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Squeezed driving induced entanglement and squeezing among cavity modes and magnon mode in a magnon-cavity QED system Ying Zhoua,b, Jingping Xua,, Shuangyuan Xiea,, Yaping Yanga aMOE Key Laboratory of Advanced Micro-Structured Materials, School of Physics Science and Engineering, Tongji University, Shanghai 200092, China bSchool of Electronics and Information Engineering, Taizhou University, Taizhou, Zhejiang, 318000, China Abstract We propose a scheme to generate entanglement between two cavity modes and squeeze magnon mode in a magnon- cavity QED system, where the two microwave cavity modes are coupled with a massive yttrium iron garnet (YIG) sphere through magnetic dipole interaction. The nonlinearity used in our system originates from a squeezed driving via parametric down-conversion process, which is the reason to cause entanglement and squeezing. By using the mean eld approximation and employing experimentally feasible parameters, we demonstrate that the system shows zero entanglement and squeezing without squeezed driving. Meanwhile, our QED system denotes that the entanglement between squeezed cavity mode and magnon mode can be transferred to the other cavity mode and magnon mode via magnon-cavity coupling interaction, and then the two cavity modes get entangled. A genuinely tripartite entangled state is formed. We also show that magnon mode can be prepared in a squeezed state via magnon-cavity beam-splitter interaction, which is as a result of the squeezed eld. Moreover, we show that it is a good way to enhance entanglement and squeezing by increasing the nonlinear gain coecient of squeezed driving. Our results denote that magnon-cavity QED system is a powerful platform for studying macroscopic quantum phenomena, which illustrates a new method to photon-photon entanglement and magnon squeezing. Keywords: Squeezed driving, Nonlinearity gain, Entanglement, Squeezed 1. Introduction In recent years, yttrium iron garnet (YIG) material, as an excellent ferrimagnetic material with high spin den- sity(about 4 :221027m3) and low dissipation rate(about 1 MHz), has attracted considerable attention [1, 2, 3]. Moreover, YIG material is ferromagnetic at both cryogenic [4, 5, 6] and room temperature [7] because its Curie tem- perature is about 559K. The magnon mode, as a collective motion of a large number of spins with zero wavevector (Kittel mode [8]) via the Holstein-Primako transforma- tion [9] in YIG sphere, possesses unique properties. It can realize strong [4, 5, 6, 7, 10, 11] and ultrastrong [12, 13] coupling to microwave cavity photons at either cryo- genic or room temperature, and then lead to magnon- cavity polaritons. Thus, a lot of meaningful development about magnons is found, including the observation of cav- ity spintronics [10, 14], bistability [15], magnon gradient memory[16], magnetically controllable slow light [17], level attraction[18], magnon-induced transparency [19, 20], and the magnon squeezed state [21]. It is noted that magnon Corresponding author Email addresses: xx_ jj_ pp@tongji.edu.cn (Jingping Xu), xieshuangyuan@tongji.edu.cn (Shuangyuan Xie)squeezed state is an important macroscopic quantum state, which can be used to improve the measurement sensitiv- ity [22] and study decoherence theories at large scales[23]. Meanwhile, by virtue of strong coupling among magnons, other interesting phenomena, including coupling the magnon mode to a single superconducting qubit [24], to photons and to phonon mode [20, 25], have also been studied. This of- fers a possibility to enable coherent information transfer between dierent information carriers. Clearly, compared to atom, the size of YIG sphere is in mesoscopic or macro- scopic scale usually with a size of 250m, which holds the potentiality for implementing quantum states, espe- cially the entanglement in more massive object. Thus, it provides a promising and completely new platform for the study of macroscopic quantum phenomena [25], which is a key step to test decoherence theories at macroscopic scale [23, 26], and probe the boundary between the quan- tum and classical worlds [27, 28, 29]. In the microwave region, one important quantum state is entangled state, which is typically produced by exploiting the nonlinearity of magnetostrictive interaction in cavity magnomechani- cal system[30], by utilizing Kerr nonlinearity results from magnetocrystalline anisotropy[31], and by using the non- linearity of quantum noise in Josephson parametric am- pliers (JPA) [32]. Meanwhile, another important macro- Preprint submitted to Physics Letters A January 25, 2022arXiv:2201.09154v1 [quant-ph] 23 Jan 2022scopic quantum state is magnon squeezed state, which is usually generated by the quantum noise of JPA process[21]. Recent interest has focused on generating entangle- ment and squeezing in a hybrid cavity magnon QED sys- tem, especially in a hybrid cavity magnomechanics sys- tem including phonons. A genuine tripartite entangle- ment is shown by using the nonlinearity of magnon-phonon coupling in a cavity magnomechanical system consisting of magnons, microwave photons and phonons [33], where the magnons couple to microwave photons and phonons via magnetic dipole interaction and magnetostrictive in- teraction, respectively. When driving the above cavity (Ref. [33]) by a weak squeezed vacuum eld generated by a ux-driven JPA process, the magnons and phonons are squeezed in succession, and larger squeezing could be real- ized by increasing the degree of squeezing of the drive eld and working at a lower temperature [21]. A hybrid cav- ity magnomechanical system includes two magnon modes in two macroscopic YIG spheres, which couple to the sin- gle microwave cavity mode via magnetic dipole interaction. By activating the nonlinear magnetostrictive interaction in one YIG sphere, realized by driving the magnon mode with a strong red-detuned microwave eld, the two magnon modes get entangled [34]. When two YIG spheres are placed inside two microwave cavities driven by a two-mode squeezed microwave eld. Each magnon mode couples to the cavity mode via magnetic dipole interaction. The quantum correlation of the two driving elds can eciently transferred to the two magnon modes and magnon-magnon entanglement can be achieved. The two cavity modes also can entangle to each other [35]. When considering the vibrational modes in the above cavity (Ref. [35]), each phonon mode couples to the magnon mode via magne- tostrictive interaction. By directly driving magnon mode with a strong red-detuned microwave eld to active the magnomechanical anti-stokes process, and further driving the two cavities by a two-mode squeezed vacuum eld as above scheme (Ref. [35]), the two phonon modes in two YIG spheres can also get entangled [36]. All the above solutions denote that magnetic dipole coupling interaction and nonlinearity are two main elements to produce entan- gled and squeezed states. The main nonlinearity used is the nonlinearity of magnetostrictive interaction[34, 30, 25, 33] and the nonlinearity generated by quantum noise of JPA process[35, 32] in cavity magnon system. In this letter, we propose a scheme to generate photon- photon entanglement and squeeze magnon in a magnon- cavity QED system. Magnon mode in a YIG sphere is coupled to two microwave elds via magnetic dipole in- teraction, respectively. Since YIG material that generates magnons is a massive object, it is considered to be a the- oretically innovation to realize the entanglement of two mesoscopic objects through a cavity mode. However, for the cavity mode or photon being a good carrier of infor- mation, we hope to entangle the two cavity modes with the magnons as a mesoscopic medium, and we think this is more important from the perspective of information.Squeezing is also a very important quantum resource, and we then emphasize the squeezing of magnons. It is found that squeezing can be transmitted to various objects in this QED system. Dierent from previous propose, the nonlinearity we used is generated by parametric down- conversion of JPA process. The intensity of that is exible tunable, resulting in a squeezed cavity mode. Meanwhile, for the phonon mode can provide another non-linearity and make the system into a more complex one, we did not take it into consideration. In our QED system, entan- glement transferred from magnon-cavity a1subsystem to magnon-cavity a2subsystem, and then transferred to two cavity modes subsystem. A genuinely tripartite entangled state is formed. Meanwhile, the squeezed driving also pre- pares the magnon mode in a squeezing state. Further, we show that increasing the nonlinearity gain coecient of squeezed driving is a good way to enhance entanglement and squeezed. Moreover, we show that the optimal entan- glement and squeezing generated when the coupling rates between the two cavity modes and magnon mode are the same. 2. The model We consider a hybrid magnon-cavity QED system, which consists of two microwave cavity modes and a magnon mode, as depicted in Fig.1. A squeezed microwave cavity 1 (with frequency !1) is implemented by parametric down- conversion in JPA process. We assume that the nonlinear gain coecient of JPA is . The second microwave cav- ity (with frequency !2) is perpendicular to the microwave cavity 1 without any non-linearity driving, the resonance frequency!2is close to that of cavity mode a1. To achieve strong couplings between the YIG sphere and these two cavity modes, we place the YIG sample at the center of both cavities. Meanwhile, the magnons are quasiparticles, a collective motion of a large number of spins spatially uniform mode (Kittel mode [8]) in a massive YIG sphere. The magnetic eld of cavity mode a1anda2are alongx andydirection, respectively. The bias magnetic eld H is alongz-axis for producing the Kittel mode. Strongly coupled is implemented via magnetic dipole interaction. Moreover, a microwave eld with angular frequency !0 and Rabi frequency "pis applied along the xdirection to drivinga1. We assume the size of the YIG sample to be much smaller than the microwave wavelengths in our QED system, so the radiation pressure on YIG sample induced by microwave elds can be neglected. The Hamiltonian of the system reads H=h=X j=1;2!jay jaj+!mmym+X j=1;2gj(ay j+aj)(m+my) +"p(a1ei!0t+ay 1ei!0t) + (a2 1e2i!0t+ay 12e2i!0t)(1) whereajanday jare, respectively, the annihilation and creation operators of cavity mode j. m(my) is annihi- 2lation (creation) operator of magnon mode [37], which represent the collective motion of spins via the Holstein- Primako transformation [9] in terms of Bosons, satisfy- ing [O;Oy] = 1 (O=a1;a2;m).!j(j= 1;2) and!m present the resonance frequency of cavity modes ajand magnon mode, respectively. The frequency of magnon mode can be adjusted by the external bias magnetic eld Hvia!m= H, where =2= 28GHz/T is the gyromag- netic ratio. gjdenotes the linear coupling rate between magnon mode and cavity mode aj, which currently can be (much) larger than the dissipation rates jandm of cavity mode ajand magnon mode, i.e. gj> j,m (j= 1;2). It denotes the magnon-cavity QED system is in the strong coupling regime, but not in the ultrastrong coupling regime, and the rotating-wave approximation can be applied for the magnon-cavity interaction terms in our magnon-cavity QED system. Fig. 1. Schematic of magnon-cavity QED system. The rst cav- ity is driven by a microwave eld with "pthe Rabi frequency and a squeezed eld with the gain coecient of parametric down- conversion, the resonance frequency of which is !1. The second cavity (with frequency !2) is perpendicular to the rst one with a close angular frequency. The magnetic eld of cavity mode a1anda2 are alongxandydirection, respectively. A YIG sphere is mounted at the center of the both microwave cavities. Simultaneously, it is also in a bias magnetic eld Halongz-axis for producing the Kittel mode, resulting in the resonance frequency !m. Here,1,2and mare the dissipation rates of cavity mode a1, cavity mode a2and magnon mode, respectively. Under the rotating-wave approximation, the magnon- photon interaction term gj(aj+ay j)(m+my) becomesgj(ajmy+ ay jm). We then switch to the rotating frame with respect to the driving frequency !0, the Hamiltonian of the system can be written as: H=h=X j=1;2
jay jaj+
mmym+X j=1;2gj(ay jm+ajmy) +"p(a1+ay 1) + (a2 1+ay 12) (2) Where
j=!j!0, and
m=!m!0are the detunings of cavity mode j and magnon mode, respectively. By including input noises and dissipations of the system, the quantum Langevin equations describing the system areas follows, _a1=(i
1+1)a1ig1mi"p2i ay 1+p 21ain 1(3) _a2=(i
2+2)a2ig2m+p 22ain 2 (4) _m=(i
m+m)mig1a1ig2a2+p 2mmin(5) Whereain jandminare input noise operators for the cavity mode ajand magnon mode m, respectively, which are zero mean value acting on the cavity and magnon modes. The Gaussian nature of quantum noises can be characterized by the following correlation function [38]: hain j(t)ainy j(t0)i= [Nj(!j) + 1](tt0),hainy j(t)ain j(t0)i= Nj(!j)(tt0)(j= 1;2), andhmin(t)miny(t0)i= [Nm(!m)+ 1](tt0),hminy(t)min(t0)i=Nm(!m)(tt0) where Nl(!l) = [exp(h!l=kBT)1]1(l= 1;2;m) are the equilib- rium mean thermal photon numbers and magnon number, respectively, with kBthe Boltzmann constant and Tthe environmental temperature. Since the rst cavity is under strong driving by the mi- crowave eld "pand squeezed eld , which results in a large amplitudejha1ij1 at the steady state. Meanwhile, due to the beam-splitter-like coupling interaction between cavity modes and magnon mode, magnon mode and cav- ity modea2are also of large amplitudes in steady state. This allows us to linearize the system dynamics around the semiclassical averages and write any mode operator asO=hOi+O(O=a1;a2;m), neglecting small second- order uctuation terms. Here, hOiis the mean value of the operatorO, andOis the zero-mean quantum uctuation. We then obtain two sets of equations for semiclassical av- erages and for quantum uctuations. The former set of equations are given by: (i
1+1)ha1iig1hmii"p2i hay 1i= 0 (6) (i
2+2)ha2iig2hmi= 0 (7) (i
m+m)hmiig1ha1iig2ha2i= 0 (8) By solving Eqs.(6)-(8), we obtain the steady-state solution for the average values ha1i=2 "p P"p
1i14 2 Pg2 1(
2i2) (
mim)(
2i2)g2 2(9) ha2i=g1g2ha1i (
mim)(
2i2)g2 2(10) hmi=g1(
2i2)ha1i (
mim)(
2i2)g2 2(11) whereP=
1+i1g2 1(
2+i2) (
m+im)(
2+i2)g2 2. Thus, we can obtain the mean photon numbers and mean magnon number from Eqs.(9)-(11). On the other hand, quantum uctuations is related to entanglement and squeezing. To study the quantum char- acteristics of the two cavity modes and magnon mode, the quadratures of quantum uctuations about cavity modes and magnon mode are as X1= (a1+ay 1)=p 2,Y1= i(ay 1a1)=p 2,X2= (a2+ay 2)=p 2,Y2=i(ay 2 3a2)=p 2,x= (m+my)=p 2, andy=i(mym)=p 2, and similarly for the input noise operators. The quantum Langevin equations describing quadrature uctuations ( X1; Y1; X 2; Y2; x; y ) can be written as _f(t) =Af(t) + (12) wheref(t) = [X1(t),Y1(t),X2(t),Y2(t),x(t),y(t)]Tand (t) = [p21Xin 1(t),p21Yin 1(t),p22Xin 2(t),p22Yin 2(t),p2mxin(t),p2myin(t)]Tare the vectors for quantum uctuations operator and noises operator, respectively. The drift matrix A is given by A=0 BBBBBB@1
12 0 0 0 g1
12 1 0 0g10 0 02
20g2 0 0
22g20 0g1 0g2m
m g1 0g20
mm1 CCCCCCA Due to the linearized dynamics and the Gaussian na- ture of the quantum noises in our system, the steady state of quantum uctuations is a continuous variable three mode Gaussian state, which is completely characterized by a 66 covariance matrix Vdened asVij=hfi(t)fj(t0) + fj(t0)fi(t)i=2 (i;j= 1;2;:::;6). In generally, the steady- state covariance matrix Vcan be obtained straightfor- wardly by solving the Lyapunov equation [39, 40] AV+VAT=D (13) whereD=diag[1(2N1+ 1),1(2N1+ 1),2(2N2+ 1), 2(2N2+ 1),m(2Nm+ 1),m(2Nm+ 1)] is the diu- sion matrix, which is dened as Dij(tt0) =hi(t)j(t0)+ j(t0)i(t)i=2. With the covariance matrix in hand, we can get the quantities related to entanglement and squeezing. To quantify entanglement between the two cavity modes and magnon mode, we adopt quantitative measures of the logarithmic negativity [41, 42] ENfor the bipartite entan- glement, which is dened as ENmax[0;ln2~] (14) where ~=min[eigji 2~V4j] is the minimum symplectic eigen- value of the ~V4=P1j2V4P1j2.V4is the 44 covariance matrix, which can be obtained by directly removing in V the rows and columns of uninteresting mode. Meanwhile, to realize partial transposition at the level of covariance matrix, we set P1j2=diag(1;1;1;1). 2is symplectic matrix with 2=2 j=1iyandyis they-Pauli matrix. A nonzero logarithmic negativity EN>0 denotes the pres- ence of bipartite entanglement in our QED system. Meanwhile, a quantication of continuous variable tri- partite entanglement is given by the minimum residual contangle [43, 44], dened as Rmin min[Rajm1m2 ;Rm1jam2 ;Rm2jam1  ] (15) whereRijjk CijjkCijjCijk0 (i;j;k =a;m 1;m2) is the residual contangle, with Cujvthe contangle of sub- systems of uandv(vcontains one or two modes), whichis a proper entanglement monotone dened as the squared logarithmic negativity. When vcontains two modes, loga- rithmic negativity Eijjkcan be calculated by the denition of Eq.(14). We only need to use 3=3 j=1iyinstead of 2=2 j=1iyand ~V6=PijjkVPijjkinstead of ~V4= P1j2V4P1j2, whereP1j23=diag(1;1;1;1;1;1),P2j13= diag(1;1;1;1;1;1) andP3j12=diag(1;1;1;1;1;1) are partial transposition matrices. Rmin 0 denotes the pres- ence of genuine tripartite entanglement in three modes Gaussian system. Meanwhile, squeezing can be calculated by the covari- ance matrix of quantum uctuations. The variances of squeezed magnon quadratures are amplitude quadrature hx(t)2i, phase quadrature hy(t)2i, and amplitude quadra- turehY2(t)2iis quadrature of cavity mode a2,x= (my+ m)=p 2,y=i(mym)=p 2, andY2=i(ay 2a2)=p 2. In our denition, hQ(t)2ivac= 1=2 (Q is a mode quadra- ture) denotes vacuum uctuations. The degree of squeez- ing can be expressed in the dB unit, which can be evalu- ated by10log10[hQ(t)2i=hQ(t)2ivac], wherehQ(t)2ivac= 1=2. 3. Results and discussion To show whether the squeezed driving can induce en- tanglement, we consider a simpler magnon-cavity QED system at rst, where no coupling interaction exists be- tween the magnon mode and cavity mode a2, i.e.,g2= 0. Fig.2(a) shows the bipartite entanglement between cav- ity modea1and magnon mode versus detunings
1and
min steady state. We employed experimentally fea- sible parameter [5] at low temperature T= 10mK, as !1=2= 10GHz, m=2= 1MHz,1=2=2=2= 5MHz,g1=2= 20MHz. Moreover, Rabi frequency of microwave eld we employed is "p= 10m. Squeezed eld used in our system is to generate nonlinear term by the JPA process with gain coecient = 2 :5m. This is the nonlinearity that causes entanglement in our QED system. Fig.2(a) shows that the photon-magnon entan- glement described by logarithmic negativity can achieve to 0.3. Meanwhile, due to the state-swap interaction be- tween the cavity mode a1and magnon mode, the squeez- ing can be transferred from squeezed cavity mode a1to the magnon mode, as shown in Fig.2(b). Note that the above results are valid only when the assumption of low-lying excitations, i.e. magnon excita- tion numberhmymi 2Ns, wheres= 5=2 is the spin number of ground-state Fe3+ion in YIG sphere. The to- tal number of spins N=Vwith= 4:221027m3 the spin density of YIG and Vthe volume of sphere. For a 250-m-diameter YIG sphere, the number of spins N'3:51016. We then calculate the mean photon numbers of cavity mode a1N1=hay 1a1i, cavity mode a2 N2=hay 2a2i, and mean magnon number Nm=hmymivia Eqs.(9)-(11), which are closely related to the input inten- sity of microwave eld and squeezed eld. Fig.2(c) and 4(d) show the mean photon number N1and mean magnon numberNmversus detunings
1and
min steady state wheng2= 0. They are drawn with logarithmic log10. We show that both the maximum number of photons and magnons are above 10, but less than 103in Fig.2(c) and (d). Meanwhile, we also get N2= 0. so the assumption of low-lying excitations is well satised. Fig. 2. (a)Density plot of photon-magnon bipartite entanglement Ea1m, (b)variance of the magnon amplitude quadrature hx(t)2i, (c)logarithm of mean photon number of cavity mode a1N1, and (d)logarithm of mean magnon number Nmversus detunings
1and
m. We choose = 2 :5m,"p= 10m. The blank area denotes hQ(t)2ivac>1=2, i.e., above vacuum uctuations. We take g2= 0 for all the plots. See text for the detail of other parameters. We then take g2into consideration. To be more gen- eral, we assume that coupling rate g2is the same as that between the cavity mode a1and magnon mode, i.e., g2= g1. In Fig.3(a)-3(c), mean photon numbers and mean magnon number, N1,N2, andNm, are plotted as func- tions of detunings
2and
m, respectively. They are also drawn with logarithmic log10. It is noted that P= 0 is the extreme value of Eqs.(9)-(11). Ignoring dissipa- tive terms and analyzing the extreme value, we can ob- tain a simple form
m= (
1g2 2+
2g2 1)=(
1
2). The black dashed curves in Fig.3(a)-(c) denote
m= (
1g2 2+
2g2 1)=(
1
2), and from which we can see that the max- imum numbers of photons and magnons are located at about this region. -1.01.0 -2.0 (a) (b) (c) 0 Fig. 3. (a)Mean photon number of cavity mode a1(squeezed cav- ity mode)N1, (b)mean photon number of cavity mode a2N2, and (c)mean magnon number Nmversus detunings
2and
m. Black dash curves indicate
m= (
1g2 2+
2g2 1)=(
1
2). All Figures are drawn with logarithmic log10. We take Rabi frequency of microwave eld"p= 10mand the nonlinear gain coecient of squeezed eld = 2:5m. We assume the coupling rate between the two cavity modes and magnon mode are the same, i.e., g2=g1. The detuning of cavity mode a1
1=20m. The other parameters are as in Fig.2.After coupling cavity mode a2to magnon mode ( g2> 0), the magnon-cavity a1entanglement Ea1mdecreased while cavity mode a2and magnon mode get entangled, as shown in Fig.4(a) and (c) with assuming g2=g1. It denotes that the quantum correlations can be trans- ferred from magnon mode and cavity mode a1to magnon mode and cavity mode a2. All results are in the steady state guaranteed by the negative eigenvalues (real parts) of the drift matrix A. We also choose the Rabi frequency of microwave eld "p= 10mand the gain coecient = 2:5m. The Black dashed curves in Fig.4(a) and (c) denote
m= (
1g2 2+
2g2 1)=(
1
2). It clearly shows that the optimal photon-magnon entanglement is achieved near the maximum of mean particle numbers. We then calculated the squeezing by the covariance ma- trix of quantum uctuations applying mean eld approxi- mation, and found that the cavity modes and the magnon mode can be squeezed. Compared with photons, it is more meaningful to study squeezed magnons, a mesoscopic ob- ject. Two quadratures of magnon mode are amplitude quadraturehx(t)2iand phase quadrature hy(t)2i, these two quadratures also obey the uncertainty relationship, i.e., when the phase (amplitude) quadrature is squeezed, the amplitude (phase) quadrature will not be squeezed. That is, the squeezed of one quadrature is at the expense of increasing the other one. Variance of magnon ampli- tude quadrature hx(t)2iand phase quadrature hy(t)2i versus detunings
2and
mare shown in Fig.4(b) and (d), respectively. The blank area denotes above vacuum uctuations, i.e., hQ(t)2ivac>1=2, (Q=x;y). Fig. 4. (a)Density plot of bipartite entanglement Ea1m, (b)variance of the magnon amplitude quadrature hx(t)2i, (c)density plot of bi- partite entanglement Ea2m, and (d)variance of the magnon phase quadraturehy(t)2iversus detunings
2and
m. We choose = 2:5m,"p= 10m. The detuning of cavity mode a1
1=20m. Black dash curves in Fig.4(a) and (c) indicate
m= (
1g2 2+
2g2 1)=(
1
2). The blank area in Fig.4(b) and (d) denoteshQ(t)2ivac>1=2, i.e., above vacuum uctuation. We takeg2=g1for all the plots. See text for the other parameters. Further, Fig.5(a) shows that the two cavity modes get entangled, which denotes that the photon-photon entan- glementEa1a2is transferred from magnon-cavity entangle- mentEa1mandEa2mdue to the state-swap interaction be- tween the two cavity modes and magnon mode. The cou- 5pling rateg2also induces the squeezing transferred from cavity mode a1to cavity mode a2via magnon mode, as shown in Fig.5(b). The blank area denotes above vac- uum uctuations, i.e., hQ(t)2ivac>1=2. Comparing to Fig.2(b), the maximum of variance of the magnon am- plitude quadratures hx(t)2iandhy(t)2idecreases, and cavity mode a2get squeezed. It denotes that the two cav- ity modes and magnon mode are all prepared in squeezed states due to the state-swap interaction between the two cavity modes and magnon mode, meaning that the mag- netic dipole interaction is an essential element to generate squeezed states. Logarithmic negativity Ea1a2as a func- tion of bath temperature is shown in Fig.5(c). It denotes that photon-photon entanglement Ea1a2is robust again bath temperature and survives up to about 200 mK. Tri- partite entanglement in terms of the minimum residual contangleRmin detunings
2and
mis shown in Fig5(d). It shows that the tripartite entanglement does exist in our QED system. The black dashed curves in Fig.5(d) denote
m= (
1g2 2+
2g2 1)=(
1
2), and from which we can see that the maximum of tripartite entanglement located at about this region. Fig. 5. (a)Density plot of photon-photon bipartite entanglement Ea1a2, and (b)variance of cavity mode a2amplitude quadrature hY2(t)2iversus detunings
2and
m. (c)Logarithmic negativ- ityEa1a2vs bath temperature T. (d) Tripartite entanglement in terms of the minimum residual contangle Rmin detunings versus
2 and
m. The blank area in Fig.5(b) denotes hQ(t)2ivac>1=2, and black dash curves indicate
m= (
1g2 2+
2g2 1)=(
1
2) in Fig.5(d). We take
2= 35m,
m= 45mfor (c), = 2 :5m, "p= 10m,
1=20mandg2=g1for all the plots. See text for the other parameters. Squeezing does not increase linearly with increasing the gain coecient. We choose
2= 0 , and nd that squeez- ing rst increases and then decreases with the increase of the gain coecient, as shown in Fig.6(a) and (b), respec- tively.The blank area denotes hQ(t)2ivac>1=2. Squeez- ing reaches the maximum near = 8 mfor the amplitude quadrature and near = 2 mfor phase quadrature. To obtain the optimal entanglement between the two cavity modes, we show photon-photon entanglement Ea1a2 versus gain coecient and the rate of magnon-cavity coupling strength g2=g1in Fig.7(a). All results are cal- culated in the steady state, and the blank area denotes Fig. 6. (a)Variance of the magnon amplitude quadrature hx(t)2i, (b)Variance of the magnon phase quadrature hy(t)2iversus gain co- ecient and detunings
m. The blank area denotes hQ(t)2ivac> 1=2. We take
1=20m,
2= 0,"p= 10mfor all the plots. See text for the other parameters. Non equilibrium state. As shown in Fig.7(a), the two cavity modes show zero entanglement in the absence of gain coecient, i.e., = 0, meaning that it is squeezed driving that induced entanglement in our QED system. It demonstrates that the nonlinearity produced by para- metric down-conversion is the reason to generate entan- glement. Bipartite entanglement Ea1mincreases with the increase of gain coecient , and then the entanglement transferred from Ea1mtoEa2mandEa1a2. But, to keep the system in steady state, the gain coecient can not be too large. Fig.7(a) denotes that increasing gain coecient is a good way to improve entanglement in our QED system. Due to the exible tunability of gain coecient, which makes large entanglement possible. Further, we show that the optimal entanglement can be generated when the rate of photon-magnon coupling strength are almost the same, i.e.,g2=g11. Meanwhile, we show variance of magnon amplitude quadraturehx(t)2iversus gain coecient and the rate of magnon-cavity coupling strength g2=g1in Fig.7(b). The blank area represents above vacuum uctuation, i.e., hQ(t)2ivac>1=2. It shows that the magnon mode can not be squeezed in the absence of squeezed eld, i.e., = 0, and the strength of squeezed magnon mode transferred from squeezed cavity mode a1can increase a lot as gain coecient increasing. It provides a good scheme to im- prove macroscopic quantum state. Further, we also show that the optimal squeezing is also located at about the regiong2=g1= 1. Fig. 7. (a)Density plot of photon-photon bipartite entanglement Ea1a2, (b)variance of the magnon amplitude quadrature hx(t)2i versus nonlinear gain coecient and the rate of magnon-cavity coupling strength g2=g1. The blank area in Fig.7(a) presents non equilibrium states and hQ(t)2ivac>1=2 in Fig.7(b), i.e., above vacuum uctuations. We take
2= 35m,
m= 45mfor (a),
2=45m,
m=15mfor (b), and
1=20m,"p= 10m for all the plots. See text for the other parameters. 64. Conclusion In summary, we have presented a scheme to gener- ate bipartite entanglement between two cavity modes and squeeze magnon mode in a magnon-cavity QED system by using a squeezed driving. With experimentally reach- able parameters, we show that without the nonlinearity induced by parametric down-conversion process, our QED system denotes zero entanglement and above vacuum uc- tuations. We also show the photon-magnon entanglement can transfer to photon-photon entanglement by state-swap interaction between cavity and magnon modes in the steady state. A genuinely tripartite entangled state is formed. Meanwhile, magnon squeezed state also can be realized due to the squeezing from squeezed driving cavity mode. Moreover, our QED system shows that increasing the non- linear gain coecient is a good way to enhance entangle- ment and squeezing. Further, the optimal entanglement and squeezing is located at about the region where the cou- pling rates between two cavity modes and magnon mode are almost the same. Our results denote that magnon- cavity QED system is a powerful platform for studying macroscopic quantum phenomena, and squeezed drive pro- vides an new method for generating macroscopic quantum state. Acknowledgements This work has been supported by the National Natural Science Foundation of China (Grant No. 12174288, Grant No. 11874287, Grant No. 11774262, Grant No. 61975154), and the Shanghai Science and Technology Committee (Grant No. 18JC1410900). References [1] Dany Lachance-Quirion, Yutaka Tabuchi, Arnaud Gloppe, Koji Usami, and Yasunobu Nakamura. Hybrid quantum systems based on magnonics. Appl. Phys. Exp. , 12(7):070101, Jun 2019. [2] Graeme Flower, Jeremy Bourhill, Maxim Goryachev, and Michael E Tobar. 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2022-01-23

We propose a scheme to generate entanglement between two cavity modes and squeeze magnon mode in a magnon-cavity QED system, where the two microwave cavity modes are coupled with a massive yttrium iron garnet (YIG) sphere through magnetic dipole interaction. The nonlinearity used in our system originates from a squeezed driving via parametric down-conversion process, which is the reason to cause entanglement and squeezing. By using the mean field approximation and employing experimentally feasible parameters, we demonstrate that the system shows zero entanglement and squeezing without squeezed driving. Meanwhile, our QED system denotes that the entanglement between squeezed cavity mode and magnon mode can be transferred to the other cavity mode and magnon mode via magnon-cavity coupling interaction, and then the two cavity modes get entangled. A genuinely tripartite entangled state is formed. We also show that magnon mode can be prepared in a squeezed state via magnon-cavity beam-splitter interaction, which is as a result of the squeezed field. Moreover, we show that it is a good way to enhance entanglement and squeezing by increasing the nonlinear gain coefficient of squeezed driving. Our results denote that magnon-cavity QED system is a powerful platform for studying macroscopic quantum phenomena, which illustrates a new method to photon-photon entanglement and magnon squeezing.

Squeezed driving induced entanglement and squeezing among cavity modes and magnon mode in a magnon-cavity QED system

2201.09154v1

Detection of entanglement by harnessing extracted work in an opto-magno-mechanics M’bark Amghar1and Mohamed Amazioug1,∗ 1Department of Physics, Ibnou Zohr University, Agadir 80000, Morocco (Dated: May 30, 2024) The connections between thermodynamics and quantum information processing are of paramount impor- tance. Here, we address a bipartite entanglement via extracted work in a cavity magnomechanical system contained inside an yttrium iron garnet (YIG) sphere. The photons and magnons interact through an interaction between magnetic dipoles. A magnetostrictive interaction, analogous to radiation pressure, couple’s phonons and magnons. The extracted work was obtained through a device similar to the Szil ´ard engine. This engine operates by manipulating the photon-magnon as a bipartite quantum state. We employ logarithmic negativity to measure the amount of entanglement between photon and magnon modes in steady and dynamical states. We explore the extracted work, separable work, and maximum work for squeezed thermal states. We investigate the amount of work extracted from a bipartite quantum state, which can potentially determine the degree of entanglement present in that state. Numerical studies show that entanglement, as detected by the extracted work and quantified by logarithmic negativity, is in good agreement. We show the reduction of extracted work by a second measurement compared to a single measurement. Also, the e fficiency of the Szilard engine in steady and dynamical states is investigated. We hope this work is of paramount importance in quantum information processing. I. INTRODUCTION Entanglement, a cornerstone of quantum mechanics pioneered in works like [1–3], holds immense potential across various fields. It has exciting applications in areas like precision measurement, quantum key distribution [4], teleporting quantum information [5], and building powerful quantum computers as explored by [6]. Entanglement creates a spooky connection between particles. Measuring one instantly a ffects the other, defying classical ideas of locality. Cavity optomechanics is a field of physics that studies the interaction between light and mechanical objects via radiation pressure at very small scales [7]. Cavity optomechanics has been used to develop new methods for generating and manipulating squeezed states of light, which are a type of quantum state. Recently this cavity has paramount importance applications in quantum information processing such as quantum entangled states [8–19], cooling the mechanical mode to their quantum ground states [20–24], photon blockade [25], enhancing precision measurements [27, 28], superconducting elements [29], and also between two massive mechanical oscillators [30, 31] have been observed. Cavity quantum electrodynamics (CQED) paved the way for the emergence of a new field called cavity optomechanics. Cavity quantum electrodynamics (CQED) o ffers control over how light (photons) interacts with atoms at the quantum level. Actually, single quanta can significantly impact the atom-cavity dynamics in the strong coupling regime, which is made possible by strong confinement. Recently, we have successfully achieved strong coupling in numerous experiments, leading to the demonstration of fascinating quantum phenomena such as quantum phase gates [32], the Fock state generation [33], and quantum nondemolition detection of a single cavity photon [34]. Building on the success of cavity QED, exploring how magnon systems interact within cavity optomechanics o ffers a promising avenue for unlocking their unique quantum properties. The first experimental demonstration of interaction between magnons, photons, and phonons has been achieved [35]. This system combines magnon-photon coupling, similar to what’s found in magnon QED, with an additional coupling between magnons and phonons. While the cavity output reflects the impact of magnon-phonon coupling, a more comprehensive understanding would require a full quantum treatment that incorporates these fluctuations. From the standpoint of cavity quantum electrodynamics (QED), ferrimagnetic systems. Particularly, the yttrium iron garnet (YIG) sphere has garnered a lot of attention. Studies have shown that the YIG sphere’s Kittel mode [36] can achieve strong coupling with microwave photons trapped within a high-quality cavity. This strong coupling leads to the formation of cavity polaritons [37] and a phenomenon known as vacuum Rabi splitting. The success of cavity QED has opened doors to applying many of its concepts to the emerging field of magnon cavity QED [38]. This new field has already seen exciting advancements, including the observation of bistability [39] and the groundbreaking coupling of a single superconducting qubit to the Kittel mode [40]. Recently, magnons (as spin waves) have been studied extensively in the field of quantum information processing [41–44]. Studies suggest that physical system information can be utilized to extract work with suitable operations [45]. This, described as information thermodynamics, explores the relationship between information theory and thermodynamics. By studying how information can be manipulated to perform physical tasks, researchers hope to uncover new ways to improve e fficiency in ∗amazioug@gmail.comarXiv:2405.19205v1 [quant-ph] 29 May 20242 various processes. Szilard’s engine [46] is a classic example of this concept. It demonstrates how information processing is capable of extracting work from a physical system. Interestingly, research has shown that the specific way information is encoded, particularly in entangled or correlated states, plays a crucial role in how much work can be extracted [47–49]. In this letter, we investigate the potential of exploiting both extracted work and e fficiency in optomagnomechanical systems. This study reveals the presence of entanglement between magnons and cavity photons in an optomechanical system. We achieve this detection by examining the extractable work. Our work breaks new ground by employing extractable work as a tool to detect entanglement between magnons and cavity photons in optomechanical systems. Our study, utilizing realistic experimental parameters, reveals excellent agreement between the entanglement region detected via extractable work and the results of Jie Li et al [50]. This highlights the validity of our method for entanglement detection in magnomechanical systems. This applies not only to steady-state conditions but also to dynamical states under thermal influence. Our study further explores the influence of various parameters, including detuning and magnon-phonon coupling, on the entanglement properties of the system. In addition, we analyze the information-work e fficiency under thermal noise, considering both steady-state and dynamical regimes. The article is outlined as follows: In Section II, we introduce the model for the optomagnomechanical system, its Hamilto- nian, and the quantum nonlinear Langevin equations for the interacting photon-magnon-phonon system. Section III tackles the linearization of the QLEs, and we then assess the covariance matrix for steady and dynamical states. Section IV delves into the connection between quantum thermodynamics and quantum entanglement in cavity magnomechanical systems. We employ logarithmic negativity to quantify the entanglement between the photon and magnon modes and investigate how it relates to the amount of work that can be extracted from the system. Besides, we have also investigated the e fficiency of a Szilard engine. The results obtained are discussed. Concluding remarks close this paper. II. MODEL In this section, we consider on a system that combines a microwave cavity with magnetic excitations (magnons) and mechan- ical vibrations (phonons). This cavity magnomechanical system is illustrated in Fig. 1. Magnons are a collective motion of numerous spins in a ferrimagnet, such as an YIG sphere (250- µm-diameter sphere used in Ref. [35]). A sphere made of YIG (Yttrium Iron Garnet) is positioned within a microwave cavity at a location with the strongest magnetic field. Additionally, a uniform bias magnetic field is applied throughout the entire system. These combined fields allow the microwave photons in the cavity to interact with the YIG sphere’s magnons through the magnetic dipole interaction. To improve the coupling between magnons and phonons, the experiment utilizes a microwave source (not shown) to directly driven the magnon mode magne- tostrictive interaction. Due to the YIG sphere’s small size compared to the microwave wavelength, we can ignore the interaction between microwave photons and phonons. We consider the three magnetic fields: a bias field pointing in the z-axis, a drive field in the y-axis, and the magnetic field of the cavity mode oriented along the x-axis, as depicted in Fig. 1. These three fields are mu- tually perpendicular at the position of the YIG sphere. The YIG sphere experiences a deformation of its geometry structure due to the creation of vibrational modes, or phonons, which influence the magnon excitations within the sphere, and vice versa [51]. The Hamiltonian writes as [50] ˆH=ℏΩcˆc†ˆc+ℏΩnˆn†ˆn+Ωd 2( ˆx2+ˆy2) +ℏgndˆn†ˆnˆx+ℏGnc(ˆc+ˆc†)(ˆn+ˆn†) +iℏΩ(ˆn†e−iΩ0t−ˆneiΩ0t),(1) where the creation and annihilation operators for the cavity (ˆ c,ˆc†) and magnon (ˆ n,ˆn†) modes satisfy the canonical commutation relation [ ˆO,ˆO†]=1 (where ˆOcan be ˆ cor ˆn). Additionally, dimensionless position ˆ xand momentum ˆ yoperators for the mechan- ical mode are included, with the commutation relation [ ˆ x,ˆy]=i. The Hamiltonian also incorporates the resonance frequencies (Ωc,Ωn, and Ωd) of the cavity, magnon, and mechanical modes, respectively. The magnon frequency Ωnis dictated by the exter- nal bias magnetic field Hand the gyromagnetic ratio γfollowing the relation: Ωn=γH. Interestingly, the magnon-microwave coupling rateGncsurpasses the dissipation rates of both the cavity λcand magnon modes λn, satisfying the condition for strong coupling:Gnc>λ c,λn[37]. The inherent coupling rate between a single magnon and the mechanical vibrations, denoted by gnd, is typically low. This limitation can be overcome by strategically applying a strong microwave field directly to the YIG sphere. This approach, employed in earlier works [39, 52], e ffectively enhances the magnomechanical interaction. The Rabi frequency Ω, derived under the assumption of low-lying excitations ( ⟨ˆn†ˆn⟩≪2Ns, where s=5/2 is the spin of the Fe3+ground state ion), characterizes the coupling strength between the driving magnetic field (amplitude B0and frequency ω0) and the magnon mode. It is expressed as Ω =√ 5 4γg√ NB0, whereγg/2π=28 GHz /T is the gyromagnetic ratio of the material and N=ϱVrepresents the total number of spins in the YIG sphere. Here, ϱ=4.22×1027m−3is the spin density and Vis the sphere’s volume. Using a rotating frame at the driving frequency Ω0and the rotating-wave approximation Gnc(ˆc+ˆc†)(ˆn+ˆn†)→G nc(ˆcˆn†+ˆc†ˆn) valid3 when Ωc,Ωn≫G nc,κc,κn[35], the system’s dynamics are described by quantum Langevin equations (QLEs). δ˙ˆc=−(iδc+λc)ˆc−iGncˆn+p 2λcˆcin, δ˙ˆn=−(iδn+λn)ˆn−iGncˆc−igndˆnˆx+ Ω +p 2λnˆnin, δ˙ˆx=ωˆy, δ ˙ˆy=−ωˆx−γdˆy−gndˆn†ˆn+η,(2) withδc= Ω c−Ω0,δn= Ω n−Ω0andγdis the mechanical damping rate. The input noise operators for the cavity and magnon modes are respectively, ˆ cinand ˆnin, with zero mean, i.e., ⟨ˆcin⟩=⟨ˆnin⟩=0, and described by the following correlation functions [53]: ⟨ˆcin(t) ˆcin†(t′)⟩=[Nc(Ωc)+1]δ(t−t′),⟨ˆcin†(t) ˆcin(t′)⟩=Nc(Ωc)δ(t−t′) ⟨ˆnin(t) ˆnin†(t′)⟩=[Nn(Ωn)+1]δ(t−t′),⟨ˆnin†(t) ˆnin(t′)⟩=Nn(Ωn)δ(t−t′),(3) we assume the mechanical mode follows a Markovian process. This means a large mechanical quality factor Q≫1, i.e., Ωd≫γd[54]. Furthermore, the noise operators for this mode possess non-zero correlation properties (with ⟨η(t)⟩=0), writes as ⟨η(t)η(t′)+η(t′)η(t)⟩/2≃γd[2Nd(Ωd)+1]δ(t−t′), (4) where Nc,NnandNdcorrespond to the equilibrium mean thermal occupation numbers for the cavity photons, magnons, and phonons, respectively. Thus Nj(Ωj)=expℏΩj kBT
−1−1(j=ˆc,ˆn,ˆd), where kBis the Boltzmann constant. III. COV ARIANCE MATRIX We consider the case where the magnon mode is highly driven. We linearize the non-linear quantum Langevin equation by assuming small fluctuations around a steady state amplitude, i.e., ˆO=ˆOss+δˆO(ˆO=ˆc,ˆn,ˆx,ˆy), where ˆ nsswrites as ˆnss=Ω(iδc+λc) G2nc+(i˜δn+λn)(iδc+λc), (5) where ¯δn=δn+gndˆxssis the e ffective magnon-drive detuning taking into account a frequency shift resulting from the interaction between magnons and phonons. This interaction is known as magnomechanical interaction. Under the condition |¯δn|,|δc|≫ λc,λn; ˆnssis given by ˆnss=iΩδc G2nc−˜δnδc. (6) and ˆxss=−gnd Ωd|ˆnss|2. The system is described by linearized quantum Langevin equations (LQLEs) δ˙ˆc=−(iδc+λc)ˆc−iGncˆn+p 2λcˆcin, δ˙ˆn=−(i¯δn+λn)ˆn−iGncˆc−igndˆnssˆx+ω+p 2λnˆnin, δ˙ˆx=ωˆy, δ ˙ˆy=−ωˆx−γdˆy−gnd(ˆnssˆn†+ˆn∗ ssˆn)+η,(7) where ˆ nss=−iGnd√ 2gndis the magnon-phonon coupling. The quadrature fluctuations ( δXc,δYc,δXn,δYn,δx,δy) are described as δXc=(δˆc+δˆc†)/√ 2, δYc=i(δˆc†−δˆc)/√ 2 δXn=(δˆn+δˆn†)/√ 2, δYn=i(δˆn†−δˆn)/√ 2 We can rewrite equation (7) as ˙v(t)=Fv(t)+χ(t), (8)4 where v(t) =δXc(t),δYc(t),δXn(t),δYn(t),δx(t),δy(t)Tis the quadrature vector, χ(t) =√2λcXin c(t),√2λcYin c(t),√2λnXin n(t),√2λnYin n(t),0,η(t)Tis the noise vector and the drift matrix Fis written as F=−λcδc 0Gnc 0 0 −δc−λc−Gnc0 0 0 0Gnc−λn˜δn−Gnd 0 −Gnc0−˜δn−λn0 0 0 0 0 0 0 ωd 0 0 0Gnd−ωd−λd, (9) The system under consideration is considered stable if all the eigenvalues of a drift matrix, have real parts that are negative [55]. By using the Lyapunov equation, the system’s state is expressed as [56, 57] FC+CFT=−L, (10) whereCi j=1 2⟨vi(t)vj(t′)+vj(t′)vi(t)⟩(i,j=1,2,...,6) is the covariance matrix and L=diagλc(2Nc+1),λc(2Nc+1),λn(2Nn+ 1),λn(2Nn+1),0,λd(2Nd+1)is the di ffusion matrix achieved through Li jδ(t−t′)=1 2⟨χi(t)χj(t′)+χj(t′)χi(t)⟩. IV . SZIL ´ARD ENGINE L´eo Szil ´ard introduced Szil ´ard’s engine as a thought experiment in 1929. This experiment simplified the famous Maxwell’s demon paradox by using just one molecule of gas and replacing the demon with a mechanical device. Szil ´ard’s engine operates in four key steps: (i) The experiment begins with a single gas molecule bouncing around freely in a container with a volume ofV. (ii) A separator is placed inside a container, dividing it into two equal chambers with a volume of V/2, ensuring no heat exchange during the process. (iii) The engine’s function relies on determining the molecule’s location in the left or right chamber. According to the measurement results, a tiny weight has been attached to the same side of the partition using a pulley system. (iv) The final stage involves connecting the entire setup to a constant temperature heat source, allowing a gas molecule to expand and fill the container, crucial for the engine’s theoretical work. Szil ´ard’s engine, a concept that challenges our understanding of thermodynamics, involves a single molecule expanding to fill a container, absorbing heat from a constant temperature bath, and converting this heat into usable work by lifting the weight attached to the partition. The amount of work extracted, can be calculated using the formula W=kBTln 2, where kBis Boltzmann’s constant and ln(2) represents the information gain from measuring the molecule’s location. This process relies on connecting a weight to the partition and allowing the single gas molecule to expand in a controlled, constant temperature, i.e., isothermal manner. Szil ´ard’s engine can theoretically extract a specific amount of work per cycle, as described in [59] W=kBTln 2[1−H(X)] (11) The uncertainty about where the molecule is situated (left or right) can be quantified using a concept called Shannon entropy. This entropy is denoted by H(X)=−P xpxlnpx, where pxis the probability of capturing the molecule in each location ( x=R orx=L). Thus, the more uncertain we are about the molecule’s location, the higher the Shannon entropy will be. Equation (11) presents a potential challenge to the second law of thermodynamics. It suggests that under specific conditions, perfect knowledge about a the microscopic information of the system state might allow for work extraction using only a single heat bath. A significant link between information processing and the physical world was suggested by physicist Rolf Landauer in 1961. He theorized that whenever a single bit of information is erased in a computer system, it leads to an increase in energy dissipation as heat. This principle suggests a fundamental link between logical operations within a computer and the laws of thermodynamics that govern physical processes [60]. Recent experiments explore innovative techniques inspired by Maxwell’s demon and Landauer’s principle, which link information processing and energy dissipation, despite the potential to defy thermodynamic laws [61, 62]. By separating two entangled particles into di fferent containers, we essentially create two Szil´ard engines, AandB. These engines are unique because their functionality is intrinsically linked. Thus, the amount of work extractable from engine Ais dependent on the specific state of its entangled partner in engine B W(A|B)=kBTlog[1−H(A|B)] (12) Due to the entanglement, any event a ffecting engine Bhas an immediate impact on our understanding of engine A. Mutual information I(A:B)=H(A)−H(A|B)≥0 quantifies the link between AandB, with a non-negative value indicating that knowing the state of Breduces uncertainty about A. The reduced uncertainty leads to a significant increase in work extraction from engine Acompared to a scenario without entangled two Szil ´ard engines, as indicated by W(A|B) being greater than or equal toW(A), i.e., W(A|B)≥W(A).5 V . NEGATIVITY LOGARITHMIC, WORK EXTRACTION AND EFFICIENCY In this section, we will quantify and harness negativity logarithmic compared to the extracted work in a two-mode Gaussian state, shared by Alice ( A: photon) and Bob ( B: magnon). The e fficiency of a Szilard engine will be adopted. A. Negativity logarithmic The covariance matrix corresponding to the photon and magnon modes in the ( δXc(t),δYc(t),δXn(t),δYn(t)) basis can be expressed as CAB=CcCcn CT cnCn, (13) CcandCndepict the covariance matrix 2 ×2, respectively, representing the photon mode and magnon mode. The correlations between photon and magnon modes in standard form are denoted by Ccn Cc=diag(α,α),Cn=diag(β,β),Ccn=diag(∆,−∆). (14) For measuring bipartite entanglement, we employ the logarithmic negativity EN[59, 63, 64], that is given by Eom=max[0,−log(2ν−)], (15) whereν−=p Y− (Y2−4 detCAB)1/2/√ 2 is the minimum symplectic eigenvalue of the CAB, whereY=detCc+detCn− 2 detCcn. B. Magnon only performs Gaussian measurement The medium under consideration is a two-mode Gaussian state, i.e., photon and magnon modes. When Bob executes a Gaussian measurement on his assigned part of the system, the measurement has an impact on Alice’s state. This measurement can be described by ˜Nn(X)=π−1˜Dn(X)˜ρNn˜D† n(X), (16) where ˜Dn(X)=eXδˆn†−X∗δˆnis the displacement operator, ˜ ρNnis a pure Gaussian state without first moment and the its covariance matrix is given by ΓNn=1 2R(ξ)diag(λ,λ−1)R(ξ)T, (17) whereλis a positive real number, R(ξ)=[cosξ,−sinξ; sinξ,cosξ] is a rotation matrix and the detection of homodyne (hetero- dyne) is suggested by λ=0 (λ=1), individually. The outcome XBob gets from his measurement, it doesn’t a ffect the state of Alice’s mode δˆc, i.e.,CNn c|X=CNnc. The constrained state of mode A’s covariance matrix can be explicitly expressed as CNnc=Cc−C cn(Cn+ ΓNn)−1CT cn. (18) Bob measurement does push the state of mode Aout of equilibrium. However, by interacting with a heat bath for long time, mode Aeventually returns to an equilibrium state Ceq c. Its average entropy is solelyR dXpXS(CNn c|X)=S(CNnc) because her state is una ffected by the result. Work can be extracted by Alice from a surrounding heat bath [65] W=kBTh S(Ceq c)−S(CNnc)i . (19) We adopt the case of the covariance matrix in a squeezed thermal state, as depicted in equation (14) and Ceq c=Cc. The entropy of the covariance matrix described by equation (18) is quantified by considering the second-order R ´enyi entropy S2(ϱ)=−lnTrϱ2 [66]. In the case of two modes, Gaussian states (see equation (14)) are written as S2(CAB)=1 2ln(detCAB). (20)6 The extracted work, Eq. (19), became W(λ)=kBT 2ln detCc detCNnc! . (21) The extractable work for both homodyne ( λ=0) and heterodyne ( λ=1) detection in the case of STSs, writes as W(0) om=kBT 2ln αβ αβ−∆2! ,W(0) omSep=kBT 2ln 4αβ 2α+2β−1! ,W(0) omMax=kBT 2ln"4αβ 1+2|α−β|# . (22) W(1) om=kBTln"2αβ+α 2αβ+α−2∆2# ,W(1) omSep=kBTln"2α(2β+1) 4α+2β−1# ,W(0) omMax=kBTln 2α ifα≤β kBTlnh2α(1+2β) 1+4α−2βi otherwise(23) The works remain independent of the measurement angle. C. Both magnon and photon perform Gaussian measurement This subsection explores the case where Alice and Bob, each make Gaussian measurements on their state. Alice now performs a second Gaussian measurement on her reduced state of the system, it can be described by ˜Nc(X)=π−1˜Dc(Y)˜ρNc˜D† c(Y), (24) where ˜Dc(Y)=eYδˆc†−Y∗δˆcis the displacement operator, ˜ ρNccorresponds to a pure Gaussian state without first moment and the its covariance matrix is given by ΓNn=1 2R(χ)diag( Λ,Λ−1)R(χ)T, (25) where R(χ) represents a rotation matrix and Λ∈[0,∞]. The probability distribution describing a Gaussian measurement on Alice mode δˆcis influenced by the measurement ˜Nn(X) performed on Bob mode δˆn. However, interestingly, the uncertainty in Alice mode δˆcremains una ffected by the outcome Xthat Bob obtains from his Gaussian measurement, i.e., CNn,Nccn=CNnc+ ΓNc, WhileCNncis provided by Eq. (18). The extracted work by Alice (photon), can be measured via the Shannon entropy of the appropriate probability distribution H(Pr(X,Y)) is similar to the entropy of the Gaussian distribution H(CNn,Nccn). Its expression writes as W(λ,Λ)(ξ,χ)=kBT 2ln detCNnc detCNn,Nccn! . (26) In the case of STSs the extractable work for both homodyne ( λ=0) and heterodyne ( λ=1), writes as W(0,0) om(ξ,χ)=kBTlns 4αβ 4αβ−2∆2[1+cos(2ξ+2χ)],W(1,1) om=kBTln"(1+2α)(1+2β) 1+2β+α(2+4β)−4∆2# . (27) D. Efficiency of the work extraction According to Zhuang et al. (2014), the information-work e fficiency of a Szilard engine can be expressed as the ratio of extracted work to erasure work [67] µ=W Weras, (28) In this case, the information contained in the system is proportionate to Weras Weras=kBT H(P) ln 2, (29) where the Shannon entropy connected to the probability Pjdistribution is expressed as ?? We exploit the density operators ρand the von Neumann entropy to serve as the quantum mechanical equivalents of probability distributions [68] S(ρ)=Tr(ρlog(ρ)), (30)7 For two modes Gaussian state ρGthe von Neumann entropy sVcan be written as S(ρG)=2X l=1sV(Φl), (31) withΦl,l=1,2, represent the symplectic eigenvalues of the matrix CAB(see equation (13)) writes as Φ±=s κ±p κ2−4 detCAB 2, (32) andsVcan be expressed sV(w)=2w+1 2 log2w+1 2 −2w−1 2 log2w−1 2 , (33) whereκ=detCc+detCn+2 detCcn. E. Results and discussions In this section, we will explore how light (photons) and magnetic excitation’s (magnons) interact and share quantum corre- lations and e fficiency in a steady and dynamical state, considering various factors. We’ve selected parameters that are suitable for experimentation [35]: Ωc/2π=10×106Hz,Ωd/2π=10×106Hz,λd/2π=102Hz,λc/2π=λn/2π=1×106Hz, gnc/2π=Gnd/2π=3.2×106Hz, and at low temperature T=10×10−3K. Under these conditions, the coupling between the magnon mode and cavity mode gncis significantly weaker than the product of the detuning between the magnon and cavity modes and the mechanical resonance frequency, i.e., g2 nc≪|˜δnδc|≃Ω2 d. In this case, we adopt the approximate of the e ffective magnomechanical coupling as Gnd≃√ 2gndΩ Ωdsee Eq. (6), whereGnd/2π=3.2×106Hz leading to|⟨n⟩|≃ 1.1×107for a 250-µm-diameter YIG sphere, is regarding the drive magnetic field B0≃3.9×10−5T for gnd/2π≃0.2 Hz and the drive power P=8.9×10−3W. In this order, one can make the Kerr e ffect negligible because of the realization of the K|⟨n⟩|3≪ω. Eom Wom(0) WomSep(0) WomMax(0) 0.00 0.05 0.10 0.15 0.200.000.050.100.150.200.250.30 T(K)(a) Eom Wom(1) WomSep(1) WomMax(1) 0.00 0.05 0.10 0.15 0.200.000.050.100.150.200.25 T(K)(b) FIG. 1: Plot of logarithmic negativity Eom, extracted work W(λ) om(in units of kBT), maximum of extractable work W(λ) omMaxand extracted work at separable state W(λ) omS epbetween photon and magnon against temperature Tfor various Gaussian measurements. (a) λ=0 (homodyne); (b) λ=1 (heterodyne). In Fig. (1), we plot the logarithmic negativity Eom, extractable work W(λ) om(in units of kBT), separable work W(λ) omsepand maximum work W(λ) ommaxbetween optical mode and magnon mode versus the temperature Tfor di fferent measurements. The extractable work W(λ) omand separable work W(λ) omsepare always bound by maximum work W(λ) ommax, as depicted in Fig. (1). We remark that photon and magnon modes are entangled in the region where W(λ) om>W(λ) omsep. This agrees with entanglement quantified by logarithmic negativity Eom[69]. This figure exhibits that W(λ) om,W(λ) omsepandW(λ) ommaxall increase with increasing temperature. Conversely, logarithmic negativity diminishes to zero around 0.17 K., i.e., the two modes photon and magnon are in separable state and W(λ) om≤W(λ) omsep, as depicted in Fig. (1)(a-b). We note that for a large value of the temperature Tthe mode corresponds to8 the optimal performance of a Szilard engine. This is for homodyne and heterodyne detection ( λ=0,1). Besides, the maximum work is larger at high temperatures. Furthermore, in homodyne detection, the maximum work W(λ) ommax(in units of kBT) achieves 0.30 at T=0.2 K (a), while in heterodyne detection it achieves 0.27, as depicted in figure (1). Thus, one can say that extractable work provides a su fficient condition to witness entanglement in generic two-mode states, which is also necessary for squeezed thermal states. Eom Wom(0) WomSep(0) WomMax(0)0.0 0.2 0.4 0.6 0.8 1.00.00.10.20.30.40.50.60.7 t(μs)(a) Eom Wom(1) WomSep(1) WomMax(1)0.0 0.2 0.4 0.6 0.8 1.00.00.10.20.30.40.50.60.7 t(μs)(b) FIG. 2: Time evolution of logarithmic negativity Eom, extracted work W(λ) om(in units of kBT), maximum of extractable work W(λ) omMaxand extracted work at separable state W(λ) omS epbetween photon and magnon for various Gaussian measurements. (a) λ=0 (homodyne) (b) λ=1 (heterodyne). In Fig. 2, we plot the time-evolution of the bipartite entanglement Eom, extractable work W(λ) om(in units of kBT), separable work W(λ) omsepand maximum work W(λ) ommaxbetween optical mode and magnon mode in homodyne measurement (a) and in hetrodyne measurement (b). This figure shows three entanglement regimes: The first regime is dedicated to classically correlated states (Eom=0), i.e., W(λ) om<W(λ) omsep. This means that the two modes (photon and magnon) are separated. Nevertheless, the extracted work increases in time while the separable work and maximum work decrease due to decoherence from the thermal bath. The second regime, W(λ) om>W(λ) omsepandEom>0, indicating entanglement sudden death between the two modes. Here, we observe the generation of oscillations, which can be explained by the Sorensen-Molmer entanglement dynamics discussed in Ref. [43, 70]. The last regime corresponds to a steady state, i.e., W(λ) omsepremains bounded by W(λ) omandEomis constant. The extracted work W(λ) om and separable work W(λ) omsepare bounded by the maximum work W(λ) ommax. Thus, the engine has the best performance for strongly squeezed vacuum states and small times of evolution. Eom Wom(0) WomSep(0) WomMax(0)-2.0 -1.5 -1.0 -0.5 0.00.000.050.100.150.200.25 δc/Ωd(a) Eom Wom(1) WomSep(1) WomMax(1)-2.0 -1.5 -1.0 -0.5 0.00.000.050.100.150.20 δc/Ωd(b) FIG. 3: Plot of logarithmic negativity Eom, extracted work W(λ) om(in units of kBT), maximum of extractable work W(λ) omMaxand extracted work at separable state W(λ) omS epbetween photon and magnon versus the normalized photon detuning for various Gaussian measurements. (a) λ=0 (homodyne) (b) λ=1 (heterodyne).9 Figure 4 presents the influence of normalized photon detuning δc/ωdon logarithmic negativity Eom, extractable work W(λ) om (in units of kBT), separable work W(λ) omsep, and maximum work W(λ) ommaxas functions of normalized magnon detuning δa/ωb. En- tanglement W(λ) om>W(λ) omsep, as expected from the logarithmic negativity Eom[69], is observed in Fig. 4, while the separable state Eom=0 and W(λ) om≤W(λ) omsepis depicted in Fig. 4(a-b). We remark that the peak in the logarithmic negativity corresponds to the dip in W(λ) omandW(λ) omsepfor both homodyne and heterodyne measurements. Eom Wom(0) WomSep(0) WomMax(0)0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.40.000.050.100.150.200.250.30 gnc/Ωd(a) Eom Wom(1) WomSep(1) WomMax(1)0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.40.000.050.100.150.200.250.30 gnc/Ωd(b) FIG. 4: Plot of logarithmic negativity Eom, extracted work W(λ) om(in units of kBT), maximum of extractable work W(λ) omMaxand extracted work at separable state W(λ) omS epbetween photon and magnon as a function of the magnon-photon coupling gcn/ωdfor various Gaussian measurements. (a)λ=0 (homodyne) (b) λ=1 (heterodyne). Figure ??explores the logarithmic negativity Eom, extractable work W(λ) om(in units of kBT), separable work W(λ) omsep, and maxi- mum work W(λ) ommaxbetween the optical mode and magnon mode versus temperature for both homodyne ( λ=0) and heterodyne (λ=1) measurements, as a function of the magnon-photon coupling gcn/ωd. As expected, entanglement W(λ) om>W(λ) omsepcoincides with non-zero logarithmic negativity Eom[69]. Conversely, the separable state W(λ) om≤W(λ) omsepandEom=0 indicates separabil- ity, as shown in Fig. ??(a-b). Interestingly, Fig. ??also shows that Eom,W(λ) om,W(λ) omsep, and W(λ) ommaxall increase with increasing magnon-photon coupling ( gcn/ωd) before gradually decreasing after reaching a maximum value. Additionally, we observe that in homodyne detection, W(λ) ommax>0 even for gcn/ωd=0, whereas in heterodyne detection, W(λ) ommax=0 for gcn/ωd=0. Wom(0) Wom(0,0) Wom(1) Wom(1,1)0.00.10.20.30.40.50.025330.025340.025350.025360.025370.025380.025390.02540 0.0 0.1 0.2 0.3 0.4 0.50.0000.0050.0100.0150.0200.0250.0300.035 T(K) FIG. 5: Plot of the extractable work W(in units of kBT) as a function of the temperature Tfor both measurement W(0,0) omandW(1,1) omand single homodyne measurement W(0) omand heterodyne measurement W(1) om.10 In Fig. 5, we represent the comparison between the extracted work from both measurement W(0,0) om(0,0) and W(1,1) omto a single homodyne measurement W(0) omand heterodyne measurement W(1) om. The decrease in extractable work observed in Fig. 5 is attributed to the second measurement introducing entropy into the system, which can be mathematically represented as a smearing of the distribution imparted by the single measurement, i.e., W(1,1) om<W(1) om, just for homodyne detection appears like identical W(0,0) om<W(0) om. λ=0 λ=1 0.00 0.05 0.10 0.15 0.200.040.050.060.070.080.09 T(K)μ(a) λ=0 λ=1 0.0 0.5 1.0 1.5 2.00.000.020.040.060.08 t(μs)μ(b) FIG. 6: Plot of the e fficiency of the work extraction as a function of (a) temperature Tand (b) time versus t(µs) for various Gaussian measurements with λ=0 (homodyne) and λ=1 (heterodyne). Fig. 6(a) shows information-work e fficiency monotonically decreasing towards zero with increasing temperature for both homodyne and heterodyne detection ( λ=0,1). The engine performs best at low temperatures. Notably, homodyne and hetero- dyne measurements achieve similar e fficiency. Fig. 6(b) explores the time-dependence of e fficiency. Here, we see e fficiency monotonically increasing to reach a steady-state value, indicating better e fficiency for longer times. Besides, the e fficiency for homodyne measurement is bound by homodyne measurement, as depicted in Fig. 6. VI. CONCLUSION In summary, we have discussed a possible strategy to measure the entanglement and separability of the two-mode Gaussian state in a steady and dynamical state by harnessing the extracted work (out of a thermal bath) by means of a correlated quantum system subjected to measurements. Our investigation focuses on a cavity magnomechanical system. Here, a microwave cavity mode is coupled with a magnon mode in a Yttrium Iron Garnet (YIG) sphere. This magnon mode further couples to a mechanical mode through the magnetostrictive interaction. We have used logarithmic negativity to quantify the entanglement between photons and magnons with experimentally reachable parameters. 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2024-05-29

The connections between thermodynamics and quantum information processing are of paramount importance. Here, we address a bipartite entanglement via extracted work in a cavity magnomechanical system contained inside an yttrium iron garnet (YIG) sphere. The photons and magnons interact through an interaction between magnetic dipoles. A magnetostrictive interaction, analogous to radiation pressure, couple's phonons and magnons. The extracted work was obtained through a device similar to the Szil\'ard engine. This engine operates by manipulating the photon-magnon as a bipartite quantum state. We employ logarithmic negativity to measure the amount of entanglement between photon and magnon modes in steady and dynamical states. We explore the extracted work, separable work, and maximum work for squeezed thermal states. We investigate the amount of work extracted from a bipartite quantum state, which can potentially determine the degree of entanglement present in that state. Numerical studies show that entanglement, as detected by the extracted work and quantified by logarithmic negativity, is in good agreement. We show the reduction of extracted work by a second measurement compared to a single measurement. Also, the efficiency of the Szilard engine in steady and dynamical states is investigated. We hope this work is of paramount importance in quantum information processing.

Detection of entanglement by harnessing extracted work in an opto-magno-mechanics

2405.19205v1

arXiv:1807.08614v2 [physics.gen-ph] 6 Aug 2018Magnon-phonon conversion experiment and phonon spin S. C. Tiwari Department of Physics, Institute of Science, Banaras Hindu Unive rsity, Varanasi 221005, and Institute of Natural Philosophy Varanasi India Recent experiment demonstrates magnon to phonon conversio n in a YIG film under the appli- cation of a non-uniform magnetic field. Light scattered from phonons is observed to change its polarization state interpreted by the authors signifying p honon spin. In this note we argue that the experimental data merely shows the exchange of angular mome ntum±¯hper photon. We suggest that it has physical origin in the orbital angular momentum o f phonons. The distinction between spin and orbital parts of the total angular momentum, and bet ween phonons and photons with added emphasis on their polarizations is explained. The mai n conclusion of the present note is that phonon spin hypothesis is unphysical. PACS numbers: 63.20.-e, 63.20.kk I. INTRODUCTION Magneto-elastic waves or magnon-phonon excitations have been of interest for various reasons, one of them be- ing the field of spintronics. A recent experimental study on the magnon-phonon conversion in the ferrimagnetic insulator YIG addresses a question of fundamental im- portance whether phonons carry spin [1]. We recall that in 1988 McLellan [2] showed that sharp angular momen- tum could be attributed to circular or elliptical phonon polarizations. Note that this angular momentum can- not be identified with the spin of phonon. In this note we discuss the recent experiment [1] and argue that the measurements show that phonons exchange angular mo- mentum with light but it is not spin of the phonons. We emphasize that this distinction is not just semantic [3] but of fundamental nature [4]. The experiment [1] first shows by time-resolved measurements that under the application of a non- uniform magnetic field on a YIG film, spin wavepackets launched by pulsed microwave signals, convert into elas- tic wavepackets, i.e. magnon-phonon conversion. Next using wavevector resolved Brillouin light scattering ex- periment the measurementsshow i) magnon-phononcon- version with constant energy and linearly varying mo- mentum, and ii) the light scattered by the phonons is circularly polarized. The meticulous data presented in Figures (4) and (5) of their paper by the authors could hardly be doubted. The question that concerns me is regarding their claim, ’that phonons created by the con- version of magnons do carry spin’. It is true that the change in the polarization state of light involves exchange of angular momentum, for ex- ample, to transform linearly polarized light to circularly polarized light an angular momentum of ±¯hper photon is required. However this angular momentum need not be associated with the spin of the medium or the light- scattering object. At the macroscopic level, Beth experi- ment [5] detected a direct mechanical effect in terms of a torque exerted by a circularly polarized beam of light ona doubly refracting medium which changes the polariza- tion state of light. The photon spin angular momentum is transferred to the body of the medium imparting or- bital rotation; the aim of the Beth experiment was, of course, to demonstrate that photons had spin. Holanda et al experiment [1], on the other hand, assumes photon spin, andinfersthatphononscarryspin frommicroscopic scattering data with photons. The crucial point is that the experiment only proves that angular momentum in the unit of ±¯his exchanged. It cannot be attributed to the phonon spin: non-zero spin of phonon does not make physical sense. We argue that the orbital angular momentum of elastic waves or phonons is responsible for angular momentum transfer. In this note we address the question: Why not spin? First photon physics is briefly reviewed in the next sec- tion to highlight the intricate relationship between polar- ization and spin. In section III elementary discussion on phononsshowsthatphoton-phononanalogyisuntenable, and spin cannot be associated with polarized phonons. FurtherelaborationconstitutessectionIV.Thenoteends with a short conclusion. II. PHOTON AND LIGHT POLARIZATION A brief review on elementary considerations on the meaning of angular momentum and its decomposition into orbital and spin parts seems necessary. For a sys- tem with rotational symmetry the angular momentum is a constant of motion; if linear momentum is pone de- fines angular momentum simply as r×p. In field theory one may construct the expression for angular momentum from the momentum density of the field or directly cal- culate the angular momentum density tensor as Noether current from the rotational invariance of the action. In general, it is useful to separate total angular momentum Jinto orbital and spin components J=L+S (1)2 For a scalar particle the spin part is zero. For a vector particle, following a textbook discussion [6], see pp197- 198, a simplified picture is obtained in terms of Lthat depends on the space or position and spin part that de- pends on the three components of the vector wavefunc- tionVwhere 3 ×3 spin matrices Sx,Sy,Szact onV, see (27.11) in [6]. Note that similar arguments hold for the second quantized field theory. Field Vbecomes an operator Vop=aλVλ+a† λV∗ λ (2) The annihilation and creation operators aλanda† λfor a modeλsatisfy the commutation rules [aλ,a† λ′] =δλλ′ (3) ThefieldoperatorsactinFockspacespannedbytheFock state vectors. The orbital angular momentum operator L=−i(r×∇) (4) acts on the space-dependence of the mode functions Vλ, whereas the spin operator S=−iǫijk (5) acts on the components of Vλ. Photonisavectorparticlewithrestmasszeroandspin one having only two projections - better understood in terms of helicity. Photon as a quantized electromagnetic radiation field continues to have fundamental questions: gauge-invariance, transversality and Lorentz covariance arecontroversialand unsettled issues, seereferencescited in [4] and also [7, 8]. Let us try to explain the problem. Classical fields E,B,Aµsatisfy the wave equation, and one assumes a plane wave representation. Introducing canonically con- jugate field variables canonical quantization is carried out. In the normal mode expansion the annihilation and creation operators can be defined, and polarization 4- vectorǫµcomprising of four mutually orthogonal unit vectors takes care of the vector nature of the field. In QED a manifest Lorentz covariant quantization results into longitudinal and time-like photons besides the phys- ical photons. HoweverincontrasttoQEDwheretheelectromagnetic potentials Aµarefundamentalfieldvariables,inquantum optics literature the utility of the electric and magnetic field operators is well known. In a simpler field quantiza- tion for the radiation field polarization index s= 1,2 for transverse fields is sufficient. The normalized eigenstate of the number operator nks=a† ksaksgives the number of photons in the mode ( k,s) as nks|nks>=nks|nks> (6) The Fock state is a direct product of number states over all possible modes |{n}>=/productdisplay ks|nks> (7)The assumption of transverse mode functions, for exam- ple,A⊥eliminates longitudinal and time-like photons in quantum optics. Physical quantities like energy, momentum and angu- lar momentum are obtained using their classical expres- sions and transforming them to the quantized field oper- ators. In the classical radiation field theory the Poynting vectorE×Brepresents the momentum density and the total angular momentum density becomes J=r×(E×B) (8) Separation of (8) into orbital and spin parts can be made similar to (1). The spin angular momentum density is identified with the expression S=E×A (9) Regarding spin angular momentum a remarkable result pointed out by van Enk and Nienhuis [9] is worth men- tioning. For a circularly polarized plane wave it is found that the spin operator corresponding to (9) has a simple form Sr=/summationdisplay kk |k|(nk+−nk−) (10) Heres=±for right and left circular polarization. The components of Srcommute with each other [Sr i,Sr j] = 0 (11) Authors [9] argue that the spin operator(10) cannot gen- erate polarization rotation of the field, and cannot be in- terpreted as spin angular momentum of photon. Note that Jauch and Rohrlich [10] define Stokes operators sat- isfying the angular momentum commutation rules which provide interpretation of the photon spin [11]. To conclude this section, in both QED and quantum optics photon spin and the role of polarization state in- volve intricate issues. One thing is, however unambigu- ous, namely that spin angular momentum is an intrinsic property associated purely with the nature of the fields. In fact, spin for electron also depends only on the Dirac field Σ= Ψ†γγ5Ψ (12) III. PHONON SPIN In the abstract of [1] the authors state that, ’while it is well established that photons in circularly polarized light carry a spin, the spin of phonons has had little attention in the literature’. Now keeping in mind the conceptual problems associated with photon physics highlighted in the preceding section the photon spin has to be inter- preted with great care. The second part of the statement is, however not correct. The condensed matter literature tacitly accepts phonon to be a zero spin boson, in spite3 of the transverse modes and the known polarization of acoustic and optical phonons. Polarization of phonon modes is not related with spin but orbital angular mo- mentum [2]. A brief discussion seems useful for the sake of clarity. Phonons are quantized lattice vibrations; phonon modes are described by wavevector k, a branch number j and the orientation of the coordinate axes [2]. The branch number has two values for crystals with two sub- lattices and there are two triplets of phonons for acoustic and optical branches. McLellan defines phonon angular momentum in terms of phonon annihilation and creation operators to be Lph=/summationdisplay kjakj×a† kj(13) This expression is, as pointed by the author [2], in agree- ment with that defined using the displacement vector ulκ L=/summationdisplay lκulκ×plκ (14) Here the index lcorresponds to the unit cell and κfor the atom on a sub-lattice. Expression (52) in [2] for the total angular momentum of the lattice includes that of the rigid body rotation of the crystal. What are the implications of above discussion? It throws light on the issue of phonon polarization and spin as follows. [1]Phononisaquasi-particlehavingnodynamicalfield equations like Maxwell field equations for photon. The most crucial point that seems to have gone unnoticed in the discussions on phonon spin and phonon-photon anal- ogy is that the displacement vector representing lattice vibrations is a real space coordinate. Canonical quanti- zation and the field operators for phonons are based on the coordinate and momentum, for example those ap- pearing in Eq.(14). On the other hand, for photon the field variable Aµis treated as a coordinate variable, and ∂L ∂˙Aµis the canonically conjugate “momentum” variable for the quantization. Here Lis the Lagrangian density of the Maxwell field. [2] Phonon polarization is physically entirely differ- ent than light or photon polarization. McLellan’s analysis clearly establishes the physical significance of phonon polarization in terms of orbital angular momen- tum. Isotropic 2D quantum oscillator best illustrates the meaning of polarization of elastic waves or phonons. In cartesian coordinates the raising and lowering operators separate into 1D oscillators; it is akin to linear polar- ization. A circular basis ( a† x±ia† y) formally resembles circular polarization. In the circular basis one gets well- defined orbital angular momentum of the oscillator. Transverse modes in paraxial optics also represent physical realizationof this example. First order Hermite- Gaussian modes HG10andHG01are not eigenstates of angular momentum operator (4). However, Laguerre-Gaussian modes LG±1 0=1√ 2(HG10±iHG01) (15) possess sharp angular momentum. Thus phonon polar- ization is related with orbital angular momentum not spin. IV. DISCUSSION Let us try to elucidate further why photon-phonon analogy is misleading. Photon as a quantized vector field has intrinsic spin one. Wigner’s group theoretical argu- ments establishthat foranymasslessorlight-likeparticle with non-zero spin there exist only two helicity states. In the classical picture the intrinsic spin is identified with the vector product of the electric field and the vector po- tential (9). Assumed transverse vector potential leads to the electric field E⊥=−∂A⊥ ∂t(16) In the field quantization assuming monochromatic light the electric and magnetic fields are obtained using (16) and∇×A⊥respectively, e. g. the expression (6) in [9]. The oscillations or vibrations around equilibrium posi- tion of ions collectively lead to the elastic waves and are analyzed in the harmonic approximation in terms of the normal modes. The mode expansion includes wave vec- tor and polarization specifications [12, 13]. Phonon field is understood in terms of the displacement of a point in the material medium u(r,t) and the corresponding mo- mentum p=/integraldisplay ρ˙u(r,t)dV (17) whereρis the mass density. Standard coordinate and momentum quantization rule, and plane wave represen- tation yield quantized phonon field. It is easy to see that expression (14) is just the orbital angular momentum. A deceptive formal analogy with the photon spin expres- sion (9) is obvious considering expression (16) and using (17) for phonon. Physical interpretation depends on the fundamentaldistinctionbetweenthe vectorpotentialand the displacement vector since the later is a real space co- ordinate variable. Thus the suggested interpretation for the phonon angular momentum corresponding to the cir- cularly polarizedmodes in [2] seems justified. We remark that in spite of the usage of phonon polarization in the literature [13], and transverse polarization in the des- ignation of creation and annihilation operators phonon spin and vector nature of the phonon field is nowhere mentioned. To avoid confusion, it has to be understood that scalar field could possess well defined orbital angu- lar momentum and laser light beams with sharp orbital angular momentum have been extensively studied in the4 literature, see references in [4]. Longitudinal modes have no spin or orbital angular momentum, however linearly polarized light could possess orbital angular momentum but not spin. Thus the conventional phonon theory has no analogy with the photon theory, and non-zero phonon spin does not make physical sense. The origin of the an- gular momentum transfer from phonon to photon in the reported experiment [1] may be logically attributed to the orbital angular momentum of phonons. In a hypothetical scenario assuming phonon has spin one it would be of interest to find its physical conse- quences. I think electron-phonon interaction and Cooper pair formation via phonon mediated electron-electron in- teraction may be re-examined: phonon creation and an- nihilation operators [13] could be generalized for the cir- cularly polarized modes in the interaction Hamiltonian and treated as spin one particles. There is another prob- leminsuperconductivityhighlightedbyPost[14], namely the angular momentum conservation in a superconduct- ing ring. Though Post sets the problem in the form of Onsager-Feynman controversyhe offers insightful discus- siononthe mechanismoftheangularmomentumbalance when supercurrent in a ring vanishes as the temperature is raised above the transition temperature. Note that Post rules out any role of lattice, therefore it may be of interest to examine the role of phonon spin in this prob- lem. We could, of course, explore new physics or unconven- tional ideas [4]. Departing from the phonon picture new kind of field excitations in the spirit of Cosserat medium was suggested in [4]. Analogy of displacement vectorwith the vector potential is not justified, however the velocity field in a rotating fluid may be treated as a vec- tor potential: postulating rotating space-time fluid with nontrivial topology of vortices we have re-interpreted the electromagnetic field tensor as the angular momentum (density or more appropriately flux) of photon fluid [8], and proposed a topological photon [15]. Note that the netangularmomentumofthemicroscopicparticlesinthe rotating fluid implies antisymmetric stress tensor. Such speculations relate spin with topological invariants. V. CONCLUSION It has been pointed out [4] that phonon angular mo- mentum discussed in [16] is ambiguous as compared to that discussed in [2]. We have shown that non-zero phonon spin hypothesis and phonon-photon analogy [17] are conceptually flawed, giving further support to the ar- gumentspresentedin [4]. Phononspin hasno experimen- tal evidence. The correct physical interpretation of the reported experiment [1] is that orbital angular momen- tum of phonons is exchanged with light beam resulting into the change in the polarization of the light. Acknowledgments I thank S. Streub for raising specific questions on the photon-phonon analogy. I also acknowledge correspon- dence with S. M. Rezende, M. Wakamatsu, and A. Hoff- mann, and conversation with D. Sa and V. S. Subrah- manyam. [1] J. Holanda et al, Nature Physics 14, 500 (2018) [2] A. G. McLellan, J. Phys. C Solid State Phys. 21, 1177 (1988) [3] S. Streub et al, Phys. Rev. Lett. 121, 027202 (2018); arXiv: 1804.07080v1 [cond-mat.mes-hall] [4] S. C. Tiwari, arXiv: 1708.07407v3 [physics.gen-ph] [5] R. A. Beth, Phys. Rev. 48, 471 (1935) [6] L. I. Schiff, Quantum Mechanics (McGraw-Hill, 1968) Third Edition [7] S. C. Tiwari, arXiv:08070.0699v1 [physics-gen.ph] [8] S. C. Tiwari, J. Mod. Optics, 46, 1721 (1999) [9] S. J. van Enk and G. Nienhuis, Europhys. Lett. 25, 497(1994) [10] J. M. Jauch and F. Rohrlich, The Theory of Photons and Electrons (Reading, Addison-Wesley, 1955) [11] S. C. Tiwari, J. Mod. Optics, 39, 1097 (1992) [12] L. D. Landau and E. M. Lifshitz, Theory of Elasticity (Pergamon, 1970) [13] D. J. Scalapino, Chapter 10in SuperconductivityVolum e 1, Edited by R. D. Parks (M. Dekker, 1969) [14] E. J. Post, Quantum Reprogramming (Kluwer, 1995) [15] S. C. Tiwari, J. Math. Phys. 49, 032303 (2008) [16] L. ZhangandQ.Niu, Phys.Rev.Lett.112, 085503 (2014) [17] D. A. Garanin and E. M. Chudnovsky, Phys. Rev. B 92, 0244421 (2015)

2018-07-17

Recent experiment demonstrates magnon to phonon conversion in a YIG film under the application of a non-uniform magnetic field. Light scattered from phonons is observed to change its polarization state interpreted by the authors signifying phonon spin. In this note we argue that the experimental data merely shows the exchange of angular momentum $\pm \hbar$ per photon. We suggest that it has physical origin in the orbital angular momentum of phonons. The distinction between spin and orbital parts of the total angular momentum, and between phonons and photons with added emphasis on their polarizations is explained. The main conclusion of the present note is that phonon spin hypothesis is unphysical.

