetic materials ( ) — in a 20-nm-thickfilm of the magnetic insulator yttrium-iron-garnet(YIG) on a ~100-nm-length scale (Fig. 1A). Ourmeasurements reveal that the magnon chemicalpotential can be effectively controlled by excitingthe system ’ s ferromagnetic resonance (FMR)(Fig. 1, B and C).Locally probing the weak magnetic fields gen-erated by the fluctuations of a spin system re-quires nanometer proximity of a magnetic fieldsensor to the system. We ensure such prox-imity by positioning diamond nanobeams ( )that contain individually addressable NV centersonto the YIG surface (Fig. 1, D and E). We use ascanning confocal microscope to optically lo-cate the NV centers and address their spin state( , ). A photoluminescence image (Fig. 1D)provides an overview of the system, showing anNV center (NV ) in a nanobeam that is locatedwithin a few micrometers from the gold andplatinum striplines used to excite magnons inthe YIG. RESEARCH Du et al ., Science , 195 –
198 (2017) 14 July 2017 Department of Physics, Harvard University, 17 OxfordStreet, Cambridge, MA 02138, USA. John A. PaulsonSchool of Engineering and Applied Sciences, HarvardUniversity, Cambridge, MA 02138, USA. Department ofPhysics and Astronomy, University of California, LosAngeles, 475 Portola Plaza, Los Angeles, CA 90095, USA. Harvard-Smithsonian Center for Astrophysics, 60 GardenStreet, Cambridge, MA 02138, USA. Department ofMaterials Science and Engineering, MassachusettsInstitute of Technology, 77 Massachusetts Avenue,Cambridge, MA 02139, USA. Koç University, Departmentof Electrical and Electronics Engineering, Sar ı yer, 34450Istanbul, Turkey. *These authors contributed equally to this work. † Corresponding author. Email: [email protected]
Fig. 1. Local control and measurement of the magnon chemical potential. ( A ) Sketch of anNV spin locally probing the magnetic fields generated by magnons in a 20-nm-thick YIG filmgrown on a Gd Ga O (GGG) substrate. ( B ) Sketch of the magnon dispersion and the magnondensity, which falls off as 1/energy (1/ E ), as indicated by the fading colors, at zero chemicalpotential. ( C ) Driving at the FMR increases the magnon chemical potential. The NV spin probesthe magnon density at the NV ESR frequencies w T . ( D ) Photoluminescence image showing adiamond nanobeam containing individually addressable NV sensor spins positioned on top ofthe YIG film. A 600-nm-thick Au stripline (false-colored yellow) provides MW control of the magnonchemical potential and the NV spin states. A 10-nm-thick Pt stripline (false-colored gray)provides spin injection through the spin Hall effect. ( E ) Scanning electron microscope imageof representative diamond nanobeams. on J u l y , h tt p :// sc i en c e . sc i en c e m ag . o r g / D o w n l oaded f r o m spin order parameter s n , where n is a unit vector:An incoming thermal magnon scatters off thetime-dependent n , generating two thermal mag-nons and transferring one unit of angular mo-mentum from the coherent spin density to theincoherent spin density ñ ( ). This process is theOnsager reciprocal of a local thermomagnonictorque – induced precession of n ( ), which isgaining increased attention in the field of spincaloritronics. By using a two-fluid phenomenology( ), we can describe the mutual dynamics of n and ñ according to the following hydrodynamicequation ( ) ~ n ! ¼ ~ a s m ħ þ h cos q n s ħ z % ð n ! ’ n Þ ð Þ Here, the first term on the right-hand side describesthe decay of thermal magnons into the lattice,with ~ a a constant related to Gilbert damping. Thesecond term describes the pumping of thermalmagnons by the FMR-induced precession of n ,with h parametrizing the local thermomagnonictorque between n and ñ and q n parametrizingthe instantaneous angle of n with respect to thesample-plane normal z . By setting ~ n ! ¼ ), we