Current Fluctuations Driven by Ferromagnetic and Antiferromagnetic Resonance
CCurrent Fluctuations Driven by Ferromagnetic and Antiferromagnetic Resonance
Arne Brataas
Center for Quantum Spintronics, Department of Physics,Norwegian University of Science and Technology, NO-7491 Trondheim, Norway ∗ (Dated: August 17, 2020)We consider electron transport in ferromagnets or antiferromagnets sandwiched between metals.When spins in the magnetic materials precess, they emit currents into the surrounding conductors.Generally, adiabatic pumping in mesoscopic systems also enhances current fluctuations. We gener-alize the description of current fluctuations driven by spin dynamics in three ways using scatteringtheory. First, our theory describes a general junction with any given electron scattering properties.Second, we consider antiferromagnets as well as ferromagnets. Third, we treat multiterminal de-vices. Using shot noise-induced current fluctuations to reveal antiferromagnetic resonance appearsto be easier than using them to reveal ferromagnetic resonance. The origin of this result is that theassociated energies are much higher as compared to the thermal energy. The thermal energy governsthe Johnson-Nyquist noise that is independent of the spin dynamics. We give results for variousjunctions, such as ballistic and disordered contacts. Finally, we discuss experimental consequences. I. INTRODUCTION
In conductors, a bias voltage generates a net current.However, the current also fluctuates. Noise exists evenat equilibrium when the bias voltage is zero. At equilib-rium, the current fluctuations are related to the conduc-tance via the fluctuation-dissipation theorem as Johnson-Nyquist noise . When there are non-zero bias voltagescomparable to or larger than the thermal energy, thefluctuation-dissipation theorem does not apply. Instead,the current fluctuates due to shot noise since the elec-tron flow is in discrete quanta of the elementary charge − e . The shot noise reveals quantum transport featuresin nanostructures .In electron transport, at low temperatures, the trans-mission probabilities of waveguide eigenmodes { T n } de-termine the shot noise of phase-coherent conductors bi-ased by a voltage V : S = 2 e h eV (cid:88) n T n (1 − T n ) . (1)The shot noise expression (1) is general and captures thenature of many contacts, such as diffusive, ballistic, andtunnel junctions. The factor 1 − T n arises from the Pauliexclusion principle; two electrons cannot simultaneouslyoccupy the same waveguide mode. The sum is over thewaveguide modes labeled by n .Ferromagnets have intriguing transport propertiescaused by the coupling between electric currents, elec-tron spin currents, and localized spin dynamics. Currentscan induce spin dynamics by spin-transfer torques and spin-orbit torques . The magnetization directioncan be switched, or magnetic oscillations can be induced.These phenomena are of a fundamental importance andmight be utilized in magnetic random access memories, ∗ [email protected] spin-torque oscillators or spin-logic devices. These de-velopments have been reviewed in Refs. 19–22. The phe-nomenon reciprocal to spin-transfer torque is spin pump-ing, the emission of spin currents into metals induced byspin excitations in adjacent magnets . Spin pump-ing exposes details of the transport properties and spindynamics.Recently, antiferromagnetic spintronics has attractedconsiderable interest because of the intrinsic high fre-quencies, new features in spin dynamics, and robustnesswith respect to external magnetic field disturbances .Many of the phenomena in ferromagnets have similaror richer behavior in antiferromagnets. For instance,currents can switch the spin configurations , andantiferromagnetic resonance excitations can pump spincurrents .Usually, bias voltages induce electric currents and shotnoise as in Eq. (1). However, out-of-equilibrium currentscan be sustained by other methods using temporal exter-nal or internal drivers that modify the conductor proper-ties. Oscillating electric and magnetic fields can inducenet currents. Such drivers also enhance the electric cur-rent noise. In magnetic systems, dynamical spin excita-tions produce spin currents .Spin pumping also causes additional magnetizationdissipation . Through the fluctuation-dissipationtheorem, this implies that fluctuating spin currents as-sociated with spin pumping and spin transfer as wellexist . In a recent study, Ref. 47 obtained an expressionfor the electric (charge) current noise caused by ferro-magnetic resonance excitations in ferromagnetic-normalmetal-ferromagnetic double tunnel barrier systems. Theparticularly nice feature is that the mechanism does notrequire spin-orbit-induced spin-to-charge conversion suchas the spin Hall and inverse spin Hall effects. It is also anew channel for detecting and characterizing ferromag-netic resonance and electron transport in magnetic con-ductors.Theoretically, scattering matrices capture electrontransport in nanostructures well . They can also de- a r X i v : . [ c ond - m a t . m e s - h a ll ] A ug scribe current-induced torques . Scattering matricesalso capture effects due to temporal external or inter-nal drivers. To the lowest order in the driver frequency,the pumped current is related to the stationary scat-tering properties . This feature considerably simpli-fies the description of adiabatic pumping such as spin-pumping . In general, scattering properties canalso describe the enhanced electric current noise due toperiodic drivers. For the case when only one waveguidemode is linked to each reservoir, Ref. 58 obtained an ex-pression for the current noise in terms of the dynamicalscattering properties of the