Magnon-phonon conversion experiment and phonon spin

1807.08614v2

arXiv:1302.1352v1 [cond-mat.mes-hall] 6 Feb 2013Theory of spin Hall magnetoresistance Yan-Ting Chen1, Saburo Takahashi2, Hiroyasu Nakayama2, Matthias Althammer3,4, Sebastian T. B. Goennenwein3, Eiji Saitoh2,5,6,7, and Gerrit E. W. Bauer2,5,1 1Kavli Institute of NanoScience, Delft University of Techno logy, Lorentzweg 1, 2628 CJ Delft, The Netherlands 2Institute for Materials Research, Tohoku University, Send ai, Miyagi 980-8577, Japan 3Walther-Meißner-Institut, Bayerische Akademie der Wisse nschaften, 85748 Garching, Germany 4University of Alabama, Center for Materials for Informatio n Technology MINT, Dept Chem, Tuscaloosa, AL 35487, USA 5WPI Advanced Institute for Materials Research, Tohoku University, Sendai 980-8577, Japan 6CREST, Japan Science and Technology Agency, Sanbancho, Tok yo 102-0075, Japan and 7The Advanced Science Research Center, Japan Atomic Energy Agency, Tokai 319-1195, Japan (Dated: February 7, 2013) We present a theory of the spin Hall magnetoresistance (SMR) in mu ltilayers made from an insulating ferromagnet F, such as yttrium iron garnet (YIG), an d a normal metal N with spin-orbit interactions, such as platinum (Pt). The SMR is induced by the simultaneous action of spin Hall and inverse spin Hall effects and therefore a non- equilibrium proximity phenomenon. We compute the SMR in F |N and F|N|F layered systems, treating N by spin- diffusion theory with quantum mechanical boundary conditions at th e interfaces in terms of the spin-mixing conductance. Our results explain the experiment ally observed spin Hall magnetoresistance in N |F bilayers. For F |N|F spin valves we predict an enhanced SMR amplitude when magnetizations are collinear. The SMR and the spin-tr ansfer torques in these trilayers can be controlled by the magnetic configuration. PACS numbers: 85.75.-d, 73.43.Qt, 72.15.Gd, 72.25.Mk I. INTRODUCTION Spin currents are a central theme in spintronics since they a re intimately associated with the manipulation and transport of spins in small structures and devices.1,2Spin currents can be gener- ated by means of the spin Hall effect (SHE) and detected by the in verse spin Hall effect (ISHE).3 Of special interest are multilayers made of normal metals (N ) and ferromagnets (F). When an electric current flows through N, an SHE spin current flows tow ards the interfaces, where it can be absorbed as a spin-transfer torque (STT) on the ferromagnet . This STT affects the magnetization damping4or even switches the magnetization.5,6The ISHE can be used to detect spin currents pumped by the magnetization dynamics excited by microwaves7–10or temperature gradients (spin Seebeck effect).11,12 Recently, magnetic insulators have attracted the attentio n of the spintronics community. Yt- trium iron garnets (YIG), a class of ferrimagnetic insulato rs with a large band gap, are interesting because of their very low magnetization damping. Their magn etization can be activated thermally to generate the spin Seebeck effect in YIG |Pt bilayers.13,14By means of the SHE, spin waves can be electrically excited in YIG via a Pt contact, and, via the ISH E, subsequently detected electrically2 in another Pt contact.15Spin transport at an N |F interface is governed by the complex spin-mixing conductance G↑↓.16The prediction of a large real part of G↑↓for interfaces of YIG with simple metals by first principles calculations17has been confirmed by experiments.18 Magnetoresistance (MR) is the property of a material to chan ge the value of its electrical resistance under an external magnetic field. In normal metal s its origin is the Lorentz force.19 The dependence of the resistance on the angle between curren t and magnetization in metallic ferromagnets is called anisotropic magnetoresistance (AM R). The transverse component of the AMR is also called the planar Hall effect (PHE), i.e.the transverse (Hall) voltage found in ferromagnets when the magnetization is rotated in the plane of the film.20,21Both effects are symmetric with respect to magnetization reversal, which di stinguishes them from the anomalous Hall effect (AHE) for magnetizations normal to the film, which c hanges sign under magnetization reversal.22The physical origin of AMR, PHE, and AHE is the spin-orbit int eraction, in contrast to the giant magnetoresistance (GMR), which reflects the cha nge in resistance that accompanies the magnetic field-induced magnetic configuration in magnet ic multilayers.23 Here we propose a theory for a recently discovered magnetore sistance effect in Pt |YIG bilayer systems.14,24,25This MR is remarkable since YIG is a very good electric insula tor such that a charge current can only flow in Pt. We explain this unusual mag netoresistance not in terms of an equilibriumstatic magnetic proximity polarization in Pt,24but rather in terms of anon-equilibrium proximity effect caused by the simultaneous action of the SHE a nd ISHE and therefore call it spin Hall magnetoresistance (SMR). This effect scales like the squ are of the spin Hall angle and is modulated by the magnetization direction in YIG via the spin -transfer at the N |F interface. Our explanationissimilartotheHanleeffect-inducedmagnetore sistanceinthetwo-dimensional electron gas proposed by Dyakonov.26Here we present the details of our theory, which is based on th e spin- diffusion approximation in the N layer in the presence of spin- orbit interactions27and quantum mechanical boundary conditions at the interface in terms of the spin-mixing conductance.16,17We also address F|N|F spin valves with electric currents applied parallel to the interface(s) with the additional degree of freedom of the relative angle between t he two magnetizations directions. This paper is organized as follows. We present the model, i.e.spin-diffusion with proper boundary conditions in Sec. II. In Sec. III, we consider an N |F bilayer as shown in Fig. 1 (a). We obtain spinaccumulation, spincurrentsandfinallythemeas uredchargecurrentsthat arecompared with the experimental SMR. We also find and discuss that the im aginary part of the spin-mixing conductance generates an AHE. F |N|F (Fig. 1 (b)) spin valves are investigated in Sec. IV, which show an enhanced SMR for spacers thinner than the spin-flip di ffusion length. We summarize the results and give conclusions in Sec. V. II. TRANSPORT THEORY IN METALS IN CONTACT WITH A MAGNETIC INSULATOR The spin current density in the non-relativistic limit ← →js=en/an}b∇acketle{t/vector v⊗/vector σ+/vector σ⊗/vector v/an}b∇acket∇i}ht/2 =/parenleftig /vectorjsx,/vectorjsy,/vectorjsz/parenrightigT =/parenleftig /vectorjx s,/vectorjy s,/vectorjz s/parenrightig (1) is a second-order tensor (in units of the charge current dens ity/vectorjc=en/an}b∇acketle{t/vector v/an}b∇acket∇i}ht), where e=|e|is the electron charge, nis the density of the electrons, /vector vis the velocity operator, /vector σis the vector of Pauli spin matrices, and /an}b∇acketle{t···/an}b∇acket∇i}htdenotes an expectation value. The row vectors /vectorjsi=en/an}b∇acketle{t/vector vσi+σi/vector v/an}b∇acket∇i}ht/2 in Eq. (1) are the spin current densities polarized in the ˆ ı-direction, while the column vectors /vectorjj s=en/an}b∇acketle{tvj/vector σ+/vector σvj/an}b∇acket∇i}ht/2 denote the spin current densities with polarization /vector σflowing in the ˆ - direction. Ohm’s law for metals with spin-orbit interactio ns can be summarized by the relation3 FIG. 1: (a) N|F bilayer and (b) F |N|F trilayer systems considered here, where F is a ferromagnetic insu lator and N a normal metal. between thermodynamic driving forces and currents that refl ects Onsager’s reciprocity by the symmetry of the response matrix:27 /vectorjc /vectorjsx /vectorjsy /vectorjsz =σ 1θSHˆx×θSHˆy×θSHˆz× θSHˆx×1 0 0 θSHˆy×0 1 0 θSHˆz×0 0 1 −/vector∇µ0/e −/vector∇µsx/(2e) −/vector∇µsy/(2e) −/vector∇µsz/(2e) , (2) where/vector µs= (µsx,µsy,µsz)T−µ0ˆ1 is the spin accumulation, i.e.the spin-dependent chemical potential relative to the charge chemical potential µ0,σis the electric conductivity, θSHis the spin Hall angle, and “ ×” denotes the vector cross product operating on the gradient s of the spin- dependent chemical potentials. The spin Hall effect is repres ented by the lower non-diagonal elements that generate the spin currents in the presence of a n applied electric field, in the following chosen to be in the ˆ x-direction /vectorE=Exˆx=−ˆx∂xµ0/e. The inverse spin Hall effect is governed by elements above the diagonal that connect the gradients of th e spin accumulations to the charge current density. The spin accumulation /vector µsis obtained from the spin-diffusion equation in the normal met al ∇2/vector µs=/vector µs λ2, (3) where the spin-diffusion length λ=√Dτsfis expressed in terms of the charge diffusion constant Dand spin-flip relaxation time τsf.28For films with thickness dNin the ˆz-direction /vector µs(z) =/vectorAe−z/λ+/vectorBez/λ, (4) where the constant column vectors /vectorAand/vectorBare determined by the boundary conditions at the interfaces. AccordingtoEq. (2), thespincurrentinNconsists of diffusio nandspinHall driftcontributions. Since we are considering a system homogeneous in the x-yplane, we focus on the spin current density flowing in the ˆ z-direction /vectorjz s(z) =−σ 2e∂z/vector µs−jSH s0ˆy, (5)4 wherejSH s0=θSHσExis the bare spin Hall current, i.e., the spin current generated directly by the SHE. The boundary conditions require that /vectorjz s(z) is continuous at the interfaces z=dNandz= 0. The spin current at a vacuum (V) interface vanishes, /vectorj(V) s= 0. The spin current density /vectorj(F) sat a magnetic interface is governed by the spin accumulation and spin-mixing conductance:16 e/vectorj(F) s(ˆm) =Grˆm×(ˆm×/vector µs)+Gi(ˆm×/vector µs), (6) where ˆm= (mx,my,mz)Tis a unit vector along themagnetization and G↑↓=Gr+iGithe complex spin-mixing interface conductance per unit area. The imagi nary part Gican be interpreted as an effectiveexchangefieldactingonthespinaccumulation. Apos itivecurrentinEq.(6)correspondsto up-spins flowing from F towards N. Since F is an insulator, thi s spin current density is proportional to the spin-transfer acting on the ferromagnet /vector τstt=−/planckover2pi1 2eˆm×/parenleftig ˆm×/vectorj(F) s/parenrightig =/planckover2pi1 2e/vectorj(F) s (7) With these boundary conditions we determine the coefficients /vectorAand/vectorB, which leads to the spin accumulation /vector µs=2eλ σ/bracketleftbigg −/parenleftig jSH s0ˆy+/vectorjz s(dN)/parenrightig coshz λ+/parenleftig jSH s0ˆy+/vectorj(F) s(ˆm)/parenrightig coshz−dN λ/bracketrightbigg /sinhdN λ,(8) where/vectorjz s(dN) = 0 for F(ˆ m)|N|V bilayers and /vectorjz s(dN) =−/vectorj(F) s(ˆm′) for F(ˆm)|N|F(ˆm′) spin valves. III. N|F BILAYERS In the bilayer the spin accumulation (8) is /vector µs(z) =−ˆyµ0 ssinh2z−dN 2λ sinhdN 2λ+/vectorj(F) s(ˆm)2eλ σcoshz−dN λ sinhdN λ, (9) whereµ0 s≡|/vector µs(0)|= (2eλ/σ)jSH s0tanh[dN/(2λ)] is the spin accumulation at the interface in the absence of spin-transfer, i.e., whenG↑↓= 0. Using Eq. (6), the spin accumulation at z= 0 becomes /vector µs(0) = ˆyµ0 s+2λ σ{Gr[ˆm(ˆm·/vector µs(0))−/vector µs(0)]+Giˆm×/vector µs(0)}cothdN λ. (10) With ˆm·/vector µs(0) =myµ0 s, (11) ˆm×/vector µs(0) =µ0 sσˆm׈y+ ˆmmy2λGicothdN λ σ+2λGrcothdN λ−/vector µs(0)2λGicothdN λ σ+2λGrcothdN λ, (12) /vector µs(0) = ˆyµ0 s1+2λ σGrcothdN λ/parenleftig 1+2λ σGrcothdN λ/parenrightig2 +/parenleftig 2λ σGicothdN λ/parenrightig2 + ˆmmyµ0 s2λ σGrcothdN λ/parenleftig 1+2λ σGrcothdN λ/parenrightig +/parenleftig 2λ σGicothdN λ/parenrightig2 /parenleftig 1+2λ σGrcothdN λ/parenrightig2 +/parenleftig 2λ σGicothdN λ/parenrightig2 +(ˆm׈y)µ0 s2λ σGicothdN λ/parenleftig 1+2λ σGrcothdN λ/parenrightig2 +/parenleftig 2λ σGicothdN λ/parenrightig2, (13)5 the spin current through the F |N interface then reads /vectorj(F) s=µ0 s eˆm×(ˆm׈y)σReG↑↓ σ+2λG↑↓cothdN λ+µ0 s e(ˆm׈y)σImG↑↓ σ+2λG↑↓cothdN λ.(14) The spin accumulation /vector µs(z) µ0s=−ˆysinh2z−dN 2λ sinhdN 2λ+[ˆm×(ˆm׈y)Re+(ˆm׈y)Im]2λG↑↓ σ+2λG↑↓cothdN λcoshz−dN λ sinhdN λ,(15) then leads to the distributed spin current in N /vectorjz s(z) jSH s0= ˆycosh2z−dN 2λ−coshdN 2λ coshdN 2λ−[ˆm×(ˆm׈y)Re+(ˆm׈y)Im]2λG↑↓tanhdN 2λ σ+2λG↑↓cothdN λsinhz−dN λ sinhdN λ. (16) The ISHE drives a charge current in the x-yplane by the diffusion spin current component flowing along the ˆ z-direction. The total longitudinal (along ˆ x) and transverse or Hall (along ˆ y) charge currents become jc,long(z) j0c= 1+θ2 SH/bracketleftigg cosh2z−dN 2λ coshdN 2λ+/parenleftbig 1−m2 y/parenrightbig Re2λG↑↓tanhdN 2λ σ+2λG↑↓cothdN λsinhz−dN λ sinhdN λ/bracketrightigg ,(17) jc,trans(z) j0c=θ2 SH(mxmyRe−mzIm)2λG↑↓tanhdN 2λ σ+2λG↑↓cothdN λsinhz−dN λ sinhdN λ, (18) wherej0 c=σExis the charge current driven by the external electric field. The charge current vector is the observable in the experimen t that is usually expressed in terms of the longitudinal and transverse (Hall) resistivities. A veraging the electric currents over the film thickness zand expanding the longitudinal resistivity governed by the current in the ( x-)direction of the applied field to leading order in θ2 SH, we obtain ρlong=σ−1 long=/parenleftbiggjc,long Ex/parenrightbigg−1 ≈ρ+∆ρ0+∆ρ1/parenleftbig 1−m2 y/parenrightbig , (19) ρtrans=−σtrans σ2 long≈−jc,trans/Ex σ2= ∆ρ1mxmy+∆ρ2mz, (20) where ∆ρ0 ρ=−θ2 SH2λ dNtanhdN 2λ, (21) ∆ρ1 ρ=θ2 SHλ dNRe2λG↑↓tanh2dN 2λ σ+2λG↑↓cothdN λ, (22) ∆ρ2 ρ=−θ2 SHλ dNIm2λG↑↓tanh2dN 2λ σ+2λG↑↓cothdN λ, (23) whereρ=σ−1is the intrinsic electric resistivity of the bulk normal met al. ∆ρ0<0 seems to imply that the resistivity is reduced by the spin-orbit interacti on. However, this is an effect of the order ofθ2 SHthat becomes relevant only when dNis sufficiently small. The spin-orbit interaction also generatesspin-flipscatteringthatincreasestheresistan cetoleadingorderaccordingtoMatthiesen’s rule. We see that ∆ ρ1(caused mainly by Gr) contributes to the SMR, while ∆ ρ2(caused mainly byGi) contributes only when there is a magnetization component n ormal to the plane (AHE), as discussed below.6 /s48 /s53 /s49/s48/s45/s49/s48/s49 /s115/s120/s61 /s106 /s115/s120/s61/s48/s40/s97/s41/s118/s97/s99/s117/s117/s109 /s70/s109 /s124/s124 /s121/s32 /s40/s114/s101/s102/s108/s101/s99/s116/s105/s110/s103/s41 /s32/s32 /s122/s32/s40/s110/s109/s41/s115/s121/s47/s48 /s115 /s106 /s115/s121/s47/s106/s83/s72 /s115 /s48 /s48 /s53 /s49/s48/s45/s49/s48/s49 /s40/s98/s41 /s109 /s124/s124 /s40/s120/s43/s121 /s41/s32/s118/s97/s99/s117/s117/s109 /s70 /s32/s32 /s122/s32/s40/s110/s109/s41/s115/s121/s47/s48 /s115 /s106 /s115/s121/s47/s106/s83/s72 /s115 /s48/s106 /s115/s120/s47/s106/s83/s72 /s115 /s48/s115/s120/s47/s48 /s115 /s48 /s53 /s49/s48/s45/s49/s48/s49 /s40/s99/s41/s118/s97/s99/s117/s117/s109 /s70/s109 /s124/s124 /s120/s32 /s40/s97/s98/s115/s111/s114/s98/s105/s110/s103/s41 /s32/s32 /s122/s32/s40/s110/s109/s41/s115/s121/s47/s48 /s115 /s106 /s115/s121/s47/s106/s83/s72 /s115 /s48/s115/s120/s61 /s106 /s115/s120/s61/s48 FIG. 2: (Color online). Normalized µsx,µsy,jsx, andjsyas functions of zfor magnetizations (a) ˆ m= ˆy, (b) ˆm= (ˆx+ ˆy)/√ 2, and (c) ˆ m= ˆxfor a sample with dN= 12 nm. We adopt the transport parameters ρ= 8.6×10−7Ωm,λ= 1.5 nm, and Gr= 5×1014Ω−1m−2. For magnetizations ˆ m= ˆyand ˆm= ˆx, both µsxandjsxare 0. A. Limit of Gi= ImG↑↓≪ReG↑↓=Gr According to first principles calculations,17|Gi|is at least one order of magnitude smaller than Grfor YIG, so Gi= 0 appears to be a good first approximation. In this limit, we p lot normalized components of spin accumulation ( µsxandµsy) and spin current ( jsx=/vectorjz s·ˆxandjsy=/vectorjz s·ˆy) as functions of zfor different magnetizations in Fig 2. When the magnetization of F is along ˆy, the spin current at the N |F interface ( z= 0) vanishes just as for the vacuum interface. By rotating the magnetization from ˆ yto ˆx, the spin current at the N |F interface and the torque on the magnetization is activated, while the spin accumulatio n is dissipated correspondingly. We note that the x-components of both spin accumulation and spin current vani sh when the magnetization is along ˆxand ˆy, and reach a maximum value at (ˆ x+ ˆy)/√ 2. ForGi= 0 the observable transport properties reduce to ρlong≈ρ+∆ρ0+∆ρ1/parenleftbig 1−m2 y/parenrightbig , (24) ρtrans≈∆ρ1mxmy, (25)7 where ∆ρ0 ρ=−θ2 SH2λ dNtanhdN 2λ, (26) ∆ρ1 ρ=θ2 SHλ dN2λGrtanh2dN 2λ σ+2λGrcothdN λ. (27) Equations (24-25) fully explain the magnetization depende nce of SMR in Ref. 25, while Eq. (27) shows that an SMR exists only when the spin-mixing conductan ce does not vanish. Since results do not depend on the z-component of magnetization, the AHE vanishes in our model w henGi= 0. B.Gr≫σ/(2λ) Here we discuss the limit in which the spin current transvers e to ˆmis completely absorbed as an STT without reflection. This ideal situation is actually n ot so far from reality for the recently found large Grbetween YIG and noble metals.17,18The spin current at the interface is then /vectorj(F) s jSH s0Gr≫σ/(2λ)= ˆm×(ˆm׈y)tanhdN λtanhdN 2λ, (28) and the maximum magnetoresistance for the bilayer is ∆ρ1 ρ=θ2 SHλ dNtanhdN λtanh2dN 2λ. (29) In Sec. IIIE we test this limit with available parameters fro m experiments. C.λ/dN≫1 When the spin-diffusion length is much larger than the thickne ss of N /vector µs(z) µ0sλ/dN≫1= ˆm×(ˆm׈y)−ˆy2z−dN dN, whilespincurrentandmagnetoresistance vanish. We can int erpretthis as multiplescattering of the spincurrent at theinterfaces; the ISHEhas both positive an dnegative charge current contributions that cancel each other. D. Spin Hall AHE Recent measurements in YIG |Pt display a small AHE-like signal on top of the ordinary Hall effect,i.e. a transverse voltage when the magnetization is normal to th e film.30As mentioned above, an imaginary part of the spin-mixing conductance Gican cause a spin Hall AHE (SHAHE). The component of the spin accumulation µsx µsx(z) µ0s=2λ σcoshz−dN λ sinhdN λ[mxmyRe−mzIm]σG↑↓ σ+2λG↑↓cothdN λ(30)8 /s50 /s52 /s54 /s56 /s49/s48/s48/s50/s52/s54/s56/s49/s48 /s32/s32/s40/s49/s48/s45/s32/s52 /s41 /s40/s110/s109 /s41/s32 /s83/s72/s61/s48/s46/s48/s50 /s32 /s83/s72/s61/s48/s46/s48/s52 /s32 /s83/s72/s61/s48/s46/s48/s54 /s32 /s83/s72/s61/s48/s46/s48/s56 /s32/s69/s120/s112/s46/s83/s97/s109/s112/s108/s101/s32/s49 /s83/s97/s109/s112/s108/s101/s32/s50/s40/s97/s41 /s50 /s52 /s54 /s56 /s49/s48/s48/s50/s52/s54/s56/s49/s48/s40/s49/s48/s45/s32/s52 /s41 /s40/s110/s109 /s41 /s32/s32 /s40/s98/s41 /s50 /s52 /s54 /s56 /s49/s48/s48/s50/s52/s54/s56/s49/s48/s40/s49/s48/s45/s32/s52 /s41 /s40/s110/s109 /s41 /s32/s32 /s40/s99/s41 /s50 /s52 /s54 /s56 /s49/s48/s48/s50/s52/s54/s56/s49/s48 /s40/s100/s41 /s40/s49/s48/s45/s32/s52 /s41 /s40/s110/s109 /s41 /s32/s32 FIG.3: (Coloronline)Calculated∆ ρ1/ρasafunctionof λfordifferentspinHallangles θSHwith(a)Gr= 1× 1014Ω−1m−2, (b)Gr= 5×1014Ω−1m−2, (c)Gr= 10×1014Ω−1m−2, and (d) the ideal limit Gr≫σ/(2λ). The Pt layers are 12-nm-thick with resistivity 8 .6×10−7Ωm (Sample 1, solid curve) and 7-nm-thick with resistivity 4 .1×10−7Ωm (Sample 2, dashed curve). Experimental results are shown as h orizontal lines for comparison.25 contains a contribution that scales with mzand contributes a charge current in the transverse (ˆ y-) direction j(SHAHE) c,trans(z) j0c=−2λθ2 SHmzsinhz−dN λ sinhdN λImG↑↓tanhdN 2λ σ+2λG↑↓cothdN λ. (31) The transverse resistivity due to this current is ρ(SHAHE) trans≈−j(SHAHE) c,trans/Ex σ2=−∆ρ2mz, (32) where ∆ρ2 ρ≈2λ2θ2 SH dNσGitanh2dN 2λ/parenleftig σ+2λGrcothdN λ/parenrightig2 +/parenleftig 2λGicothdN λ/parenrightig2≈2λ2θ2 SH dNσGitanh2dN 2λ/parenleftig σ+2λGrcothdN λ/parenrightig2. E. Comparison with experiments There are controversies about the values of the material par ameters relevant for our theory, i.e. the spin-mixing conductance G↑↓of the N|F interface, as well as spin-flip diffusion length λand spin Hall angle θSHin the normal metal.9 Experimentally, Burrows et al.18found for an Au|YIG interface with G0=e2/h. Gexp r G0= 5.2×1018m−2;Gexp r= 2×1014Ω−1m−2. (33) On the theory side, the spin-mixing conductance from scatte ring theory for an insulator reads16 G↑↓ G0=NSh−/summationdisplay nr∗ n↑rn↓=NSh−/summationdisplay nei(δn↓−δn↑), (34) wherern↑(↓)=eiδn↑(↓)is the reflection coefficient of an electron in the quantum chan nelnon a unit area at the N |F interface with unit modulus and phase δn↑(↓)for the majority (minority) spin, andNShis the number of transport channels (per unit area) at the Fer mi energy, i.e.NShis the Sharvin conductance (for one spin). Therefore Gr G0≤2NSh;|Gi| G0≤NSh, (35) Jiaet al.17computed Eq. (34) for a Ag |YIG interface by first principles. Theaverage of different crystal interfaces G(0) r= 2.3×1014Ω−1m−2, (36) is quite close to the Sharvin conductance of silver ( NShG0≈4.5×1014Ω−1m−2). For comparison with experiment we have to include the Schep d rift correction:31 1 ˜Gr/G0=1 G(0) r/G0−1 2NSh, (37) which leads to ˜Gr≈3.1×1014Ω−1m−2. (38) One should note that the mixing conductance of the Pt |YIG interface can then be estimated to be ˜Gr≈1015Ω−1m−2since the Pt conduction electron density and Sharvin conduc tance are higher than those of noble metals. Using parameters ρ=σ−1= 8.6×10−7Ωm,dN= 12 nm, and λ= 1.5 nm,29we see that the absorbed transverse spin currents with Gr=˜GrandGr=Gmax robtained from above for a Ag |YIG interface are 44% and 70% of the value for a perfect spin sink Gr→∞, respectively. For a Pt |YIG interface this value should be even larger. In order to compare our results with the observed SMR, we have to fill in or fit the parameters. The values of the spin-diffusion length and the spin Hall angle differ widely.29In Fig. 3 we plot the SMR for three fixed values of Gr. We observe that the experiments can be explained by a sensib le set of transport parameters ( Gr,λ,θSH) that somewhat differ for the two representative samples reported in Ref. 25. Generally, the SMR increases with a larg er value of Grbut decreases when λ is getting longer. These features are in agreement with the d iscussion of the simple limits above. Sample 1 in Ref. 25 has a larger resistivity but a smaller SMR ( ratio), implying a smaller spin Hall angle and/or smaller spin-diffusion length. When we fix th e spin Hall angle θSH= 0.06 and the spin-mixing conductance Gr= 5×1014Ω−1m−2, the corresponding estimated spin-diffusion lengths of Samples 1 and 2 are λ1≈1.5nm and λ2≈3.5nm, respectively. Finally we discuss the AHE equivalent or SHAHE. From experim ents ∆ρ2/ρ≈1.5×10−5for ρ= 4.1×10−7Ωm and dN= 7 nm.30Choosing θSH= 0.05,λ= 1.5nm, and Gr= 5×1014Ω−1m−2, we would need a Gi= 6.2×1013Ω−1m−2to explain experiments, a number that is supported by first principle calculations.1710 IV. SPIN VALVES In this section we discuss F(ˆ m)|N|F(ˆm′) spin valves fabricated from magnetic insulators with magnetization directions ˆ mand ˆm′. The general angle dependence for independent rotations of ˆm and ˆm′is straightforward buttedious. We discussinthe following two representative configurations in which the two magnetizations are parallel and perpendicu lar to each other. We disregard in the following the effective field due to Gisuch that the parallel and antiparallel configurations ˆm=±ˆm′are equivalent. Moreover, we limit the discussion to the sim ple case of two identical F |N and N|F interfaces, i.e., the spin-mixing conductances at both interfaces are the sa me. A. Parallel Configuration ( ˆm·ˆm′=±1) When the magnetizations are aligned in parallel or antipara llel configuration, the boundary condition /vectorj(z) s(dN) =−/vectorj(F) sapplies. We proceed as in Sec. III to obtain the spin accumula tion /vector µs µ0s=−/bracketleftigg ˆy+ ˆm×(ˆm׈y)2λGrtanhdN 2λ σ+2λGrtanhdN 2λ/bracketrightigg sinh2z−dN 2λ sinhdN 2λ, (39) and the spin current /vectorjz s jSH s0= ˆy/parenleftigg cosh2z−dN 2λ coshdN 2λ−1/parenrightigg + ˆm×(ˆm׈y)2λGrtanhdN 2λ σ+2λGrtanhdN 2λcosh2z−dN 2λ coshdN 2λ. The spin currents at the bottom and top of N are absorbed as STT s and read /vectorjz s(0) jSH s0=/vectorjz s(dN) jSH s0= ˆm×(ˆm׈y)2λGrtanhdN 2λ σ+2λGrtanhdN 2λ, (40) leading to opposite STTs at the bottom ( /vector τ(B) stt) and top ( /vector τ(T) stt) ferromagnets /vector τ(B) stt=/planckover2pi1 2e/vectorj(z) s(0) =−/vector τ(T) stt (41) since/vectorj(F) s(ˆm) =/vectorjz s(0) =/vectorjz s(dN) =−/vectorj(F) s(ˆm′). The longitudinal and transverse (Hall) charge currents are jc,long j0c= 1+θ2 SH/bracketleftigg 1−/parenleftbig 1−m2 y/parenrightbig2λGrtanhdN 2λ σ+2λGrtanhdN 2λ/bracketrightigg cosh2z−dN 2λ coshdN 2λ, (42) jc,trans j0c=−θ2 SHmxmy2λGrtanhdN 2λ σ+2λGrtanhdN 2λcosh2z−dN 2λ coshdN 2λ. (43) and the longitudinal and transverse resistivities read ρlong=ρ+∆ρ0+∆ρ1/parenleftbig 1−m2 y/parenrightbig , (44) ρtrans= ∆ρ1mxmy, (45) where ∆ρ0 ρ=−θ2 SH2λ dNtanhdN 2λ, (46) ∆ρ1 ρ=θ2 SH dN4λ2Grtanh2dN 2λ σ+2λGrtanhdN 2λ. (47)11 /s50 /s52 /s54 /s56 /s49/s48/s48/s46/s48/s48/s46/s50/s48/s46/s52/s48/s46/s54/s48/s46/s56 /s78/s47/s70 /s32/s32/s32/s32/s50 /s83/s72 /s110/s109/s70/s47/s78/s47/s70 FIG. 4: (Color online) Calculated ∆ ρ1//parenleftbig ρθ2 SH/parenrightbig in an F|N|F spin valve as a function of spin-diffusion length λwithdN= 12 nm, Gr= 5×1014Ω−1m−2, andρ= 8.6×10−7Ωm chosen from Sample 1 in Ref. 25. ∆ρ1//parenleftbig ρθ2 SH/parenrightbig in an N|F bilayer is plotted as a dotted line for comparison. Figure 4 shows ∆ ρ1//parenleftbig ρθ2 SH/parenrightbig with respect to the spin-diffusion length in an F |N|F spin valve with parallel magnetization configuration. Compared to N |F bilayers, the SMR in spin valves is larger and does not vanish in the limit of long spin-diffusion lengths . B. Limit λ/dN≫1 The spin accumulation for weak spin-flip reads /vector µs µ0sλ/dN≫1=−/bracketleftbigg ˆy+dNGr σ+dNGrˆm×(ˆm׈y)/bracketrightbigg2z−dN dN, (48) leading to the spin current /vectorjz s jSH s0λ/dN≫1=dNGr σ+dNGrˆm×(ˆm׈y). (49) In contrast to the bilayer, we find a finite SMR in this limit for spin valves: jc,long j0cλ/dN≫1= 1+ θ2 SH/bracketleftbigg 1−dNGr σ+dNGr/parenleftbig 1−m2 y/parenrightbig/bracketrightbigg Gr≫σ/dN= 1+ θ2 SHm2 y, (50) jc,trans j0cλ/dN≫1=−θ2 SHdNGr σ+dNGrmxmyGr≫σ/dN=−θ2 SHmxmy (51) or ∆ρ0 ρ=−θ2 SH, (52) ∆ρ1 ρ=θ2 SHdNGr σ+dNGrGr≫σ/dN=θ2 SH. (53) Here we find the maximum achievable SMR effects in metals with sp in Hall angle θSHby taking the limit of perfect spin current absorption. Clearly this r equires spin valves with sufficiently thin spacer layers. We interpret these results in terms of spin an gular momentum conservation: The finite SMR is achieved by using the ferromagnet as a spin sink t hat suppresses the back flow of spins and the ISHE. This process requires a source of angular momentum, which in bilayers can only be the lattice of the normal metal. Consequently, the SM R is suppressed in the F |N system when spin-flip is not allowed. In spin valves, however, the se cond ferromagnet layer can act as a spin current source, thereby allowing a finite SMR even in the absence of spin-flip scattering.12 C. Perpendicular Configuration ( ˆm·ˆm′= 0) We may consider two in-plane magnetizations ˆ m= (cosα,sinα,0) and ˆm′= (−sinα,cosα,0), which are perpendicular to each other. When α= 0, the first layer maximally absorbs the SHE spin current, while ˆ m′is completely reflecting, just as the vacuum interface in the bilayer. For generalα: µsx(z) µ0s=2λGr σ+2λGrcothdN λ/parenleftigg coshz−dN λ sinhdN λ+coshz λ sinhdN λ/parenrightigg cosαsinα, (54) µsy(z) µ0s=−sinh2z−dN 2λ sinhdN 2λ−2λGr σ+2λGrcothdN λ/parenleftigg coshz−dN λ sinhdN λcos2α−coshz λ sinhdN λsin2α/parenrightigg ,(55) µsz(z) = 0, (56) which leads to the components of spin current normal to the in terfaces jsx(z) jSH s0=−2λGrtanhdN 2λ σ+2λGrcothdN λ/parenleftigg sinhz−dN λ sinhdN λ+sinhz λ sinhdN λ/parenrightigg cosαsinα, (57) jsy(z) jSH s0=cosh2z−dN 2λ−coshdN 2λ coshdN 2λ+2λGrtanhdN 2λ σ+2λGrcothdN λ/parenleftigg sinhz−dN λ sinhdN λcos2α−sinhz λ sinhdN λsin2α/parenrightigg .(58) The total current is the sum of those from the two ferromagnet s at the top and bottom; in contrast to the parallel ˆ m=±ˆm′configuration, they do not feel each other. We can extend the d iscussion from the previous subsection: the second F can be a spin curre nt source, and we can switch this source on by rotating the magnetization from perpendicular to (anti)parallel configuration. The longitudinal and transverse electric currents read jc,long(z) j0c= 1+θ2 SHcosh2z−dN 2λ coshdN 2λ+θ2 SH2λGrtanhdN 2λ σ+2λGrcothdN λ/parenleftigg sinhz−dN λ sinhdN λcos2α−sinhz λ sinhdN λsin2α/parenrightigg , (59) jc,trans(z) j0c=θ2 SH2λGrtanhdN 2λ σ+2λGrcothdN λ/parenleftigg sinhz−dN λ sinhdN λ+sinhz λ sinhdN λ/parenrightigg cosαsinα. (60) Since the angle-dependent contributions vanish upon integ ration over z, there is no magnetoresis- tance in the perpendicular configuration. D. Controlling the spin-transfer torque Like the SMR, the STT at the N |F interface depends on the relative orientation between ˆ m and ˆm′, too. We may pin ˆ m= ˆxand observe how the STT at the bottom magnet, /vector τ(B) stt(ˆm,ˆm′), changes with rotating ˆ m′= ˆxcosα+ ˆysinα. Figure 5 displays the ratio βdefined as β(α)≡/vextendsingle/vextendsingle/vextendsingle/vector τ(B) stt(ˆx,ˆx)−/vector τ(B) stt(ˆx,ˆxcosα+ ˆysinα)/vextendsingle/vextendsingle/vextendsingle /vextendsingle/vextendsingle/vextendsingle/vector τ(B) stt(ˆx,ˆx)/vextendsingle/vextendsingle/vextendsingle, (61) as a function of αfor some spin-diffusion lengths. Only when λ≪dN,βremains constant under rotation of ˆ m′. A larger spin-mixing conductance and smaller dNenhances the SMR as well as angle dependence of β. This modification of the STT should lead to complex dynamics of the spin valve in the presence of an applied current and will be the sub ject of a subsequent study.13 /s48/s46/s48 /s48/s46/s53/s48/s49/s50 /s32/s32 /s32 /s61/s50 /s110/s109 /s32 /s61/s52 /s110/s109 /s32 /s61/s54 /s110/s109 /s32 /s61/s56 /s110/s109 /s32 /s61/s49/s48 /s110/s109 FIG. 5: (Color online) The ratio β(α) which characterize how /vector τ(B) sttchanges with respect to the relative orientation between ˆ mand ˆm′. We adopt the transport parameters dN= 12 nm, ρ= 8.6×10−7Ωm, and Gr= 5×1014Ω−1m−2. V. SUMMARY We developed a theory for the SMR in N |F and F|N|F systems that takes into account the spin-orbit coupling in N as well as the spin-transfer at the N |F interface(s). In a N |F bilayer system, the SMR requires spin-flip in N and spin-transfer at t he N|F interface. Our results explain the SMR measured in Ref. 25 both qualitatively and quantitat ively with transport parameters that are consistent with other experiments. The degrees of s pin accumulation in N that can be controlled by the magnetization direction is found to be v ery significant. In the presence of an imaginary part of the spin-mixing conductance Giwe predicted a AHE-like signal (SHAHE). Such a signal was observed in Ref. 30 and can be explained with values of Githat agree with first principles calculations.17We furthermoreanalyzed F |N|F spin valves for parallel and perpendicular magnetization configurations. A maximal SMR ∼θ2 SHis found for a collinear magnetization configuration in the limit that the spin-diffusion length is mu ch larger than the thickness of the normal spacer. TheSMR vanishes when rotating the two magnet izations into a fixed perpendicular constellation. The SMR torques under applied currents in N a re expected to lead to magnetization dynamics of N|F and F|N|F structures. Acknowledgments This work was supported by FOM (Stichting voor Fundamenteel Onderzoek der Materie), EU- ICT-7 “MACALO,” the ICC-IMR, DFG Priority Programme 1538 “S pin-Caloric Transport” (GO 944/4), and KAKENHI (Grant-in-Aid for Scientific Research) C (22540346). 1S. D. Bader and S. S. P. Parkin, Ann. Rev. Cond. Matt. Phys. 1, 71 (2010). 2J. Sinova and I ˇZuti´ c, Nature Mater. 11, 368 (2012). 3For a review see: T. Jungwirth, J. Wunderlich, and K. Olejn´ ık, Natu re Mater. 11, 382 (2012). 4K. Ando, S. Takahashi, K. Harii, K. Sasage, J. Ieda, S. Maekawa, a nd E. Saitoh, Phys. Rev. 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2013-02-06

We present a theory of the spin Hall magnetoresistance (SMR) in multilayers made from an insulating ferromagnet F, such as yttrium iron garnet (YIG), and a normal metal N with spin-orbit interactions, such as platinum (Pt). The SMR is induced by the simultaneous action of spin Hall and inverse spin Hall effects and therefore a non-equilibrium proximity phenomenon. We compute the SMR in F$|$N and F$|$N$|$F layered systems, treating N by spin-diffusion theory with quantum mechanical boundary conditions at the interfaces in terms of the spin-mixing conductance. Our results explain the experimentally observed spin Hall magnetoresistance in N$|$F bilayers. For F$|$N$|$F spin valves we predict an enhanced SMR amplitude when magnetizations are collinear. The SMR and the spin-transfer torques in these trilayers can be controlled by the magnetic configuration.

Theory of spin Hall magnetoresistance

1302.1352v1

1 Submitted Oct 11, 2012 Intrinsic Spin Seebeck Effect in Au/YIG D. Qu1, S. Y. Huang1,2, Jun Hu3, Ruqian Wu3, and C. L. Chien1* Affiliations: 1Department of Physics and Astronomy, Johns Hopkins University, Baltimore Maryland 21218, USA 2Department of Physics, National Tsing Hua University, Hsinchu 300, Taiwan 3 Department of Physics and Astronomy, University of California, Irvine , California 92697, USA *clc@pha.jhu.edu Abstract: The acute magnetic proximity effects in Pt/YIG compromise the suitability of Pt as a spin current detector. We show that Au/YIG, with no anomalous Hall effect and a negligible magnetoresistance, allows the measurements of the intrinsic spin Seebeck effect with a magnitude much smaller than that in Pt/Y IG. The experiment results are consistent with the spin -polarized density -functional calculations for Pt with a sizable and Au with a negligible magnetic moment near the interface with YIG. PACS numbers : 72.15.Jf, 72.20.Pa, 85.80. -b, 85.75. -d 2 The expl oration of spintronic phenomena has been advanced towards the manipulation of a pure spin current without a charge current. A pure spin current can be realized by compelling electrons of opposite spins to move in opposite directions, or be carried by spin waves (magnons). Pure spin current is beneficial for spintronic operations with the attributes of maximal angular momentum and minimal charge current thus with much reduced Joule heating, circuit capacitance and electromigration. In the spin Hall effect (S HE), a charge current driven by a voltage gradient can generate a transverse spin current [ 1]. Using the spin Seebeck effect (SSE), a temperature gradient can also generate a spin current. Consequently, the SSE, within the emerging field of “spin caloritronics”, where one exploits the interplay of spin, charge, and heat, has attracted much attention. SSE has been reported in a variety of ferromagnetic (FM) materials (metal [2], semiconductor [3], or insulator [4]), where the pure spin current is detected in the Pt strip patterned onto the FM material by the inverse spin Hall effect (ISHE) with an electric field of ESHE = DISHE jS, where DISHE is the ISHE efficiency, jS is the pure spin current density diffusing into the Pt strip and is the spin direction. Consider a FM layer sample in the xy-plane, there are two ways to observe SSE using either the transverse or the longitudinal SSE configuration with a temperature gradient applied either in the sample plane (∇xT) or out -of-plane (∇zT) respectively. Various potential applications of SSE have already been proposed [ 5]. However, the SSE is not without controversies and complications. One fundamental mystery is that SSE has been reported in macroscopic structures on the mm scale whereas the spin diffusion length within which the spin coh erence is preserved is only on the nm scale [2-4]. Furthermore, the previous reports of SSE have other 3 unforeseen complications. In the transverse geometry of SSE with an intended in- plane∇xT, due to the overwhelming heat conduction through the substrate, there exists also an out -of-plane ∇zT, which gives rise to the anomalous Nernst effect (ANE) with an electric field of EANE zT m, where m is the magnetization direction [ 6]. The ANE is very sensitive in detecting ∇zT in a manner similar to the high sensitivity of the anomalous Hall effect to perpendicular magnetization in FM layer s less than 1 nm in thickness. As a result, ESHE jS due t o SSE with jS in the z -direction and EANE zT m due to ANE are both along the y -direction and asymmetric in magnetic field. The voltages of SSE due to ∇xT and ANE due to ∇zT are additive, entangled, and inseparable [6]. In the longitudinal SSE using Pt on a FM insulator (e.g., YIG), while the temperature gradient ∇zT is unequivocally out -of-plane, one encounters a different issue of magnetic proximity effects (MPE) in Pt in contact with a FM material. As a result , in the longitudinal configuration there is also entanglement of SSE and ANE [7]. These complications, when present, prevent the unequivocal establishment of SSE in either the transverse or the longitud inal configuration. The characteristics of the intrinsic SSE including its magnitude, remain outstanding and unresolved issues. In this work, we report the measurements of intrinsic SSE in gold (Au) using the longitudinal configuration with an unambiguo us out -of-plane (zT) gradient near room temperature . It is crucial to identify metals other than Pt that can unequivocally detect the pure spin current without MPE . Gold offers good prospects since it has been successfully used as a substrate or underlaye r for ultrathin magnetic films. We use polished 4 polycrystalline yttrium iron garnet (YIG=Y 3Fe5O12) a well -known FM insulator with low loss magnons as the substrate . A large spin -mixing conductance at Au/YIG interface has been reported using ferromagnetic r esonance [ 8]. Our results show that Au( t)/YIG does not have the large anomalous Hall effect and large MR that plagued Pt( t)/YIG but exhibits an unusual thickness dependence in the thermal transport. These resul ts allow us to place an upper limit for the intrinsic SSE of about 0.1 µ V/K much smaller than the thermal effect in Pt/YIG [ 7]. Thin Au films have been made by magnetron sputtering on YIG and pattern ed into parallel wires and Hall bars. As shown in the inset of Fig. 1(a), the xyz axes are parallel to the edges of the YIG substrates. The parallel wires with ascending order of thickness from 4 nm to 12 nm are in the xy-plane and oriented in the y-direction, where each wire is 4 mm long, 0.1 mm wide, and 2 mm apart. The Hall bar samples (inset of Fig. 1b) consist of one long segment along the x-direction and several short segments along the y-direction. For the MR measurements current is along the x-axis, for the thermal transport measurement ∇zT is along the z-axis, and the magnetic field is in the xy-plane in both cases. We use 4 -probe and 2 -probe measurements for MR and thermal voltage respectively. The multiple wires facilitate a systematic study of the thickness dependence of electric transport and thermal measurement under the same uniform thermal gradient with a temperature difference of Tz ≈ 10 K. The sample was sandwiched between, and in thermal contact with, two large Cu blocks kept at constant temperatures differing by 10 K. We first describe the thickness dependence of electrical resistivity ( ρ) of the Au wires. As expected ρ increases with decreasing film thickness as shown in Fig. 1(a). The 5 results can be well described by a semi classical theoretical model in the frame of Fuchs - Sondheimer (FS) theory [9], which includes the contributions from thickness ( t) as well as surface scattering ( p) and grain boundary scattering ( ), ρ = ρ∞{1-(1/2+3λ/4t)[1 -pexp(- tξ/λ)]exp( -t/λ)}-1 for t/ > 0.1 . Using bulk resistiv ity (ρ∞ = 2.2 cm) and the mean free path (λ=37 nm) [ 10], we find the data can be well described by p = 0.89 and = 0.37 as shown by the solid line in Fig. 1(a) . The anomalous Hall effect (AHE) is an essential measurement for assessing MPE . Hall measurements of the Au/YIG Hall bar samples have been made from 2K to 300K as shown in Fig. 1(b). The Hall resistance of Au/YIG is linea r in magnetic field at all temperatures (2 -300 K) showing only the ordinary Hall effect (OHE) with no observable AHE. In contrast, strong AHE has been observed in Pt/YIG due to the acute MPE [ 7]. The Hall const ant ( RH = 1/ne) of Au/YIG indicates the carrier concentration n ≈ 6×1022 cm-3 as shown in Fig. 1(c), essentially constant from 2 K to 300 K, is in good agreement with the bulk carrier concentration of n =5.9× 1022 cm-3 [11]. The spin polarized moment induced in Au, is very small, if any, i.e., Au is not appreciably affected by MPE and will be further discussed below. We employ the longitudinal configuration with spin current along the out -of-plane temperature gr adient zT to determine the thickness dependence of the thermal transport of Au/YIG, and to compare the results with those of the Pt/YIG. As shown in Fig. 2(a), the transverse thermal voltage (in the y-direction) across the Au strip is asymmetrical when the magnetic field is along the x-axis with the same sense as that for the Pt strip. The same sign of the thermal spin -Hall voltage between Pt and Au is consistent with the theoretical calculation of positive values of spin Hall conductivity in Pt and Au [ 12]. 6 However, there are several distinct differences between the thermal results of Au/YIG and Pt/YIG. We take Vth as the magnitude of spin -dependent thermal voltage between the positive and the negative switchin g fields. As shown in Fig. 2(b), the value of Vth of the Pt( t)/YIG is far larger, increasing sharply and unabatedly with decreasing t to a value of 64 µ V at t = 2.2 nm , due to the strong MPE at the interface between Pt and YIG. In contrast, the thermal v oltage Vth of the Au/YIG samples is much smaller than that of Pt/YIG and it varies with thickness ( t) in a non -monotonic manner as shown in Fig. 2(c). The value of Vth is negligible (less than 0.2 µ V) for t ≤ 7 nm , increasing to a maximum of 1.3 µ V at t = 8 nm before decreasing at larger thicknesses. This contrasting behavior shows that there is much smaller, perhaps negligible, MPE in Au/YIG. Consequently, the measured thermal voltage may be attributed enti rely to intrinsic SSE. With a maximal (Vth)max ≈1.3 µV at t = 8 nm at T of 10 K, the strength of the intrinsic SSE in Au/YIG is about 0.1 µ V/K, far smaller than the values in Pt( t)/YIG of 6 µ V/K at t = 2.2 nm, and 1 µ V/K at t = 10 nm, by one to two order s of magnitude. This suggests most of the spin - dependent thermal voltage in Pt/YIG is due to ANE and not SSE. From the value of Sxy ≈ 610-3 μV/K (Sxy=E xy/∇T=(Vth/l)/(T/d), where Vth is the thermal voltage, l is the distance between the voltage leads, T is the temperature difference and d is the thickness of Au/YIG sample) we measured and using the Seebeck coefficient Sxx ≈1.9 μ V/K of Au at 300 K [ 13], we obtain a spin Nernst angle of N = Sxy/Sxx ≈ 0.003, wh ich is very close to the spin Hall angle H = 0.0016, defined as the ratio of spin Hall and charge conductivities, from spin pumping measurement [ 14]. 7 However, we have observed MR, albeit with very small but c lear signals, in Au(t)/YIG. The MR result of Au(7 nm)/YIG Hall bar sample is shown in Fig. 3 (a). It is of a very small magnitude of ≈ - 4 x 10-6, where = || - T, about two orders of magnitude smaller than those of Pt( t)/YIG as shown in Fig. 3(b). Nevertheless all the Au(t)/YIG with 4 nm ≤ t ≤ 11 nm show similarly small but measurable More unexpectedly, the MR of Au(t)/YIG has the opposite angular dependence as that of the usual anisotropic MR (AMR). In the AMR of most 3d ferromagnetic met als of Fe, Co, Ni, and their alloys, the common behavior is positive , that is || > T, the resistivity with current parallel to, is higher than that with the current perpendicular, to the magnetization aligned by a magnetic field. The MR observed in P t(t)/YIG also has the same behavior of > 0. In contrast, the small MR in Au( t)/YIG is opposite with T > ||, or inverse AMR, The mechanism of this up behavior in Au( t)/YIG is not yet fully understood, but probably due to spin -dependent scattering at i nterface between Au and YIG, supported by the fact that | | increases with decreasing Au films thickness. One notes that inverse AMR has occasionally been reported in thin Co films. The s -d scattering influenced by spin -orbital and electron -electron interactions may be enhanced by the disorder in thin films [ 15]. To assess the magnetic moments of Pt and Au near the interface with YIG, spin - polarized density functional calculations have been carried out with th e Vienna ab initio simulation package (VASP), [16,17] at the level of the generalized gradient approximation (GGA) [18] with a Hubbard U correction for Fe -3d orbitals in YIG. We use the projector augmented wave (PAW) method for the description of the core -valence interaction [19,20]. The YIG structure has two Fe sites: tetrahedral Fe t and octahedral Fe o. 8 To model the Pt/YIG and Au/YIG interfaces, we construct a superlattice structure with a slab of YIG(111) of about 6 Å thick along with a 4 -layer Pt or Au film of about 7 Å thick. In the initial configuration, the Fe o atoms match the hcp sites of Pt(111) or Au(111) slab. During the relaxation process, the in -plane lattice constant has been fixed at the experimental value of the bulk YIG, with a dimension of 17.5× 17.5 Å2, and thickness of superlattice [notated as c in Fig. 4 (a)] is allowed to change. All atoms are fully relaxed until the calculated force on each atom is smaller than 0.02 eV/Å. For this lar ge unit cell with 274 atoms, we find that a single Γ point is enough to sample the Bril louin zone. The optimized atom ic structure of Pt/YIG in Fig. 4(a) shows significant reconstructions in both Pt and YIG layers. The average bond lengths are: d Pt-O ~ 2.2 Å, d Pt-Fe ~ 2.6 Å. Au/YIG has a similar structure. It is important to note that all fo ur Pt layers are significantly spin polarized as shown in Fig. 4(b) . The Pt layers adjacent to the interfaces [labeled by 1 and 4 in Fig. 4(b)] tend to cou ple ferromagnetically to their neighboring Fe atoms in YIG, as found in most studies for Pt on magnet ic substrates. The local spin moments of Pt atoms in the Pt 2 and Pt 3 layers can still be as large as 0.1 μB. By integrating spin density in the lateral planes, we can obtain the z -dependent spin density as shown in Fig. 4(c). Clearly, the spin polarization in all Pt layers is significant for the measurement of SSE. In particular, the total spin moments of th e Pt 2 and Pt 3 layers (each has 36 Pt atoms) are about 0.8 μ B and 1.1 μ B, respectively, even after the mutual cancelation with the intra -layer antiferromagnetic ordering. In contrast, spin polarizations induced in the Au layers are much weaker , with the max imum local spin moment smaller than 0.05 μB and the integrated spin moment in the entire Au layers smaller than 0. 1 μ B. Therefore, one can 9 view Au as nearly “nonmagnetic” in contact with YIG, in contrast to Pt. The sizable magnetic moments of Pt near the interface from the theoretical calculations is consisting with the strong MPE shown in Pt( t)/YIG by the electric transport . Therefore, the ANE and SSE are not only entangled but with ANE dominating in Pt/YIG. In contrast, the negligible Au moments from the oretical calculations is also consistent with no apparent AHE in Au( t)/YIG . The only noticeable magnetic characteristic is the inverse AMR of Au( t)/YIG but with a magnitude two orders smaller than that of Pt/YIG. This indicates that most, if not all, of the thermal voltage measured in Au/YIG is due to the intrinsic SSE as a result of the pure spin current injected from YIG. As shown in Fig. 2(c), the measurement of the thickness dependence is essential in revealing the non -monotonic dependence of intrin sic SSE voltage in Au/YIG due to the spin diffusion length SF. For very thin Au layer with t < 6 nm, SF is short due to the large resistivity from interface and boundary scattering, thus no appreciable spin current could survive intact , and this results in negligible Vth. As the Au film thickness increases, t he value of Vth exhibits a rapid rise reaching a maximum of 1.3V at t~8 nm and then decreases owing to the spin flip relaxation mechanism. Using the expression xx sf F sf / h/e/ k =l ) ( )23()2/(2 2 including the Fermi wave vector kF, the conductivity xx, the mean time between collisions and the mean time between spin - flipping collision sf, we estimate SF ≈ 40 nm. [ 21] The critical thickness of 8 nm is close to spin diffusion length 10.5 nm evaluated from weak localization [ 10]. Given the weak inverse AMR and the nonexistent AHE, the thermal signal of 0.1 µ V/K measured in Au/YIG at an optimal thickness of 8 nm should be considered as an upper limit of the 10 intrinsic S SE effect. The spin Hall angle between Au and YIG might be further enhanced by chemical modification on the YIG surface at high temperature . But a careful surface treatment is very important to avoid the metallic state of Fe developed, which could result in a reduction of spin mixing conductance and contamination in SSE [ 22]. In summary, we use Au(t)/YIG with no anomalous Hall signals and a very weak inverse MR results with non -monotonic dependence of spin -thermal voltage to show that the acute magnetic proximity effects that plagued Pt/YIG do not affect Au/YIG . The thermal voltage in Au/YIG is thus due to primarily intrinsic spin Seebeck effect with an upper limit of 0.1 µ V/K. Although the spin Hall angle of Au is smaller th an that of Pt, Au is a good spin current detector, far better than Pt. Acknowledgments: The work is supported at Johns Hopkins University by US NSF (DMR 05 -20491) and Taiwan NSC (99 -2911 -I-007- 510), and at University of California by DOE -BES (Grant No: D E-FG02 -05ER46237) and by NERSC for computing time. References : 1. J. E. Hirsch, Phys. Rev. Lett. 83, 1834. (1999). 2. K. Uchida, S. Takahashi, K. Harii, J. Ieda, W. Koshibae, K. Ando, S. Maekawa, and E. Saitoh, Nature 455, 778 (2008). 3. C. M. Jaworski, J. Yang, S. Mack, D. D. Awschalom, J. P. Heremans, and R. C. Myers, Nat. Mater. 9, 898 (2010). 4. K. Uchida, J. Xiao, H. Adachi, J. Ohe, S. Takahashi, J. Ieda, T. Ota, Y. Kajiwara, H. Umezawa, H. Kawai, G. E. W. Bauer, S. Maekawa, and E. Saitoh, Nat. Mater. 9, 894 ( 2010). 5. A. Kirihara, K. Uchida, Y. Kajiwara, M. Ishida, Y. Nakamura 1, T. Manako 1, E. Saitoh and S. Yorozu, Nature Mater . 11, 686 (2012). 6. S. Y. Huang, W. G. Wang, S. F. Lee, J. Kwo, and C. L. Chien, Phys. Rev. Lett. 107, 216604 (2011). 7. S. Y. Huang, X. F an, D. Qu, Y. P. Chen, W. G. Wang, J. Wu, T. Y. Chen, J. Q. Xiao and C. L. Chien, Phys. Rev. Lett. 109, 107204 (2012). 11 8. B. Heinrich, C. Burrowes, E. Montoya, B. Kardasz, E. Girt, Young -Yeal Song, Yiyan Sun, and Mingzhong Wu, Phys. Rev. Lett. 107, 066604 (2011). 9. P. Fan, K. Ti. J. D. Shao, and Z. X. Fau, J. Appl. Phys. 95, 2527 (2004). 10. J. Bass and W. P. Pratt Jr., J. Phys.: Condes. Matter 19, 183201 (2007). 11. C. L. Chien and C. R. Westgate, The Hall effect and its applications. (Plenum Press, New York, 1980). 12. T. Tanaka, H. Kontani, M. Naito, T. Naito, D. S. Hirashima, K. Yamada, and J. Ionoue, Phys. Rev. B 77, 165117 (2008). 13. D. M. Rowe , CRC handbook of thermoelectrics (CRC Press, New York , 1995) 14. O. Mosendz , J. E. Pearson, F. Y. Fradin , G. E. W. Bauer, S. D. Bade r, and A. Hoffmann, Phys. Rev. Lett. 104, 046601 (2010). 15. T. Y. Chung and S. Y. Hsu, J Phys.: Conf. Ser. 150, 042063 (2009). 16. G. Kresse and J. Furthmuller, Comput. Mater. Sci. 6, 15 (1996). 17. G. Kresse and J. Furthmuller, Phys. Rev. B 54, 11169 (1996). 18. J. P. Perdew, K. Burke, and M. Ernzerhof, P hys. Rev. Lett. 77, 3865 (1996). 19. P. E. Blochl, Phys. Rev. B 50, 17953 (1994). 20. G. Kresse and D. Joubert, Phys. Rev. B 59, 1758 (1999). 21. M. Gradhand, D. V. Fedorov, P. Zahn, and I. Mertig, Phys. Rev. B. 81, 245109 (2010). 22. C. Burrowes, B. Heinrich, B. Kardasz, E. A. Montoya, E. Girt, Yiyan Sun, Young -Yeal Song, and Mingzhong Wu, Appl. Phys. Lett. 100, 092403 (2012). 12 Fig. 1 (color online). (a) Resistivity as a function of Au thickness t for Au /YIG. The solid line represent s semiclassical theoretic fittings . Inset is schematic diagram of multiple patterned strips with ascending thickness. (b) Field dependence of Hall resistance RH at different temperatures for Au(7 nm)/YIG. Inset is schematic diagr am of patterned Hall bar. (c) Carrier concentration as a function of temperature for Au(7 nm)/YIG. 13 Fig. 2 (color online). (a) Field -dependent thermal voltage for Pt(5.1nm)/YIG and Au(10nm)/YIG ). Thermal voltage (left scale) and (right scale) for multiple strips as a function of Pt thicknesses (b) and Au thicknesses (c) on YIG. All thermal results are under a temperature difference of T ≈ 10 K. 14 Fig. 3 (color online). (a) Magnetor esistance (MR) result of Au(7nm)/YIG as a function of magnetic fi eld H at ||) and 90 (T). (b) AMR ratio as a function of metal layer thickness t for Pt/YIG (open triangles) and Au/Pt (open squares). 15 Fig. 4 (color online). (a) The optimized structural model of Pt/YIG. The teal, coral, purple, cyan and red spheres represent for Pt, Fe o (center of octahedron), Fe t (center of tetrahedron), Y and O atoms, respectively. The thickness of the superlattice, denoted as c, is 15.6 Å after relaxation . The numerals in the left side label the Pt layers for the convenien ce of discussions. (b) Isosurfaces of spin density (at 0.03 e/ Å3) of Pt/YIG. The blue and yellow isosurfaces are positive and negative spin polarizations. (c) Planar averaged spin density along c axis. The vertical dashed lines indicate the average z - coordinates of Pt and Au layers. Arrows ↑ and ↓ stand for the majority spin and minority spin contributions, respectively.

2013-01-25

The acute magnetic proximity effects in Pt/YIG compromise the suitability of Pt as a spin current detector. We show that Au/YIG, with no anomalous Hall effect and a negligible magnetoresistance, allows the measurements of the intrinsic spin Seebeck effect with a magnitude much smaller than that in Pt/YIG. The experiment results are consistent with the spin-polarized density-functional calculations for Pt with a sizable and Au with a negligible magnetic moment near the interface with YIG.