obtain m ¼ k h ~ a B cos q n ð Þ where q n is the average magnetization angle withrespect to the sample-plane normal and k is aparameter resulting from averaging over the el-liptical motion of n .A key prediction of this model, resulting fromsymmetry considerations ( ), is that the couplingbetween n and ñ vanishes for an in-plane orienta-tion of the magnetization (i.e., for q n ¼ p = B ext (Fig. 3D), as changing B ext changes q n in awell-defined way ( ). We find that the dependenceof the drive efficiency d m = dB on B ext in the low-power regime is accurately described by ourtheoretical prediction given by Eq. 4 (Fig. 3E),further supporting our conclusion that we areextracting the chemical potential correctly. Wehighlight that the precise knowledge of the in situdrive amplitude B AC provided by our NV sensor( ) is essential for this comparison. From a fit,we extract h ≈ − ( ), which is comparable tothe measured YIG Gilbert damping parameter a ≈ − ( ). According to the theoretical modelstudied in ( ), h describes the purely magnoniccontribution to Gilbert damping and may thusbe expected to be bounded by a . The compara-bility of h and a suggests that thermal magnonscan exert a torque large enough to induce amagnetization precession.Finally, we illustrated the power of our tech-nique by characterizing the chemical potentialthat results from electrically controlled spin injec-tion via the spin Hall effect (SHE). The SHE isa phenomenon originating from spin-orbit inter- action, in which a charge current generates atransverse spin current. Such a spin current canbe injected into a magnetic system, a techniquewidely used to study nonequilibrium magnonproperties ( , , , , ). Fig. 4A shows themeasured relaxation rate of NV , located ~1.7 m mfrom the edge of the Pt stripline (Fig. 1D), as afunction of the electrical current density J c in thePt. We observed a clearly asymmetric dependencethat is well described by a second-order polyno-mial (blue solid line) G ð m Þ ¼ G ð Þ þ G þ G ð Þ with G º J c the linear part and G º J c2 thequadratic part.Intuitively, we may expect the quadratic partof this polynomial to result from heating due toOhmic dissipation in the Pt wire and the linearpart to result from the SHE. We checked thisexpectation by exploiting the capability of the NVsensor to determine the temperature at the NVsite through measurements of changes in thezero-field splitting of the NV spin states ( , ).We assumed this temperature to be equal to thelocal YIG temperature because of the high thermalconductivity of diamond and the relatively insulat-ing properties of air. We then used Eq. 1 to cal-culate the expected change in NV relaxationover the experimentally determined relevant tem-perature range of ~40 K ( ). A comparison ofthis calculation with the data shows excellentquantitative agreement (Fig. 4B), illustrating the Du et al ., Science , 195 –
198 (2017) 14 July 2017
Fig. 3. Magnon chemical potential ( m ) underFMR excitation. ( A ) NV relaxation rate G as afunction of the on-chip power B of a magneticdrive field applied at the FMR frequency, fordifferent values of B ext . The gray line is the fitfrom Fig. 2A. Top, measurement sequence. ( B ) m as a function of B and B ext . m saturates at theminimum of the magnon band set by the FMRfrequency. ( C ) Field dependence of the saturatedvalue of the chemical potential m sat calculatedfrom averaging m in the region 0.05 mT < B <0.1 mT [see (B)]. The black curve is the FMR.The red and blue points are measured by usingthe NV m s = 0 ↔ – m s = 0 ↔ +1transitions, respectively. B ext is oriented alongthe NV axis, at a q = 65° angle with respect to thesample-plane normal for both NVs and a f = 52°( f = 6°) in-plane angle with respect to the Austripline for NV (NV ). The NV to YIG distanceis 65 ± 10 nm ( ). ( D ) At low B , m increaseslinearly at a rate d m = dB that depends on B ext .( E ) Field dependence of d m = dB extracted from(D). A comparison to theory yields the localthermomagnonic torque (see text). Fig. 3, A, B,and D, are shown with error bars in ( ). In Fig. 3,C and E, the error bars are comparable to orsmaller than the symbol size. RESEARCH |
