Dynamically stable negative-energy states induced by spin-transfer torques
DDynamically stable negative-energy states induced by spin-transfer torques
J. S. Harms, ∗ A. R¨uckriegel, and R. A. Duine
1, 3 Institute for Theoretical Physics, Utrecht University, 3584CC Utrecht, The Netherlands Institut f¨ur Theoretische Physik, Universit¨at Frankfurt,Max-von-Laue Strasse 1, 60438 Frankfurt, Germany Department of Applied Physics, Eindhoven University of Technology,P.O. Box 513, 5600 MB Eindhoven, The Netherlands
We investigate instabilities of the magnetic ground state in ferromagnetic metals that are inducedby uniform electrical currents, and, in particular, go beyond previous analyses by including dipolarinteractions. These instabilities arise from spin-transfer torques that lead to Doppler shifted spinwaves. For sufficiently large electrical currents, spin-wave excitations have negative energy withrespect to the uniform magnetic ground state, while remaining dynamically stable due to dissipativespin-transfer torques. Hence, the uniform magnetic ground state is energetically unstable, but is notable to dynamically reach the new ground state. We estimate this to happen for current densities j (cid:38) (1 − D/D c )10 A / m in typical thin film experiments, with D the Dzyaloshinskii-Moriya interactionconstant, and D c the Dzyaloshinskii-Moriya interaction that is required for spontaneous formationof spirals or skyrmions. These current densities can be made arbitrarily small for ultrathin filmthicknesses at the order of nanometers, due to surface- and interlayer effects. From an analoguegravity perspective, the stable negative energy states are an essential ingredient to implement eventhorizons for magnons – the quanta of spin waves – giving rise to e.g. Hawking radiation and can beused to significantly amplify spin waves in a so-called black-hole laser. I. INTRODUCTION
Unruh’s 1981 paper ”Experimental black hole evap-oration” [1] proposed that following the argument forthermal black-hole radiation [2] a sonic analogue eventhorizon can be created by considering sound waves in aflowing medium. This sonic event horizon emits a ther-mal spectrum of sound waves and opens up possibilitiesfor the experimental observation of Hawking radiation.The event horizon for sound waves is created by a tran-sition from subsonic to supersonic background flow, suchthat sound waves incoming from the subsonic region can-not escape the supersonic region once they have passedthe event horizon. Motivated by Unruh’s work, theoreti-cal proposals of analogue event horizons based on differ-ent systems were put forward [3–5]. These include phaseoscillations in a Bose-Einstein condensate [6], slow lightin dielectric media [7, 8], trapped ion rings [9, 10], Weylsemi-metals [11] and, as discussed in this article, metallicmagnets [12]. Although Unruh’s original proposal con-siders waves in water which can not be pushed into thequantum regime, the existence of classically stimulatedHawking emission has been observed in Ref. [13]. Fur-thermore, thermal Hawking radiation in a Bose-Einsteincondensate, a system which might be driven to the quan-tum regime, has been observed in Ref. [14].Moreover, the combination of a black-hole and white-hole horizon – the time-reversed partner of a black-holehorizon – is proposed to lead to huge amplitude enhance-ments at specific resonant frequencies [15], thereby actingas a black-hole laser. The resonance frequencies occur ∗ [email protected] due to constructive interference of particle-hole couplingat each horizon, which gives rise to Hawking radiation inthe quantum regime. An implementation of the latter isthe spin-wave laser proposed in Ref. [16], which providesa way of injecting spin angular momentum into a mag-netic sample through amplification of spin waves, drivenby current induced spin-transfer torques [17].Spin waves are collective excitations that occur in mag-netically ordered systems and correspond, at the semi-classical level, to the precession of spatially separatedspins where the phase difference between them is deter-mined by the wavelength. Using spin waves for infor-mation transport and processing is the goal of magnon-ics [18] – the spin-wave analogue of electronics. A diffi-culty towards realizing