Intrinsic Spin Seebeck Effect in Au/YIG

1301.6164v1

Giant Enhancement of Vacuum Friction in Spinning YIG Nanospheres Farhad Khosravi,1,2Wenbo Sun,2Chinmay Khandekar,2Tongcang Li,3,2and Zubin Jacob2,∗ 1Department of Electrical and Computer Engineering, University of Alberta, Edmonton, Alberta T6G 1H9, Canada 2Elmore Family School of Electrical and Computer Engineering, Birck Nanotechnology Center, Purdue University, West Lafayette, Indiana 47907, USA 3Department of Physics and Astronomy, Purdue Quantum Science and Engineering Institute, Purdue University, West Lafayette, Indiana 47907, USA (Dated: January 19, 2024) Experimental observations of vacuum radiation and vacuum frictional torque are challenging due to their vanishingly small effects in practical systems. For example, a rotating nanosphere in free space slows down due to friction from vacuum fluctuations with a stopping time around the age of the universe. Here, we show that a spinning yttrium iron garnet (YIG) nanosphere near aluminum or YIG slabs exhibits vacuum radiation eight orders of magnitude larger than other metallic or dielectric spinning nanospheres. We achieve this giant enhancement by exploiting the large near- field magnetic local density of states in YIG systems, which occurs in the low-frequency GHz regime comparable to the rotation frequency. Furthermore, we propose a realistic experimental setup for observing the effects of this large vacuum radiation and frictional torque under experimentally accessible conditions. I. INTRODUCTION The physics of rotating nanoparticles is gaining more attention as recent technological advancements provide experimental platforms for rotating levitated nanoparti- cles at GHz speeds [1–8]. Besides having implications in the fields of quantum gravity [9], dark energy de- tection [10], and superradiance [11], rotating nanopar- ticles are crucial for studying the effects of quantum vac- uum fluctuations [12–17]. Rotating nanoparticles can emit real photons and experience frictional torques from the fluctuating quantum vacuum even at zero temper- ature [18, 19]. Although Casimir forces between static objects have been measured extensively [20–22], the ex- perimental sensitivity is only starting to reach the limit needed to measure the frictional torque exerted on ro- tating nanoparticles from the vacuum [23]. Meanwhile, direct observation of vacuum radiation from rotating nanoparticles remains challenging due to the extremely low number of radiated photons. In the specific case of moving media or rotating par- ticles, a unique regime of light-matter interaction occurs when the material resonance frequency becomes compa- rable to the mechanical motion frequency [24–26]. In particular, a giant enhancement or even a singularity is possible in vacuum fluctuation effects [24–26]. Re- cently, world record rotation frequencies were achieved for optically levitated nanospheres [2, 3, 6]. This immedi- ately opens the question of whether unique material res- onances comparable to this rotation frequency can help enter a new regime of light-matter interaction. Here, we show that gyromagnetic yttrium iron garnet (YIG) ex- hibits the magnon polariton resonance at GHz frequen- ∗zjacob@purdue.educies[27,28]comparabletothelevitatednanoparticle’sro- tation frequency, providing a unique opportunity for en- hancing vacuum fluctuation effects on rotating nanopar- ticles. Inthisarticle,weputforthanapproachtoenhanceand observe the vacuum radiation and frictional torques by leveraging a YIG nanosphere spinning at Ω = 1 GHz in the vicinity of a metallic or YIG interface. Our proposal exploits an asymmetry between the electric and magnetic local density of states (LDOS) which was previously re- ported in Ref. [29]. In particular, near conventional met- als, the electric LDOS is enhanced at optical frequen- cies, whereas the magnetic LDOS becomes dominant at GHz frequencies. Therefore, our proposal exploits mag- netic materials with magnon polaritons to enhance the magnetic local density of states beyond those of con- ventional plasmonic metals. Due to the large magnetic LDOS and YIG magnetic resonance at GHz frequencies, the fluctuating magnetic dipoles of the YIG nanosphere canstrongly coupleto alargedensityofevanescentwaves inthe near-fieldofmetallic andmagneticinterfaces, lead- ing to colossal vacuum radiation. We demonstrate that a spinning YIG nanosphere gen- erates vacuum radiation eight orders of magnitude larger than other metallic or dielectric nanospheres in the vicin- ity of a metallic or magnetic slab. We show that, near magnetic materials, most of this radiated energy can be transferred to surface magnon polaritons. Furthermore, we reveal that the large vacuum radiation and vacuum frictionhaveexperimentallyobservableeffectsonthebal- ance rotation speed, stopping time, and balance temper- ature of the spinning YIG nanospheres under experimen- tally accessible rotation speeds, particle sizes, tempera- tures, and vacuum pressures. Therefore, the setup pro- posed in this article based on spinning YIG nanospheres represents a unique tool for detecting and analyzing vac- uum radiation and frictional torques.arXiv:2401.09563v1 [quant-ph] 17 Jan 20242 II. GIANT VACUUM RADIATION FROM SPINNING YIG NANOSPHERES We first consider the vacuum radiation from a spinning YIG nanosphere with a radius of 200 nm, as illustrated in Fig. 1(a, b). A stationary nanosphere at the equilibrium temperature exhibits zero net radiation since the num- ber of photons emitted by the fluctuating dipoles of the nanosphere is equal to the number of photons absorbed by the nanosphere from the fluctuating electromagnetic fields in the vacuum. However, for rotating nanospheres, the balance between the emitted and absorbed photons is broken. A net radiated power from the nanosphere arises even at zero temperature due to the extra boost of mechanical rotational energy [30]. The source of this vacuum radiation energy is the non-inertial motion of the nanosphere, which is transferred to generate real photons from vacuum fluctuations [19]. Based on fluctuational electrodynamics (see derivations in Appendix A), we find thetotalradiatedpowerfromaspinningYIGnanosphere Prad=R∞ 0dωℏω ΓH(ω)−ΓH(−ω) can be determined from ΓH(ω), which is the spectral density of the radiation power arising from magnetic dipole fluctuations. In the absence of any interface, vacuum radiation from a spinning YIG nanosphere does not exhibit any sub- stantial enhancement. However, metallic or magnetic in- terfaces can drastically change this observation. Metal- lic nanospheres are known to possess higher radiation rates compared to dielectric nanospheres near material interfaces [30, 31]. Here, we observe that magnetic nanospheres exhibit even larger radiation rates, which are about eight orders of magnitude compared to metal- lic nanospheres near metallic or magnetic interfaces, as shown in Fig. 1(c, d). We demonstrate that radiated photons per second per frequency expressed through ΓH(ω)−ΓH(−ω)from spinning YIG nanospheres (blue curves) are much more than those from the aluminum nanospheres(orangecurves)nearAlinterfaces(Fig.1(c)) andYIGinterfaces(Fig.1(d)). Furthermore,wefindthat a spinning YIG nanosphere radiates about 6femtowatts of power, in stark contrast to the Al sphere, which ra- diates about 6×10−7femtowatts near Al interfaces (Fig. 1(c)). In the vicinity of YIG interfaces (Fig. 1(d)), we find about 61.3femtowatts and 4.63×10−7fem- towatts of radiated power from YIG and Al nanospheres, respectively. The radiated energy mostly goes into the lossy surface waves in both metallic and magnetic mate- rials [32]. However, if the magnetic material is properly biased, as is the case studied here with a bias magnetic field of 812Oe for the YIG slab, the magnetic resonance in the magnetic slab can become resonant with the mag- netic resonance in the magnetic sphere. As a result, most of the radiated energy is transferred to surface magnon polaritons. These results clearly show the advantage of YIG over Al nanospheres for probing vacuum radiation. The above results are explained by the YIG magnon polariton resonance at GHz frequencies and differences in the low-frequency electric and magnetic LDOS near FIG. 1. (a) A YIG sphere trapped in the laser beam and spinning at 1 GHz rotation frequency in the vacuum. The stopping time for the sphere is on the order of the age of the universe. (b) YIG sphere spinning in the vicinity of an Aluminum or YIG interface exhibits colossal vacuum radi- ation. The stopping time, due to the presence of the in- terface, is reduced to about 1 day. (c, d) Number of pho- tons emitted per second per radiation frequency, defined as 1 ℏωdP/dω = Γ(ω)−Γ(−ω), for a YIG (blue solid curve) or Aluminum (dashed orange curve) nanosphere of radius 200 nm at distance d= 0.5µmfrom (c) an aluminum slab or (d) a YIG slab at room temperatures. For the Al slab, a non-local model has been used. The YIG slab in panel (d) is biased along the y direction (panel (a)) with a magnetic field ofH0= 812Oe. metallic and magnetic interfaces. Vacuum fluctuation effects on rotating nanoparticles can be significantly en- hanced when the rotation frequency is comparable to res- onance frequencies. In addition, as shown by Joulain et al.[29], LDOS near metals is dominated by the magnetic LDOS at wavelengths above a few microns. Here, we ex- tend this observation to magnetic materials and take into account the effects of non-local electromagnetic response in Al [32] (also see Appendix F). Higher magnetic LDOS thanelectricLDOSatlowfrequenciesoriginatesfromdif- ferences in the reflection of the s- and p-polarized evanes- cent waves. The near-field electric LDOS is mainly in- fluenced by p-polarized evanescent waves since their con- tributions to the electric LDOS are strongly momentum- dependent and dominate the high momentum contribu- tions crucial for near-field LDOS. In contrast, the oppo- siteistrueforthenear-fieldmagneticLDOS,andthecon- tributions from the s-polarized evanescent waves dom- inate. At GHz frequencies, the imaginary part of the reflection coefficient for evanescent s-polarized waves is much larger than that for evanescent p-polarized waves. Thus, the spolarization contributes more to the LDOS3 than the ppolarization, leading toa more dominant mag- netic LDOS near metallic and magnetic interfaces. These near-fieldLDOScanbefurtherenhancedbymaterialres- onances [24–26, 33, 34]. To this end, we discuss the spectral density ΓH(ω)that determines the vacuum radiation. Through a sim- ilar approach as the methods used by Abajo and Man- javacas [18], our result for the radiation spectral den- sityΓH(ω)of a spinning gyromagnetic nanosphere due to magnetic dipole fluctuations is (see derivations in Ap- pendix A): ΓH(ω) = (ωρ0/8)(h gH ⊥,2(ω) + 2gH ∥(ω) + 2gH g,2(ω)i Im αm,⊥(ω−) −Re αm,g(ω−)  n1(ω−)−n0(ω) +gH ⊥,1(ω)Im αm,∥(ω) [n1(ω)−n0(ω)]) ,(1) where ρ0=ω2/c2π3is the vacuum density of states, gH ⊥,1 ,gH ⊥,2are the two components of the magnetic Green’s function in the plane of the interface (the xxandzz components for the setup shown in Fig. 1(b)), gH ∥is the component normal to the interface (the yycomponent here), and gH g,2is the off-diagonal component between the in-plane and normal directions (the xycomponent here), all normalized by πωρ 0/8.αm,⊥(ω),αm,g(ω), and αm,∥(ω)are the xx(oryy),xy, and zzcomponents of the YIG nanosphere magnetic polarizability tensor in the ro- tating sphere frame (see Appendix D for derivations). Ω is rotating frequency of the nanosphere and ω−=ω−Ω. n1(ω)andn0(ω)are the Bose-Einstein distribution func- tions pertinent to the sphere temperature T1and the en- vironment temperature T0, respectively. Detailed deriva- tions for all these quantities and discussions of various YIG interface orientations and bias magnetic field direc- tions are provided in Appendix B. When the sphere is stationary ω−=ω, and the sphere temperature is equal tothetemperatureoftheenvironment T1=T0,theterms n1(ω−)−n0(ω)andn1(ω)−n0(ω)become zero; thus, the radiation becomes zero as expected. Here, we emphasize one important aspect of ΓH(ω)re- garding the rotation-induced magnetization of the YIG nanosphere, which can occur without any external mag- netic field. This is known as the Barnett effect and origi- natesfromtheconservationofangularmomentum, where the mechanical angularmomentumof the sphereistrans- ferred to the spin of the unpaired electrons in the mag- netic material [35]. Assuming the magnetic field is paral- lel to the rotation axis, the Larmor precession frequency ω0of the electrons inside the sphere is [36] (also see Ap- pendix E): ω0= Ω + µ0γH0, (2) for the electron gyromagnetic ratio γ, vacuum permeabil- ityµ0, and applied external magnetic field H0. We in- corporate this effect on ω0to find the magnetic response of the spinning YIG nanosphere.III. ENHANCEMENT OF VACUUM FRICTIONAL TORQUE We now discuss the vacuum frictional torque exerted on the rotating YIG nanosphere in the vicinity of YIG and Al interfaces. We use a similar approach to find the vacuum torque exerted on the spinning gyromagnetic YIG sphere due to magnetic dipole and magnetic field fluctuations (detailed derivations are provided in Ap- pendix G). The torque along the axis of rotation is given byMz=R∞ 0dωℏ ΓH M(ω) + ΓH M(−ω) , where the expres- sion for ΓH M(ω)is similar to the expression for ΓH(ω)in Eq. (1), with the difference being that the last term on thesecondlineisnotpresentin ΓH M(ω)(seeAppendixG). Additionally, wefindthatothercomponentsofthetorque (MxandMycomponents) are not necessarily zero in the vicinity of the YIG interface, in contrast to the Al slab. Due to the anisotropy of the YIG slab, MxandMydo not vanish for some directions of the bias magnetic field. We provide further discussions of these cases in the sup- plementary material. In Fig. 2, we compare vacuum torques exerted on spin- ning YIG nanospheres (Fig. 2(a, c)) and spinning Al nanospheres (Fig. 2(b, d)), on nanospheres spinning in the vicinity of YIG slabs (Fig. 2(a, b)) and Al slabs (Fig. 2(c, d)), as well as on nanospheres spinning in the vicinity of slabs (solid colored curves) and spinning in vacuum(dashedblackcurves). Wedemonstratethatvac- uum torques exhibit more than 10 orders of magnitude enhancement in the vicinity of YIG and Al slabs com- pared to the vacuum, and about 4 orders of magnitude enhancement due to employing YIG nanospheres instead ofAlnanospheres. Theseresultsunraveltheadvantageof utilizing YIG nanospheres for probing vacuum frictional torques at GHz frequencies. In Fig. 2, we consider non- local electromagnetic response [32] for Al interfaces and incorporate effects from the magnetic and electric dipole and field fluctuations on vacuum torques. We notice that the vacuum torque is dominated by magnetic rather than electric fluctuations in all cases (see Appendix G). In ad- dition, we have taken into account the effect of recoil4 FIG. 2. The negative vacuum frictional torque experienced by the YIG and aluminum nanosphere with a radius of 200 nm at room temperature. (a) Torque experienced by a YIG sphere in the vicinity of the YIG slab (solid blue curve) and in vacuum (dashed black curve). (b) Torque exerted on an Al sphere in the vicinity of the YIG slab (solid orange curve) and in vacuum (dashed black curve). (c), (d) the same as (a) and (b) with the YIG slab replaced by an Al slab. The YIG slab is biased along the ydirection with H0= 812Oe (see Fig. 1(a)). A non-local model is used for the Al slabs. The distance between the spinning spheres and slabs is d= 0.5µm for all cases. Placing the YIG or Al interface in the vicinity of spinning nanospheres results in about 12 orders of magnitude increase in the exerted vacuum torque. torque [37] – the torque exerted on the sphere due to the scatteringofvacuumfieldfluctuationsofftheparticle. As discussed in Appendix G, we find that effects from this second-order torque are negligible compared with the ef- fects of magnetic fluctuations in the studied cases. IV. OBSERVABLE OUTCOMES OF GIANT VACUUM FRICTION IN SPINNING YIG NANOSPHERES The observable effects of the colossal vacuum radia- tion and frictional torques come down to changes in ex- perimentally measurable parameters when the spinning nanosphereisbroughtclosertothevicinityofAl/YIGin- terfaces. In Fig. 3(a), we show the proposed experimen- tal setup for this observation where a YIG nanosphere is trapped inside an Al or YIG ring. We note that the size of the ring is much larger than that of the nanosphere, and it does not lead to any resonant behavior. However, for smaller ring sizes, LDOS can be further enhanced compared to the slab interface case due to the presence FIG. 3. Experimental considerations of the setup. (a) Pro- posed experimental setup with nanosphere trapped inside a ring. (b) Balance rotation speed Ωbfor Al sphere (red curve) and YIG sphere in the presence of Al (blue curve) and YIG (pink curve) interfaces, as a function of distance dfrom the interfacefora 200nmradiussphereat 10−4Torrvacuumpres- sure. The values are normalized by the vacuum balance rota- tion speed Ω0. (c) Characteristic stopping time as a function of distance from the interface at 10−6Torr vacuum pressure. (d) Balance temperature of the YIG sphere Tsatd= 500 nm distance from Al (blue curve) and YIG (pink curve) in- terfaces as a function of lab temperature T0, at10−4Torr vacuum pressure. For Al spheres, there is no final tempera- ture as the temperature keeps rising with time. of interfaces on all sides. We evaluated some observable experimental outcomes due to large vacuum radiation and friction. This analy- sis is based on the experimentally accessible parameters from Refs. [3, 38, 39]. In Fig. 3(b), we show the balanced rotation speed Ωbof the spinning nanosphere normalized bytherotationspeed Ω0intheabsenceofanyinterfaceas a function of distance dfrom the interface. The balance rotation speed is defined as the sphere’s stable, perpetual rotation speed and occurs when the driving force due to the laser is equal to the drag force due to the vacuum chamber. In the absence of any interface, due to the neg- ligible value of vacuum radiation, the balance rotation speed Ω0is obtained when the torque from the trapping laser balances the frictional torque from air molecules in the imperfect vacuum [3] (also see Apeendix H). We as- sume the laser driving torque is constant and the drag force from air molecules has a linear dependence on rota- tional speeds [3]. In Fig. 3(b), we show that the balance rotation speed of the YIG nanosphere is reduced when it is closer to Al (blue curve) or YIG (pink curve) in- terfaces, as a result of the large frictional torques from vacuum fluctuations. Remarkably, we notice that there5 is no observable change in the balance speed for spinning Al nanospheres in the vicinity of Al or YIG interfaces (red curve). In Fig. 3(c, d), we further demonstrate outcomes of the large vacuum radiation in other experimental observ- ables, such as the stopping time as a function of distance (Fig. 3(c)) and the balance temperature as a function of the vacuum temperature T0(Fig. 3(d)). Stopping time is the time constant of the exponential decrease of the nanosphere rotation velocity after the driving torque is turned off. The torque can be switched off by chang- ing the polarization of the trapping laser from circular to linear without having to switch off the trapping laser. The balance temperature refers to the nanosphere tem- perature Ts, at which the loss of mechanical rotational energy due to vacuum frictional torque stops heating the nanospheres. As shown in Fig. 3(c, d), YIG nanospheres exhibit distinct behaviors in the stopping time and bal- ance temperature compared to Al nanospheres near YIG and Al interfaces. The results of Fig. 3 show that the vacuum radiation and frictional torque can be experimentally measured through the balance speed, balance temperature, and stopping time of the YIG nanosphere. In stark contrast, the Al nanosphere (or any other metallic nanospheres) may not experience enough vacuum friction to exhibit observable outcomes unless it is in a sensitive setup with very low vacuum pressure [3, 23]. V. DISCUSSION AND CONCLUSION Our results show that due to YIG magnon polariton resonance and the dominance of magnetic LDOS over electric LDOS in the vicinity of metallic or magnetic ma- terials at GHz frequencies, spinning YIG nanospheres can exhibit orders of magnitude larger vacuum radia- tion and frictional torque compared to any metallic or dielectric nanosphere. By investigating the case of a YIG nanosphere spinning at 1 GHz speed, we have shown that the effect of colossal vacuum fluctuations can be observed in an experimentally accessible setup. Our re- sults set a new perspective for observing and understand- ing radiation and frictional torques from vacuum fluctu- ations. Furthermore, our discussions of magnetic LDOS near YIG interfaces under various bias fields pave the way for magnetometry [40] and spin measurement [41] applications. ACKNOWLEDGEMENTS This research was supported by the Army Research Office under grant number W911NF-21-1-0287 and the Office of Naval Research under award number N000142312707.Appendix A: Radiation Power due to Magnetic Fluctuations In this appendix, we provide detailed derivations of the radiation power Pradfrom a spinning YIG nanosphere and its spectral density ΓH(ω)due to magnetic fluctua- tions. Using an approach similar to that taken by Abajo et. al[18, 30], we can write the radiated power due to the magnetic fluctuations of dipoles and fields as, Pmag=−⟨Hind·∂mfl/∂t+Hfl·∂mind/∂t⟩,(A1) where Hindis the induced magnetic field due to the mag- netic dipole fluctuations mflof the particle and mindis the induced magnetic dipole in the particle due to the fluctuations of the vacuum magnetic field Hfl. Note that all of these quantities are written in the lab frame. For the sphere spinning at the rotation frequency Ω, we can write, mfl x=m′fl xcos Ω t−m′fl ysin Ωt, mfl y=m′fl xsin Ωt+m′fl ycos Ω t, mfl z=m′fl z,(A2) where the primed quantities are written in the rotat- ing frame. Performing a Fourier transform as m′fl(t) =Rdω 2πe−iωtm′fl(ω), we can write in the frequency domain mfl x(ω) =1 2h m′fl x(ω−) +m′fl x(ω+) +im′fl y(ω+)−im′fl y(ω−)i , mfl y(ω) =1 2h im′fl x(ω−)−im′fl x(ω+) +m′fl y(ω+) +m′fl y(ω−)i , (A3) where ω±=ω±Ω. We can similarly write for the mag- netic fields H′fl x(ω) =1 2 Hfl x(ω+) +Hfl x(ω−)−iHfl y(ω+) +iHfl y(ω−) , H′fl y(ω) =1 2 iHfl x(ω+)−iHfl x(ω−) +Hfl y(ω+) +Hfl y(ω−) . (A4) Thus, using the fact that, m′ind(ω) =¯αm(ω)·H′fl(ω), (A5) with ¯αm(ω) = αm,⊥(ω)−αm,g(ω) 0 αm,g(ω)αm,⊥(ω) 0 0 0 αm,∥(ω) (A6) being the magnetic polarizability tensor of the YIG sphere biased along the zaxis, we find in the lab frame mind(ω) =¯αeff m(ω)·Hfl(ω), (A7)6 where ¯αeff m= αeff m,⊥(ω)−αeff m,g(ω) 0 αeff m,g(ω)αeff m,⊥(ω) 0 0 0 αeff m,∥(ω) , (A8a) αeff m,⊥(ω) =1 2 αm,⊥(ω+) +αm,⊥(ω−) +iαm,g(ω+)−iαm,g(ω−) , (A8b) αeff m,g(ω) =−i 2 αm,⊥(ω+)−αm,⊥(ω−) +iαm,g(ω+) +iαm,g(ω−) , (A8c) αeff m,∥=αm,∥(ω). (A8d) Note that we have used an expression similar to Eq. (A3) but written for the induced magnetic dipole moments. Expression for αm,⊥(ω)andαm,g(ω)are given in Appendix D. Using the fluctuation-dissipation theorem (FDT) [42], ⟨Hfl i(ω)Hfl j(ω′)⟩= 4πℏ[n0(ω) + 1]( GH ij(ω)−GH∗ ji(ω) 2i) δ(ω+ω′), (A9) with GH ij(ω) =GH ij(r,r′=r, ω)defined as the equal-frequency magnetic Green’s function of the environment defined through the equation, Hi(r,r′, ω) =GH ij(r,r′, ω)mj(r′, ω), (A10) we find the second term in Eq. (A1) employing Eqs. (A4) and (A5): ⟨Hfl i(ω)∂mind i(ω′)/∂t⟩=−iω′2πℏh n0(ω) + 1i δ(ω+ω′)× ( Im GH xx(ω) +Im GH yy(ω)
h αm,⊥(ω′+) +αm,⊥(ω′−) +iαm,g(ω′+)−iαm,g(ω′−)i + Re GH xy(ω) −Re G∗H yx(ω)
h αm,⊥(ω′+)−αm,⊥(ω′−) +iαm,g(ω′+) +iαm,g(ω′−)i +2Im GH zz(ω) αm,∥(ω′)) , (A11) where n0(ω) = 1 /(eℏω/kBT0−1)is the Planck distribution at the temperature of the lab T0. Writing FDT for the fluctuating dipoles, ⟨m′fl i(ω)m′fl j(ω′)⟩= 4πℏ[n1(ω) + 1]αm,ij(ω)−α∗ m,ji(ω) 2i δ(ω+ω′), (A12)7 we find the first term in Eq. (A1) employing Eq. (A3) and Hind i(ω) =GH ij(ω)mfl j(ω): ⟨Hind i(ω)∂mfl i(ω′)/∂t⟩=−2πℏiω′ n1(ω−) + 1( δ(ω+ω′)h GH xx(ω)Im αm,⊥(ω−) −GH xx(ω)Re αm,g(ω−) −GH yy(ω)Re αm,g(ω−) +GH yy(ω)Im αm,⊥(ω−) +iGH xy(ω)Im αm,⊥(ω−) −iGH xy(ω)Re αm,g(ω−) +iGH yx(ω)Re αm,g(ω−) −iGH yx(ω)Im αm,⊥(ω−) i) −2πℏiω′ n1(ω+) + 1( δ(ω+ω′)h GH xx(ω)Im αm,⊥(ω+) +GH xx(ω)Re αm,g(ω+) +GH yy(ω)Re αm,g(ω+) +GH yy(ω)Im αm,⊥(ω+) −iGH xy(ω)Im αm,⊥(ω+) −iGH xy(ω)Re αm,g(ω+) +iGH yx(ω)Re αm,g(ω+) +iGH yx(ω)Im αm,⊥(ω+) i) −4πℏiω′[n1(ω) + 1]( δ(ω+ω′)GH zz(ω)Im αm,∥(ω) ) ,(A13) where n1(ω)is the Planck distribution at the sphere temperature T1. Taking the inverse Fourier transform, adding Eqs. (A11) and (A13), taking the real part of the radiated power, and changing integral variables, we find Pmag=ℏ πZ+∞ −∞ωdω(  n1(ω−)−n0(ω) Im GH xx(ω) +Im GH yy(ω) +Re GH xy(ω) −Re GH yx(ω)
×  Im αm,⊥(ω−) −Re αm,g(ω−)  + [n1(ω)−n0(ω)]Im GH zz(ω) Imn αH m,∥(ω)o) .(A14) In this derivation, we have used the property αm(−ω) =α∗ m(ω). The expressions for Green’s functions in different YIG and aluminum interface arrangements are given in Appendix B. Plugging these expressions into Eq. (A14), we obtain Eq. (1) in the main text. Appendix B: Green’s Function Near an Anisotropic Magnetic Material In this appendix, we provide the Green’s function near a half-space of magnetic material, which would change due to the anisotropy of the material. We study two cases when the interface is the x−yplane and x−zplane, as shown in Fig. 4. We can write the electric and magnetic fields in the vacuum as E=Ei+Er,H=Hi+Hr, (B1a) Ei= (E0sˆs−+E0pˆp−)eik−·r, (B1b) Er= (E0srssˆs++E0prppˆp++E0srpsˆp++E0prspˆs+)eik+·r, (B1c) Hi=1 η0(−E0sˆp−+E0pˆs−)eik−·r, (B1d) Hr=1 η0(−E0srssˆp++E0prppˆs++E0srpsˆs+−E0prspˆp+)eik+·r, (B1e) where ˆs±,ˆp±, and ˆk±/k0form a triplet with ˆk±=k0(κcosϕˆx+κsinϕˆy±kzˆz),ˆs±= sin ϕˆx−cosϕˆy,ˆp±=−(±kzcosϕˆx±kzsinϕ−κˆz),(B2)8 FIG. 4. Schematic of the problem for the two cases of when the interface is in (a) x−yplane and (b) x−zplane. andη0=p µ0/ϵ0,k0=ω/c,κ2+k2 z= 1, and k0kzis the zcomponent of the wavevector. Similarly, we can write the electric and magnetic fields inside the magnetic material as E′=Et,H′=Ht, (B3a) Et= E0stssˆs′ −+E0ptppˆp′ −+E0stpsˆp′ −+E0ptspˆs′ −
eik′ −·r, (B3b) ¯¯µHt=p κ2+k′2z η0 −E0stssˆp′ −+E0ptppˆs′ −+E0stpsˆs′ −−E0ptspˆp′ − eik′ −·r, (B3c) where k′ ±=k0ˆk′ ±=k0(κcosϕˆx+κsinϕˆy±k′ zˆz),ˆs′ ±= sin ϕˆx−cosϕˆy,ˆp′ ±=−±k′ zcosϕˆx±k′ zsinϕˆy−κˆzp κ2+k′2z.(B4) Note that κis the same in the two media due to the boundary conditions. Also ˆk′ ±×ˆp′ ±= ˆs′ ±. We can write Maxwell’s equations in the magnetic material in matrix form as [43] (M+Mk)ψ=¯¯ϵ0 0¯¯µ + 0¯¯κ −¯¯κ0 Et η0Ht = 0, (B5) where ¯¯κ= 0 −k′ zκsinϕ k′ z 0−κcosϕ −κsinϕ κcosϕ 0 . (B6) Setting the det (M+Mk) = 0we get the solutions for k′ zin terms of κandϕ[43]. From these solutions and applying the boundary conditions, we can find the values of rss,rsp,rps,rppfor a given κandϕ. Note that different bias directions for the magnetic field of the YIG slab change the ¯¯µtensor and thus change the reflection coefficients rss, rsp,rps,rpp. In the following, we first provide the expression for the magnetic dyadic Green’s function ¯GHfor a source at z′=d when the interface is in the x−yplane (Fig. 4(a)). Here, we take the spinning sphere to be at the origin to simplify the derivations and move z= 0toz′=d. This would not change the Fresnel reflection coefficients. The incident magnetic Green’s function at the location of the source is thus, ¯GH i(z=z′, ω) =ik2 0 8π2ϵmZdkxdky kz(ˆsˆs+ ˆp−ˆp−)eikx(x−x′)+iky(y−y′). (B7) The reflected magnetic Green’s function at the location of the source is ¯GH r(z=z′, ω) =ik2 0 8πϵmZdkxdky kz(ˆsrppˆs+ ˆp+rspˆs+ ˆsrpsˆp−+ ˆp+rssˆp−)e2ikzd, (B8) where kx=κcosϕandky=κsinϕ. NotethatheretheFresnelreflectioncoefficientsgenerallydependontheincidence angle ϕ. For the special case of magnetization along the zaxis, they become independent of ϕ. Using Eq. (B4) and9 dropping the terms that vanish after integration over ϕ, we can write the total magnetic Green’s function at the location of source as, ¯GH(r,r, ω) =ik3 0 8π2Z2π 0dϕZ+∞ 0κdκ p(  sin2ϕˆxˆx+ cos2ϕˆyˆy−sinϕcosϕ(ˆxˆy+ ˆyˆx) 1 +rppe2ikzd
+p2cos2ϕˆxˆx+p2sin2ϕˆyˆy+κ2ˆzˆz +e2ik0pdrss −p2cos2ϕˆxˆx−p2sin2ϕˆyˆy+κ2ˆzˆz−p2cosϕsinϕ(ˆxˆy+ ˆyˆx) −pκcosϕ(ˆxˆz−ˆzˆx)−pκsinϕ(ˆyˆz−ˆzˆy) +e2ik0pdrps psinϕcosϕ(ˆxˆx−ˆyˆy) +psin2ϕˆxˆy−pcos2ϕˆyˆx+κsinϕˆxˆz−κcosϕˆyˆz +e2ik0pdrsp −pcosϕsinϕ(ˆxˆx−ˆyˆy) +pcos2ϕˆxˆy−psin2ϕˆyˆx+κsinϕˆzˆx−κcosϕˆzˆy) .(B9) Note that the electric Green’s function can be obtained by changing rsstorpp,rpptorss,rpstorspandrsptorps and dividing by ϵ0. In general, the non-diagonal parts of the Green’s function are not zero. Using this equation, we find, Im GH xx(ω) =πωρ 0 8gH ⊥,1(ω), (B10a) Im GH yy(ω) =πωρ 0 8gH ⊥,2(ω), (B10b) Re GH xy(ω) −Re GH yx(ω) =πωρ 0 4gH g,1(ω), (B10c) Im GH zz(ω) =πωρ 0 4, gH ∥(ω) (B10d) where ρ0=ω2/π2c3is the vacuum density of states and, gH ⊥,1(ω) =1 πZ2π 0dϕ(Z1 0κdκ ph 1 + sin2ϕRe rppe2ik0pd −κ2cos2ϕ+ cos2ϕ κ2−1
Re rsse2ik0pd +psinϕcosϕRe e2ik0pd(rps−rsp) i +Z∞ 1κdκ |p| sin2ϕIm{rpp}+ cos2ϕ κ2−1
Im{rss}+|p|sinϕcosϕRe{rps−rsp} e−2k0|p|d) , (B11a) gH ⊥,2(ω) =1 πZ2π 0dϕ(Z1 0κdκ ph 1 + cos2ϕRe rppe2ik0pd −κ2sin2ϕ+ sin2ϕ κ2−1
Re rsse2ik0pd −psinϕcosϕRe e2ik0pd(rps−rsp) i +Z∞ 1κdκ |p| cos2ϕIm{rpp}+ sin2ϕ κ2−1
Im{rss} − |p|sinϕcosϕRe{rps−rsp} e−2k0|p|d) , (B11b) gH g,1(ω) =−1 πZ2π 0( dϕZ1 0κdκ sin2ϕIm rpse2ik0pd + cos2ϕIm rspe2ik0pd  +Z∞ 1κdκ sin2ϕIm{rps}+ cos2ϕIm{rsp} e−2k0|p|d) ,(B11c)10 gH ∥(ω) =1 2πZ2π 0dϕ(Z1 0κ3dκ p 1 +Re rsse2ik0pd
+Z∞ 1κ3dκ |p|e−2k0|p|dIm{rss}) . (B11d) Plugging Eq. (B10) into Eq. (A14), we find, Pmag=Z∞ −∞dωℏωΓH(ω), (B12) with, ΓH(ω) = (ωρ0/8)(  gH ⊥,1(ω) +gH ⊥,2(ω) + 2gH g,1(ω) Im αm,⊥(ω−) −Re αm,g(ω−)  n1(ω−)−n0(ω) +2gH ∥(ω)Im αm,∥(ω) [n1(ω)−n0(ω)]) .(B13) For the case when the YIG interface is the x−zplane (Fig. 4(b)), we find the radiated power by exchanging the axes ˆx→ˆz,ˆy→ˆx, and ˆz→ˆyin Eq. (B9). In this case, we have Im GH xx(ω) =πωρ 0 8gH ⊥,2(ω), (B14a) Im GH yy(ω) =πωρ 0 4gH ∥(ω), (B14b) Im GH zz(ω) =πωρ 0 8gH ⊥,1, (B14c) where gH ⊥,1,gH ⊥,2, and gH ∥given by Eq. (B11). For the xyandyxcomponent of the Green’s function, however, we get Re GH xy(ω) −Re GH yx(ω) =πωρ 0 4gH g,2(ω), (B15) with gH g,2(ω) =1 πZ2π 0dϕ(Z1 0κ2dκ p psinϕIm rsse2ik0pd +cosϕ 2Im (rps−rsp)e2ik0pd  +Z∞ 1κ2κ |p| |p|sinϕIm{rss} −cosϕ 2Re{rsp−rps} e−2k0|p|d) ,(B16) and thus we have for the case when the YIG interface is the x−zplane, ΓH(ω) = (ωρ0/8)(h gH ⊥,2(ω) + 2gH ∥(ω) + 2gH g,2(ω)i Im αm,⊥(ω−) −Re αm,g(ω−)  n1(ω−)−n0(ω) +gH ⊥,1(ω)Im αm,∥(ω) [n1(ω)−n0(ω)]) ,(B17) with gH ⊥,1,gH ⊥,2, and gH ∥given by Eq. (B11) and gH g,2by Eq. (B16). This is the same as Eq. (1) in the main manuscript. Appendix C: Dominance of Magnetic Local Density of States Although the expressions found in the previous sections for the radiated power Pradare not, in general, exactly proportional to the local density of states (LDOS), they are proportional to terms of the same order as the LDOS. The expression for LDOS is given by [29], ρ(r, ω) =1 πωTr ϵ0Im GE(r,r, ω) +Im GH(r,r, ω)  , (C1)11 where the Tr represents the trace operator. Using the expressions of the previous section, it is easy to see that the LDOS at the location of the nanosphere is given by, ρ(ω) = (ρ0/8)h ϵ0(gE ⊥,1+gE ⊥,2+ 2gE ∥) +gH ⊥,1+gH ⊥,2+ 2gH ∥i , (C2) wheretheexpressionsfor gH ⊥,1,gH ⊥,2, and gH ∥aregivenbyEq.(B11)andtheexpressionfortheelectricGreen’sfunctions are found from the magnetic ones by replacing s→pandp→sand dividing by ϵ0. As discussed before, the magnetic Green’s functions are about eight orders of magnitude larger than the electric ones at GHz frequencies, and thus, the LDOS is dominated by the magnetic LDOS. This shows that the magnetic field fluctuations dominate the vacuum radiation, vacuum torque, and LDOS simultaneously. Appendix D: Magnetic Polarizability Tensor of YIG In the appendix, we provide derivations of the YIG polarizability tensor. We consider the Landau-Lifshitz-Gilbert formula to describe the YIG permeability tensor [36], ¯¯µ= µ⊥−µg0 µgµ⊥0 0 0 µ∥ , (D1) where µ⊥(ω) =µ0(1 +χ⊥) =µ0( 1 +ω0ωm(ω2 0−ω2) +ω0ωmω2α2+i αωω m ω2 0+ω2(1 +α2) [ω2 0−ω2(1 +α2)]2+ 4ω2 0ω2α2) ,(D2a) µg(ω) =µ0χg=µ0−2ω0ωmω2α+iωωm ω2 0−ω2(1 +α2) [ω2 0−ω2(1 +α2)]2+ 4ω2 0ω2α2, (D2b) µ∥=µ0, (D2c) andω0=µ0γH0is the Larmor precession frequency with γbeing the gyromagnetic ratio and H0the bias magnetic field (assumed to be along ˆzdirection), ωm=µ0γMswith Msbeing the saturation magnetization of the material, andαis the YIG damping factor related to the width of the magnetic resonance through ∆H= 2αω/µ 0γ. In the main text, we considered Ms= 1780 Oe and∆H= 45 Oe [36] in our calculations. When the magnetic field is reversed (along −ˆzdirection), we can use the same results by doing the substitutions ω0→ −ω0, ω m→ −ωm, α→ −α, (D3) which gives µ⊥→µ⊥, µ g→ −µg. (D4) Using the method in Ref. [44] for the polarizability tensor of a sphere with arbitrary anisotropy, we find the polarizability tensor of YIG with the permeability tensor described by Eq. (D1), ¯¯αm= 4πa3 (µ⊥−µ0)(µ⊥+2µ0)+µ2 g (µ⊥+2µ0)(µ⊥+2µ0)+µ2g−3µ0µg (µ⊥+2µ0)(µ⊥+2µ0)+µ2g0 3µ0µg (µ⊥+2µ0)(µ⊥+2µ0)+µ2g(µ⊥−µ0)(µ⊥+2µ0)+µ2 g (µ⊥+2µ0)(µ⊥+2µ0)+µ2g0 0 0µ∥−µ0 µz+2µ0 . (D5) Therefore the magnetic polarizability terms in Eqs. (B13) and (B17) are given by, αm,⊥(ω) = 4 πa3(µ⊥−µ0)(µ⊥+ 2µ0) +µ2 g (µ⊥+ 2µ0)(µ⊥+ 2µ0) +µ2g, (D6a)12 αm,g(ω) = 4 πa3 3µ0µg (µ⊥+ 2µ0)(µ⊥+ 2µ0) +µ2g, (D6b) where µ⊥andµgare frequency dependent terms give by Eq. (D2). It is important to note that magnetostatic approximation has been assumed in the derivation of the magnetic polarizability. This is similar to the electrostatic approximation used for the derivation of the electric polarizability [45], where, using the duality of electromagnetic theory, the electric fields and electric dipoles have been replaced by the magnetic fields and magnetic dipoles. In this approximation, the fields inside the sphere are assumed to be constant. One can apply the Mie theory to find the magnetic polarizability to the first order in the Mie scattering components. This, however, is mathematically challenging due to the anisotropy of the magnetic material. For the purpose of our study, the magnetostatic assumption is enough to find the polarizability properties of YIG since the size of the sphere is much smaller compared to the wavelength, and the polarizability is dominated by the magneto-static term. For metals, however, higher order terms are important for finding the magnetic polarizability since the magneto- static terms are zero and only higher order terms due to electric dipole fluctuations give rise to the magnetic polar- izability of metals [30]. We provide derivations based on Mie theory for the polarizability constant of an aluminum particle in Section S1 in the supplementary material. Appendix E: Barnett Effect In the simplest models of magnetic materials, electrons are assumed to be magnetic dipoles with the moments µBspinning about the magnetization axis determined by the applied magnetic field H0with the Larmor precession frequency ω0=µ0γH0, where γis the gyromagnetic ratio of the material [36]. Barnett showed that the spontaneous magnetization of a material with the magnetic susceptibility of χis given by [35] Mrot=χΩ/γ, (E1) where Ωis the rotation frequency of the magnetic material. This magnetization can be assumed to be caused by an applied magnetic field Hrotwhich is Hrot=Mrot/χ=Ω γµ0. We thus get the Larmor frequency due to rotation, ω0,rot= Ω. (E2) Therefore, the Larmor frequency of a spinning magnetic material is the same as the rotation frequency. We thus can write the total Larmor frequency of spinning YIG as ω0= Ω + µ0γH0. (E3) We use this expression to find the permeability tensor of a spinning YIG nanosphere discussed in Appendix D. Appendix F: Non-local Model for Aluminum Since the sphere is spinning in close proximity to material interfaces, the non-local effects in aluminum electromag- netic response can become important. Here, we employ the non-local Fresnel reflection coefficients from Ref. [46]. rss=Zs−4π cp Zs+4π cp, r pp=4πp/c−Zp 4πp/c +Zp, (F1) where p=√ 1−κ2, and Zs=8i cZ∞ 0dq1 ϵt(k, ω)−(q2+κ2), (F2a) Zp=8i cZ∞ 0dq1 q2+κ2q2 ϵt(k, ω)−(q2+κ2)+κ2 ϵl(k, ω) , (F2b) with the longitudinal and transverse dielectric permittivities given by ϵl(k, ω) = 1 +3ω2 p k2v2 F(ω+iΓ)fl(u) ω+iΓfl(u), (F3a)13 ϵt(k, ω) = 1−ω2 p ω(ω+iΓ)ft(u), (F3b) with k2= (ω/c)2 q2+κ2
,u= (ω+iΓ)/(kvF), and fl(u) = 1−1 2ulnu+ 1 u−1, f t(u) =3 2u2−3 2u(u2−1) lnu+ 1 u−1. (F4) These expressions give the non-local reflection coefficients at a metallic interface for the semi-classical infinite barrier (SCIB) model. The SCIB model is accurate as long as z=k 2kF∼0, where kF=mvF/ℏwith mbeing the free-electron mass. For example, for aluminum with vF≃2.03×106m/s, we have kF≃1.754×1010andk=ω/c≃20, which shows that for our case the SCIB model is valid. Appendix G: Vacuum Frictional Torque In this section, we provide the derivations of the vacuum frictional torque exerted on the spinning YIG nanosphere due to vacuum fluctuations. The torque on a magnetic dipole is given by M=m×H. (G1) Since we are interested in the torque along the rotation axis ( zdirection), we can write the torque as Mz=ˆz· ⟨mfl×Hind+mind×Hfl⟩ =⟨mfl xHind y−mfl yHind x+mind xHfl y−mind yHfl x⟩,(G2) using the Fourier transform, we get Mz=Zdωdω′ (2π)2e−i(ω+ω′)th ⟨mfl x(ω)Hind y(ω′)⟩ − ⟨mfl y(ω)Hind x(ω′)⟩+⟨mind x(ω)Hfl y(ω′)⟩ − ⟨mind y(ω)Hfl x(ω′)⟩i .(G3) Through a similar approach as that used in Appendix A, after some algebra, we find Mz=ℏ 2πZ∞ −∞dω( Im GH yy(ω) +Im GH xx(ω) +Re GH yx(ω) −Re GH xy(ω)
×  Im αm,⊥(ω+) +Re αm,g(ω+)  n1(ω+)−n0(ω) − Im GH yy(ω) +Im GH xx(ω) −Re GH yx(ω) +Re GH xy(ω)
×  Im αm,⊥(ω−) −Re αm,g(ω−)  n1(ω−)−n0(ω)) ,(G4) which can be written as Mz=−Z+∞ −∞dωℏΓH M(ω). (G5) For an interface in the x−yplane ΓH Mis given by ΓH M(ω) = (ωρ0/8) gH ⊥,1(ω) +gH ⊥,2(ω) + 2gH g,1(ω) Im αm,⊥(ω−) −Re αm,g(ω−)  n1(ω−)−n0(ω) ,(G6) which is the same expression for the radiated power minus the term related to the axis of rotation z. For an interface in the x−zplane, on the other hand, ΓH Mwe have ΓH M(ω) = (ωρ0/8)h gH ⊥,2(ω) + 2gH ∥(ω) + 2gH g,2(ω)i Im αm,⊥(ω−) −Re αm,g(ω−)  n1(ω−)−n0(ω) .(G7) This expression is the same as Eq. (1) in the main manuscript, with the difference that it does not have the last term involving the term n1(ω)−n0(ω). Compared to the vacuum radiation expression, vacuum torque has an extra minus sign in Eq. (G5), indicating that this torque acts as friction rather than a driving force, as expected.14 1. Other components of torque In the previous section, we only derived the zcomponents of the torque exerted on the nanosphere. The xandy components can be written as Mx=⟨mfl yHind z−mfl zHind y+mind yHfl z−mind zHfl y⟩, (G8a) My=⟨mfl zHind x−mfl xHind z+mind zHfl x−mind xHfl z⟩. (G8b) UsingasimilarapproachasthatusedintheprevioussectionandsectionA,incorporatingthetorqueduetotheelectric field fluctuations of vacuum and the magnetic dipole fluctuations of the YIG sphere, we find for the xcomponent of torque, Mx=ℏ 4πZ∞ −∞dω(  2n1(ω−) + 1 Im αm,⊥(ω−) −Re αm,g(ω−)  2Im GH zx(ω) + 2Re GH zy(ω)  −4 [n1(ω) + 1]Im αm,∥(ω) Re GH yz(ω) + [2n0(ω) + 1](  Re{αm,⊥(ω−)}+Im{αm,g(ω−)} Re{GH xz(ω)} −Re{GH zx(ω)}+Im{GH yz(ω)}+Im{GH zy(ω)}
+ Im{αm,⊥(ω−)} −Re{αm,g(ω−)} −Im{GH xz(ω)} −Im{GH zx(ω)}+Re{GH yz(ω)} −Re{GH zy(ω)}
) + [n0(ω) + 1]( −2Re{αm,∥(ω)} Im{GH zy(ω)}+Im{GH yz(ω)}
+ 2Im{αm,∥(ω)} −Re{GH zy(ω)}+Re{GH yz(ω)}
) , (G9) and for the ycomponent, My=ℏ 4πZ∞ −∞dω(  2n1(ω−) + 1 Im αm,⊥(ω−) −Re αm,g(ω−)  −2Re{GH zx(ω)}+ 2Im{GH zy(ω)} +4 [n1(ω) + 1]Im αm,∥(ω) Re{GH xz(ω)}) −[2n0(ω) + 1](  Re{αm,⊥(ω−)}+Im{αm,g(ω−)} Im{GH xz(ω)}+Im{GH zx(ω)} −Re{GH yz(ω)}+Re{GH zy(ω)}
+ Im{αm,⊥(ω−)} −Re{αm,g(ω−)} Re{GH xz(ω)} −Re{GH zx(ω)}+Im{GH yz(ω)}+Im{GH zy(ω)}
) −[n0(ω) + 1]( −2Reαm,∥(ω) Im{GH zx(ω)}+Im{GH xz(ω)}
+ 2Imαm,∥(ω) −Re{GH zx(ω)}+Re{GH xz(ω)}
) . (G10) We can find the xandycomponents of frictional torque by plugging magnetic Green’s function expressions into Eqs. (G9) and (G10). Remarkably, we find that the spinning YIG nanosphere can experience a large torque along the x or y direction when the YIG interface is biased by external magnetic fields in the x or y direction. This means that in these cases, the sphere can rotate out of the rotation axis and start to precess. This will change the validity of the equations found for the vacuum radiation and frictional torque along the zaxis since it has been assumed that the sphere is always rotating around the zaxis and is also magnetized along that axis. However, this torque is still small enough compared to the driving torque of the trapping laser and it will still give enough time to make the observations of vacuum fluctuation effects. In Section S2 in the supplementary material, we present the plots of these torques when the interface is the x−yorx−zplane and provide more detailed discussions.15 2. Recoil torque Another contribution to the torque comes from the case when the induced dipole moments on the YIG sphere re-radiate due to the vacuum electric field fluctuations. This causes a recoil torque on the sphere and can be written as Mrec=⟨mind×Hsc⟩, (G11) where Hscis the scattered fields from the dipole and are given by, Hsc(r, ω) =¯GH(r,r′, ω)·mind(r′, ω), (G12) which shows that this term is of higher order contribution. We find that this recoil torque is much smaller than the torque derived in Eq. (G5) for YIG spheres spinning near YIG or Al interfaces and can thus be ignored in all studied cases. We provide detailed derivations of Mrecand quantitative comparisons in Section S2 in the supplementary material. Appendix H: Experimental Analysis In this section, we present the analytical steps for finding the experimental prediction plots provided in the last section of the main text. 1. Effects of drag torque due to imperfect vacuum In the real system of a spinning sphere, the environment is not a pure vacuum. This causes an extra torque on the spinning sphere from air molecules in the imperfect vacuum. The steady-state spin of the sphere happens when the driving torque of the trapping laser is equal to the drag and vacuum friction torques. In the case when there is no interface present, the only important counteracting torque is the drag torque given by [47] Mdrag=2πµa4 1.497λΩ, (H1) where ais the sphere radius, µis the viscosity of the gas the sphere is spinning in, λis the mean free path of the air molecules, and Ωis the rotation frequency. We further have for gases [48], λ=µ pgasr πKBT 2m, (H2) where pgasandmare the pressure and the molecular mass of the gas, respectively. Thus, we get the drag torque, Mdrag=2a4pgas 1.479r 2πm kBTΩ. (H3) For 1 GHz rotation of a sphere, the balance between the drag torque and the optical torque Mopthappens at about pgas= 10−4torr. Therefore we get, at room temperature and for a molecular mass of 28.966gram /mol, r 2πm KBT= 8.542×10−3, (H4) and thus [3], Mopt= 1.568×10−21N·m, (H5) This is important for studying the effects of vacuum torque on the rotation speed of the sphere. As shown in the main text, we find that for vacuum pressures of about 10−4torr, changes in the balance speed of the YIG nanoparticle when it is closer to material interfaces are detectible in the power spectral density (PSD) of the nanosphere [3].16 2. Effects of negative torque and shot noise heating due to the trapping laser When the trapping laser is linearly polarized, it can exert a negative torque on the spinning particle. The torque on the sphere due to the laser is given by Mopt=1 2Re{p∗×E}[3], where pis the dipole moment of the sphere, given by p=¯αeff·E, with ¯αeffbeing the effective polarizability of the sphere as seen in the frame of the lab, and E is the electric field from the laser. As derived in Section S3 in the supplementary material, in the case when the laser is linearly polarized, the negative torque from the laser is proportional to Im {α(ω0+ Ω)} −Im{α(ω0−Ω)}, where ω0= 1.21×1016is the frequency of the laser, and Ω = 6 .28×109is the rotation frequency. Since Ω≪ω0, we get α(ω+)≃α(ω−)and thus the second term is negligible. We can thus ignore the negative torque coming from the laser when the laser is linearly polarized. Another effect from the trapping laser is the heating of nanoparticles due to the shot noise. 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Vincenti, John Wiley & Sons , 414 (1965).SUPPLEMENTAL MATERIAL FOR ‘GIANT ENHANCEMENT OF VACUUM FRICTION IN SPINNING YIG N ANOSPHERES ’ Farhad Khosravi1,2, Wenbo Sun2, Chinmay Khandekar2, Tongcang Li2,3, and Zubin Jacob2,∗ 1Department of Electrical and Computer Engineering, University of Alberta, Edmonton, Alberta T6G 1H9, Canada 2Elmore Family School of Electrical and Computer Engineering, Birck Nanotechnology Center, Purdue University, West Lafayette, Indiana 47907, USA 3Department of Physics and Astronomy, Purdue Quantum Science and Engineering Institute, Purdue University, West Lafayette, Indiana 47907, USA ∗zjacob@purdue.edu Contents S1 Non-Electrostatic Limit and Magnetic Polarizability due to Electric Fluctuations 1 S2 Vacuum Frictional Torque 4 S2.1 Other components of torque . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 S2.2 Recoil torque . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 S2.3 Plots of torque terms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 S3 Experimental Considerations 8 S3.1 Effect of torque due to the trapping laser . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8 S3.2 Effect of heating due to the shot noise . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10 S1 Non-Electrostatic Limit and Magnetic Polarizability due to Electric Fluctuations In this section, we provide derivations for the magnetic polarizability of metallic nanoparticles due to the electric dipole terms based on Mie theory. If a sphere is placed in the direction of a plane wave polarized along ˆxdirection and propagating along zdirection Ei=E0eik0rcosθˆx, (S1) The scattered fields are given by [2], Es=−∞/summationdisplay n=1En/parenleftig ianN(1) e1n−bnM(1) o1n/parenrightig , (S2) Hs=−k0 ωµ∞/summationdisplay n=1En/parenleftig ibnN(1) o1m+anM(1) e1n/parenrightig , (S3) where Memn=−m sinθsinmϕPm n(cosθ)zn(kr)ˆθ−cosmϕdPm n(cosθ) dθzn(kr)ˆϕ, (S4a) Momn=m sinθcosmϕPm n(cosθ)zn(kr)ˆθ−sinmϕdPm n(cosθ) dθzn(kr)ˆϕ, (S4b) Nemn=zn(kr) krcosmϕn(n+ 1)Pm n(cosθ)ˆr+ cos mϕdPm n(cosθ) dθ1 krd d(kr)[krzn(kr)]ˆθ −msinmϕPm n(cosθ) sinθ1 krd d(kr)[krzn(kr)]ˆϕ,(S4c)Nomn=zn(kr) krsinmϕn(n+ 1)Pm n(cosθ)ˆr+ sin mϕdPm n(cosθ) dθ1 krd d(kr)[krzn(kr)]ˆθ +mcosmϕPm n(cosθ) sinθ1 krd d(kr)[krzn(kr)]ˆϕ,(S4d) the superscripts (1)forMandNindicate that the Bessel functions are the Hankel functions of the first kind h(1)(kr), En=inE0(2n+ 1)/n(n+ 1) , and anandbnare the Mie scattering coefficients. On the other hand, the radiated fields due to an electric dipole are given by Ed=k3 0 4πϵm/braceleftbigg (ˆr×p)׈reikr kr+ [3ˆr(ˆr·p)−p]/parenleftbigg1 (kr)3−i (kr)2/parenrightbigg eikr/bracerightbigg , (S5a) Hd=ck2 0 4π(ˆr×p)eikr r/parenleftbigg 1−1 ikr/parenrightbigg . (S5b) Using the facts that P1 1(cosθ) =−sinθ,dP1 1(cosθ) dθ=−cosθ, (S6) h(1) 1(kr) =−eikr/parenleftbiggi (kr)2+1 kr/parenrightbigg ,1 krd d(kr)/bracketleftig krh(1) 1(kr)/bracketrightig =−eikr/parenleftbigg −i (kr)3−1 (kr)2+i kr/parenrightbigg . (S7) The scattered fields to the first order of nbecome Es=3 2E0/braceleftigg ia1/bracketleftbigg eikr/parenleftbigg1 (kr)3−i (kr)2/parenrightbigg 2 cosϕsinθˆr−(cosϕcosθˆθ−sinϕˆϕ)eikr/parenleftbigg1 (kr)3−i (kr)2−1 kr/parenrightbigg/bracketrightbigg −b1/bracketleftbigg (cosϕˆθ−sinϕcosθˆϕ)eikr/parenleftbigg−1 (kr)2+i kr/parenrightbigg/bracketrightbigg/bracerightigg . (S8) Assuming that the dipole is along xdirection p=p0ˆx, the dipole fields become Ed=p0k3 0 4πϵm/braceleftbigg (cosθcosϕˆθ−sinϕˆϕ)eikr kr+ (2ˆrsinθcosϕ−ˆθcosθcosϕ+ˆϕsinϕ)/parenleftbigg1 (kr)3−i (kr)2/parenrightbigg eikr/bracerightbigg =p0k3 0 4πϵm/braceleftbigg 2ˆrsinθcosϕ/parenleftbigg1 (kr)3−i (kr)2/parenrightbigg eikr−(ˆθcosθcosϕ−ˆϕsinϕ)/parenleftbigg1 (kr)3−i (kr)2−1 kr/parenrightbigg eikr/bracerightbigg . (S9) In the low-frequency limit when kr=2πr λ≪1, the scattered fields are dominated by terms of the order (kr)−3. Thus, we can neglect the contribution from the Mterms or the b1terms in Eq. (S8). In this limit, the fields of the dipole and the scattered fields become equivalent, if we take p0=6πϵmia1 k3 0E0, (S10) or in other words, the sphere takes the polarizability αe=6πϵmc3 ω3ia1, (S11) where an=ϵ1jn(x1)[x0jn(x0)]′−ϵ0jn(x0)[x1jn(x1)]′ ϵ1jn(x1)[x0h(1) n(x0)]′−ϵ0h(1) n(x0)[x1jn(x1)]′, (S12) withx0=k0a,x1=k1a, and k1=ω√µ1ϵ1, and µ1andϵ1being properties of the sphere. Now, we look at the scattered magnetic fields. We have to the first order Hs=3 2k0 ωµ0E0/braceleftigg ib1/bracketleftbigg 2ˆrsinϕsinθ/parenleftbigg1 (kr)3−i (kr)2/parenrightbigg eikr−(ˆθsinϕcosθ+ˆϕcosϕ)/parenleftbigg1 (kr)3−i (kr)2−i kr/parenrightbigg eikr/bracketrightbigg −an/bracketleftbigg (ˆϕcosθcosϕ+ˆθsinϕ)/parenleftbiggi (kr)2+1 kr/parenrightbigg eikr/bracketrightbigg/bracerightigg . (S13) 2Again, we can ignore the second line or, in other words, anin this expression for low frequencies. Then, comparing this expression with the magnetic fields of a magnetic dipole polarized along ˆydirection m=m0ˆy, Hm=m0k3 0 4π/braceleftbigg 2ˆrsinθsinϕ/parenleftbigg1 (kr)3−i (kr)2/parenrightbigg eikr−(ˆθcosθsinϕ+ˆϕcosϕ)/parenleftbigg1 (kr)3−i (kr)2−1 kr/parenrightbigg eikr/bracerightbigg , (S14) Taking H0=k0 ωµ0E0, we find that the two are equivalent if we have m0=6πib1 k3 0H0, (S15) or if the sphere takes the magnetic polarizability αm=6πc3 ω3ib1, (S16) where bn=µ1jn(x1)[x0jn(x0)]′−µ0jn(x0)[x1jn(x1)]′ µ1jn(x1)[x0h(1) n(x0)]′−µ0h(1) n(x0)[x1jn(x1)]′. (S17) In the low-frequency limit, we have lim x→0jn(x) =2nn! (2n+ 1)!xn, (S18) and lim x→0yn(x) =−(2n)! 2nn!1 xn+1. (S19) Therefore, we have in this limit j1(x)≃x/3,y1(x)≃ −1/x2,[xj1(x)]′≃2x/3and[xy1(x)]′≃1/x2which gives a1≃ϵ1x1 32x0 3−ϵ0x0 32x1 3 ϵ1x1 3/parenleftig 2x0 3+i x2 0/parenrightig −ϵ02x1 3/parenleftig x0 3−i x2 0/parenrightig≃2k3 0a3 3iϵ1−ϵ0 ϵ1+ 2ϵ0, (S20a) b1≃2k3 0a3 3iµ1−µ0 µ1+ 2µ0. (S20b) We thus get for the polarizabilities αe≃4πϵ0a3ϵ1−ϵ0 ϵ1+ 2ϵ0, α m≃4πa3µ1−µ0 µ1+ 2µ0, (S21) which are exactly equal to the results derived using the electro-static and magneto-static approximations method. For a non-magnetic material, b1becomes b1≃x3 0 45ix2 0/parenleftbiggϵ1 ϵ0−1/parenrightbigg , (S22) which gives for the magnetic polarizability, αm≃2π 15k2 0a5/parenleftbiggϵ1 ϵ0−1/parenrightbigg =8π3 15a3/parenleftiga λ/parenrightig2/parenleftbiggϵ1 ϵ0−1/parenrightbigg . (S23) 3S2 Vacuum Frictional Torque S2.1 Other components of torque In this section, we provide further discussions of components of the torque other than the zcomponent exerted on a spinning nanosphere near YIG slabs under different bias fields. The xcomponent of torque, Mx=ℏ 4π/integraldisplay∞ −∞dω/braceleftigg /bracketleftbig 2n1(ω−) + 1/bracketrightbig/bracketleftbig Im/braceleftbig αm,⊥(ω−)/bracerightbig −Re/braceleftbig αm,g(ω−)/bracerightbig/bracketrightbig/bracketleftbig 2Im/braceleftbig GH zx(ω)/bracerightbig + 2Re/braceleftbig GH zy(ω)/bracerightbig/bracketrightbig −4 [n1(ω) + 1] Im/braceleftbig αm,∥(ω)/bracerightbig Re/braceleftbig GH yz(ω)/bracerightbig + [2n0(ω) + 1]/braceleftigg /bracketleftbig Re{αm,⊥(ω−)}+Im{αm,g(ω−)}/bracketrightbig/parenleftbig Re{GH xz(ω)} −Re{GH zx(ω)}+Im{GH yz(ω)}+Im{GH zy(ω)}/parenrightbig +/bracketleftbig Im{αm,⊥(ω−)} −Re{αm,g(ω−)}/bracketrightbig/parenleftbig −Im{GH xz(ω)} −Im{GH zx(ω)}+Re{GH yz(ω)} −Re{GH zy(ω)}/parenrightbig/bracerightigg + [n0(ω) + 1]/braceleftigg −2Re{αm,∥(ω)}/parenleftbig Im{GH zy(ω)}+Im{GH yz(ω)}/parenrightbig + 2Im{αm,∥(ω)}/parenleftbig −Re{GH zy(ω)}+Re{GH yz(ω)}/parenrightbig/bracerightigg , (S24) and for the ycomponent, My=ℏ 4π/integraldisplay∞ −∞dω/braceleftigg /bracketleftbig 2n1(ω−) + 1/bracketrightbig/bracketleftbig Im/braceleftbig αm,⊥(ω−)/bracerightbig −Re/braceleftbig αm,g(ω−)/bracerightbig/bracketrightbig/bracketleftbig −2Re{GH zx(ω)}+ 2Im{GH zy(ω)}/bracketrightbig +4 [n1(ω) + 1] Im/braceleftbig αm,∥(ω)/bracerightbig Re{GH xz(ω)}/bracerightigg −[2n0(ω) + 1]/braceleftigg /bracketleftbig Re{αm,⊥(ω−)}+Im{αm,g(ω−)}/bracketrightbig/parenleftbig Im{GH xz(ω)}+Im{GH zx(ω)} −Re{GH yz(ω)}+Re{GH zy(ω)}/parenrightbig +/bracketleftbig Im{αm,⊥(ω−)} −Re{αm,g(ω−)}/bracketrightbig/parenleftbig Re{GH xz(ω)} −Re{GH zx(ω)}+Im{GH yz(ω)}+Im{GH zy(ω)}/parenrightbig/bracerightigg −[n0(ω) + 1]/braceleftigg −2Reαm,∥(ω)/parenleftbig Im{GH zx(ω)}+Im{GH xz(ω)}/parenrightbig + 2Imαm,∥(ω)/parenleftbig −Re{GH zx(ω)}+Re{GH xz(ω)}/parenrightbig/bracerightigg . (S25) In the case when the interface is in the x−yplane, we have Re/braceleftbig GH xz(ω)/bracerightbig = (πωρ 0/8)1 π/integraldisplay2π 0dϕ/braceleftigg/integraldisplay1 0κ2dκ p/parenleftbig Im/braceleftbig rsse2ik0pd/bracerightbig pcosϕ−Im/braceleftbig rpse2ik0pd/bracerightbig sinϕ/parenrightbig +/integraldisplay∞ 1κ2dκ |p|e−2k0|p|d(Im{rss}|p|cosϕ+Re{rps}sinϕ)/bracerightigg ,(S26a) Re/braceleftbig GH zx(ω)/bracerightbig = (πωρ 0/8)1 π/integraldisplay2π 0dϕ/braceleftigg/integraldisplay1 0κ2dκ p/parenleftbig −Im/braceleftbig rsse2ik0pd/bracerightbig pcosϕ−Im/braceleftbig rspe2ik0pd/bracerightbig sinϕ/parenrightbig +/integraldisplay∞ 1κ2dκ |p|e−2k0|p|d(−Im{rss}|p|cosϕ+Re{rsp}sinϕ)/bracerightigg ,(S26b) Im/braceleftbig GH xz(ω)/bracerightbig = (πωρ 0/8)1 π/integraldisplay2π 0dϕ/braceleftigg/integraldisplay1 0κ2dκ p/parenleftbig −Re/braceleftbig rsse2ik0pd/bracerightbig pcosϕ+Re/braceleftbig rpse2ik0pd/bracerightbig sinϕ/parenrightbig +/integraldisplay∞ 1κ2dκ |p|e−2k0|p|d(−Re{rss}|p|cosϕ+Im{rps}sinϕ)/bracerightigg ,(S26c) 4Im/braceleftbig GH zx(ω)/bracerightbig = (πωρ 0/8)1 π/integraldisplay2π 0dϕ/braceleftigg/integraldisplay1 0κ2dκ p/parenleftbig Re/braceleftbig rsse2ik0pd/bracerightbig pcosϕ+Re/braceleftbig rspe2ik0pd/bracerightbig sinϕ/parenrightbig +/integraldisplay∞ 1κ2dκ |p|e−2k0|p|d(Re{rss}|p|cosϕ+Im{rsp}sinϕ)/bracerightigg ,(S26d) and Re/braceleftbig GH zy(ω)/bracerightbig = (πωρ 0/8)1 π/integraldisplay2π 0dϕ/braceleftigg/integraldisplay1 0κ2dκ p/parenleftbig −Im/braceleftbig rsse2ik0pd/bracerightbig psinϕ+Im/braceleftbig rspe2ik0pd/bracerightbig cosϕ/parenrightbig +/integraldisplay∞ 1κ2dκ |p|e−2k0|p|d(−Im{rss}|p|sinϕ−Re{rsp}cosϕ)/bracerightigg ,(S27a) Re/braceleftbig GH yz(ω)/bracerightbig = (πωρ 0/8)1 π/integraldisplay2π 0dϕ/braceleftigg/integraldisplay1 0κ2dκ p/parenleftbig Im/braceleftbig rsse2ik0pd/bracerightbig psinϕ+Im/braceleftbig rpse2ik0pd/bracerightbig cosϕ/parenrightbig +/integraldisplay∞ 1κ2dκ |p|e−2k0|p|d(Im{rss}|p|sinϕ−Re{rps}cosϕ)/bracerightigg ,(S27b) Im/braceleftbig GH zy(ω)/bracerightbig = (πωρ 0/8)1 π/integraldisplay2π 0dϕ/braceleftigg/integraldisplay1 0κ2dκ p/parenleftbig Re/braceleftbig rsse2ik0pd/bracerightbig psinϕ−Re/braceleftbig rspe2ik0pd/bracerightbig cosϕ/parenrightbig +/integraldisplay∞ 1κ2dκ |p|e−2k0|p|d(Re{rss}|p|sinϕ−Im{rsp}cosϕ)/bracerightigg ,(S27c) Im/braceleftbig GH yz(ω)/bracerightbig = (πωρ 0/8)1 π/integraldisplay2π 0dϕ/braceleftigg/integraldisplay1 0κ2dκ p/parenleftbig −Re/braceleftbig rsse2ik0pd/bracerightbig psinϕ−Re/braceleftbig rpse2ik0pd/bracerightbig cosϕ/parenrightbig +/integraldisplay∞ 1κ2dκ |p|e−2k0|p|d(−Re{rss}|p|sinϕ−Im{rps}cosϕ)/bracerightigg .(S27d) And for the case when it is in the x−zplane Re/braceleftbig GH xz(ω)/bracerightbig = (πωρ 0/8)1 π/integraldisplay2π 0dϕ/braceleftigg/integraldisplay1 0κdκ 2p/bracketleftig sin 2ϕIm/braceleftbig rppe2ik0pd/bracerightbig +p2sin 2ϕIm/braceleftbig rsse2ik0pd/bracerightbig +2pIm/braceleftbig rpse2ik0pd/bracerightbig cos2ϕ+ 2pIm/braceleftbig rspe2ik0pd/bracerightbig sin2ϕ/bracketrightig +/integraldisplay∞ 1κdκ 2|p|/bracketleftig −sin 2ϕ/parenleftig 1 +Re{rpp}e−2k0|p|d/parenrightig −p2sin 2ϕRe{rss}e−2k0|p|d +2Im{rps}|p|e−2k0|p|dcos2ϕ+ 2Im{rspe−2k0|p|d}sin2ϕ/bracketrightig/bracerightigg , (S28a) Re/braceleftbig GH zx(ω)/bracerightbig = (πωρ 0/8)1 π/integraldisplay2π 0dϕ/braceleftigg/integraldisplay1 0κdκ 2p/bracketleftig sin 2ϕIm/braceleftbig rppe2ik0pd/bracerightbig +p2sin 2ϕIm/braceleftbig rsse2ik0pd/bracerightbig −2pIm/braceleftbig rpse2ik0pd/bracerightbig sin2ϕ−2pIm/braceleftbig rspe2ik0pd/bracerightbig cos2ϕ/bracketrightig +/integraldisplay∞ 1κdκ 2|p|/bracketleftig −sin 2ϕ/parenleftig 1 +Re{rpp}e−2k0|p|d/parenrightig −p2sin 2ϕRe{rss}e−2k0|p|d −2Im{rps}|p|e−2k0|p|dsin2ϕ−2Im{rsp}|p|e−2k0|p|dcos2ϕ/bracketrightig/bracerightigg , (S28b) 5Im/braceleftbig GH xz(ω)/bracerightbig = (πωρ 0/8)1 π/integraldisplay2π 0dϕ/braceleftigg/integraldisplay1 0κdκ 2p/bracketleftig −sin 2ϕ/parenleftbig 1 +Re/braceleftbig rppe2ik0pd/bracerightbig/parenrightbig −p2sin 2ϕRe/braceleftbig rsse2ik0pd/bracerightbig −2pRe/braceleftbig rpse2ik0pd/bracerightbig cos2ϕ−2pRe/braceleftbig rspe2ik0pd/bracerightbig sin2ϕ/bracketrightig +/integraldisplay∞ 1κdκ 2|p|e−2k0|p|d/bracketleftig −sin 2ϕIm{rpp} −p2sin 2ϕIm{rss} −2Re{rps}|p|cos2ϕ−2Re{rsp}|p|sin2ϕ/bracketrightig/bracerightigg , (S28c) Im/braceleftbig GH zx(ω)/bracerightbig = (πωρ 0/8)1 π/integraldisplay2π 0dϕ/braceleftigg/integraldisplay1 0κdκ 2p/bracketleftig −sin 2ϕ/parenleftbig 1 +Re/braceleftbig rppe2ik0pd/bracerightbig/parenrightbig −p2sin 2ϕRe/braceleftbig rsse2ik0pd/bracerightbig +2pRe/braceleftbig rpse2ik0pd/bracerightbig sin2ϕ+ 2pRe/braceleftbig rspe2ik0pd/bracerightbig cos2ϕ/bracketrightig +/integraldisplay∞ 1κdκ 2|p|e−2k0|p|d/bracketleftig −sin 2ϕIm{rpp} −p2sin 2ϕIm{rss} +2Re{rps}|p|sin2ϕ+ 2Re{rsp}|p|cos2ϕ/bracketrightig/bracerightigg , (S28d) and the expressions for the real and imaginary parts of GH zyandGH yzare the same as the ones for GH xzandGH zx, respectively, for when the interface is in the x−yplane as given in Eq. (S26). We can find the xandycomponents of torque by plugging these expressions into Eqs. (S24) and (S25) for the two cases when the interface is the x−yor x−zplane. We present the plots of these torques at the end of this section. S2.2 Recoil torque There is also another contribution to the torque from the case when the induced dipole moments on the YIG sphere re-radiate due to the vacuum electric field fluctuations. This causes a recoil torque on the sphere and can be written as Mrec=⟨mind×Hsc⟩, (S29) where Hscis the scattered fields from the dipole and are given by, Hsc(r, ω) =¯GH(r,r′, ω)·mind(r′, ω), (S30) which shows that this term is of higher order contribution and is thus smaller than the torque discussed in the main text. Repeating a similar procedure used before and plugging in all of the induced terms and writing them in terms of the fluctuations, we find after some algebra, Mrec z=ℏ π/integraldisplay∞ −∞dω[n0(ω)+1] /braceleftigg Im{Gxx}/bracketleftbig Re{Gyx}αeff ⊥⊥−Re{Gxy}αeff gg+Re{α⊥g}Re{Gyy−Gxx}+Im{α⊥g}Im{Gyy+Gxx}/bracketrightbig +Im{Gyy}/bracketleftbig Re{Gyx}αeff gg−Re{Gxy}αeff ⊥⊥+Re{α⊥g}Re{Gxx−Gyy}+Im{α⊥g}Im{Gyy+Gxx}/bracketrightbig +Re{Gyx−Gxy}/bracketleftbigg Re{Gyx−Gxy}Im{α⊥g}+1 2Im{Gyy+Gxx}(α⊥⊥+αgg)/bracketrightbigg +Im{Gyx+Gxy}/bracketleftbigg −Re{Gyx+Gxy}Re{α⊥g}+1 2Re{Gxx−Gyy}(αgg−α⊥⊥)/bracketrightbigg +1 2Im/braceleftig αeff∗ m,∥/bracketleftbig (Gxz−G∗ zx)/parenleftbig G∗ yzαeff m,⊥−G∗ xzαeff m,g/parenrightbig −/parenleftbig Gyz−G∗ zy/parenrightbig/parenleftbig G∗ yzαeff m,g+G∗ xzαeff m,⊥/parenrightbig/bracketrightbig/bracerightig/bracerightigg ,(S31) 6where we have defined αeff m,⊥⊥(ω) =αeff m,⊥(ω)αeff m,⊥(−ω), αeff m,gg(ω) =αeff m,g(ω)αeff m,g(−ω), αeff m,⊥g(ω) =αeff m,⊥(ω)αeff m,g(−ω), αeff m,g⊥(ω) =αeff m,⊥(−ω)αeff m,g(ω),(S32) and have used the facts that αeff m,⊥⊥(ω)andαeff m,gg(ω)are real, and αeff m,⊥g(ω) =/bracketleftig αeff m,g⊥(ω)/bracketrightig∗ . Note that we have dropped the frequency dependence as well as the H superscript of the Green’s function in Eq. (S31) for simplicity. For the special case when the substrate material is isotropic, the non-diagonal elements of the Green’s function become zero, and we get Mrec z=ℏ π/integraldisplay∞ −∞dω[n0(ω) + 1]/braceleftigg Im{Gxx−Gyy}Re{Gyy−Gxx}Re{α⊥g}+[Im{Gxx+Gyy}]2Im{α⊥g}/bracerightigg .(S33) Note that the expressions for the real and imaginary parts of GxzandGyzare given by Eqs. (S26),(S27), and (S28) for the two possible interface directions while the imaginary parts of GxxandGyyare given by equations in Appendix B. Also note that Re/braceleftbig GH yx/bracerightbig for when the interface is the x−yplane is the same as Re/braceleftbig GH xz/bracerightbig for when the interface is in thex−zplane given by Eq. (S28). Also Re/braceleftbig GH yx/bracerightbig for when the interface is the x−zplane is the same as Re/braceleftbig GH zy/bracerightbig for when the interface is in the x−yplane given by Eq. (S27). Thus, the only new term is Re {Gyy−Gxx}which is given by Re/braceleftbig GH yy(ω)−GH xx(ω)/bracerightbig =(πωρ 0/8)1 π/integraldisplay2π 0dϕ/braceleftigg/integraldisplay1 0κdκ p/bracketleftig −cos 2ϕIm/braceleftbig rppe2ik0pd/bracerightbig −cos 2ϕIm/braceleftbig rsse2ik0pd/bracerightbig +2psinϕcosϕIm/braceleftbig (rps−rsp)e2ik0pd/bracerightbig/bracketrightig +/integraldisplay∞ 1κdκ |p|/bracketleftig cos 2ϕ/parenleftig κ2+Re{rpp}e−2k0|p|d/parenrightig +p2cos 2ϕRe{rss}e−2k0|p|d +2|p|sinϕcosϕIm{rps−rsp}e−2k0|p|d/bracketrightig/bracerightigg , (S34) when the interface is the x−yplane, and Re/braceleftbig GH yy(ω)−GH xx(ω)/bracerightbig =(πωρ 0/8)1 π/integraldisplay2π 0dϕ/braceleftigg/integraldisplay1 0κdκ p/bracketleftig cos2ϕIm/braceleftbig rppe2ik0pd/bracerightbig −/parenleftbig κ2+p2sin2ϕ/parenrightbig Im/braceleftbig rsse2ik0pd/bracerightbig −psinϕcosϕIm/braceleftbig (rps−rps)e2ik0pd/bracerightbig/bracketrightig +/integraldisplay∞ 1κdκ |p|/bracketleftig −cos2ϕ/parenleftig 1 +Re{rpp}e−2k0|p|d/parenrightig −p2sin2ϕ+κ2+/parenleftbig κ2+p2sin2ϕ/parenrightbig Re{rss}e−2k0|p|d −|p|sinϕcosϕIm{rps−rsp}e−2k0|p|d/bracketrightig , (S35) when the interface is the x−zplane. S2.3 Plots of torque terms In this section, we present the components of torque derived in previous sections for YIG slabs with various bias magnetic fields and for the two cases when the slab is the x−yandx−zplanes. Figure S2 shows the plots of Mx,My,Mz, and Mrecderived in the previous sections for the magnetic and electric fluctuations. The expressions for the torques due to the electric fields and dipoles fluctuations are found by changing s topandptosinrss, rpp, rsp, andrps, in the expressions for the Green’s functions. Moreover, magnetic polarizability is replaced by a simple isotropic electric polarizability, assuming a simple dielectric polarizability scalar for the YIG and Al interfaces. The results are for three directions of the bias magnetic field for the YIG interface labeled as x−, y−, and z−bias. The meaning of these bias directions is demonstrated in Fig. S1 when the YIG slab is the x−yand x−zplanes. 7YIG YIG yz x x-bias(a) YIG YIG yz x y-bias (b) YIG YIG yz x z-bias (c) YIG YIG yz x x-bias (d) YIG YIG yz x y-bias (e) YIG YIG yz x z-bias (f) Figure S1: Schematics of different bias directions for the YIG interface for the two cases of the interface being the x−y(top row) and x−zplanes (bottom row). The green arrow shows the direction of the bias magnetic field applied to the slab of YIG. It is interesting to note that in Figs. (S2a), (S2e), and (S2g), the sphere can experience a large value of torque along xorydirections for the x−ory−biases. This means that in these cases, the sphere can rotate out of the rotation axis and start to precess. This will, of course, change the validity of the equations found for the vacuum radiation and frictional torque along the zaxis since it has been assumed that the sphere is always rotating around the zaxis and is also magnetized along that axis. This torque is still small enough compared to the driving torque of the trapping laser and it will still give enough time to make the observations. A more careful investigation of these components of torque is out of the scope of this study and will be explored in the future. Figures S2i-S2p show the axial torque Mzas well as the recoil torque Mrecfor all orientations of the bias magnetic field and YIG slab. As expected, the recoil torque is much smaller than Mzsince it is a second-order term. Figure S3 shows the results for MzandMrecfor the case when the Al interface is placed in the vicinity of the spinning sphere. Because Al is an isotropic material, MxandMyvanish for both orientations of the interface and thus are not included in the plots of the torques. Note that similar to the YIG interface results, Mrecis much smaller than the Mz for all cases of the Al interface. These results show that the recoil torque Mreccan be ignored in all studied cases. S3 Experimental Considerations In this section, we present details of the experimental analysis regarding negative torque and shot noise heating due to the trapping laser discussed in Appendix H. S3.1 Effect of torque due to the trapping laser When the trapping laser is linearly polarized, it can exert a negative torque on the spinning particle. The torque on the sphere due to the laser is given by Mopt=1 2Re{p∗×E}[1], where pis the dipole moment of the sphere, given by p=¯αeff·E, with ¯αeffbeing the effective polarizability of the sphere as seen in the frame of the lab, and Eis the electric field from the laser. As shown in Appendix A, the polarizability tensor of the sphere when it is spinning in the x−yplane is given by ¯αeff(ω) = αeff ⊥(ω)−αeff g(ω) 0 αeff g(ω)αeff ⊥(ω) 0 0 0 αeff ∥(ω) , (S36) where αeff ⊥(ω) =1 2/bracketleftbig α(ω+) +α(ω−)/bracketrightbig , α g(ω) =−i 2/bracketleftbig α(ω+)−α(ω−)/bracketrightbig , α∥(ω) =α(ω), (S37) withα(ω)being the electric polarizability of YIG at the laser frequency. Note that here, we have assumed that the polarizability of the YIG is scalar in the range of frequencies around 1550 nm. Plugging these into the equation for 80 2 4 6 8 10 Frequency (GHz)-15-10-5051015YIG Magnetic Torque Mx - Z bias My - Z bias Mx - X bias My - X bias Mx - Y bias My - Y bias(a) 0 2 4 6 8 10 Frequency (GHz)-0.500.511.522.510-4YIG Electric Torque Mx - Z bias My - Z bias Mx - X bias My - X bias Mx - Y bias My - Y bias (b) 0 2 4 6 8 10 Frequency (GHz)-6-4-2024610-6Al Magnetic Torque Mx - Z bias My - Z bias Mx - X bias My - X bias Mx - Y bias My - Y bias (c) 0 2 4 6 8 10 Frequency (GHz)-505101510-12Al Electric Torque Mx - Z bias My - Z bias Mx - X bias My - X bias Mx - Y bias My - Y bias (d) 0 2 4 6 8 10 Frequency (GHz)-2024681012YIG Magnetic Torque Mx - Z bias My - Z bias Mx - X bias My - X bias Mx - Y bias My - Y bias (e) 0 2 4 6 8 10 Frequency (GHz)-2.5-2-1.5-1-0.500.510-4YIG Electric Torque Mx - Z bias My - Z bias Mx - X bias My - X bias Mx - Y bias My - Y bias (f) 0 2 4 6 8 10 Frequency (GHz)02468Al Magnetic Torque Mx - Z bias My - Z bias Mx - X bias My - X bias Mx - Y bias My - Y bias (g) 0 2 4 6 8 10 Frequency (GHz)-15-10-50510-12Al Electric Torque Mx - Z bias My - Z bias Mx - X bias My - X bias Mx - Y bias My - Y bias (h) 0 2 4 6 8 10 Frequency (GHz)-12-10-8-6-4-20YIG Magnetic Torque Mz - Z bias Mrec - Z bias Mz - X bias Mrec - X bias Mz - Y bias Mrec - Y bias (i) 0 2 4 6 8 10 Frequency (GHz)-20-15-10-5010-4YIG Electric Torque Mz - Z bias Mrec - Z bias Mz - X bias Mrec - X bias Mz - Y bias Mrec - Y bias (j) 0 2 4 6 8 10 Frequency (GHz)-0.2500.05Al Magnetic Torque Mz - Z bias Mrec - Z bias Mz - X bias Mrec - X bias Mz - Y bias Mrec - Y bias (k) 0 2 4 6 8 10 Frequency (GHz)-3-2-101210-7Al Eelctric Torque Mz - Z bias Mrec - Z bias Mz - X bias Mrec - X bias Mz - Y bias Mrec - Y bias (l) 0 2 4 6 8 10 Frequency (GHz)-15-10-50YIG Magnetic Torque Mz - Z bias Mrec - Z bias Mz - X bias Mrec - X bias Mz - Y bias Mrec - Y bias (m) 0 2 4 6 8 10 Frequency (GHz)-20-15-10-5010-3 YIG Electric Torque Mz - Z bias Mrec - Z bias Mz - X bias Mrec - X bias Mz - Y bias Mrec - Y bias (n) 0 2 4 6 8 10 Frequency (GHz)-0.30Al Magnetic Torque Mz - Z bias Mrec - Z bias Mz - X bias Mrec - X bias Mz - Y bias Mrec - Y bias (o) 0 2 4 6 8 10 Frequency (GHz)-4-2024681010-5 Al Electric Torque Mz - Z bias Mrec - Z bias Mz - X bias Mrec - X bias Mz - Y bias Mrec - Y bias (p) Figure S2: Plots of MxandMy(first two rows) and MzandMrec(second two rows) in the vicinity of the YIG slab when the slab is the x−yplane (first and third rows), and when the slab is x−zplane (second and fourth rows). The plots show the results for various magnetic field directions. The meanings of x−,y−, andz−bias are demonstrated in Fig. S1 for the two orientations of the interface. the exerted torque, we find the zcomponent of the torque Mopt=1 2Re/braceleftbig αeff∗ ⊥(ω)E∗ xEy−αeff∗ g(ω)E∗ yEy−αeff∗ g(ω)E∗ xEx−αeff∗ ⊥(ω)E∗ yEx/bracerightbig =1 2/bracketleftbig Im{αeff ⊥(ω)}Im{E∗×E} −Re{αeff g(ω)}/parenleftbig |Ex|2+|Ey|2/parenrightbig/bracketrightbig =1 2/bracketleftbig Im{α(ω+) +α(ω−)}Im{E∗×E} −Im{α(ω+)−α(ω−)}/parenleftbig |Ex|2+|Ey|2/parenrightbig/bracketrightbig .(S38) 90 2 4 6 8 10 Frequency (GHz)-3.5-3-2.5-2-1.5-1-0.50YIG Magnetic Torque Mz - local Mrec - local Mz - nonlocal Mrec - nonlocal(a) 0 2 4 6 8 10 Frequency (GHz)-1.4-1.2-1-0.8-0.6-0.4-0.2010-10 YIG Electric Torque Mz - local Mrec - local Mz - nonlocal Mrec - nonlocal (b) 0 2 4 6 8 10 Frequency (GHz)-0.150Al Magnetic Torque Mz - local Mrec - local Mz - nonlocal Mrec - nonlocal (c) 0 2 4 6 8 10 Frequency (GHz)-6-5-4-3-2-10110-16Al Electric Torque Mz - local Mrec - local Mz - nonlocal Mrec - nonlocal (d) 0 2 4 6 8 10 Frequency (GHz)-5-4-3-2-10YIG Magnetic Torque Mz - local Mrec - local Mz - nonlocal Mrec - nonlocal (e) 0 2 4 6 8 10 Frequency (GHz)-20-15-10-5010-11 YIG Electric Torque Mz - local Mrec - local Mz - nonlocal Mrec - nonlocal (f) 0 2 4 6 8 10 Frequency (GHz)-0.2500.05Al Magnetic Torque Mz - local Mrec - local Mz -nonlocal Mrec - nonlocal (g) 0 2 4 6 8 10 Frequency (GHz)-8-6-4-2010-16 Al Electric Torque Mz - local Mrec - local Mz - nonlocal Mrec - nonlocal (h) Figure S3: Plots of MzandMrecin the vicinity of the YIG slab when the slab is the x−yplane (first row) and when the slab is x−zplane (second row). Note that due to the isotropy of Al, the other components of torque, including MxandMy, vanish. The first term is proportional to the spin of the electromagnetic field and causes a positive torque on the particle. This is the term for the transferring of angular momentum from the laser to the particle. The second term is negative and thus causes a negative torque on the sphere. In the case when the laser is linearly polarized, this negative term is proportional to Im {α(ω0+ Ω)} −Im{α(ω0−Ω)}where ω0= 1.21×1016is the frequency of the laser, and Ω = 6 .28×109is the rotation frequency. Since Ω≪ω0, we get α(ω+)≃α(ω−)and thus the second term is negligible. We can thus ignore the negative torque coming from the laser when the laser is linearly polarized. S3.2 Effect of heating due to the shot noise The particle can heat up due to the shot noise of the trapping laser [4]. In this section, we calculate the rate of temper- ature change due to the shot noise and vacuum radiation, respectively. The rate of energy change in the nanosphere due to the shot noise is [4], ˙ETR=ℏk MIL cσ, (S39) where ωL=ckis the laser frequency, ILis the power of the laser per unit area, Mis the mass of the particle, andσis the cross section of scattering where, which is equal to σ=/parenleftbig8π 3/parenrightbig/parenleftig αk2 4πϵ/parenrightig2 for Rayleigh particles with the polarizability α= 4πϵ0a3/parenleftig ϵ−1 ϵ+2/parenrightig . For the range of wavelengths around visible and infrared, the Rayleigh limit is valid for particles of radii asmaller than 50nm. Since the radius of the particle in our case is 200nm, this expression may not be valid. Mie scattering parameters should be used to evaluate the scattering cross section. Assuming the trapping laser wavelength of λ= 1550 nm and using the Mie theory, the rate of energy change of YIG with refractive index n= 2.21[3] is close to that of the diamond with n = 2.39 in the Rayleigh limit [4]. Therefore, we get the energy change rate in the sphere ˙ETR=2ℏω0 ρc2AP0a3k4/parenleftbiggn2−1 n2+ 2/parenrightbigg2 , (S40) where A=πR2 Lis the area of the beam where the laser with the power P0is focused on, and ρis the mass density which for YIG is ρ= 5110 kg/m3. For a laser power of 500mW focused on an area of radius 0.7566µm, we find ˙TL= 15.45K/s. (S41) This is a very small temperature change compared to the time scale of the rotation, which is 1ns. Therefore, the thermodynamic equilibrium condition for the FDT is still valid. This temperature change gets damped by the radiated 10power of the sphere due to the rotation. For a YIG sphere spinning at about 0.5µm from the aluminum interface, the rate of change due to vacuum radiation at the equilibrium temperature T0= 300 K is, ˙TR=−362.973K/s, (S42) which is much larger than the temperature rise due to the shot noise of the laser, and this shows that the sphere will cool down. Note that this energy heats the aluminum instead. In this derivation, we have not included the heating due to the noise in the aluminum or YIG interface. The value found in Eq. (S42) is much smaller at lower temperatures. References [1] J. Ahn, Z. Xu, J. Bang, Y .-H. Deng, T. M. Hoang, Q. Han, R.-M. Ma, and T. Li. Optically levitated nanodumbbell torsion balance and ghz nanomechanical rotor. Physical review letters , 121(3):033603, 2018. [2] C. F. Bohren and D. R. Huffman. Absorption and scattering of light by small particles . John Wiley & Sons, 2008. [3] T. Seberson, P. Ju, J. Ahn, J. Bang, T. Li, and F. Robicheaux. Simulation of sympathetic cooling an optically levitated magnetic nanoparticle via coupling to a cold atomic gas. J. Opt. Soc. Am. B , 37(12):3714–3720, Dec 2020. [4] T. Seberson and F. Robicheaux. Distribution of laser shot-noise energy delivered to a levitated nanoparticle. Phys. Rev. A , 102:033505, Sep 2020. 11

2024-01-17

Experimental observations of vacuum radiation and vacuum frictional torque are challenging due to their vanishingly small effects in practical systems. For example, a rotating nanosphere in free space slows down due to friction from vacuum fluctuations with a stopping time around the age of the universe. Here, we show that a spinning yttrium iron garnet (YIG) nanosphere near aluminum or YIG slabs exhibits vacuum radiation eight orders of magnitude larger than other metallic or dielectric spinning nanospheres. We achieve this giant enhancement by exploiting the large near-field magnetic local density of states in YIG systems, which occurs in the low-frequency GHz regime comparable to the rotation frequency. Furthermore, we propose a realistic experimental setup for observing the effects of this large vacuum radiation and frictional torque under experimentally accessible conditions.