REPORT on J u l y , h tt p :// sc i en c e . sc i en c e m ag . o r g / D o w n l oaded f r o m Magnons generate a characteristic magneticfield spectrum reflecting the occupation of themagnon density of states. We probe this spectrumusing the sensitivity of the NV spin relaxationrates G T to magnetic field fluctuations at theNV electron spin resonance (ESR) frequencies w T (Fig. 1, B and C) ( ). For a system at thermalequilibrium, these rates can be expressed as ( ) G T ð m Þ ¼ n ð w T ; m Þ ∫ D ð w T ; k Þ f ð k ; d Þ d k þ G T ð Þ Here, n ð w ; m Þ ¼ k B T ℏ w % m is the Rayleigh-Jeans dis-tribution (which is the Bose-Einstein distributionin the high temperature limit appropriate for ourroom-temperature measurements), m is the chem-ical potential, k B is Boltzmann ’ s constant, T is thetemperature, ħ is the reduced Planck ’ s constant, D ð w ; k Þ is the magnon spectral density, k is themagnon wave vector, f ð k ; d Þ is a transfer func-tion describing the magnon-generated fields atthe NV site, d is the distance of the NV to theYIG, and G T is an offset relaxation rate that isindependent of the magnon spectrum. From Eq.1, it is clear that, when G T ð m Þ ≫ G T , the chemicalpotential can be extracted in a way that is in-dependent of many details of both sensor andsystem: Normalizing the relaxation rate mea-sured at m = 0 by the relaxation rate measured at m > 0 yields m ¼ ħ w T ð % G T ð Þ G T ð m ÞÞ ð Þ As a first step in gaining confidence in thisprocedure, we probed the magnetic-noise spectrumof the YIG film in the absence of external drivefields. We measured the NV spin relaxation ratesas a function of an external magnetic field B ext applied along the NV axis [( ), fig. S4] andfound excellent agreement with the model describedby Eq. 1 (Fig. 2A). Qualitatively, the observed fielddependence can be understood by noting the highdensity of magnons just above the FMR frequency,which induces a peak in the m s = 0 ↔ – relaxation rate when the corresponding ESRfrequency crosses this region (Fig. 2B), where m s labels the electron-spin eigenstate. A fit allows usto extract the distance of the NV to the YIG film( ). In extracting the NV relaxation rates, we as-sumed the direct relaxation rate between the m s =±1 states to be negligible, as this transition is spinforbidden and insensitive to magnetic noise.Next, we studied the magnetic noise generatedby the system under the application of a microwave(MW) drive field of amplitude B AC ( ) using thesensitivity of the NV photoluminescence to magneticfields at the ESR frequencies. Fig. 2C shows thephotoluminescence of NV as a function of thedrive frequency and B ext . The straight lines resultfrom the expected decrease in NV fluorescencewhen the drive frequency matches one of the NVESR frequencies. In addition, the fluorescencedecreases when the MW excitation frequencymatches the calculated FMR condition of theYIG film. This effect results from an FMR-induced increase in the magnon density andassociated magnetic field noise at the NV ESRfrequencies ( , , ). A line cut at B ext =14.4 mT shows a typical FMR linewidth of 8 MHzthat we observe in these measurements (Fig.2C, inset).If the magnon occupation under the applicationof an FMR drive field can be described by theRayleigh-Jeans distribution — as may be expectedbecause magnon thermalization is mainly drivenby the exchange interaction, which is by far thelargest energy scale in our system (~THz) — Eq. 2 allows us to extract the chemical potential m bymeasuring the NV relaxation rates. We measuredthe power dependence of the relaxation rate G % of NV at several values of the external magneticfield (Fig. 3A). We found that G % increases withdrive power B , consistent with the FMR-induceddecrease of NV photoluminescence shown inFig. 