spin wave based technology is thefinite lifetime of spin waves resulting from processes thatlead to decay of spin angular momentum. The spin-wavelaser gives a potential way to compensate relaxation ofspin waves by injection of spin angular momentum.In this article, we investigate energetic and dynamicinstabilities of spin waves in metallic ferromagnetic thinfilms, induced by spin-transfer torques, i.e., torques aris-ing from the interaction of the spin-polarized current andthe magnetization dynamics [17, 19–23]. More specif-ically, spin waves are Doppler shifted in the presenceof an electrical current [21, 24], with an effective spin-drift velocity proportional to the electrical current. Thisspin wave Doppler shift was experimentally observedby Vlaminck and Bailleul [25]. The spin-drift velocity,if large enough, can lead to instabilities in the ferromag-netic ground state [21, 23]. For the existence of ana-logue horizons it is important to distinguish energetic anddynamic instabilities. Energetic instabilities are charac-terized by the existence of negative energy excitations,while dynamical instabilities lead to exponential growth a r X i v : . [ c ond - m a t . m e s - h a ll ] J a n of small amplitude excitations. Contrary to most physi-cal systems, these instabilities do not necessarily coincidefor spin waves in a ferromagnetic metal, due to dissipa-tive spin-transfer torques [19]. We find that magnons –the quanta of spin waves – can be dynamically stablefor a wide range of currents that make the ferromagneticground state energetically unstable.In the context of analogue gravity, the magnonic eventhorizon is defined by the transition from a region of pos-itive energy states to a region with dynamically stablenegative energy states. For linearly dispersing soundwaves, such as waves in water, the negative energy re-gion corresponds to unidirectional movement of soundwaves. In general, an event horizon is a region whichcouples positive energy states to negative energy states.For non-linearly dispersing sound waves one can still de-fine the event horizon as the region that couples posi-tive energy states and dynamically stable negative energystates. These generalized event horizons are referred toas dispersive horizons [26].The ferromagnetic thin film set-up we consider in thisarticle is similar to Ref. [16], but treated more gener-ally, including effects of surface- and volume anisotropies,Dzyaloshinskii-Moriya interaction, dipole-dipole interac-tions and finite thickness of the thin film. We find thatthe current density needed to create energetically un-stable, but dynamically stable, states is of the order j (cid:38) (1 − D/D c )10 A / m for typical thin film experi-ments, with D the Dzyaloshinskii-Moriya constant, and D c the Dzyaloshinskii-Moriya interaction that is requiredfor spontaneous formation of spirals or skyrmions. Thecritical current density can be made arbitrarily smallfor thin film thicknesses at the order of a nanometer.This decrease is primarily due to the effects of surfaceanisotropy and interfacial Dzyaloshinskii-Moriya interac-tion.The remainder of this article is organized as follows.We put foreward our model and discuss spin wave so-lutions in Section II. Furthermore, the critical currentneeded for energetic instabilities to exist and the regionof dynamical stability are derived in Section III. Addi-tionally, we derive the critical thickness at which the fer-romagnetic ground state becomes unstable due to sur-face and interfacial effects in Appendix A. A derivationof the lowest energy dipole-exchange spin wave mode ispresented in Appendix B. We conclude with a discussionand outlook. II. METALLIC THIN FILM FERROMAGNETA. Model and set-up
We consider a ferromagnetic metallic thin film of thick-ness L in the z direction with the surfaces correspondingto z = ± L/