Giant Enhancement of Vacuum Friction in Spinning YIG Nanospheres

2401.09563v1

arXiv:2009.04162v1 [cond-mat.mes-hall] 9 Sep 2020Sub-pico-liter magneto-optical cavities J. A. Haigh,1,∗R. A. Chakalov,2and A. J. Ramsay1 1Hitachi Cambridge Laboratory, Cambridge, CB3 0HE, United K ingdom 2Cavendish Laboratory, University of Cambridge, Cambridge , CB3 0HE, United Kingdom (Dated: September 10, 2020) Microwave-to-optical conversion via ferromagnetic magno ns has so-far been limited by the optical coupling rates achieved in mm-scale whispering gallery mode resonat ors. Towards overcoming this limitation, we pro- pose and demonstrate an open magneto-optical cavity contai ning a thin-film of yttrium iron garnet (YIG). We achieve a 0.1 pL (100 µm3) optical mode volume, ∼50 times smaller than previous devices. From this, we estimate the magnon single-photon coupling rate is G≈50Hz. This open cavity design offers the prospect of wavelength scale mode volumes, small polarization splitti ngs, and good magneto-optical mode overlap. With achievable further improvements and optimization, efficie nt microwave-optical conversion and magnon cooling devices become a realistic possibility. I. INTRODUCTION Magnetic-field tunable ferromagnetic modes can be easily strongly coupled to microwave resonators [ 1–3]. Further cou- pling to optical photons offers the prospect of useful trans duc- tion of microwave quantum signals to telecoms optical wave- lengths [ 4]. For this reason, the interaction of magnons and optical photons has been explored recently in the whisperin g gallery modes (WGM) of yttrium iron garnet (YIG) spheres [5–7]. However, despite the high Q-factor of the magnetic and optical modes [ 8], the optomagnonic coupling rates achieved in mm-scale YIG spheres have been limited to ∼1 Hz. If the coupling rate can be increased significantly, in turn raisin g the conversion efficiency, this would open a wide range of tech- nological opportunities [ 9], as well as the ability to coherently modify the magnetization dynamics, for example cooling or dynamical driving the magnon mode [ 10,11]. The low coupling rate for optical whispering gallery modes is due to the poor mode overlap and the large volume of mag- netic material involved. To overcome the poor overlap, it ma y be possible to exploit magnon whispering gallery modes in YIG spheres, with almost ideal overlap with the optical WGM [12]. A simpler strategy is to explore more compact struc- tures, as very recently shown in rib waveguide devices [ 13]. In that case, the mm-long structure confines both the magnons and photons, yielding excellent overlap inside the structu re, enabling a coupling rate of 17 Hz. The estimated maximum coupling rate for a YIG optical resonator is ≈0.1MHz [ 14], based on a mode volume of the order of the resonant wavelength cubed λ3. There are sev- eral candidate wavelength-scale optical resonators [ 15], which could get close to this maximum coupling rate. The choice of resonator, however, must take into account the significan t challenges of micro-patterning YIG [ 16]. We note that, while sub-wavelength mode confinement is possible with plasmonic devices [ 17–19], this typically comes with high optical losses in the metal components. A simple optical resonator design, with wavelength scale mode volumes combined with large Q-factors, is an open mi- ∗jh877@cam.ac.ukcrocavity [20]. These are typically hemispherical resonators where a reflection-coated microlens is positioned in close proximity to a mirror surface. Devices can be fiber-based [21,22], or fabricated on planar surfaces [ 23], and have previ- ously been used to obtain large coupling rates to single atom s [24], N-V centers [ 25], single organic dye molecules [ 26], and excitons in 2D materials [ 27]. The advantage of this structure is that any transferable material can be easily embedded [ 28], and the modes are tunable by the position of the lens. Optical mode volumes as small as 1 fL (1 µm3) have been achieved, withQ-factors in excess of 10,000 [ 29]. In this article, we demonstrate a viable route to low mode volume, high coupling rate magneto-optical cavities with mode volumes limited by the optical wavelength. We embed single crystal YIG layers in an open microcavity, and show a two orders of magnitude increase in the coupling rate over whispering gallery mode devices. With further reduction in mode volume, it is expected that the strong-coupling limite d can be reached. This work, therefore, shows a path towards efficient microwave-optical conversion, and optical magno n cooling. II. COUPLING RATE We first briefly review the enhanced scattering process in cavity optomagnonics. Magnetic Brillouin light scatterin g is an inelastic process where a photon is scattered from an input mode ˆaiinto an output mode ˆao, with absorption or emission of a magnon in mode ˆm. Brillouin light scatter- ing is most efficient between orthogonally polarized opti- cal modes, as this compensates the angular momentum lost or gained to the magnon mode, conserving angular momen- tum. To enhance the BLS significantly, we require two op- tical resonances, enhancing both the input and output optic al fields. These should be orthogonally polarized, and with fre - quency separation matching the magnon frequency. This is thetriple resonance condition , which has been observed pre- viously in whispering gallery mode resonators [ 7], and in a recent waveguide device [ 13]. The interaction Hamiltonian that governs the scattering is of the formHint=G(ˆa† iˆaoˆm+ˆaiˆa† oˆm), with interaction strength2 quantified by the coupling rate, G=−iθfc n/radicalbigg 4gµB Ms/radicalbiggηmagηopt Vopt. (1) The numerical constant µBis the Bohr magneton and cthe speed of light. The factors affecting this rate can be separa ted in two parts. Firstly, the materials parameters of the embed - ded magnetic material: the Faraday coefficient θf, refractive indexn, gyromagnetic ratio gand saturation magnetization Ms. These parameters can be optimized by materials devel- opment, finding new materials and improving the quality of those available. Secondly, the geometry of the optical cavi ty affects the coupling rate through the volumes of the optical modesVi≈Vo=Vopt=/integraltext |ui,o(r)|2, whereui,o(r)is the mode function with normalization max(|ui,o(r)|2) = 1 . The overlap is contained in the fill-factors ηmag=Vint/Vmagand ηopt=Vint/Vopt, which are the proportion of the magnetic Vmagand optical Voptmodes volumes that contribute to the coupling through the triple-mode overlap, Vint=/integraldisplay drum(r)·[u∗ i(r)×uo(r)]. (2) Here, the mode function umis also normalized such that max(|um(r)|2) = 1 . This expression includes the effect of the mode polarization. To maximize the geometric factors we would like a low optical mode volume resonator, with excel- lent overlap with the magnon mode, and orthogonal polariza- tion of all three modes. In this paper, we focus exclusively on the minimization of the optical mode volume. The design is such that lateral pat- terning of the continuous YIG layer to confine the magnon mode can incorporated at a later stage. III. DESIGN A schematic of our proposed device is shown in Fig. 1(a). The mirrors forming the open microcavities are purchased from Oxford HiQ [ 30]. They consist of one planar surface and one microlens array. Both surfaces have a high qual- ity reflective coating consisting of 22 layers of SiO 2and Ta2O5, deposited by sputtering. This distributed Bragg reflec- tor (DBR) has a reflectivity of 99.8 %at its design wavelength of 1300 nm. The microlens array contains 16 lenses with four different radii of curvature, from 100 µm down to 20 µm, fab- ricated by focused-ion-beam milling [ 31]. The lens array is situated on a raised pedestal to allow alignment of the two mirrors. To embed a high quality, single crystal YIG layer in the mi- crocavity, we avoid deposition techniques such as pulsed la ser deposition and sputtering, because the non-lattice matche d DBR substrate would lead to poly-crystalline YIG growth [32]. Furthermore, post growth annealing to improve the crys- tallinity of those layers requires temperatures above 700◦C, which has been found to be detrimental to the DBR [ 33]. In- stead, we use a lift-off technique [ 34] to remove a layer of Au antennalens array crack50
ma) YIG/GGG layer Au antenna DBRGGG (7m) YIG (2m) BCB (~1 m) DBR sapphire substrate lens array DBRsYIG/GGG c) substrate DBR magnet material b)i) ii) iii)~1m Au antenna Figure 1. (a) Schematic of the magneto-optical cavity desig n. The open microcavity consists of two parts: a concave lens mille d into a substrate and coated with a DBR, and a planar mirror with magn etic layer. Left: Cross section of design to show relative dimens ions of the lens and beam waist. Right: 3D representation of the devi ce structure. (b)Fabrication of magneto-optical microcavit y. (i) Flat side of open microcavity. A gold microwave antenna is patter ned on the DBR surface before the YIG/GGG layer is bonded. (ii) Cros s section of flat mirror, showing two-layer BCB bonding polyme r. (iii) Cross-section section of open microcavity, showing microl ens array. (c) Optical image through cavity structure, showing lens ar ray and microwave antenna. single-crystal YIG from a lattice matched GGG growth sub- strate. We later bond this layer to the mirror surface with a spin-on polymer. The advantage of the open microcavity design is that the polarization splitting is minimized due to the cylindrical sym- metry. In principle, a minor asymmetry can tune the splittin g to match the magnon frequency, given the precise control of lens profile that has been demonstrated [ 31]. For an optical resonator with an asymmetrical cross-section, the splitti ng is typically a fixed fraction of the free spectral range. There- fore, as the mode volume shrinks, the frequency separation can become too large. This can be seen in the whispering gallery mode resonators, where a 1 mm diameter YIG sphere was chosen to match the splitting to the magnon frequency [7]. It is not possible to decrease the size of the sphere, be- cause the splitting would become too large. This effect can also be seen in the rib waveguide geometry [ 13], where the length of the cavity must be long ( ≈4mm) to keep the polar- ization splitting small.3 IV . FABRICATION We start with a YIG film of thickness 2 µm grown by liq- uid phase epitaxy on a gadolinium gallium garnet (GGG) sub- strate [ 35]. The lift-off is achieved by inducing spontaneous delamination as follows. The sample is first subjected to a high dose ion implantation 5×1016cm−2with He ions at 3.5 MeV [ 36]. The penetration depth is approximately 9 µm, with straggle ≈1µm, creating a narrow layer in which the lat- tice is substantially damaged [ 37]. Annealing at 470◦C for 1 min leads to delamination of a bilayer consisting of the 2 µm of YIG and around 7 µm of GGG . This delamination occurs due to the slight lattice mismatch and different coefficient of expansion of YIG and GGG [ 38]. The lattice mismatch leads to a membrane with ∼10-mm radius of curvature at room tem- perature. The YIG wafer is diced into 1 mm square chips post im- plantation, but prior to delamination. Post delamination, the thin membrane is manipulated using a small piece of 25 µm thick Kapton film, where it is held in place by static. To bond the YIG/GGG membrane to the mirror surface, we use BCB cyclotene [ 39], a polymer used as a dielectric in mi- croelectronics, adhesive wafer bonding [ 40] and planarization applications [ 41]. It has excellent optical properties [ 42], al- lowing its use, for example, in bonding active III-V devices to silicon photonic wafers [ 43]. Prior to bonding, a strip-line antenna is patterned on to the surface of the mirror using photo-lithography and lift-off as shown in Fig. 1(b). A titanium adhesion layer of 7 nm is de- posited under 100 nm of gold, with a final 7 nm of titanium above. This final layer is required to avoid the poor adhesion of BCB cyclotene to gold [ 44]. We prepare the mirror surface with solvent cleaning in an ultrasonic bath. The device is then soaked in DI water, be- fore a 2 min plasma cleaning process in a reactive ion etcher. This is followed by a further 2 min soak in DI water. The sur- face is primed with an adhesion promoter AP3000 [ 44]. The BCB cyclotene is deposited and spun for 30s at 6000 RPM, followed by 1 min on a hot plate at 150◦C to remove the sol- vent. We use a double layer of BCB cyclotene [ 43]. The first layer is partially cured with a 2 min anneal at 250◦C on a strip annealer. This layer remains ‘tacky’ and bonds well to a second layer of BCB cyclotene, but is viscous enough to pre- vent pinch-through under the membrane during curing, where there can be significant re-flow of the polymer [ 43]. The sec- ond layer of BCB is spun under the same conditions. Separately, the YIG/GGG layer is prepared with a 2 min plasma ash to activate the surface, before a 12 hr evaporatio n of AP3000 is performed in a desiccator. The membrane is removed from the desiccator immediately prior to bonding. The bonding is performed in a simple spring-loaded clamp. The Kapton tape bearing the YIG/GGG membrane is placed on the mirror, with the YIG layer in contact with the BCB cyclotene. The clamp is closed to the point where the layer is held in place with minimal pressure and then heated on a hot plate to 150◦C. The pressure is then increased to the required load. The assembly is then transferred to an oven at 150◦C under nitrogen flow to prevent oxidation of the BCB cyclotene linear polarizerpol. beam splitterhalf-wave platephoto- diode electromagnetxyz tilt/roll obj. lenshalf-wave platephoto- diodes vector network analyserport 1port 2MW amps.tunable laser obj. lens H0 Figure 2. Experimental setup. The output polarization of a 1 270- 1370 nm tunable laser is controlled via a linear polarizer an d a half- wave plate, before being separated into a local oscillator a nd cavity drive. After passing through the cavity, the optical signal orthogonal to the input polarization is recombined with the local oscil lator and measured on a high frequency photodiode. The transmission t hrough the cavity is measured via the light with the same polarizati on as the input on a dc photodiode. A vector network analyzer drive s the magnetic modes and measures the microwave signal from the fa st photodiode. at elevated temperatures. The oven temperature is ramped to 250◦C at 1◦/min, for a 1 hr soak. After allowing the oven to cool to room temperature, the clamp is removed and the Kapton film peeled from the mirror surface. This leaves the YIG/GGG layer secured to the device. During the bonding process, there is some re-flow of the BCB cyclotene to the top surface of the YIG/GGG membrane. To remove this, a 3 min Ar/CF 4reactive ion etch descum is performed. V . EXPERIMENTAL SETUP For measurement, the planar mirror is glued over an aper- ture on a PCB patterned with input and output coplanar waveguides, which are connected to semi-rigid coaxial cabl es. The on-chip strip-line antenna is then wire-bonded to the PC B waveguides for microwave measurement and excitation of the magnon modes in the YIG. The PCB is mounted on a circu- lar stub which sits in an xyz-translation lens mount. The lens array is similarly mounted on a circular stub in a tilt-yaw le ns holder, for full control of the cavity geometry. The device is mounted in an electromagnet, with magnetic field applied orthogonal to the cavity length. Light is focus ed into and out-of the cavity using two aspheric lenses mounted onxyz stages. The cavity is selected by scanning the laser to the correct position. The input laser is an external cavit y diode laser with linewidth ≈1 MHz. The input polarization is set with a rotatable Glan-Thompson prism. On the output, a rotatable half-wave plate is used to select the measuremen t basis on a polarizing beam splitter. From the beam splitter, the transmitted signal with the same polarization as the inp ut light field is measured with a dc photodiode. The polariza- tion scattered light is focused into a single mode fiber, and combined with a local oscillator directly from the laser in a 50:50 fiber coupler. One output of this coupler is measured on a fast photodiode (12 GHz bandwidth) connected via a mi- crowave amplifier to a vector network analyzer (VNA). The VNA is also used to drive the magnetization dynamics via the4 1280 1300 1320 1340 13600.00.10.20.30.40.50.6transmitted intensity (arb. units) input laser wavelength (nm)transmitted intensity (arb.units) -100 -50 0 50 1000.00.10.20.30.40.5 laser detuning (GHz) a) b) c) -50 -25 0 25 50 laser detuning (GHz)0.00.10.20.30.4FSR Figure 3. Transmission spectroscopy of optical modes. (a) W ide wavelength scan, showing free spectral range. Insets show m ode profile imaged in transmission. (b) Measurement of polariza tion of modes. The linear polarization can be set so that only one mod e is excited. This device had a smaller polarization splitting ≈16 GHz. (c) Measurement of polarization splitting and optical line width of device used in BLS measurements. The linear polarization is set so that both modes are probed. This device is also measured in (a ). microwave antenna. VI. CHARACTERIZATION We first characterize the optical modes of the microcavi- ties. The transmitted intensity is measured as a function of input laser wavelength, and angle of input linear polarizat ion, as shown in Fig. 3. A measurement over a wide wavelength range (Fig. 3(a)) is used to determine the free spectral range ∆ωFSR/2π≈6.7THz. A number of spatial modes result- ing from the lateral confinement of the microlens are visible . These can be identified by imaging in transmission, see inset s of Fig. 3(a). The coupling to these higher order modes is min- imized by optimizing the transmitted intensity on resonanc e through the lowest order mode. By measuring the transmitted intensity as a function of the angle of linear polarization, we can find the axes of the ortho g- onal, linearly polarized modes, and the splitting between t he two. An example of this measurement is shown in Fig. 3(b), where the polarization splitting is 16 GHz. This splitting varies with different lens arrays, and is related to slight a sym- metries in the nominally-cylindrical fabricated lens. In t he device used for BLS measurements shown in this paper, the splitting is 32 GHz, as shown in Fig. 3(c). Because the ap- plied magnetic field from the electromagnet is limited to <1 T, we are unable to reach the triple resonance condition. We not e that the frequency splitting due to the magnetic linear bire frin- gence in YIG [ 45] is estimated as ∼900 MHz. This is not largeenough to explain the observed splittings. We note that by fa b- ricating arrays of lenses with varying ellipticity, it woul d be possible to obtain microcavities with a specific splitting. This would enable the triple resonance condition to be achieved. We extract the total dissipation of the optical mode from the linewidth of peak (Fig. 3(c))κ/2π= 11 GHz. This corre- sponds to a Q-factor of 20,000 and Finesse of 600. The expected external loss rate can be estimated from the DBR reflectivity R= 0.9986 asκext=−2∆ωFSRlogR [46], giving κex/2π≈3 GHz. Using these values, and κ=κext+κint, we can estimate the internal dissipation rate κint/2π=8 GHz. This is consistent with the transmitted in- tensity on resonance κ2 ext/κ2≈0.07. If this internal dissipa- tion were solely due to absorption in the YIG layer, we would expectκint=κabs= (αc/n YIG)(tYIG/L)≈1GHz. The dis- crepancy suggests that other dissipation mechanisms play a role. A likely source is the surface roughness on the GGG top surface, where the crack propagates during lift-off. This c ould be alleviated by post-bonding polishing. The choice of mirror reflectivity was conservative to en- sure good coupling to the cavity. If the scattering losses ca n be eliminated, then the mirror reflectivity could be increas ed, while keeping the system over-coupled. In this case, the min - imum possible dissipation rate would be κabs∼1GHz - as achieved in WGM cavities [ 6,8]. VII. BRILLOUIN LIGHT SCATTERING Next, we use homodyne detection to measure the magnon- scattered light, emitted from the microcavity with opposit e linear polarization to the input. The input laser wavelengt h is fixed and set to the lower wavelength optical mode, with frequency ωi. The VNA is used to drive the magnon modes via the microwave antenna, as well as detect the signal at the same frequency from the fast photodiode, where the scattere d light is combined with a local oscillator taken from the inpu t laser. A measurement using this method is shown in Fig. 4(a), as a function of microwave drive frequency and applied magnetic field. When the microwave drive is resonant with a magnon mode with frequency ωmwithin the microcavity, the magnons created scatter with the input optical photons to create opt ical photons at a frequency ωi±ωm. When combined with the local oscillator ωLO=ωi, and mixed on the photodiode, this results in a microwave signal at ±ωm, resulting in the bright lines in Fig. 4(a). The power plotted is the optical power at the photodiode, using the responsivity of the photodiode an d amplification of the amplifier chain to convert from the mea- sured microwave power at the VNA. To check this conversion, we measure the noise equivalent power of the photodiode in darkness, and compare to its specified value. To confirm that the modes result from the embedded mag- netic material, we compare the optical measurement to a stan - dard inductive ferromagnetic resonance (FMR). This is made via the reflected microwave power to the output port of the VNA, and is shown in Fig. 4(b). The change in microwave re- flection coefficient ∆|S11|with magnetic field is plotted over5 a) b) Figure 4. (a) Brillouin light scattering signal. The mixed p ower with the local oscillator, incident on the fast photodiode. A mag netic field independent background has been subtracted. (b) Microwave mea- surement of magnetic modes, via |S11|using the vector network an- alyzer. the same range as Fig. 4(a). We have confirmed that the res- onances in Fig. 4(b) results from the YIG layer in FMR mea- surements over a wider magnetic field. The fact that the slope with magnetic field is the same in both measurements con- firms that the optical signal results from Brillouin light sc at- tering in the YIG. The band of resonances also has the same upper limit in both measurements. The differences in the re- sponse – in particular, that the microwave reflection spectr a has more resonances than the optical BLS – can be explained by the fact that, in the inductive measurement, the entire st rip- line is probed, whereas the optical measurement is only sens i- tive to the region of the YIG film below the lens. The large number of resonances in the inductive measurement is due to strain inhomogeneity across the film from the film trans- fer process. The magnon modes observed in the optical measurement depend on overlap with both the microwave and optical fields [47]. The Kittel mode has the correct symmetry to fulfill these requirements, and we tentatively assign the strongest scat ter- ing to this uniform mode. There are two other modes at higher frequency visible in Fig. 4(a). The mode spacing of these is too large to be due to perpendicular standing spin waves, giv en the thickness of the YIG film [ 48]. A possible candidate for these modes would be magneto-static surface spin waves [ 49] with wavevector set by the width of the microstrip antenna [50]. However, the robust identification of these modes re- quires further measurement and will be the subject of future work. A fit to the Kittel mode in the BLS measurement gives a linewidth of Γ≈20MHz, a value larger than is typical for high quality YIG thin films [ 51]. This is expected, because the current device has imperfections in the YIG layer due to the ion-implantation process, and strain disorder from the bon d- ing process, such as the cracks visible in Fig. 1(c). These im- perfections can be improved by further fabrication process es. Firstly, the damage from ion-implantation can be alleviate d via annealing [ 52]. Secondly, the strain disorder can be re- duced by polishing the GGG from the back of the YIG/GGG bi-layer. It has also been possible to transfer a YIG layer crack-free.The peak measured optical power of the BLS signal is ≈1.2 nW. Given the local oscillator power PLO= 65µW and input microwave power 1 mW, the total conversion efficiency is calculated to be 8×10−16. This low value is to be ex- pected, since the microwave coupling and magnon mode over- lap in this devices have not been engineered. Therefore, to show the value of design we separate the coupling rate Eq. 1 into an optical part Goptand the magnetic fill-factor ηmag, G=Gopt√ηmag, and estimate the obtained rate for the fab- ricated cavity. The optical mode volume for a Gaussian beam can be esti- mated as [ 29] Vopt=πw2 1L/4, (3) whereLis the cavity optical path length and w1is the beam waist on the flat mirror surface. We estimate w2 1= (λ/π)√βL(1−L/β)in the parallax approximation, where βis the radius of curvature of the lens. With the parameters of the measured device, L≈12µm andβ= 70µm, this yields Vopt≈100µm3. This corresponding to Gopt≈50kHz. The magnon mode overlap in the device measured is poor. Taking the whole area of the cracked film part, we estimate η∼10−3, reducing the coupling rate to G≈50Hz. Compared to the whispering gallery mode Vopt≈ 5000µm3, and the waveguide device of Ref. 13Vopt≈ 105µm3, the optical mode volume achieved here is a signif- icant improvement. However, the microwave coupling and magnon confinement are lacking, severely limiting the con- version efficiency. The waveguide device [ 13] has optimized magnon modes ηmag≈1and optimized microwave coupling through an microwave resonator, and even WGM mode de- vices (ηmag≈10−5) benefit from impedance matched mi- crowave coupling to the Kittel mode [ 5]. VIII. CONCLUSIONS We propose and demonstrate an open magneto-optical cav- ity device with optical mode volume limited by the thickness of an embedded magnetic layer. This design is tunable, has the correct polarized modes, and a significantly reduced opt i- cal mode volume compared to previous devices [ 5–7,13]. We envisage that simple improvements in the demonstrated device design should enable the strong-coupling regime to b e reached. By removing the GGG from the device via polish- ing, the cavity length can be reduced to 3 um, and using the lowest radius of curvature lens β= 22µm, the resulting mode volume would be Vopt≈7µm3(from Eq. 3). Combining this with lateral patterning of the YIG layer to confine a magnon mode to a disk with diameter 5 µm, it should be possible to achieveG= 200 kHz using the open microcavity design. If we combine this with the discussed improvements in the magnon and optical linewidth to Γ = 1 MHz and κ= 1GHz, respectively, this would lead to a single photon cooperativ ity ofC= 4G2/Γκ= 10−4. We would then require an opti- cal pump power of ≈5mW to achieve the strong coupling regime√nG > κ, Γ. In order to achieve cooling of magnetic6 mode via optical damping, Γopt= 4nG2/κ[53,54] compa- rable to the magnetic damping would require only ≈1µW input power [ 10]. Finally, it will be necessary to couple microwaves effi- ciently into the resulting small volume of magnetic materia l. Elsewhere, we have demonstrated that this is possible using low impedance microwave resonators [ 55]. With careful mi- crowave circuit optimization it is possible to achieve coup ling to femtolitre magnetic volumes [ 56], to match that possible with open optical microcavities [ 29]. As well as demonstrating progress towards microwave- optical conversion [ 4], it is expected that the enhancement of the magnon-photon interaction demonstrated could have sig - nificant impact in magnonics [ 57], through the increased mea- surement sensitivity and in optical modification of the magn on dynamics. The versatility of the fabrication method meansthat antiferromagnetic materials could also be embedded in the microcavity in order to explore the interaction of optic al photons with THz magnon modes [ 18,58]. ACKNOWLEDGEMENTS We are grateful to Aurilien Trichet and Jason Smith (Ox- ford HiQ) for advice on open microcavities, Roger Webb (Surrey ion beam centre) for assistance with ion implantati on, and Miguel Levy, Dries Van Thourhout, Koji Usami, Andreas Nunnenkamp and Paul Walker for useful discussions. This work was supported by the European Union’s Horizon 2020 research and innovation programme under grant agreement No 732894 (FET Proactive HOT). 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2020-09-09

Microwave-to-optical conversion via ferromagnetic magnons has so-far been limited by the optical coupling rates achieved in mm-scale whispering gallery mode resonators. Towards overcoming this limitation, we propose and demonstrate an open magneto-optical cavity containing a thin-film of yttrium iron garnet (YIG). We achieve a 0.1 pL (100 $\mu$m$^{3}$) optical mode volume, $\sim$50 times smaller than previous devices. From this, we estimate the magnon single-photon coupling rate is $G\approx50$ Hz. This open cavity design offers the prospect of wavelength scale mode volumes, small polarization splittings, and good magneto-optical mode overlap. With achievable further improvements and optimization, efficient microwave-optical conversion and magnon cooling devices become a realistic possibility.

Sub-pico-liter magneto-optical cavities

2009.04162v1

arXiv:1905.07278v1 [physics.optics] 17 May 2019Keywords : Optomagnonic Cavity, Voigt Geometry, Magnetostatic Spin Waves , Inelastic Light Scattering, Time Floquet Method High-efficiency triple-resonant inelastic light scatterin g in planar optomagnonic cavities Petros Andreas Pantazopoulos,∗Kosmas L. Tsakmakidis, Evangelos Almpanis, Grigorios P. Zouros, and Nikolaos Stefanou Section of Solid State Physics, National and Kapodistrian U niversity of Athens, Panepistimioupolis, GR-157 84 Athens, Greece (Dated: May 20, 2019) Abstract Optomagnonic cavities have recently been emerging as promi sing candidates for implementing coherent microwave-to-optical conversion, quantum memor ies and devices, and next generation quantum networks. A key challenge in the design of such cavit ies is the attainment of high efficien- cies, which could, e.g., be exploited for efficient optical in terfacing of superconducting qubits, as well as the practicality of the final designs, which ideally s hould be planar and amenable to on-chip integration. Here, on the basis of a novel time Floquet scatt ering-matrix approach, we report on the design and optimization of a planar, multilayer optomag nonic cavity, incorporating a Ce:YIG thin film, magnetized in-plane, operating in the triple-res onant inelastic light scattering regime. This architecture allows for conversion efficiencies of abou t 5%, under realistic conditions, which is orders of magnitude higher than alternative designs. Our re sults suggest a viable way forward for realizing practical information inter-conversion betwee n microwave photons and optical photons, mediated by magnons, with efficiencies intrinsically greate r than those achieved in optomechanics and alternative related technologies, as well as a platform for fundamental studies of classical and quantum dynamics in magnetic solids, and implementation of futuristic quantum devices. PACS numbers: 1I. INTRODUCTION Optomagnoniccavitiesarejudiciouslydesigneddielectricstructure sthatincludemagnetic materials capable of simultaneously confining light and spin waves in the same region of space. This confinement leads, under certain conditions, to stron g enhancement of the inherently weak interaction between the two fields, which allows for a n efficient microwave- to-optical transduction, enabling, e.g., optical interfacing of sup erconducting qubits1,2. Theoptomagnonic interactionis expected to belarger when theso- called triple-resonance condition is met, i.e., when the frequency of a cavity magnon matches a photon transition between two resonant modes. This implies that the cavity must supp ort two well-resolved optical resonances (in the hundred terahertz range) separate d by a few gigahertz, which requires quality factors at least of the order of 105, as schematically depicted in figure 1. A (sub)millimeter-sized sphere, made of a low-loss dielectric magnetic material, consti- tutes a simple realization of an optomagnonic cavity. The sphere sup ports densely spaced long-lifetime optical whispering gallery modes3–7, and infrared incident light evanescently coupled to these modes can be scattered by a uniformly precessing (so-called Kittel) spin wave toa neighbouring optical whispering gallery mode. Inthe prosp ect of achieving smaller modal volumes and larger spatial overlap between the interacting fi elds, higher-order mag- netostatic modes8–10, magnetically split optical Mie resonances in small spheres11, as well as particles of different shapes12have been proposed. However, these proposals currently face appreciable challenges in the fabrication of high-quality particle s and/or the efficient excitation of the spin waves. A promising alternative design of optomagnonic cavities is based on planargeometries, which can exhibit even stronger magnon-to-photon conversion effi ciencies13, while at the same time allowing integration into a hybrid opto-microwave chip using m odern nanofab- rication methods. To this end, optomagnonic cavities formed in a mag netic dielectric film bounded by two mirrors14–16, or in a defect layer in a dual photonic-magnonic periodic lay- ered structure17, have also been investigated. However, the studies reported so f ar refer to the Faraday configuration, with out-of-plane magnetized films, wh ere it is challenging to obtain two optical resonances in the required close proximity to eac h other. In this work we show that, by using in-plane magnetized films in the so- called Voigt con- figuration, wecanovercometheafore-describedshortcomingso fprevious schemes anddesign 2/g90out /g90in /g39/g90/g58 /g900/g900~ 10 -4 FIG. 1: Schematic of inelastic light scattering through mag non absorption in an optomagnonic cavity. The frequency of a cavity magnon, Ω, matches a photon transition between two resonant optical modes: Ω = ∆ ω≡ωout−ωin(triple-resonance condition). efficient optomagnonic cavities operating in the triple-resonance re gime. In section II we de- scribe our statically magnetized structure and discuss its optical r esponse. In section III we summarize our recently developed fully dynamic time Floquet metho d for layered opto- magnonic structures16and in section IV we present details of our attained numerical result s. The last section concludes the article. II. STRUCTURE DESIGN We propose a simple design of planar optomagnonic cavity, simultaneo usly confining light and spin waves in the same subwavelength region of space. It consis ts of an iron garnet thin film bounded symmetrically by two-loss, dielectric Bragg mirrors, in air , as schematically illustrated in figure 2(a). Iron garnets are ferrimagnetic materials exhibiting important func tionalities for bulk 3and thin-film device applications that require magnetic insulators, ow ing to their unique physical properties such as high optical transparency in a wide ran ge of wavelengths, high Curie temperature, ultra-low spin-wave damping, and strong magn eto-optical coupling18. In our work, we consider cerium-substituted yttrium iron garnet ( Ce:YIG) which, at the telecom wavelength of 1 .5µm, has a relative electric permittivity ǫ= 5.10+i4×10−4and a Faraday coefficient f=−0.0119, while its relative magnetic permeability equals unity. The Ce:YIG film extends from −d/2 tod/2 and is magnetically saturated to M0by an in-plane bias magnetic field H0oriented, say, along the xdirection. Therefore, the corresponding relative electric permittivity tensor, neglecting the small Cotton-M outon contributions, is of the form20 ǫ= ǫ0 0 0ǫ if 0−if ǫ . (1) We consider the Voigt geometry with light propagating in the y-zplane. The struc- ture in this geometry, with the magnetic field parallel to the surface and also perpendicular to the propagation direction, remains invariant under reflection wit h respect to the plane of incidence. Consequently, contrary to the Faraday configurat ion studied in our previous work15–17, the transverse magnetic (TM) and transverse electric (TE) pola rization modes, i.e., modes with the electric field oscillating in and normal to the plane of in cidence, respec- tively, are eigenmodes of the system. Interestingly, in the chosen geometry, the magnetic film behaves as isotropic, with permittivity ǫ−f2/ǫandǫfor TM- and TE-polarized waves, respectively. In other words, only TM-polarized light is affected by t he (magnetic) polariza- tion field. Each Bragg mirror consists of an alternate sequence of six SiO 2and six Si quarter-wave layers, i.e., dm/radicalig (2πnm/λ)2−q2y=π/2, where dm(m: SiO 2or m: Si) is the layer thickness andnmthe corresponding refractive index ( nSiO2= 1.47 andnSi= 3.5) at the operation wavelength λ≈1.5µm21,22. Due to translation invariance parallel to the x-yplane, the in-plane component of the wave vector, qy= 2πsinθ/λ, whereθis the angle of incidence, remains constant. Taking, for instance, qy= 3µm−1, which corresponds to an angle of incidence of about 45o, we obtain dSiO2= 290 nm and dSi= 110 nm. Accordingly, we choose a thickness d= 350 nm for the Ce:YIG film to satisfy the half-wave condition that corresponds to transmission maxima. 4M z /s48/s46/s48/s48/s46/s53/s49/s46/s48/s49/s46/s52/s57/s55/s51 /s49/s46/s52/s57/s55/s50 /s50/s48/s48/s46/s50/s50 /s50/s48/s48/s46/s50/s52/s40 /s109/s41 /s84/s69 /s47/s50 /s40/s84/s72/s122/s41/s84/s84/s77 /s32(a) (b) FIG. 2: (a) Schematic view of the optomagnonic cavity under s tudy. It is formed by a 350- nm-thick Ce:YIG film magnetized in-plane (along the xdirection), bounded symmetrically by two Bragg mirrors, each consisting of6periodsofalternating S iO2andSilayers ofthickness 290nmand 110 nm, respectively, grown along the zdirection. For light incident with qy= 3µm−1, the cavity supports two localized resonant modes, one of TM and the othe rs of TE polarization, manifested in the corresponding transmission spectrum shown in (b), with the dotted and solid curves referring to the lossless and lossy structure, respectively. A snapsh ot of the associated electric field profiles along the zdirection in the lossless case is illustrated in (a). This design provides two (one TM and one TE) high-quality-factor re sonances within the lowest Bragg gap, at a wavelength of about 1.5 µm, separated by a frequency difference ∆f= 9.5 GHz that matches the frequency of magnetostatic spin waves18,20. These resonant modes are strongly localized in the region of the Ce:YIG film, which can b e considered as a defect in the periodic stacking sequence of the Bragg mirrors. Abs orption losses reduce the transmittance peak. In particular, the long-lifetime TE resonance is strongly suppressed in the presence of dissipative losses, as shown by the solid line blue line in fi gure 2(b). It should be pointed out that the position and width of the optical re sonances can be tailored at will by appropriate selection of the materials, and by prop erly adjusting the geometric parameters of the structure and the angle of incidence . 5III. THEORY FOR LAYERED OPTOMAGNONIC STRUCTURES The magnetic Ce:YIG film supports magnetostatic spin waves where t he magnetization precesses in-phase, elliptically, throughout the film (uniform prece ssion mode) with angular frequencyΩ =/radicalbig ΩH(ΩH+ΩM), whereΩ H=γµ0H0andΩ M=γµ0M0,γbeingthegyromag- netic ratio and µ0the magnetic permeability of vacuum20. The corresponding magnetization field profile is given by M(r,t)/M0=/hatwidex+ηAysin(Ωt)/hatwidey+ηAzcos(Ωt)/hatwidez, (2) whereAy=/radicalbig (ΩH+ΩM)/(2ΩH+ΩM),Az=/radicalbig ΩH/(2ΩH+ΩM), andηis an amplitude factor that defines the magnetization precession angle. Under the action of the spin wave, the magnetic film and, consequen tly, the entire struc- ture can be looked upon as a periodically driven system because the m agnetization field, given by Eq. (2), induces a temporal perturbation15 δǫ(t) =1 2/bracketleftbig δǫexp(−iΩt)+δǫ†exp(iΩt)/bracketrightbig (3) in the permittivity tensor of the statically magnetized material, wher e δǫ=fη 0iAzAy −iAz0 0 −Ay0 0 . (4) The solutions of the underlying Maxwell equations are Floquet modes F(r,t) = Re{F(r,t)exp(−iωt)}, withF(r,t+T) =F(r,t),T= 2π/Ω, where by Fwe denote electric field, electric displacement, magnetic field, and magnetic indu ction, while ωis the Floquet quasi-frequency, similarly to the Floquet quasi-momentum ( or else the Bloch wave vector) when there is spatial periodicity23,24. Seeking Floquet modes in the form of plane waves with given qyand expanding all time-periodic quantities into truncated Fourier se ries in the basis of complex exponential functions exp( inΩt),n=−N,−N+1,...,N, leads to an eigenvalue-eigenvector equation, which has 4(2 N+ 1) physically acceptable solutions16. We characterize them by the following indices: s= +(−) that denotes waves propagating or decaying in the positive (negative) zdirection, p= 1,2 that indicates the two eigen- polarizations, and ν=−N,−N+ 1,···,Nwhich labels the different eigenmodes. These eigenmodes are polychromatic waves, each composed of 2 N+1 monochromatic components 6of angular frequency ω−nΩ,n=−N,−N+ 1,...,N16. We note that, in a static ho- mogeneous medium, the corresponding eigenmodes of the electrom agnetic (EM) field are monochromatic waves characterized by the indices s,p, andn. Scattering of an eigenmode occurs at an interface between two diff erent homogeneous media. For such a planar interface between a static and a time-perio dic medium, the relative complex amplitudes of the transmitted (reflected) waves, denote d byQI pν;p′n′(QIII pn;p′n′) for incidence intheforwarddirection or QIV pn;p′ν′(QII pν;p′ν′)for incidence inthebackward direction in the configuration shown in figure 3, are obtained in the manner des cribed in Ref.16. Primed indices refer to the incident wave. For an interface between two static homogeneous media, the Qmatrices relate monochromatic waves and are diagonal in n, which reflects frequency conservation. We note that, in order to evaluate the s cattering properties of layered optomagnonic structures in a straightforward manner, t he waves on each side of a given interface are expressed around different points, at a dista nce−d1andd2from the center of the interface (see figure 3), so that all backward a nd forward propagating or evanescent waves in the region between two consecutive interf aces refer to the same (arbitrary) origin. Of course, because of translation invariance p arallel to the x-yplane, the choice of the x-ycomponents of d1andd2are immaterial; thus, for simplicity, we choose d1 andd2along the zdirection. The transmission and reflection matrices of a pair of consecutive int erfaces, i and i+1, are obtained by properly combining those of the two interfaces so a s to describe multiple scattering to any order. This leads to the following expressions aft er summing up the infinite geometric series involved, as schematically illustrated in figure 3, i.e., QI(i,i+1)=QI(i+1)[I−QII(i)QIII(i+1)]−1QI(i) QII(i,i+1)=QII(i+1)+QI(i+1)QII(i)[I−QIII(i+1)QII(i)]−1QIV(i+1) QIII(i,i+1)=QIII(i)+QIV(i)QIII(i+1)[I−QII(i)QIII(i+1)]−1QI(i) QIV(i,i+1)=QIV(i)[I−QIII(i+1)QII(i)]−1QIV(i+1). (5) It should benotedthat thewaves onthe left (right) ofthepair of in terfaces arereferred toan origin at a distance −d1(i) [d2(i+1)] from the center of the i-th [(i+1)-th] interface. We also recall that, though the choice of d1andd2associated to each interface is to a certain degree arbitrary, it must besuch that d2z(i)+d1z(i+1) equals thethickness ofthelayer between the 7Dynamic Static QI pν ;p'n' QIV pn ;p'ν'QII pν ;p'ν'QIII pn ;p'n' O1 O2 d1d2 z+ + +... + + +...+ + +...+ + +... Q :I Q :II Q :III Q :IV FIG. 3: Left-hand diagram: Transmission and reflection matr ices for a planar interface between a static and a dynamic medium, defined with respect to appropri ate origins, O1andO2, at distances −d1andd2from a center at the interface, respectively. Right-hand di agram: The transmission and reflection matrices of two consecutive interfaces are ev aluated by summing up all relevant multiple-scattering processes. i-thand(i+1)-thinterfaces. It is obvious that onecan repeat the above process to obtainthe transmission and reflection matrices Qof three consecutive interfaces, by combining those of the pair of the first interfaces with those of the third interface , and so on, by properly combining the Qmatrices of component units, one can obtain the Qmatrices of a slab which comprises any finite number of interfaces23,24. This method applies to an arbitrary slab which comprises periodically time-varying layers, provided that a ll dynamic media have the same temporal periodicity. It is then straightforward to calcu late the transmittance, T, and reflectance, R, of the slab as the ratio of the transmitted and reflected, respec tively, energy flux to the energy flux associated with the incident wave. TandRare given by the sum of the corresponding quantities over all scattering channels ( p,n):T=/summationtext p,nTpnand R=/summationtext p,nRpn. It is worthnoting that, because of the time variation of the permit tivity tensor, the EM energy is not conserved even in the absence of diss ipative (thermal) losses. In this case, A= 1−T −R >0(<0) means energy transfer from (to) the EM to (from) 8the spin-wave field. We close this section by pointing out a useful polarization selection ru le, which can be readily derived in the linear-response approximation. To first order , the coupling strength associated to the photon-magnon scattering is proportional to t he overlap integral G= /an}bracketle{tout|δǫ|in/an}bracketri}ht, where/an}bracketle{trt|in/an}bracketri}ht=Ein(z)exp[i(q/bardbl·r−ωt)] and/an}bracketle{tout|r′t′/an}bracketri}ht=Eout⋆(z)exp[−i(q′ /bardbl· r−ω′t)] denote appropriate incoming and outgoing monochromatic time-h armonic waves in the static magnetic layered structure. Using Eq. (3) we obtain G= 4π3fηδ(q/bardbl−q′ /bardbl)/bracketleftbig δ(ω−ω′−Ω)g−+δ(ω−ω′+Ω)g+/bracketrightbig (6) whereg±=/integraldisplay dzu±·/bracketleftbig Eout⋆(z)×Ein(z)/bracketrightbig , withu±=∓Ay/hatwidey+iAz/hatwidez. The delta functions in Eq. (6) express conservation of in-plane momentum and energy in in elastic light scattering processes that involve emission and absorption of one magnon by a p hoton, as expected in the linear regime. Obviously, the amplitude of transition between two optical eigenmodes of the same polarization, TM or TE, is identically zero because the corre sponding eigenvectors are real. In other words, one-magnon processes change the linea r polarization state of a photon. IV. RESULTS AND DISCUSSION We now assume continuous excitation of a uniform-precession spin- wave mode in the magnetic film, with a relative amplitude η= 0.06, which induces a periodic time variation in thecorresponding electricpermittivitytensor, givenbyEq. (3). T heoptomagnonicstructure is illuminated from the left by TM-polarized light with qy= 3µm−1at the corresponding resonance frequency, which corresponds to an angle of incidence of about 45o. The dynamic optical response of the structure is calculated with sufficient accu racy by considering a cutoff ofN= 5 in the Fourier series expansions involved in our time Floquet scatte ring-matrix method outlined in section III. figure 4(a) shows the total (transmitted plus reflected) intensit ies,In=/summationtext p(Tpn+Rpn), as a function of the spin-wave frequency Ω /2π. It can be seen that inelastic light scattering is negligible when the allowed final photon states fall within a gap, wher e the optical density of states is very low, and we essentially have only the elastic outgoing beam. On the contrary, when the spin-wave frequency matches the frequenc y difference ∆ f= 9.5 GHz 9/s53 /s49/s48 /s49/s53/s48/s46/s48/s48/s46/s53/s49/s46/s48 /s45/s48/s46/s51/s48/s46/s48 /s53 /s49/s48 /s49/s53/s48/s46/s51/s48/s46/s52/s40/s100/s41/s40/s99/s41/s40/s98/s41 /s110 /s50/s110 /s49 /s32/s32/s73 /s110 /s47/s50 /s40/s71/s72/s122/s41/s110 /s48 /s32/s32 /s32/s49/s48/s52 /s65 /s32 /s32/s65 /s47/s50 /s40/s71/s72/s122/s41 FIG. 4: The structure of figure 2(a), under continuous excita tion of a uniform precession spin-wave mode of angular frequency Ω with a relative amplitude η= 0.06, is illuminated from the left by TM-polarized light with qy= 3µm−1at the corresponding resonance frequency [see figure 2(b)]. Variation of the dominant elastic and inelastic total outgo ing light intensities versus the spin-wave frequency (a) and corresponding optical absorption (b) and (c). Dotted and solid curves refer to the lossless and lossy structure, respectively. between the two optical resonances [see figure 2(a)], the triple-r esonance condition is fulfilled and one-magnon absorption processes are favoured, leading to e nhanced intensities of the corresponding ( n=−1) inelastically transmitted and reflected light beams, with conversio n efficiency of the order of 30% if dissipative losses are neglected. At t he same time, the elastic beam intensity is considerably reduced while the other inelastic proce sses are also resonantly affected, though to a much lesser degree, as shown in figure 4(a) a nd also in figure 5. Overall, there is an excess number of magnons absorbed, which can be accounted for by our fully dynamic time Floquet scattering-matrix method16. This is manifested as a small negative absorption peak [see figure 4(b)], which clearly indicates a r esonant energy transfer from the magnon to the photon field. Considering a saturation magnetization M0= 150 emu /cm3for Ce:YIG19, the triple- resonance condition (Ω /2π= 9.5 GHz) is achieved with a bias magnetic field H0= 2.5 kOe. Inthiscase, theconeangleofmagnetizationprecession(ellipticalin thechosenconfiguration) attains a maximum of 2 .75o, which is a tolerable value for linear spin waves. 10/s53 /s49/s48 /s49/s53/s49/s48/s45/s53/s49/s48/s45/s51/s49/s48/s45/s49/s49/s48/s48 /s53 /s49/s48 /s49/s53/s49/s48/s45/s57/s49/s48/s45/s55/s49/s48/s45/s53/s49/s48/s45/s51/s49/s48/s45/s49 /s40/s98/s41/s32 /s32/s73 /s84/s77/s59 /s110 /s40/s97/s41 /s47/s50 /s40/s71/s72/s122/s41 /s47/s50 /s40/s71/s72/s122/s41/s110 /s51/s32 /s32/s73 /s84/s69/s59 /s110/s110 /s48 /s110 /s50/s110 /s49 /s110 /s49 /s110 /s51 /s110 /s50 FIG. 5: Polarization-converting (a) and polarization-con serving (b) contributions to the spectrum of the figure 4(a). The peak in (a) indicated by the arrow corre sponds to the resonant transition when accomplished by absorption of three magnons. It is interesting to note that the triple-resonance condition can be accomplished by many- magnon absorption processes as well ( mΩ/2π= ∆f), provided that the number of magnons, m, isoddinordertochangethepolarizationstateofthephoton, fro mTMtoTE, asrequired in our case. We recall that our method of calculation is not restricte d to the first-order Born approximation and thus it can describe nonlinear effects that are us ually relatively weak. Forexample, such a three-magnonabsorptionprocess ismanifest ed asa peakintheintensity of then=−3 outgoing beam, for Ω /2π= ∆f/3≈3.2 GHz, as pointed out by the arrow in figure 5(a). As can be seen in figure 4(a), when dissipative losses are taken into a ccount, the elastic beamintensity is uniformlyby about 30%, inagreement withthe result s shown infigure 2(b) for the TM mode. Here, when the triple-resonance condition is satis fied, the corresponding drop in the n=−1 beam is considerably larger because of the longer lifetime of the fina l (TE) state but, nonetheless, the optical conversion efficiency is s till as high as 5%. We note that, because of the high quality factor of the final (TE) sta te and the presence of non-negligible losses in this case, we overall obtain resonant optical absorption (instead of gain in the lossless case), as shown in figure 4(c). 11V. CONCLUSIONS To conclude, we have presented a detailed analysis and optimization o f a planar opto- magnonic structure operating in the triple-resonance regime and a llowing for optical con- version efficiencies of the order of 5% [cf. figure 4(a)] under realist ic conditions, mediated by a uniformly precessing spin wave. The outlined time Floquet multiple-sc attering methodol- ogy was able to resolve absorption and emission of multiple magnons, in dicating that under special conditions the attained conversion efficiencies mediated by m ultiple magnons can be comparable to those mediated by a single magnon [cf. orange and pink dotted lines in fig- ure 5(a)]. We have also found that the absorption or emission of a ma gnon leads to a change in the polarization of the optical conversion process. An interestin g further objective would be to extend the current approach to the full spatio-temporal Floquet scattering-matrix methodology, which should allow for investigating, among others, su rface Dammon-Eshbach and backward volume waves with an in-plane propagation wave vecto r that can lead to more exotic physical behavior, including emergence of a paraxial ou tgoing scattered beam and bandgap formation. Acknowledgments P.A.P. was supported by the General Secretariat for Research an d Technology (GSRT) andtheHellenic FoundationforResearch andInnovation(HFRI) th rougha PhD scholarship (No. 906). K.L.T., E.A., and G.P.Z. were supported by HFRI and GSRT un der Grant 1819. References ∗Electronic address: pepantaz@phys.uoa.gr 1Tabuchi Y, Ishino S, Noguchi A, Ishikawa T, Yamazaki R, Usami K and Nakamura Y 2015 Coherent coupling between a ferromagnetic magnon and a supe rconducting qubit Science349, 405408 122Lachance-QuirionD,TabuchiY,IshinoY,NoguchiA,Ishikaw aT,Yamazaki RandNakamuraY 2017Resolvingquantaofcollective spinexcitations inami llimeter-sized ferromagnet Sci. Adv. 3, e1603150 3Osada A, Hisatomi R, Noguchi A, Tabuchi Y, Yamazaki R, Usami k , Sadgrove M, Yalla R, Nomura M and Nakamura Y 2016 Cavity optomagnonics with spin- orbit coupled photons Phys. Rev. Lett. 116, 223601 4Zhang X, Zhu N, Zou C.-L. and Tang H X 2016 Optomagnonic whispe ring gallery microres- onatorsPhys. Rev. Lett. 117, 123605 5Haigh J A, Nunnenkamp A, Ramsay A J and Ferguson A J 2016 Triple -resonant Brillouin light scattering in magneto-optical cavities Phys. Rev. Lett. 117, 133602 6Viola-Kusminskiy S, Tang H X and Marquard F 2016 Coupled spin -light dynamics in cavity optomagnonics Phys. Rev. A 94, 033821 7Sharma S, Blanter Y M and Bauer G E W 2017 Light scattering by ma gnons in whispering gallery mode cavities Phys. Rev. B 96, 094412 8Haigh J A, Lambert N J, SharmaS, Blanter Y M, Bauer G E W and Rams ay A J 2018 Selection rules for cavity-enhanced Brillouin light scattering from magnetostatic modes Phys. Rev. B 97, 214423 9Osada A, Gloppe A, Hisatomi R, Noguchi A, Yamazaki R, Nomura M , Nakamura Y and Usami K 2018 Brillouin light scattering by magnetic quasivo rtices in cavity optomagnon- icsPhys. Rev. Lett. 120, 133602 10Osada A, Gloppe A, Nakamura Y and Usami K 2018 Orbital angular momentum conservation in Brillouin light scattering withina ferromagnetic spher eNew J. Phys. 20, 103018 11Almpanis E 2018 Dielectric magnetic microparticles as phot omagnonic cavities: Enhancing the modulation of near-infrared light by spin waves Phys. Rev. B 97, 184406 12Graf J, Pfeifer H, Marquardt F, and Viola Kusminskiy S 2018 Ca vity optomagnonics with magnetic textures: Coupling a magnetic vortex to light Phys. Rev. B 98, 241406(R) 13Kostylev M and Stashkevich A A 2019 Proposal for a microwave p hoton to optical photon converter based on traveling magnons in thin magnetic films J. Magn. Magn. Mat. 484, 329– 344 14Liu T Y, Zhang X F, Tang H X and Flatt´ e M E 2016 Optomagnonics in magnetic solids Phys. Rev. B 94, 060405(R) 1315PantazopoulosPA,StefanouN,AlmpanisEandPapanikolaou N 2017Photomagnonicnanocav- ities for strong light–spin-wave interaction Phys. Rev. B 96, 104425 16Pantazopoulos P A and Stefanou N 2019 Layered optomagnonic s tructures: Time Floquet scattering-matrix approach Phys. Rev. B 99, 144415 17Pantazopoulos P A, Papanikolaou N and Stefanou N 2019 Tailor ing coupling between light and spin waves with dual photonic-magnonic resonant layered st ructures J. Opt.21, 015603 18ZvezdinAKandKotov VA1997 Modern Magnetooptics and Magnetooptical Materials (Bristol: Institute of Physics Publishing) 19Onbasli M C, Beran L, Zahradn´ ık M, Kuˇ cera M, Antoˇ s R, Mistr ´ ık J, Dionne G F, Veis M and Ross K A 2016 Optical and magneto-optical behavior of Cerium Yttrium Iron Garnet thin films at wavelengths of 200–1770 nm Sci. Rep. 6, 23640 20Stancil D D and Prabhakar A 2009 Spin Waves-Theory and Applications (Boston: Springer) 21Pierce D T and Spicer W E 1972 Electronic structure of amorpho us Si from photoemission and optical studies Phys. Rev. B 5, 307 22Gao L, Lemarchand F and Lequime M 2012 Exploitation of multip le incidences spectrometric measurements for thin film reverse engineering Opt. Express 20, 15734–15751 23Stefanou N, Yannopapas V and Modinos A 1998 Heterostructure s of photonic crystals: fre- quency bands and transmission coefficients Comput. Phys. Commun. 113, 49–77 24Stefanou N, Yannopapas V and Modinos A 2000 MULTEM 2: A new ver sion of the program for transmissionandband-structurecalculations ofphotonic crystalsComput. Phys. Commun. 132, 189–196 14

2019-05-17

Optomagnonic cavities have recently been emerging as promising candidates for implementing coherent microwave-to-optical conversion, quantum memories and devices, and next generation quantum networks. A key challenge in the design of such cavities is the attainment of high efficiencies, which could, e.g., be exploited for efficient optical interfacing of superconducting qubits, as well as the practicality of the final designs, which ideally should be planar and amenable to on-chip integration. Here, on the basis of a novel time Floquet scattering-matrix approach, we report on the design and optimization of a planar, multilayer optomagnonic cavity, incorporating a Ce:YIG thin film, magnetized in-plane, operating in the triple-resonant inelastic light scattering regime. This architecture allows for conversion efficiencies of about 5%, under realistic conditions, which is orders of magnitude higher than alternative designs. Our results suggest a viable way forward for realizing practical information inter-conversion between microwave photons and optical photons, mediated by magnons, with efficiencies intrinsically greater than those achieved in optomechanics and alternative related technologies, as well as a platform for fundamental studies of classical and quantum dynamics in magnetic solids, and implementation of futuristic quantum devices.

High-efficiency triple-resonant inelastic light scattering in planar optomagnonic cavities

1905.07278v1

1 Efficient g eometrical control of spin waves in microscopic YIG waveguides S. R. Lake1 , B. Divinskiy2*, G. Schmidt1,3, S. O. Demokritov2, and V . E. Demidov2 1Institut für Physik, Martin -Luther -Universität Halle -Wittenberg, 06120 Halle, Germany 2Institute for Applied Physics, University of Muenster, 48149 Muenster, Germany 3Interdisziplinäres Zentrum für Materialwissenschaften , Martin -Luther -Universität Halle - Wittenberg, 06120 Halle, Germany We study experimentally and by micromagnetic simulations the propagation of spin waves in 100-nm thick YIG waveguides , where the width linearly decreases from 2 to 0.5 m over a transition region with varying length between 2.5 and 10 m. We show that this geometry result s in a down conversion of the wavelength , enabling efficient generation of waves with wavelengths down to 350 nm. We also find that th is geometry leads to a modification of the group velocity , allow ing for almost -dispersionless propagation of spin -wave pulses . Moreover , we demonstrate that the in fluence of energy concentration outweighs that of damping in these YIG waveguides , resulting in an overall increase of the spin -wave intensity during propagation in the transition region. These findings can be utilized to improve the efficiency and functio nality of magnonic devices which use spin waves as an information carrier. *Corresponding author, e -mail: b_divi01@uni -muenster.de 2 Spin waves propagating in microscopic magnetic waveguides present a flexib le and highly functional tool for transmission and processing of information on the nano -scale1-4. Among the most important advantages provided by spin waves is their controllabili ty by the magnetic field, which , for example, enables efficient control of t heir propagation characteristics by electric current s. This controllability also forms the basis of using spatially non -uniform , dipolar magnetic fields to manipulate spin waves5-7. Because these fields are determined by the waveguide’s geometry , varying its spatial parameters enable s different mode transformations and wavelength conversion7-15. Although tuning spin waves by using geometrical effects provides many opportunities for the implementation of magnonic devices, the functionality of this approach is strongly limited by the spatial attenuation. Indeed, in metallic waveguides with a small decay length, the passage through a conversion region can lead to a massive loss of spin-wave intensity16. The restrictions imposed by the fast spatial decay of spin waves can be overcome by using high-quality , nanometer s-thick films of the low-damping magnetic insulator , yttrium iron garnet (YIG)17-19, where the decay length of spin waves can surpass many tens of micrometers20-22. Recently it was shown that these films can be structured on the micrometer and the sub -micrometer scale without significantly increasing the magnetic damping23-25. Additionally, magnetic dynamics in these films can be controlled by sp in-torque effects, which can be used to enhance further the propagation characteristics26,27 and generate propagating spin waves by dc electric currents without the need to use energy -inefficient microwave excitation28. These features make ultrathin YIG fi lms an excellent candidate for magnonic applications where spin waves are steered via geometrical parameters. In this Letter , we study the control of spin -wave propagation characteristics in microscopic , ultrathin -YIG waveguides in which the width linearly decreases along the 3 propagation direction . By using spatially -, temporally -, and phase -resolved measurements and micromagnetic simulations , we show that the spatial variation of the demagnetizing field caused by the narrowing of the waveguide results in a robust decrease of the spin-wave wavelength. Due to the minimal spatial attenuation, this wavelength conversion occurs without the decrease of the spin-wave intensity during propagation in the transition region. On the contrary , due to spatial compression, the intensity exhibits a noticeable increase, which becomes particularly pronounced for shorter transition lengths . These effects can be utilized to implement highly efficient excitation of short -wavelength spin waves . Additionally, we show that the geometrical control can be used to tune the propagation velocity of spin -wave pulses and reach a regime where the velocity is almost independent of the spin -wave frequency . Our findings demonstrate a simple and robust method to control spin -wave propag ation which can enhance the functionality of nano scale magnonic devices. Figure 1(a) shows the schematics of our experiment. We study a microscopic spin-wave waveguide patterned from a 100-nm thick YIG film grown by pulsed -laser deposition (PLD) . The YIG film is characterized by a saturation magnetization of 4πM = 1.75 kG and a Gilbert damping constant α = 4×10-4, as determined from ferromagnetic -resonance measurements. The width of the waveguide, w, linearly decreases from 2 m to 0.5 m over a 10 - m long transition region. The spin waves are excited by using a 500 -nm wide and 150 -nm thick inductive Au antenna perpendicular to the waveguide, with its right -hand edge located at the beginning of the transition region. The structures were patterned on a GGG <111> substrate using a two -layer PMMA resist and subsequent electron beam lithography. After development in isopropanol , 110 nm of YIG was deposited via PLD, following a recipe published by Hauser et al. (Ref. 19). The sample was then 4 placed in aceto ne for lift -off of extraneous material and afterwards annealed in a pure oxygen atmosphere19. Next, 10 nm of YIG were etched using phosphoric acid in order to remove seams that can appear at the edges of the structures due to the mobility of the deposited atoms during PLD. Finally, the overlying antenna was patterned using a tri-layer PMMA resist, evaporation of Ti (10 nm) and Au (150 nm), and lift -off. The YIG waveguide is magnetized to saturation by an in-plane , static magnetic field , H0, applied along the Au antenna . Because of demagnetization effects, the internal magneti c field Hint is smaller than H0. It is not uniform across the waveguide width and strongly differs in the waveguide’s wide st and narrow est parts (see the distribution in Fig. 1(b) calculated by using the micromagnetic simulation software MuMax3 (Ref. 29)). At H0 = 1000 Oe, the maximum internal field is 945 and 785 Oe in the wide st and the narrow est part, respectively. As seen from Fig. 1(c), this difference results in a shift of approx imately 0.7 to 0.8 GHz in the dispersion curves . We note that the dispersion curves calculated by using MuMax3 (curves in Fig. 1(c)) coincide well with those obtained from phase -resolved measurements (symbols in Fig. 1(c)) described in detail below. This good agreement allows us to rely on results of simulations to obtain the information about the propagation of spin waves which cannot be obtained from direct measurements. To analyze the propagation of spin waves experimentally , we use the time- and phase - resolved micro -focus Brillouin light scattering (BLS) spectroscopy16. We focus the probing laser light with the wavelength of 473 nm and a power of 0.25 mW into a diffraction -limited spot on the surface of the YIG waveguide (see Fig. 1(a)) and analyze the light inelastically scattered from spin waves. The measured signal , or BLS intensity , is proportional to the intensity of spin waves at the position of the probing spot, which allows us to record two-dimensional spin-wave intensity 5 maps. Additionally, by using the interference of the scattered light with the probing light modulated by the microwave excitation signal, we measure the spatial maps of cos( ), where is the phase difference between the spin wave and the signal applied to the antenna. The Fourier analysis of the se maps provides direct information about the wavelength of spin waves at a given excitatio n frequency . Figures 2(a) and 2(b) show representative phase and intensity maps recorded at the excitation frequency f=4.5 GHz. The left edge of the maps corresponds to the position x=0.5 m which is selected to avoid measuring where the probing light is partially blocked by the antenna . The data of Fig. 2(a) indicate that the w avelength of spin waves gradually decreases during propagation in the transition region, reflecting the frequency shift in the dispersion spectrum due to the continuous reduction of the internal static magnetic field as seen in Figs. 1(b) and 1(c). We note that the phase profiles are slightly disturbed in the vicinity of the antenna, which is caused by the weak excitation of higher -order transverse waveguide mode s16. The slight periodic transverse modulation of the intensity distribution seen in Fig. 2(b) also demonstrates this effect . Analysis of the experimental maps shows that in the transition region the wavelength of the spin wave decreases from about 4 m to 0.5 m, i.e., by a factor of 8, (point -down triangles in Fig. 2(c)). This is in quantitative agreement with the results obtained from micromagnetic simulations ( point -up triangles in Fig. 2(c)). The wavelength -conversion process is characterized thoroughly in Fig. 2(d), which shows the spin -wave wavelength at the end of the transition region , OUT, as a function of the wavelength of the spin wave excited by the antenna , EXC. We note that the efficiency of the inductive excitation by the 500-nm wide antenna quickly decreases for waves with wavelength s smaller than 1 m (Ref. 16), limiting the interval of EXC accessible in the experiment . This restriction is a significant drawback of the inductive excitation mechanism, 6 which strongly limits its use in magnonic devices operating with short -wavelength spin waves. As seen from the da ta of Fig. 2(d), the observed conversion of the wavelength allows one to extend the range of usable wavelengths down to 350 nm. From the point of view of technical applications, the d emonstrated wavelength conversion is advantageous only if it is not acco mpanied by a strong decrease of the intensity of the spin wave in the transition region. On one hand, one expects a spatial decrease of the intensity due to the damping and/or wave reflections. On the other hand, the narrowing of the waveguide is expected to result in an increase of the wave’s intensity due to its energy being concentrated into a smaller cross section. To prove which of these mechanisms dominate s in the studied waveguide , we analyze spatial dependencies of the spin -wave intensity obtained from the measurements and micromagnetic simulations (Fig. 3(a)) . The experimental curve in Fig. 3(a) exhibits an almost constant intensity in the interval x=0.5-7 m. This indicates that the energy -concentration effect approximately compensates the effect s of the damping. However, at larger x, the intensity quickly decreases . We emphasize that this observation cannot be related to wave reflection s because the intensity profile shows no signature of the formation of a standing wave. We also note that the observed decrease is not reproduced in the micromagnetic simulations. The calculated intensity coincides well with the experimental one in the range x=0.5-7 m. However, contrary to experimental data, the calculated intensity noticeably increases in the vicinity of the end of the transition region. We associate this discrepancy with the wavelength -dependent sensitivity of the measurement apparatus. Indeed, the sensitivity of magneto -optical techniques is known to decrease with decreasing wavelength of spin waves , and vanish when the latter becomes equal to the diameter of the probing light spot, d. Assuming d=0.3 m and decreasing from 4 to 0.5 m, one can estimate that the experimental sensitivity decrease s approximately by a factor of 4. This is in 7 good agreement with the ratio between the calculated and experimental intensities at x>10 m in Fig. 3(a) . Taking this into account , we base our further analysis on the results of simulations. Considering the calculated intensity curve (Fig. 3(a)) and comparing the intensit y of spin waves at the beginning of the transition region with the intensity at a point 0.5 m beyond the end of this region, we conclude that the conversion process is accompanied by an increase of the spin - wave intens ity by approximately a factor of 1.5. This result clearly proves the suitability of the proposed conversion mechanism for practical applications. Additionally, as shown by the data of Fig. 3(b), the intensity enhancement can be further improved by reducing the length of the transition region. Note here, that this reduction also leads to stronger reflections of the wave at the end of the transition, as seen from the increasing intensity drop at that point (marked by arrows in Fig. 3(b)) . However, this advers e effect does not compromise the overall increase of the intensity enhancement (see the inset in Fig. 3(b)). Finally, we analyze the effects of the spatial variation of the waveguide geometry on the propagation of spin -wave pulses. As can be seen in Fig. 1(c), the reduction of the waveguide width affects not only the wavelength of a spin wave at a given frequency, but also the slope of the dispersion curve, which determines the group velocity. In other words, during the propagation in the transition region, the dispersion of a spin-wave pulse is expected to change as well. In order to address this phenomenon, we perform time-resolved BLS measurements using an excitation signal in the form of 20 -ns long pulses and determin e the temporal delay of the spin-wave pulses at different spatial positions. In agreement with the above arguments, the found dependence of the propagation delay (Fig. 4(a)) is not linear within the transition region and clearly exhibits a gradual decelera tion of the pulse. The observed deceleration can be used, for example, to implement controllable compression of the spin -wave pulses in the space domain. 8 From the local slope of the dependence shown in Fig. 4(a) , we determine the spin-wave group velocities at the beginning of the transition region ( x=0.5 m) and at its end ( x=10 m). Figure 4(b) summarizes the se results obtained at different excitation frequencies. These data show that the initial -stage group velocity depends strongly on the carrier frequen cy, f, while the velocity at x=10 m is almost independent of frequency. The latter observation indicates that the spin -wave pulses experience nearly dispersionless propagation in the narrow ( w=0.5 m) waveguide30. In narrow waveguides, the spectral region where the group velocity has weak frequency dependence corresponds to spin waves with relatively short wavelengths30. These are difficult to excite by an inductive antenna but, by the down conversion presented here , can be easily achieved . The data of Fig. 4(b) show that the demonstrated approach makes it possible to achieve propagation of spin - wave pulses over longer distances without pulse broadening by dispersion effects . In conclusion, we show that the geometrical control in ultrathin -YIG waveguides p rovides practical opportunities for spin-wave manipulation s, such as the down conversion of the wavelength and the tuning of the propagation velocity. We demonstrate that the intensity of spin waves can be maintained while passing through the control region , due to the small damping in YIG, and can in fact noticeably increase due to spatial compression. These findings can be used for implementation of energy -efficient magnonic devices that exploits sub-micrometer wavelengths. This work was supported in part by the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) – Project -ID 433682494 – SFB 1459 and TRR227 TP B02. 9 Data availability The data that support the findings of this study are available from the corresponding author upon reasonable request. References 1. S. Neusser and D. Grundler, Adv. Mater. 21, 2927 (2009). 2. V. V. Kruglyak, S. O. Demokritov, and D. Grundler, J. Phys. D: Appl. Phys. 43, 264001 (2010). 3. B. Lenk, H. Ulrichs, F. Garbs, and M. Münzenberg, Phys. Rep. 507, 107–136 (2011). 4. A. V. Chumak, V. I. Vasyuchka, A. A. Serga, and B. Hillebrands, Nat. Phys. 11, 453 (2015). 5. V. E. Demidov, S. O. Demokritov, K. Rott, P. Krzysteczko, and G. Reiss , Appl. Phys. Lett. 92, 232503 (2008). 6. J. Topp, J. Podbielski, D. Heitmann, and D. Grundler, Phys. Rev. B 78, 024431 (2008). 7. V. E. Demidov, J. Jersch, S.O. Demokritov, K. Rott, P. Krzysteczko, and G. Reiss , Phys. Rev. B 79, 054417 (2009). 8. V. E. Demidov, M. P. Kostylev , K. Rott, J. Münchenberger, G. Reiss, and S. O. Demokritov, Appl. Phys. Lett. 99, 082507 (2011). 9. K. Vogt, H. Schultheiss, S. Jain, J. E. Pearson, A. Hoffmann, S. D. Bader, and B. Hillebrands, Appl. Phys. Lett. 101, 042410 (2012). 10. C. S. Davies, A. Francis, A. V. Sadovnikov, S. V. Chertopalov, M. T. Bryan, S. V. Grishin, D. A. Allwood, Y. P. Sharaevskii, S. A. Nikitov, and V. V. Kruglyak, Phys. Rev. B 92, 020408(R) (2015). 10 11. Z. Zhang, M. Vogel, J. Holanda, M. B. Jungfleisch, C. Liu, Y. Li, J. E. Pearson, R. Diva n, W. Zhang, A. Hoffmann, Y. Nie, and V. Novosad, Appl. Phys. Lett. 115, 232402 (2019). 12. D. V. Kalyabin, A. V. Sadovnikov, E. N. Beginin, and S. A. Nikitov , J. Appl. Phys. 126, 173907 (2019). 13. S. Mieszczak, O. Busel, P. Gruszecki, A. N. Kuchko, J. W. Kłos, and M. Krawczyk, Phys. Rev. Applied 13, 054038 (2020 ). 14. A. Haldar and A . O. Adeyeye , Appl. Phys. Lett. 119, 060501 (2021) . 15. H. Yu, J. Xiao , and H. Schultheiss , Physics Reports , 905, 1 (2021). 16. V. E. Demidov and S. O. Demokritov, IEEE Trans. Mag. 51, 0800215 (2015). 17. Y. Sun, Y. Y. Song, H. Chang, M. Kabatek, M. Jantz, W. Schneider, M. Wu, H. Schultheiss, and A. Hoffmann, Appl. Phys. Lett. 101, 152405 (2012). 18. O. d’Allivy Kelly, A. Anane, R. Bernard, J. Ben Youssef, C. Hahn, A. H. Molpeceres, C. Carretero, E. Jacquet, C. Deranlot, P. Bortolotti, R. Lebourgeois, J. -C. Mage, G. de Loubens, O. Klein, V. Cros, and A. Fert, Appl. Phys. Lett. 103, 082408 (2013). 19. C. Hauser, T. Richter, N. Homonnay, C. Eisenschmidt, M. Qaid, H. Deniz, D. Hesse, M. Sawicki, S. G. Ebbinghaus, and G. Schmidt, Sci. Rep. 6, 20827 (2016). 20. H. Yu, O. d’Allivy Kelly, V. Cros, R. Bernard, P. Bortolotti, A. Anane, F. Brandl, R. Huber, I. Stasinopoulos, and D. Grundler, Sci. Rep. 4, 6848 (2014). 21. M. Collet, O. Gladii, M. Evelt, V. Bessonov, L. Soumah, P. Bortolotti, S. O. Demokritov, Y. Henry, V. Cros, M. Bailleul, V. E. Demidov, and A. Anane, Appl. Phys. Lett. 110, 092408 (2017). 11 22. C. Liu, J. Chen, T. Liu, F. Heimbach, H. Yu, Y. Xiao, J. Hu, M. Liu, H. Chang, T.Stueckler, S. Tu, Y. Zhang, Y. Zhang, P. Gao, Z. Liao, D. Yu, K. Xia, N. Lei, W. Zhao, and M. Wu, Nat. Commun. 9, 738 (2018). 23. S. Li, W. Zhang, J. Ding, J. E. Pearson, V. Novosad, and A. Hoffmann, Nanoscale 8, 388 (2016). 24. B. Heinz, T. Br ächer, M. Schneider, Q. Wang, B. L ägel, A. M. Friedel, D. Breitbach, S. Steinert, T. Meyer, M. Kewenig, C. Dubs, P. Pirro, and A. V. Chumak, Nano Letters 20, 4220−4227 (2020). 25. G. Schmidt , C. Hauser, P. Trempler, M. Paleschke, and E.T. Papaioannou, Phys. Stat. Sol. B 257, 1900644 (2020). 26. M. Evelt, V. E. Demidov, V. Bessonov, S. O. Demokritov, J. L. Prieto, M. Munoz, J. Ben Youssef, V. V. Naletov, G. de Loubens, O. Klein, M. Collet, K. Garcia -Hernandez, P. Bortolotti, V. Cros, and A. Anane, Appl. Phys. Lett. 108, 172406 (2016). 27. T. Wimmer, M. Althammer, L. Liensberger, N. Vlietstra, S. Geprägs, M. Weiler, R. Gross, and H. Huebl , Phys. Rev. Lett. 123, 257201 (2019). 28. M. Evelt, L. Soumah, A. B. Rinkevich, S. O. Demokritov, A. Anane, V. Cros, J. Ben Youssef, G. de Loubens, O. Klein, P. Bortolotti, and V. E. Demidov, Phys. Rev. Appl. 10, 041002 (2018). 29. A. Vanstee nkiste, J. Leliaert, M. Dvornik, M. Helsen, F. Garcia -Sanchez, and B. Van Waeyenberge, AIP Adv. 4, 107133 (2014). 30. B. Divinskiy, H. Merbouche, K. O. Nikolaev, S. Michaelis de Vasconcellos, R. Bratschitsch, D. Gouéré, R. Lebrun, V. Cros, J. Ben Youssef, P. Bortolotti, A. Anane, S. O. Demokritov, and V. E. Demidov, Phys. Rev. Appl. 16, 024028 (2021). 12 FIG. 1. (a) Schematics of the experiment. Inset shows the scanning -electron micrograph of the sample recorded under a n angle of 70°. (b) Distribution of the internal static magnetic field in the waveguide calculated by using micromagnetic simulations. (c) Dispersion curves of spin waves in the wide st (w=2 m) and the narrow est (w=0.5 m) parts of the waveguide obtained from micromagnetic si mulations (curves) and from phase -resolved measurements (symbols). The data are obtained at H0=1000 Oe. 13 FIG. 2. Representative maps of the spin -wave phase (a) and intensity (b) recorded by BLS at the excitation frequency f=4.5 GHz. The left edge of the maps corresponds to the displacement x=0.5 m from the edge of the antenna. (c) Spatial dependence of the wavelength of the spin wave with the frequency f=4.5 GHz . Vertical dashed line marks the end of the transition region. (d) Spin - wave wavelength at the end of the transition region , OUT, as a function of the wavelength of the spin wave excited by the antenna , EXC. Dashed curve – guide for the eye. In ( c) and ( d): point - down triangles – experimental data, point -up triangl es – results of micromagnetic simulations. The data are obtained at H0=1000 Oe. 14 FIG. 3. (a) Spatial dependences of the spin -wave intensity integrated over the width of the waveguide obtained from the measurements and micromagnetic simulations, as labelled. Vertical dashed line marks the end of the transition region. (b) Spatial dependence of the spin -wave intensity calculated for the waveguides with the transition length of 10 and 5 m, as labelled. Arrows mark the intensity drop du e to reflections at the end of the transition region. Inset shows the ratio between the intensity detected at a point 0 .5 m beyond the end of the transition region and the intensity detected at its beginning as a function of the transition length. The dat a are obtained at H0=1000 Oe and f=4.5 GHz . 15 FIG. 4. Spatial dependence of the propagation delay measured for a 20 -ns long spin -wave pulse at the carrier frequency f=4.5 GHz . Vertical dashed line marks the end of the transition region. Frequency dependence of the group velocity at the beginning of the transition region ( x=0.5 m) and at its end ( x=10 m), as labelled. Symbols – experimental data. Curves – guide for the eye. The data are obtained at H0=1000 Oe.