2C. Notably, G % saturates as a function ofdrive power; moreover, the corresponding chem-ical potential saturates exactly at the minimumof the magnon band set by the FMR ( ) (Fig. 3B).Because the band minimum is the maximum al-lowed value for the chemical potential of a bosonsystem in thermal equilibrium ( ), it provides adistinct reference point that is independent of anyassumptions or experimentally unknown parame-ters. The precise match between the saturated valueof the extracted chemical potential and the bandminimum therefore underscores the validity ofour method to extract the chemical potential. Weconfirmed this saturation behavior using two dif-ferent NV centers over a broad range of magneticfields (Fig. 3, B and C), providing compellingevidence that the magnon density is describedby a finite chemical potential in the spectralregion probed in this measurement. We inde-pendently verified that the magnon temperatureincreases by less than 5 K at the highest drivepower used in the measurements of Fig. 3B anddoes not significantly influence the extractedchemical potentials [see section S6 of ( )].Another notable feature of our data is the ini-tial slow increase of chemical potential ob-served at small B ext and low drive power (Fig. 3,B and D).The build-up of chemical potential under theapplication of an FMR drive field can be under-stood as a pumping process of thermal magnonsby the FMR-induced precession of the coherent Du et al ., Science , 195 –
198 (2017) 14 July 2017
Fig. 2. Tools for characterizing the magnonchemical potential. ( A ) Measured spin relaxa-tion rates G T corresponding to the m s = 0 ↔ ±1transitions of NV as a function of an externalmagnetic field B ext . A fit to Eq. 1 yields thedistance d of NV to the YIG film. The measure-ment sequence is depicted on top: The NVspin is prepared in the m s = 0 state by using a~3- m s green-laser pulse and left to relax for atime t . At the end of this time, we characterize theoccupation probabilities of the m s = 0, –
1, and+1 states by using MW pi pulses ( p ) on theappropriate ESR transitions and measuringthe spin-dependent photoluminescence duringthe first ~600 ns of a green-laser readout pulse.We extract G T by fitting the resulting data witha three-level model, as further detailed in ( )and fig. S4. B ext is applied along the axis of NV , ata q = 65° angle with respect to the sample-plane normal and a f = 52° in-plane angle withrespect to the Au stripline (see inset). ( B ) Sketchof the magnon density and the NV ESR fre-quencies versus B ext . ( C ) Normalized photoluminescence (PL) of NV as a function of B ext and the frequency of a 0.17-mT MW drive field. The labeledand unlabeled straight lines correspond to the NV ESR transitions in the electronic ground and excited states, respectively ( ). Inset: Linecut showing the8-MHz linewidth of the YIG FMR at B ext = 14.4 mT. RESEARCH |
R EPORT on J u l y , h tt p :// sc i en c e . sc i en c e m ag . o r g / D o w n l oaded f r o m (a)
NV centers in diamond are point defects where one carbonatom in the diamond’s crystal lattice is replaced by a nitrogenatom (N) and an adjacent lattice site is left empty [61]. Dueto their exceptional sensitivity to magnetic fields, they haverecently emerged as a minimally-invasive probe of magneticsystems, which provides a frequency resolution not achievableby other techniques [60]. For the purpose of this tutorial, itsuffices to model a NV center as a three-level spin system( | S | =