2. We consider the set-up in Fig. 1 that in-volves a thin film subject to a static external field H e applied in the y direction and a uniform charge current FIG. 1. Sketch of the set-up. We consider a metallic ferro-magnetic thin film of thickness L which is subjected to anexternal magnetic field H e pointing in the y direction and anelectric current driven along the x direction. Furthermore, θ is the angle of the steady state magnetisation M with theplane and φ H is the angle between the spin wave propagationdirection and the y axis. j pointing in the − x direction. For temperatures far be-low the Curie temperature, amplitude fluctuations in themagnetization are negligible. In this case the dynamics ofthe magnetization direction n = M /M s is described bythe Landau-Lifschitz-Gilbert (LLG) equation, with spin-transfer torques (STTs), and Maxwell’s equations in themagnetostatic limit. The LLG equation with STTs isgiven by [19]( ∂ t + v s · ∇ ) n = − γ n × H eff + α n × (cid:18) ∂ t + βα v s · ∇ (cid:19) n , (1)provided that spin-orbit coupling is not very strong sothat spin-orbit torques are negligible. Inclusion of spin-orbit torques in our discussion is straightforward butomitted here to reduce the number of parameters. Inthe above equation, the adiabatic spin-transfer torque isparametrized by the velocity v s = − gP µ B j / eM s thatis referred to as spin-drift velocity, which is proportionalto the current density j . Here g is the Land´e factor, µ B the Bohr magneton, e the elementary charge, P the spinpolarization of the current and M s the saturation mag-netization. The LLG equation describes damped pre-cession of the magnetization around the effective field H eff = − δE/ ( M s δ n ) . Here, E [ n ] is the magnetic energyfunctional, which we consider to be of the general form E = M s (cid:90) dV (cid:26) − J n · ∇ n − µ H · n − K v n z − D (cid:2) ˆ y · ( n × ∂ x n ) − ˆ x · ( n × ∂ y n ) (cid:3)(cid:27) . (2)In the above J is the spin stiffness, D the Dzyaloshinskii-Moriya interaction (DMI) constant that in this particu-lar set-up may result from interfacing the magnet with aheavy metal, and K v is the volume anisotropy constant– this type of anisotropy is e.g. typical in the Co layerspin wave spectroscopy experiments in Ref. [27]. The di-mensionless parameters α and β characterise the strengthof the Gilbert damping parameter and the non-adiabaticspin-transfer torques, respectively. Usually these dissi-pative constants are comparable, α ∼ β , and of the or-der 10 − [28]. For now, we neglect surface anisotropy inthe energy functional, which we discuss in Appendix B.Additionally, dipole-dipole interactions are taken into ac-count by considering the magnetostatic Maxwell’s equa-tions [29] ∇ × H = j , ∇ · B = 0 . (3)Here H is the magnetic field strength B = µ ( H + M )the total magnetic field. In the steady state, the internalmagnetic field H and the magnetization M are par-allel. For an external magnetic field pointing in the y direction with, jL (cid:28) H e , the internal magnetic fieldand magnetization are related to the external magneticfield by µ jz sin( θ ) (cid:39) ( µ M s − K v ) sin(2 θ ) / µ H e sin( θ ) , (4)with θ the angle between the magnetization direction andthe x − y plane. We find that the steady state magnetiza-tion points along the y axis if K v < µ ( H e + M s − jL/ y axis if K v > µ ( H e + M s − jL/ z direction. From this point onward weassume K v < µ ( H e + M s − jL/
2) such that the steadystate magnetization is pointing in the y direction. Exper-imentally, this may be achieved by applying a sufficientlylarge external magnetic field. B. Dipole-exchange spin wave modes
The dipole-exchange spin wave modes [27, 30–37] aregenerated by dynamical fluctuations of both the magne-tization direction and the demagnetizing field, which aresmall compared to M and H , M = M + m ( t ) , H = H + h D ( t ) . (5)Notice that up to linear order in the dynamical fluctua-tions, m is perpendicular to M , lying in the x − z planesince we consider the magnitude of the magnetization tobe constant | M | = M s . Both the static and dynamicpart of the magnetization and magnetic field strength should satisfy the magnetostatic Maxwell equations (3).We accordingly require ∇ × h D = 0 , ∇ · b = 0 , with b = µ ( h D + m ). The first Maxwell equation allows us towrite the dynamic demagnetizing field in terms of a scalarpotential h D = ∇ Φ D . The second Maxwell equation ac-cordingly gives ∇ Φ D = −∇· m , where the magnetization m outside the film is zero. The Landau-Lifschitz-Gilbert-and magnetostatic Maxwell equations may be rewrittenby means of n (cid:39) ˆ z √ x √ y (cid:16) − | Ψ | (cid:17) ,with the complex field Ψ = (1 / √