2021-11-03

We study experimentally and by micromagnetic simulations the propagation of spin waves in 100-nm thick YIG waveguides, where the width linearly decreases from 2 to 0.5 micrometers over a transition region with varying length between 2.5 and 10 micrometers. We show that this geometry results in a down-conversion of the wavelength, enabling efficient generation of waves with wavelengths down to 350 nm. We also find that this geometry leads to a modification of the group velocity, allowing for almost-dispersionless propagation of spin-wave pulses. Moreover, we demonstrate that the influence of energy concentration outweighs that of damping in these YIG waveguides, resulting in an overall increase of the spin-wave intensity during propagation in the transition region. These findings can be utilized to improve the efficiency and functionality of magnonic devices which use spin waves as an information carrier.

Efficient geometrical control of spin waves in microscopic YIG waveguides

2111.02236v1

Tunable sign change of spin Hall magnetoresistance in Pt/NiO/YIG structures Dazhi Hou,1, 2Zhiyong Qiu,1, 2,Joseph Barker,3Koji Sato,1Kei Yamamoto,3, 4, 5Sa ul V elez,6Juan M. Gomez-Perez,6Luis E. Hueso,6, 7F elix Casanova,6, 7and Eiji Saitoh1, 2, 3, 8 1WPI Advanced Institute for Materials Research, Tohoku University, Sendai 980-8577, Japan 2Spin Quantum Rectication Project, ERATO, Japan Science and Technology Agency, Sendai 980-8577, Japan 3Institute for Materials Research, Tohoku University, Sendai 980-8577, Japan 4Institut f ur Physik, Johannes Gutenberg Universit at Mainz, D-55099 Mainz, Germany 5Department of Physics, Kobe University, 1-1 Rokkodai, Kobe 657-8501, Japan 6CIC nanoGUNE, 20018 Donostia-San Sebastian, Basque Country, Spain 7IKERBASQUE, Basque Foundation for Science, 48011 Bilbao, Basque Country, Spain 8Advanced Science Research Center, Japan Atomic Energy Agency, Tokai 319-1195, Japan Spin Hall magnetoresistance (SMR) has been investigated in Pt/NiO/YIG structures in a wide range of temperature and NiO thickness. The SMR shows a negative sign below a temperature which increases with the NiO thickness. This is contrary to a conventional SMR theory picture applied to Pt/YIG bilayer which always predicts a positive SMR. The negative SMR is found to presist even when NiO blocks the spin transmission between Pt and YIG, indicating it is governed by the spin current response of NiO layer. We explain the negative SMR by the NiO 'spin- op' coupled with YIG, which can be overridden at higher temperature by positive SMR contribution from YIG. This highlights the role of magnetic structure in antiferromagnets for transport of pure spin current in multilayers. Magnetoresistance plays essential roles in providing both a fundamental understanding of electron transport in magnetic materials and in various technological ap- plications. Anisotropic magnetoresistance (AMR) [1, 2], giant magnetoresistance [3, 4], and tunneling magnetore- sistance [5{8] underpin technologies in sensors, memo- ries, and data storage. Recent studies of thin lm bi- layer systems comprised of a normal metal (NM) and a ferromagnetic insulator (FI) revealed a new type of mag- netoresistance called spin Hall magnetoresistance (SMR) [9{11], originating from the interplay between the spin accumulation at the NM/FI interface and the magnetiza- tion of the FI layer. When the NM layer has a signicant spin-orbit interaction, e.g. Pt, an in-plane charge current jcinduces a spin current via the spin Hall eect, which in turn generates a spin accumulation near the NM/FI interface. At the same time, this spin accumulation is aected by the orientation of the magnetization in the ferromagnet. The conductivity of the NM layer is thus subject to a magnetization dependent modication to the leading order in 2 SHE, whereSHEis the spin Hall angle in the NM layer. Since the discovery of SMR, experimental studies were instigated in various systems [12{19]. The amplitude of SMR is dened as the dierence of the resistivities with an applied eld, H, parallel (k) and perpendicular ( ?) tojc:SMR=k?. This is predicted to be always positive because when Hkjc, the FI can absorb more spin current, by which the back ow required to ensure the stationary state is reduced at the FI/NM interface, in turn causing less secondary forward charge current, and therefore gives : k> ?[9, 10]. Positive SMR is found in most experimental observations. Very recently, a negative SMR ( k<?) was reportedwhen an antiferromagnetic (AFM) insulator, in this case NiO, is inserted between Pt and YIG [20]. The negative SMR was also found to revert to the conventional positive sign at higher temperatures. Signal contamination from other magnetoresistances such as AMR was excluded by a systematic eld angle dependence measurement. This re- sult challenges the present understanding of SMR. Since the SMR does not change its sign in the Pt/YIG bilayer structure, the NiO layer must be the cause. However, it is not clear why NiO should give a negative SMR since antiferromagnets are thought only to aect the eciency of the spin communication between Pt and YIG [21{26]. In this letter, we report the temperature dependence of SMR in Pt/NiO/YIG structures with dierent thick- nesses of NiO. The temperature at which the SMR be- comes negative is found to depend on the NiO thickness. The anomalous negative SMR at low temperatures is ex- plained from a `spin- op' conguration whereby the N eel order of the NiO is perpendicularly coupled to the mag- netization of YIG [27]. As the spin conductivity of NiO increases with increasing temperature [24{26], the mo- ments of the YIG beneath have an increasing in uence on the total SMR signal. The positive SMR contribution from YIG competes the negative SMR from NiO. At the sign change, the competition leads to a vanishing SMR. Above, in the high temperature regime, the positive SMR of the YIG dominates. We introduce a phenomenological model to describe the competition between the positive and negative SMR contributions, which reproduces the NiO thickness dependent SMR sign change behaviors in Pt/NiO/YIG. An epitaxial YIG lm of thickness 3 m was grown on a gadolinium gallium garnet (111) substrate prepared by the liquid phase epitaxy. NiO lms of dierent thick-arXiv:1610.07362v2 [cond-mat.mes-hall] 8 Mar 20172 30 40 50 60 70 80 YIG(444) NiO(222)Intensity (a.u.) 2θ (deg)NiO(111) Pt NiO YIG 3 nm FIG. 1. X-ray diraction patterns of a 50 nm NiO lm on YIG(111). Inset shows the cross section TEM photo for a Pt/NiO/YIG trilayer measured in the transport experiment. nesses were grown by sputtering onto the YIG at 400C. The lm was then covered with 4 nm of sputtered Pt. The X-ray diraction patterns of a 50 nm NiO lm on YIG is plotted in Fig. 1, which only shows (111) and (222) NiO peaks of narrow line width. It suggests that the NiO lm is of high crystallinity and a (111) preferred orienta- tion. The inset in Figure 1 shows a representative cross- section TEM picture for a Pt/NiO/YIG sample, which conrms a good thickness uniformity and clean interface. Figure 2(a) shows the illustration of the magnetore- sistance (MR) measurement setup and the denition of magnetic eld angles. Standard four-probe method is employed for the MR observation at current density 108A/m2, and MR can be detected either by sweep- ingHalong a xed direction or by rotating Hof the same magnitude [9]. Figure 2(b) shows the MR measured by Hsweeping in a Pt/NiO(2.5 nm)/YIG sample at eld angle= 0for various temperatures. The range of magnetic eld over which the magnetoresistance occurs, coincides with that of the switching process of YIG [28]. The MR data for T >140 K is consistent with the predic- tionk> ?of the SMR theory. When T=140 K, the MR nearly vanishes. For T <140 K, a sign change of MR is observed and the MR amplitude increases with decreas- ing temperature. The MR data from the same sample at eld angle= 90is plotted in Fig. 2(c), which shows the same feature of the sign change. The SMR ratio
SMR=xxextracted from Fig. 2(b) and 2(c) are plot- ted in Fig. 2(d). Figure 2(e) and 2(f) show the eld angle dependence of resistance in Pt/NiO(2 nm)/YIG at 260 K and 20 K, which not only reproduces the MR sign change behaviour, but conrms the SMR-type eld angle depen- dence symmetry as well [20]. Thus, it looks reasonable to claim that SMR is the dominant contribution for the MR in Pt on NiO/YIG, since other mechanisms such as anisotropic magnetoresistance will cause a dierent eld angle dependence [29]. However, the sign change of the magnetoresistance in the low temperature regime seems to be at odds with SMR which, conventionally, can only be positive [10]. 0 100 200 300 400-40-2002040 α = 0˚ ΔSMR/xx (10-6) T (K)α = 90˚ Rxx (Ω)(d) 2e-5ΔSMR/2xx= 0˚ αt = 2.5 nm NiO -400 -200 0 200 400 H (Oe)(b) 0 -400 -200 0 200 40 H (Oe) 20 K40 K 60 K100 K140 K180 K220 K260 K300 K340 K 2e-5= 90˚ α(c) 20 K40 K 60 K100 K140 K180 K220 K260 K300 K340 K/ xx(H)xx(400 Oe) / xx(H)xx(400 Oe) (a) Hβ xz (a) γ x y,J z H e(a) Hαx y,J ez y,J e(a) 98.29898.29998.30098.30198.302Rxx (Ω) 62.305262.305662.306062.3064 α β γ -90 0 90 180 270 α, β, γ (deg ) 20K260Kα β γ -90 0 90 180 270 α, β, γ (deg )(e) (f) Pt NiO YIGFIG. 2. (a), The illustration for the magnetoresistance mea- surement setup for various magnetic eld ( H) orientations. , and are the eld angles dening the Hdirections when H is applied in the x-y,x-zandy-zplanes, respectively. (b), (c), Magnetoresistance measured by Hsweeping for a Pt/NiO(2.5 nm)/YIG at = 0and 90for various temperatures. (d), Temperature dependence of the SMR ratio
SMR=xxfor Pt/NiO(2.5 nm)/YIG at = 0and 90. (e), (f), Field an- gle dependent resistance measured for Pt/NiO(2 nm)/YIG at 260 K and 20 K with jHj= 20000 Oe, which shows positive and negative SMR, respectively. Fig. 3(a) shows the temperature dependence of the SMR ratio measured in Pt/NiO/YIG devices with dif- ferent NiO thicknesses, dNiO. The change in sign of the SMR occurs at higher temperatures in larger dNiOsam- ples. ThedNiOdependence proves to be a key piece of in- formation for understanding the negative SMR. Further- more, the SMR ratios have (positive) maxima at higher temperatures for thicker NiO samples. These dNiOde- pendent characteristics show a quantitative eect of the NiO on the SMR modulation, rather than a nuanced in- terface eect [30]. To gain further insight into the temperature depen- dence of spin transport in NiO, we carried out spin pump- ing measurements for the same samples, in which spin current is injected from YIG through NiO to generate a voltage in Pt via the inverse spin Hall eect (ISHE) [22]. The Pt/NiO/YIG device is placed on a coplanar waveguide which serves as a 5 GHz microwave source at 14 dbm, and the details of the experimental setup can be found elsewhere [24]. The ISHE voltage VISHE from all the samples is plotted against Tin Fig. 3(b), the be- haviour of which is very similar to the result we found in Pt/CoO/YIG [24]: spin transmission is nearly zero for3 0 100 200 300 400024680 100 200 300 400-20020406080 T (K) 2.0 nm 2.2 nm 2.5 nm 2.7 nm 4.0 nm 5.4 nm 7.0 nm 15 nm 30 nm T (K) ΔSMR/xx (10-6 ) VISHE (μV) 0 200 40001V V(b)(a) FIG. 3. (a), The SMR ratio measured in Pt/NiO( dNiO)/YIG devices with dierent NiO thickness dNiOat various temper- atures, which shows that the SMR sign change temperature is lower for a thinner NiO sample. The SMR ratio peak posi- tions are marked by arrows. Negative SMR at low tempera- tures can be observed for all the NiO thickness except dNiO= 30 nm. The dashed curves are the tting based on Eq. (2). (b),VISHE in Pt/NiO/YIG devices versus temperature from spin pumping measurement. The peak positions are marked by arrows, which are found to be close to the SMR ratio peak positions marked in Figure 2a. The inset shows the normal- izedVISHE temperature dependence. low temperature limit and increases with temperature to reach the maximum around the N eel point. At room temperature, VISHE shows a non-monotonic dNiOdepen- dence, which is consistent with previous result. Fig. 3(b) inset shows the normalized VISHE temperature depen- dence, in which the data for dNiO= 5.4 nm, 7 nm and 15 nm collapse into a single curve. This conrms that the VISHE is governed by the NiO spin conductivity, which shows the same Tdependence when NiO is thick enough to exhibit bulk property. For dNiO= 30 nm,VISHE is below our measurement sensitivity 5 nV. An important conclusion can be drawn by combining the results from SMR and spin pumping measurements: the negative SMR does not rely on the spin transmis- sion between Pt and YIG, because it reaches the largest magnitude for the lowest temperature at which NiO spin conductivity vanishes. This argument can be further sup-ported by the fact that the negative SMR is present even fordNiO= 15 nm, where the NiO spin conductivity is nearly zero throughout the entire temperature range. It indicates that the negative SMR is not caused by the magnetic moment of the YIG layer but that of the NiO layer, which is beyond any model based on spin commu- nication between YIG and Pt [10, 31]. Let us next provide an explanation for the negative SMR. The SMR in the trilayer system in this experiment is governed by the spin current through the Pt/NiO in- terface, which also re ects the eect of the presence of the NiO/YIG interface. The sign change and the thick- ness dependent behavior can be understood by assum- ing a `spin- op' coupling between NiO and YIG [27, 32], which means the antiferromagnetic axis (N eel vector unit nAFM) in NiO is perpendicular to the YIG magnetization unit vectormFIas illustrated in Fig. 4(a). Although a perpendicular coupling has not yet been conrmed ex- perimentally for NiO on YIG, spin- op coupling between NiO and other ferromagnets is quite common and well understood[27, 33, 34]. For dNiObelow the domain wall width of NiO (15 nm) [35], which is the case for nearly all the samples, nAFM tends to be uniform in NiO, which is strongly coupled with YIG and can be manipulated by magnetic eld [36]. Thus, nAFM is always perpendicu- lar toHbelow the N eel temperature, because the mFI is parallel to H. In the low temperature limit, e.g. 10 K, the spin current generated in Pt can not penetrate through the NiO, thus the SMR signal is only caused by the NiO layer. The NiO local moments perpendicu- lar toHgives rise to a 90-degree phase shift in the SMR eld angular dependence with respect to the conventional SMR [9]. Such a 90-degree phase shift in a four-fold SMR eld angular dependence is equivalent to a sign reversal in the conventional denition of MR, which explains the negative SMR in Pt/NiO/YIG at low temperatures. For dNiO= 30 nm which is beyond the domain wall width, nAFM at the Pt/NiO interface decouples with mFIand does not respond to H, which explains the vanishing of the negative SMR. At higher temperatures, but below the N eel point, antiferromagnetic order is maintained but the spin cur- rent from Pt has some transmission through NiO, which makes the eect of the YIG more visible as illustrated in Fig. 4(b). The negative SMR contribution from NiO and positive SMR contribution from YIG compete with each other. With increasing temperature, NiO becomes more transparent to the spin current, so the SMR con- tribution from YIG is enhanced. The SMR from NiO may also be suppressed because of the attenuation of the antiferromagnetic order at elevated temperatrues. As a result, the zero point of the SMR occurs at a temperature where the antiferromagnet is still in the ordered phase. Thinner NiO layers have a lower N eel point due to the - nite size eect [37], hence the SMR also changes the sign at lower temperatures in thinner-NiO samples, which is4 (d) ΔSMR/xx T0T( = 0)ΔSMR T 0 H H H(a) ) c ( ) b ( mFInAFM mFInAFM mFIPt YIGNiO FIG. 4. Illustrations for the magnetic structure and spin transport in Pt/NiO/YIG at dierent temperatures. The red and green arrows represent the phenomenologically described spin currents, j1and j2in Eq. (1), respectively. The length of the arrow describes the penetration depth of the spin current. (a),Tclose to the low temperature limit. (b), Tfar above the low temperature limit and lower than the N eel temperature. (c),Thigher than the N eel temperature. (d), Illustration of T-dependent SMR in which the temperatures corresponding to the conditions in Fig. 4(a), (b) and (c) are marked with red circle. in accordance with our observation shown in Fig. 3(a). Around the N eel point as illustrated in Fig. 4(c), the spin transparency of NiO are maximized [24], where the SMR contribution from YIG reaches its peak value and the SMR contribution from NiO vanishes. As ex- plained above, all the main features of the SMR data in Pt/NiO/YIG, such as negative SMR at low tempera- tures,dNiOdependent sign change temperature and peak temperature, can be interpreted by the `spin- op' con- guration. Figure 4(d) shows an illustration of SMRtemperature dependence, in which the temperature cor- responding to these features are marked. We note that negative SMR has also been reported in bilayers of Pt on gadolinium iron garnet and Ar-sputtered YIG, in which the garnet interface moments can align perpendicularly toH[30, 38]. A simple phenomenological model based on the picture discussed above can also provide a quantitative descrip- tion of the observed SMR temperature dependence. Let us consider a NM/AFM/FI trilayer system. The key as- sumption is that we can describe the spin current through the NM/AFM interface by ejs=GAFnAFM(nAFMs) +t(T)mFI(mFIs) =ej1+ej2; (1) GAFis the real part of the spin mixing conductance at NM/AFM interface. sis the spin accumulation at the same interface. The rst term, which we denote by ej1, is what is expected for NM/AFM bilayer systems as seen in the case studied in Ref. [39]. We have introduced the second term, which is denoted by ej2, to phenomenologi- cally capture the eect of the FI layer. t(T) encapsulates the temperature dependent transparency of the AFM to the spin current. In the case that the AFM is completely transparent the NM/FI bilayer result mFI(mFIs) is recovered. The linear combination of the NM/AFM and NM/FI terms has been chosen in an attempt to em- ulate our SMR data in the NM/AFM/FI system, seen in Fig. 3(a), which seems to indicate a crossover from NM/AFM bilayer like behavior at low temperatures to NM/FI bilayer like behavior for higher temperatures. Once we admit the form of the interfacial spin current in Eq. (1), we can calculate the SMR by employing the diusion equation and the Onsagar principle, according to Refs. [10, 39]. The SMR contribution to the longitu- dinal resistivity then is given by  0=22 SHE2 N dNGAFcos2n+t(T) cos2m+t(T)GAFsin2(mn) 1 +GAF+t(T) +2t(T)GAFsin2(mn)tanh2dN 2N ; (2) where we dened = (2N=) coth(dN=N) withN andSHE being the spin diusion length and the spin Hall angle in NM, respectively, and =1 0is the con- ductivity of the NM layer. Here, n(m)denotes the angle betweennAFM(mFI) and the applied current jcin NM. Now we set out a hypothesis that the crossover between the negative and positive SMR is of the same origin as the temperature dependence of the spin pumping signal (Fig. 3(b)). In order to support it, the temperature de- pendence of t(T) is obtained by tting to the spin pump-ing data. The resulting function is then used alongside the other parameters in Eq. (1) to t the SMR data to test the validity of our model. Based on the observation that the ISHE signal in Fig. 3(b) is roughly exponential in the intermediate tem- perature regime, we employ VISHE/t(T)/eaT1 to re- produce the temperature dependence of both spin pump- ing and SMR. The exponential behavior may not apply near the N eel temperature and the data points near and above the N eel temperature have been excluded from the5 tting. Under these assumptions, acan be determined from the spin pumping data (TABLE I). We then t =jm=0=jm==2based on Eq. (2) to the experimentally obtained SMR ratio
SMR=xx in Fig. 3(a) using the tted value of afrom theVISHE data. We x N= 1:5nm,dN= 4:0nm,0=1= 860 nm, andSHE = 0:05, which are taken to be relevant values to the present experiment, and we further deter- mineGAFandGFfrom the data, where the latter two are dened by t(T) =GF(eaT1);nm==2, re- spectively. The temperature dependence of 0andSHE is ignored since they scale in some powers of T, which is wiped out by the exponential change in t(T). The t- ting curves can quantitatively reproduce the SMR sign change behavior as shown in Fig. 3(a), and the tting parameters are summarized in TABLE I. dNiOa[K1]102GAF GF 2:0 1:830:22 3:580:3210128:390:571011 2:2 1:380:19 4:480:1710127:780:261011 2:7 1:420:10 3:670:0910123:010:081011 4:0 1:160:09 2:460:1310122:220:141011 TABLE I. The results of the tting with the data from the SMR and spin pumping signals. The parameters are dened in the main text. The units of the last two columns are both [ 1m2]. 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2016-10-24

Spin Hall magnetoresistance (SMR) has been investigated in Pt/NiO/YIG structures in a wide range of temperature and NiO thickness. The SMR shows a negative sign below a temperature which increases with the NiO thickness. This is contrary to a conventional SMR theory picture applied to Pt/YIG bilayer which always predicts a positive SMR. The negative SMR is found to persist even when NiO blocks the spin transmission between Pt and YIG, indicating it is governed by the spin current response of NiO layer. We explain the negative SMR by the NiO 'spin-flop' coupled with YIG, which can be overridden at higher temperatures by positive SMR contribution from YIG. This highlights the role of magnetic structure in antiferromagnets for transport of pure spin current in multilayers.

Tunable sign change of spin Hall magnetoresistance in Pt/NiO/YIG structures

1610.07362v2

arXiv:2112.01727v1 [quant-ph] 3 Dec 2021Microwave Amplification in a PT-symmetric-like Cavity Magnomechanical System Hua Jin1, Zhi-Bo Yang1, Jing-Wen Jin1, Jian-Yu Liu1, Hong-Yu Liu1,∗and Rong-Can Yang2,3,4† 1Department of Physics, College of Science, Yanbian Univers ity, Yanji, Jilin 133002, China 2College of Physics and Energy, Fujian Normal University, Fujian Provincial Key Laboratory of Quantum Manipulation a nd New Energy Materials, Fuzhou, 350117, China 3Fujian Provincial Engineering Technology Research Center of Solar Energy Conversion and Energy Storage, Fuzhou, 350117, Chin a and 4Fujian Provincial Collaborative Innovation Center for Adv anced High-Field Superconducting Materials and Engineering, Fuzhou, 35011 7, China (Dated: December 6, 2021) We propose a scheme that can generate tunable magnomechanic ally induced amplification in a double-cavityparity-time-( PT-)symmetric-likemagnomechanical systemunderastrongco ntrol and weak probe field. The system consists of a ferromagnetic-mat erial yttrium iron garnet (YIG) sphere placed in a passive microwave cavity which is connected with another active cavity. We reveal that ideally induced amplification of the microwave probe signal may reach the maximum value 106when cavity-cavity, cavity-magnon and magnomechanical coupli ng strengths are nonzero simultaneously. The phenomenon might have potential applications in the fiel d of quantum information processing and quantum optical devices. Besides, we also find the phenom ena of slow-light propagation. In this case, group speed delay of the light can achieve 3 .5×10−5s, which can enhance some nonlinear effect. Moreover, due to the relatively flat dispersion curve , the proposal may be applied to sensitive optical switches, which plays an important role in storing p hotons and quantum optical chips. I. INTRODUCTION The interaction between light and matter is an impor- tant subject in the field of quantum optics. The study of light toward the perspective of quantum leads to some interesting phenomena different from classical ones. One of the most famous phenomenon is induced transparency (such as electromagnetically/optomechanically induced transparency)[ 1–16], aswellasinducedabsorption[ 3,14] and induced amplification [ 15–20] which has been widely studied in current decades. Besides, signal amplification whose aim is to increase signal-to-noise ratio is signifi- cantly crucial in the field of quantum information and quantum optics. It is known that optical amplification usually results from the inversion of particle numbers under the action of a pumping field and stimulated ra- diation. It can directly amplify optical signals without converting them into electrical ones so as to possess a high degree of transparency on the format and rate of signals, making the whole optical fiber communication transmission system more simple and flexible [ 15,16]. It is noted that there are many mechanisms of light ampli- fication, such as adding external drive and changing de- tuning conditions. Through the coupling effect of strong photon tunneling, double-cavityOMS not only showsthe characteristics of photomechanically-induced absorption, photomechanically-inducedamplificationandsimplenor- mal mode splitting (NMS), but also adjusts the photon tunneling intensity. The transformation from photome- chanical induced absorption to photomechanical induced amplification can be further realized. In this article we ∗liuhongyu@ybu.edu.cn †rcyang@fjnu.edu.cnbuild our mechanism by adding active cavities. In ad- dition, the added gain scheme is widely used in quan- tum information and quantum communication due to its excellent characteristics of convenience and easy adjust- ment and may very useful for optical and microwave am- plifiers [21]. Parity-time( PT) symmetry, the non-Hermitian Hami- tonian, which has a real spectra was proposed by Beb- der in 1998 firstly and attracted wide attention [ 22–30]. SincePT-symmetry requires a strict balance between loss and gain. However the balance condition may be too difficult for the realistic implementation, especially when tiny disturbances are inevitable. PT-symmetric- like system not requiring the strict balance can still fol- low the predictions of the PT-symmetry in many cases and thus attract considerable attention [ 31,32]. At the exceptional point, where the system undergoes the transition from the PT-symmetric-like phase and PT- symmetric-like broken, pairs of eigenvalues collide and become complex has manifested in various physical sys- tem, such as photonics, electronics, acoustics, phonon- ics. And the OMIA of the PT-symmetric OMS has been achieved in the whispering-gallery-mode microtoroidal cavities. Common PT symmetric systems have two- cavity systems, but there are also examples of single cavities achieving effective gain by introducing external drives or other means [ 27]. In the past few years, cavity magnonics, a new inter- discipline, attracted much attention. It mainly explores the interaction between confined electromagnetic fields and magnons, especially Yttrium iron garnet (YIG) [ 33– 41]. The reason is that the Kittel mode within YIG has a low damping rate and holds great magnonic nonlinear- ities [39]. In addition, the high spin density of magnons allows strong coupling between magnons and photons,2 Probe Field Control Field YIG may xz b a FIG. 1. Schematic of the setup studied in this paper. A cavity magnomechanical system consists of one ferromagnet ic yttrium iron garnet (YIG) sphere placed inside a passive mi- crowave cavity, which connected with an auxiliary cavity. A bias magnetic field is applied in the zdirection on the sphere toexcites themagnon modes, which are strongly coupled with the cavity field. In the YIG sphere, bias magnetic field ac- tivates the magnetostrictive interaction. The magnetic co u- pling strength of a magnon depends on the diameter of the sphere and the direction of the external bias field [ 45]. We assumed that the YIG’s magnomechanical interactions were directly enhanced by microwave driving (in the ydirection) its magnon mode. Cavity, phonon, and magnon modes are labeledai,b,m(i= 1,2). giving rise to quasiparticles, i.e. the cavity-magnon po- laritons. Then strong coupling between magnons and cavity photons can be observed at both low and room temperature. In this case, a large number of quantum- information-related problems have been studied by this method, including the coupling of magnons with super- conductingqubits, observationofbistability [ 42,43], cav- ity spintronics, energy level attraction of cavity mag- netopolaron, magnon dark modes. Other interesting phenomena including magneton-induced transparency (MIT), magnetically induced transparency (MMIT), and magnetically controlled slow light have also been stud- ied [44]. In this paper, we utilize a cavity-magnomechanical system, which consists of a YIG sphere placed inside a three-dimensional microwave cavity that is connected with anpassivecavityto realizemicrowaveamplification. Through the discussion the properties of absorption and transmission, we obtain the amplification in the context ofPT-symmetric-like cavity magnomechanical system. The remaining parts are organized as follows. In Sec.II,weintroducethemodelofourproposal. InSec. II, we plot the magnomechanically induced transparency window profiles. In Sec. III, we explore magnomechan- ically induced amplification of the PT-symmetric-like cavity magnomechanical system and slow light propega- tion Sec. IV, we present the conclusion of our work.II. MODEL AND HAMILTONIAN We use a hybrid cavity magnomechanical system that consists of one high-quality YIG sphere placed inside a microwave cavity which connects with another empty cavity, as shown in Fig. 1. The YIG sphere has 250 µm in diameter and ferric ions Fe+3of density ρ= 4.22× 1027m−3. This causes a total spin S= 5/2ρVm= 7.07×1014, where Vmis the volume of the YIG and Sis the collective spin operator which satisfies the alge- bra i.e., [ Sα,Sβ] =iεαβγSγ. A uniform bias magnetic field (along z direction) is applied on the sphere, exciting the magnon mode that is then coupled to the first cav- ity field via magnetic-dipole interaction. In addition, the excitation of the magnon mode (i.e. Kittel mode) inside the sphere leads to a variable magnetization that results in the deformation of its lattice structure. The magne- tostrictiveforcecausesvibrationsofthe YIG, resultingin magnon-phonon interaction within YIG spheres [ 45]. It is noted that the single-magnon magnomechanical cou- pling strength depended on sphere diameter and direc- tion of the external bias field is very weak. In this case, magnomechanical interaction of YIG can be enhanced by directly driving its magnon mode via an external mi- crowave field. Furthermore, the first cavity is not only coupled to the second cavity, but also driven by a weak probe field. With consideration of the situation, the Hamiltonian for the whole system reads [ 44,46] H//planckover2pi1=ωmˆm†ˆm+ωa1ˆa† 1ˆa1+ωa2ˆa† 2ˆa2+ωbˆb†ˆb +g1(ˆm†ˆa1+ ˆmˆa† 1)+g2ˆm†ˆm(ˆb+ˆb†) +J(ˆa† 1ˆa2+ˆa† 2ˆa1)+iΩ(ˆm†e−iωput−ˆmeiωput) +iεpr(ˆa† 1e−iωprt−ˆa1eiωprt),(1) where ˆa† j(j= 1,2), ˆm†andˆb†(ˆaj, ˆmandˆb) are the cre- ation (annihilation) operators of the jth cavity, magnon and phonon, respectively. They all satisfy the stan- dard commutation relations for bosons. ωaj,ωm,ωb represent the resonance frequencies for the jth cavity, magnonandphonon, respectively. g1(J)denotesthecou- pling strength between the first cavity mode and magnon (the second cavity), and g2is the coupling constant be- tween magnon and phonon. It is noted that the fre- quencyωmis determined by the gyromagnetic ratio γ and external bias magnetic field Hi.e.,ωm=γHwith γ/2π= 28GHz. In addition, Ω =√ 5/4γ√ NB0is the Rabi frequency, which is dependent of the coupling strength of the driving field with amplitude B0and fre- quencyωpu. Andωpris the probe field frequency having amplitude εpr=/radicalbig 2Ppκ1//planckover2pi1ωpr. It should be noted that wehaveignoredthenonlinearterm Kˆm†ˆm†ˆmˆminEq.(1) that may arise due to strongly driven magnon mode [ 43] so as to K|/angbracketleftm/angbracketright|3≪Ω. With the rotating wave approxi-3 mation, we can rewrite the whole Hamiltonian as H//planckover2pi1= ∆mˆm†ˆm+∆a1ˆa† 1ˆa1+∆a2ˆa† 2ˆa2+ωbˆb†ˆb +g1(ˆm†ˆa1+ ˆmˆa† 1)+g2ˆm†ˆm(ˆb+ˆb†) +J(ˆa† 1ˆa2+ˆa† 2ˆa1)+iΩ(ˆm†−ˆm) +iεpr(ˆa† 1e−iδt−ˆa1eiδt),(2) with ∆ aj=ωaj−ωpu(j= 1,2), ∆m=ωm−ωpu, and δ=ωpr−ωpu. In order to obtain the evolution of aj(t),m(t) andb(t), we use quantum Heisenberg-Langevin equations, which can be expressed by ˙ˆa1=−i∆a1ˆa1−ig1ˆm−κ1ˆa1+εpre−iδt +√2κ1ˆain 1(t)−iJˆa2, ˙ˆa2=−i∆a2ˆa2−κ2ˆa2+√ 2κ2ˆain 2(t)−iJˆa1, ˙ˆm=−i∆mˆm−ig1ˆa1−κmˆm−ig2ˆm(ˆb+ˆb†) +√2κmˆmin(t)+Ω, ˙ˆb=−iωbˆb−ig2ˆm†ˆm−κbˆb+√2κbˆbin(t)(3) whereκ1(κ2),κbandκmare the decay rates of the cav- ities, phonon and magnon modes, respectively. ˆ ain 1(t), ˆain 2(t),ˆbin(t) and ˆmin(t) are the vacuum input noise operators which have zero mean values and satisfies/angbracketleftbig ˆqin/angbracketrightbig = 0(q=a1,a2,m,b). The magnon mode m is strongly driven by a microwave field that causes a large steady-state amplitude corresponds to |/angbracketleftms/angbracketright| ≫1. Moreover, owing to the magnon coupled to the cavity modethroughthebeam-splitter-typeinteraction, thetwo cavity fields also exhibit large amplitudes |/angbracketleftajs/angbracketright| ≫1. Then we can linearize the quantum Langevin equations around the steady-state values and take only the first- order terms in the fluctuating operator:/angbracketleftBig ˆO/angbracketrightBig =Os+ ˆO+e−iδt+ˆO−eiδt[43], where ˆO=a1,a2,b,m.the steady- state solutions are given by a1s=−(ig1ms+iJa2s) i∆a1+κ1,a2s=−iJa1s i∆a2+κ2, bs=−ig2|ms|2 iωb+κb, ms=−ig1a1s+Ω i/tildewide∆m+κm, /tildewide∆m= ∆m+g2(bs+bs∗)(4) In ordertoachieveourmotivationofsignalamplification, we neglect off resonance terms to let ˆO−= 0, but ˆO+ safisfying the relations (iλ−κ1)ˆa1+−ig1ˆm+−iJˆa2++εpr= 0, (iλ−κ2)ˆa2+−iJˆa1+= 0, (iλ−κm)ˆm+−ig1ˆa1+−iGˆb+= 0, (iλ−κb)ˆb+−iG∗ˆm+= 0,(5)0 0.5 1 1.5 2 δ/ωb0100200300400|tp|2 (a) 0 0.5 1 1.5 2 δ/ωb05101520|tp|2 (b) 0 0.5 1 1.5 2 δ/ωb11.21.41.61.8|tp|2 (c) 0 0.5 1 1.5 2 δ/ωb11.21.41.6|tp|2 (d) FIG. 2. The transmission |tp|2spectrum of probe field as function of δ/ωbwhen only interaction between two cavities is nonzero, (a) J/2π= 0.6MHz, (b) J/2π= 0.8MHz, (c) J/2π= 2.0MHz and (d) J/2π= 6MHz. where we have set G=g2ms,λ=δ−ωb,ωai≫κi (i= 1,2), and ∆ a1= ∆a2=/tildewide∆m=ωb. In this case, we can easily obtain ˆa1+=εpr κ1−iλ+J2 κ2−iλ+g12 κm−iλ+|G|2 κb−iλ.(6) By use of the input-output relation for the cavity field εout=εin−2κ1/angbracketlefta1+/angbracketrightand setting εin= 0, the amplitude of the output field can be written as ε′ out=εout εpr=2κ1ˆa1+ εpr. (7) The real and imaginary parts of the output field are Re [ε′ out] =κ1(ˆa1++ ˆa∗ 1+)/εprand Im [ ε′ out] =κ1(ˆa1+− ˆa∗ 1+)/εpr. These factors describe the absorption and dis- persion of the systems, respectively. III. INDUCED AMPLIFICATION AND SLOW LIGHT PROPEGATION IN PT-SYMMETRIC-LIKE MAGNOMECHANICALLY SYSTEMS For the numerical calculation, we use parameters cho- sen from a recent experiment on a hybrid magnome- chanical system, where ωa1/2π=ωa2/2π= 10GHz, ωb/2π= 10MHz, κb/2π= 100Hz, ωm/2π= 10GHz, κ1/2π= 2.0MHz,κm/2π= 0.1MHz,g1/2π= 1.0MHz, G/2π= 3.5MHz,∆ a1= ∆a2=/tildewide∆m=ωb,ωd/2π= 10GHz are set [ 26,33,34].4 0 0.5 1 1.5 2 δ/ωb11.11.21.31.41.5|tp|2 (a) 0 0.5 1 1.5 2 δ/ωb11.11.21.31.41.5|tp|2 (b) 0 0.5 1 1.5 2 δ/ωb11.11.21.31.41.5|tp|2 (c) 0 0.5 1 1.5 2 δ/ωb123456|tp|2 (d) FIG. 3. The transmission |tp|2spectrum of probe field as function of δ/ωbwhen only coupling between magnon and phonon is absent means G= 0,J/2π= 3.0MHz (a) g1/2π= 1.0MHz, (b) g1/2π= 1.2MHz, (c) g1/2π= 1.5MHz and (d) g1/2π= 2.0MHz. At first, we consider the transmission rate |tp|2as a function of the probe detuning δ/ωbin the context of parity-time-( PT-) symmetric-like magnomechanical sys- tem. FromEq.( 7), the rescaledtransmissioncorrespond- ing to the probe field can be expressed as tp= 1−2κ1ˆa1+ εpr. (8) We first depict the transmission spectrum of the probe field against the scaled detuning δ/ωb, for different val- ues ofJin Fig.2, where the phonon-magnon coupling rate and photon-magnon interaction parameters are set to zero, i.e. G=g1= 0. From Fig. 2(a), we can observe that the transmission peak near δ=ωbwhich is asso- ciated with the coupling rate of two cavities can much be larger than 1. The reason is that the gain cavity can scatter photons into the dissipative cavity. From Fig2(a)-(d), transmission coefficient decreases with the increase of coupling strength between two cavities. And we got a downward dip with two peaks From Fig 2(c)- (d), amplification area becomes wider when Jgetting lager simultaneously. This means that we can adjust the transmission coefficient and the size of the amplification region by changing the coupling between the two cavities when the system is double-cavity PT-symmetric-likeand the cavity contains no magnon. Next, we introduced one more coupling constant only set the coupling between magnon-phonon G= 0. We got another peak near δ=ωbcompared with Fig 2(c)-(d), which was caused by coupling between magnon-photon in Fig3. This is because the magnon can scatter the0 0.5 1 1.5 2 δ/ωb012345|tp|2×106 (a) 0 0.5 1 1.5 2 δ/ωb01000300050007000|tp|2 (b) 0 0.5 1 1.5 2 δ/ωb040010001200|tp|2 (c) 0 0.5 1 1.5 2 δ/ωb0100200300400|tp|2 (d) FIG. 4. The transmission |tp|2spectrum of probe field as function of δ/ωbwhenG/2π= 2.0MHz,g1/2π= 6.0MHz (a) J/2π= 0.64MHz, (b) J/2π= 0.8MHz, (c) J/2π= 2MHz and (d)J/2π= 4MHz. photons of the active field into the probe field via indi- rect interaction. However, from Fig 3(b)-(d), the middle peak became taller when g1increases. Hence the effect of light amplification caused by the interaction of magnon- photon get better as g1increase. And with the increasing ofmiddlepeak, theheightoftwopeaksonbothsidesstay the same, that is, the light amplification caused by the coupling between two cavities not affected by g1. How- ever, the amplification effect is not ideal. We show the transmission spectrum when three cou- pling constants are nonzero simultaneously and coupling betweenmagnon-photonlagerthanmagnon-phonon g1> Gin Fig4(a)-(d). We got only one amplification peak when the coupling between two cavities J/2π= 0.64MHz, another upward peak appeared with the in- creasing of J, and the height of two peaks is the same and the amplification effect induced by the interaction of magnon-phonon and magnon-photon were superior at this time, this is because magnon and phonon can also scatter the photons of active cavity field into the probe field. And since the excited states of the cavity field are pumped into higher energy levels, they stay long enough can also be amplified by stimulated radiation. Amplifi- cation area becomes wider when Jgetting lager simul- taneously. These results provide an effective way to re- alize continuous optical amplification and have practical significance for the construction of quantum information processing enhancement signal based on cavity magnetic system. Finally, we plotted the transmission spectrum of the probe field against the scaled detuning δ/ωb, for different values of /tildewide∆m. From Fig 5(a)-(d), The obvious displace-5 0 0.5 1 1.5 2 δ/ωb020406080|tp|2 (a) 0 0.5 1 1.5 2 δ/ωb020406080|tp|2 (b) FIG. 5. The transmission |tp|2spectrum of probe field as function of δ/ωbwhen three coupling constants are nonzero, (a)/tildewide∆m= 0.5ωb, (b)/tildewide∆m= 1.5ωb. 0 0.5 1 1.5 2 δ/ωb-0.501233.5τg(s)×10-5 FIG. 6. The group delay τgas functions of δ/ωbwhenG= 2MHz,J/2π= 6.3MHz,g1/2π= 6.1MHz. ment ofthe twopeaksmeansthat wecannotonly change the value of amplification and the size of the amplifica- tion region by adjusting the coupling strength, but also flexibly change the location of the amplification region. Moreover, the phase φtof the output field can be givenas φt= arg[εout] (9) And the rapid phase dispersion of output field can cause the group delay, which can expressed as τg=∂φt ∂ωpr(10) From Fig. 6shows that the group delay τgas a func- tion of the detuning δ/ωbwhen three coupling constants are present. We can observe double peaks and double dips, peaks corresponding positive group delay i.e., slow light propagation, dips corresponding negative group de- lay means fast light propagation. And we can realize group speed delay of 3 .5×10−5s, a tunable switch from slow to fast can be achieved by adjusting the gain of ac- tive cavity or coupling constants. IV. CONCLUSION In conclusion, we study the transmission of probe field inthesituationof PT-symmetric-likeunderastrongcon- trol field in a hybrid magnomechanical system in the microwave regime and realized ideal induced amplifica- tion when three coupling constants are nonzero simulta- neously, which due to gain cavity, magnon and phonon can also scatter the photons into the dissipative cavity. Therefore, our results are not only providing rich scien- tific insight in terms of new physics but also potentially have important long-term technological implications, in- cluding the development of on-chip optical systems that support states of light that are immune to back scat- ter, are robust against perturbation and feature guar- anteed unidirectional transmission. Then we achieved a group delay of 3 .5×10−5seconds. Slowing down the en- ergyspeed oflight allowsphotonsto interact with matter enough to enhance some nonlinear effects. And because the dispersion curve is relatively flat, a small change in frequency will also cause a large change in photon mo- mentum, so it can be made into a more sensitive opti- cal switch. Finally, the slow light effect slows down the energy speed of light, which can play a role of storing photons and quantum optical chips. ACKNOWLEDGMENTS This work is supported by the National Natural Sci- ence Foundation of China (Grant No. 62165014) and the Fujian Natural Science Foundation (Grant No. 2021J01185). [1] S.E. Harris, J.E. Field, Phys.Rev.Lett. 64, 1107 (2019). [2] R. Thomas, C. Kupchak, G. S. Agarwal and A. I.Lvovsky, Opt. Express. 21, 6880 (2013). [3] H. Fredrik, S. Albert, J. K. Tobias and H. Hans, New. J.6 Phys.14, 123037 (2012). [4] H. Y. Ma and L. Zhou, Phys. Rev. B. 22, 024204 (2013). [5] K. J. Boller, A. Imamoglu and S. E. Harris, Phys. Rev. Lett.66, 2593 (1991). [6] M. Fleishhauer, A. Imamoglu and J. P. Marangos, Rev. Mod. Phys. 77, 633 (2005). [7] S. M. Huang and G. S. 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2021-12-03

We propose a scheme that can generate tunable magnomechanically induced amplification in a double-cavity parity-time-(PT -) symmetric-like magnomechanical system under a strong control and weak probe field. The system consists of a ferromagnetic-material yttrium iron garnet (YIG) sphere placed in a passive microwave cavity which is connected with another active cavity. We reveal that ideally induced amplification of the microwave probe signal may reach the maximum value 1000000 when cavity-cavity, cavity-magnon and magnomechanical coupling strengths are nonzero simultaneously. The phenomenon might have potential applications in the field of quantum information processing and quantum optical devices. Besides, we also find the phenomena of slow-light propagation. In this case, group speed delay of the light can achieve 0.000035s, which can enhance some nonlinear effect. Moreover, due to the relatively flat dispersion curve, the proposal may be applied to sensitive optical switches, which plays an important role in storing photons and quantum optical chips.

Microwave Amplification in a PT -symmetric-like Cavity Magnomechanical System

2112.01727v1

Determination of the origin of the spin Seebeck eect - bulk vs. interface eects Andreas Kehlberger,Ren e R oser, Gerhard Jakob, and Mathias Kl aui Institute of Physics, University of Mainz, 55099 Mainz, Germany Ulrike Ritzmann, Denise Hinzke, and Ulrich Nowak Department of Physics, University of Konstanz, D-78457 Konstanz, Germany Mehmet C. Onbasli, Dong Hun Kim, and Caroline A. Ross Department of Materials Science and Engineering, Massachusetts Institute of Technology, Cambridge, Massachusetts 02139, USA Matthias B. Jung eisch and Burkard Hillebrands Fachbereich Physik and Landesforschungszentrum OPTIMAS, Technische Universit at Kaiserslautern, Kaiserslautern 67663, Germany (Dated: November 2, 2021) The observation of the spin Seebeck eect in insulators has meant a breakthrough for spin caloritronics due to the unique ability to generate pure spin currents by thermal excitations in insulating systems without moving charge carriers. Since the recent rst observation, the under- lying mechanism and the origin of the observed signals have been discussed highly controversially. Here we present a characteristic dependence of the longitudinal spin Seebeck eect amplitude on the thickness of the insulating ferromagnet (YIG). Our measurements show that the observed behavior cannot be explained by any eects originating from the interface, such as magnetic proximity eects in the spin detector (Pt). Comparison to theoretical calculations of thermal magnonic spin currents yields qualitative agreement for the thickness dependence resulting from the nite eective magnon propagation length so that the origin of the eect can be traced to genuine bulk magnonic spin currents ruling out parasitic interface eects. PACS numbers: 72.20.Pa, 72.25.Mk, 75.30.Ds, 85.80.-b Graduate School Materials Science in Mainz, Staudinger Weg 9, 55128, GermanyarXiv:1306.0784v1 [cond-mat.mtrl-sci] 4 Jun 20132 INTRODUCTION In the fast evolving eld of spin caloritronics[1] many interesting discoveries have been made, such as the magneto Seebeck eect[2], and the spin Seebeck eect (SSE) in metals[3], and semiconductors[4]. One of the most interesting eects is the SSE in ferromagnetic insulators (FMI)[5], such as yttrium iron garnet (YIG). Even in insulators this eect oers the possibility to generate a pure spin current by just thermal excitation. Hence this spin current excited in an insulating system is not carried by moving charge carriers but by excitations of the magnetization, known as magnons. Common theories explain this magnonic SSE, being due to a dierence between the phonon- and magnon temperatures TNandTm[6, 7], while other theories rely on a strong local magnon-phonon[8] coupling. Up to now no experimental method has been capable of directly observing this temperature dierence of magnons and phonons[9] so that the origin of the genuine SSE in the transverse conguration is still unclear. Furthermore, in the transverse conguration a thermal gradient is generated in the lm plane, while the detection layer is on top of the ferromagnetic lm. Due to dierences in the thermal conductivity of the substrate and ferromagnetic lm and the thickness dierence between lm and substrate as well as the temperature dierences between sample and environment, it is challenging to generate only a pure in-plane thermal gradient without an out-of-plane component[10]. In conductors this out-of-plane component of the thermal gradient will unavoidably lead to parasitic eects, such as the anomalous Nernst eect, that superimpose with any genuine SSE signals in the transverse geometry. For insulators an alternative geometry provides a better controlled conguration. The so called longitudinal conguration[11] establishes the thermal gradient in the out-of-plane direction across the substrate, the ferromagnetic thin lm and the detection layer on top. This opens the possibility to study the SSE independently of the thermal conductivity of the substrate, which does not aect the direction of the thermal gradient. Another key point, which complicates the interpretation of the SSE experiments, is the detection method for the thermally excited spin currents. Most spin caloric experiments rely on the indirect measurement by the inverse spin Hall eect (ISHE)[12] to detect the spin current pumped by the SSE. The measured inverse spin Hall voltage[6, 13] is predicted to be: VISHE =  SHElNN Ipump s SHElNN ~g"#kB MsVaA TmTN : (1) Here,  SHEis the spin Hall angle of the spin detector material, lNthe length between the voltage contacts, and N the resistance of detection layer. The underlying spin current Ipump s itself depends on the gyromagnetic ratio , the saturation magnetization MS, the spin mixing conductance g"#, the Boltzmann constant kB, the coherence volume of the magnetic system Va, the contact area A, and the temperature dierence of the magnons of the FMI and the phonons of the normal metal (NM) at the interface TmTN
. Unfortunately the spin mixing conductance g"#and with that the pumped spin currents Ipump s are very sensitive to the interface quality[14]. This interplay of the interface and the ISHE makes it necessary to carefully maintain the properties of the interface if one wants to compare dierent samples. Furthermore parasitic eect may be caused by the detection layer itself: Most experiments use platinum3 Table I. Thickness of YIG and Pt, crystalline orientation and number of samples for each series. Series Number YIG (nm) Pt (nm) Orientation in-situ etching 1 70,130,200 8.5 100 yes 2 20,70,130,200,300 8.5 100 yes 3 40,80,100,130,150 10.3 100 no (Pt) for this layer, due to the high spin Hall angle, making it an ecient spin current detector[15]. Many experiments have shown that the paramagnetic Pt shows a measureable magnetoresistance eect in contact with YIG[16, 17]. The YIG/Pt interface has been investigated more closely by X-ray magnetic circular dichroism measurements that reveal possibly a small induced magnetic moment of the Pt[18]. In combination with a thermal gradient, this proximity eect can cause an additional parasitic thermoelectric eect, the anomalous Nernst eect. For this reason one needs even in insulating ferromagnets to clearly distinguish between such parasitic interface eects and the genuine spin Seebeck eects due to spin currents[19]. Other recent measurements[20, 21] attribute such magnetoresistance eects to a spin Hall magnetoresistance and observe no proximity eect[22]. So given these dierent contradicting claims there is a clear need to distinguish whether the observed signals originate from a parasitic interface eect or a real bulk spin Seebeck eect. Here we present a detailed study of the relevant length scales of the longitudinal SSE in in YIG/Pt by varying the thickness of the ferromagnetic insulator. The obtained results show an increasing and saturating SSE signal with increasing YIG lm thickness. By determining also the dependence of the magnetoresistive eect and the saturation magnetization on the thickness, we can exclude an interface eect as the source of the measured signal. By atomistic spin simulation of the propagation of exchange magnons in temperature gradients, we are able to explain this behavior as being due to a nite eective propagation length of the thermally excited magnons. RESULTS All Y 3Fe5O12samples presented in this paper were grown by pulsed laser deposition with lm thicknesses ranging from 20 nm to 300 nm as shown in table I. The samples have been sorted into three series, where the interface conditions are identical for samples within one series. Details are given in the Methods section. We rst determined the intrinsic magnetic properties of every sample by SQUID magnetometry. Fig. 1a shows the saturation magnetization ( MS) as a function of lm thickness. Apart from very thin lms (20 nm), we nd values of approximately 120 kA/m 25 kA/m, which is close to the literature value of 140 kA/m for YIG thin lm samples[23]. For the very thin lms, a decrease of the moment has been previously observed for other thin YIG lms produced by PLD[24]. To estimate the in uence of the YIG/Pt interface coupling, which was previously claimed to be the origin of the measured SSE signals[16], we checked the magnetic eld dependence of the Pt resistivity. The magnetoresistive eect,4 05 0100150200250300040801201600 5 01001502002503000123 Series 1 Series 2 Series 3Ms (kA/m)Y IG thickness (nm) Series 2 Series 3Δρ/ρY IG thickness (nm)ba Figure 1. Thickness dependence of saturation magnetization and magnetoresistance eect. (a) Saturation magnetization ( MS) as a function of YIG lm thickness. Each series is marked in dierent colors. The literature value of 140 kA/m is indicated by a grey dashed line. The error of 25 kA/m takes into account that the active magnetic volume of the lm had to be estimated and the subtraction of the paramagnetic substrate signal. The uncertainty in the thickness determination translates into an error of the active magnetic volume and therefore an error of MS.(b)
=as a function of YIG-layer thickness measured for series 2 and 3. The y-axis error represents one standard deviation combined with a systematic error considering the temperature variability of the measurement. which was observable in every sample, showed a dependence on the magnetization direction as well as an in-plane angular dependence that can be explained by the novel spin Hall magnetoresistance eect[20, 21]. To determine the correlation with YIG lm thickness, we measured the in-plane resistivity for = 90and= 0in a four-point contact conguration. From this data we calculated
== 2 (090)=(0+90), as shown in Fig. 1b. For each series
=remained constant, and largely independent of the YIG lm thickness. Due to the identical interface conditions for samples of one series, we can assume that the magnetoresistive eect amplitude exhibits no signicant dependence on the YIG-lm thickness as expected for an interface eect. The changes of the absolute magnetoresistance signal between the dierent series can be explained by the change of the Pt-thickness and a residual variation of the interface quality. With the knowledge of the thickness dependence of these material and interface-related parameters, we can now ascertain whether the spin Seebeck eect is correlated to one of those parameters. Three series of YIG lms have been investigated in terms of the spin-Seebeck coecient (SSC), covering a thickness range from 20 nm to 300 nm. A more detailed explanation of the SSC measurements is given in the Methods section. Fig. 2 shows the measured YIG-layer thickness dependence of the SSC for each series. Below 100 nm, lms of each series showed an increase of the signal amplitude with increasing thickness. For larger thicknesses the signal starts to saturate. This saturation behavior could be observed in all our series that consist of epitaxial single crystalline lms. The samples of series 3 generated signals a factor of two lower than the signals of the other series, due to no in-situ interface etching prior to the Pt deposition, which leads to a less transparent interface5 0501001502002503000,00,20,40,60,8 SSC (µV/K)Y IG thickness (nm) Series 1 Series 2 Series 3 Figure 2. Spin Seebeck coecient as a function of YIG layer thickness . SSC as a function of YIG-layer thickness. The samples are sorted into dierent series. Samples of one series have been processed under identical conditions. Data points of each series are connected for clarity. The error in y-axis corresponds to one standard deviation of the measurement data combined with a systematic error taking into account the uncertainty of the mechanical mounting. for the magnons and therefore a smaller spin mixing conductance[14]. This observation underlines the importance of the interface conditions for the comparison of dierent samples, but the absolute trend of the thickness dependence was not aected by this. DISCUSSION In order to understand the origin of the signal, we compare the thickness dependence of the SSC with the thickness dependence of possible underlying mechanisms: When comparing this thickness scaling with that of the saturation magnetization MS, shown in Fig. 1a, we can exclude a direct correlation. We would expect a constant SSC for lms thicker than 40 nm, since only lms below 40 nm showed a MSdependence on the YIG thickness. Secondly we compare the thickness dependence of the magnetoresistive eect, shown in Fig. 1b, with the one of the SSC. Again one would expect a constant contribution to the measured signal independent of the YIG lm thickness when comparing with the thickness dependence of the magnetoresistive eect. For this reason we can exclude that any interface coupling eect leads to the observed thickness dependence of the SSC. Even if the magnetoresistive eect in combination with a thermal gradient leads to a Nernst eect, the signals produced by it deliver a constant oset for each series, which cannot be the source of the signal with the thickness dependence that we observe. This is of major importance as it allows us to conclude that the source of the observed signals is not the currently discussed proximity eects at the6 interface[16]. The clear thickness dependence points to an origin in the bulk of the YIG. In the following analysis we assume that the role of the YIG thickness for the SSE might be due to a nite length scale for magnon propagation in the YIG material. In order to investigate this we simulate the propagation of thermally excited magnons in a temperature gradient using an atomistic spin model. The model is generic and not intended to describe YIG quantitatively. It contains a ferromagnetic nearest-neighbor exchange interaction Jand an uniaxial anisotropy with easy-axis along x-direction and anisotropy constant dx= 0:1J. We investigate a cubic system with 51288 spins, which are initialized parallel to the x-axis. The dynamics of the spin system is calculated by solving the stochastic Landau-Lifshitz-Gilbert equation numerically with the Heun-Method[25]. The phonons provide a heat-bath for the spin system where we assume a linear temperature gradient over the length Linx-direction as shown in Fig. 3. This temperature prole remains constant during the simulation. After an initial relaxation, the local, reduced magnetization m(x) depends on the space coordinate xand its prole is determined as an average over all spins Siin the corresponding y-z-plane and additionally as an average over time. Due to the temperature gradient, magnons propagate from the hotter towards the colder region of the system and this magnonic spin current leads to deviations of the local magnetization m(x) from its local equilibrium value m0(x) which would follow from the local temperature Tp(x) of the phonon system. A temperature dependent calculation of the equilibrium magnetization m0(T) for a system with constant temperature allows us to describe this deviation, which we dene as magnon accumulation
m(x)[26] via
m(x) =m(x)m0(x;Tp(x)) . (2) Fig. 3 shows this magnon accumulation
mxas a function of space coordinate xin a system with a damping constant of = 0:01 and a temperature prole with a linear temperature gradient of
T= 105J=(kBa), whereais the lattice constant of the cubic system, for two dierent lengths Lof the temperature gradient. At the hotter end of the gradient magnons propagate towards the cooler region of the system and this reduces the number of the local magnons and increases the local magnetization. On the other side at the cold end of the gradient magnons arriving from hotter parts of the system decrease the local magnetization. The resulting magnon accumulation is symmetric in space and changes its sign in the center of the temperature gradient. The spatial dependence of the magnon accumulation as well as the height of the peaks at the hot and cold end are aected by the mean eective propagation length of the magnons[26] in the system. If the length Lof the gradient is smaller than the mean propagation length of the magnons, the magnon accumulation depends linearly on the space coordinate x. For higher values of the length Lthe accumulation at the center of the gradient vanishes and appears only at the edges of the temperature gradient. This is in agreement with simulation by Ohe et al. of the transverse spin Seebeck eect[7]. In their simulation they modify the mean propagation length of the magnons by changing the damping constant and obtain comparable results. The eective mean propagation length of the magnons can be estimated by tting it to the function.7 -100- 500 5 01 00-3-2-10123T p (J/kB) Δm • 105x [# spins] 20a 100a0,000,01 Figure 3. Magnon accumulation in a spin system with a temperature gradient. Magnon accumulation
mas a function of the space coordinate xfor a given phonon temperature Tpincluding a temperature gradient of two dierent lengths L= 20a;100a The magnitude of the magnon accumulation at the cold end of the gradient increases with increasing length Lup to a saturation value depending on the mean propagation length of the magnons. The magnon accumulation can be understood as the averaged sum of the magnons, which can reach the end of the gradient. As illustrated in Fig. 4 only those magnons from distances smaller than their propagation length contribute to the resulting magnon accumulation at the cold end of the temperature gradient. Xiao et al. showed that the resulting spin current from the ferromagnet into the non-magnetic material is proportional to the temperature dierence between the magnon temperature Tm in the ferromagnet and the phonon temperature of the non-magnetic material TN[6]. Here, for simplicity, we assume that the temperature of the non-magnetic material is TN= 0 K and no back ow from the non-magnetic material exists. The magnon temperature Tmat the cold end of the gradient can be calculated from the local magnetization m(x). The resulting magnon temperature dependence on the length Lof the temperature gradient saturates due to the mean propagation length of the magnons as shown in Fig. 5 for two dierent damping constants . The variation of the damping constant leads to variation of the mean magnon propagation length and, consequently, dierent length scales where saturation for the magnon temperature sets in. Tcold m/ 1exp L  . (3)8 ferromagnetic insulator normal metal Figure 4. Origin of SSC thickness dependence. Illustration of the saturation eect of the measured voltage due to a nite propagation length of the excited magnonic spin currents. 05 01 001 502 000,00,20,40,60,81,00 1 002 003 004 000,00,20,40,60,81,0bTm/ Tc • 104L [# spins] α 0.01 α 0.05a normalized SSCY IG thickness (nm) Series 1 Series 2 Series 3 Figure 5. Comparison between the theoretical and the experimental results. (a) Magnon temperature Tmat the cold end of the temperature gradient as a function of the length Lof the temperature gradient for two dierent damping constants shows a saturation eect depending on the propagation length of the thermally excited magnons. (b)Normalized SSC data and corresponding t functions plotted against the YIG thickness. SSC data have been normalized to the saturation value for an innitely large system. From the t we obtain an eective magnon propagation length of 101 nm 5 nm for series 1, 127 nm 44 nm for series 2 and 89 nm 19 nm for series 3. The resulting ts are shown as solid lines in Fig. 5. The calculated values are comparable to other calculations of the mean propagation length of thermally induced magnons [26]. This mean propagation length depends on the9 frequency spectra of the excited magnons, with that on the model parameters and the damping process during their propagation. The latter depends on the damping constant as well as the frequency !and the group velocity of the magnons@!=@ q[27]. The proportionality between the magnon temperature and the measured ISHE-voltage[6] allows us now to evaluate the SSC data points using eq. 3. Each series was evaluated separately as can be seen in Fig. 2b. We obtain a mean eective magnon propagation length of 101 nm 5 nm for series 1, 127 nm 44 nm for series 2, and 89 nm 19 nm for series 3. By this we derive, independent of the dierent interface qualities between the series, an eective mean propagation length for thermally excited magnons of the order of 110 nm 16 nm from all our series. Based on our model, we can now explain the behavior of the SSC data qualitatively: The increase of the SSC with increasing YIG lm thickness below 110 nm can be attributed to an increasing magnon accumulation at the interface, while the accumulation starts to saturate and therefore the ISHE-voltage in thicker lms. Consequently we can assume that the magnon emitting source is the ferromagnetic thin lm and thus we can pinpoint the origin of the observed signal to the magnonic spin Seebeck eect. In conclusion, we have observed an increasing and saturating spin Seebeck signal with increasing YIG lm thickness. This behavior can neither be explained by the thickness dependence of the saturation magnetization nor magnetore- sistive eects in the Pt detection layer or any other interface eect. Instead we present a model based on atomistic simulations that attributes this characteristic behavior to a nite propagation length of thermally excited magnons, which are created in the bulk of the ferromagnetic material. From the evaluation of our data we obtain an eective mean propagation length of the order of 110 nm for thermally excited magnons, which is in agreement with other studies predicting a nite propagation length of thermally excited magnons of the order of 100 nm [27]. Our results thus clearly allow us to rule out parasitic interface eects and identify thermal magnonic spin currents as the source of the observed signals and thus identify unambiguously the longitudinal spin Seebeck eect. METHODS Thin lm Y 3Fe5O12samples were grown by pulsed laser deposition from a stoichiometric powder target, using a KrF excimer laser ( = 248 nm) with a uence of 2 :6 J/cm2, and repetition rate of 10 Hz[28]. Monocrystalline 10 mm10 mm0:5 mm gadolinium gallium garnet (Gd 3Ga5O12,GGG) substrates in the (100) crystalline orientation were used to ensure an epitaxial growth of the lms, due to the small lattice mismatch below 0 :06 %. The optimal deposition conditions were found for a substrate temperature of 650C30C and an oxygen partial pressure of 6:67103mbar. In order to improve the crystallographic order and to reduce oxygen vacancies, every lm was ex-situ annealed at 820C30C by rapid temperature annealing for 300 s under a steady ow of oxygen. X-ray re ectometry (XRR) and prolometer (Tencor P-16 Surface Prolometer) measurements were done to determine the lm thickness, while the crystalline quality was measured by X-ray diraction. The samples are sorted into series to highlight the dierent platinum deposition conditions and therefore interface10 qualities, which have been used to study interface in uence. This in uence is minimized for samples within a series by sputtering and cleaning these samples at the same time. Therefore the Pt thickness and the interface preparation were kept identical, while the YIG lm thickness varied. Between the series the interface preparations and thus qualities dier and lead to dierent spin mixing conductance and thus dierent signal amplitudes for a given thermally excited spin current. For the Pt deposition the samples had been transferred at atmosphere and may therefore have suered from surface contamination. In order to enhance the interface quality, an in-situ low power ion etching of the YIG surface was performed for some series prior the deposition. DC-magnetron sputtering was used for a homogeneous deposition of the Pt-lm under an argon pressure of 1 102mbar at room temperature. XRR measurements were done afterwards to control the Pt thickness. In the last fabrication step, the Pt-layer was structured by optical lithography and ion etching in order to reduce in uences on the ISHE-voltage by slight variations of the sample geometry. Fig. 6a shows a sketch of the nal sample stack. V 𝐻 𝛻𝑇 Θ 5 mm GGG (100) YIG (100) x z 500 µm -60 -30 0 30 60-8-6-4-202468 V Voltage (µV) H-Field (Oe) 11K 9K 7K 5K 3K 1K 036912-6-3036 T|| -z T|| -z T|| z VISHE (µV) Tz(K) T|| -za b Figure 6. Experimental conguration and measured signals. (a) Sketch of the sample conguration geometry. The structured grey layer, indicates the 4 mm long platinum stripes with 1 :2 mm large triangular shaped contact pads. The YIG layer is indicated by yellow, the GGG substrate by light grey. For a further understanding of the spin caloric measurements the direction of thermal gradient and the in-plane magnetic eld have also been marked. (b): Recorded voltage signals for the 200 nm thick YIG lm of series 2. Each color represents a dierent stable temperature dierence. The inset shows the evaluated ISHE-voltage VISHE for both directions of the thermal gradient
Tz. For the thermoelectric transport measurements a setup was constructed that is able to generate a temperature dierence up to 15 K at room temperature in the parallel and anti-parallel out-of-plane direction. Two copper blocks can either serve as heat source or cooling bath to establish a temperature dierence between both blocks, while the sample is mounted in between. The relative temperature dierence, which was used in the graphs and for the calculation of the spin Seebeck coecient, was determined by the dierence between those two copper blocks. To ensure a good thermal connection, each sample was mounted with thermally conductive adhesive transfer tape. In addition the tape compensated misalignments of the sample mounting. Due to the pressure sensitive heat conduction of the tape, springs were used to mechanically press the two copper blocks together with a constant force in order to reproduce the same conditions for every measurement. Magnons, generated by the thermal gradient in the ferromagnetic layer, will now propagate, depending on the11 orientation of the thermal gradient, to the FMI/NM interface. An exchange interaction of the local moments of the FMI and the conduction electrons of the NM leads to a spin transfer torque, which creates spin-polarized charge carrier in the NM[13]. Due to the inverse spin-Hall eect in Pt, a charge carrier separation, based on the spin orientation, is taking place, leading to a measureable potential dierence at the edges of the stripe geometry[12, 13]. Our setup used a two point-conguration, shown in Fig. 6a, to detect this voltage by a nanovoltmeter (Keithley 2182A). By sweeping the in-plane magnetic eld with = 90with respect to the platinum stripes in addition to an out-of-plane thermal gradient, one is able to measure magnetic eld dependence of the measured voltage as shown as in Fig. 6b. To exclude in uences of a ground oset, the ISHE-voltage VISHE needs to be extracted from these voltages signals by dividing the dierence of the voltage in saturation for positive and negative H-eld of the signals,
V, by two. If one now plots the ISHE-Voltage against the corresponding out-of-plane thermal dierence
Tz, as done in the inset of Fig. 6b the SSC can be derived from the slope by a linear t. Dependent on the direction of the thermal gradient the SSC will switch its sign, while the absolute value should be the same. The values of the order of 0 :54V=K (200 nm thick YIG lm with 8 :5 nm of Pt) derived for the SSC, in our setup and sample preparation, are similar to experiments of other groups[5, 8, 11, 17, 29]. A SSC data point for one sample, as shown in Fig. 2, is the result of an average of the measurements of the two stripes per sample and for both directions of the thermal gradient. [1] G. E. W. Bauer, E. Saitoh, and B. J. van Wees, Nature Mat. 11, 391 (2012). [2] M. Walter, J. Walowski, V. Zbarsky, M. M unzenberg, M. Sch afers, D. Ebke, G. Reiss, A. Thomas, P. Peretzki, M. Seibt, J. S. Moodera, M. Czerner, M. Bachmann, and C. Heiliger, Nature Mat. 10, 742 (2011). [3] K. Uchida, S. Takahashi, K. Harii, J. Ieda, W. Koshibae, K. Ando, S. Maekawa, and E. Saitoh, Nature 455, 778 (2008). [4] C. M. Jaworski, J. Yang, S. Mack, D. D. Awschalom, J. P. Heremans, and R. C. Myers, Nature Mat. 9, 898 (2010). [5] K. Uchida, J. Xiao, H. Adachi, J. Ohe, S. Takahashi, J. Ieda, T. Ota, Y. Kajiwara, H. Umezawa, H. Kawai, G. E. W. Bauer, S. Maekawa, and E. Saitoh, Nature Mat. 9, 894 (2010). [6] J. Xiao, G. E. W. Bauer, K. Uchida, E. Saitoh, and S. Maekawa, Phys. Rev. B 81, 214418 (2010). [7] J. Ohe, H. Adachi, S. Takahashi, and S. Maekawa, Phys. Rev. B 83, 115118 (2011). [8] H. Adachi, K. Uchida, E. Saitoh, J.-i. Ohe, S. Takahashi, and S. Maekawa, Appl. Phy. Lett. 97, 252506 (2010). [9] M. Agrawal, V. I. Vasyuchka, A. A. Serga, A. D. Karenowska, G. A. Melkov, and B. Hillebrands, arXiv:1209.3405. [10] S. Y. Huang, W. G. Wang, S. F. Lee, J. Kwo, and C. L. Chien, Phys. Rev. Lett. 107, 216604 (2011). [11] K. Uchida, H. Adachi, T. Ota, H. Nakayama, S. Maekawa, and E. Saitoh, Appl. Phy. Lett. 97, 172505 (2010). [12] E. Saitoh, M. Ueda, H. Miyajima, and G. Tatara, Appl. Phy. Lett. 88, 182509 (2006). [13] Y. Kajiwara, K. Harii, S. Takahashi, and J. Ohe, Nature 464, 262 (2010). [14] M. B. Jung eisch, V. Lauer, R. Neb, A. V. Chumak, and B. Hillebrands, arXiv:1302.6697. [15] O. Mosendz, V. Vlaminck, J. E. Pearson, F. Y. Fradin, G. E. W. Bauer, S. D. Bader, and A. Homann, Phys. Rev. B 82, 214403 (2010). [16] S. Y. Huang, X. Fan, D. Qu, Y. P. Chen, W. G. Wang, J. Wu, T. Y. Chen, J. Q. Xiao, and C. L. Chien, Phys. Rev. Lett. 109, 107204 (2012).12 [17] M. Weiler, M. Althammer, F. Czeschka, H. Huebl, M. Wagner, M. Opel, I.-M. Imort, G. Reiss, A. Thomas, R. Gross, and S. Goennenwein, Phys. Rev. Lett. 108, 106602 (2012). [18] Y. M. Lu, Y. Choi, C. M. Ortega, X. M. Cheng, J. W. Cai, S. Y. Huang, L. Sun, and C. L. Chien, Phys. Rev. Lett. 110, 147207 (2013). [19] D. Qu, S. Y. Huang, J. Hu, R. Wu, and C. L. Chien, Phys. Rev. Lett. 110, 067206 (2013). [20] H. Nakayama, M. Althammer, T. Y. Chen, K. Uchida, Y. Kajiwara, D. Kikuchi, T. Ohtani, S. Gepr ags, M. Opel, S. Takahashi, R. Gross, G. E. W. Bauer, S. T. B. Goennenwein, and E. Saitoh, arXiv:1211.0098. [21] N. Vliestra, J. Shan, V. Castel, B. J. van Wees, and J. B. Youssef, arXiv:1301.3266v1. [22] S. Geprags, S. Meyer, S. Altmannshofer, M. Opel, F. Wilhelm, A. Rogalev, R. Gross, and S. T. B. Goennenwein, Appl. Phy. Lett. 101, 262407 (2012). [23] A. A. Serga, A. V. Chumak, and B. Hillebrands, J. Phys. D: Appl. Phys. 43, 264002 (2010). [24] N. Kumar, D. S. Misra, N. Vankataramani, S. Prasad, and R. Krishnan, J. Magn. Magn. Mater. 272-276 , e899 (2004). [25] U. Nowak, \Handbook of magnetism and advanced magnetic materials," (John Wiley & Sons, 2007) Chap. Classical Spin-Models. [26] U. Ritzmann, D. Hinzke, and U. Nowak, (in preparation). [27] A. A. Kovalev and Y. Tserkovnyak, Europhys. Lett. 97, 67002 (2012). [28] L. Bi, J. Hu, P. Jiang, D. H. Kim, G. F. Dionne, L. C. Kimerling, and C. A. Ross, Nature Photon. 5, 758762 (2011). [29] A. Kirihara, K. Uchida, Y. Kajiwara, M. Ishida, Y. Nakamura, T. Manako, E. Saitoh, and S. Yorozu, Nature Mat. 11, 686 (2012). Author contributions M.K. and G.J. conceived and supervised the research. A.K. and R.R. performed the experiments and analyzed the data. M.C.O., D.H.K. and C.A.R. provided and characterized the YIG lms A.K. and R.R. structured and characterized the samples. U.R., D.H. and U.N. worked on the theoretical modelling. B.J. and B.H. performed additional sample analysis. A.K. organized and wrote the paper with all authors contributing to the discussions and preparation of the manuscript. ACKNOWLEDGMENTS The authors would like to thank the Deutsche Forschungsgemeinschaft (DFG) for nancial support via SPP 1538 "Spin Caloric Transport" and the Graduate School of Excellence Materials Science in Mainz (MAINZ) GSC 266, the EU (IFOX, NMP3-LA-2012246102, MAGWIRE, FP7-ICT-2009-5 257707, MASPIC, ERC-2007-StG 208162) and the National Science Foundation.