1) subjected to an external field B ( r nv ) . The externalfield accounts for the magnetic noise generated by an adjacentsystem and for a static field applied to split the degeneracyof the m s = ± S , set at theposition r nv , is oriented along its anisotropy axis ˆ n nv , withˆ n nv · ˆ z = cos θ . The corresponding Hamiltonian can be written as H nv = DS z + γ e S · B nv ( r nv ) , (31)where D = .
87 GHz is the ground state zero-field splittingbetween the m s = m s = ± γ e =
28 GHz · T − is the gyromagnetic ratio of the electronicspin. Here, we have introduced B nv ( r nv ) = R x ( θ ) B ( r nv ) ,where R x ( θ ) is a rotation matrix that allows us to easily dis-tinguish between the longitudinal and transverse componentsof the field with respect to the NV anisotropy axis.NV magnetometry and relaxometry are used as probes of,respectively, static and dynamical properties of magnetic sys-tems [60]. The quantity measured in a magnetometry ex-periment is the projection of the magnetic field onto the NVanisotropy axis, B nv , z , to which the NV electron spin reso-nance splitting is first-order sensitive. The full vector field canbe reconstructed from the field component B nv , z and comparedwith the one generated by a given magnetic texture [60].In a relaxometry setup, the relevant quantities are the fieldcomponents transverse to the NV anisotropy axis, B nv , x and B nv , y . Up to leading order in perturbation theory, the Zee-man coupling between the NV spin and the transverse fieldcomponents induces NV transitions between the spin states m s = ↔ ± ± ω . For a NV-center spin set at a height d above a U (1)- and translationally-symmetric magnetic system, the relaxation rate at frequency ω can be written as [62] Γ ( ω ) = f ( θ ) (cid:90) ∞ dk k e − kd [ C xx ( k , ω ) + C zz ( k , ω )] , (32)with f ( θ ) = ( γγ e ) ( − cos 2 θ ) / π . Here, C αβ ( k , ω ) is theFourier transform of the spin-spin correlator C αβ ( r , r (cid:48) ; t ) = (cid:104){ ˆ s α ( r (cid:48) , t ) , ˆ s β ( r , ) }(cid:105) of the magnetic system, with α , β = x , y , z . Equation (32) shows that the NV relaxation rate is ameasure of the magnetic noise transverse, C xx , and longitudi-nal, C zz , to the equilibrium orientation of the order parame-ter n (cid:107) ˆ z of the magnetic system. Invoking the HP transfor-mations (1) and (10) for, respectively, a ferromagnetic (FM)and an antiferromagnetic (AFM) system, it is easy to see thatthe transverse spin-spin correlator accounts for one-magnonprocesses, i.e., the creation (or annihilation) of a magnon atfrequency ω . Thus, it is proportional to the thermal magnondistribution function at frequency ω , i.e., C FM xx ( k , ω ) ∝ n BE (cid:18) ¯ h ω − µ k B T (cid:19) , (33) C AFM xx ( k , ω ) ∝ n BE (cid:18) ¯ h ω ± µ k B T (cid:19) . (34)Equations (32), (33) and (34) show that the NV relaxationrate can provide a direct measurement of the magnon chemi-cal potential, which has proven to be hardous to perfom in spintransport setups [27]. The first measurement of the magnonchemical potential via NV-center relaxometry was performedby Du and coauthors [29]. In their room-temperature setup,sketched in Fig. 6(a), a NV center is set above a YIG film. Astripline drives the ferromagnetic resonance of the YIG film,i.e., it excites the coherent spin dynamics obeying Eqs. (4), (8)and (9). The coherent spin dynamics, in turn, pumps spin an-gular momentum into the thermal magnon cloud. This mech-anism increases thermal magnon density and, consequently,the magnon chemical potential (9), as depicted in Fig. 6(b).Figure 6(c) shows the dependency of the NV-center relax-ation frequencies and of the YIG ferromagnetic resonance fre-quency on the static external field. The transverse noise cor-responding to single-magnon processes can be probed by aNV-center relaxation rate at