device.We consider a magnet that is in contact with normalmetal leads. Our purpose is to obtain a general expres-sion for how spin excitations in ferromagnetic and anti-ferromagnetic structures generate the thermal and shotnoise of the electric current. To this end, we generalizethe results of Ref. 47 in three ways: 1) the formalism isvalid for arbitrary junctions, 2) the theory applies to spindynamics in ferromagnets and antiferromagnets, and 3)multiterminal devices are treated. In this way, we obtaingeneral results for electric current noise driven by spindynamics in magnetic materials in arbitrary junctions.We will find that, when spin angular momentum is con-served, the noise vanishes when the magnet is insulating.Our results are therefore most relevant for conductingsystems.In antiferromagnets, the deduced expressions for thenoise are entirely new to the best of our knowledge. Ourgeneral results in the case of ferromagnetic excitationsare also new. In the limited case of two-terminal double-barrier tunnel ferromagnetic junctions, our general re-sults agree with the results of Ref. 47 by taking into ac-count random disorder in our formulation. Since we usean entirely different approach, this agreement establishesthe consistency of both treatments in this limit. We dis-cuss the fact that other junctions have different behaviorsin ferromagnets.We have organized the presentation as follows. Our pa-per first gives the main results and consequences beforeproceeding section by section with more details of thederivations. The next section II introduces the modeland presents the main results. We will find that four fac-tors determine the shot noise: i) The electron-transport-related shot noise coefficients, ii) the driver frequency,iii) the thermal energy, and iv) the spin-dynamics fac-tor. Section III discusses the specifics of the influenceof ferromagnetic and antiferromagnetic dynamics drivenby magnetic fields that govern the spin-dynamics factor.Then, in section IV, we discuss the shot noise coefficientsin various junctions, such as ballistic and disordered con-tacts, both in antiferromagnets and ferromagnets. Wepresent the general theory of adiabatic pumping-inducedelectric current noise in section V. Section VI applies thegeneral theory in section V to derive the spin dynamics-driven shot noise in section III. We conclude our pre-sentation in section VII. Finally, we derive the generalscattering theory of adiabatic driven enhanced electric current noise in Appendix A. II. MODEL AND MAIN RESULTS
We consider a magnet embedded between metals (orsemiconductors) in an open circuit. At equilibrium, elec-tric currents fluctuate in the metals. In the magnet, therecan be thermally induced spin fluctuations or coherentspin precessions caused by external forces. We considerthe latter case that the spin dynamics is coherent anddominated by external drivers as in ferromagnetic reso-nance or antiferromagnetic resonance. It is straightfor-ward to generalize our results to explicitly include con-tributions from incoherent spin dynamics relevant whenthe external drive is weak or absent.When the spins in the magnet precess, the fluctuationsare enhanced. Fig. 1 schematically depicts the system ina two-terminal configuration. Our results are also validfor many terminals. The unit vector aligned with the
Metal MetalMagnet δ j l (t) δ j r (t)n(t) FIG. 1. Schematic description of a metal-magnet-metal sys-tem. An open circuit (not shown) is connected to the system.The electric currents fluctuate. The spin dynamics in themagnet described by precession of the temporal unit vectoralong the order parameter n ( t ) enhance the current fluctua-tions. order parameter, the magnetization in ferromagnets andthe staggered field in antiferromagnets, n , is homoge-neous. When external magnetic fields or currents drivethe system, the order parameter n precesses around anequilibrium direction. While we subsequently developa formulation describing general junctions that may in-clude the spin-orbit interaction and magnetic impurityscattering, our first and primary focus is on systems withthe conservation of spin angular momentum. Giant mag-netoresistance, tunnel magnetoresistance, spin-transfertorques, and spin pumping are examples of central phe-nomena in such systems.In systems with the conservation of spin angular mo-mentum, two independent scattering matrices, S ↑ and S ↓ , for spin-up and spin-down electrons govern electrontransport. The scattering matrices contain all details ofthe junctions related to the interfaces between the metalsand magnets, band structure, and bulk and surface im-purity scattering. We evaluate the electric current andthe associated noise in the metallic leads. The electriccurrent direction is towards the magnet. While our for-malism is valid irrespective of the magnet‘s conductingproperties, it is most relevant for metallic systems sincewe will demonstrate that, in the absence of spin-orbitcoupling, the electric noise vanishes when there is no flowof electric charge between the leads. Furthermore, weconsider systems where itinerant electrons carry the cur-rent and spin currents carried by localized spins can bedisregarded. When the spin angular momentum is con-served, our main result is that the low-frequency electriccurrent noise in the presence of coherent spin excitationshas two contributions: p ζη = p (th) ζη + p (sn) ζη , (2)where ζ and η label the leads. Electric current conserva-tion ensures that (cid:80) ζ p ζη = 0 = (cid:80) η p ζη .In Eq. (2), the first term describes thermal Johnson-Nyquist noise, which is independent of the spin dynamicsand determined by the conductance tensor G and thethermal energy k B