2013-06-04

The observation of the spin Seebeck effect in insulators has meant a breakthrough for spin caloritronics due to the unique ability to generate pure spin currents by thermal excitations in insulating systems without moving charge carriers. Since the recent first observation, the underlying mechanism and the origin of the observed signals have been discussed highly controversially. Here we present a characteristic dependence of the longitudinal spin Seebeck effect amplitude on the thickness of the insulating ferromagnet (YIG). Our measurements show that the observed behavior cannot be explained by any effects originating from the interface, such as magnetic proximity effects in the spin detector (Pt). Comparison to theoretical calculations of thermal magnonic spin currents yields qualitative agreement for the thickness dependence resulting from the finite effective magnon propagation length so that the origin of the effect can be traced to genuine bulk magnonic spin currents ruling out parasitic interface effects.

Determination of the origin of the spin Seebeck effect - bulk vs. interface effects

1306.0784v1

Quantum drives produce strong entanglement between YIG samples without using intrinsic nonlinearities Jayakrishnan M. Prabhakarapada Nair1,and G. S. Agarwal1, 2,y 1Institute for Quantum Science and Engineering, Texas A &M University, College Station, TX 77843, USA 2Department of Biological and Agricultural Engineering, Department of Physics and Astronomy, Texas A &M University, College Station, TX 77843, USA (Dated: July 15, 2019) We show how to generate an entangled pair of yttrium iron garnet (YIG) samples in a cavity-magnon system without using any nonlinearities which are typically very weak. This is against the conventional wisdom which necessarily requires strong Kerr like nonlinearity. Our key idea, which leads to entanglement, is to drive the cav- ity by a weak squeezed vacuum field generated by a flux-driven Josephson parametric amplifier (JPA). The two YIG samples interact via the cavity. For modest values of the squeezing of the pump, we obtain significant en- tanglement. This is the principal feature of our scheme. We discuss entanglement between macroscopic spheres using several di erent quantitative criteria. We show the optimal parameter regimes for obtaining entanglement which is robust against temperature. We also discuss squeezing of the collective magnon variables. Yttrium iron garnet (YIG), an excellent ferrimagnetic sys- tem, has attracted considerable attention during the past few years. The Kittel mode [1] in YIG possesses unique properties including rich magnonic nonlinearities [2] and a low damping rate [3] and in addition the high spin density in YIG allows strong coupling between magnons and microwave cavity pho- tons giving rise to quasiparticles, namely the cavity-magnon polaritons [3–8]. Strong coupling between the YIG sphere and the cavity photons have been observed at both cryogenic and room temperatures [8]. Aided by these superior proper- ties, YIG is reckoned to be the key ingredient in future quan- tum information networks [9]. Thus a variety of intriguing phenomena have been investigated in the context of magnons. This include the observation of bistability [10], cavity spin- tronics [7, 11], level attraction for cavity magnon-polaritons [12], magnon dark modes [13], the exceptional point [14] etc. By virtue of the strong coupling among magnons, a multi- tude of quantum information aspects have been investigated including the coupling of magnons to a superconducting qubit [15] and phonons [16]. Other interesting phenomena involve magnon induced transparency [17], magnetically controllable slow light [18] etc. Owing to the diverse interactions of magnons with other in- formation carriers, YIG o er a novel platform in the analysis of macroscopic quantum phenomena. The coherent phonon- magnon interactions due to the radiation pressure like mag- netostrictive deformation [19] was studied. The nonlinear interaction between magnons and phonons can give rise to magnomechanical entanglement which further transfers to photon-magnon and photon-phonon subsystems, generating a tripartite entangled state [20]. Another recent work pro- posed a scheme to create squeezed states of both magnons and phonons in a hybrid magnon-photon-phonon system [21]. The squeezing generated in the cavity was transferred to the magnons via the cavity-magnon beamsplitter interaction. There is not much work on the coupling of two macroscopic YIG samples in a cavity. Recently the spin current genera- tion in a YIG sample due to excitation in another YIG sam- ple has been investigated [11]. This arises from the cavitymediated coupling between the two samples. It is thus nat- ural to consider the possibility of quantum entanglement be- tween two YIG samples as there has been significant interest in the study of quantum entanglement between macroscopic systems. Recently there has been remarkable success in the observation of quantum entanglement between macroscopic mechanical oscillators [22, 23] with photonic crystal cavities and with superconducting quibits. In addition entanglement between cavity field and mechanical motion has been reported [24]. The conventional wisdom of producing entanglement involves nonlinearities in the system. The well known nonlin- earities are the magnetostrictive interaction [16] and the Kerr eect [2]. The magnetostrictive force allows the magnons to couple to the phonons and can be used to generate magnon- phonon entanglement [20]. The Kerr nonlinearity arises from the magnetocrystalline anisotropy and has been used to pro- duce bistability in magnon-photon systems. In recent publica- tions, these nonlinearities have been used to produce entangle- ment between two magnon modes in a magnon-cavity system [25–27]. Here we present a scheme to generate an entangled pair of YIG spheres in a cavity-magnon system without using any nonlinearities. In addition, we also investigate the squeezed states of the coupled system of two YIG spheres. Two YIG spheres are coupled to the cavity field and the cavity is driven by a squeezed vacuum field [30, 31], resulting in a squeezed cavity field. A flux-driven Josephson parametric amplifier (JPA) is used to generate the squeezed vacuum microwave field. The squeezing in the cavity will be transferred to the two YIG samples due to the cavity-magnon beamsplitter in- teraction. Based on experimentally attainable parameters, we show that significant bipartite entanglement can be generated between the YIG samples. The entanglement is robust against temperature. Our results can be extended to other geometries of YIG. Further the method that we propose is quite generic and can be used for other macroscopic systems. We consider the cavity-magnon system [16, 19, 20] which consists of cavity microwave photons and magnons, as shown in figure 1. The magnons are quasiparticles, a collective ex-arXiv:1905.07884v2 [quant-ph] 11 Jul 20192 Signal Pump Output YIG YIG YIG FIG. 1: Two YIG spheres are placed inside a microwave cav- ity near the maximum magnetic field of the cavity mode, and si- multaneously in a uniform bias magnetic field. The cavity is driven by a week squeezed vacuum field generated by a flux-driven JPA. The magnetic field of the cavity mode is in the xdirec- tion and the bias magnetic field is applied along the zdirection. citation of a large number of spins in a YIG sphere. They are coupled to the cavity photons via the magnetic dipole interac- tion. The Hamiltonian of the system reads [32] H=~=!aaya+!m1my 1m1+!m2my 2m2 +gm1a(a+ay)(m1+my 1)+gm2a(a+ay)(m2+my 2);(1) where a(ay) are the annihilation (creation) operator of cav- ity mode, m1,m2(my 1,my 2) are the annihilation (creation) operators of the two magnon modes and they represent the collective motion of spins via the Holstein-Primako trans- formation [33] in terms of Bosons. The parameters !a,!mi (i=1,2) are the resonance frequencies of the cavity and the magnon modes. Hereafter, wherever we use a subscript ‘ i’ it can take values from 1 to 2. The magnon frequency is given by the expression !mi= Hi, where =2=28 GHz /T is the gyromagnetic ratio and Hiare the external bias magnetic fields. The gmiain Eq.(1) are the linear photon-magnon cou- pling strengths. The cavity is driven by a week squeezed vac- uum field generated by a flux driven JPA. JPAs can in prin- ciple amplify a single signal quadrature without adding any extra noise. The squeezed vacuum is generated by degenerate parametric down-conversion using the nonlinear inductance of Josephson junctions [34–44] and a squeezing down to 10% of the vacuum variance has been produced [36]. The oper- ation of generating squeezed vacuum is depicted in figure 1. Vacuum fluctuations are at the signal port and the pump field is applied at frequency 2 !s. The pump photon splits into a signal and an idler photon. Strong quantum correlations be- tween the signal and idler photons are generated which re- sult in squeezing. The output is at the frequency !s[37, 40]. The Hamiltonian described by Eq.(1) does not contain terms involving the input drive field. We use standard quantum Langevin formalism to model the system and the equations describing the evolution of the system operators will contain the input drive terms. Applying the rotating-wave approxima-tiongmia(a+ay)(mi+my i) becomes gmia(amy+aym) [3–7, 16]. In the rotating frame at the frequency !sof the squeezed vac- uum field, the quantum Langevin equations (QLEs) describ- ing the system can be written as follows ˙a=(i
a+ka)aigm1am1igm2am2+p 2kaain; ˙m1=(i
m1+km1)m1igm1aa+p 2km1min 1; (2) ˙m2=(i
m2+km2)m2igm2aa+p 2km2min 2; where
a=!a!s,
mi=!mi!s,kais the dissipation rate of the cavity, kmiare the dissipation rates of the magnon modes, andain,min iare the input noise operators of the cavity and magnon modes, respectively. The input noise operators are characterized by zero mean and the following correlation re- lations [45],hain(t)ainy(t0)i=(N+1)(tt0),hainy(t)ain(t0)i= N(tt0),hain(t)ain(t0)i=M(tt0),hainy(t)ainy(t0)i= M(tt0), whereN=sinh2r,M=eisinhrcosh rwith r andbeing the squeezing parameter and the phase of the input squeezed vacuum field, respectively. We have the other input correlations for the magnon as hmin i(t)miny i(t0)i= [Nmi(!mi)+1](tt0),hminy i(t)min i(t0)i=Nmi(!mi)(tt0), where Nmi(!mi)=[exp(~!mi kBT)1]1are the equlibrium mean thermal magnon numbers of the two magnon modes. We now show that the YIG spheres can be entangled by resonantly driving the cavity with a squeezed vacuum field. We write down the field operators as their steady state values plus the fluctuations around the steady state. The fluctuations of the system can be described by the QLEs ˙a=(i
a+ka)aigm1am1igm2am2+p 2kaain; ˙m1=(i
m1+km1)m1igm1aa+p 2km1min 1; (3) ˙m2=(i
m2+km2)m2igm2aa+p 2km2min 2: The quadratures of the cavity field and the two magnon modes are given by X=(a+ay)=p 2,Y=i(aya)=p 2,xi= (mi+my i)=p 2 andyi=i(my imi)p 2, and similarly for the input noise operators. The QLEs describing the quadrature fluctuations ( X;Y;x1;y1;x2;y2) can be written as ˙u(t)=Au(t)+n(t); (4) where u(t)=[X(t);Y(t);x1(t);y1(t);x2(t);y2(t)]T, n(t)=[p2kaXin;p2kaYin;p 2km1xin 1;p 2km1yin 1;p 2km2xin 2;p 2km2yin 2]Tand A=2666666666666666666666664ka
a 0 gm1a 0 gm2a
akagm1a0gm2a0 0 gm1akm1
m1 0 0 gm1a0
m1km10 0 0 gm2a 0 0km2
m2 gm2a0 0 0
m2km23777777777777777777777775: (5) The system is a continuous variable (CV) three- mode Gaus- sian state and it can be completely described by a 6 6 covariance matrix (CM) Vdefined as V(t)=1 2hui(t)uj(t0)+3 (a) (b) FIG. 2: Density plot of bipartite entanglement Em1m2between the two magnon modes versus
aand
m1(a) with
m2=
m1, r=1,=0,T=20 mK, (b) with
m2=
m1,r=2, =0,T=20 mK. Other parameters are given in the text. 00.050.10.150.20.250.3 Temperature (K)0.40.50.60.70.80.9Em1m2 FIG. 3: Plot of bipartite entanglement Em1m2be- tween the two magnon modes against temperature with
a=
m1=
m2=0,r=2 and=0: uj(t0)ui(t)i, (i,j=1, 2....6). The steady state CM Vcan be obtained by solving the Lyapunov equation [46, 47] AV+VAT=D; (6) where Dis the di usion matrix defined as hni(t)nj(t0)+ nj(t0)ni(t)i=2=Di j(tt0). We use logarithmic negativ- ity [48] as the quantitative measure to investigate the bi- partite entanglement Em1m2between the two magnon modes. It can be obtained from Em1m2=max[0;ln(2)] where =min[eig(i P12VP 12)], = iyL iy,P12=1L z andy,zare the Pauli matrices [49]. Figure 2(a)-(b) shows the bipartite entanglement between the two magnon modes at two di erent squeezing parameters. We use a set of ex- perimentally feasible parameters [16]: !a=2=10 GHz, ka=2=5kmi=2=5 MHz, gm1a=gm2a=4kaandT=20 mK, Nm1=Nm20 at 20 mK. The YIG sphere has a diam- eter 250-mand the number of spins N3:51016. We have adopted the parameters so that the two magnon modes are identical. We observe that
a=
m1=
m2=0, in other words!a=!s,!mi=!sare optimal for the entanglement between the two YIG samples. At resonance we observe the maximum amount of entanglement and it increases with the increase in the squeezing parameter. Figure 3 shows that the (a) (b) FIG. 4: (a)hM2 xi+hm2 yiagainst
aand
m1with
m2=
m1, r=2,=0,T=20 mK. (b)hM2 xiagainst
aand squeez- ing parameter rwith
m1=
m2=0,=0, T=20 mK. bipartite entanglement is quite robust against temperature. We observe significant amount of entanglement even at T=0.5 K which is quite remarkable for the system of two YIG spheres. We have chosen identical coupling between photon and the two magnon modes. In the case of unequal coupling the en- tanglement goes down. Although we have chosen two identi- cal YIG spheres, one can have two cuboidal YIG samples as in [11] with an angle between the external magnetic field and the local microwave magnetic field at one YIG sample. This makes the resonance frequencies of the two samples di erent. To compare our results with the protocols using nonlinear methods, a recent work [26] produced an entanglement close to 0.25 between the magnon modes at a temperature 10 mK through a Kerr nonlinearity introduced by a strong classical drive. The use of a di erent kind of nonlinearity, namely the magnetostrictive interaction in one YIG sphere produces sim- ilar entanglement [25] at a temperature 10 mK. The entan- glement vanishes as the temperature approaches 20 mK. In contrast our scheme for entanglement generation produces a steady and strong entanglement between 0 to 100 mK and a significant amount of entanglement is present even at 500 mK. The mechanism of the entanglement generation will become clear from the discussion below. Next we discuss two di erent criteria for entanglement in a two mode CV system. The advantage of these criteria over logarithmic negativity is that the former can be easily exam- ined through experiments [22, 23], though in a qualitative way. The first inseparability condition proposed by Simon [49] and Duan et al. [50] is the su cient condition for entan- glement in a two mode CV system. We define a new set of operators M=(m1+m2)=p 2,m=(m1m2)=p 2. The cri- terion suggests that if the two modes are separable then they should satisfy the following inequality hM2 xi+hm2 yi1; (7) whereMxandmyare the fluctuations in the quadratures Mx andmydefined as Mx=(M+My)=p 2,my=i(mym)=p 2. In other words, violation of the inequality in Eq.(7) means the existance of entanglement between the two YIG samples. Fig- ure 4(a) shows that there is region around
a=0 and
m1=0 (resonance) in which hM2 xi+hm2 yiis less than one and it is4 a clear manifestation of the entanglement present between the YIG samples. Mancini et al. [51] derived another inequality which is useful in characterizing separable states. It suggests that if the two mode CV system is separable, then it should satisfy the following inequality hM2 xihm2 yi1=4: (8) Hence the violation of Eq.(8) implies that the YIG samples are entangled. We use identical coupling strengths between the cavity and the two YIG samples. Therefore when
m1=
m2=0 the Hamiltonian of the system in the rotating frame of the drive can be written as H=~=
aaya+p 2gm1a(a+ay)(M+My): (9) The Hamiltonian does not contain a term involving mandmy. Hence the fluctuations in mwill be equal to the fluctuations at time t=0. Since matt=0 is in the vacuum state (at low temperature 20 mK), we have hm2 yi=1=2. Figure 4(b) shows that there is a region close to resonance where the quan- tityhM2 xiis less than 1 =2. This violates the inequality in Eq.(8) and hence the two YIG samples are entangled. This further corroborates our results. As a byproduct of our results we investigate the squeezing of the two magnon modes and show that it can be acheived by resonantly driving the cavity with a squeezed vacuum field. We are interested in the vari- ances of the cavity and magnon mode quadratures and they are given by diagonal elements of the time-dependent CM V(t) as defined previously. The amount of squeezing in a mode quadrature Xcan be expressed in decibels (dB). It is obtained from the expression 10log10[hX(t)2i=hX(t)2ivac], wherehX(t)2ivac=1 2. As discussed in [21] when the cav- ity and the two magnon modes are decoupled, the cavity field is squeezed as a result of the squeezed driving field and the magnon modes possesses vacuum fluctuations. As we in- crease the coupling strength, squeezing is partially trans ered to the two identical YIG samples. The blue region in figure 5(a)-(b) represents the region of squeezing. For r=2 the input squeezing is about 17.35 dB. We observed a squeezing of about 2.27 dB for each of the two magnon modes at res- onance with T=20 mK. Note that figure 5 give the magnon quadrature when both the YIG samples are present. Figure 6(a) shows that the magnon squeezing is robust against tem- perature. We observe moderate squeezing for both spheres even at T=0.35 K. At resonance we also find a squeezing of about 7.28 dB for the Mxquadrature of the collective variable M. This is comparable to the results when one had only one YIG sample present and clearly manifested in figures 6(b)-(c). In conclusion, We have presented a scheme to generate an entangled pair of YIG samples in a cavity-magnon system. Entanglement of magnon modes can be generated through res- onantly driving the cavity by a squeezed vacuum field and it can be realized using experimentally attainable parameters. The entanglement produced is robust against temperature. We observe considerable amount of entanglement even at T=0:5K. We have also discussed possible strategies to mea- sure the generated entanglement. We have also showed that (a) (b) FIG. 5: (a) Variance of the first magnon quadrature hx1(t)2iver- sus
aand
m1. (b) Variance of the first magnon quadrature against squeezing parameter rand phase. The other parameters in (a) are r=2,=0,
m1=
m2,T=20 mK. Other parameters in (b) are
a=
m1=
m2=0 and T=20 mK.hx2(t)2iis identical tohx1(t)2i. (a) (b) (c) FIG. 6: (a) Variance of the first magnon quadrature hx1(t)2i against squeezing parameter rand temperature Twhen both YIG samples are present. (b) Variance of the first magnon quadrature hx1(t)2iagainst squeezing parameter rand temperature Twith only one YIG sample is present. (c) Variance hM2 xiof the col- lective variable Magainst squeezing parameter rand temperature T. The other parameters are
a=
m1=
m2=0,=0. by employing the same method squeezed states of magnons in two di erent modes can be achieved. For an input squeez- ing of 17.35 dB we have observed a squeezing of about 2.27 dB for the magnon modes at T=20 mK. Our scheme for entangling YIG samples does not require any nonlinearities and hence goes against the conventional wisdom of producing entanglement. This provides an entirely new method for entangling macroscopic systems, which can be used in other macroscopic systems. ACKNOWLEDGMENTS Jayakrishnan would like to thank Jie Li for helpful discus- sions, carefully reading the article and providing constructive feedback.5 jayakrishnan00213@tamu.edu ygirish.agarwal@tamu.edu [1] C. Kittel, On the theory of ferromagnetic resonance absorption , Phys. Rev. 73, 155 (1948). [2] Y .-P. Wang, G.-Q. Zhang, D. Zhang, X.-Q. Luo, W. Xiong, S.-P. Wang, T.-F. Li, C.-M. Hu, and J. Q. You, Magnon Kerr e ect in a strongly coupled cavity-magnon system , Phys. Rev. B 94, 224410 (2016). [3] X. 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2019-05-20

We show how to generate an entangled pair of yttrium iron garnet (YIG) samples in a cavity-magnon system without using any nonlinearities which are typically very weak. This is against the conventional wisdom which necessarily requires strong Kerr like nonlinearity. Our key idea, which leads to entanglement, is to drive the cavity by a weak squeezed vacuum field generated by a flux-driven Josephson parametric amplifier (JPA). The two YIG samples interact via the cavity. For modest values of the squeezing of the pump, we obtain significant entanglement. This is the principal feature of our scheme. We discuss entanglement between macroscopic spheres using several different quantitative criteria. We show the optimal parameter regimes for obtaining entanglement which is robust against temperature. We also discuss squeezing of the collective magnon variables.

Quantum drives produce strong entanglement between YIG samples without using intrinsic nonlinearities

1905.07884v2

Nonreciprocal Pancharatnam-Berry Metasurface for Unidirectional Wavefront Manipulation Hao Pan1, Mu Ku Chen2, Din Ping Tsai2, and Shubo Wang1,3 * 1Department of Physics, City University of Hong Kong, Tat Chee Avenue, Kowloon, Hong Kong, China. 2Department of Electrical Engineering, City University of Hong Kong, Tat Chee Avenue, Kowloon, Hong Kong, China. 3City University of Hong Kong Shenzhen Research Institute, Shenzhen, Guangdong 518057, China. *Corresponding author: shubwang@cityu.edu.hk ABSTRACT Optical metasurfaces have been widely used for manipulating electromagnetic waves due to their low intrinsic loss and easy fabrication. The metasurfaces employing the Pancharatnam- Berry (PB) geometric phase, called PB metasurfaces, have been extensively applied to realize spin-dependent functionalities, such as beam steering, focusing, holography, etc. The demand for PB metasurfaces in complex environments has brought about one challenging problem, i.e., the interference of multiple wave channels that limits the performance of PB metasurfaces. A promising solution is developing nonreciprocal PB metasurfaces that can isolate undesired wave channels and exhibit unidirectional functionalities. Here, we propose a mechanism to realize nonreciprocal PB metasurfaces of subwavelength thickness by using the magneto- optical effect of YIG material in synergy with the PB geometric phase of spatially rotating meta-atoms. Using full-wave numerical simulations, we show that the metasurface composed of dielectric cylinders and a thin YIG layer can achieve nearly 92% and 81% isolation of circularly polarized lights at 5.5 GHz and 6.5 GHz, respectively, attributed to the enhancement of the magneto-optical effect by the resonant Mie modes and Fabry-Pérot cavity mode. In addition, the metasurface can enable efficient unidirectional wavefront manipulations of circularly polarized lights, including nonreciprocal beam steering and nonreciprocal beam focusing. The proposed metasurface can find highly useful applications in optical communications, optical sensing, and quantum information processing. Ⅰ. INTRODUCTION The recent decade has witnessed significant progress in the design and fabrication of artificial optical structures working at microwave [1,2], terahertz [3,4], infrared [5,6], and visible optical bands [7,8], which can exhibit intriguing electromagnetic (EM) properties not existing in nature [9,10]. One important type is an ultrathin layer of structures known as metasurfaces. Metasurfaces can induce strong light-matter interaction in the nanoscale, and they benefit from small intrinsic loss and easy fabrication compared to conventional bulky metamaterials. By carefully designing the subwavelength elements (i.e., meta-atom) in each unit cell, metasurfaces can give rise to various fascinating wavefront-manipulation functionalities, such as perfect absorption [11-13], structural colors [14,15], anomalous reflection or refraction [16- 18], surface wave excitation [19,20], metalens [21-23], metaholograms [24-26], and many others [27,28]. Different from conventional diffractive optical devices, metasurfaces not only can employ the resonance phase and propagation phase (also called dynamic phase) but also can utilize the PB geometric phase derived from the spatial rotation of meta-atoms [29-33], leading to the so-called PB metasurfaces. Therefore, the PB metasurfaces can acquire an extra degree of freedom to control the wavefront of circularly polarized (CP) light besides the resonance and dynamic phases, giving rise to many intriguing phenomena such as photonic spin Hall effect [34,35], vortex beam generation [36,37], etc. Meanwhile, the PB metasurfaces can serve as a powerful platform for developing CP-light-associated applications, e.g., the CP wave control of the motions of biomolecules exhibiting chiral structures [38,39]. Consequently, the PB metasurfaces have great application potential in the next-generation photonic devices with multifunctionalities. Despite the PB metasurfaces’ unprecedented performance in wavefront manipulation, their functionalities are intrinsically restricted by the Lorentz reciprocity [40]. Introducing additional mechanisms to break the reciprocity of PB metasurfaces can generate new functionalities that are essential to many applications, such as invisible sensing, full-duplex communication, and noise-tolerant quantum computation, etc., where nonreciprocity can prevent the backscattering from defects or boundaries [41-43]. In addition, nonreciprocity allows metasurfaces to exhibit different properties for the opposite propagating waves, thus giving rise to Janus-type functionalities. One effective way to achieve nonreciprocity is by using gyroelectric or gyromagnetic materials sensitive to magnetic-field biasing, which exhibit asymmetric permittivity or permeability tensor accounting for the Faraday-magneto-optical (FMO) effect [44]. Qin et al. proposed a set of self-biased nonreciprocal magnetic metasurfaces to achieve bidirectional wavefront modulation based on the different hybrid resonant-dynamic phase profiles for bidirectional CP waves [45]. However, the local resonant phase can be easily affected by the coupling from neighboring meta-atoms, resulting in an undesired phase profile that limits the performance of the metasurface. In contrast, the PB geometric phase only determined by the rotation angle of meta-atoms is a better mechanism for realizing the stable nonreciprocal wavefront manipulation. Zhao et al. presented an interesting metadevice combining a PB metasurface and an anisotropic metasurface, which can simultaneously realize phase modulation and nonreciprocal isolation [46]. The metadevice involves a complex multilayer structure with a large thickness that inevitably affects its diffraction efficiency. Therefore, simple and thin PB metasurfaces capable of achieving high-efficiency nonreciprocal wavefront manipulation are highly desirable. In this article, we report a nonreciprocal PB metasurface composed of elliptical dielectric cylinders and a thin YIG layer to simultaneously realize PB-phase-based wavefront manipulation and microwave isolation. The thin YIG layer under a magnetic field bias can give rise to strong spin-selective isolation due to the FMO effect and Fabry-Pérot (FP) resonance. Meanwhile, the resonant coupling between the Mie modes in the dielectric cylinder and the FP mode in the YIG layer can effectively tune the nonreciprocal band by enhancing the FMO effect, thus achieving a high isolation ratio of nearly 92% and 81% at 5.5 GHz and 6.5 GHz, respectively. The PB phase of each meta-atom can be individually controlled via the corresponding optical-axis rotation and is unaffected by the FMO effect. Following a digital coding metasurface design methodology, the proposed nonreciprocal PB metasurfaces can offer multiple functionalities with high isolation, which are demonstrated by the tailored metadeflector with nonreciprocal beam steering and the metalens with nonreciprocal focusing. Our proposed all-dielectric nonreciprocal PB metasurface can find applications in multiple fields, e.g., EM wave isolation, nonreciprocal antennas, optical sensing, quantum information processing, etc. Ⅱ. RESULTS AND DISCUSSIONS A. Unidirectional spin-selective nonreciprocal metasurface The metasurface consists of subwavelength meta-atoms arranged in a square lattice with period l, as depicted in Fig. 1(a). Each meta-atom comprises a dielectric cylinder sitting on a YIG substrate of thickness t. Under the external biased magnetic field along the + z-direction, the YIG is characterized by a permeability tensor with asymmetric off-diagonal elements [44] 00 0 0 0 1r r r ri i , (1) where0 2 2 01m r m ,2 2 0m r , 0m sM , 0 0 0H i , γ = 1.579×1011 C/kg is the gyromagnetic ratio, 4π Ms = 1780 G is the saturation magnetization, B0 = μ0H0 = 0.05 T is the external magnetic field, α = 0.002 is the damping factor, and μ0 is the vacuum permeability. The relative permittivity of YIG is εr1 = 15. The dielectric cylinder has relative permittivity εr2 = 24 and relative permeability μr2 = 1. Its major and minor axis are a and b, respectively, and its height is h. The orientation of the cylinder is denoted by the angle θ. In this configuration, the metasurface exhibits different refractive indices for the incident CP waves with the wavevector parallel and antiparallel to the biased magnetic field, attributed to the FMO effect. Thus, the metasurface can give rise to spin- and direction-dependent manipulation of EM waves. To illustrate the physical mechanism, we first analyze the spin-selective transmission of normally incident CP waves in the thin infinite YIG layer with the + z magnetic field bias. The reflection and transmission coefficients for the left-hand circularly polarized (LCP) and right- hand circularly polarized (RCP) EM waves with wavevectors antiparallel to the biased magnetic field can be derived straightforwardly (See Appendix A for the derivations), and their expressions are 2 2 2 221 1 1 1f L f Li k t f Lf RLi k t f f L LY e r Y e Y , 0( ) 2 224 1 1f L f Li k k tf f L LLi k t f f L LY et Y e Y , 2 2 2 221 1 1 1f R f Ri k t f Rf LRi k t f f R RY e r Y e Y , 0( ) 2 224 1 1f R f Ri k k tf f R RRi k t f f R RY et Y e Y , (2) where 1f L r r rY and 1f R r r rY are the relative wave admittances for the LCP and RCP waves forward propagating in the YIG layer, 1 0( )f L r r rk k and 1 0( )f R r r rk k are the corresponding LCP and RCP wavevectors in the YIG layer, and k0 is the wavevector in free space. Here, the superscript “+” denotes the + z-biased magnetic field, “f” denotes forward incidence (i.e., - z-direction), and the subscript “ RL” (“LR”) denotes the CP conversion from LCP to RCP (RCP to LCP). It can be noted from Eq. (2) that the off-diagonal element κr in the permeability tensor results in the different impedances and wavevectors for the LCP and RCP waves, which leads to the differences in the co-polarized transmission (2| |f LLtand 2| |f RRt) and cross-polarized reflection (2| |f RLrand2| |f LRr). Figure 1(b) shows the transmission spectra given by Eq. (2) (denoted by the blue symbol lines). For the considered thin YIG layer, the FP cavity resonance can enhance the FMO effect and increase the transmission difference (i.e., 2 2| | | |f f RR LLt t ) for the LCP and RCP incident waves. This transmission difference can reach a maximum of nearly 94% around 6.8 GHz. To verify the analytical results, we conducted full-wave finite-element simulations by using COMSOL and computed the transmission spectra. The numerical results are denoted by the red symbol lines in Fig. 1(b), which is consistent with the analytical results. In addition, as the forward (− z- direction) normally incident LCP wave is equivalent to the backward (+ z-direction) normally incident RCP wave for the infinite YIG layer with the + z-biased magnetic field, Fig. 1(b) also indicates that the thin YIG layer can exhibit an evident nonreciprocal-transmission feature, i.e., the large transmission contrast between the forward and backward CP waves (See Appendix A for details). Figure 1(c) shows the transmission spectra of the PB metasurface under the incidence of the LCP and RCP plane waves. We set θ=0° for the rotation angle of all the dielectric cylinders. Due to the breaking of cylindrical symmetry by the meta-atoms, the helicity of the wave is not conserved, and the transmitted wave generally contains both LCP and RCP components. We notice that the transmission is dominated by the cross-polarized components f RLt and b LRt for the forward LCP and backward RCP incidence, respectively, which have two resonance peaks at 5.5 GHz and 6.5 GHz with differences (2 2| | | |f b RL LRt t ) of 92% and 81%, respectively. The large isolation ratio can be attributed to the FMO effect of the YIG layer enhanced by the Mie resonance in the dielectric cylinders. To understand the effect of the Mie resonant modes of the cylinders, we show in Fig. 1(d) the numerically calculated multipole decomposition of the cylinder scattering power under the excitation of the forward incident LCP wave (See Appendix C for the multipole decomposition). It is noted that the two resonances at 5.5 GHz and 6.5 GHz are mainly attributed to the magnetic dipole mode and the hybrid magnetic dipole- electric quadrupole mode, respectively. The resonant electric and magnetic field amplitudes are shown in the insets of Fig.1(d). We notice that the magnetic field inside the cylinder is strongly enhanced at 5.5 GHz, while both the electric and magnetic fields are strongly localized in the cylinder at 6.5 GHz due to the resonant electric quadrupole and magnetic dipole resonances. The resonant coupling between these hybrid Mie resonances and the FP cavity resonance in the YIG layer can enhance the interaction between the wave and the magnetic material, leading to the enhanced FMO effect and thus the strong nonreciprocity of the metasurface [47]. We further investigate the relationship between the nonreciprocal properties of the PB metasurface and various system parameters, including the cylinder height h, the biased magnetic field B0, and the incident angle. Figure 2(a) shows the numerically simulated isolation ratio 2 2| | | |f b RL LRt t as a function of the cylinder height h for the system in Fig. 1(a). As seen, the isolation peaks undergo redshift as h increases, which is expected since the eigenfrequencies of the Mie modes in the cylinder are generally inversely proportional to the geometric size of the cylinder. Specifically, as h varies, the spectral profile of the first resonance maintains a Lorentz shape, where the local maximum of the isolation remains above 85%. In contrast, the spectral profile of the second resonance undergoes dramatic variation due to the interference with other multipoles, as evidenced by the sharp transition of isolation from negative to positive values. Figure 2(b) shows the isolation ratio 2 2| | | |f b RL LRt t of the proposed metaisolator when the external magnetic field is B0 = 0.05 T, B0 = 0 T, and B0 = −0.05 T (corresponding to red, magenta, and blue symbol lines, respectively). We notice that the results for different biasing directions are nearly antisymmetric with respect to the case of B0 = 0 T which induces zero isolation. This can be understood as follows. The transmission coefficients follow the relationships f b RL RLt t and b f LR LRt t because the forward normally incident LCP (RCP) wave is converted to RCP (LCP) wave by the elliptical cylinder and the resulting RCP (LCP) wave is equivalent to the LCP (RCP) wave backward normally incident on the YIG layer (similar to the property of single YIG layer mentioned above). In addition, the magnetic field bias direction decides the spin-selective transmission of the metaisolator. For the opposite magnetic biasing, we can obtain the relationships b b RL LRt t and f f LR RLt t (See Appendix B for details). Consequently, we have the relationships f b RL LRt t and b f LR RLt t , and thus 2 2 2 2| | | | | | | |f b f b RL LR RL LRt t t t , i.e., reversing the direction of biased magnetic field leads to a sign change of the isolation value in Fig. 2(b). Figure 2(c) shows the dependence of the isolation on the magnitude of the external magnetic field. We notice that the isolation peaks at 5.5 GHz and 6.5 GHz are blue-shifted without obvious reduction of the isolation ratio, demonstrating the robust performance of the proposed metasurface isolator. We also investigate the effect of the incident angle of CP waves on the isolation. At large incident angles, higher-order diffractions can appear, and we only consider the isolation for the 0th-order cross-polarized transmission under the forward LCP and backward RCP wave incidence with the same incident angle. As depicted in Fig. 2(d), the isolation at the resonance frequency of 5.5 GHz will slightly shift with the increase of the incident angle. At large incident angles, the isolation at 5.5 GHz is reduced owing to the combined effect of the resonance shift and change of CP conversion efficiency in the elliptical cylinder. Notably, the isolation can still reach above 80% for the incident angle as large as 45°. Interestingly, the isolation ratio at 6.5 GHz is insensitive to the variation of the incident angle, and it can maintain a large value above 80% for the incident angle within [0°, 60°]. Therefore, the proposed nonreciprocal metaisolator can achieve a stable and high isolation ratio at the targeted frequencies for a wide range of incident angles, which lays the foundation for further nonreciprocal wavefront manipulations. In addition to manipulating the wave amplitude, the metasurface can also be applied to achieve unidirectional phase manipulation for the transmitted CP wave. This is done by varying the orientational angle θ of the dielectric cylinder to induce PB geometric phases, as shown by the inset in Fig. 3. For CP waves normally forward incident on the metasurface, the output waves can be expressed as 2 ( , ) 2 ( , )0 0out f i x y in L LR L out f i x y in R RL RE t e E E t e E , (3) where f LRt and f RLt are the cross-polarized transmission coefficients for the forward incident RCP and LCP waves, respectively. The superscript “±” denotes the direction of the external biased magnetic field B0. The dielectric cylinder can induce a PB phase shift φ = 2σθ, where σ = +1 (σ = –1) for the LCP (RCP) wave. Figure 3 shows the simulated amplitude of the transmitted electric field (blue symbol line) and the PB phase (red symbol line) for different orientation angles of the cylinder. As seen, the orientation angle θ of the cylinder has a negligible impact on the transmission amplitude, which is around 96% for different rotation angles. Meanwhile, the PB phase agrees with the relationship φ= 2σθ. The stable high CP transmission and the PB phase of 2π range lay the foundation for designing wavefront- manipulation metasurfaces. B. Nonreciprocal PB metadeflector for beam steering Owing to the superior nonreciprocal isolation under the large-angle incidence and the stable PB phase of the meta-atoms, it is possible to construct a nonreciprocal metadeflector with an on-demand phase profile to manipulate the propagation direction of the incident CP beam. Figure 4(a) schematically shows the concept of the nonreciprocal PB-phase-based metadeflector with the + z-biased magnetic field. The meta-atoms are invariant along y direction, but they are orientated differently in the x direction to induce the PB geometric phase profile. At 5.5 GHz, the metasurface can convert the forward incident LCP wave into the RCP wave and deflect it away from the normal direction. Meanwhile, the metasurface can isolate the backward RCP wave incident along the opposite deflection direction, i.e., the time-reversed wave of the deflected RCP wave. The transmitted wavevector and the incident wavevector satisfy the phase-matching condition in the periodic structure [48]: out in PBk k mk , (4) where kout = 2πsinθout/λ, kin = 2πsinθin/λ, kPB = 2π/P, θin and θout are the incident and deflected angles, respectively, λ is the incident wavelength, P is the period size of the supercell (covering 2π phase range) along the y-direction, and m is the deflection order. For the normally incident wave (θin = 0°), Eq. (4) can be simplified as sin θout = mλ/P where the supercell period P = Nl with N being the meta-atom number in the supercell and l being the meta-atom period. The discrete PB phase profile in the supercell can be expressed as φ(n) = 2πn/N where n denotes the n-th meta-atom in the supercell, thus requiring a rotation angle distribution θ(n) = πn/N. Following this principle, we design four different metadeflectors working at 5.5 GHz with the supercells consisting of 4, 6, 8, and 12 meta-atoms, respectively. These metasurfaces induce the 1st-order diffraction at the angles 74.64°, 40°, 28.82°, and 18.75°, respectively. Figure 4(b) shows the simulated electric field ( Ey) profiles at 5.5 GHz for the four metasurfaces. The deflection angles of the output beam are consistent with analytical values given by Eq. (4). Under the forward normal incidence, the 1st-order diffraction efficiency in these four cases is 66.44%, 92.1%, 95.59%, and 96.08%, respectively. Under the backward incidence, the transmission efficiency in the four cases is 6.64%, 0.79%, 0.026%, and 0.002%, respectively. Accordingly, the isolation ratios are 59.8%, 91.31%, 95.564%, and 96.078%, which demonstrate the highly efficient nonreciprocal beam steering function of the proposed metadeflectors. Additionally, we note that for 6-, 8-, and 12-cell cases, the deflected beams are mainly composed of the 1st-order diffraction, while higher-order diffraction components begin to appear in the output beam of the 4-cell case, which can be attributed to the large wavevector component parallel to the metasurface. The emergence of the higher-order diffractions in this case decreases the isolation ratio and leads to a complex output wavefront. C. Nonreciprocal PB metalens for beam focusing The PB-phase-based planar metalenses with excellent performance, e.g., high numerical aperture (NA), have been widely proposed and fabricated, generating broad applications in imaging [49,50], microscopy [51], and spectroscopy [52,53]. However, the effect of backscattering is usually neglected in conventional PB metalenses, thus limiting their applications in the platforms requiring anti-echo and anti-reflection functions. Introducing nonreciprocity to PB metalenses can be a solution to this problem. This corresponds to the concept of nonreciprocal PB metalens for unidirectional beam focusing, as illustrated in Fig. 5(a). The forward normally incident LCP wave passes through the nonreciprocal metalens with the +z-biased magnetic field and is focused into one spot, but the RCP wave radiated from the focusing spot, i.e., the time-reversal excitation, will be blocked by the metalens, thus realizing the nonreciprocal beam focusing. The PB phase profile φ(x,y) of the metalens should follow [49] 2 2 2 2( , )x y f x y f , (5) where λ is the wavelength, f is the focal length, x and y are the coordinates of each meta-atom. Similar to the metadeflector mentioned above, we consider the metalens with invariant phase profile in x-direction. The rotation angle profile of the meta-atoms in this case is 2 2( )y f y f , which has the discretized form 2 2 2( )n f n l f , where n denotes the n-th meta-atom, and l is the period of each meta-atom. To demonstrate the nonreciprocal focusing functionality, we design three metalenses with different focal lengths 1.5λ, 2λ, and 3λ (λ=54.5 mm at 5.5 GHz), respectively. We conduct numerical simulations for the nonreciprocal focusing realized by the three metalenses. Figure 5(b) depicts the simulated electric-field distributions in the yz-plane with the forward incident LCP (the upper panels) and the backward RCP radiation from the focal point (the bottom panels). It is noticed that the forward incident LCP waves are focused into spots at different focal points. The corresponding focal lengths are determined to be 80.62 mm, 108.85 mm, and 149.83 mm, respectively. The discrepancy between the theoretical and simulated focal lengths can be attributed to the coupling effect between the adjacent meta-atoms. Figures 5(c)-(e) show the intensity on the focal planes with the diffraction-limited ( λ/(2×NA)) full width at half-maximum (FWHM) of 30 mm, 30.27 mm, and 31.47 mm, respectively. The corresponding NA of the metalenses is 0.908, 0.9, and 0.866, respectively. To understand the nonreciprocity of the metalenses, we calculate the light transmission under the forward incidence, which reaches 79.9%, 85.1%, and 89.64% for the three cases, respectively. Meanwhile, the focusing efficiency is found to be 68.18%, 73.14%, and 74.51% for the three cases, respectively, where the focusing efficiency is defined as the fraction of the incident light that passes through a circular aperture in the focal plane with a diameter equal to three times of the FWHM spot size [54]. Additionally, we find that the backward RCP radiation from the focal point only gives rise to the transmission of 12.9%, 14.67%, and 10.9%, respectively. Therefore, the isolation ratios of the three metalenses are 55.28%, 58.47%, and 63.61%, respectively. The contrast between the focusing efficiency under forward incidence and the transmission under backward radiation clearly demonstrates the nonreciprocal focusing functionality of the designed PB metalenses. Ⅲ. CONCLUSION To summarize, we have demonstrated that high-performance nonreciprocal wavefront manipulation of CP beams can be achieved by using the magnetic-biased PB metasurfaces consisting of elliptical dielectric cylinders and a thin magnetic YIG layer. Due to the strong resonant coupling between the Mie modes in the cylinders and the FP cavity mode in the thin YIG layer, the FMO effect can be greatly enhanced near the resonant frequencies, thus giving rise to significant spin-selective nonreciprocal isolation. Meanwhile, the stable PB phase and the large isolation ratio over a wide range of incident angles can guarantee efficient nonreciprocal wavefront manipulation. By designing the PB phase gradient profile, we have demonstrated two types of nonreciprocal functional metasurfaces: the metadeflectors that can realize nonreciprocal beam steering with different deflection angles, and the high-NA metalenses that can realize nonreciprocal focusing with different focal lengths. The proposed nonreciprocal PB metasurfaces can simultaneously achieve high-efficiency wavefront manipulation and large isolation ratio, which pave the way to the applications in wave multiplexing for high-capacity communications and optical imaging with anti-reflection functions. ACKNOWLEDGEMENTS The work described in this paper was supported by grants from the Research Grants Council of the Hong Kong Special Administrative Region, China (Projects No. AoE/P-502/20 and No. CityU 11308223). APPENDIX A: SPIN-SELECTIVE TRANSMISSION OF AN INFINITE YIG LAYER The yttrium iron garnet (YIG) material is a common magnetic material that can show obvious asymmetry spin characteristics, e.g., the different propagation constants and impedances for the orthogonal CP states, due to the large off-diagonal elements in the permeability tensor under the external magnetic field biasing. For the considered thin YIG layer, the FP cavity resonance can enhance the FMO effect. To understand this property, we analytically determine the transmission and reflection for different CP waves propagating through the YIG layer. Consider an LCP wave backward (+ x direction) normally incident onto the YIG layer of thickness t and with the + x-biased magnetic field, as shown in Fig. 6, the electric and magnetic fields in regions 1, 2, and 3 can be expressed as: Region 1: 0 00 1ik xE e i incE, 0 0 00 1ik xY E i e incH, 0' '0 ik x y zE e E refE, 0' 0 '0 ik x z yY E e E refH. (A1) Region 2: 10 1b Lik x LE e i LCPE, 1 00 1b Lik x b L LE YY i e LCPH , 10 1b Rik x RE e i RCPE, 1 00 1b Rik x b R RE YY i e RCPH , (A2) 20 1f Lik x LE e i LCPE, 2 00 1f Lik x f L LE YY i e LCPΗ, 20 1f Rik x RE e i RCPE , 2 00 1f Rik x f R RE YY i e RCPH . (A3) Region 3: 0 " "0 ik x y zE e E tranE . (A4) where 0 0 0Y is the wave admittance in free space, 1( )b f L R r r rY Y and 1( )b f R L r r rY Y are the relative wave admittances for LCP and RCP waves normally forward (- x-direction) and backward (+ x-direction) propagating in the YIG material, 1 0( )b f L R r r rk k k and 1 0( )b f R L r r rk k k are the wave vectors of the LCP and RCP waves normally forward (- x-direction) and backward (+ x-direction) propagating in the YIG material, k0 is the wave vector in free space. Furthermore, according to the boundary conditions ( 20 n 1e H H , 0 n 1 2e Ε E ) between regions 1 and 2 ( x=0), and between regions 2 and 3 ( x=t), we can get eight equations to solve for the eight unknowns in Eq. (A1- A4), which can be expressed as below: At x=0: ' 0 1 1 2 2 ' 0 1 1 2 2 ' 0 0 1 0 1 0 2 0 2 0 ' 0 0 1 0 1 0 2 0 2 0( ) ( )y L R L R z L R L R b b f f z L L R R L L R R b b f f y L L R R L L R RE E E E E E iE E iE iE iE iE Y iE E iE YY iE YY iE YY iE YY Y E E E YY E YY E YY E YY (A5) At x=t: 0 0 0" 1 1 2 2 " 1 1 2 2 1 0 1 0 2 0 2 0 " 0 1 0f f b b L R L R f f b b L R L R f f b b L R L R b Lik t ik t ik t ik t ik t L R L R y ik t ik t ik t ik t ik t L R L R z ik t ik t ik t ik tb b f f L L R R L L R R ik t z ik tb L LE e E e E e E e E e iE e iE e iE e iE e E e iE YY e iE YY e iE YY e iE YY e Y E e E YY e 01 0 2 0 2 0 " 0f f b R L R ik t ik t ik tb f f R R L L R R ik t yE YY e E YY e E YY e Y E e (A6) By solving the Eq. (A5-A6), we can get the solutions: 2 0 122 2 1 2 0 222 2 22 ' 0 22 2 22 ' 0 22 2 " 02 ( 1) ( 1) ( 1) 0 0 2 ( 1) ( 1) ( 1) (1 )( 1) ( 1) ( 1) (1 )( 1) ( 1) ( 1) 4b L b L f R b L b L b L b Li k t b L Li k t b b L L R L f R Ri k t f f R R i k tb L yi k t b b L L i k tb L zi k t b b L L yE Y eE Y e Y E E E YE Y e Y E Y eE Y e Y E Y eE i Y e Y E YE 0 0( ) 22 2 ( ) " 0 22 2( 1) ( 1) 4 ( 1) ( 1)b L b L b L b Li k k tb L i k t b b L L i k k tb L zi k t b b L Le Y e Y E Y eE i Y e Y (A7) From Eq. (A7), we can observe that there exists the LCP wave along the + x-direction and the RCP wave along the - x-direction in the YIG layer, thus inducing coherent interference and the FP cavity resonance. Additionally, we note that the reflected and transmissive waves are always the RCP and LCP waves, respectively, and their coefficients can be represented as 22 22 2(1 )( 1) ( 1) ( 1)b L b Li k tb b L RLi k t b b L LY er Y e Y , 0( ) 22 24 ( 1) ( 1)b L b Li k k tb b L LLi k t b b L LY et Y e Y , (A8) where the superscript “+” indicates the + x-directional magnetic biasing, “ b” represents the backward normal incidence (+ x-direction), “ RL” symbolizes the CP state conversion from LCP to RCP, and of course “ LL” stands for the CP state conservation for LCP wave. Similarly, we also can get the solutions of the reflected and transmitted coefficients for the case of the RCP wave normally backward (+ x-direction) passing through the t-thick YIG layer with + x- directional magnetic biasing: 22 22 2(1 )( 1) ( 1) ( 1)b R b Ri k tb b R LRi k t b b R RY er Y e Y , 0( ) 22 24 ( 1) ( 1)b R b Ri k k tb b R RRi k t b b R RY et Y e Y . (A9) Comparing Eq. (A8) with Eq. (A9), it can be concluded that the difference in spin-dependent reflection and transmission is determined by the off-diagonal element κr. Furthermore, when the external magnetic field reverses the direction, the off-diagonal element in the permeability tensor will change from κr to –κr, thus the intrinsic admittance and wave vector of the LCP (RCP) wave in the case of + x-biased magnetic field case will be equal to those of the RCP (LCP) wave in the case of – x-biased magnetic field. Therefore, the corresponding reflection and transmissive coefficients can be expressed by , ,b b b b RL LR LR RL b b b b LL RR RR LLr r r r t t t t (A10) Meanwhile, due to the symmetry feature of YIG layer relative to the yz-plane, the reverse of the applied magnetic field is equivalent to the reverse of the incident direction of CP waves, thus getting b f RL LRr r , b f LR RLr r , b f LL RRt t , and b f RR LLt t . Thus, the spin-selective transmission and reflection also depend on the incident direction in addition to the magnetic field biasing. APPENDIX B: DEPENDENCE OF TRANSMISSION ON THE DIRECTIONS OF INCIDENCE AND BIASED MAGNETIC FIELD FOR METAISOLATOR The relationship between the CP states, wave propagation direction, and the biased magnetic field direction for the proposed metaisolator is numerically verified in Fig. 7. It can be noted in Fig. 7(a) that the cross-polarized transmissions for normally forward and backward incident LCP waves are nearly identical (i.e., f b RL RLt t ). This is also true for the RCP wave (i.e., f b LR LRt t ). A similar phenomenon can also be found in the case of − z-biased magnetic field shown in Fig. 7(b). This can be understood as follows. For the forward LCP wave passing through the metasurface with the + z-biased magnetic field, the cross-polarized transmission can be expressed as f L R f RL c Rt t t , where L R ctdenotes the conversion from LCP wave to RCP wave and f Rt is the forward RCP transmission for the YIG layer under the resonant coupling of the dielectric cylinder. Similarly, for the backward LCP wave, the cross-polarized transmission can be represented by b b L R RL L ct t t , where b Lt is the backward LCP transmission for the YIG layer under the resonant coupling of the dielectric cylinder. It should be noted that the forward RCP transmission f Rt is equal to the backward LCP transmission b Lt for the YIG layer in the presence of the Mie resonances of the cylinder, similar to property of the single YIG layer (discussed in Appendix A). Since the Mie resonances in the elliptical cylinder are spin-independent, the efficiency of its coupling to the YIG layer is unaffected by the CP state. Therefore, f b RL RLt t can be concluded. Meanwhile, their co-polarized transmission can also be expressed as (1 )f L R f LL c Lt t t and (1 )b b L R LL L ct t t , respectively. Since f b L Lt t owing to the nonreciprocal characteristic of YIG, we can obtain f b LL LLt t . Additionally, the magnetic-biased direction determines the spin-selective property of the metaisolator due to the electromagnetic characteristic of YIG. As demonstrated by the equal transmission of different CP states passing through the metaisolators with the opposite magnetic biasing, i.e., f f LR RLt t and f f RL LRt t . To summarize, these relationships can be described by f b f b RL RL LR LRt t t t and f b f b LR LR RL RLt t t t . APPENDIX C: ELECTROMAGNETIC MULTIPOLE EXPANSION The external field can induce the charge density ρ and current density J in the metasurface, which give rise to electromagnetic multipoles. Therefore, the resonance response of the metastructure can be understood based on the multipole decompositions. The multipole moments can be evaluated using the current density J(r) within the unit cell ( α, β, γ=x, y, z) as [55-57]: 31d rip J , (C1) 3 1 2d rc m r J , (C2) 2 3 1210r d rc T r J r J , (C3) 3 , ,1 2[ ( )]2 3eQ r J r J d ri r J, (C4) 3 3 ,1[( ) ] [( ) ]3mQ r d r r d rc r J r J , (C5) 2 2 3 , ,1[4 ( ) 5 ( ) 2 ( )]28TQ r J r r J r J r d rc r J r J , (C6) where p, m, T, Qe, Qm, and QT represent the electric dipole, magnetic dipole, toroidal dipole, electric quadrupole, magnetic quadrupole, and toroidal quadrupole, respectively, c is the light speed. The total scattered power Is of the metasurface can be expressed as [58] 4 4 5 2 2 3 3 4 6 6 62 2 2 , , 5 5 5 52 2 4 3 3 3 2 1 3 5 40s e mIc c c Q Q Oc c c c p m p T T (C7) We evaluated each term on the right-hand side of Eq. (C7) for the metaisolator, and the results are shown in Fig. 1(d). REFERENCES [1] K. Chen, Y. Feng, F. 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E 65, 046609 (2002). [56] V. Savinov, V. A. Fedotov, and N. I. Zheludev, Toroidal dipolar excitation and macroscopic electromagnetic properties of metamaterials, Phys. Rev. B 89, 205112 (2014). [57] G. N. Afanasiev, and Y. P. Stepanovsky, The electromagnetic field of elementary time- dependent toroidal sources, J. Phys. A: Gen. Phys. 28, 4565 (1995). [58] P. C. Wu, C. Y. Liao, V. Savinov, T. L. Chung, W. T. Chen, Y. W. Huang, P. R. Wu, Y. H. Chen, A. Q. Liu, N. I. Zheludev, and D. P. Tsai, Optical Anapole Metamaterial, ACS Nano 12, 1920 (2018). FIG. 1. The PB metasurface isolator and its nonreciprocal properties. (a) The schematics of the metaisolator and the meta-atom. The meta-atom is composed of an elliptical dielectric resonator and a YIG layer with geometric parameters l=14 mm, h=20 mm, a=12 mm, b=6 mm, and t=4.5 mm. The external biased magnetic field B0 is along + z direction. (b) The simulated (red symbol lines) and analytical (blue symbol lines) nonreciprocal transmission spectra of the YIG layer with B0=0.05 T pointing in the + z direction. 2| |f LLtand 2| |f RRt are the co-polarized transmission for the forward (- z-direction) normally incident LCP and RCP waves, respectively. (c) The transmission spectra of the metaisolator with B0=0.05 T pointing in the + z direction. 2| |f LLtand 2| |f RLt are the co- and cross-polarized transmission for the forward (- z-direction) normally incident LCP waves, and 2| |b RRtand 2| |b LRtare those for the backward (+ z-direction) normally incident RCP waves. (d) The normalized multipole scattering power of the dielectric elliptical cylinder under the excitation of the forward-incident LCP wave. p, m, T, Q e, and Q m are the electric dipole, magnetic dipole, toroidal dipole, electric quadrupole, and magnetic quadrupole, respectively. The inner image shows the corresponding electric and magnetic fields in the meta-atom at the resonant frequencies 5.5 GHz and 6.5 GHz. FIG. 2. The nonreciprocal characteristics of the PB-phase-based metaisolator. (a) The isolation ratio (i.e. 2 2| | | |f b RL LRt t ) of the metaisolator as a function of the frequency and the height of the dielectric elliptical cylinder. (b) The isolation ratio of the metasurface with the ± z-biased magnetic field of 0.05 T or without the magnetic field for the normally incident CP light (i.e., 2 2| | | |f b RL LRt t ). (c) The isolation ratio (i.e., 2 2| | | |f b RL LRt t ) as a function of the external + z- biased magnetic field strength B0 and the frequency of normally incident CP waves. (d) The isolation ratio (i.e., 2 2| | | |f b RL LRt t ) as a function of the incident angle and the frequency of the CP waves. The external magnetic field is in the + z-direction with a magnitude of 0.05 T. FIG.3. The cross-polarized transmission amplitude and phase shift for the metaisolator composed of elliptical dielectric cylinders with different orientation angles. The incident wave is LCP working at 5.5 GHz, and it normally incidents on the metasurface. FIG. 4. Nonreciprocal metadeflector for beam steering. (a) The schematic for the nonreciprocal beam steering by the PB metadeflector with the magnetic-field biasing pointing in + z-direction. (b) The simulated normalized electric-field profiles for the supercells with different meta-atoms. The incident wave is LCP for the upper-row panels and RCP for the bottom-row panels, and their propagation directions are denoted by the white arrows. The frequency is at 5 GHz. The metadeflectors with the supercells consisting of 4, 6, 8, and 12 cells can achieve the deflection angle of 74.64°, 40°, 28.82°, and 18.75°. FIG. 5. The nonreciprocal PB metalens for beam focusing. (a) The schematic for the nonreciprocal focusing of the proposed PB metalens. The incident LCP beam is converted to the RCP beam and focused at one point, while the RCP radiation from the focal point cannot pass through the metalens. (b) The normalized electric-field distribution in the yz-plane when the forward incident LCP beam passes through the metalenses with different focal lengths 1.5 λ, 2λ, and 3λ (the upper panels), and when the RCP wave radiated from the focal points backward propagates into the metalenses (the bottom panels). The normalized intensity distribution at the focal planes z=−1.5λ (c), z=−2λ (d), and z=−3λ (e), respectively, corresponding to the three cases in the upper panels of (b). FIG. 6. The schematic for the thin YIG layer with + x-biased magnetic field under the normal incidence of a LCP plane wave. FIG. 7. The comparison of cross-polarized transmission of different CP waves with different propagation directions through the metaisolator with magnetic biasing along +z (a) and -z (b) directions shown in the insets. The magnitude of the magnetic field is 0.05 T. The subscript “LR” (“RL”) represents the CP state conversion from RCP to LCP (LCP to RCP). The superscripts “+” and “−” indicate the + z- and −z-biased magnetic field; “ f” and “b” stand for the forward (− z-direction) and backward (+ z-direction) incidence.

2024-01-19

Optical metasurfaces have been widely used for manipulating electromagnetic waves due to their low intrinsic loss and easy fabrication. The metasurfaces employing the Pancharatnam-Berry (PB) geometric phase, called PB metasurfaces, have been extensively applied to realize spin-dependent functionalities, such as beam steering, focusing, holography, etc. The demand for PB metasurfaces in complex environments has brought about one challenging problem, i.e., the interference of multiple wave channels that limits the performance of PB metasurfaces. A promising solution is developing nonreciprocal PB metasurfaces that can isolate undesired wave channels and exhibit unidirectional functionalities. Here, we propose a mechanism to realize nonreciprocal PB metasurfaces of subwavelength thickness by using the magneto-optical effect of YIG material in synergy with the PB geometric phase of spatially rotating meta-atoms. Using full-wave numerical simulations, we show that the metasurface composed of dielectric cylinders and a thin YIG layer can achieve nearly 92% and 81% isolation of circularly polarized lights at 5.5 GHz and 6.5 GHz, respectively, attributed to the enhancement of the magneto-optical effect by the resonant Mie modes and Fabry-P\'erot cavity mode. In addition, the metasurface can enable efficient unidirectional wavefront manipulations of circularly polarized lights, including nonreciprocal beam steering and nonreciprocal beam focusing. The proposed metasurface can find highly useful applications in optical communications, optical sensing, and quantum information processing.