2) (ˆ z + i ˆ x ) · n . Inthese coordinates the linearised LLG and magnetostaticMaxwell equations becomeˆΩΨ = − (cid:0) Ω H − ∆ v − Λ ∇ (cid:1) Ψ+ ∆ v Ψ ∗ + ( ∂ z + i∂ x ) √ M s Φ D , (6a) ∇ Φ D M s = ( ∂ z − i∂ x ) √ M s Ψ + ( ∂ z + i∂ x ) √ M s Ψ ∗ . (6b)Additionally, the exchange boundary conditions for thinfilms [38] require ± ∂ z Ψ − ( K s /J ) (Ψ + Ψ ∗ ) (cid:12)(cid:12) z = ± L/ = 0 , (7)with K s the surface anisotropy constant. In the above,we defined the following dimensionless operators andvariables [39]: dimensionless magnetic field Ω H = µ H e /µ M s , dimensionless volume anisotropy ∆ v = K v / µ M s , exchange length Λ = (cid:112) J/µ M S and the di-mensionless frequency operator ˆΩ = i [(1 − iα ) ∂ t + (1 − iβ ) v s · ∇ + γD∂ x ] / ( γµ M s ) . Using the Bogoliubov ansatz, and taking v s inthe x direction, we write Ψ( x , t ) = u ( x ) e − iωt + v ∗ ( x ) e iω ∗ t and Φ D ( x , t ) = w ( x ) e − iωt + w ∗ ( x ) e iω ∗ t , where (cid:0) u ( x ) , v ( x ) , w ( x ) (cid:1) ∝ e i k · r (cid:107) (cid:0) u ( k , z ) , v ( k , z ) , w ( k , z ) (cid:1) , with k = (cid:0) k x , k y (cid:1) and r (cid:107) = (cid:0) x, y (cid:1) . The above plainwave ansatz gives rise to a spectrum of spin wave so-lutions. The lowest energy dipole-exchange spin wavedispersion relation is obtained in Appendix B for thinfilms with thicknesses comparable to the exchange length L ∼ O (Λ). Up to linear order in α and β the lowestenergy dipole-exchange spin wave dispersion relation isgiven by( ω k − v s k x ) (cid:39) ω k − iκαω k − iκ ( α − β ) v s k x , (8)where (cid:0) ω k − γDk x (cid:1) / ( γµ M s ) = (cid:113) [Ω H − ∆ + Λ k − / ( φ H ) f ( k )] − (cid:2) ∆ + 1 / { ( φ H ) } f ( k ) (cid:3) , (9)is the real part of the dispersion in the absence of anelectrical current, which is plotted in Fig. 2. Here, f ( k ) = 1 − (1 − e − kL ) /kL is the form factor, φ H the angle between the spin wave propagation direction and the y axis and ∆ ≡ ∆ v + ∆ s − /
2, with ∆ s = (Λ /µ M s L ) K s the dimensionless parameter corresponding to the sum of - Λ k f r e q u e n c y ( ω / γ μ M s ) FIG. 2. Dispersion relation (9) of the lowest energy spin wavemode including dipolar interactions and anisotropy. Here, weconsider Ω H = 1, D/µ M s = 0, ∆ = − . L ∼ φ H = π/ surface anisotropies [33, 38]. In the above, κ is an overallfactor of the imaginary part of the dispersion relation,stemming from the fact that the isotropic Gilbert damp-ing only enters in the diagonal part of Eq. (6a). Thisterm is not of importance for the stability analyses, sinceit remains positive in the region of interest. The preciseform of κ can be found in Appendix B. III. ENERGETIC AND DYNAMICAL SPINWAVE INSTABILITIES
Motivated by theoretical predictions of magnonicblack/white-hole horizons [12] and black-holes lasers [15,16], we investigate energetic and dynamic instabilities inthe spin wave spectrum, due to a spin-polarized elec-trical current [21], including effects of dipole-dipole in-teractions, volume- and surface anisotropies, and DMI.A negative real part of the spin wave dispersion rela-tion, Eq. (8), indicates energetic instabilities, necessaryfor analogue black/white-hole setups [3–5]. Dynamicalinstabilities on the other hand are characterized by a pos-itive imaginary part of the spin wave dispersion relationand classically lead to an exponential growth of unsta-ble modes. In contrast to most physical systems, thesetwo types of instabilities do not necessarily coincide forthe magnetization dynamics in a metallic magnetic sys-tem, due to the dissipative spin-transfer torques charac-terised by the parameter β . Accordingly, we investigatethe regime in which the system is energetically unstable,but dynamically stable, see Fig. 3. From Eq. (8) we findthat the system is dynamically stable if | ( α − β ) v s k x + αγDk x | < α ( γµ M s )Ω k (10)is satisfied for all k , with γµ M s Ω k = ω k − γDk x theinversion symmetric part of the dispersion relation (9).Energetic instabilities on the other hand are present ( a ) - - - - kL f r e q u e n c y ( ω / γ μ M s ) v s < v c v s > v c v s ≫ v c ( b ) - - - - - - - - kL f r e q u e n c y ( ω / γ μ M s ) v s < v c v s > v c v s ≫ v c FIG. 3. Dispersion relation (8) for different values of the spindrift-velocity v s , with α = 0 . β = 0 .