T : p (th) ζη = ( G ζη + G ηζ ) k B T . (3)Microscopically, the conductance tensor is a sum over thescattering properties of two spin components: G ζη = e h Tr o (cid:104) δ ζη − S ↑† ηζ S ↑ ζη (cid:105) + e h Tr o (cid:104) δ ζη − S ↓† ηζ S ↓ ζη (cid:105) . (4)The trace, Tr o , is over orbital (”o”) degrees of freedomonly, a sum over the waveguide modes in the leads.The second and more interesting contribution to thenoise in Eq. (2) is the shot noise driven by the coherentspin dynamics. We obtain our main result for the zero-frequency shot noise: p (sn) ζη = A ζη + A ηζ (cid:20) (cid:126) ω coth (cid:126) ω k B T − k B T (cid:21) D ( ω ) , (5)where the spin-dynamics control the spin-dynamics fac-tor D ( ω ) that is independent of the electron transportproperties. We discuss D ( ω ) further below. D ( ω ) is apositive definite quantity. ω is the frequency of the spinexcitations. The diagonal components of the shot noiseof Eq. (5) are positive definite quantities as are the di-agonal components of the shot noise coefficients A . Theshot noise coefficients A ζη depend on the electron trans-port via products of the scattering matrices of spin-upand spin-down electrons: A ζη = e h Tr o δ ζη − (cid:88) αβ S ↑ ζα S ↓† αη S ↓ ηβ S ↑† βζ + e h Tr o δ ζη − (cid:88) αβ S ↓ ζα S ↑† αη S ↑ ηβ S ↓† βζ . (6) In the case of a two-terminal device, as in Fig. 1: A ll = 2 e h Tr o (cid:104) − ( r ↑ ll r ↓† ll + t ↑ lt t ↓† rl )( r ↓ ll r ↑† ll + t ↓ lt t ↑† rl ) (cid:105) , (7)where l means left and r means right, r is a reflectioncoefficient matrix, and t is a transmission coefficient ma-trix.In general, the shot noise parameter A of Eq. (6) differsfrom the conductance G of Eq. (4), as does the voltage-biased shot noise of Eq. (5) compared to the average cur-rent governed by the conductance G . As is well knownfor the latter case , signatures of the junctions and con-ductors can be distinguished by the ratio between thevoltage-biased shot noise parameter and the conductancevia the so-called Fano factor, F = (cid:80) n T n (1 − T n ) / (cid:80) n T n .For instance, in tunnel junctions F = 1, and in diffusivewires F = 1 / . The spin dynamics-driven shot noisereveals more aspects of the electron transport in spinmaterials. We will compute the central shot noise pa-rameter of Eq. (7) for ballistic and disordered junctionsin ferromagnets and antiferromagnets in Section IV.Ref. 58 found that the adiabatic pumping driven en-hanced noise was related to the behavior of two parti-cles injected into the system. In agreement with this,the shot noise parameter A of Eq. (6) contains productsof four scattering matrices describing two-particle pro-cesses. The new aspect of the shot noise parameter A of Eq. (6) is that two of the scattering matrices relateto spin-up electrons, and two relate to spin-down elec-trons. In contrast, spin pumping is a one-particle process.The spin-mixing conductance is a product of onespin-up scattering matrix and one spin-down scatteringmatrix, two scattering matrices in total. This is becausethe pumped spin current has spin along the directiontransverse to the magnetization direction, a linear com-bination of spin-up and spin-down states along the spinquantization axis that is parallel to the order parame-ter. Similarly, we note that the shot noise coefficientsof Eq. (6) have combinations of spin-up and spin-downproperties related to the same lead. Since spin dynamicsproduce electric (charge) current noise, a natural inter-pretation is that the fluctuations arise due to temporalfluctuations of the emissions of spin currents. While theemitted spin currents are instantaneously transverse tothe order parameter, they can be reconverted to electric(charge) currents at later times due to the spin-filteringeffect in magnetic materials.We observe that the shot noise parameter A vanisheswhen no transmission occurs between the left and rightreservoirs. This can be seen by letting t ↑ → t ↓ → . Beyondthe formulation in this section that is based on spin con-servation, spin-orbit coupling in heavy metals provides aconversion between charge and spin currents so that evenmagnetic insulators can become noisy . Such small ef-fects are proportional to the square of the small spin Hallangle.In the expression for the shot noise (5), the spin dy-namics solely determine the spin-dynamics factor D ( ω ).This quantity is small and related to the power ab-sorbed in resonance experiments , which implies thatit can be independently measured. At equilibrium andat sufficiently low temperatures, n ( t ) = n . Oscillat-ing transverse magnetic fields at frequency ω , H ( t ) = H + exp iωt + H − exp − iωt , induce small transverse ex-citations of the order parameter, δ n = n + exp iωt + n − exp − iωt . In the linear response, the changes in theorder parameter and the (external or current-induced)magnetic fields are related by the frequency-dependentspin susceptibility χ ( ω ), a 2 × i , so that n i ± = χ ij ± H j ± . In terms of the spin susceptibilities and theoscillating magnetic fields: D ( ω ) = (cid:88) i n i + n i − = (cid:88) ijk χ ij + χ ik − H j + H k − . (8)The spin susceptibilities χ ( ω ) have peaks at the reso-nance frequencies, as does D ( ω ). In section III, we givegeneric examples for central classes of anisotropies in fer-romagnets and antiferromagnets. In the linear response,the transverse excitations are small. Therefore, the factor D is small. Nevertheless, distinguishing the shot noisefrom the thermal noise should be possible because theformer has a strong dependence on the frequency of thedriver, while the latter has no such features. Subtract-ing the frequency-independent background thermal noisereveals the shot noise.The