Nonreciprocal Pancharatnam-Berry Metasurface for Unidirectional Wavefront Manipulation

2401.10772v2

arXiv:2306.04390v1 [physics.optics] 7 Jun 2023Gain assisted controllable fast light generation in cavity magnomechanics Sanket Das,1Subhadeep Chakraborty,2and Tarak N. Dey1,∗ 1Department of Physics, Indian Institute of Technology Guwa hati, Guwahati-781039, Assam, India. 2Centre for Quantum Engineering Research and Education, TCG Centres for Research and Education in Science and Techno logy, Sector V, Salt Lake, Kolkata 70091, India (Dated: June 8, 2023) We study the controllable output field generation from a cavi ty magnomechanical resonator system that consists of two coupled microwave resonators. T he first cavity interacts with a ferromagnetic yttrium iron garnet (YIG) sphere providing t he magnon-photon coupling. Under passive cavities configuration, the system displays high ab sorption, prohibiting output transmission even though the dispersive response is anamolous. We replac e the second passive cavity with an active one to overcome high absorption, producing an effecti ve gain in the system. We show that the deformation of the YIG sphere retains the anomalous disp ersion. Further, tuning the exchange interaction strength between the two resonators leads to th e system’s effective gain and dispersive response. As a result, the advancement associated with the a mplification of the probe pulse can be controlled in the close vicinity of the magnomechanical r esonance. Furthermore, we find the existence of an upper bound for the intensity amplification a nd the advancement of the probe pulse that comes from the stability condition. These finding s may find potential applications for controlling light propagation in cavity magnomechanics. I. INTRODUCTION Cavity magnonics [1, 2], has become an actively pur- sued field of research due to its potential application in quantum information processing [3, 4]. The key con- stituent to such systems is a ferrimagnetic insulator with high spin density and low damping rate. It also supports quantized magnetization modes, namely, the magnons [5, 6]. With strongly coupled magnon-photon modes, cavity magnonics is an excellent platform for studying all the strong-coupling cavity QED effects [7]. Besides originating from the shape deformation of the YIG, the magnon can also couple to a vibrational or phonon mode [5]. This combined setup of magnon-photon-phonon modes, namely the cavity magnomechanics, has already demonstrated magnomechanically induced transparency [5], magnon-induced dynamical backaction [8], magnon- photon-phonon entanglement [9, 10], squeezed state gen- eration [11], magnomechanical storage and retreival of a quantum state [12]. Recently, PT-symmetry drew extensive attention to elucidate the dynamics of a coupled system character- ized by gain and loss [13, 14]. Here, Pstands for the parity operation, that results in an interchange between the twoconstituentmodes ofthe system. Thetime rever- sal operator Ttakesito−i.PT-symmetry demands the Hamiltonian is commutative with the joint PToperators i.e.,[H,PT] = 0. This system possesses a spectrum of entirely real and imaginary eigenvalues that retain dis- tinguishable characteristics [15]. The point separating these two eigenvalues is the exceptional point (EP) [16] where the two eigenvalues coalesce, and the system de- generates. Anaturaltestbed for PT-symmetricHamilto- ∗tarak.dey@iitg.ac.innian is optical as well as quantum optical systems [17–19] which alreadyled to the demonstrationof someofthe ex- oticphenomena,likenonreciprocallightpropagation[20], unidirectional invisibility [21, 22], optical sensing and light stopping [23]. Very recently, a tremendous effort has been initiated to explore non-Hermitian physics in magnon assisted hybrid quantum systems. The second- order exceptional point is detected in a two-mode cavity- magnoic system, where the gain of the cavity mode is ac- complished by using the idea of coherent perfect absorp- tion [24]. The concept of Anti- PTsymmetry has been realized experimentally [25], where the adiabatic elim- ination of the cavity field produces dissipative coupling between two magnon modes. Beyond the unique spectral responses, these non-Hermitian systems can manipulate the output microwavefield transmission[26, 27]. Theun- derlying mechanism behind such an application is mag- netically induced transparency[5, 28], where the strong magnon-photoncouplingproducesanarrowspectral hole inside the probe absorption spectrum. Further studies in this direction establish the importance of the weak magnon-phonon coupling to create double transmission windows separated by an absorption peak. Moreover, manipulating the absorption spectrum is also possible by varying the amplitude and phase of the applied magnetic field [29]. It is well established over the past decade that op- tomechanically induced transparency (OMIT) [30–32] is an essential tool for investigating slow light [33] and light storage [34, 35] in cavity. In addition, incorporating PT- symmetry in optomechanical systems, provides a better controllability of light transmission [36, 37] and produces subluminal to superluminal light conversion. Nonethe- less, their proposals may find experimental challenges as the gain of the auxiliary cavity can lead the whole system to instability [38]. An eminent advantage of the2 magnomechanical system over the optomechanical sys- tem is that it offers strong hybridization between the magnon-photon mode. The magnomechanical systems offer better tunability as an external magnetic field can vary the magnon frequency. Exploiting these advan- tages, aPT-symmetry-like magnomechanicalsystem can be constructed by resonantly driving the YIG sphere to an active magnon mode [39]. The controllable sideband generation with tunable group delay can be feasible by changing the power of the control field. This paper investigates a controllable advancement and transmission of the microwave field from a coupled cavity magnomechanical system. Optical coupling be- tween a passive cavity resonator containing YIG sphere and a gain-assisted auxiliary cavity can form a coupled cavity resonator. An external drive has been used to deform the YIG sphere’s shape, resulting in the magnon- phonon interaction in the passive cavity. We show how the gain of the auxiliarycavityhelps to overcomeabsorp- tive behaviourin our hybrid system. As a result, the out- put microwave field amplifies at the resonance condition. Moreover, the weak magnon-phonon interaction exhibits anomalous dispersion accompanied by a gain spectrum, demonstrating superluminal light. We also examine how the slope of the dispersion curve can be controlled by tuning the photon hopping interaction strength between the two cavities. The paper is organized as follows. In Section II, a the- orical model for the coumpound cavity magnomechanical system with PT-symmetric resonator is described. The Heisenserg equations of motion to govern the expecta- tion values of operators of every system are derived in this Section. In Section IIIA, we analyse the stability criteria of the model system and examine the effect of the auxiliary cavity gain on the absorptive and disper- sive response of the system in Section IIIB. Section IIIC discusses the output probe field transmission. Further, the group velocity of the optical probe pulse has been studied analytically and verified numerically in Section IIID. Finally, we draw our conclusions in Section IV. II. THEORETICAL MODEL Recently, there has been a growing interest in real- izing a gain in different components of cavity magnon- ics systems [24, 39]. In this work, we investigate the effect of medium gain on the probe response and its transmission. The system under consideration is a hy- brid cavity magnomechanical system that consists of two coupled microwave cavity resonators. One of the res- onators is passive and contains a YIG sphere inside it. We refer to this resonator as a cavity magnomechanical (CMM) resonator. Applying a uniform bias magnetic field to the YIG sphere excites the magnon mode. The magnon mode, in turn, couples with the cavity field by the magnetic-dipole interaction. Nonetheless, the exter- nalbiasmagneticfieldresultsinshapedeformationofthea1a2J εc,ωl εp,ωp κ1 κ2B0 FIG. 1. The schematic diagram of a hybrid cavity mag- nomechanical system. The system consists of two coupled microwave cavities. One of them is passive, and another one is active. The passive cavity contains a ferromagnetic YIG sphere inside it. The applied bias magnetic field pro- duces the magnetostrictive interaction between magnon and phonon. The coupling rates between the magnon-photon and magnon-phonon are gmaandgmb, respectively. Strong con- trol field of frequency ωland a weak probe field of frequency ωpare applied to the passive cavity. YIG sphere, leading to the magnon-phonon interaction. The second resonator (degenerate with the first one) is coupled to the first resonator via optical tunnelling at a rateJ. Two input fields drive the first resonator. The amplitude of the control, εl, and probe fields, εp, are given byεi=/radicalbig Pi/ℏωi,(i∈l,p) withPiandωibeing the power and frequency of the respective input fields. The Hamiltonian of the combined system can be written as H=ℏωca† 1a1+ℏωca† 2a2+ℏωmm†m+ℏωbb†b +ℏJ(a† 1a2+a† 2a1)+ℏgma(a† 1m+a1m†) +ℏgmbm†m(b†+b)+iℏ/radicalbig 2ηaκ1εl(a† 1e−iωlt−a1eiωlt) +iℏ/radicalbig 2ηaκ1εp(a† 1e−iωpt−a1eiωpt), (1) wherethe firstfourtermsoftheHamiltoniandescribethe free energy associated with each system’s constituents. The constituents of our model are characterized by their respective resonance frequencies: ωcfor the cavity mode, ωmfor the magnon mode, ωbfor the phonon mode. The annihilationoperatorsforthecavity,magnonandphonon modes are represented by ai, (i= 1,2),mandb, respec- tively. The fifth term signifies the photon exchange in- teraction between the two cavities with strength, J. The sixth term of the Hamiltonian corresponds to the inter- action between the magnon and photon modes, charac- terized by a coupling rate gma. The interaction between the magnon and phonon modes is described by the sev- enth term of the Hamiltonian and the coupling rate be- tween magnon and phonon mode is gmb. Finally, the last two terms arise due to the interaction between the cav- ity field and two input fields. The cavity, magnon and phonon decay rates are characterized by κ1,κmandκb, respectively. The coupling between the CMM resonator and the output port is given by ηa=κc1/2κ1, where3 κc1is the cavity external decay rate. In particular, we will consider the CMM resonator to be working in the critical-coupling regime where ηais 1/2. At this point, it is convenient to move to a frame rotating at ωl. Fol- lowing the transformation Hrot=RHR†+iℏ(∂R/∂t)R† withR=eiωl(a† 1a1+a† 2a2+m†m)t, the Hamiltonian in Eq. (1) can be rewritten as Hrot=/planckover2pi1∆a(a† 1a1+a† 2a2)+/planckover2pi1∆mm†m+/planckover2pi1ωbb†b +/planckover2pi1J(a† 1a2+a† 2a1)+/planckover2pi1gma(a† 1m+a1m†) +ℏgmbm†m(b†+b)+i/planckover2pi1/radicalbig 2ηaκ1εl(a† 1−a1) +i/planckover2pi1/radicalbig 2ηaκ1εp(a† 1e−iδt−h.c), (2) where ∆ a=ωc−ωl(∆m=ωm−ωl) andδ=ωp−ωlare, respectively, the cavity (magnon) and probe detuning. The mean response of the system can be obtained by the Heisenberg-Langevinequationas /angbracketleft˙O/angbracketright=i//planckover2pi1/angbracketleft[Hrot,O]/angbracketright+ /angbracketleftN/angbracketright. Further, we consider the quantum fluctuations ( N) as white noise. Then starting form Eq. 2, the equations of motion of the system can be expressed as /angbracketleft˙a1/angbracketright= (−i∆a−κ1)/angbracketlefta1/angbracketright−igma/angbracketleftm/angbracketright−iJ/angbracketlefta2/angbracketright +/radicalbig 2ηaκ1εl+/radicalbig 2ηaκ1εpe−iδt, /angbracketleft˙m/angbracketright= (−i∆m−κm)/angbracketleftm/angbracketright−igma/angbracketlefta1/angbracketright −igmb/angbracketleftm/angbracketright(/angbracketleftb†/angbracketright+/angbracketleftb/angbracketright), /angbracketleft˙b/angbracketright= (−iωb−κb)/angbracketleftb/angbracketright−igmb/angbracketleftm†/angbracketright/angbracketleftm/angbracketright, /angbracketleft˙a2/angbracketright= (−i∆a+κ2)/angbracketlefta2/angbracketright−iJ/angbracketlefta1/angbracketright, (3) whereκ2andκbrespectively denote the gain of the sec- ond resonator and phonon damping rates. We note that κ2>0 corresponds to a coupled passive-active CMM resonators system and κ2<0 describes a passive-passive coupled CMM resonators system. Assuming the control field amplitude εlto be larger than the probe field εp, each operator expectation values /angbracketleftO(t)/angbracketrightcan be decom- posedintoitssteady-statevalues Osandasmallfluctuat- ing termδO(t). The steady-state values of each operator are a1s=(−i∆a+κ2)(−igmams+√2ηaκ1εl) (i∆a+κ1)(−i∆a+κ2)−J2,(4a) ms=−igmaa1s i∆′m+κm, (4b) bs=−igmb|ms|2 iωb+κb, (4c) a2s=iJa1s (−i∆a+κ2). (4d) While the fluctuating parts of Eq. 3can be expressed as δ˙a1=−(i∆a+κ1)δa1−iJδa2−igmaδm +/radicalbig 2ηaκ1εpe−iδt, δ˙m=−(i∆′ m+κm)δm−igmaδa1−iGδb−iGδb†, δ˙b=−(iωb+κb)δb−iGδm†−iG∗δm, δ˙a2=−(i∆a−κ2)δa2−iJδa1, (5)where∆′ m= ∆m+gmb(bs+b∗ s)istheeffectivemagnonde- tuning and G=gmbmsis the enhanced magnon-phonon coupling strength. For simplicity, we express these fluc- tuation equations as id dt|ψ/angbracketright=Heff|ψ/angbracketright+F, (6) where the fluctuation vector |ψ/angbracketright= (δa1,δa† 1,δa2,δa† 2,δb,δb†,δm,δm†)T, input field F= (√2ηaκ1εpe−iδt,√2ηaκ1εpeiδt,0,0,0,0,0,0)T. Next, we adopt the following ansatz to solve Eq. 5: δa1(t) =A1+e−iδt+A1−eiδt, δm(t) =M+e−iδt+M−eiδt δb(t) =B+e−iδt+B−eiδt, δa2(t) =A2+e−iδt+A2−eiδt. (7) HereAi+andAi−correspond to the ithcavity generated probe field amplitude and the four-wavemixing field am- plitude, respectively. By considering h1=−i∆a+iδ− κ1, h2=−i∆a−iδ−κ1, h3=−i∆a+iδ+κ2, h4= −i∆a−iδ+κ2, h5=−iωb+iδ−κb, h6=−iωb−iδ− κb, h7=−i∆′ m+iδ−κm, h8=−i∆′ m−iδ−κm, we obtainA1+which corresponds to the output probe field amplitude from the CMM resonator as A1+(δ) =C(δ) D(δ), (8) where C(δ) =−/radicalbig 2ηaκaεph3(h5h7h∗ 6(J2h∗ 8+h∗ 4(g2 ma+h∗ 2h∗ 8)) +|G|2(h5−h∗ 6)(J2(h7−h∗ 8)−h∗ 4(gma2+h∗ 2(h∗ 8−h∗ 7)))), D(δ) =h5h∗ 6(g2 mah3+h7(h1h3+J2)) (J2h∗ 8+h∗ 4(g2 ma+h∗ 2h∗ 8))+|G|2(h5−h∗ 6) (J2(g2 mah3+(h1h3+J2)(h7−h∗ 8))−h∗ 4 ((h1h3+J2)(g2 ma−h∗ 2(h7−h∗ 8))−h∗ 2h3g2 ma)).(9) The output field from the CMM resonator is obtained by the cavity input-output relation εout=/radicalbig 2ηaκ1/angbracketlefta1/angbracketright−εl−εpe−iδt.(10) By substituting Eq. 7into Eq. 10, we obtain the normal- izedoutputprobefieldintensityfromtheCMMresonator as T=|tp|2=/vextendsingle/vextendsingle/vextendsingle/vextendsingle√2ηaκ1A1+ εp−1/vextendsingle/vextendsingle/vextendsingle/vextendsingle2 . (11) In order to numerically simulate the transmitted output probefield spectrum, weusethe followingexperimentally realizable set of parameter values [5, 40]. The degenerate microwave cavities of frequency ωc/2π= 7.86 GHz. The decay rate of the first cavity is κ1/π= 3.35 MHz. The spin density ρ= 4.22×1027m−3and the diameter of the4 YIG sphere D= 25µm. It results in 3 ×1016number of spins (Nm) present in the YIG sphere. The phonon mode has frequency ωb/2π= 11.42 MHz with decay rateκb/π= 300 Hz, and the magnon-phonon coupling strengthgmb/2πis 1 Hz. The Kittel mode frequency of the YIG sphere is ωm=γeB0,i, with gyromagnetic ratio,γe/2π= 28 GHz/T and B0,iis the input bias magnetic field amplitude. The magnon decay rate is κm= 3.52 MHz. Magnon-photon coupling strength gma=γeBvac√5Nm/2 can be controlled by changingthe vacuum magnetic field amplitude as Bvac=/radicalbig 2π/planckover2pi1ωc/V. III. RESULTS A. Stability Analysis Initially we consider the two coupled cavities which are operating under a balanced gain-loss condition. The Hamiltonian describing such coupled resonator system (gma=gmb= 0) can be written as Hcav=/planckover2pi1(∆a−iκ1)δa† 1δa1+/planckover2pi1(∆a+iκ1)δa† 2δa2 +/planckover2pi1J(δa† 1δa2+δa† 2δa1). (12) The eigenvalues of Hcavareλ±= ∆a±/radicalbig J2−κ2 1. Note that the above Hamiltonian remains invariant under the simultaneous parity P:a1↔a2and time-reversal operation T:i→ −ioperations, and, its eigenvalues are entirely real and complex for J > κ 1andJ < κ 2. The pointJ=κ1, which marks this transition from PT symmetric to the PTbreaking phase, is known as the exceptional point (EP). One must understand the com- petitive behaviour between the inter-cavityfield coupling and the loss/gain rates to get insight into this transition. ForJ > κ 1the intracavity field amplitudes can be coherently exchanged and thus give rise to a coherent oscillation between the field amplitudes. However, for J < κ 1the intracavity field can not be transferred to the other one, resulting in a strong field localization or in other words exponential growth. A quick look at Eq. 4(a) also suggests such gain-induced dynamic instability ina1atJ=κ1for ∆a= 0. This situation becomes more complicated in the presence of magnon-photon coupling. Now, the combined system ( gma,gmb/negationslash= 0) ceases to become PTsymmetric. However, the effect of an additional gain cavity ( κ2>0) can be understood by analyzing the stability diagram of the whole system. In the following, we derive the stability condition by invoking the Routh-Hurwitz criterion which requires all the eigenvalues of Heffhave negative real parts. The magenta region of Fig. 2suggests that when gma is small the instability threshold remains close to the J=κ1(the conventional EP for a binary PTsymmetric system). However, with increasing gmathe system reaches instability at a larger exchange interaction J. Such a restriction over the choice of the photon exchange FIG. 2. The stable and unstable regions are determined as a function of normalized evanescent coupling strength ( J/κ1) and the cavity-magnon coupling strength ( gma) when the loss of the CMM-resonator is perfectly balanced by the gain of the auxiliary cavity ( κ1=κ2). We consider the control field intensity to be 10 mW. The other parameters are ωc= 2π× 7.86 MHz, ωb= 2π×11.42 MHz, ∆ a= ∆′ m=ωb= 2π×11.42 MHz,κ1=κ2=π×3.35 MHz, κm=π×1.12 MHz, κb= π×300 Hz, and gmb= 2πHz. rateparameter Jwill be followedthroughoutthis paper. B. Absorption and dispersion spectrum Themagnomechanicalsystemunder considerationcor- responds to the level diagram of Fig. 3. Application of a probe field excites the passive cavity mode and allows the transition between |1/angbracketrightand|2/angbracketright. The exchange in- teraction,J, couples two degenerate excited states |2/angbracketright and|5/angbracketright. The presence of the strong control field dis- tributes the population between the two states |2/angbracketrightand |3/angbracketright. The magnon-phonon coupling, gmb, couples both the metastable ground states |3/angbracketrightand|4/angbracketright. Here, we con- sider both the microwave cavities to be passive ( κ2<0) under a weak magnon-photon coupling strength, gma. In this situation, the magnon-photon hybridization is in- significant. The absorptive and dispersive response can be quantified by the real and imaginary components of (tp+1) that will be presented as αandβ, respectively. In Fig.4(a), we present the absorptive response of the system as a function of normalized probe detuning. The black-solid curve depicts a broad absorption spectrum of the probe field when the exchange interaction is much weaker than the cavity decay rate. One can explain it by considering the level diagram of Fig. 3, where the ini- tial population stays in the ground state |1/angbracketright. Applying a probefieldtransfersthepopulationfromthegroundstate to the excited state, |2/angbracketright. In addition, the weak magnon-5 εp,ωpεl,ωl gmb /C2 |1/angbracketright|2/angbracketright |3/angbracketright |4/angbracketright|5/angbracketright |n1,n2,m,b/angbracketright|n1+1,n2,m,b/angbracketright|n1,n2+1,m,b/angbracketright |n1,n2,m+1,b/angbracketright |n1,n2,m,b+1/angbracketright FIG. 3. Level diagram of the model system. |ni/angbracketright,|m/angbracketrightand|b/angbracketright represents the photon number state of ithcavity, magnon mode and phonon mode, respectively. The application of strong control field to the CMM resonator couples |n1+ 1,n2,m,b/angbracketrightand|n1,n2,m+ 1,b/angbracketright, whereas, the pres- ence of a weak probe field increases the photon number of CMM resonator by unity. gmbcouples |n1,n2,m+ 1,b/angbracketrightand|n1,n2,m,b+ 1/angbracketright. The hopping interaction be- tween the two cavities directly couples |n1+ 1,n2,m,b/angbracketright ↔ |n1,n2+1,m,b/angbracketright. photon coupling (with respect to κ1) restricts a signifi- cant transition from |2/angbracketrightto|3/angbracketright. As a result, it allows the transfer of a fraction of the excited state’s population by invoking the exchange interaction J. The increase in the exchange interaction strength causes a gradual decrease in|2/angbracketright’s population. It reduces the absorption coefficient around the resonance condition except for δ=ωb. This phenomenon is shown by the red-dashed and the blue dotted-dashed curve of Fig. 4(a). We observe a nar- row absorption peak inside the broad absorption peak forJ= 0.4κ1. The sharp absorption peak, exactly at δ=ωb, occurs due to the magnomechanical resonance. Furtherincreasingtheexchangeinteractionvirtuallycuts off the population distribution from |2/angbracketrightto|3/angbracketright. As a result, the effect of magnon-phonon resonance also de- creases, and the absorption peak at δ=ωbeventually diminishes. In Fig. 4(b), we present the dispersion spec- trum as a function of normalized detuning δ/ωb. For the time being, we neglect the effect of magnomechanical couplingandobservetheoccurrenceofanomalousdisper- sion around δ=ωbforJ= 0.4κ1. Further, increasing the exchange interaction strength more significant than the cavity decay rate can alter the dispersive response from anomalous to normal, as shown by the red-dashed and blue-dot-dashed curves. In the inset of Fig. 4(b), we plot the slope of the temporal dispersion dβ/dδat the extreme vicinity of the magnon-phonon resonance con- dition. The positive values of the slope of the temporal dispersion signify anomalous dispersion due to the mag- nomechanical coupling. However, the steepness of the dispersion curve can be reduced by increasing the ex- change interaction strength, as shown by the red-dashed and blue-dot-dashed curves. Note that this dispersion curve is accompanied by absorption. Output transmis- sionoftheprobefieldisprohibitedinthepresenceofhuge0.5 0.75 1 1.25 1.5 δ/ωb00.20.40.60.81αJ = 0.4 κ1 J = 1.1 κ1 J = 1.3 κ1(a) 0.5 0.75 1 1.25 1.5 δ/ωb-0.4-0.200.20.4βJ = 0.4 κ1 J = 1.1 κ1 J = 1.3 κ1 0.999 1.0008δ/ωb 0123 dβ/dδ(b) x10-6 FIG. 4. (a) Absorption and (b) dispersion spectrum of the model system. The slope of the dispersion curve is shown in the inset. Here we consider both the microwave cavities are passive, with identical decay rates ( κ1=−κ2). The magnon- photon couplong strength, gmais taken as 2 MHz. All the other parameters are mentioned earlier. absorption. Therefore, reducing absorption or introduc- ing the gain to the system is mandatory for observingthe group velocity phenomena. To achieve reasonable transmission at the output, we replace the auxiliary passive cavity with an active one where the second cavity’s gain ( κ2>0) completely bal- ances the first cavity’s loss. In this scenario, the sta- bility criterion for the hybrid system allows us to con- sider the exchange interaction strength Jto be greater than 1.053κ1forgma= 2 MHz. We present the ab- sorptive response of the model system in Fig. 5(a). The black solid curve of Fig. 5(a) illustrates the occurrence of a double absorption peak spectrally separated by a broad gain regime. The graphical nature is determined by the roots of D(δ), which are, in general, complex. The real parts of the roots determine the spectral peak position, and the imaginary parts correspond to their widths. To illustrate this, we consider J= 1.30κ1with all other parameters remaining the same as earlier. The real parts of the root of D(δ) present two distinct nor- mal mode positions at δ/ωbvalues 0.88 and 1.12. The other two normal modes are spectrally located at the same position δ/ωb= 1. The interference between these6 0.8 0.9 1 1.1 1.2 δ/ωb-40-2002040α J = 1.30 κ1 J = 1.10 κ1 J = 1.07 κ1(a) 0.8 1 1.2 δ/ωb-40-2002040Im (tp+1) J = 1.30 κ1 J = 1.10 κ1 J = 1.07 κ10.96 11.04δ/ωm -25025Im (tp+1) (b) FIG. 5. (a) Absorption and (b) dispersion spectrum of the model system. The slope of the dispersion curve is shown in the inset. Here we consider the second cavity as a gain cavity, with κ2=κ1. All the other parameters are the same as before. two normal modes becomes significant while approach- ing the stability bound as depicted by the red dashed and blue dot-dashed curve of Fig. 5(a). In turn, it re- duces the overall gain of the composite system. Further, we investigate the effect of a gain-assisted auxiliary cav- ity on the medium’s dispersive response in Fig. 5(b). For J= 1.30κ1, the two absorption peaks produce two dis- tinct anomalous dispersion regions separated by a broad normal dispersive window. Weakening the exchange in- teraction strength reveals prominent normal dispersion around the resonance condition except for δ=ωb, and the window shrinks. In the inset of Fig. 5(b), we present the slope of the dispersive response due to the magnome- chanical resonance. The black solid curve of Fig. 5(b) suggests the occurrence of anomalous dispersion at the magnon-phonon resonance condition. Moreover, one can increase the steepness of the dispersion curve by simply approaching the instability threshold, as delineated by the red-dashed and blue-dotted-dashed curve of the inset of Fig.5(b). In the consecutive section, we will discuss how the change in the dispersion curve can produce con- trollable group velocity of the light pulses through the medium and investigate the role of the exchange interac- tion.C. Output probe transmission The output probe intensity from the system depends onitsabsorptiveresponse. Equation 11dictatethetrans- mission of the probe field and is presented in Fig. 6. For Fig.6(a), we consider both the microwave cavities as passive ones with identical decay rates, i.e.,κ1=−κ2. The black solid curve shows a broad absorptive response forJ= 0.40κ1. Increasing the exchange interaction strengthcausesgradualenhancementintheoutputprobe transmission, as delineated by the red-dashed and blue dot-dashed curve of Fig. 6(a), and the absorption win- dow splits into two parts. A precise observation confirms the presence of extremely weak transmission dip exactly atδ=ωbfor all the three exchange interaction strengths under consideration. In Fig. 6(b), we present the advan- tage of using a gain-assistedauxiliary cavity along with a CMM resonator to obtain a controllable amplification of the output probe field. We begin our discussion consid- ering the photon hopping interaction, J= 1.30κ1. The black solid curve of Fig. 6(b) estimates the normalized probe transmission of 6 .03. Here the normalization is done with respect to the input probe field intensity. By decreasingthe parameter J, weapproachthe unstablere- 0.5 0.75 1 1.25 1.5 δ/ωb00.20.40.60.81TJ = 0.4 κ1 J = 1.1 κ1 J = 1.3 κ1(a) 0.96 0.98 1 1.02 1.04 δ/ωb0200400600800 TJ = 1.30 κ 1 J = 1.10 κ1 J = 1.07 κ1(b) FIG. 6. Exchange interaction Jdependent normalized out- put probe transmission is plotted as a function of normalize d detuning between the control and the probe field when (a) both the cavities are passive ones, and (b) one is active and another one is passive.7 gion and observe the occurrence of a double transmission peak separated by a sharp and narrow transmission dip. The amplitude of the double transmission peak demon- stratestheprobepulseamplificationbyafactorof830,as presented by the blue dotted-dashed curve. However, an explicit observation suggests the output probe field am- plification by a factor of 67 at the resonance condition δ=ωb. The physics behind the probe field amplifica- tion can be well understood as: Introduction of gain to the second cavity compensates a portion of losses in the first cavity through J. This leads to an enhanced field amplitude in the first cavity. In the presence of mod- erate magnon-photon coupling it increases the effective magnon-photon coupling strength. Hence, we observe a higher transmission at the two sidebands but also find a large transmission dip at δ=ωb. D. Group delay Controllable group delay has gained much attention due to its potential application in quantum information processing and communication. The dispersive nature of the medium is the key to controlling the group delay of the light pulse under the assumption of low absorptionor gain. The pulse with finite width in the time domain is produced by superposing severalindependent waveswith different frequencies centered around a carrier frequency (ωs). The difference in time between free space propa- gation and a medium propagation for the same length can create a group delay. The analytical expression for the group delay can be constructed by considering the envelope of the optical pulse as f(t0) =/integraldisplay∞ −∞˜f(ω)e−iωt0dω, where˜f(ω) corresponds to the envelope function in the frequency domain. Accordingly, the reflected output probe pulse can be expressed as fR(t0) =/integraldisplay∞ −∞tp(ω)˜f(ω)e−iωt0dω, (13) =e−iωst0/integraldisplay∞ −∞tp(ωs+δ)˜f(ωs+δ)e−iδt0dδ, =tp(ωs)e−iωsτgf(t0−τg). (14) This expression can be obtained by expanding tp(ωs+δ) in the vicinity of ωsby a Taylor series and keeping the terms upto first order in δ. An expression for time-delay is obtained as [31, 41] τg= Re/bracketleftBigg −i tp(ωs)/parenleftbiggdtp dω/parenrightbigg/vextendsingle/vextendsingle/vextendsingle/vextendsingle ωs/bracketrightBigg , (15) which can be further simplified as τg=(α(ωs)−1)dβ dω/vextendsingle/vextendsingle/vextendsingle/vextendsingle ωs−β(ωs)dα dω/vextendsingle/vextendsingle/vextendsingle/vextendsingle ωs |tp(ωs)|2.(16)0.98 0.99 1 1.01 1.02 δ/ωb-18-12-60612τg (µs) J = 1.30 κ1 J = 1.10 κ1 J = 1.07 κ1 FIG. 7. Time delay of the probe pulse for different evanescent coupling strength Jhave been plotted against the normalized probe detuning δ/ωb, while the control power is 10 mW. All other parameters are taken as the same as in Fig. 5. From Eq. 16, the slope of the medium’s absorption and dispersion curves determine the probe pulse propagation delay or advancement. However, Fig. 5(b) suggests that the value of βis negligibly small near the magnomechan- ical resonance. Hence, the group delay depends on the first term of the numerator of Eq. 16. In Fig. 7, we ex- amine the effect of photon-photon exchange interaction on the probe pulse propagation delay when both cavities operate under balanced gain-loss condition. The system produces anomalous dispersion accompanied by a gain response. The black solid curve of Fig. 7depicts the probe pulse advancement of 2 .4µs for the photon hop- ping interaction strength, J= 1.3κ1. Moreover, one can enhance the effective gain and the slope of the anomalous dispersion curve by approaching the instability thresh- old. That, in turn, brings out the super luminosity of the output probe pulse, characterized by the advancement of 17.9µs as shown by the blue dotted-dashed curve of Fig. 7. To verify the above results, we consider a Gaussian probepulsewith afinite width aroundthe resonancecon- dition,i.e.,δ=ωb, and numerically integrate it by using Eq.13. The shape of the input envelope is considered as, ˜f(ω) =εp√ πΓ2e−(ω−ωs)2 Γ2, (17) where Γ is the spectral width of the optical pulse. We consider Γ to be 7 .17 kHz, such that the Gaussian enve- lope is well-contained inside the gain-window around the resonance condition ( δ=ωb), as depicted in Fig. 5(a). The dispersive, absorptive as well as gain response of the system can be demonstrated by examining the effect of the probe transmission coefficient ( tp) on the shape of the input envelope. The gain of the auxiliary cavity manipulates the probe transmission coefficient in such a way that it amplifies the intensity of the output probe pulse. The black solid curve depicts the output probe pulse amplification of 6 .2 for photon-hopping interaction strength,J= 1.30κ1. A decrease in the Jvalue8 gradually enhances the effective gain in the system. It -3 -1.5 0 1.5 3 Normalized time Γt020406080|tp|2J = 1.30 κ1 J = 1.10 κ1 J = 1.07 κ1-0.2 -0.1 0 0.1 0.2Γt 0.9960.99750.999 Normalized intensity FIG. 8. The relative intensity of the output probe pulse is plotted against the normalized time (Γt) for different photo n- photon exchange interaction strength when both cavities ar e operating under balanced gain-loss condition. amplifies the output probe transmission as presented by the red dashed and blue dotted-dashed curves of Fig. 8. We observe that the output field amplification can reach to a factor of 65.3 while considering the exchange interaction strength to be 1 .07κ1. Further decreasing the exchange interaction will lead to dynamical insta- bility in our model system. Interestingly, the temporal width of the probe pulse is almost unaltered during the propagation through the magnon-assisted double cavity system. This numerical result agrees with our analytical results for the output probe transmission, as shown in Fig. 6(b). Moreover, the importance of the photon-photon exchange interaction on the probe pulse propagation advancement can be observed from the inset of Fig. 8. The peak separation between the input pulse ( t= 0) and the output pulse for J= 1.30κ1estimates the probe pulse advancement of 2 .34µs. The red dashed, and blue dashed-dotted curve of the inset estimates the probe pulse advancement of 8 .75 µsand 13.30µsforJ= 1.10κ1and 1.07κ1, respectively. IV. CONCLUSION In conclusion, we have theoretically investigated the controllable output field transmission from a critically coupled cavity magnomechanical system. We drive the first cavity with a YIG sphere inside it, establishing the magnon-photon coupling. The photon exchange interac- tion connects the second microwave cavity with the first. An external magnetic field induces the deformation ef- fect of the YIG sphere. In this study, the interaction between the magnon and photon modes lies under the weak coupling regime. The medium becomes highly ab- sorbent when both cavities are passive, and the output probe transmission is prohibited. We introduce a gain to the auxiliary cavity to overcome this situation. It is noteworthy that the instability threshold must be close to the conventional exceptional point for a binary PT- symmetric system. At the magnomechanical resonance, the auxiliary cavity produces an effective gain associ- ated with anomalous dispersion. Further, decreasing the photon exchange interaction strength causes gradual en- hancement of the effective gain and the steepness of the dispersion spectrum. As a result, we observe a control- lable superluminal microwave pulse propagation associ- atedwithamplificationbyafactorof67. Bystudyingthe propagationdynamics ofa Gaussianprobepulse ofwidth 7.17 kHz, we confirm that the numerical study is con- sistent with the analytical results. 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2023-06-07

We study the controllable output field generation from a cavity magnomechanical resonator system that consists of two coupled microwave resonators. The first cavity interacts with a ferromagnetic yttrium iron garnet (YIG) sphere providing the magnon-photon coupling. Under passive cavities configuration, the system displays high absorption, prohibiting output transmission even though the dispersive response is anamolous. We replace the second passive cavity with an active one to overcome high absorption, producing an effective gain in the system. We show that the deformation of the YIG sphere retains the anomalous dispersion. Further, tuning the exchange interaction strength between the two resonators leads to the system's effective gain and dispersive response. As a result, the advancement associated with the amplification of the probe pulse can be controlled in the close vicinity of the magnomechanical resonance. Furthermore, we find the existence of an upper bound for the intensity amplification and the advancement of the probe pulse that comes from the stability condition. These findings may find potential applications for controlling light propagation in cavity magnomechanics.

Gain assisted controllable fast light generation in cavity magnomechanics

2306.04390v1

Design of an optomagnonic crystal: towards optimal magnon-photon mode matching at the microscale Jasmin Graf,1,2Sanchar Sharma,1Hans Huebl,3,4,5and Silvia Viola Kusminskiy1,2 1Max Planck Institute for the Science of Light, Staudtstraße 2, 91058 Erlangen, Germany 2Department of Physics, University Erlangen-Nürnberg, Staudtstraße 7, 91058 Erlangen, Germany 3Walther-Meißner-Institut, Bayerische Akademie der Wissenschaften, Walther-Meissner-Straße 8, 85748 Garching, Germany 4Physik-Department, Technische Universität München, James-Franck-Straße 1, 85748 Garching, Germany 5Munich Center for Quantum Science and Technology (MCQST), Schellingstraße 4, 80799 München, Germany We put forward the concept of an optomagnonic crystal: a periodically patterned structure at the microscale based on a magnetic dielectric, which can co-localize magnon and photon modes. The co-localization in small volumes can result in large values of the photon-magnon coupling at the single quanta level, which opens perspectives for quantum information processing and quantum conversionschemeswiththesesystems. Westudytheoreticallyasimplegeometryconsistingofaone- dimensional array of holes with an abrupt defect, considering the ferrimagnet Yttrium Iron Garnet (YIG) as the basis material. We show that both magnon and photon modes can be localized at the defect, and use symmetry arguments to select an optimal pair of modes in order to maximize the coupling. We show that an optomagnonic coupling in the kHz range is achievable in this geometry, and discuss possible optimization routes in order to improve both coupling strengths and optical losses. I. INTRODUCTION Progress in fundamental quantum physics has by now established a basis for developing new technologies in the fields of information processing, secure communication, and quantum enhanced sensing. In order to perform these tasks, physical systems are needed which are ca- pable of processing, storing, and communicating infor- mation in a quantum coherent manner and with a high fidelity. Similar to the classical realm, accomplishing this goal requires different degrees of freedom and efficient couplings between them, giving rise to hybridsystems. In this context, systems at the mesoscopic scale (with di- mensions ranging from nanometers to microns) are spe- cially interesting since their collective degrees of freedom can be tailored [1]. An important and successful exam- ple of these mesoscopic hybrid systems are optomechan- ical systems [2], where light couples to mechanical mo- tion. Seminal experiments in these systems have demon- strated extra-sensitive optical detection of small forces and displacements [3–10], manipulation and detection of mechanical motion in the quantum regime with light [11– 13], and the creation of nonclassical light and mechanical motion states [11, 14, 15]. In the recent years, the family of hybrid quantum sys- tems has been extended by incorporating magnetic ma- terials, where the collective spin degree of freedom can be exploited. For example in spintronics [16], information is carried by spins (as opposed to electrons) in order to re- move Ohmic losses and to increase memory and process- ing capabilities [17, 18]. An ultimate form of spintronics is the new field of quantum magnonics [19, 20], where superconducting quantum circuits couple, via microwave yxz(b) (c)(a)zxyd2rDefect areaaMagnetic modeOptical modeG YIG-SiN-heterostructure YIGSiNSiN |Hz| 01|δmz| Optical modeMagnetic modeFigure 1. Investigated geometry: (a) Optomagnonic crystal with an abrupt defect at its center for localizing an optical and a magnon mode at the same spot in the defect area. (b) Optomagnonic crystal from the side representing a het- erostructure. (c) Mode profiles of the localized optical and magnon mode discussed in the main text. Note: all mode shape plots are normalized to their corre- sponding maximum value. fields in a cavity, coherently to magnetic collective ex- citations (magnons) [21, 22]. Such systems are promis- ing for generating and characterizing non-classical quan- tum magnon states [19, 23–25], quantum thermometry protocols [26], and for developing microwave-to-optical quantum transducers for quantum information process- ing [27, 28]. The coherent coupling of magnons to op- tical photons has also been demonstrated in recent ex-arXiv:2012.00760v2 [cond-mat.mes-hall] 25 Mar 20212 periments [29–33], in what have been denominated op- tomagnonic systems [28, 34–40]. In current experiments exploring optomagnonics, the ferrimagnetic dielectric Yttrium Iron Garnet (YIG) is used as the magnetic element, since YIG presents low absorption and a large Faraday constant in the infrared (= 0:069 cm1andF= 240 deg=cm@1:2µm[29, 30, 41–43]). The coupling between spins and optical pho- tons is an second order processes involving spin-orbit coupling and it is generally small. This can be enhanced using a well polished sphere that acts as an optical cav- ity, trapping the photons by total internal reflection in order to effectively enhance the spin-photon coupling. The coupling, however, still remains too small for ap- plications. This is due to the large size of the used YIG spheres (the coupling increases as the volume of the cavity decreases [34]), with radius of the order of hun- dreds of microns, and, concomitantly, the large differ- ence between the optical and the magnetic mode volume (VmagVopt), by which most of the magnetic mode vol- ume does not participate in the coupling. This can be partially mitigated by making smaller cavities [44, 45], but care has to be taken both to obtain a good mode matching and to retain a good confinement of the optical mode in order to minimize radiation losses. Recent pro- posals have investigated one dimensional layered struc- tures to this end [37, 46]. In order to tackle these issues, we propose an opto- magnonic array at the microscale, which acts simultane- ously as a photonic crystal [47], determining the opti- cal properties of the structure, and as a magnonic crys- tal [48–50] with tailored magnetostatic modes. Our pro- posal is inspired in the success of this approach for op- tomechanical crystals, which can be designed such as to enhance the phonon-photon coupling by many orders of magnitude [51–74]. In our case, we use similar concepts in order to design the coupling between photonic and magnonic modes. Although similar conceptually, mag- netic materials present new challenges for the design, due to the complexity of the magnon modes. Photonic crystals are the basis for many novel appli- cations in quantum information, and are of high interest due to their ability to guide [75–79] and confine [80–85] light, allowing for example to enhance non-linear optical interactions [86–89]. In turn, magnonic crystals can be designed to create reprogrammable magnetic band struc- tures [90], to act as band-pass or band-stop filters, or to create single-mode and bend waveguides [91–94]. Addi- tionally these crystals can be used for spin wave comput- ing via logical gates [95–97]. An advantage of magnonic crystalsistheirscalability, theirlowenergyconsumption, and possibly faster operation rates [49, 97, 98]. Together withthestateoftheartinoptomagnonicsdetailedabove, thisprovidesagreatincentivetoexplorethepossibilityof anoptomagnonic crystal, combining both photonic and magnetic degrees of freedom.Specifically, we consider an optomagnonic crystal con- sisting of a dielectric magnetic slab (YIG in our simula- tions) with a periodic array of holes along the slab and with an abrupt defect in the middle. The repeated holes at each side of the defect act as a Bragg mirror for both optical and magnetic modes, localizing them in the re- gion of the defect (see Fig. 1(a)+(c)). We show that this structure can co-localize photonic and magnonic modes, and explore how the symmetry of the modes can be used to optimize their coupling. We find that coupling rates in the range of kHz are achievable in these structures, and that optimization of the geometry can lead to higher coupling values, indicating the promise of this approach. Further optimization is nevertheless needed to improve the decay rates, in particular the optical quality factor is low compared to the state of the art in non-magnetic structures (where Silicon is used as the dielectric). This manuscript is organized as follows. In Sec. II we derive the general expression for the coupling of magnons to optical photons and discuss the normalization of the modes required to find the photon-magnon coupling at the single quanta level, denominated optomagnonic cou- pling. The remaining sections refer to the numerical method for evaluating this coupling. For our simula- tions we choose YIG as the magnetic material, in line with the material of choice in experiments. In Sec. III we discuss the properties of the proposed structure as a photonic crystal. In Sec. IV, in turn, we investigate its properties as a magnonic crystal. Sec. V combines the re- sults in order to numerically evaluate the optomagnonic coupling for appropriately chosen confined modes. For concreteness, we focus on the coupling between one sin- gle magnon mode and one single optical mode. Sec. VI is devoted to a discussion on how the structure can be op- timized and presents results for an optimized geometry. TheconclusionsandanoutlookarepresentedinSec.VII. The supplementary material contains further details of the analytic calculations and of the simulations. II. OPTOMAGNONIC COUPLING In this section we derive the theoretical expression for the coupling rate between magnons and photons. The instantaneous electromagnetic energy is [99] Etot em=1 2 dr D(r;t)
E(r;t) +B(r;t)
H(r;t) (1) withDthe displacement field, Ethe electric field, B the magnetic induction, and Hthe magnetic field. In complex representation D= (D+D)=2andE= (E+E)=2, and similar for the magnetic induction and field. The effect of the magnetization Mis to modify the displacement field as D(r;t) = "[M(r;t)]
E(r;t)where3 the components of the permittivity tensor "are [100, 101] "ij(M) ="0 "rijifFX kijkMk+fCMiMj! ; (2) with"0the vacuum permittivity, "rthe relative per- mittivity,ijkthe Levi-Cevita tensor, and ffF;fCgma- terial dependent magneto-optical constants. At opti- cal frequencies the second term in Eq. (1) can be ne- glected [101, 102], being smaller than the first by the fine structure constant squared, and the permeability of the material can be set to the vacuum permeability 0. The magneto-optical constants can be related to the Faraday rotation Fand the Cotton-Mouton ellip- ticityCper unit length as F=!=(2cp"r)fFMsand C=!=(2cp"r)fCM2 s, withcthe speed of light in vac- uum andMsthe saturation magnetization. We are interested in how light couples to the fluctua- tionsofthemagnetizationaroundthestaticgroundstate. We consider norm-preserving small fluctuations, M(r;t) =M0(r)s 1M(r;t) Ms2 +M(r;t);(3) where the ground state satisfies M0
M0=M2 sand the fluctuations are perpendicular to local equilibrium magnetization M
M0= 0. In complex notation M= [M+M]=2. The correction to the electromag- netic energy stemming from the interaction between the light field and the magnetization can be rewritten as Eem=1 8 dr E(r;t)
D(r;t)+E(r;t)
D(r;t) ;(4) ignoring E(r;t)
D(r;t)andE(r;t)
D(r;t)in the rotating wave approximation. Inserting the relation be- tween the displacement and the electric field along with the permittivity in Eq. (2) gives Eem=EF em+EC emwhere EF em="0fF 8 dr[i(EE)
M+h.c.](5) is the Faraday contribution and EC em=0fC 8 dr[E
(MM 0+M0M)
E+h.c.] (6) the Cotton-Mouton one. We have used the dyadic no- tation and neglected all terms that represent a constant energy shift or that are higher order in M. Quantizing this expression leads to the optomagnonic coupling Hamiltonian. By assuming that the magnetic material acts as an optical cavity, the electric field of the light can be quantized by using the annihilation (cre- ation) operator ^a(y) of one photon E(r;t)!2iX E(r) ^a(t); (7)withE() the mode shape, and the mode index. We note that we identified E(r;t)with 2E+(r;t)from the well known quantization expression of the electric field [103] E(r;t) =E+(r;t) +E(r;t) =iX h E(r) ^a(t)E (r) ^ay (t)i :(8) In order to find the coupling per photon, we normalize the electromagnetic field amplitude to one photon over the electromagnetic vacuum [104] ~! 2"0= dr"r(r)jE(r)j2: (9) The spin waves can be quantized as M(r;t)!2MsX m (r)^b (t);(10) where ^b(y) annihilates (creates) one magnon, m() is the mode shape, and the mode index. We note that as in the optical case we identified M(r;t)with 2M+(r;t) from the magnetic quantization expression M(r;t) =M+(r;t) +M(r;t) =MsX h m (r)^b (t) +m (r)^by (t)i : (11) In order to normalize the amplitude of the magnetic fluc- tuations to one magnon, we use the following expression derived in the supplementary material (see appendix A) gB Ms= drim0
 m m  (12) withgthe g-factor, Bthe Bohr magneton, and m0= M0=Ms. This expression is valid for arbitrary magnetic textures and it is consistent with the normalization de- rived previously for a uniform ground state [105]. The quantized optomagnonic energy, neglecting the constant energy shifts, leads to the coupling Hamiltonian ^Hom=~X  h G ^ay ^a^b +h.c.i (13) with the coupling constant G =GF  +GC  , where GF  =i"0"r ~Fn  dr E E
m ;(14) GC  ="0"r ~Cn  dr[m0
E] [E 
m ] +"0"r ~Cn  dr[m0
E ] [E
m ](15)4 are, respectively, the Faraday and Cotton-Mouton com- ponents of the optomagnonic coupling constant, being n the light wavelength inside the material. The coupling between optical photons and magnons, as can be seen from Eq. 13, involves a three-particle pro- cess in which a magnon is created or annihilated by a two-photon scattering process. This is an example of parametric coupling, andreflectsthefrequencymismatch between the excitations. The coupling can be enabled by a triple-resonance, where the frequency of the magnon matches the frequency difference between two photonic modes [29–31, 35], or, in the case of scattering with a single photon mode, by an external driving laser at the right detuning [34]. If the laser is red (blue) detuned by the magnon frequency, implying a lower (higher) driving frequency than the photon resonance, it will annihilate (create) magnons. In the red detuned regime this can be used, for example, to actively cool the magnon mode to its ground state [2, 40, 106]. In this work we focus on the coupling between a given magnon mode and a given optical mode hosted by the 1D optomagnonic crystal shown in Fig. 1(a)+(b). Hence we set=and drop the indices in the following. For con- creteness, we focus on GFas the analysis for GCis anal- ogous.GFis proportional to the overlap of the magnon’s spatial distribution with the electric component of the optical spin density defined as [107] Sopt(r) ="0 2i!opt EE : (16) The optical spin density is finite only for fields with cer- tain degree of circular polarization, and points perpen- dicular to the plane of polarization. III. PHOTONIC CRYSTAL Photonic crystals are engineered structures which, by proper shape design, can confine light to a specific re- gion. These are formed by low-loss media exhibiting a periodic dielectric function "(r), with a discrete transla- tional symmetry "(r) ="(r+R)for any R=nawith nan integer and athe lattice constant given by the im- posed periodicity. Photonic band gaps arise at the edges of the Brillouin zone (BZ) k==adue to the periodicity imposed by the susceptibility of the crystal on the electric field, with wavelength = 2a(correspondingtotheedgeoftheBZ). For example, in a 1D photonic crystal (see Fig. 2(a)) the symmetry of the unit cell around its center implies that the nodes of the standing light wave must be centered either at each low- "layer or at each high- "layer. The latter necessarily has lower energy than the former, re- sulting in a band gap. The position of the photonic band gap is given by the mid-gap frequency at the BZ edge. In the case of two materials with refractive indices n1 (b)(a)1D photonic crystal1D photonic crystal with defectɛ2ɛ2ɛ2ɛ2ɛ1ɛ1ɛ1ɛ2ɛ2ɛ2ɛ2ɛ1ɛ1ɛ1localization area“mirror”“mirror”d1d2aFigure 2. General structure of a 1D photonic crystal and mode localization at a defect: (a) 1D photonic crystal con- sisting of periodic layers alternated by the lattice constant awith different dielectric constants "1> " 2and widths d1 andd2. (b) A defect breaks the symmetry and can pull a band-edge mode into the photonic band gap. Since a mode in the band gap cannot propagate into the structure, the light is Bragg-reflected and is thus localized (see e.g. [47]). andn2and thicknesses d1andd2=ad1, the normal incidence gap is maximized for n1d1=n2d2. In this case the mid-gap frequency is given by [47] !mg=n1+n2 4n1n2
2c a(17) withn1=p"1,n2=p"2. The corresponding vacuum wavelength mg= (2c)=!mgthereby satisfies the rela- tionsmg=n1= 4d1andmg=n2= 4d2meaning that the individual layers are a quarter-wavelength thick. An input at frequencies within the photonic band gap is reflected entirely except for an exponentially decaying tail inside the crystal. Thus, two of such crystals can be used to create a Fabry-Perot like cavity. More concretely, as shown in Fig. 2(b), a defect in the form of a layer with a different width breaking the symmetry of the crystal may permit localized modes in the band gap by consec- utive reflection on both sides. Since the light is localized in a finite region, the modes are quantized into discrete frequencies. We note that the degree of localization is the largest for modes with frequencies near the center of the gap [47]. For our purposes, we consider a geometry in which the permittivity can take two distinct values, attained by holes carved into a dielectric slab (see Fig. 1(a)). The typical material used for photonic crystals is silicon due toitshighrefractiveindexatopticalfrequencies, "r= 12. We use instead YIG for our study, which is a dielectric magnetic material transparent in the infrared range with "r= 5[108]. The lower dielectric constant reduces the confinement of the optical modes along the height of the slab, which is reflected in low optical quality factors as discussed below. This structure is a 1D photonic crystal, periodic in one direction (chosen to be the ^x-direction), with a band gap along this direction and which confines light through index guiding [47] (a generalization of total internal reflection) in the remaining directions. In order to localize an optical mode in this structure we create a defect by increasing the spacing between the two middle holes, which pulls a mode into the band gap. We note that due to the (discrete) periodicity, the crystal only5 possesses an incomplete band gap and the localized mode can scatter to air modes [47]. We search for a localized mode in the infrared fre- quency range where YIG is transparent and presents low absorption [42, 109]). Thus, the geometrical parameters of the crystal need to be chosen in such a way that the band gap lies in the desired frequency range. We choose a lattice constant of a= 450 nm which gives a mid- gap frequency of !mg= 2240 THz (corresponding to1250 nm), using Eq. (17) with refractive indices of YIGn1=nYIG=p 5and of airn2=nair= 1. Note that we choose a lattice constant that allows us to work in the transparency frequency range for the optics, and which at the same time is small enough in order to reduce the computational cost of the micromagnetic simulations of the corresponding magnonic crystal in the next section. Using the relation dairnair=dYIGnYIGfor a maximized normal incidence gap we find the optimal radius of the air holes as rair=nYIG nYIG+ 1
a 2(18) withdair= 2rair=adYIG, from which we obtain rair= 155:25 nm. In order to find the mode with the least losses, the defect width is optimized in order to lo- calize the desired mode most effectively to the defect. We find the optimal defect size, defined as the center- to-center distance between the two bounding holes (see Fig. 1(a)), to be d= 731 nm , obtained by evaluating the transmission spectra as a function of the defect size (we used the electromagnetic simulation tool MEEP to this end [110]). In order to get a good quality factor of the localized mode we need to insert as many air holes as possible. For creating a compromise between short com- putational evaluation time (especially important for the magnetic simulations discussed later) and a good quality factor we chose N= 12holes at each side of the defect. Therefore the investigated crystal is in total l= 11:75µm long. The overall width of the wave guide is w= 600 nm and its height is h= 60 nm, again to keep the magnetic simulations (which we detail in the next section) feasible. Such a thin slab will not be good at confining the opti- cal modes along its height, since it is much smaller than the light wavelength in the material. In order to increase the optical quality factor without influencing the mag- netics we sandwich the crystal with two Si 3N4layers (see Fig. 1(b)) with a height of hSi3N4= 200 nm as proposed in [111]. Si 3N4has an index of refraction similar to YIG (nSi3N4=p 4), so that the combined structure acts ap- proximately as a single cavity for the light and its height is roughly half a wavelength, enough to provide a rea- sonable confinement. The presented simulations include these two extra layers. We now turn to categorizing the photonic modes of the crystal by using its three mirror symmetry planes (see Fig. 3(a)). This imposes several restrictions on the (b)(a)z=0TE-likeTM-like Mirror symmetry (y=0)Mirror symmetry (x=0)Symmetries of the photonic crystalMirror symmetry (z=0)Symmetry of TE and TM modesyzEEFigure 3. Symmetries of the optical modes in a periodic waveguide: (a) Symmetry planes of the investigated 1D pho- tonic crystal shown in Fig. 1(a). (b)Symmetry of a transverse electric (TE)-like and a transverse magnetic (TM)-like opti- cal mode in a thin 3D structure. The red arrows indicate the electric field vector Ewhich forz= 0(middle of the crystal along the height) lie in plane for TE-like modes and point out of plane for TM-like modes. For z6= 0this is not fulfilled anymore (see e.g. [47]). mode shape and the mode polarization. We define the three mirror symmetry operations ^E zE(r) =0 @Ex(x;y;z) Ey(x;y;z) Ez(x;y;z)1 A; ^E yE(r) =0 @Ex(x;y;z) Ey(x;y;z) Ez(x;y;z)1 A; ^E xE(r) =0 @Ex(x;y;z ) Ey(x;y;z ) Ez(x;y;z )1 A:(19) In the following we restrict the discussion to transverse electric modes with an in-plane electric field, which are the modes of interest for the magnetic configuration we consider, as it will be clear from the next section. We note that structures made of a high- "material with air holesfavourabandgapfortransverseelectricmodes[47], which is advantageous for our purposes. Unlike in 2D, in three dimensions we cannot generally distinguish be- tween transverse electric (TE) and transverse magnetic (TM) modes. However, provided that the crystal has a mirrorsymmetryalongitsheight(under ^E z), andthatits thickness is smaller than the mode wavelength, the fields are mostly polarized in TE-like and TM-like modes [47] (see Fig. 3(b)). Since a TE-like mode has a non-zero elec- tric field in the plane of the crystal ( xy-plane), both Ex andEycannot be odd as a function of z(see Fig. 3(b)). From Eq. (19), this implies that the mode must be even under ^E z. Similar symmetry considerations [47] show that a TE-like mode must satisfy ^E zE(r) =E(r); ^E yE(r) =E(r); ^E xE(r) =E(r):(20)6 zxyzxy -11(b)(a) |Ey||Ex||Hz|Mode shape 01Optical spin density ~ i(E* x E)z Band diagramSymmetriesRe[Ex]Re[Ey]Extended states 0𝞹/akxfno defectDiscrete states1. band2. band ~ 200 Hz~ 270 HzDefect state07 x 10-3kx [1/nm] -11 -11 01(c) Re[δmz]Im[δmz]Mode profileBand diagram|δmz| SymmetriesLocalized mode @ 13.12 GHz016f [GHz]FT[mz]normMode spectrumkx = 𝞹/aδmzkx [1/nm] f [GHz]016𝛅mzdefect07 x 10-3 0𝞹/akxzxy Figure 4. Optical (a and b) and magnetic modes (c): (a) Band diagram (obtained with MEEP) for TE-like modes within the irreducible BZ with a state that was pulled into the gap from the upper band-edge state by the insertion of a defect (note that the gap-state was not obtained by band diagram simulations). The bands in the green shaded area representing the light cone are leaky modes which couple with radiating states inside the light cone [112]. From the shape of the localized defect mode (obtained with Comsol) with a frequency of !opt= 2246 THz (middle layer in the xy-plane) we see that this mode is odd with respect to x= 0andy= 0(and even with respect to z= 0). (b) Optical spin density (middle layer in the xy-plane) of the localized mode, fulfilling the same symmetries, see main text. (c) Band diagram of backward volume waves within the irreducible BZ showing magnetic modes with extended k-values but preferring wave vectors at the edge of the BZ. The highest excited localized mode has a frequency of !mag= 213:12 GHzand is odd along the mirror symmetry planes for x= 0 andy= 0(and additionally even with respect to the plane for z= 0). The dashed line in the middle inset shows the mode spectrum in case of no defect. Note: all mode shape plots are normalized to their corresponding maximum value. For evaluating the optical modes we used the two finite elementtoolsMEEP[110]andComsolMultiphysics[113] (see appendix B). The simulated band structure for TE- like modes in the considered photonic crystal shows a broad band gap in the infra-red frequency range with a nicely pulled defect band (see Fig. 4(a)) which is ex- tended in frequency space resulting from the confinement in real space. The defect mode in the gap at the edge of the BZ has a frequency of !opt2246 THz (obtained by Comsol, 205 THz/1550 nm according to MEEP, note that the difference is due to a reduction of the sim- ulation geometry to 2D in order to save simulation time) with a damping factor of opt20:1 THzwhich gives a linewidth (FWHM) of opt= 2
opt20:2 THz. Using the values obtained by Comsol this gives an optical quality factor ofQ=!opt=(4opt) = 1250 , which is in the expected range for this kind of geometry [114] (note that MEEP gives a roughly three times larger value due to the 2D simulation which effectively resembles a sim- ulated system of infinite height). The obtained quality factor is however small compared to 1D crystals made of silicon with a smooth defect, where quality factors in the order of 104106can be achieved [53, 55, 57]. The corresponding mode shape in real space is shown in Fig. 4(a). We see that the mode is nicely localized at the defect. Furthermore the Excomponent is even (odd) as a function of x(y), whereas the Eyis even as a function of both xandyfulfilling the symmetry require- ments for a TE-like mode given in Eqs. (20). Due to thissymmetry, we can disregard the Ezcomponent here since Ez0. FortheFaradaycomponentoftheoptomagnonic coupling, the relevant quantity is the electric component of the optical spin density, Sopt/EE[see Eq. (14)]. Soptpoints mostly along z-direction, is odd as a function ofxandy, and is even along z(see Fig. 4(b) and 6(b)). IV. MAGNONIC CRYSTAL Asphotoniccrystalscontroltheflowoflight, magnonic crystals can be used to manipulate the spin wave dynam- ics in magnetic materials. In general a magnonic crystal is made of a magnetic material with a periodic distribu- tion of material parameters. Examples include the mod- ulation of the saturation magnetization or the magne- tocrystalline anisotropy, a periodic distribution of differ- ent materials, or the modulation by external parameters, such as an applied magnetic field [48–50]. Historically, magnonic crystals precede photonic crystals [115, 116]. Unlike in photonic or phononic crystals, the band struc- ture in magnonic crystals depends not only on the peri- odicity of the crystal but also on the spatial arrangement of the ground state magnetization, resulting in an ad- ditional degree of freedom. Hence the band structure depends on the applied external magnetic field, the rela- tive direction of the wave vector, the shape of the mag- net, and the magnetocrystalline anisotropy of the mate- rial [48–50]. In this section we study the properties of the crystalpresentedinSec.III(seeFig.3(a)), asamagnonic7 crystal. In the following we consider magnetic excitations which are non homogeneous in space, and we focus only on systems in the presence of an external magnetic field saturating the magnetization in a chosen direction. In this case spin waves can be divided into three classes: if all spins precess uniformly in phase, the mode is homo- geneous and denominated the Kittel mode. If the dis- persion is dominated by dipolar interactions (which is usually the case for wavelengths above 100 nm) the exci- tationsarecalleddipolarspinwaves. Forwavelengthsbe- low100 nmthe exchange interaction dominates instead, giving rise to exchange spin waves. The frequencies of the dipolar spin waves lie typically in the GHz-regime, whereas the exchange spin waves have frequencies in the THz-regime. Since the size of the structure considered in this work is in the micrometre range, we will focus on dipolarspinwaves. Forthiscase, themodescanbeclassi- fiedfurtherbytheirpropagationdirectionwithrespectto the magnetization. For an in-plane magnetic field, modes with a frequency higher than the frequency of the uni- form precession tend to localize at the surface and have a wave vector pointing perpendicular to the static magne- tization M0and thus the external field, k?M0kHext (seeFig.5(a)). ThesemodesarecalledsurfaceorDamon- Eshbach modes [117, 118]. If the wave vector is parallel to the external field such that kkM0kHextholds, the waves are called backward volume waves and their fre- quency is smaller than the frequency of the Kittel mode (see Fig. 5(a)). Finally, if the external field and the mag- netization are normal to the crystal’s plane and the wave vector lies in plane k?M0kHextthese waves are called forward volume waves (see Fig. 5(a)) [50, 119]. In the fol- lowing we restrict the discussion to external fields which are applied in the plane of the crystal. Similar to light modes in photonic crystals, magnon modes can also be localized within a certain region in the magnonic crystal. It is well known that the two di- mensional periodic modification of a continuous film, for example by the insertion of holes (denominated antidot arrays) can drastically change the behavior of the spin waves [120, 121]. In this case the modes have either a localized or extended character. The localized mode is a consequence of non-uniform demagnetization fields cre- ated by the antidots. These fields change abruptly at the edges of the antidots and act as potential wells for the spin waves [50]. Thus, the above designed crystal, which localizes the optical mode by the insertion of a defect, is also a good candidate for acting as a magnonic crys- tal localizing magnetic modes via the holes. Although the geometry of the crystal is optimized for the optics, it should be able to host and localize magnetic modes due to its shape and material (YIG). Therefore we do not change the crystal further and use this structure as a proof of principle. This implies that we expect con- siderable room of improvement with respect to the opto- (b)(a)BWVWHextkFWVWkHextHextkSWDipolar spin waves 𝞹-rotation symmetry (around x-axis)Mirror symmetry (x=0)Symmetries of the magnonic crystalFigure 5. Dipolar spin wave types and symmetries in the magnonic crystal: (a) Dipolar spin waves can be divided into three types: backward volume waves (BWVW) with their wave vector parallel to the external field which both lie in the plane of the structure ( kkm0kHext). Forward volume waves (FWVW) with their wave vector in plane and perpen- dicular to the external field which lies normal to the struc- ture’s plane ( k?m0kHext). Surface waves are also forward volume waves but they have their wave vector in plane and perpendicular to the external field which also lies in plane of the structure. ( k?m0kHext). (b) Symmetries of the inves- tigated 1D magnonic crystal shown in Fig. 1(a). Since the external magnetic field breaks two mirror symmetry planes only the mirror symmetry plane normal to the saturation di- rection remains. Additionally a -rotation symmetry around the saturation axis is present. magnonic coupling rates obtained in this structure. YIG is a good choice for magnonics since it has the lowest spin wave damping when compared to other materials commonly used [49]. It is however difficult to pattern at the microscale, but recent advances in fabrication show great promise in this respect [122, 123]. For concreteness, in the following we proceed to design the Faraday part of the optomagnonic coupling GF, see Eq. (14). Since GFis proportional to the overlap inte- gral between the optical spin density and the magnon mode, we search for a magnon mode with the same symmetries as the optical spin density, in order to get the highest possible overlap. Like in the optical case, the magnonic crystal has three mirror symmetry planes (z= 0; y= 0; x= 0). However, the external applied magnetic field saturating the magnetization breaks two of these symmetries and thus only the mirror symmetry w.r.t. the plane perpendicular to the external field re- mains (see Fig. 5(b)). Note that the magnetization is a pseudo vector and its components perpendicular to the mirror does not change. Thus, the mirror operation is inverted from Eq. (19), ^M xm(r) =0 @mx(x;y;z ) my(x;y;z ) mz(x;y;z )1 A=m(r):(21) Since the optical spin density pointing along ^zis odd as a function of x, we require mzto be odd as well and consequently mto be even under ^M x. Additionally, a8 -rotationaroundthe ^x-axissymmetryremainsunbroken ^R xm(r) =0 @mx(x;y;z) my(x;y;z) mz(x;y;z)1 A=m(r):(22) Invoking again the symmetries of the optical spin den- sity (odd as a function of yand even with z) we consider modes with even rotational symmetry. We note that due to the different symmetries respected by the photon and magnon modes, we choose the symmetries of the modes in such a way that they preferably match in the xy-plane, which is the most relevant dimension for thin structures. In this case, the symmetries of the optical and the mag- netic mode along the height do not necessarily match. For thin films however they do, see Fig. 6(b). Since spin waves are excited by an external magnetic pulse which controls the direction of the wave vector k, the pulse also breaks the mirror symmetries of the crys- tal. Therefore we focus on a setup which conserves the relevant mirror symmetry, and only excite backward vol- ume waves where the external saturation field and the wave vector of the mode are parallel and lie in the plane of the crystal. We note that this configuration is also the most favourable one from an experimental point of view, and additionally the configuration most likely used in magnonic devices [119]. We evaluated the magnetization dynamics numerically bymeansofthefinitedifferencetoolMuMax3[124]which solves for the Landau-Lifshitz-Gilbert equation of motion for the local magnetization vector (see appendix C). In order to excite magnon modes with the desired symme- try, we use a 2D antisymmetric sinc-pulse which should moreover avoid spurious effects in the spectrum [125] Hpulse =Hpulsesin2(!ct) !ctsin2(kcx) kcxsin2(kcy) kcyey;(23) pointing along the ^y-direction in order to excite back- ward volume waves [119]. The cut-off frequency was chosen to be !c= 216 GHzand the cut-off wave vector to be kc==ain order to concentrate all the excitation energy in the first BZ. Since this pulse is cen- tered in the middle of the crystal, we only excite modes around the crystal’s center. The external saturation field was set to H ext= 400 mT (found by hysteresis) and the pulse field to H pulse = 0:4 mT. We note that the pulse strength should be a small perturbation of the saturation field in order to minimize non linear effects. We used the material parameters for YIG, Ms= 140 kA=m(sat- uration magnetization), Aex= 2 pJ=m(exchange con- stant),Kc1=610 J=m3(anisotropy constant) with the anisotropy axis along ^z[126]. In order to accelerate the simulations, we used an increased Gilbert damping pa- rameter = 0:008(compare to 105104for YIG) [127, 128]. In the following considerations we focus only on the mzcomponent of the magnetization dynamics, since the yxzzxy -11 01 -11(b)(a) Different symmetries(E*xE)zMirror symmetryRe[δmz] Rotational symmetrySpatial shape of the couplingRe[G]Im[G]GFigure 6. Spatial shape of the coupling and different sym- metries of the optical and the magnetic mode: (a) Spatial shape of the coupling similar to the magnon mode shape. (b) Different symmetries along the crystal’s height of the opti- cal spin density and the magnon mode. Due to the external magnetic field the mirror symmetry along the height is broken andonlya-rotationsymmetryremains, resultingindifferent mode shapes along the height of the crystal. For thin films this difference is rather small. Note: all mode shape plots are normalized to their corre- sponding maximum value. optical spin density of a TE-like mode mostly points into the^z-direction, rendering mxandmyirrelevant for GF (see Eq. (14)). We find that the optical defect also acts as confinement of the magnetic mode, resulting in the defect like dispersion relation presented in Fig. 4(c). The obtained band structure shows modes around the edge of the BZ with extended wave vector character, imply- ing that the modes are highly localized in space. The frequency of the highest excited localized mode at the BZ edge is !mag= 213:12 GHzwith an estimated linewidth (FWHM) of mag= !mag= 2131:2 kHz whereweusedtherealGilbertdampingofYIG = 105. Note that the simulated linewidth shown in Fig. 4c is larger due to the different choice of the Gilbert damping in order to speed up the simulation. As we see from its mode shape, this mode is nicely localized at the holes attached to the defect and is odd with respect to x= 0andy= 0, and hence has the same symmetry as the optical spin density as we aimed for (see Fig. 4c). V. OPTOMAGNONIC CRYSTAL As shown above, the crystal in Fig. 1a can host both optical and magnetic modes and therefore can be consid- ered anoptomagnonic crystal . In this section, we eval- uate the optomagnonic coupling GFgiven in Eq. (14) (GCis briefly discussed at the end of the section) for the modes found in Secs. III and IV shown in Fig. 4. Numerically evaluating Eq. (14) gives a Faraday con- tribution to the optomagnonic coupling per magnon and per photon ofjGF numj= 20:5 kHz(spatial shape of the coupling see Fig. 6a). In order to gauge this value we want to compare it to the analytical estimate derived in [39]. In the optimal case, the magnetic mode volume and the optical mode volume coincide, VmagVopt. In9 this case, we estimate the coupling as jGF optimalj=Fn 2!optrgB Ms1p Vmag;(24) which evaluates to jGF optimalj= 20:6 MHzusing the material parameters of YIG ( (Fn)=(2) = 4
105; Ms= 140 kA=m) and the optical frequency found in Sec. III, !opt= 2249 THz. The magnetic mode volume is defined as the one where the magnon intensity is above a certain threshold, giving Vmag= 2:8
102µm3 (see appendix D). The coupling is bounded by the magnon mode volume, since in the investigated structure itissmallerthantheopticalmodevolume(seeFig.4). In order to take the mismatch in the mode volume into ac- count, we introduce the following overlap measure which is also known as filling factor O=Voverlap Vopt; (25) whereVoverlaprepresents the volume where the magnon and photon modes overlap. The volumes are estimated similar to the case of magnons to be Voverlap = 9:7
103µm3andVopt= 0:7µm3(see appendix D). Note that for the optical volume it was taken into account that the mode leaks out of YIG into the Si 3N4layer and air, which is not shown in Fig. 4 (a)+(b). Thus the over- lap measure evaluates to O= 0:01, shrinking the opti- mal coupling toO
jGF optimalj26 kHz. Hence, even though the optomagnonic crystal localizes both modes in the same region, the overlap measure is rather small due to the much larger optical mode volume (see Fig. 4 and Fig 6(a)), which is detrimental for the coupling strength. Furthermore by looking at the fine structure of the op- tical spin density and the magnon mode we see that the amplitude peaks of both do not coincide (see Fig. 7): the magnonic peaks are localized nearer to the center than the optical ones. This results in a smaller overlap volume which would be Vmagif the peaks of the modes would be at the same position. Sincethecouplingalsostronglydependsontherelative direction between the vectors of the modes, we addition- ally introduce a ‘directionality’ measure D=
drm(r)
[E(r)E(r)]
drjm(r)jjE(r)E(r)j(26) evaluating toD= 51%using the numerical results pre- sented above. As we see, although the symmetries of the optical spin density and the magnon mode match, the vectors of the modes do not perfectly align in the defect area (see Figs. 4). Taking also this sub-optimal alignment into account the coupling estimate reduces to jGF expectedj=O
D
jGF optimalj= 23 kHzwhichcoincides well with the numerically obtained value. We conclude that the coupling in the investigated structure is mostly Fine structure |δm|[E*×E]0L01Normalized strengthLengthFigure 7. Fine structure of the optical spin density and the magnon mode along the length of the crystal for a fixed height and width. affected by the large difference between the optical and the magnetic mode volumes, shrinking the coupling value by two orders of magnitude. We remind the reader that the obtained values are for a proof of principle struc- ture which has been only partially optimized, since we started from a fixed photonic crystal structure. In the next section we discuss a possible optimization from the magnonics side. We now proceed to briefly discuss the Cotton-Moutton effect for the results found in Secs. III and IV. For YIG, the Cotton-Moutton coefficient (Cn)=(2) =2 105[41] is of the same order of magnitude as the corre- sponding Faraday coefficient, determined by F. Since in the Voigt configuration both effects are of leading order in the magnetization fluctuations (see Eqs. (14) and (15)), it is important to take its contribution into account. Moreover, since the coefficients GFandGCare complex, it is difficult to estimate a priori the total cou- plingjGF+GCj, duetotheunknownpossibleinterference effects. Numerically evaluating Eq. (15) gives an interac- tion value ofjGC numj= 21:6 kHz. This large value can be explained by the symmetry of the integrand which re- duces tomx 0[ExE ymy+E xEymy]due to the backward volume wave setup and the TE-like character of the opti- calmode. Thisintegrandisfullyevensince ExE yhasthe same symmetry as my. The full optomagnonic coupling jGnumj=GF num+GC num (27) is found to be Gnum= 21:3 kHz. Compared to the optomechanical coupling in simi- lar 1D crystals, where coupling values (per photon and phonon) up to 2950 kHzcan be obtained [53–58], the optomagnoniccouplingobtainedhereisstillrathersmall. However, this is large compared to other optomagnonic systems. As we argued above, the coupling is limited10 Height dependence of the coupling|G|𝓓304050607080902.02.22.42.62.83.0 49.049.550.050.551.051.552.0|Optomagnonic coupling|/2𝛑 [kHz]Overlap [%] Height [nm] Figure 8. Height dependence of the Faraday component of the optomagnonic coupling: The coupling shows ap Vmag dependence since the optical mode volume in the YIG and the Si 3N4slab is constant. The decrease with larger height can be explained by the shrinking directionality measure (see Eq. 26) between the optical and the magnetic mode. by the imperfect spatial matching of magnons and pho- tons with overlap O= 0:01while it is enhanced due to small volumes, Vmag0:01µm3andVopt1µm3. In the standard setups involving spheres [29–31], typically optical volumes are very large 105µm3with low op- tomagnonic overlap 103, resulting in low couplings 1 Hz. It was theoretically shown that >75%over- lap in such systems is achievable [129] but the couplings would still be2500 Hz. The miniaturization of an optical cavity to 100µm3was demonstrated in [44], where the coupling is however still small, 250 Hz, in this case due to the large magnon volume involved. Animportantprerequisiteforapplicationsinthequan- tum regime such as magnon cooling, wavelength conver- sion, and coherent state transfer based on optomagnonics isahighcooperativity. Thecooperativityperphotonand magnon is an important figure of merit which compares the strength of the coupling to the lifetime of the coupled modes, and is given by C0=4G2 num optmag; (28) where optis the optical linewidth (FWHM), and mag is the magnonic linewidth (FWHM). To evaluate the theoretical cooperativity of the struc- ture proposed in this manuscript, we use mag= !mag where = 105is the Gilbert constant and !mag= 213:12 GHz. The optical linewidth is found from sim- ulations to be opt= 20:2 THz. Using the corresponding parameters the cooperativity per photon and magnon of the optomagnonic crystal is Ccrystal 02:5
1010. The single-particle cooperativity can be enhanced by the photon number in the cavity,C=nphC0. Experimentally, there is a bound on the photon density that can be supported by the cavity with- out undesired effects due to heating, and it is empirically given by 5
104photons per m3[130]. In our structure, considering the effective mode volume Voptthis gives an enhanced cooperativity at maximum photon density of Ccrystal1
105, whichistwoordersofmagnitudelarger than the current experimental state of the art [44, 130]. Since our model does not account for fabrication im- perfections, this number is expected to be lower in a physical implementation, indicating that optimization is needed. Results for similar 1D optomechanical crys- tals indicate that optimization can lead to larger co- operativity values (at maximum photon density), e.g. 10[54]. The small cooperativity obtained in our struc- ture is a combination of a reduced coupling due to mode- mismatch, plus the very modest quality factor of the op- tical mode in this simple geometry. For boosting the coupling strength we investigate briefly in the following the influence of the optomagnonic crystal’sheightonthecoupling, asproposedinRef.[111]. Therefore we increase the height of the YIG layer from 30 nmto90 nmwithout changing the other parameters of the geometry (including the Si 3N4layer in the optical simulations). As we see from the result (see Fig. 8) the coupling exhibits ap Vmagdependence. We find that the optical mode volume does not change substantially in the modified geometry, and therefore the observed behavior is consistent with the expectedp Vmag=Voptdependence for a constant optical mode volume. The slight decrease forlargerheightscanbeexplainedbytheshrinkingofthe directionality measure D, stemming from the difference in symmetries obeyed by the magnetization (rotational) and the electric field (mirror). VI. OPTIMIZATION So far, we optimized the crystal in order to minimize optical losses for the given geometry. In this section we investigate how to optimize the geometry for magnonics. The optical optimization was achieved by fixing the hole radius and intra-hole distance, which are both along the zxyd2raOptimized crystal Figure 9. Optimization of the geometry: Through increasing the parameters along the width of the crystal we create more spaceforthemodeswithouttouchingtheopticaloptimization of the original crystal (dashed line). We note that we also increased the defect size, not shown here.11 zxyzxy 01 -11 -11 (b)(a)|Ey||Ex||Hz|Mode shape Optical spin density ~ i(E* x E)zBand diagramSymmetriesRe[Ex]Re[Ey]Extended states 0𝞹/akxfno defectDiscrete states1. band2. band ~ 230 Hz~ 310 HzDefect state07 x 10-3kx [1/nm] -11 01 (c) Re[δmz]Im[δmz]Mode profileBand diagram|δmz| SymmetriesLocalized mode @ 13.17 GHz016f [GHz]FT[mz]normMode spectrumkx = 𝞹/aδmz kx [1/nm]f [GHz]016𝛅mzdefect07 x 10-3 0𝞹/akxzxy Figure 10. Optical (a and b) and magnetic modes (c) of the optimized crystal: (a) Band diagram for TE-like modes within the irreducible BZ with a defect mode in the photonic band gap which was pulled from the upper band-edge state into the gap by the insertion of a defect. From the mode shape of the localized mode with a frequency of !opt=2= 279 THz (middle layer in thexy-plane) we see that this mode is odd with respect to x= 0andy= 0(and even with respect to (z=0)). (b) Optical spin density of the localized mode (middle layer in the xy-plane) which is odd with respect to x= 0andy= 0(and even with respect to z= 0). (c) Band diagram of backward volume waves within the irreducible BZ showing magnetic modes with extended k-values but preferring wave vectors at the edge of the BZ. The highest excited localized mode has a frequency of !mag= 213:17 GHzand is odd along the mirror symmetry planes for x= 0andy= 0(and additionally even with respect to the plane for z= 0). The dashed line in the middle inset shows the mode spectrum in case of no defect. Note: all mode shape plots are normalized to their corresponding maximum value. length of the crystal. In the following we tune instead only the parameters along the width of the crystal ( ^y- direction), in order to perturb as little as possible the optical optimization. We found a promising structure by increasing the width of the crystal and considering ellip- tical holes, see Fig. 9. From a set of trials, we found that a width of w= 900 nm and a radius of the holes along the width of rw= 380 nm give the highest coupling. An increased defect size of d= 1201:5 nmis also beneficial for decreasing the optical losses in this case, it nicely lo- calizes the optical defect mode in the middle of the band gap and thus does not drastically change the localization behavior of the photonic crystal. For evaluating the photonic band structure and the optical modes we use the same procedure as described in Sec. III. We obtain a similar band structure for TE- like modes and also a similar localized mode with a frequency of !opt= 2279 THz (obtained by Com- sol,235 THz according to MEEP) and a damping of opt= 23 THzwhich gives an optical linewidth (FWHM) of opt= 26 THz(see Fig. 10(a)). Us- ing the values obtained by Comsol this results in a re- duced optical quality factor of Q= 93(note that MEEP gives a twice as large value). This rather low optical quality factor is a trade off for the magnetic optimiza- tion achieved by elliptical holes. Moreover, the optical spin density compared to the original crystal is mostly localized within the defect which is advantageous for our purposes (see Fig. 10(b)). Similarly, for evaluating the magnon modes we used the parameters and procedurespresented in Sec. IV. In the following we focus on the Faraday part of the optomagnonic coupling and there- fore consider only the mzcomponent of the magnon modeduetothestructureoftheopticalspindensity. The Cotton-Mouton term is discussed briefly at the end of the section. The simulated band diagram for backward vol- ume waves again shows extended magnon modes but in thiscaseweobtainonebroadband, mostlikelystemming from a fusion of several bands due to the larger width of the crystal (see Fig. 10(c)). The frequency of the highest excited localized mode is !mag= 213:17 GHzwith an estimated linewidth of !mag= 2131:7 kHzwhere we used the Gilbert damping of YIG = 105. As in the previouscase, thesimulatedlinewidthislargerduetothe larger Gilbert damping used in the simulations. As we see from its mode shape, this mode is nicely localized at the holes attached to the defect and has approximately the same shape and symmetry as the optical spin density (see Fig. 10(c)). Using the results discussed above, the Faraday com- ponent of the optomagnonic coupling of Eq. 14 for the optimized crystal evaluates to jGF numj= 22:9 kHz. Therefore the optimized coupling is one order of mag- nitude larger than in the crystal discussed in Sec. V. As before we want to gauge this value by comparing it to the analyticalestimategiveninEq.24. Theoptimalcoupling in the optimized crystal is jGF optimalj= 20:5 MHz. Again the magnetic mode volume bounds the coupling due to the smaller size of the magnetic mode compared to the optical mode which also extends to the Si 3N4lay-12 ers. This results in a overlap measure (see Eq. 25) of O= 0:04. Therefore the mode overlap is increased by 25%compared to the un-optimized crystal. Evaluating the directionality measure given in Eq. 26 gives D= 53% whichisjustslightlylargerthanintheun-optimizedcase. Taking both measures into account the analytical cou- pling estimate shrinks to jGF expectedj=O
D
jGF optimalj 210 kHzwhich lies slightly above the numerically ob- tained value. Although the fine structure peaks of the optical spin density and the magnon mode still do not coincide (see Fig. 11), the coupling values are improved by “pulling" the optical and magnetic modes completely into the defect area by the insertion of elliptical holes, creating an overlap area with high density of both modes. The Cotton-Moutton effect in this structure evaluates tojGC numj= 21 kHzand results in a total coupling of jGnumj=jGF num+GC numj= 22 kHz. We can conclude that in this case both effects interfere constructively for the total coupling. The cooperativity per photon and magnon in this case isCop 02
1011, which can be enhanced to Cop0:5
106by the number of photons trapped in the cavity. Thus the cooperativity at maximum photon density is slightly lower as in the crystal presented above, a consequence of the reduced quality factor of the optical mode. VII. CONCLUSION We proposed an optomagnonic crystal consisting of a one-dimensional array with an abrupt defect. We showed that this structure acts as a Bragg mirror both for pho- ton and magnon modes, leading to co-localization of the modes at the defect. By proper design and taking into account the required symmetries of the modes in order to optimize the coupling, we showed that coupling values in the kHz range are possible in these structures. This value is orders of magnitude larger than the experimental state of the art in the field, but still rather small compared to the theoretically predicted optimal value for micron sized structures, which is in the range of 101MHz[34]. We showed that the strength of the coupling in our proposed structure is still limited largely by the sub- optimal mode overlap, <5%. Further optimization in design is moreover needed in order to boost the coop- erativity value, which is limited mainly by the optical losses. The simultaneous optimization is challenging, due to the complexity of the demagnetization fields in patterned geometries. Whereas it is well known that a tapered defect (that is, a smooth defect) can highly in- crease the optical quality factors, its effect on the mag- netic modes is non-trivial and is disadvantageous for localizing the magnon modes of the kind used in this work. Other magnon modes, however, could be explored in this case. More complex geometries, including one- Fine structure |δm|[E*×E] 0L01Normalized strengthLengthFigure 11. Fine structure of the optical spin density and the magnon mode along the length of the optimized crystal for a fixed height and width. dimensional crystals combining tapering and an abrupt defect, or two-dimensional crystals, are good candidates to be explored in order to improve quality factors and coupling. The first results shown in this work point to the promise of designing the collective excitations in op- tomagnonic systems via geometry, in order to boost the coupling strength and minimize losses, paving the way for applications in the quantum regime. VIII. ACKNOWLEDGEMENTS We thank Clinton Potts and Tahereh Parvini for in- sightful discussions. J.G. acknowledges financial support from the International Max Planck Research School - Physics of Light (IMPRS-PL). H.H. acknowledges fund- ing from the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) under Germany’s Excel- lence Strategy EXC-2111-390814868. S.S. and S.V.K. acknowledge funding from the Max Planck Society through an Independent Max Planck Research Group. S.V.K also acknowledges funding from the Deutsche Forschungsgemeinschaft (DFG, German Research Foun- dation) through Project-ID 429529648 – TRR 306 QuCoLiMa (“Quantum Cooperativity of Light and Mat- ter"). APPENDIX A: NORMALIZATION OF MAGNON MODES In this section, we discuss the normalization of magnonsoverageneralmagnetizationtexture. Themag- netization satisfies the Landau-Lifshitz (LL) equation dM dt=gB0 ~M(Hex+He[M]);(29)13 where Hexis an external field and Heis a linear func- tional which can be interpreted as the effective field gen- erated by spin-spin interactions such as exchange, dipo- lar, etc. Let the static solution, i.e. putting dM=dt= 0, beMsm0(r)with saturation magnetization Msand unit vector m0
m0= 1. This magnetization generates an effective field of the form He[Msm0(r)] =H0(r)m0(r)Hex(r);(30) where the function H0(r)depends on the nature of spin- spin interactions. The magnon modes m (r)ei! tare found by the linearized LL equation i! m =gB0 ~[m0h +H0m m0];(31) whereh =He[m ]. The LL equation can be derived from the Hamiltonian H=0 dVM
He[M] 2+Hex
M :(32) Up to quadratic terms in m, we expand the magnetiza- tion MMs 1m
m 2 m0+Msm;(33) and the effective field He[M] 1m
m 2 (H0m0Hex) +h;(34) where A=X  A  +A   ; (35) withAbeing morhand being magnon amplitudes, i.e. classical counterpart of ^b defined in Eq. (10) in the main text. The above form ensures M
M=M2 sup to second order in m. The Hamiltonian becomes (ignoring a constant term) H=0Ms 2 dV H0m
m+m
h +m
Hex+m0
h :(36) The last two terms are linear in m and thus should be zero for m to correspond to a magnon mode. We can simplify the second term by finding the component ofh perpendicular to m0using Eq. (31), h m0(m0
h ) =H0m i~! gB0m0m :(37) Usingthisandignoringthelinearterms, theHamiltonian simplifies to H=0Ms 2X  i~! gB0 dVm0
h mm  
 m  +m 
i : (38)As the eigenmodes should diagonalize the Hamiltonian toP~! j j2, we should have  dVm0
(m m) = 0; (39) and iMs dVm0
mm
=gB :(40) For= , this gives the normalization for magnons. For circularlypolarizedmagnonswith m=m(y+iz)=p 2 andm0=x, the normalization becomes  dVm2=gB Ms: (41) APPENDIX B: NUMERICAL SETTINGS - OPTICAL SIMULATIONS In the following we shortly discuss how the optical band structure and the optical modes can be evaluated numerically. In our work we used two different computa- tional methods: for calculations done in the time domain we use the electromagnetic simulation tool MEEP [110], whereas for calculations done in the frequency domain we use the finite element solver Comsol [113]. We use two simulation tools since with MEEP it is much easier to ob- tain the band structure of the crystal, and with Comsol the exact mode shape. 1. MEEP MEEP in general solves for Maxwell’s equations in the time domain within some finite composite volume. Therefore it essentially performs a kind of numerical ex- periment [110]. We use MEEP for simulating the band structure of a YIG crystal without defect in order to find its band gap and its corresponding mid-gap frequency. Furthermore we use a transmission spectrum simulation to optimize the defect size in order to get the least lossy localized mode. For finding the exact frequency of the localized mode we also simulate its spatial shape in the time domain. For simplicity we simulate the YIG crystal in a 2D model (see [47, 112]). Its material parameters are set by the relative permittivity of "= 5. In order to account for the leakage of the electromagnetic field the investigated crystal is surronded by a finite size air region large enough so that the leaking electric field de- cays before it reaches the boundaries, in order to avoid spurious reflection effects. This is achieved by choosing the distance between the surface of the crystal and the boundary of the air region as dair= 3
0where0is the vacuum wavelength ( dair= 33:6µmin our case). Fur- thermore, we need boundary conditions along the out- side of the air region that are transparent to the leaking14 field such that the truncated air region represents a rea- sonable approximation of free space. Therefore we use a perfectly matched layer at the boundaries of the air region which absorbs all outgoing waves. The thickness of this layer should be at least a vacuum wave length [131]. The whole geometry is meshed by one single reso- lution parameter which discretizes the structure in time and space and gives the number of pixels per distance unit. For all band simulations we used a resolution of 40 pixels, whereas in case of the transmission spectrum we used a resolution of 20pixels and a resolution of 50pixels in case of the mode shapes [112]. Band structure simulations For obtaining the band structure we use a YIG crystal without defect ( d=a) and therefore we can simulate only one single unit cell with a side length of acontain- ing one air hole, and apply an infinite repetition of this cell at each side in ^x-direction. Since we expect the mid- gap frequency of the crystal with defect to be around 240 THz, we excite the crystal with a gaussian pulse with a center-frequency of 225 THz and a width of 450 THz to cover all modes around the band gap. We center the pulse peak at an arbitrary postion (x= 0:00123;y= 0) in order to couple the pulse to an arbitrary mode. Since we want to simulate only TE-like modes, in order to save computational time the pulse only has a Hzcomponent. For decreasing the computation time even more, we ap- ply an odd mirror symmetry plane for y= 0. The mirror symmetry for x= 0is broken by a boundary condition for0<kx<[112]. Transmission spectrum simulations For optimizing the defect size we simulate a transmission spectrum for frequencies at the band gap by measuring the flux at the end of the waveguide stemming from a source at the other end. The measured flux then is nor- malized to the flux of a waveguide without holes. We therefore simulate the transmission spectrum as a func- tion of different defect sizes and use the defect size which gives the highest transmission. In order to consider only TE-like modes where the electric field lies in plane we need to excite the system with a Jy-current source trans- verse to the propagation direction which is achieved by a gaussian pulse with only a Ey-component. Its cen- ter frequency thereby is 222 THz (simulated mid-gap fre- quency) and its width is 90 THz(>band width). Also in this case we apply an odd mirror symmetry for y= 0for decreasing the simulation time. We note that the mirror symmetry for x= 0is broken by the source since it is located at the edge of the waveguide [112]. Mode shape simulations For evaluating the mode frequency of the localized mode within the band gap we simulate the time evolution of this single mode by exciting it by a gaussian pulse with a center frequency of 203 THz (frequency of the peak in thetransmission spectrum) and a width of 15 THz. Since in this simulation no symmetry is broken we also apply an odd mirror symmetry for x= 0andy= 0for obtaining only a TE-like mode [112]. 2. Comsol We use Comsol to find the spatial mode shape. There- fore we use the “Electromagnetic waves, Frequency do- main" package of COMSOL’s “RF module" which solves for the Helmholtz equation of the form r1 r(rE)k2 0 "ri !" E= 0;(42) wherek0indicates the vacuum wave number, !the an- gular frequency, rthe relative permeability and "0the vacuum permittivity. Contrary to the MEEP simula- tions above, we simulate the full geometry composite of a YIG layer sandwiched by two Si 3N4layers. The used material parameters thereby are "YIG= 5,"Si= 4, YIG=Si= 0, andYIG=Si= 0withthe relative permeabilty and the conductivity. Again we also need to simulate an truncated air region around the crystal which is able to absorb the outgoing radiation. The cor- responding material parameters are "air=air= 1and air= 0. Besides perfectly matched layers we also can use second order scattering boundary conditions at the air surfaces given by the expression [131] n
rEz+ik0Ezi 2k0r2 tEz= 0 (43) withnthe normal vector to the considered plane. For large enough air regions both approaches are almost equivalent as long as the leaking field is propagating normal to the air surfaces. In order to account for a large enough air region we choose the distance between the surfaces of the crystal and the air boundaries as 4:5µm. For reducing the simulation time we use the symmetry requirements of a TE-like mode. Therefore, we cut the geometry into an eighth of the whole struc- ture and apply perfect electric conductor boundary con- ditions ( nE= 0) at the cut surfaces along x= 0and y= 0and a perfect magnetic conductor boundary condi- tion (nH= 0) at the cut surface along z= 0. The full solution is then obtained by using the symmetry require- ments of a TE-like mode. The whole geometry is meshed by a physics-controlled tetrahedral mesh with a maxi- mum element size of 0=50:3µm[132]. We note that in case of a physics-controlled mesh Comsol automati- cally meshes the material areas of interest with a finer mesh and uses a coarser mesh e.g. for the air regions.15 APPENDIX C: NUMERICAL SETTINGS - MAGNETIC SIMULATIONS In this section we briefly discuss how the magnetic band structure and magnetic mode shape is obtained nu- merically. For evaluating the magnetization dynamics we use the finite difference tool MuMax3 [124] which solves for the Landau-Lifshitz-Gilbert equation of the form @m @t=g1 1 +2[mBeff+(m(mBeff))](44) withm=M=Msthe local reduced magnetization of one simulation cell, gthe gyromagnetic ratio, the damp- ing parameter, and Beffan effective field which contri- butions can be found in [124]. As material parameters we used the parameters for YIG, Ms= 140 kA=m(sat- uration magnetization), Aex= 2 pJ=m(exchange con- stant),Kc1=610 J=m3(anisotropy constant) with the anisotropy axis along ^z. In order to accelerate the simu- lations, weusedanincreasedGilbertDampingparameter = 0:008(compare to 105for YIG) [126]. The used meshgrid had (1024;50;5)cells in the (^x;^y;^z)- direction what guarantees to take the exchange interac- tion into account ( lex13 nmfor YIG). In general, in all our simulations the spin wave dy- namics is excited via an external pulse field and the time evolution is recorded for all three magnetization com- ponents. For post processing the output of the form mi(x;y;z )withi= (x;y;z )saved for all simulated time steps separately we create for each magnetization com- ponenti= (x;y;z )a 4D-array of the form mi(t;x;y;z ). Band structure simulations In order to obtain the band structure along a spe- cific direction jwithj= (x;y;z )e.g. chosen to be the^x-direction, we reduce the four dimensional array to a two dimensional array of the form mi(t;x) =Pny mPnz nmi(t;x;ym;zn)and perform a 2D Fourier transform on this array mi(f;kx) =FT2D[mi(t;x)]re- sulting in the band diagram along the chosen direction. For increasing the resolution in the band diagram we plot the quantityp jmi(t;x)j=max (jmi(t;x)j). Mode shape simulations In order to obtain the mode shape we perform a space- dependent Fourier transform in time on each array entry separatelymi(f;x;y;z ) =FT1D[mi(t;x;y;z )]. APPENDIX D: EVALUATION OF THE MODE VOLUMES For evaluating the mode volume numerically we first, due to numerical errors, need to identify all cells of the simulated array (either containing m(r)in case of the magnetic mode volume or E(r)in case of the opticalmode volume) which contribute to the volume by a high enough mode density. This means we need to define a threshold which determines if a cell should contribute to the mode volume or not. We define this threshold by T=max(jxj)min(jxj) 5; (45) where xis the content of the cell (either x=morx= E). For being able to count the cells which contribute to the volume we create an additional array matching the simulated arrays in size. This array then contains ones if the absolute value of the cell ( jxj) in the original array is larger than the threshold given in Eq. (45) and zeros ifjxjis smaller. The mode volume is then obtained by summing over this array (giving the number of cells contributing to the volume) and by multiplying this such calculated number by the volume of one cell sx
sy
sz. For evaluating the overlap volume between the mag- netic mode ( x=m) and in this case the optical spin density ( EE) we create the same additional arrays as above identifying the cells which contribute to their cor- responding mode volume. For identifying the cells where the magnon mode and the optical spin density overlap we create a third array again matching the size of the original array. But this array now contains ones if the corresponding cells of the “threshold arrays" both in the magnetic and optical case contain a one, otherwise we set the cell value to zero. The overlap volume is then obtained by summing over this third array (giving the number of cells contributing to the overlap) and by mul- tiplying this such calculated number by the volume of one cellsx
sy
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2020-12-01