frequencies for which the magnonthermal population is finite. The increase in the magnonchemical potential was found to be directly proportional tothe driving power, and, as predicted by Bose-Einstein statis-tics, to saturate at the ferromagnetic resonance frequency, asshown in Fig. 6(c). A further analysis of the experimentaldata [29] lead to the extrapolation of the parameter η (8).The chemical potential of a U (1)-symmetric antiferromag-netic system might be probed in a similar fashion [36]. Thetwo antiferromagnetic normal modes undergo precessionswith opposite handedness, as shown in Fig. 1(b); thus, theycan be selectively excited by an ac field with matching po-larization. Resonantly driving the coherent spin dynamics ofthe antiferromagnetic modes α and β increases their thermalmagnon populations, as dictated by Eqs. (20). The increasein the thermal population of α ( β ) leads to a increase of anegative (positive) chemical potential, which can be measuredaccording to Eqs. (32) and (34). FIG. 7. The interaction between the NV-center spin and a nearbymagnetic system, here depicted as gas of magnons with spin ¯ h andfrequency ω k (with ω k = = ∆ ), leads to a NV-center transition rate Γ ( ω ) with emission of energy ω . When ω > ∆ , the latter can resultin the creation of a magnon at frequency ω k = ω or in a magnonscattering with energy gain ¯ h ω . These events are accounted for, re-spectively, by the transverse, i.e., C xx ( k , ω ) , and by the longitudinal,i.e., C zz ( k , ω ) , spin-spin correlation function. When ω > ∆ , the re-laxation rate is typically dominated by one-magnon processes, i.e., C xx ( k , ω ) (cid:29) C zz ( k , ω ) . Conversely, for ω < ∆ , one-magnon eventsare suppressed, i.e., C xx ( k , ω ) → In order to probe the transverse noise, the NV transitionfrequency ω must be larger than the gap ∆ of the spin-wave dispersion. At lower frequencies, there are no avail-able magnon states and, thus, Eqs. (33) and (34) vanish. For ω > ∆ , the transverse noise dominates over the longitudinalnoise [29,62], while for ω < ∆ the longitudinal noise rep-resents the leading contribution to the relaxation rate (32),as sketched in Fig. 7. By invoking the HP transforma-tions (1) and (10), for, respectively, a ferromagnetic and anantiferromagnetic system, the longitudinal spin-spin correla-tor C zz ( k , ω ) can be expressed in terms of two-magnon scatter-ing processes, i.e., a magnon with frequency ω + ω scattersinto a magnon state with frequency ω , or vice versa , emitting(or absorbing) magnetic noise at frequency ω . The detailsof how two-magnon scattering processes occur depend on thespin transport properties of the system. For diffusive magnontransport in the absence of a heat gradient (7, 15), the imagi-nary part of the spin susceptibility can be found as [62] χ (cid:48)(cid:48) zz ( k , ω ) = χ ¯ h D ω k ( k + /(cid:96) s ) + ( ω / D ) , (35)where χ is the static uniform longitudinal susceptibility and D = σ / χ the spin-wave diffusion coefficient. In thermal equi-librium, the fluctuation-dissipation theorem dictates [63] C zz ( k , ω ) = coth (cid:18) ¯ h ω k B T (cid:19) χ (cid:48)(cid:48) zz ( k , ω ) . (36)Wang and coauthors [64] recently measured the relaxationrate (32) of a YIG film at frequencies below the magnetic gap.Using Eqs. (35) and (36) they extracted a spin diffusion lengthof (cid:96) s ∼ . µ m, which was further corroborated by a non-localspin transport measurement performed on the same sample.An analogous method can be used to probe non-invasively thespin transport properties of collinear antiferromagnets. Wangand coauthors [55] probed the time-dependent fluctuations ofthe longitudinal spin density of α -Fe O . They estimated the0spin diffusion length (cid:96) s to be 3 µ m at 200 K, which is in agree-ment with the values reported by non-local spin transport ex-periments [17]. IV. SUMMARY AND PERSPECTIVES