shot noise of Eq. (5) takes a different form depend-ing on the ratio between the energy quantum associatedwith the time dynamics, (cid:126) ω , and the thermal energy, k B T . At low temperatures, when (cid:126) ω (cid:29) k B T , the shotnoise becomes: p (sn) ζη ≈ A ζη + A ηζ | (cid:126) ω | D ( ω ) . (9)The shot noise can be distinguished from direct heat-ing by the different frequency dependence. The low-temperature shot noise of Eq. (9) is linear in the absolutevalue of the excitation frequency ω relative to the spin-dynamics factor D ( ω ) that can be independently mea-sured.We find below that A ∼ G in many systems. Theratio between the shot noise of Eq. (5) and the thermalnoise of Eq. (3) at low temperatures is then p sn ζη /p th ζη ∼| (cid:126) ω | D ( ω ) /k B T . Since the transverse precession angle issmall, typically D ( ω ) ∼ − at resonance, but the pos-sibly large prefactor | (cid:126) ω | /k B T will increase the ratio be-tween the shot noise and the thermal noise from thisvalue. Stronger external drives can also enhance D ( ω ). In contrast, at high temperatures, k B T (cid:29) (cid:126) ω , the shotnoise is smaller. We can expand the shot noise in thesmall parameter (cid:126) ω and obtain: p (sn) ζη ≈ A ζη + A ηζ (cid:126) ω ) k B T D ( ω ) . (10)At high temperatures, the shot noise of Eq. (10) is sup-pressed by a factor | (cid:126) ω | / k B T with respect to the lowtemperature limit of the shot noise of Eq. (9).Ferromagnets typically have resonance frequencies lessthan 100 GHz. These frequencies correspond to a lowtemperature of less than 1 K. Transport measurementsin this temperature range can reveal the low-temperatureshot noise (9). Such and considerably lower-temperaturemeasurements are standard in the study of the fractionalquantum Hall effect and require sophisticated cryogenicinstrumentation. At the temperature of liquid helium,approximately 4 K, the ratio between the resonance en-ergy and the thermal energy is approximately 0.2.The resonance frequencies in antiferromagnets can beone to two orders of magnitude higher than those inferromagnets. Therefore, detecting the low-temperaturelimit of the shot noise of Eq. (9) appears to be easierfor antiferromagnets. Antiferromagnets can have reso-nance frequencies in the THz range. We can then expectto observe low-temperature shot noise (9) at tempera-tures below 10 K when an antiferromagnet precesses atits resonance frequency. At room temperature, the ra-tio between the high-temperature shot noise of Eq. (10)and the low-temperature shot noise of Eq. (9) is on theorder 2 × − . Such corrections are small, but their mea-surement might be possible since corrections due to, e.g.,the spin Hall magnetoresistance (SMR), are of a similarmagnitude and routinely probed .We conclude that detection of low-temperature shotnoise (9) should be possible in antiferromagnets and, withcryogenic techniques, in ferromagnets. Measurement ofthe high-temperature shot noise (10) is possible in bothsystems. III. SPIN DYNAMICS
In this section, we will compute the spin-dynamics fac-tor D ( ω ) in ferromagnets and antiferromagnets.Consider a uniaxial ferromagnet with the easy axisalong the z direction. The free energy density is f F = − M γ ω A m z + δf F , (11)where m is a unit vector along the magnetization withmagnitude M and ω A is the anisotropy energy. A trans-verse oscillating magnetic field drives the spin dynam-ics via the additional contribution to the free energy, δf F = ω H ⊥ ( m x cos ωt + m y sin ωt ) M/γ , where ω H ⊥ isthe magnitude of transverse magnetic field in units of fre-quency. We compute the spin susceptibility that governsthe spin dynamics factor (8) from the Landau-Lifshitz-Gilbert equation ∂ m ∂t = − m × ω eff + α m × ∂ m ∂t , (12)where the effective field ω eff depends on the free energydensity (11) as ω eff = − γδf F /M δ m and α is the Gilbertdamping constant. In linear response, the spin dynamicsfactor (8) then becomes D F ( ω ) = ω H ⊥ ω − ω A ) + α ω ] . (13)As in Eq. (8), the spin dynamics factor of Eq. (13) isquadratic in the transverse fields, represented by theirmagnitudes ω H ⊥ in units of frequency. At resonance, D F ( ω A ) = ( ω H ⊥ /αω A ) / z direction. The free energydensity is: f AF = L γ (cid:2) ω E m − ω A n z (cid:3) + δf AF , (14)where n is a unit vector along the staggered field, m is the dimensionless small magnetic moment, L isthe magnitude of the staggered magnetization, γ isthe gyromagnetic ratio, ω E is the exchange energy,and ω A is the anisotropy energy. A transverse os-cillating magnetic field drives the spin dynamics viathe additional contribution to the free energy, δf AF = ω H ⊥ ( m x cos ωt + m y sin ωt ) L/γ . The coupled Landau-Lifshitz-Gilbert equations for the staggered field n andthe magnetic moment m are ∂ n ∂t = − n × ω m,eff − m × ω n,eff + α n × ∂ m ∂t + α m × ∂ n ∂t , (15) ∂ m ∂t = − n × ω n,eff − m × ω m,eff + α n × ∂ n ∂t + α m × ∂ m ∂t , (16)where the effective fields ω n,eff and ω m,eff depend on thefree energy density (14) as ω n,eff = − γδf AF /Lδ n and ω m,eff = − γδf AF /Lδ m .In linear response, and in the exchange approximation, ω E (cid:29) ω A , the spin dynamics factor (8) becomes: D AF M ( ω ) = ω ω H ⊥ ω − ω r ) + 8 α ω ω E , (17)where ω r = √ ω A ω E is the resonance energy and α is the Gilbert damping constant. As in Eqs. (8) and(13), the spin dynamics factor of Eq. (17) is quadraticin the transverse fields, represented by their magnitudes ω H ⊥ in units of frequency. At resonance, D AF M ( ω r ) =( ω H ⊥ /αω E ) /
8. Generalizations to other anisotropies and the inclu-sion of effects arising from external magnetic fields arestraightforward in both antiferromagnets and ferromag-nets.