01. (a) Real part ofthe dispersion relation. (b) Imaginary part of the dispersionrelation. For small v s < v c , spin waves are energetically anddynamically stable (red). For spin drift-velocities larger thanthe critical velocity, v s > v c , we obtain energetically unstablebut dynamically stable spin waves (blue). For very large v s (cid:29) v c spin waves are both energetically and dynamically unstable(yellow). if Re ω k <
0, for some k . By considering minimaof the dispersion relation we find the critical currentabove which energetic instabilities exist should satisfy ∂ k c Re ω k c | v s = v c = 0 and Re ω k c | v s = v c = 0, for some k c . Thus, energetic instabilities are present for currents v s > v c and do not exist for v s < v c , which characterisesthe critical velocity v c . The above constraints that de-termine the critical current are equivalent to ∂ k c Ω k c = 2Ω k c /k c , (11a) v c /γ = [ µ M s / sin( φ H )]Ω k c /k c − D. (11b)For spin waves travelling perpendicular to the externalmagnetic field, φ H = π/ H − ∆] − Λ k c ∼ [∆ + f ( k c )] (cid:2) ∆ + f ( k c ) (cid:3) , (12)where we used f ( k ) − kf (cid:48) ( k ) ∼ f ( k ) . We note that f ( k ) ∈ [0 ,
1] and typically ∆ = ∆ v + ∆ s − / (cid:38) − / H ∼
1. Accordingly, we assume [∆ + f ( k c )][∆ + f ( k c ) ] / [Ω H − ∆] to be small compared to unity aroundthe critical wavelength k c . Next, we expand k c = κ c + δk c around Λ κ c = Ω H − ∆, up to linear order in δk c and[∆ + f ( k c )][∆ + f ( k c ) ] / [Ω H − ∆] in Eq. (12). This givesΛ δk c ∼ −
12 [∆ + f ( κ c )] (cid:2) ∆ + f ( κ c ) (cid:3) Ω H − ∆ . (13)Similarly, we find that Ω k c , up to first order in δk c and[∆ + f ( k c )] / [Ω H − ∆] , is given byΩ k c ∼ H − ∆] + Λ δk c −
14 [∆ + f ( κ c )] Ω H − ∆ . (14)Finally, using Eq. (11b) we find that the critical currentthat generates energetic instabilities is up to linear orderin δk c and [∆ + f ( k c )] / [Ω H − ∆] given by( v c /γµ M s ) (cid:39) (cid:18) Ω k c κ c (cid:19) (cid:18) − δk c κ c (cid:19) − ( D/µ M s ) . (15)This can be rewritten as v c = γ ( D c − D ) , (16)where( D c / µ M s Λ) (cid:39) (cid:113) [Ω H − ∆] − (1 /
2) [∆ + f ( κ c )] (17)is the critical DMI constant above which the groundstate becomes both energetically and dynamically un-stable. Once DMI reaches this value, the homogeneousground state becomes unstable towards the formation oftextures, typically spirals and skyrmions. Additionally,we note that the contribution of δk c drops out of thecritical DMI, up to first order.Finally, we find from Eq. (10) that the region in whichelectrical currents generate energetically unstable but dy-namically stable spin waves is given by (cid:26) γ ( D c − D ) < v s < γ ( D c − D ) | − β/α | − β < α,γ ( D c − D ) < v s < γ ( D c + D ) | − β/α | − β > α. (18)This provides a large window of stability, given that usu-ally α ∼ β . We note that this region is determined bysolely considering spin waves travelling along the x axis– perpendicular to the external magnetic field. This is aconsequence of the fact that the critical current for en-ergetic and dynamic instabilities increases as spin wavestravel at increasing angles | φ H − π/ | with respect tothe x axis. In Fig. 4, we plotted the angular depen-dence of the critical current. Additionally, in the casewhere β/α > D > D c (1 − α/β ) for energetically unstable, but dynam-ically stable states to exist in the region where β/α > β (cid:29) α . For instancein Ref. [40] a value of β ∼ α was found.Taking typical values for the saturation magnetiza-tion µ M s ∼ µ H e ∼ γ/ π ∼ − and exchange length Λ = (cid:112) J/µ M s ∼ j c ∼ M s | e | v c /µ b ∼ A / m , where wetook g ∼ P ∼ M s /µ B ∼ nm − and v c ∼ ϕ H ( radians ) v c / γ μ M s Λ FIG. 4. Numeric solution of the dimensionless critical veloc-ity ( v c /γµ M s Λ) in Eq. (11) – for dispersion relation (9) –against the angle φ H in radians. We took the typical val-ues Ω H ∼ DL/µ M s Λ ∼ .