In this tutorial, we have outlined the fundamental proper-ties of collective spin excitations, i.e., magnons, in collinearmagnetic insulating systems. We have reviewed their statis-tical and transport properties and discussed the key ingredi-ents of the coupling between magnetic and lattice degrees offreedom. Making use of this theoretical framework, we haveintroduced the reader to two magnon sensing techniques, i.e.,spin transport setups and NV-center relaxometry.Here we have restricted our discussion to simplemonoatomic ferromagnetic and two-sublattice antiferromag-netic systems. However, there is a variety of collinear insula-tors with different crystalline structures and spin interactionsthat continue to be discovered and that can be addressed withthe methods presented here.Finally, while non-collinear magnetic systems do not sup-port long-range diffusive spin transport, their spin non-conserving interactions might endow the spin-wave bandswith a nontrivial topological structure. The emergenceof topologically-protected long-range propagating magnonmodes and their non-Hermitian topology represent an emer-gent promising platform to investigate spin transport andnovel magnetic phenomena. Igor Zuti´c, Jaroslav Fabian, and S. Das Sarma, Rev. Mod. Phys. , 323(2004). J. Sinova and Igor Zuti´c, Nat. Mat. , 368 (2012). M. Z. Wu and A. Hoffmann,
Recent Advances in Magnetic Insulators -From Spintronics to Microwave Applications (Elsevier, Amsterdam, 2013). A. V. Chumak, V. I. Vasyuchka, A. A. Serga, and B. Hillebrands, Nat. Phys. , 453 (2015). M. I. Dyakonov, and V. I. Perel, Phys. Lett. , 459 (1971). M. N. Tkachuk, B. P. Zakharchenya, V. G. Fleisher, Z. Eksp. Teor. Fiz.Pis’ma , 47 (1986); Sov. Phys. JETP Lett. , 59 (1986). J. E. Hirsch Phys. Rev. Lett. , 1834 (1999). Y. K. Kato, R. C. Myers, A. C. Gossard, and D. D. Awschalom, Science , 1910 (2004). J. Wunderlich, B. Kaestner, J. Sinova, and T. Jungwirth, Phys. Rev. Lett. , 047204 (2005). A. Hoffmann, IEEE Trans. Magn. , 5172 (2013). J. Sinova, S. O. Valenzuela, J. Wunderlich, C. H. Back, and T. Jungwirth,Rev. Mod. Phys. , 1213 (2015). K. Uchida, S. Takahashi, K. Harii, J. Ieda, W. Koshibae, K. Ando, S.Maekawa, and E. Saitoh, Nature , 778 (2008); K.-i. Uchida, H. Adachi,T. Ota, H. Nakayama, S. Maekawa, and E. Saitoh, Appl. Phys. Lett. ,172505 (2010) G. E. W. Bauer, E. Saitoh, and B. J. van Wees, Nat. Mater. , 391 (2012);H. Adachi, K. Uchida, E. Saitoh, and S. Maekawa, Rep. Prog. Phys. ,036501 (2013). L. J. Cornelissen, J. Liu, R. A. Duine, J. B. Youssef, and B. J. Van Wees,Nat. Phys. , 1022 (2015). K. An, K. S. Olsson, A. Weathers, S. Sullivan, X. Chen, X. Li, L. G. Mar-shall, X. Ma, N. Klimovich, J. Zhou, L. Shi, and X. Li, Phys. Rev. Lett. , 107202 (2016). B. L. Giles, Z. Yang, J. S. Jamison, J. M. Gomez-Perez, S. Vélez, L. E.Hueso, F. Casanova, and R. C. Myers, Phys. Rev. B , 180412(R) (2017). R. Lebrun, A. Ross, S. A. Bender, A. Qaiumzadeh, L. Baldrati, J. Cramer,A. Brataas, R. A. Duine, and M. Klaui, Nature , 222 (2018); R. Lebrun, A. Ross, O. Gomonay, V. Baltz, U. Ebels, A.-L. Barra, A. Qaiumzadeh, A.Brataas, J. Sinova, and M. Klaui, Nat. Comm. , 6332 (2020). T. Holstein and H. Primakoff Phys. Rev. , 1098 (1940). H. A. Bethe, Z. Physik , 20 (1931); L. Hulthén, Arkiv Mat. Astron. Fysik , 1 (1938); F. J. Dyson Phys. Rev. , 1217 (1956). T. Gilbert, IEEE Trans. Magn. , 3443 (2004). M. Lakshmanan, Phil. Trans. R. Soc. A , 1280 (2011). B. Flebus, R. A. Duine, and H. M. Hurst, Phys. Rev. B 102, (2020). S. O. Demokritov, V. E. Demidov, O. Dzyapko, G. A. Melkov, A. A. Serga,B. Hillebrands, and A. N. Slavin, Nature , 430 (2006). E. B. Sonin, Advances in Physics, 59(3), (2010). S. Takei, B. I. Halperin, A. Yacoby, and Y. Tserkovnyak, Phys. Rev. B ,094408 (2014). W. Yuan, Q. Zhu, T. Su, Y. Yao, W. Xing, Y. Chen, Y. Ma, X. Lin, J. Shi, R.Shindou, X. C. Xie, and W. Han, Sci. Adv. , 1098 (2018). L. J. Cornelissen, K. J. H. Peters, G. E. W. Bauer, R. A. Duine, and B. J.van Wees, Phys. Rev. B , 014412 (2016). V. E. Demidov, S. Urazhdin, B. Divinskiy, V. D. Bessonov, and A. B. Rinke-vich, Nat. Commun. , 1579 (2017). C. R. Du, T. Van der Sar, T. X. Zhou, P. Upadhyaya, F. Casola, H. Zhang, M.C. Onbasli, C. A. Ross, R. L. Walsworth, Y. Tserkovnyak, and A. Yacoby,Science , 195 (2017). B. Flebus, P. Upadhyaya, R. A. Duine, and Y. Tserkovnyak, Phys. Rev. B , 214428 (2016). T. Jungwirth, X. Marti, P. Wadley, and J. Wunderlich, Nat. Nanotechnol. , 231 (2016). V. Baltz, A. Manchon, M. Tsoi, T. Moriyama, T. Ono, and Y. Tserkovnyak,Rev. Mod. Phys. , 015005 (2018). V. Cherepanov, I. Kolokolov, and V. L’vov, Phys. Rep. , 81 (1993). S. M. Rezende, R. L. Rodríguez-Suárez, and A. Azevedo, Phys. Rev. B ,014425 (2016). R. M. White, M. Sparks, and I. Ortenburger, Phys. Rev. , A450 (1965). B. Flebus, Phys. Rev. B , 064410 (2019). F. Keffer and C. Kittel, Phys. Rev. , 329 (1952). S. M. Rezende, A. Azevedo, and R. L. Rodríguez-Suárez, J. Appl. Phys. , 151101 (2019). A. Kamra, R. E. Troncoso, W. Belzig, and A. Brataas, Phys. Rev. B ,184402 (2018). L. Onsager, Phys. Rev. (4), 405 (1931). H. B. Callen, Physical Review (11), 1349 (1948). F. Reif,
Fundamentals of Statistical and Thermal Physics (Mc Graw-Hill,New York, 2008). Y. Tserkovnyak, A. Brataas, G. E. W. Bauer Phys. Rev. Lett. (2002),117601 (2002). Y. Tserkovnyak, A. Brataas, G. E. W. Bauer, Phys. Rev. B , 224403(2002). Y. Tserkovnyak, A. Brataas, G. E. W. Bauer, B.I. Halperin Rev. Mod. Phys. (2005). R. Cheng, J. Xiao, Q. Niu, and A. Brataas, Phys. Rev. Lett. , 057601(2014). A. Ruckriegel, P. Kopietz, D. A. Bozhko, A. A. Serga, and B. Hillebrands,Phys. Rev. B , 184413 (2014); S. Streib, N. Vidal-Silva, K. Shen, and G.E. W. Bauer, Phys. Rev. B , 184442 (2019). E. Abrahams and C. Kittel, Phys. Rev. 88, (1952); C. Kittel and E.Abrahams, Rev. Mod. Phys. 25, (1953); C. Kittel, Phys. Rev. , 836(1958). M. I. Kaganov and V. M. Tsukernik, Sov. Phys. JETP , 151 (1959). T. Kikkawa, K. Shen, B. Flebus, R. A. Duine, K. Uchida, Z. Qiu, G. E. W.Bauer, and E. Saitoh, Phys. Rev. Lett. , 207203 (2016), B. Flebus, K. Shen, T. Kikkawa, K. Uchida, Z. Qiu, E. Saitoh, R. A. Duine,and G. E. W. Bauer, Phys. Rev. B , 144420 (2017). A. S. Sukhanov, M. S. Pavlovskii, Ph. Bourges, H. C. Walker, K. Manna,C. Felser, and D. S. Inosov Phys. Rev. B , 214445 (2019). H. T. Simensen, R. E. Troncoso, A. Kamra, and A. Brataas, Phys. Rev. B , 064421(2019). J. Li, H. T. Simensen, D. Reitz, Q. Sun, W. Yuan, C. Li, Y. Tserkovnyak, A.Brataas, and J. Shi, Phys. Rev. Lett. , 217201 (2020). H. Wang, S. Zhang, N. J. McLaughlin, B. Flebus, M. Huang, Y. Xiao, E. E.Fullerton, Y. Tserkovnyak, C. R. Du, arXiv:2011.03905 (2020). F. L. Bakker, A. Slachter, J.-P. Adam, and B. J. van Wees, Phys. Rev. Lett. , 136601 (2010). A. Prakash, B. Flebus, J.Brangham, F. Yang, Y. Tserkovnyak, and J. P.Heremans, Phys. Rev. B , 020408(R) (2018). M. Agrawal, V. I. Vasyuchka, A. A. Serga, A. Kirihara, P. Pirro, T. Langner,M.B. Jungfleisch, A.V. Chumak, E.T. Papaioannou, and B. Hillebrands,Phys. Rev. B , 224414 (2014). S. M. Wu, W. Zhang, A. KC, P. Borisov, J. E. Pearson, J. S. Jiang, D. Le-derman, A. Hoffmann, and A. Bhattacharya, Phys. Rev. Lett. , 097204 (2016). F. Casola, T. van der Sar, and A. Yacoby, Nat. Rev. Mat. , 17088 (2018). E. Abe and K. Sasaki, J. Appl. Phys. , 161101 (2018). B. Flebus and Y. Tserkovnyak Phys. Rev. Lett. , 187204 (2018). R. Kubo, Rep. Prog. Phys. , 255 (1966). X. Wang, B. Flebus, Y. Xiao, H. Wang, C. Liu, E. Lee-Wong, M. Wu, H.Wang, Y. Tserkovnyak, E. E. Fullerton, and C. R. Du, in preparationin preparation