IV. JUNCTIONS
In this section, we compute the shot noise coefficients A for simple models of ballistic and disordered junc-tions. Beyond the scope of the present paper, exten-sions of these calculations are feasible. Generalizationsto consider the effects of the band structure with ab ini-tio calculations and more complicated models of junc-tions and disorder are possible. Similar calculations havebeen successfully carried out for interface resistances ,spin-transfer torques , spin pumping , and Gilbertdamping .In ferromagnets, the potential landscapes for spin-upand spin-down electrons strongly differ. Therefore, thereflection and transmission amplitudes as well as prob-abilities are spin dependent. In antiferromagnets, thereflection and transmission probabilities are the same forspin-up and spin-down electrons under compensation ofthe localized spins. Nevertheless, the quantum mechan-ical phases associated with reflection and transmissiondiffer for the two spin directions. A. Clean metal
In clean, ballistic systems, the waveguide modes ex-perience either perfect transmission or perfect reflec-tion. In a simple semiclassical model of a normal metal-ferromagnet-normal metal junction, we can assume N ↑ propagating channels for spin-up electrons and N ↓ prop-agating channels for spin-down electrons. Then, A ll = 2 e h P N , (18)where P = ( N ↑ − N ↓ ) / ( N ↑ + N ↓ ) is the polarization and N = N ↑ + N ↓ is the total number of conducting channels.Similarly, the two-terminal conductance becomes G ll = e N/h so that the ratio between the shot-noise coefficientand conductance is A ll /G ll = 2 P .In a similar model of compensated antiferromagnets, N ↑ = N ↓ , and thus, A ll = 0 (19)while G ll = e N/h so that A ll /G ll = 0.Therefore, for this simple semiclassical model, the shotnoise vanishes in antiferromagnets. However, this isgenerically not the case for other kinds of junctions. Morerealistic models of clean junctions will probably result ina small but finite shot noise coefficient in antiferromag-nets as well.The semiclassical results for clean junctions illustratethat the shot noise coefficients can strongly differ in an-tiferromagnets and ferromagnets. B. Disordered Metals
When sufficient disorder exists, either because of bulkimpurity scattering or scattering at boundaries, we canuse random matrix theory to evaluate the average of scat-tering matrices. In the semiclassical regime, the phases ofthe reflection and transmission coefficients are random.They are also statistically independent for spin-up andspin-down electrons. The averages of the transmissionand reflection probabilities are : T σij = 1 N
11 + π σ (20)and R σij = 1 N π σ π σ , (21)where π σ = ρ σ dN e /Ah , N is the number of waveguidemodes, ρ σ is the resistivity for each spin direction, d isthe width of the junction, A is the cross section of thejunction, and the spin directions are σ = ↑ and σ = ↓ . Wecan then compute that the spin-dependent conductanceis G σ = ( e /h ) (cid:80) ij T ij = G dσ / (1 + G dσ /G sh ), where theconductance of a diffusive conductor is G dσ = A/ρ σ d and the Sharvin conductance is G sh = e N/h . A moreintuitive expression is that the resistance consists of theSharvin resistance in series with the diffusive resistance,1 /G σ = 1 /G sh + 1 /G dσ . The total conductance is G = G ↑ + G ↓ .In ferromagnets, the conductances for spin-up andspin-down electrons differ. Based on Eqs. (20) and (21),we can now obtain the average: (cid:104) A ll (cid:105) = 2 (cid:18) G ↑ + G ↓ − G ↑ G ↓ G sh (cid:19) . (22)In the diffusive regime, G ↑ , G ↓ (cid:28) G sh , and we obtain A ll = 2 G . This result agrees with the results computedin Ref. 47 for a double barrier tunnel junction system.While the transport regimes in our approach and Ref.47 are not identical, the treatments seem to share thecommon feature that strong randomization of the elec-tron trajectories occurs. It is, therefore, natural that theresults agree in this limited case.In compensated antiferromagnets, spin-up and spin-down electrons have the same conductance, G ↑ = G ↓ = G/
2. However, the phases of the reflection and trans-mission coefficients for the spin-up and spin-down elec-trons remain statistically independent, as in ferromag-nets. Then, the shot noise coefficient is: (cid:104) A ll (cid:105) = 2 e h (cid:18) G − G G sh (cid:19) , (23) and in the diffusive limit, we obtain the same result asfor a ferromagnet, A ll = 2 G .We conclude that for disordered ferromagnets and anti-ferromagnets, the ratio between the shot noise coefficientand the two-terminal conductance is A ll /G ll = 2. V. THEORY OF PUMPING-INDUCED NOISE
We will, in this section, present our general resultsfor noise enhancements by adiabatic pumping. We de-rive these results from the general scattering theory withmulti-terminals and an arbitrary number of waveguidemodes in Appendix A. The results in this section arevalid for any periodic drive and are not limited to spin-dynamics driven noise discussed in section II. We will,in the next section VI, use the results in this section toobtain the results for the spin-dynamics drive noise thatwe presented in section II.We consider phase-coherent conductors attached toreservoirs via leads. Within the conductors, scatteringby spin-conserving impurities, the spin-orbit interaction,and the exchange field arising from localized spins canoccur. Above, in section II, we have assumed that spinangular momentum is conserved and that the magneti-zation in ferromagnets or staggered fields in antiferro-magnets is homogeneous. However, we do not use theseassumptions here when presenting the general formulafor pumping-induced noise. Appendix A gives details ofthe derivation of the formulas presented in this section.While we consider a general setup with many reser-voirs, we give an example of a three-terminal device inFig. 