1, ∆ v = K v / µ M s ∼ . s L = K s /µ M s Λ ∼ . L/ Λ = 3. L / Λ v c / γ μ M s Λ FIG. 5. Dimensionless critical velocity v c /γµ M s Λ againstdimensionless thickness L/ Λ, taking typical values Ω H ∼ DL/µ M s Λ ∼ .
1, ∆ v = K v / µ M s ∼ .
2, ∆ s L/ Λ = K s /µ M s Λ ∼ .
4. The dashed line corresponds to the linearapproximation in Eq. (16) and the solid line corresponds tothe numerically obtained solution of Eq. (11) with dispersionrelation (9). γµ M s Λ ∼ m / s. Furthermore, for typical valuesof DMI DL/µ M s Λ ∼ . v = K v / µ M s ∼ . s L/ Λ = K s /µ M s Λ ∼ . IV. DISCUSSION AND OUTLOOK
We have investigated the occurrence of energeticallyunstable but dynamically stable spin wave excitations,due to spin-transfer torques, including effects of dipole-dipole interactions, anisotropies and DMI. We haveshown that in typical thin film experiments [25, 27, 33],the critical current needed to create energetically un-stable, but dynamically stable states is of the order j (cid:38) (1 − D/D c )10 A / m . If one could experimentallyenhance the DMI to be near the critical DMI, abovewhich the homogeneous ground state becomes unstabletowards the formation of textures, such as spirals andskyrmions, then a relatively small current should be suf-ficient to create the dynamically stable negative energystates. Additionally, we found that the critical currentdensity becomes arbitrarily small for thin film thicknessesof the order of nanometers. This decrease is primarily dueto the cumulative effect of DMI and surface anisotropies,which become dominant in ultrathin films.Furthermore, the region in which dynamically stablenegative energy spin wave excitations exist is found tobe large, given that typically α ∼ β . In the case where β (cid:29) α we note that energetically stable, dynamicallyunstable states could occur. Hence, dynamically stablenegative energy states are difficult to create in materialswhen β (cid:29) α .For the typical values considered in Section III, wesee a slight deviation of the first order critical velocitywith respect to the numerical critical velocity at ultra-thin film thicknesses, see Fig. 5. This is due to the surfaceanisotropy contribution becoming larger in the ultrathinfilm limit, where the increased inaccuracy stems from thefact that we determine the critical velocity in Eq. (16)up to first order assuming [∆ + f ( k c )][∆ + f ( k c ) ] and[∆ + f ( k c )] / H − ∆] . Thisapproximation is accurate when anisotropies are smallcompared to the external magnetic field, but describesthe critical velocity less accurately when anisotropies be-come relatively large – especially volume anisotropy –approaching 2∆ v (cid:46) Ω H + 1. Additionally, the crit-ical momentum k c becomes small for ultrathin filmthicknesses – if surface anisotropies are dominating –,which makes the expansion of 1 /k c less accurate in thisrange. When dealing with relatively large anisotropies,it is more appropriate to expand the k c around κ c = (cid:112) [Ω H − ∆] − [∆ v + ∆ s ] in Eq. (12). In this casethe critical DMI constant is given by ( D c / √ µ M s ) (cid:39) (cid:113) (Ω H − ∆ + κ c ) − ˜ f (∆ v + ∆ s + ˜ f ) /κ c , with ˜ f = f ( κ c ) − / ACKNOWLEDGMENTS
This work is part of the research programme FluidSpintronics with projectnumber 182.069, which is(partly) financed by the Dutch Research Council (NWO).R.D. is member of the D-ITP consortium, a program ofthe Dutch Organization for Scientific Research (NWO)that is funded by the Dutch Ministry of Education, Cul-ture and Science (OCW).
Appendix A: Critical thickness of energeticinstabilities at zero current
In this Appendix, we determine the critical thickness atwhich spin wave excitations become energetically unsta-ble at zero electrical current. These energetic instabilitiesare due the increase in magnitude of surface anisotropiesand DMI in the ultra thin film limit and are dynami-cally unstable by Eq. (10). Additionally, the range ofelectrical currents that generate energetically unstablebut dynamically stable spin wave excitations decreaseswhen approaching the critical thickness. This is a directconsequence of decreasing the critical current, see Sec-tion III. If surface anisotropies are dominant at smallthicknesses, the critical thickness at which instabilitiesappear, at zero current and vanishing DMI, may be ap-proximated at zero’th order by closing the spin-wave gapin Eq. (9), givingΩ H − ≡ Ω H + 1 − v − ∗ s /L ∼ , (A1)with k c L (cid:28) ∗ s = ∆ s L constant. For non-vanishing DMI, up to first order in f ( κ c ) ∼ κ c L/ (cid:28) H − ∆] ∼ [∆ + f ( k c )] ∆ , (A2) k c (cid:2) H − ∆) − ( D ∗ /L ) (cid:3) ∼ [∆ + f ( k c )] f (cid:48) ( k c ) , (A3)where D ∗ = ( D/µ M s Λ) L is constant. We expand theabove equations around ∆ = ∆ + δ ∆, with δ ∆ = − (∆ ∗ s /L ) δL + O ( δL ). Up to first order Eq. (A2) gives k c L / ∼ − δ ∆ . (A4)We find that Eq. (A3) in combination with the aboveequation leads to δL ∼ Ω H L ∗ s (cid:44)(cid:40) Ω H (cid:20) L + 16 − L ∗ s (cid:21) − − (cid:18) D ∗ L (cid:19) (cid:41) , (A5)with L = 2∆ ∗ s / (2Ω H + 1 − v ) given by Eq. (A1).We thus find that the critical thickness for spin waveinstabilities is given by L ∼ L + δL , in the case thatsurface anisotropies dominate in the ultrathin film. Appendix B: Approximate dipole-exchange mode inthin films
Here, we discuss an analytic approximation of the low-est energy spin wave dispersion relation for the setupdiscussed in Section II, using the thin film magnetostaticGreens function [32, 33, 36, 37, 43]. We start by express-ing the demagnetizing field h D in Eqs. (5), (6a) and (6b)in terms of the magnetization by using the magnetostaticGreens function. It is explicitly given by h D ( k , z ) = (cid:90) L/ − L/ dz (cid:48) G ( k ; z − z (cid:48) ) m ( k ; z (cid:48) ) . (B1)Where the magnetostatic Greens function [36, 43] satis-fies the magnetostatic Maxwell equation Eq. (6b) alongwith the appropriate boundary conditions [44]. It is ex-plicitly given by, G ˆ ζ ˆ η ˆ z ( k ; z, z (cid:48) ) = − G p iG q iG q G p − δ ( z − z (cid:48) ) , (B2)with ˆ ζ ∝ k the in-plane direction of spin wave propaga-tion, ˆ η the orthogonal in-plane direction, ˆ z the thick-ness direction, G p = | k | exp( −| k || z − z (cid:48) | ) and G q = G p sign( z − z (cid:48) ) . By substituting Eq. (B1) into the lin-earised Landau-Lifschitz-Gilbert equation Eq. (6a) weobtain the effective linearised LLG equationˆΩΨ = − (cid:2) Ω H − ∆ v/s − Λ ∇ (cid:3) Ψ + ∆ v/s Ψ ∗ + (cid:90) L/ − L/ (cid:104) A k Ψ k + B k Ψ ∗− k (cid:105) dz (cid:48) , (B3)where A k = cos ( φ H ) G p − δ ( z − z (cid:48) ) , B k = { ( φ H ) } G p − φ H ) G q − δ ( z − z (cid:48) ) and φ H the an-gle between wave vector k and the y axis. Addition-ally we introduced the dimensionless anisotropy con-stant ∆ v/s = K v/s / µ M s , where we took K v/s = K v + K − s δ ( z − L/