2. Currents can flow between the reservoirs, arising lead lead l e a d scatteringregionreservoir r e s e r v o i r r e s e r v o i r FIG. 2. Schematic example of a three-terminal device. A scat-tering region (red area) is connected via leads (green areas)to particle reservoirs (blue areas). Currents can flow betweenthe reservoirs. from either differences in bias voltages therein or time-dependent changes within the scattering region. Above,we have considered the latter case when spin excitationsdrive the scattering region. Our focus is on the currentfluctuations when all of the reservoirs are at equilibrium.We consider a general phase-coherent conductor. Scat-tering matrices then describe transport between thereservoirs. All orbital waveguide modes and spin quan-tum numbers span these scattering matrices. In general,the matrices have diagonal and off-diagonal componentsin orbit and spin. In our case, since the scattering regionchanges in time, the scattering matrices also have a com-plex temporal dependence. However, when the temporalchanges are slow compared to the typical electron trans-port time, knowing the temporal behavior of the frozenscattering matrix is sufficient (see Appendix A). We eval-uate the frozen scattering matrix at a snapshot in timewhen the driver has a constant value. This scatteringmatrix is S αnγl ( t, (cid:15) ), where α is the outgoing lead, n isthe outgoing waveguide mode (including spin), γ is theincoming lead, l is the incoming waveguide mode (includ-ing spin), t is the time, and (cid:15) is the electron energy.The current fluctuations are P ζη ( t , t ) = 12 (cid:104) ∆ I ζ ( t )∆ I η ( t )+∆ I η ( t )∆ I ζ ( t ) (cid:105) , (24)where ∆ I ζ ( t ) = I ζ ( t ) − (cid:104) I ζ (cid:105) ( t ) is the deviation of thecurrent I ζ ( t ) from its expectation value (cid:104) I ζ ( t ) (cid:105) in lead ζ .The period of the driver is T = 2 π/ω . Following Ref.58, apart from a factor of 2, we define the zero frequencynoise as: p ζη = (cid:90) T T (cid:90) ∞−∞ dτ P ζη ( t + τ / , t − τ / . (25)Our first central step is that we compute a general ex-pression for the noise induced by a slowly and periodicvarying change in the scattering region. When the elas-tic transport properties are weakly energy dependent, thecurrent cross correlations are: p ζη = (cid:88) q X ( s ) ζη ( (cid:126) ω q ) k B T + 12 (cid:88) q X ( s ) ζη ( (cid:126) ω q ) (cid:20) (cid:126) ω q coth (cid:126) ω q k B T − k B T (cid:21) , (26)where the first term represents the thermal noise contri-bution and the second represents the shot noise contribu-tion. The frequency quantum (cid:126) ω q relates to the period T of the driver by (cid:126) ω q = 2 πq/T , where q is an inte-gral number. The coefficients X ( s ) ζη ( (cid:126) ω ) = [ X ζη ( (cid:126) ω ) + X ζη ( − ω q )] / X ζη ( (cid:126) ω q ) = e h (cid:88) n ζ n η (cid:88) βmγl Φ ζn ζ βmγl ( ω q )Φ ηn η γlβm ( − ω q ) , (27)whereΦ ζn ζ βmγl ( ω q ) = 1 T (cid:90) T dte − iω q t Φ ζn ζ βmγl ( t ) , (28) Φ αnβmγl ( t ) = (cid:2) δ αnβm δ αnγl − S ∗ αnβm ( t ) S αnγl ( t ) (cid:3) , (29)and the static (”frozen”) scattering matrices are to beevaluated at the Fermi energy.The result for the thermal and shot noise of Eq. (26)are general for any drivers and valid when the elastictransport properties are weakly energy-dependent. Inthe next section VI, we use this general result to find thenoise driven by spin excitations. VI. SPIN DYNAMICS-DRIVEN NOISE
In this section, we explain how we can use the generalresult of the pumping-driven noise in the previous sec-tion V to obtain the shot noise when the pumping is dueto spin dynamics. We consider homogeneous spin dy-namics relevant to ferromagnetic resonance and antifer-romagnetic resonance. Now, we assume the conservationof spin angular momentum as in the phenomena of spin-transfer torques and spin pumping. We do not explicitlyconsider the spin-orbit coupling instrumental relevant fore.g. spin-orbit torques, but further investigations usingthe same formalism can elucidate its role.Since the degrees of freedom of the orbital are indepen-dent of the spin degrees of freedom, we use the notationthat the states n consist of orbital quantum numbers n o and spin quantum numbers s , n → n o s . When spin an-gular momentum is conserved, we separate the frozenS-matrix into spin-independent (labelled by superscript”(c)”) and spin-dependent terms (labelled by superscript”(s)”): S ηn o sζm o s ‘ = S ( c ) ηn o ζm o δ ss ‘ + σ ss ‘ · n ( t ) S ( s ) ηn o ζm o , (30)where n is a unit vector in the direction of the orderparameter, n = 1. The calculations in this section arevalid for both ferromagnets where the order parameteris the magnetization and for antiferromagnets where theorder parameter is the staggered field.As the spins precess, only the spin-dependent part ofthe scattering matrix acts as a pump. Inserting the spin-dependent scattering matrix into Eq. (27) and using theunitarity of the scattering matrices and the normalization n = 1, after considerable algebra, we obtain that thefactor that appears in the general expression for the noiseof Eq. (26) becomes: X ( s ) ζη = (cid:90) T dt T e − iω q t q (cid:90) T dt T e iω q t × (cid:20) G ζη + G ηζ + A ζη + A ηζ n ( t ) − n ( t )] (cid:21) , (31)where the conductance tensor G is defined in Eq. (4) andthe shot noise coefficients A are defined in Eq. (6).To proceed to find the expression for the spin-dynamicsdriven shot noise of Eq. (2) with the thermal contributionof Eq. (3) and the shot noise contribution of Eq. (5), weneed to evaluate the following integral appearing in thelast term of Eq. (31): W = (cid:90) dt T e − iω q t q (cid:90) dt T e iω q t [ n ( t ) − n ( t )] (32)to the second order in the deviation of the order parame-ter from equilibrium. To this end, expanding to the linearorder is sufficient: n ( t ) − n ( t ) = (cid:88) ± δ n ± (cid:2) e ± iωt − e ± iωt (cid:3) , (33)where δ n + and δ n − are transverse to the equilibriumspin directions n .Inserting the linear expansion of Eq. (33) into Eq. (32),we then obtain that W ( (cid:126) ω q = 0) = 2 δ n + · δ n − and W ( (cid:126) ω q = ± (cid:126) ω ) = − δ n + · δ n − . As a consequence, wefind X ( s ) ζη ( (cid:126) ω q = 0) = ( G ζη + G ηζ ) + A ζη + A ηζ δ n + · δ n − (34)and X ( s ) ζη ( (cid:126) ω q = ± (cid:126) ω ) = − A ζη + A ηζ δ n + · δ n − . (35)For both ferromagnets and antiferromagnetrs, we cannow insert the expressions for X ( s ) of Eqs. (34) and (35)into the general expression for the noise of Eq. (26). Thethermal contribution to the noise is then Eq. (3) andis independent of the spin oscillations. The shot noisecontribution is given in Eq. (5). VII. CONCLUSIONS
In conclusion, we have presented general expressionsfor the noise driven by ferromagnetic and antiferromag-netic resonance. The noise consists of thermal and shotnoise contributions. Conductances determine the ther-mal noise. Shot noise coefficients and the frequency-dependent magnitude of the spin excitations determinethe shot noise. The shot noise attains its maximum atferromagnetic resonance in ferromagnets and antiferro-magnetic resonance in antiferromagnets.The shot noise parameter can be evaluated for arbi-trarily junctions. We have given examples for ballisticsystems and disordered systems. The ratio between thespin dynamics-driven shot noise parameter and the con-ductance is smaller for ballistic systems than for disor-dered systems. This feature is similar to the behavior ofthe Fano factor associated with voltage-driven shot noise.Our formalism can be generalized to treat the spin-orbit coupling related to spin-orbit torques and electric(charge) pumping . Such extensions will shed furtherlight on spin-charge conversions related to spin dynamics.