2) + K + s δ ( z + L/ K s corre-spond to the surface anisotropies of the thin film. Follow-ing Gladii et al. [33] we add the term ( K ± s / δ ( z ± L/ n z in the energy functional Eq. (2) to account for sur-face anisotropies. This differs from the approach usedby Kalinikos and Slavin [37] where surface anisotropiesdetermine the exchange boundary conditions of the thinfilm [38].Using the Bogoliubov ansatz Ψ( x , t ) = u ( x ) e − iωt + v ∗ ( x ) e iω ∗ t , with (cid:0) u ( x ) , v ( x ) (cid:1) = (cid:82) d k π e i k · r (cid:107) (cid:0) u ( k , z ) , v ( k , z ) (cid:1) , the linearised LLGequation becomes (cid:18) ˆ F + 1 / / − ∆ v/s ∆ v/s − / − ˆ F ∗ − / (cid:19) (cid:18) u ( k, z ) v ( k, z ) (cid:19) + (cid:90) L/ − L/ dz (cid:48) (cid:18) − C ( s ) D + ( s ) − D − ( s ) C ( s ) (cid:19) (cid:18) u ( k, z (cid:48) ) v ( k, z (cid:48) ) (cid:19) = 0 , (B4)with s = z − z (cid:48) , ˆ F = Ω + (cid:0) Ω H − ∆ v + Λ k − Λ ∂ z (cid:1) ,ˆ F ∗ = − Ω ∗ + (cid:0) Ω H − ∆ v + Λ k − Λ ∂ z (cid:1) , C ( s ) =(1 /
2) cos ( φ ζ ) G p and D ± ( s ) = − (1 / { ( φ ζ ) } G p ±| sin( φ ζ ) | G q . Furthermore, γµ M s Ω = (1 + iα ) ω + (1 + iβ ) v s k x + γDk x and γµ M s Ω ∗ = (1 − iα ) ω +(1 − iβ ) v s k x + γDk x . The magnetization profile in the thickness directionmay be expanded in eigenfunctions of the unpinned ex-change boundary conditions, which form a complete ba-sis [36]. We approximate the magnetization profile of thelowest mode by the lowest Fourier mode, for thicknessesof the order L ∼ O (Λ), (cid:18) u ( k, z ) v ( k, z ) (cid:19) ∼ u ( k ) (cid:114) L (cid:18) (cid:19) + v ( k ) (cid:114) L (cid:18) (cid:19) , (B5)which is the uniform mode approximation. Using theabove ansatz the linearised LLG equation Eq. (B4) be-comes (cid:18) Ω + Ω d − Ω i Ω i Ω ∗ − Ω d (cid:19) = 0 , (B6)whereΩ d = Ω H − ∆ + Λ k − / ( φ H ) f ( k ) , (B7a)Ω i = ∆ + 1 / { ( φ H ) } f ( k ) . (B7b)With f ( k ) = 1 − (1 − e − kL ) /kL , ∆ = ∆ v + ∆ s − / s = (Λ / µ M s L ) ( K − s + K + s ). Hence, the lowest modedispersion relation, up to first order in α and β , is givenby ( ω k − v s k x ) (cid:39) ω k − iκαω k − iκ ( α − β ) v s k x , (B8)with (cid:0) ω k − γDk x (cid:1) / ( γµ M s ) = Ω k , κ = (Ω d / Ω k ) andΩ k = (cid:2) Ω H − ∆ + Λ k − / ( φ H ) f ( k ) (cid:3) − (cid:2) ∆ + 1 / { ( φ H ) } f ( k ) (cid:3) . (B9) [1] W. G. Unruh, Physical Review Letters , 1351 (1981),publisher: American Physical Society.[2] S. W. Hawking, Nature , 30 (1974), number: 5443Publisher: Nature Publishing Group.[3] D. Faccio, F. Belgiorno, S. Cacciatori, V. Gorini,S. Liberati, and U. Moschella, eds., Analogue Grav-ity Phenomenology , Lecture Notes in Physics, Vol. 870(Springer International Publishing, 2013).[4] M. Novello, M. Visser, and G. Volovik,