ACKNOWLEDGMENTS
This work was supported by the Research Councilof Norway through its Centres of Excellence fundingscheme, project number 262633, ”QuSpin”. We wouldlike to thank Akashdeep Kamra, Sebastian Goennenwein,and Thomas Tybell for comments on the manuscript.
Appendix A: Derivation of General Theory of Noisedue to Adiabatic Pumping
Using Floquet scattering theory, Ref. 58 consideredpumping-driven noise in a multiterminal configurationwith one-dimensional leads. In other words, each leadonly had one waveguide mode. The purpose of thepresent section is to generalize this description to findequations for arbitrary two- and three-dimensional leadsthat can also capture the effects of impurities and bound-ary scattering. To this end, we include many waveguidemodes. In our derivation, we also found that an alter-native path without explicitly using Floquet scatteringstates could be easily followed. We will demonstratethat our results agree with the results in Ref. 58 for one-dimensional leads.In metallic systems, the energy quantum associatedwith the pump oscillations is typically much smaller thanthe Fermi energy. In this regime, the relevant previousresult is Eq. (27) in Ref. 58 when the scattering matrixis weakly energy dependent due to the noise: p M ζη = p (th) ζη + p (sh)M ζη . (A1)The expression for the thermal noise p (th) ζη is the sameas that in Eq. (3) in the limit of only one mode in allleads. The one-dimensional shot noise contribution inthe notation of Ref. 58 is: p (sh)M ζη = 2 e h ∞ (cid:88) q =1 C (sym) ζη,q (cid:20) (cid:126) ω coth (cid:126) ω q k B T − k B T (cid:21) , (A2)where C (sym) ζηq = [ C ζηq + C ζη − q ] / C αβq = (cid:88) γδ (cid:2) S ∗ αγ S αδ (cid:3) q (cid:2) S ∗ βδ S βγ (cid:3) − q , (A3)and the (frozen) scattering matrices should be evaluatedat the Fermi energy E F . The Fourier transform of theproduct of the (frozen) scattering matrices (at the Fermienergy) is defined as follows: (cid:2) S ∗ αγ S αδ (cid:3) q = (cid:90) T dtT e iqωt (cid:2) S ∗ αγ ( t ) S αδ ( t ) (cid:3) . (A4)We reproduce the result in Ref. 58 represented by Eqs.(A1), (A2), (A3), and (A4) for one-dimensional leads andobtain generalizations to leads with an arbitrary numberof waveguide modes.The starting point for our derivation is the expressionfor the current operator in lead α :ˆ I α ( t ) = 2 π (cid:126) e (cid:88) n (cid:104) ˆ a † αn ( t )ˆ a αn ( t ) − ˆ b † αn ( t )ˆ b αn ( t ) (cid:105) , (A5)where α denotes the lead and n denotes the transversewaveguide mode (orbital and spin). The outgoing oper-ators ˆ b are related to the incoming operators ˆ a via thetime-dependent scattering matrix S :ˆ b αn ( t ) = (cid:88) βm (cid:90) ∞−∞ dt S αnβm ( t , t )ˆ a βm ( t ) . (A6)We use the Fourier transform as follows:ˆ a βm ( t ) = 12 π (cid:126) (cid:90) d(cid:15)e − i(cid:15)t ˆ a βm ( (cid:15) ) (A7)and the corresponding inverse Fourier transform. Atthermal equilibrium, the thermal averages are: (cid:104) ˆ a † αn ( (cid:15) )ˆ a βm ( (cid:15) ) (cid:105) eq = δ αβ δ nm δ ( (cid:15) − (cid:15) ) f ( (cid:15) ) , (A8)where f ( (cid:15) ) is the Fermi-Dirac distribution function thatdepends on the chemical potential µ and the thermalenergy k B T . The fluctuations are: (cid:104) ˆ a † αk ( (cid:15) )ˆ a βl ( (cid:15) )ˆ a † γm ( (cid:15) )ˆ a δn ( (cid:15) ) (cid:105)−(cid:104) ˆ a † αk ( (cid:15) )ˆ a βl ( (cid:15) ) (cid:105)(cid:104) ˆ a † γm ( (cid:15) )ˆ a δn ( (cid:15) ) (cid:105) = δ αkδn δ βlγm f ( (cid:15) )[1 − f ( (cid:15) )] δ ( (cid:15) − (cid:15) ) δ ( (cid:15) − (cid:15) ) . (A9)We express the scattering matrix in terms of theWigner representation : S ( t, t (cid:48) ) = 12 π (cid:126) (cid:90) ∞−∞ d(cid:15)S (cid:18) t + t (cid:48) , (cid:15) (cid:19) e − i(cid:15) ( t − t (cid:48) ) / (cid:126) . (A10)The inverse transform is: S ( t, (cid:15) ) = (cid:90) ∞−∞ dτ S ( t + τ / , t − τ / e i(cid:15)τ/ (cid:126) . (A11)By Taylor expanding the S-matrix S (( t + t (cid:48) ) / , (cid:15) ) around S ( t, (cid:15) ) in the Wigner representation of Eq. (A10), we ob-tain: S ( t, t (cid:48) ) = 12 π (cid:126) (cid:90) ∞−∞ d(cid:15)e − i(cid:15) ( t − t (cid:48) ) / (cid:126) e i (cid:126) ∂ (cid:15) ∂ t / S ( t, (cid:15) ) . (A12)The current operator of Eq. (A6) can then be expressedas: I α ( t ) = e π (cid:126) (cid:88) nβmγl (cid:90) d(cid:15) (cid:90) d(cid:15) e i ( (cid:15) − (cid:15) ) t/ (cid:126) × φ αnβmγl ( t, (cid:15) , (cid:15) )ˆ a † βm ( (cid:15) )ˆ a γl ( (cid:15) ) , (A13)where φ αnβmγl ( t, (cid:15) , (cid:15) ) = δ αnβm δ αnγl − e − i (cid:126) ∂ (cid:15) ∂ t / S ∗ αnβm ( t, (cid:15) ) e i (cid:126) ∂ (cid:15) ∂ t / S αnγl ( t, (cid:15) ) . (A14) The current fluctuations are defined in Eq. (24) andcan be expressed as: P ζη ( t , t ) = 12 [ F ζη ( t , t ) + F ηζ ( t , t )] (A15)in terms of F ζη ( t , t ) = (cid:104) I ζ ( t ) I η ( t ) (cid:105) − (cid:104) I ζ ( t ) (cid:105)(cid:104) I η ( t ) (cid:105) . (A16)Using the expectation value of the fluctuations of Eq.(A9), we find: F ζη = e (2 π (cid:126) ) (cid:88) n ζ n η βmγl (cid:90) d(cid:15) (cid:90) d(cid:15) e i ( (cid:15) − (cid:15) )( t − t ) / (cid:126) × φ ζn ζ βmγl ( t , (cid:15) , (cid:15) ) φ ηn η γlβm ( t , (cid:15) , (cid:15) ) f ( (cid:15) )[1 − f ( (cid:15) )] . (A17)We follow Ref. 58 (Eq. (9)), apart from a factor of 2,and define the zero-frequency noise as in Eq. (25). Wetherefore introduce: f ζη = (cid:90) T dtT (cid:90) ∞−∞ dτ F ζη ( t + τ / , t − τ /
2) (A18)so that p ζη = ( f ζη + f ηζ ) / . (A19)We therefore first consider quantities of the form: λ = (cid:90) T dtT (cid:90) ∞−∞ dτ e i ( (cid:15) − (cid:15) ) τ/ (cid:126) A ( t + τ / B ( t − τ / , (A20)where A ( t + τ /
2) and B ( t − τ /
2) are periodic functionswith period T that depend on the energies (cid:15) and (cid:15) . TheFourier transforms of the periodic functions are: A ( t ) = (cid:88) n e iω n t A n (A21)and similarly for B ( t ), where ω n = n π/T and n is anintegral number. The inverse transforms are defined incorresponding ways. We then obtain: λ = 2 π (cid:126) (cid:88) n A n B − n δ ( (cid:126) ω n + ( (cid:15) − (cid:15) )) . (A22)Using Eqs. (A17) and (A18), the low-frequency noise isof the form: κ = (cid:90) d(cid:15) (cid:90) d(cid:15) f ( (cid:15) ) [1 − f ( (cid:15) )] 2 π (cid:126) × (cid:88) n A n B − n δ ( (cid:126) ω n + ( (cid:15) − (cid:15) ) , (A23)where A n and B − n depend on the energies (cid:15) and (cid:15) .Carrying out the integral over the energy (cid:15) : κ = 2 π (cid:126) (cid:88) n (cid:90) d(cid:15) f ( (cid:15) ) [1 − f ( (cid:15) + (cid:126) ω n )] A n B − n , (A24)0where the energy (cid:15) = (cid:15) + (cid:126) ω n in the coefficients A n and B − n .In metallic systems, the Fermi energy and exchange in-teraction are typically much larger than the driving fre-quency. In this case, we can approximate the scatteringmatrix as independent of the driving frequency (cid:126) ω n andthe temperature k B T . We can then evaluate the scatter-ing matrices at the Fermi energy and obtain: κ n = A n ( (cid:15) F ) B − n ( (cid:15) F )2 π (cid:126)(cid:126) ω n [1 + f BE ( (cid:126) ω n , k B T )] , (A25)where the Bose-Einstein distribution function is: f BE = 1exp (cid:126) ω n /k B T − . (A26)We then obtain: f ζη = (cid:88) q Y ζη ( ω q ) (cid:126) ω q [1 + f BE ( (cid:126) ω q )] , (A27)where Y ζη ( (cid:126) ω q ) = e h (cid:88) n ζ n η (cid:88) βmγl φ ζn ζ βmγl ( ω q ) φ ηn η γlβm ( − ω q ) , (A28)and we have defined the Fourier transform as: φ ζn ζ βmγl ( ω q ) = 1 T (cid:90) T dte − iω q t φ ζn ζ βmγl ( t, (cid:15) F , (cid:15) F ) . (A29) We see that Y ζη ( − (cid:126) ω q ) = Y ηζ ( (cid:126) ω q ). Consequently, usingEq. 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