We put forward the concept of an optomagnonic crystal: a periodically patterned structure at the microscale based on a magnetic dielectric, which can co-localize magnon and photon modes. The co-localization in small volumes can result in large values of the photon-magnon coupling at the single quanta level, which opens perspectives for quantum information processing and quantum conversion schemes with these systems. We study theoretically a simple geometry consisting of a one-dimensional array of holes with an abrupt defect, considering the ferrimagnet Yttrium Iron Garnet (YIG) as the basis material. We show that both magnon and photon modes can be localized at the defect, and use symmetry arguments to select an optimal pair of modes in order to maximize the coupling. We show that an optomagnonic coupling in the kHz range is achievable in this geometry, and discuss possible optimization routes in order to improve both coupling strengths and optical losses.

Design of an optomagnonic crystal: towards optimal magnon-photon mode matching at the microscale

2012.00760v2

1 Competing effects at Pt/YIG interfaces: spin Hall magnetoresistance, magnon excitations and magnetic frustration Saül Vélez1,*, Amilcar Bedoya -Pinto1,‡, Wenjing Yan1, Luis E. Hueso1,2, and Fèlix Casanova1,2,† 1CIC nanoGUNE, 20018 Donostia -San Sebastian, Basque Country, Spain 2IKERBASQUE, Basque Foundation for Science, 48011 Bilbao, Basque Country, Spain ‡ Present address: Max Planck Institute of Microstructure Physics, D -06120 Halle, Germany * s.velez@nanogune.eu † f.casanova@nanogune.eu We study the spin Hall magnetoresistance (SMR) and the magnon spin transport (MST) in Pt/ Y3Fe5O12(YIG) -based devices with intentionally modified interfaces. Our measurements show that the surface treatme nt of the YIG film results in a slight enhancement of the spin -mixing conductance and an extraordinary increase in the efficiency of the spin -to-magnon excitations at room temperature . The surface of the YIG film develop s a surface magnetic frustration at low temperatures, causing a sign chang e of the SMR and a dramatic suppression of the MST. Our results evidence that SMR and MST could be used to explore magnetic properties of surfaces, including those with complex magnetic textures , and stress the critical importance of the non -magnetic/ferro magnetic interface properties in the performance of the resulting spintronic devices. I. Introduction Insulating spintronics [1] has emerged as a promising, nove l technological platform based o n the integration of ferroma gnetic insulators (FMIs) in devices as a media to generate, process and transport spin information over long distances [1–30]. The advantage of using FMIs against metall ic ones is that the flow of c harge currents is avoided, thus preventing ohmic losses or the emergence of undesired spurious effects. Some phenomena explored in insulating spintronics include the spin pumping [2–5], the spin Hall magnetoresistance (SMR) [5–15], the spin Seebeck effect [5,16 –18], the spin Peltier effect [19], the magnetic gating of pure spin currents [20,21] or the magnon spin transport (MST) [2,22 –30]. The fundamental building block structure employed to explore these phenomena is formed by a FMI layer –typically Y 3Fe5O12 (YIG) due to its small damping, soft ferrimagnetism and negligible magnetic anisotropy – and a non -magnetic (NM) metal with strong spin -orbit coupling (SOC) such as Pt or Ta placed next to it, which is essentially used to either generate or detect spin currents via the spin Hall effect (SHE) or its inverse [ 31–35]. Since these spintronic phenomena are based on the transfer of spin currents a cross the NM/FMI interface, it plays a key role in the properties and the performance of the resulting devices. It is well established that the most relevant parameter that determines the spin -current transport across the interface is the spin -mixing conductance 𝐺↑↓=(𝐺𝑟+ 𝑖𝐺𝑖) [5,36,37] . However, it is still under debate whether other interface effects could 2 also be relevant in these hybrid systems. Some examples are the magnetic proximity effect (MPE) [38–43], the Rashba -Edelstein effect [44–47], the anomalous Nerns t effect [38,48,49] or the spin -dependent interfacial scattering [50]. Therefore, understanding the role of the NM/FMI interface and the impact of its proper ties on the resulting spintronic phenomena is of outmost importance. In this work, we show that different spin -dependent phenomena in Pt/YIG -based devices (SMR and MST) are dramatically altered when the YIG surface is treated with a soft Ar+-ion milling. At room temperature, while the SMR effect in the treated samples is slightly larger than in the non -treated ones , the MST signal is fourfold increased. This extraordinary increase in the MST amplitude indicates that the spin-to-magnon conversion in Pt /YIG interfaces is strongly dependent on the magnetic details of the atomic layer of the YIG beyond the change in 𝐺↑↓. In addition, at low temperature, we observe a sign change of the SMR and a strong suppression of the MST signal in the treated samples, indicating the emergence of a surface magnetic frus tration of the treated YIG at low temperature. Our experimental res ults point out SMR and MST to be powerful tools to explore magnetic properties of surfaces and show that care should be taken when treating the surface of YIG, especially when used for studying spin - dependent phenomena originati ng at interfaces. II. Exper imental details Two different types of device structures were studied. In the first design, Pt/YIG samples were prepared by patterning a Pt Hall bar (width W=100 m, length L=800 m and thickness dN=7 nm ) on top of a 3.5 -m-thick YIG film [51] via e-beam lithography, sputtering deposition of Pt and lift -off, as fabricated in Ref. 52. In some samples, the YIG top surface was treated with a gentle Ar+-ion milling [53] prior the Pt deposition (Pt/YIG+ samples). In the second design, non-local NL -Pt/YIG and NL-Pt/YIG+ lateral nano structures were prepared on top of a 2.2 -m-thick YIG film [51] by patterning two long Pt strip lines ( W=300 nm, L1=15.0 m, L2=12.0 m and dN=5 nm) separated by a gap of ~500 nm –similar to the device structure used in Refs. 25 and 29–, following the same fabrication procedure used for the Hall bar . For each device structure , the Pt for both treated and non -treated samples was deposited in the same run . Here, for the sake of clarity, we present data taken for one sample of each type (Pt/YIG, Pt/YIG+, NL- Pt/YIG and NL-Pt/YIG+), although more samples were fabricated and measured, all showing reproducible results . Magnetotransport measurements wer e performed using a Keithley 6221 sourcemeter and a Keithley 2182 A nanovoltmeter operating in the dc -reversal method [54–56]. These measureme nts were performed at different temperatures between 10 and 300 K in a liquid -He cryostat that allows applying magnetic fields H of up to 9 T and to rotate the sample by 360º degrees . No difference in the magnetic properties between YIG and YIG+ substrates were observed via VSM magnetometry measurements (not shown). III. Results and Discussion IIIa. Spin Hall magnetoresistance First, we explore the angular -dependent magnetoresistance (ADMR) in Pt/YIG and Pt/YIG+ at room temperature. Figures 1( a)-1(c) show the longitudinal ( RL) ADMR 3 curves obtained for both samples in the three relevant H-rotation planes. The transverse (RT) ADMR curves taken in the plane are plotted in Fig. 1(d). The measurement configuration, the definition of the axes, and the rotation angles ( ) are defined in the sketches next to each panel. Note that for the magnetic fields applied, the magnetization of the YIG film is saturated [see Ref. 52 for the characterization of the YIG films]. The angular dependences are the same in both milled and non -milled samples and show the expected behaviour for the SMR effect, in agreement with measurements reported earlier in Pt/YIG bilayers [5–7,11,52] . FIG. 1 (color online). (a) -(c) Longitudinal ADMR measurements performed in Pt/YIG (dashed lines) and Pt/YIG+ (solid lines) samples at 300 K in the three relevant H-rotation planes (). (d) Transverse ADMR measurements taken in the same samples and temperature in the plane. Sketches on the right side indicate the definition of the angles, the axes, and the measurement configuration. The applied magnetic field is denoted in each panel . RL0 and RT0 are the subtracted base resistances. The SMR arises from the interaction of the spin currents generated in the NM layer due to the SHE with the magnetic moments of the FMI. According to the SMR theory [8,52] , the longitudinal and transverse resistivities of the Pt layer are given by 𝜌𝐿=𝜌0+∆𝜌0+∆𝜌1 (1−𝑚𝑦2), 𝜌𝑇=∆𝜌1𝑚𝑥𝑚𝑦+∆𝜌2𝑚𝑧, (1) where 𝐦(𝑚x,𝑚y,𝑚z)=𝐌/𝑀s are the normalized projections of the magnetization of the YIG film to the three main axes, 𝑀s is the saturated magnetization of the YIG and 𝜌0 is the Drude resistivity. ∆𝜌0 accounts for a number of corrections due to the SHE [52,57,58] , ∆𝜌1 is the main SMR term, and ∆𝜌2 accounts for an anomalous Hall - like contribution. Considering that these magnetoresistance (MR) corrections are very small, we identify the base resistivity of our longitudinal ADMR measurements as 𝜌𝐿0(𝑚𝑦=1)=𝜌0+∆𝜌0≃𝜌0. Since H is rotated in the plane of the film in our transverse measurements, the ∆𝜌2 contribution does not appear. Note that , in ADMR measurements, the amplitude of 𝜌𝐿(𝛽), 𝜌𝐿(𝛼) and 𝜌𝑇(𝛼) are equal and given by ∆𝜌1. Therefore, these measurements are equivalent when only the SMR contributes to the MR. The SMR term is quantified by 4 Δ𝜌1 𝜌0=𝜃𝑆𝐻2 λ 𝑑𝑁Re 2λ𝐺↑↓𝜌0tanh2(𝑑𝑁 2λ⁄ ) 1+2λ𝐺↑↓𝜌0coth (𝑑𝑁 λ⁄) , (2) where isthe spin diffusion length and SH the spin Hall angle of the Pt layer. According to Eq s. (1) and (2), the difference in the SMR amplitude observed between the two samples (see Fig. 1) can be interpreted as an enhanced 𝐺↑↓ at the Pt/YIG+ interface –with respect to Pt/YIG – due to the Ar+-ion milling process . Note that the spin transport properties for both Pt layers are expected to be the same because the measured resistivity is the same [59–61]. As the spin relaxation is governed by the Elliott -Yafet mechanism in Pt [59–61], we can calculate its spin diffusion length using the relation ×10-15Ωm2)/ρ [61]. Following Ref. 61, the spin Hall angle in the moderately dirty regime can be calculated using the intrinsic spin Hall conductivity 𝜎𝑆𝐻𝑖𝑛𝑡 (𝜃𝑆𝐻= 𝜎𝑆𝐻𝑖𝑛𝑡𝜌), which for Pt is 1600 Ω-1cm-1 [61,62] . In our films, L0 ~ 63 cm at 300 K, which thus corresponds to ~1.0 nm and SH~0.097. Using these andSH value s, dN=7 nm, Δ𝜌𝐿/𝜌0~5.310-5 and ~7.0610-5 (for Pt/YIG and Pt/YIG+, respectively, at 300 K) , and that Gi<<Gr [63], Eq. (2) yields Gr ~3.31013 -1m-2 and ~4.41013 -1m-2 for the Pt/YIG and Pt/YIG+ samples , respectively, which is within the range of values reported using the same bilayer structure [2,5–7,9–11,52,64,65] . This increase in Gr is in agreement with previous studies, where it was shown that an Ar+-ion milling process can improve the NM/YIG int erface quality by removing residues that might remain over the YIG substrate before the deposition of the NM layer [65,66] . However, it has been observed that an Ar+-ion milling process might also affect the YIG structure [49,64] . In the following, we proceed to study the temperature dependence of the SMR effect in these samples. Figures 2(a) and 2(b) show the measured temperature dependence of RT() for Pt/YIG and Pt/YIG+, respectively, in the angular range 0 -180º and for H=0.1 T. In both samples, the angular dependence predicted by the SMR effect is preserved when decreasing the temperature, following a sin( )cos() dependence [see Eq. (1)]. However, the polarity of the ADMR amplitude reverses the sign for Pt/YIG+ at low temperatures (crossing zero around T~45 K ), which is a completely unexpected behavior. According to the SMR theory , this amplitude is given by the term Δ𝜌1/𝜌0 in Eq. (2), which is a positive magnitude by definition. In Fig. 2(c), we plot the temperature dependence of the normalized amplitude of the transverse ADMR T/0T/L0=[[RT(45º)-RT(135º)]/RL0]·[L/W] for Pt/YIG (black squares) and Pt/YIG+ (red circles) . The weak temperature dependence of the SMR effect observed in our Pt/YIG sample is very similar to the one reported by other s using the same bilayer structure and it can be well understood with the temperature evolution of the spin transpor t properties in Pt [13,14,59,61] . In contrast, the different temperature dependen ce observed in Pt/YIG+ [see red dashed line in Fig. 2(c), which shows a scaling of the MR measured in Pt/YIG], having a sharp drop below 140 K and even a sign change at low temperatures, suggests the emergence of an additiona l interface effect. Systematic ADMR measurements are required to address its origin. Figure 2(d) show s the temperature dependence of the normalized amplitude of the longitudinal ADMR L/0L/L0=[RL(0º)-RL(90º)]/ RL0 measured in Pt/YIG+ for the three relevant H-rotation planes at H=1 T. We can see that both L()/0 and 5 L()/0 follow the same trend and that L()/0 remains zero, except for T~10 K. At very low temperatures, weak anti -localization effects emerge in Pt thin films [52,67 – 69], resulting in an extra out -of-plane vs in-plane MR, giving an explanation for the very small signal detected at 10 K. These measurements show that the sudden drop and the change in sign of the MR observed in Pt/YIG+ when decreasing temperature preserve the symmetry given by the polarization ( s) of the spin current produced in the Pt layer via the SHE , i.e., the measured MR has the symmetry of the SMR effect, which is distinct to the anisotropic MR that would appear if MPE were present. Therefore, this excludes MPE to be at the origin of the sign change of the MR at low temperatures in Pt/YIG+. FIG. 2 (color online). (a), (b) Transverse ADMR curves measured in Pt/YIG and Pt/YIG+, respectively, at different temperatures for H=0.1 T in the plane (see sketch). Data in the 180º - 360º range reproduce the same curves. RT0 is the subtracted base resistance at the corresponding temperature. (c) Temperature dependence of the normalized amplitude of the transverse ADMR, T/0, for the Pt/YIG (black squares) and Pt/YIG+ (red circles) samples extracted from (a) and (b), respectively. The red dashed line in (c) is a scaling of the temperature dependence of the amplitude measured in Pt/YIG to overlap with the amplitude obtained in Pt/YIG+ in the high temperature range (from ~150 to 300 K). (d) Temperature dependence of the normalized amplitude of the longitudinal ADMR, L/0, obtained in Pt/YI G+ at H=1 T and for the three H- rotation planes ( ). (e), (f) Transverse magnetic -field-dependent MR curves measured in Pt/YIG and Pt/YIG+, respectively, at 10 K with H in the plane of the film and for =45º and =135 º [see sketch in (e) for the color code of the magnetic field direction]. The vertical dashed lines show the saturation field of the YIG film obtai ned via magnetometry measurements . It is important to point out that , in hybrid systems of this kind , the interaction of s with the magneti zation M of the FMI leads to a resistance modulation not only due to the SMR , but also due to the excitation of magnons [25,29] . While the amplitude of the SMR is maximum when s and M are perpendicular, the resistance modulation due to magnon excitation is maximized when s and M are collinear. This implies that the MR modulation obtained in NM/FMI hybrids via ADMR measurements must actually be the result of the competition of these two spin-dependent MR effects, having the same angular dependences, but with reversed polarity. However, the MR expected from magnon excitation s is much smaller than from the SMR for the range of temperatures explored here. It has been estimated to be ~16 % at room temperature with respect to the SMR [19,21,25] , and that it should vanish at zero temperature [29]. Therefore, this rules out the excitation of magnons as responsible for the unexpected MR measured in H Jc H x y z VT H 0 50100150200250300-404812 H = 0.1 TPt/YIG Pt/YIG+T /(10-5) T(K)-3.0-1.50.01.53.0 0 45 90 135 180-4.0-2.00.02.04.0 0 50100150200250300-404812 60 K 30 K 10 K RT-RT0 (m) 300 K 200 K 100 K Pt/YIG+ 40K 30K 10K 80K 70K 60K 50K 300K 200K 130K 100K c)Pt/YIG Pt/YIG+ Pt/YIG/(10-5) T(K)0 50100150200250300-404812 H = 1 T Pt/YIG+L/0 (10-5 T (K)c) d) RT -RT0 (m) e) -240-120 0 120240-2.0-1.00.01.02.0 Pt/YIG H (Oe) = 45 = 135T=10 K -240-120 0 120240-1.0-0.50.00.51.0 Pt/YIG+ 45 135 H (Oe)T=10 KH Jc H x y z VT f) a) b) Fig. 2 6 Pt/YIG+ at low temperatures [see Fig. 2(b) and 2c)]. However, note that the excitation of magnons may lead to a larger correction in the ADMR amplitude at very high temperatures. This could give an alternative explanation to the measured temperature dependence of the MR in Pt/YIG bilayers close to the Curie temperature of the YIG film [15]. Figure s 2(e) and 2(f) show the magnetic -field-dependent MR curves measured in Pt/YIG and Pt/YIG+, respectively, at 10 K with the magnetic field applied in the plane of the film and along two representative directions ( =45º and =135º). The peaks and dips correspond to the magnetization reversal of the YIG film as reported earlier [6,9– 11]. Note that the saturation field of the YIG film obtained via magnetometry measurements (denoted as vertical dashed line s) matches perfectly with the one obtained through MR measurements in both samples. Moreover, the signs of the MR signals (for 45º and 135º) are reversed in Pt/YIG+ with respect to the ones measured in Pt/YIG, which is in agreement with the sign change observed in the ADMR at low temperatures [see Figs. 2(b) and 2(c)]. Because the SMR effect is basically sensitive to the magnetic properties of the first magnetic layer, having an estimated penetration depth of just a few Å [36], all previous measurements indicate that the ma gnetic moments of the surface of the YIG+ film are perpendicularly coupled to the ones of the bulk at low temperatures. The emergence of this surface magnetic frustration in our treated samples could be caused by a competing ferromagnetic and antiferromagn etic coupling of the modified complex stoichiometry of the YIG film due to the Ar+-ion milling process. In fact, magnetic frustration has already been observed in some ferrimagnets at low temperatures [70–73]. The angle between the magnetic moments of the surface and the bulk magnetization would be maximum (up to 90º) at low temperatures . The fact t hat the external magnetic field H aligns the bulk M but the SMR is sensitive to the magnetic moments of the surface yields a negative amplitude of the ADMR. A rise in the temperature would lead to a reduction of the angle due to the increase of the thermal energy in the magnetically coupled system . Considering our measurements, both surface and bulk magnetizations would lie together above ~140 K , recovering the expected positive amplitude of the ADMR. According to this physical picture, when the magnetic field (with H>HS) rotates in a particular H-rotation plane, the magnetic frustration forces the surface magnetization to point to a perpendicular direction. Due to the degeneracy in the orientation where the surface magnetization could point to, the angular dependences of the ADMR signals are preserved . As for the magnetic -field-dependent MR curves, when H<Hs, our YIG bulk film breaks in domains [74–76], resulting in the peaks and dips observed [see Fig. 2(e)]. The fact that the estimated HS of the surface magnetization via MR measur ements is the same for both samples [see Figs. 2(e) and 2(f)] and correlates with the measured HS of the film indicates that the magnetic moments of the surface of the YIG+ must be coupled to the bulk. The fact that the peaks and dips in the MR curves are reversed confirms that the angle between the magnetizations of the frustrated surface and the bulk should approach 90º at very low temperatures. In this scenario, one may think that , by applying a large enough magnetic field , we should be able to exert enough canting to the frustrated surface magnetic moments to shift the ADMR amplitude to positive values (i.e., reduce . Positive ADMR values have actually been measur ed for H>2T at low temperatures. However, the large Hanle 7 magnetoresistance (HMR) effect [52] present in our samples (the measured HMR amplitude at 300 K and 9 T is L/0~16·10-5) dominates the MR at large fields , preventing us from quantifying the canting exerted to the frustrated magnetic moments via MR measurements. An alternative interpretation of the temperature dependence of the SMR, motivated by the results obtained exploring a Pt/NiO/YIG system [77], is that the magnetic moments of the treated YIG+ surface are perpendicularly coupled to the magnetization of the YIG film at any temperature. In this situation, the frustrated magnetization of the surface dominates the SMR at low temperature, which is negative. When increasing the temperature, the frustrated surface becomes more transparent to the spin currents due to the thermal fluctuations and the YIG magnetization progressively dominates the SMR, which becomes positive. In other words, the spin current generate d by the Pt reaches the bulk YIG and the usual SMR in Pt/YIG is detected. This competition would lead to a decrease in the SMR amplitude below ~140 K , a comp ensation at an intermediate temperature (i.e., zero SMR amplitude, which oc curs around 45 K in our system), and a negative amplitude at low temperature s, when the frustrated Pt/YIG+ interface dominates. Our model allows us to qualitatively show that the emergence of a surface magnetic frustration can be well identified via SMR measurements. Note that magnetic frustration at the first atomic layer of a film cannot be detected by means of standard surface techn iques such as ma gneto -optical Kerr effect , magnetic force microscopy, or X -ray magnetic circular dichroism because of the relatively long penetration depth. Other surface sensitive techniques such as spin -polarized scanning tunneling microscopy or scanning electron microscopy with polarization analysis cannot be used in magnetic insulators either. Only complex, depth sensitive techniques such as polarized neutron reflectometry might resolve the surface magnetization independently of bulk. In other words, t he magnetic properties of the very first layer of a n insulating film will generally remain hidden by the large magnetic response of its bulk. Remarkabl y, unlike other techniques, the SMR can be applied to FMI films, is sensitive to only the first atomic layer [36], and its response is associated to the relative direction of the magnetic moments of the FM with respect to the spins of the NM layer (whether they are parallel or perpendicular), but not to their orientation (up or down). This highlights the potential of the SMR to explore complex surface magnetic properties [78]. IIIb. Magnon spin transport We now move to study the magnon spin transport in the non-local NL-Pt/YIG and NL- Pt/YIG+ samples. Figure 3(a) shows an optical image of one of the devices fabricated. In these samples, the current is injected in the central wire and both the local resistance (RL=VL/I) and the non -local resistance ( RNL=VNL/I) are measured as schematically drawn in Fig. 3(a). Note that RNL is measured using the dc -reversal method [54–56], which is equivalent to the first harmonic signal in ac lock -in type measurements [79]. 8 FIG. 3 (color online). (a) Optical image of the NL -Pt/YIG sample. Grey wires are the Pt stripes and the yellow areas correspond to additional Au pads. The black background is the surface of the YIG film. Both the local and non -local measurement configurations are schematically shown. (b) and (c) are the local ( RL) and non -local ( RNL) ADMR signals, respectively, measured in the NL -Pt/YIG sample at 150 K and for H=1 T rotating in the plane. Note that, along this rotation angle, M changes its relative orientation with s (being parallel for =90º and 270º and perpendicular for =0º and 180º). In (b), the bias current was 100 A. In (c), non -local ADMR measurements performed at I=100 (black line) and 300 (red line) are shown. The arrows in (b) and (c) indicate the sign convention used for the amplitude of the local ( RL) and non -local (RNL) resistance plotted in Figs. 4(a) and 4(b), respectively. Figure s 3(b) and 3(c) show an example of the local and non -local ADMR measurements, respectively, performed in our samples. The data correspond to the NL- Pt/YIG sample measured at 150 K with H=1 T rotating in the plane [see Fig. 1(b) for the definition]. Simila r ADMR curves were obtained in the NL-Pt/YIG+ sample. The local resistance RL [Fig. 3(b)] shows the expected cos2() dependence for the SMR effect . Taking into account that in these samples 𝜌𝐿0(300 K)~54 𝜇Ωcm –which according to Ref. 61 corresponds to ~1.2 nm and SH~0.083 for the Pt film –, that the measured SMR amplitudes at the same temperature are Δ𝜌𝐿/𝜌0~6.210-5 and ~7.610-5 (for the NL -Pt/YIG and NL-Pt/YIG+ samples , respectively) , dN=5nm, and that Gi<<Gr [63], Eq. (2) yields Gr ~3.21013-1m-2 and ~4.01013 -1m-2 for the Pt/YIG and Pt/YIG+ interfaces , respectively, which is in very good agreement with our previous results. 9 The non -local resistance RNL [Fig. 3(c)] shows a sin2() dependence, which is expected for the excitation, transpor t and detection of magnon spin information through the YIG film [25,29,30] . The physical description of this phenomenon is the following. The current applied in the central Pt wire (injector) produces a transverse spin current ( via the SHE ) that flows along the z axis [being s parallel to the y axis; see Fig. 3(a) for the definition of the axes]. When these spins reach the Pt/YIG interface, they can excite (annihilate ) magnons in the YIG film when s is parallel (antiparallel) to M [25], which produce a change in the magnon population below the Pt injector. These non - equilibrium magnons diffuse throug h the YIG film and, when they reach the nearby Pt wire (detector), the reciprocal process takes place. Therefore, the non -equilibrium magnons below the Pt detector transform into a non -equilibrium spin imbalance at the Pt/YIG interface, which produces the flow of a pure spin current perpendicular to the interface that is ultimately converted into a perpendicular charge current (along the Pt wire) via the ISHE. The combination of all these processes generates the non -local resistance RNL shown in Fig. 3(c) [80]. The angular dependence observed in Fig. 3(c) confirms that the excitation and absorption of propagating magnons in the YIG film are maxima when s and M are collinear, which occurs for =90º and 270º (note that the sign of the signal captured agrees with th e sign convention chosen for our experiments [25,29] ). Moreover, VNL should be linear with I for moderate applied currents [25]. This is confirmed in Fig. 3(c), where it is shown that the same RNL() curve is obtained for I=100 (black) and 300 A (red). The amplitude of the RNL() curve measured in our sample is consistent with results reported using YIG films with similar thicknesses [29]. Figure 4 shows the temperature dependence of the amplitude of (a) the SMR and (b) the MST measured in both the NL-Pt/YIG (black squares) and NL-Pt/YIG+ (red circles) samples. The sign of the amplitude of the SMR (local) and the MST (non -local measurements) is indicated with the arrows drawn in Figs. 3(b) and 3(c), respectively. The SMR data is presented normalized to the base resistance, followin g the same procedure used in the previous case . In Fig. 4(a), we see that the temperature dependence of the SMR i n these samples is qualitatively similar to the one observed in the previous experiments [see Figs. 2(c) and 2(d)], which confirms once again the emergence of a surface magnetic frustration in the treated YIG+ substrate at low temperatures. Interestingly, while the amplitude of the SMR in the temperature range ~150 -300 K is only slightly larger in the NL-Pt/YIG+ sample than in the NL-Pt/YIG one (i.e., slight enhancement of 𝐺r), the amplitude of the MST is about four times larger [see Fig. 4( b)]. This indicates that in this temperature range the efficiency of the spin -to-magnon conversion (and its reciprocal process) in the treated Pt/YIG+ interface is much higher than in the non-treated Pt/YIG interface , but not related to the change in 𝐺r. Instead , it must be associated to the different magnetic properties of the treated YIG+ surface compared to the YIG bulk for temperatures above the emergence of the magnetic frustration. Further studies will be needed in order to fully understand the role of this surface enhancement . 10 FIG. 4 (color online). Temperature dependence of the amplitude of (a) the SMR and (b) the MST measured in NL -Pt/YIG (black squares) and NL -Pt/YIG+ (red circles). The amplitude is extracted from ADMR measurements performed in the plane at H=1 T. Measurements in (a) and (b) are independent of I (at least) to up to 300 A. The inset in panel (b) shows a zoom of the measured RNL at low temperatures. Black solid line is a fit to the experimental points to the power law dependence T3/2. The temperature dependence of the amplitude of the MST follows a remarkabl y different trend than the SMR, which is in agreement with recent report s [29]. In fact, we found that the MST amplitude in the NL-Pt/YIG sample at low temperatures follows a ~T3/2 dependence [see inse t in Fig. 4(b)], expected for thermally induced diffusive magnons in the limit of large magnon diffusion lengths (i.e., weak magnon -phonon interactions) [27,29,81,82] . Importantly, the temperature dependence decays more abruptly for the NL-Pt/YIG+ sample, and no MST signal is detected at low temperatures (within the noise level), evidencing that the emergence of the surface magnetic frustration r esults in the suppression of non -equilibrium diffusive magnons at the surface of the YIG+ film. In other words, the frustrated magnetic surface, which may host a magnon dispersion relation different from the YIG bulk, is preventing the efficient spin -to-magnon conversion (and viceversa) at the Pt/YIG+ interface. IV . Conclusions We demonstrate via SMR and MST measurements in Pt/YIG -based devices that an Ar+- ion milling treatment of the YIG surface has a profound impact in the resulting spintronic phenomena. Beyond a slight increase in the spin-mixing conductance observed for the treated samples at room temperature, which accounts for a better interface quality, we show tha t the MST is fourfold increased. This elucidates the higher sensitivity of the magnon excitations to fine details in the magnetic properties of the magnetic surface. Moreover, we show that the treated surface of YIG develops a magnetic frustration at low temperature, which makes the SMR signal to reverse the 11 sign below ~45K and dramatically suppresses the spin -to-magnon excitations in these interfaces. 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2016-06-09

We study the spin Hall magnetoresistance (SMR) and the magnon spin transport (MST) in Pt/Y3Fe5O12(YIG)-based devices with intentionally modified interfaces. Our measurements show that the surface treatment of the YIG film results in a slight enhancement of the spin-mixing conductance and an extraordinary increase in the efficiency of the spin-to-magnon excitations at room temperature. The surface of the YIG film develops a surface magnetic frustration at low temperatures, causing a sign change of the SMR and a dramatic suppression of the MST. Our results evidence that SMR and MST could be used to explore magnetic properties of surfaces, including those with complex magnetic textures, and stress the critical importance of the non-magnetic/ferromagnetic interface properties in the performance of the resulting spintronic devices.

Competing effects at Pt/YIG interfaces: spin Hall magnetoresistance, magnon excitations and magnetic frustration

1606.02968v2

1 Planar Hall effect in Y3Fe5O12 (YIG) /IrMn films X. Zhang1 1 Beijing National Laboratory for Condensed Matter Physics, Institute of Physics, Chinese Academy of Sciences, Beijing 100190, China Correspondence and requests for materials should be addressed to X. Z. (zhangxu1986@ iphy.ac.cn ) ABSTART The planar Hall effect of IrMn on an yttrium iron garnet (YIG = Y 3Fe5O12) was measured in the magnetic field rotating in the film plane . The magnetic field angular dependence of planar Hall resistance (PHR) has been observed in YIG/IrMn bilayer at different temperatures , while the GGG/IrMn (GGG= Gd3Ga5O12) shows constant PHR for different magnetic field angles at both 10 K and 300 K . This provides evidence that IrMn has interfacial spins which can be led by FM in YIG/IrMn structure. A hysteresis can be observed in PHR- magnetic field angle loop of YIG/IrMn films at 10 K , indicating the irreversible switching of IrMn interfacial spins at low temperature. I. INTRODUCTION The coupling between a ferromagnetic (FM )-antiferromagnetic ( AFM ) bilayer at the interface can lead to a shift of the hysteresis loop along the magnetic field axis due to unidirectional anisotropy. This phenomenon is referred as exchange bias (EB) , which often accompanies with the enhancement of coercivity. Due to the pinning e ffect of magnetic moment in the adjacent FM layer, exchange bias effect has become an integral part in modern spintronics devices such as giant magnetoresistive heads 1. Although EB was discovered more than 60 years ago2, the mechanism of it is still a controversial and attracting object to be comprehended as one fundamental physic al issue. To 2 understand the mechanism of EB effect, much effort has been dedicated to investigate the behaviors of FM layer in EB systems ,3-5 while the attention on the essential feature of AFM is relatively less due to the notorious difficulties of requirement of large -scale facilities, such as neutron diffraction or XMCD. The theory has predicted that the unidirectional anisotropy in EB system derives from the interfacial uncompensated AFM spin, which does not reverse with the magnetization reversal of the FM layer. However, there is still no solid evidence to prove the existence of it even in the intensively studied Co (-Fe)/IrMn (111) in -plane exchange -biased system .6-8 Compared to the measuring methods above, there is a more common way to investigate EB system , which is through the electronic transport properties. In the past years, EB effect measured by planar Hall effect has become a more interesting subject due to higher signal -to-noise ratio compared with the magnetoresistance or spin valve configuration .3,9,10 In a FM metallic film, the magneto resistance can be separated into longitudinal and transverse part due to the anisotropic scattering of conduction elections. For a single domain model, the electric field can be described as :11 2( ) cosxEj jρ ρρ α⊥⊥=+− (1) ( )sin cosyEjρρ α α⊥= − (2) where the current density j is assumed along the x- axis direction, the magnetization of the single domain is at angle α with respect to x -axis, while ρand ρ⊥are the resistivity parallel and perpendicular to the magnetization, respectively. Equ. (1) is for anisotropic magnetoresistance (AMR) , while Equ. (2) is f or planar Hall effect ( PHE ). By using AMR effect and PHE, Guohong Li .etc have performed significant work on investigating the NiFe/NiMn EB multilayer and spin- valve.10 And recently, researches have shown that the reversal of AFM magnetization can lead to the AMR effect in the tunnel junction 3 structures .12-14 One significant work is the observation of large AF M-tunneling anisotropic magnetores istance (TAMR ) in the MTJs based on NIFe/IrMn/MgO/Pt structure by B.G. Park group,12 which emphasizes the significant role of AFM in generating the TAM R. However, these works mentioned above contains both conductive FM and AFM, suggesting that the magnetoresistance signal is consisted of the contribution from both layers. In this condition, the FM may provide an important part of the AMR signal since the reversing of its magnetization can cause a large variation of magnetoresistance. This prevent s us from directly observing the magnetization state of AFM layer and the proportion of AMR contribut ed by AFM is also not cle ar. In spite of this , these works shed the light on studying the AFM properties in EB system by AMR and PHE . In this work , in order to study the behavior of AFM with other disturbance excluded, an almost isolated FM material Y 3Fe5O12 (YIG) was chosen as FM layer. Through this way, all the magnetoresistance signal would derive from IrMn, thus the result would give out the more specific and directly information of AFM IrMn. By usin g isolated FM layer in the FM -AFM structure, it provide s evidence that there exists uncompensated spins at the interface of IrMn that can be reversed by FM in the YIG/IrMn system at room temperature, and the anisotropy of A FM interfacial spins get enhanced down to the low temperature region. II. Experimental In this study, the YIG substrates is consist of 4 μm thick single crystalline (111) YIG layers grown by liquid phase epitaxy on (111) Gd 3Ga5O12 (GGG) substrates , and single crystalline (111) GGG substrates are also used in this work . Films with structure of YIG/ Ir25Mn 75 (1.8-15 nm)/Ta (2 nm) and GGG/Ir 25Mn 75 (5 nm)/Ta (2 nm) were deposited by DC magnetron sputtering, where (111) texture of the substrates is to promote (111) texture in Ir 25Mn 75 layer , and Ta serves as a cap layer to protect the sample from oxidation/contamination. The vacuum of the sputtering system was better than 4×10-5 Pa, and the working Ar pressure was 0.5 Pa. Sets of up to 18 samples were prepared at a time. Composite Ir -Mn target by placing Mn chips symmetrically on the Ir target 4 was used to deposit Ir 25Mn 75 film. The composition was determined by inductively coupled plasma- atomic emissio n spectroscopy (ICP -AES). The samples were patterned into four terminal Hall bars. For simplicity, Ir 25Mn 75 hereinafter is denoted as IrMn. M -H, PHR –θ and PHR- H curves were measured by a vibrating sample magnetometer (VSM) and physical property measurement system (PPMS) at different temperature s, respectively , where PHR is planar Hall resistivity and θ is t he external magnetic field angl e. III. RESULTS AND DISCUSSION Magnetic hysteresis loops measured by VSM s hows that the YIG films are magnetically soft and isotropic in the film plane. As shown in the uppe r panel of Fig. 1(a ), at room temperature, the YIG film has an in -plane saturation field of about 60 Oe , and the loop remains the same as the sample rotates 90° in the same plane (not shown) . The YIG film has a spontaneous magnetization (4πM S) of 1.73 kG, which can be obtained in the out -of-plane loop due to the shape anisotropy , as indicated in the bottom panel of Fig. 1(a) . These magnetic properties is in agreement with the YIG films reported before .15 Fig. 1(b) shows the hysteresis loop of a typical exchange bias structure of CoFe (8 nm)/IrMn (10 nm), which displays a saturation field smaller than 300 Oe and an exchange bias field of about 120 Oe, proving that the IrMn deposited by our composite target is reliable antiferromagnetic material . Fig. 1(c ) and ( d) exhibit the magnetic field angular dependence of PH R for mon o-CoFe (8 nm) layer and CoFe (8 nm)/IrMn (10 nm), respectively. In the angular -dependent measurement, the sample was rotated first from θ=0° to 360° and backwards from to θ= 360° to 0° in a fixed magnetic field. The data in Fig. 1 ( c) was measured under 2 kOe magnetic field at 300 K, which is larger than the saturation field of CoFe . The behavior of CoFe layer illustrates that the PHR of single FM layer would exhibit a sinusoidal dependence of magnetic field angular with period of 180º, and also no difference between the clockwise and anticlockwise curve is observed. Note that the curve is not symmetric of zero resistance point as expected for uniaxial anisotropy of FM material s, which 5 can be explained by the misalignment of the Hall bar that measures the voltage, leading to an additional resistance along the current direction . When coupled to IrMn, the CoFe (8 nm)/IrMn (10 nm) (referred as CoFe/IrMn hereinafter ) structure develops exchange bias effect due to the coupling between FM and AFM layer , and an interfacial unidirectional anisotropy would be induced into the system as a result . Fig. 1( b) shows the hysteresis loop of CoFe/IrMn , which displays a saturation field smaller than 300 Oe and an exchange bias field of about 120 Oe. In Fig. 1( d), for the magnetic field below 300 Oe , the angular -dependent curve of CoFe/IrMn shows a distortion compared to mono- CoFe layer data . Since 300 Oe is larger than the saturation field of CoFe/IrMn structure based on the loops i n Fig. 1(b) , the PHR- theta curve of 300 Oe indicat es that the unidirec tional anisotropy would lead to a magnetic moment rotating processes different to that of uni axial anisotropy . When magnetic field increases to 2 kOe, which is much larger than the saturation field obta ined in Fig. 1( b), although no obvious distortion can be observed in the PHR curve, the re is still a slightly deviation when fitting PHR- θ data by a sine function. This suggests that most of the CoFe layer synchronously rotates with the external magnetic field of 2 kOe. It is worth mentioning that because of the coupling between CoFe and IrMn, some of the interfacial IrMn spins should rotates certain degree with CoFe, which also should have contribution to PHR . However , owning to the much larger signal of 8 nm thick CoFe layer, this PHR from IrMn can hardly be confirmed . In addition, it is worth noting that the PHR -θ curve under 50 Oe field shows a hysteresis behavior. Since 50 Oe is not large enough to reverse the CoFe spins pinned by IrMn, and also PHR- θ curve at this field shows a smaller amplitude when compared with the curves of 300 Oe and 2 kOe , which suggests that the FM moments still rotates a small angle and this spin flo pping follow s a different route in the clockwise and anticlockwise magnetic field cycling procedure , i.e. the FM moment shows an irreversible switching under 50 Oe field. Recently, several researches have reported the AMR derives from AFM in the FM-AFM EB systems, proposing a new possibility to obtain large magnetoresistance in relatively small fields .12-14 However, these studies probed AFM with conductive FM layer, which 6 would lea d to difficulty of distinguishing AFM signal from FM signal, thus the specific behavior of AFM is still not clearly revealed. In order to investigate the interfacial magnetization of AFM in a n FM-AFM coupled system more independently , antiferromagnetic material IrMn was deposit on the isolated ferromagnetic material YIG. Figure 2 shows the angular dependence of PHE for YIG/ IrMn (1.8 nm) and ( 5 nm) at different temperatures . The sample was first ly field cooling to 10 K under a constant field of 3 kOe in the same direction as the magnetic field applied during the film growth , and then measured at increasing temperatures in the magnetic field of 2 kOe, which is larger than the saturation field of common exchange bias system . At each temperature, t he measurement s were also performed form 0 ° to 360° and cycling back to 0° for both samples . Clearly , magnetic field angle dependence of PHR can be observed in both YIG/IrMn sampl es from 10 K to 300 K, indicating an anisotropic magnetoresist ance. Since 2 kOe is not large enough to rotate bulk magnetic moments of IrMn as well as YIG is a nearly isolated material , the anisotropy signal c an only derive from the rotation of IrMn interfacial moment, confirm ing that there are uncompensated spins existing in the interface of IrMn that can be rotated by a small magnetic field. In Fig. 2(a) and (e), a hysteresis of magnetic field angle can be observed for both samples at 10 K, suggesting that the IrMn interfacial moment cannot be fully reversed by the magnetic field of 2 kOe at this temperature. It was reported that in an EB structure of Co/YMnO 3 with i nsulated YMnO 3 as antiferromagnetic layer, a similar hysteresis can be observed in the angular -dependent magnetic field measurement of Co at low temperatures5, where EB effect is more remarkable. However, in contrast to Co/YMnO 3 system, the contributions to PHE are all from the AFM layer in YIG/IrMn film , which indicates that the coupled spins of AFM may exhibit the same behavior as adjacent FM spins. Furthermo re, according to CoFe/IrMn in Fig. 1(c) and Co/YMnO 3 result s, this irreversible of IrMn interfacial magnetization implie s a combination of uniaxial and unidirectional anisotr opy in IrMn interfacial moment at low temperature s. On the other hand, at 7 higher temperatures, the hysteresis and distortion gradually disappear, and the curves are sinusoidal at 300 K, indicating that the IrMn interfacial moment is uniaxial at room temperature. Thus, it can be deduced that the unidirectional anisotropy is the cause of not -fully reversed moment at 10 K, which is analogous to the CoFe/IrMn- 50 Oe curve in Fig. 1(d) . Similar to the behavior of NiFe/IrMn stack ,13 the fast loss of the unidirectional anisotropy is probably due to the decrease of IrMn anisotropy as temperature approaches to the Neel temperature of IrMn .16 Therefore, it can be conjectured that the uncompensated spins at IrMn interface has relatively large coupling with IrMn spins around it at low temperature, which leads to the unidirectional anisotropy. When the temperature increases, this exchange interaction becomes weak and it vanishes at room temp erature. Due to the coupling interaction between FM and AFM, the rotation of interfacial magnetic moment is supposed to be le d or promoted by FM moments. To verify this scenario , the angular dependence of GGG/IrMn (5 nm) sample was papered , where GGG is a non- magnetic insulated substrate of (111) Gd 3Ga5O12. As shown in Fig ure 2(i) and (j) , the sample with GGG substrate shows isotropy of PH R for the field below 2 kOe at both 10 K and 300 K , this proves that the promotion of FM to the reversing of interfacial moment of AFM. Therefore , the sinusoidal curves of YIG/IrMn samples at 300 K suggests that the coupling between of YIG and IrMn still exist s at room temperature , leading to the rotation of some IrMn interfacial spins with uniaxial anisotropy behavior . Besid es, in Fig. 2(d) and (h), the curves of 200 Oe are almost identical to those measured under 2 kOe field, indicating that both IrMn- 1.8 nm and IrMn -5 nm samples have reached saturation under field as low as 200 Oe. This phenomenon also suggests a very small anisotropy of interfacial IrMn moments , which is different to the large unidirectional anisotropy observed in CoFe/IrMn at low magn etic field (Fig. 1(c)) . Whereas there is no solid evidence to explain this contradiction, one possible assumption is that the roughness of IrMn interfaces may be the reason for this difference. Owning to the fact that the series of YIG/IrMn samples are not in- situ grown, which could induce defects and impurities at the interface to cause 8 the interface of YI G become much rougher than that of CoFe. Therefore, a small part of uncompensated spins of the net magnetic moment at IrMn interface may has relatively weak coupling with the bulk than the other s due to the roughness of the interface , resulting in the presentation of a smaller anisotropy, these spins are referred as free spins (F spins) . At low temperature, the F spins exhibit unidirectional anisotropy due to their exchange interaction of the bulk IrMn, which decreases at room temperature, leading to the YIG/IrMn sinusoidal dependence at 300 K. And the rest of uncompensated pinned spins (P spins) of IrMn net magnetic moment still has strong coupling with IrMn since the Neel (blocking) temperature is much higher than the room tem perature , these P spins cannot be rotated or can only be rotated for a small angle under 2 kOe field . And because CoFe/IrMn is grown in-situ, the roughness of the CoFe/IrMn interface should be much smaller than YIG/IrMn interface. Thus the EB effect of CoFe/IrMn are mostly developed by t he coupling between P spins and FM layer , which provides the unidirectional anisotropy for CoFe/IrMn at room temperature. Another evident phenomenon is that both the IrMn- 1.8 nm and IrMn- 5 nm sample s show dramatic deducti on at 90 K and 130 K, as well as the overall trend of low field curve phase shifts 90° as the temperature increases, indicating the shift of easy axis. This phenomenon can be explained by the existence of strongly coupled Fe3+ and weakly coupled Y3+ in YIG, of which Y3+ and the ferrimagnetic component of Fe3+ magnetic moments are antiparallel to each other17. Because the interactions between Y3+ ions are weak, Y3+ shows paramagnetic properties in the exchange field generated by Fe3+ spins. In low temperature region, the total magnetic moments parallel to the Y3+ ions due to its large magnetic moments. As the temperature increases, the magnetic moments of the sub -lattice of Y3+ quickly decrease, leading to the domination of Fe3+ magnetic moments at high temperature. Therefore, the decreasing of PHR at 90 K and 130 K is probably as a result of the competition of the two kinds of moments, and the change of easy axis is because the exchange interaction only exits between Fe 3+ in the high temperature region. Additionally, as shown in Fig ure 2, apart from the larger amplitude of PHR, the YIG/IrMn 9 (1.8 nm) sample shows basically the same behavior of the YIG/IrMn (5 nm) at different temperatures . Since the anisotropy of AFM decreases with decreasing thickness in ultrathin AFM film,18 it can be argued that the spins in IrMn is easier to be reversed by FM, which leads to a larger PHE of YIG/IrMn (1 .8 nm). By a ssuming the interface of IrMn is a single domain film structure , this observed angular dependence of the PHR at 10 K can be described by 2 01 2 3 cos ( ) sin[2( )] cos( )Mj Mj Mj PHR R R R R ϕϕ ϕϕ ϕϕ =+ −+ − + − (3) where 𝜑𝑀 and 𝜑𝑗 are the angle s of IrMn interfacial magnetization and current direction relative to the sample position of zero degree, respectively . The first and second term represent the ARM caused by the misaligning of Hall bar which lead s to a voltage difference along the current direction , while the third and fourth term are classic PH R and an extra PH R generating from unidirectional anisotropy, respectively. The curve of IrMn -1.8 nm and IrMn- 5 nm fitted by Eq. (3) is shown in Figure 3, exhibiting good agreement with the experiment da ta. The parameters obtained from fitting are shown in Table. I . It should be noted that R3 become smaller as the temperature increases, which consists with the decreasing tendency of unidirectional anisotropy observed in Figure 2. TABLE I . Fitting parameters for YIG/IrMn (1.8 nm) and YIG/IrMn (5 nm) film by Equ. (3) at 10 K and 300 K Sample Temperature (K) R0 (ohm) R1 (ohm) R2 (ohm) R3 (ohm) IrMn -1.8nm 10 5.91278 3.93735× 10-11 0.00413 0.00101 300 4.94969 3.023× 10-10 0.00541 0.00013 IrMn -5nm 10 4.3297 -5.47 ×10-4 -0.00171 0.000469 300 3.97303 1×10-5 0.00361 6.09181× 10-5 A further investigation on the IrMn thickness dependence of Hall angle is summarized in Fig. 4(a). T he measurement was performed at 300 K under 200 Oe, and Δρxy and the ρ xx was obtained by the PHR -H curve while the angle between magnetic field and current is 45°. The PHR- H loops of representative samples, IrMn -1.8 nm and IrMn- 5 nm are shown in Fig. 4(b) and 10 (c). Despite the IrMn -1.8 nm sample shows a larger PH R amplitude than IrMn -5 nm sample in Figure 2, it can be observed in Fig. 4(a) that the Hall angle almost remains constant when IrMn is not thicker than 5 nm. When the thickness of IrMn is larger than 5 nm, the Hall angle begin to decrease dramatically and reaches a constant as IrMn is thicker than 10 nm. This confirms the interface nature of PHE in IrMn, and th is interfacial effect does not quickly decrease until IrMn is not thick enough. IX. CONCLUSION In conclusion, the PHR of YIG/IrMn samples with different IrMn thickness es were investigated in this work. The magnetic f ield angle dependence of PHR can be observed from 10 K to 300 K below 2 kOe , suggesting the existence of uncompensated spins at YIG/IrMn interface. And also a hysteresis derived from unidirectional anisotropy can be observed in the PHR- θ curves at 10 K in YIG/IrMn films . By comparing with GGG/IrMn sample, which PHR- θ curve shows isotropic behavior at both 10 K and 300 K, it can be considered that the interfacial IrMn spins are led by the magnetic moment of YIG. ACKNOWLEDGEMENTS This work was supported by the National Key Basic Research Program of China under Grant No. 2014CB921002, and the National Natural Science Foundation of China under Grant Nos.51171205 and 11374349. REFERENCES 1 C. Chappert, A. Fert, and F. N. Van Dau, Nature Mater. 6, 813 (2007). 2 W. H. Meiklejohn and C. P . Bean, Phys. Rev. 105, 904 (1957). 3 J. Barzola -Quiquia, A. Lessig, A. Ballestar, C. Zandalazini, G. Bridoux, F. Bern, and P . Esquinazi, J. Phys. : Condens. M atter 24, 366006 (2012). 4 J. Sort, V. Baltz, F. Garcia, B. Rodmacq, and B. Dieny, Phys. Rev. B 71, 054411 (2005). 5 Y . F. Liu, J. W. Cai, and a. S. L. He, J. Phys. D: Appl. Phys. 42, 115002 (2009). 11 6 H. Ohldag, A. Scholl, F. Nolting, E. Arenholz, S. Maat, A. T. Young, M. Carey, and J. Stöhr, Phys. Rev. Lett. 91, 017203 (2003). 7 S. Doi, N. Awaji, K. Nomura, T. Hirono, T. Nakamura, and H. Kimura, Appl. Phys. Lett. 94 (2009). 8 H. Takahashi, Y . Kota, M. Tsunoda, T. Nakamura, K. Kodama, A. Sakuma, and M. Taka hashi, J. Appl. Phys. 110 (2011). 9 G. Li, Z. Lu, C. Chai, H. Jiang, and W. Lai, Appl. Phys. Lett. 74, 747 (1999). 10 G. Li, T. Yang, Q. Hu, H. Jiang, and W. Lai, Phys. Rev. B 65, 134421 (2002). 11 T. R. McGuire and R. I. Potter, IEEE Trans. Magn. 11, 1018 (1975). 12 B. G. Park, J. Wunderlich, X. Martí, V. Holý, Y . Kurosaki, M. Yamada, H. Yamamoto, A. Nishide, J. Hayakawa, H. Takahashi, A. B. Shick, and T. Jungwirth, Nature Mater. 10, 347 (2011). 13 X. Martí, B. G. Park, J. Wunderlich, H. Reichlová, Y . Kurosaki, M. Yamada, H. Yamamoto, A. Nishide, J. Hayakawa, H. Takahashi, and T. Jungwirth, Phys. Rev. Lett. 108, 017201 (2012). 14 Y . Y . Wang, C. Song, B. Cui, G. Y . Wang, F. Zeng, and F. Pan, Phys. Rev . Lett. 109, 137201 (2012). 15 Y . M. Lu, Y . Choi, C. M. Ortega, X. M. Cheng, J. W. Cai, S. Y . Huang, L. Sun, and C. L. Chien, Phys. Rev. Lett. 110, 147207 (2013). 16 G. Vallejo -Fernandez, L. E. Fernandez -Outon, and K. O’Grady, Appl. Phys. Lett. 91 (2007). 17 L. Neel, Comp. Rend. 239, 8 (1954). 18 Y . Xu, Q. Ma, J. W. Cai, and L. Sun, Phys. Rev. B 84, 054453 (2011). Figure captions: FiG. 1. (a) The in-plane (top panel) and out -of-plane (bottom panel) hysterias loops of YIG, and (b) the in-plane hysterias loop of CoFe (8 nm)/IrMn (10 nm), and (c) the PHR-θ curve of CoFe measured under 2 kOe magnetic field at room temperature, and (d) the PHR -θ of CoFe (8 nm)/IrMn (10 nm) measured under different magnetic fields at room temperature. FIG. 2. PHR of (a -d) YIG/IrMn (1.8 nm) and (e -h) YIG/IrMn (5 nm) sample when rotating in the applied magnetic field of 2 kOe at different temperatures. The measurements were also performed under 200 Oe for both YIG/IrMn samples at 300 K. (i) and (j) are the magnetic field angu lar dependence of GGG/IrMn (5 nm) film at 10 K and 300 K, respectively. At each temperature, the PHR -θ curves were measured under different magnetic fields of 200 Oe and 2 kOe, both of which exhibit no obvious variation with magnetic field angular θ. FIG. 3. PHR -θ cure and its fitting curve of YIG/IrMn (1.8 nm) sample at (a) 10 K and (b) 300 K, and PHR -θ cure and its fitting curve for YIG/IrMn (5 nm) sample at (c) 10 K and (d) 300 K. The fitting curve uses the data of magnetic field clockwisely rotating from 0° to 360°. The data of 10 K and 300 K are fitted by Equ. (3). FIG. 4. (a) The IrMn thickness dependence of Hall angle, and the PHR-H curve for (b) YIG/IrMn (1.8 nm) and (c) YIG/IrMn (5 nm) measured at 300 K. The angle between magnetic field and curre nt is 45°. Figures: 12 FIG.1. 13 FIG. 2. FIG. 3. 14 FIG. 4.

2014-10-05

The planar Hall effect of IrMn on an yttrium iron garnet (YIG = Y3Fe5O12) was measured in the magnetic field rotating in the film plane. The magnetic field angle dependence of planar Hall resistance (PHR) has been observed in YIG/IrMn bilayer at different temperatures, while the GGG/IrMn (GGG= Gd3Ga5O12) shows constant PHR for different magnetic field angles at both 10 K and 300 K. This provides evidence that IrMn has interfacial spins which can be led by FM in YIG/IrMn structure. A hysteresis can be observed in PHR-magnetic field angle loop of YIG/IrMn films at 10 K, indicating the irreversible switching of IrMn interfacial spins at low temperature.

Planar Hall effect in Y3Fe5O12(YIG)/IrMn films

1410.1112v1

1 Thermally Driven Long Range Magnon Spin Currents in Yttrium Iron Garnet due to Intrinsic Spin Seebeck Effect Brandon L. Giles1, Zihao Yang2, John S. Jamison1, Juan M. Gomez -Perez4, Saül Vélez4, Luis E. Hueso4,5, Fèlix Casanova4,5, and Roberto C. Myers1-3 1Department of Materials Science and Engineering, The Ohio State University, Columbus, OH, 43210, USA 2Department of Electrical and Computer Engineering, The Ohio State University, Columbus, OH, 43210, USA 3 Department of Physics, The Ohio State University, Columbus, OH, USA 4CIC nanoGUNE, 20018 Donostia- San Sebastian, Basque Country, Spain 5IKERBASQUE, Basque Foundation for Science, 40813 Bilbao, Basque Country, Spain Email: myers.1079@osu.edu , Web site: http://myersgroup.engineering.osu.edu The longitudinal spin Seebeck effect refers to the generation of a spin current when heat flows across a normal metal/magnetic insulator interface. Un til recently, most explanations of the spin Seebeck effect use the interfacial temperature difference as the conversion mechanism between heat and spin fluxes. However, recent theoretical and experimental works claim that a magnon spin current is generated in the bulk of a magnetic insulator even in the absence of an interface. This is the so -called intrinsic spin Seebeck effect. Here, by utilizing a non -local spin Seebeck geometry, we provide additional evidence that the total magnon spin current in the ferrimagnetic insulator yttrium iron g arnet (YIG ) actually contains two distinct terms: one proportional to the gradient in the magnon chemical potential (pure magnon spin diffusion) , and a second proportional to the gradient in magnon 2 temperature (𝛁𝛁𝑻𝑻𝒎𝒎). We observe two characteristic decay lengths for magnon spin currents in YIG with distinct temperature dependences: a temperature independent decay length of ~ 10 𝝁𝝁m consistent with earlier measurements of pure ( 𝛁𝛁𝑻𝑻𝒎𝒎=𝟎𝟎) magnon spin diffusion, and a long er decay length ranging from about 20 𝝁𝝁m around 250 K and exceeding 80 𝝁𝝁m at 10 K. The coupled spin- heat transport processes are modeled using a finite element method revealing that the longer range magnon spin current is attributable to the intrinsic spin Seebeck effect (𝛁𝛁𝑻𝑻𝒎𝒎≠𝟎𝟎), whose length scale increases at lower temperatures in agreement with our experimental data . Recently, s ignificant effort s have focused on understanding magnon spin diffusion arising from the spin Seebeck effect [1,2] . In particular , the effective magnon spin diffusion length in YIG has been experimentally measured using many different methods, including the systematic variation of YIG sample thickness to observe the effect on the longitudinal spin Seebeck signal [3 – 5], and by the use of a non- local geometry to directly measure the magnon spin diffusion length of electrically and thermally excited magnons [6–8]. Both methods demonstrated that the magnon spin diffusion length in YIG is only minimally dependent on film thickness and also that the magnon spin diffusion length is around 10 𝜇𝜇m at low temperatures. However, the studies report contradictory results near room temperature. T he thickness dependence study carried out by Kehlberger et. al. [3] found that the magnon spin diffusion length gradually decreases from 10 to 1 𝜇𝜇m as the temperature is increased to room temperature, while the non- local measurement carried out by Cornelissen et. al. [7] found that the magnon spin diffusion length is only very slightly dependent on temperature . These discrepancies might be expected due to variation in the temperature profile between experiments with different sample sizes and geometries, and the variation in the relative impact of the intrinsic (bulk) spin Seebe ck effect. The need to include the se 3 bulk temperature gradient driven magnon currents to fully explain room temperature nonlocal spin transport in thin film YIG has recently been discussed in detail in Ref. [8]. In this Rapid Communication, we further demonstrate the central role of the intrinsic spin Seebeck effect in the generation of long -range spin signals in bulk YIG that emerge at low temperatures . For this purpose, we carry out two independent experiment s to measure diffusive magnon spin currents in bulk single crystal YIG as a function of temperature using the nonlocal opto- thermal [9] and the nonlocal electro -thermal [6] techniques . For both measurements , magnons carrying spin angular momentum are thermally excited beneath a Pt injector resulting in a measureable voltage induced in an electrically isolated Pt spin detect or. In both the opto- thermal and electro -thermal measureme nts, two independent magnon spin current decay lengths are observed. The shorter decay length ~ 10 𝜇𝜇m is roughly temperature independent and in agreement with Cornelissen et al . [6]. In addition to this shorter decay length, we also identify a longer range magnon spin decay length at lower temperatures that reaches values in excess of 80 𝜇𝜇m at 10 K . The longer magnon spin decay length originates from magnon s generat ed by heat flow within the bulk YIG itself, and represents the intrinsic spin Seebeck effect. Finite element modeling (FEM) is used to solve coupled spin- heat transport equations in YIG that describe both the pure magnon spin diffusion that is driven by a gradient in the magnon chemical potential, ∇𝜇𝜇𝑚𝑚, and also the magnon spin current that is driven by a thermal gradient in the YIG itself, ∇𝑇𝑇𝑚𝑚. Microscope image s of typical devices used for opto- thermal measurements and electro - thermal measurements are shown in Fig. 1(a) and Fig. 1(c) . The opto- thermal device consists of 10 nm of Pt that was sputter deposited onto a 500 𝜇𝜇m <100> single crystal YIG that was purchased commercially from Princeton Scientific . Standard lithography techniques we re used to pattern the Pt into a 50×50 𝜇𝜇m detection pad surrounded by electrically isolated 5× 5 𝜇𝜇m injector pads with 3 4 𝜇𝜇m between them . The electro -thermal device consists of 5 nm of Pt that was sputter deposited onto a 500 𝜇𝜇m <100> single crystal YIG from the same wafer . Each electro -thermal device was fabricated via high-resolution e-beam lithography using a negative resist and Ar -ion milling to pattern one Pt injector and two Pt detectors (width W = 2.5 µm and length L = 500 𝜇𝜇 m). Injector - detector distances range from 12 to 100 𝜇𝜇 m. FIG 1. Optical images of the devices used in the opto- thermal and electro -thermal measurements. (a) In the opto- thermal measurement, a laser is used to thermally excite magnons in YIG beneath a Pt injector. The magnons diffuse laterally and are con verted into a measureable voltage in the Pt detector. (b) A typical hysteresis loop showing the measured voltage as a function of magnetic field. 𝑉𝑉𝑁𝑁𝑁𝑁,𝑂𝑂 is defined as the magnitude of the hysteresis loop. (c) In the electro -thermal measurement, current flowing through the injector causes resistive heating, resulting in the excitation of magnons into YIG. The non- equilibrium magnons produced diffuse to the re gion beneath a non- local Pt detector, where can be detected due to the inverse spin Hall voltage induced. (d) The measured voltage depends sinusoidally on the angle α of the applied in- plane magnetic field. The maximum detected voltage is defined as 𝑉𝑉 𝑁𝑁𝑁𝑁,𝐸𝐸. 𝑑𝑑 represents the distance the magnons have diffused from the injection to the detection site. In the opto- thermal experiment a diffraction -limited 980-nm-wavelength laser is used to thermally excite magnons beneath a Pt injector whose center is located at a distance d from the closest edge of the Pt detector . The experiments were carried out in a Montana Instruments C2 5 cryostat at temperatures between 4 and 300 K. The laser is modulated at 10 Hz and a lock- in amplifier referenced to the laser choppin g frequency is used to measure t he inverse spin Hall effect voltage , defined as 𝑉𝑉𝐼𝐼𝐼𝐼𝐼𝐼𝐸𝐸 ,𝑂𝑂, across the detector . An in -plane magnetic field is applied along the x axis and is swept from - 200 mT to 200 mT while 𝑉𝑉𝐼𝐼𝐼𝐼𝐼𝐼𝐸𝐸 ,𝑂𝑂 is continuously recorded. A representative hysteresis loop taken at 89.5 K and for d = 21 𝜇𝜇m is shown in Fig. 1(b). The detector signal proportional to nonlocal magnon spin diffusion, defined as 𝑉𝑉𝑁𝑁𝑁𝑁,𝑂𝑂, is obtained by taking half the difference between saturated 𝑉𝑉𝐼𝐼𝐼𝐼𝐼𝐼𝐸𝐸 ,𝑂𝑂 values at positive and negative fields, i.e. the height of the hysteresis loop. For the electro -thermal experiment, m agnetotransport measurements were carried out using a Keithley 6221 sourcemeter and a 2182A nanovoltmeter operating in delta mode . In contrast to the standard current -reversal method, where one obtains information about the electrically excited magnons in devices of this kind [10] , here a dc- pulsed method is used where the app lied current is continuously switched on and off at a frequency of 20 Hz. This measurement provides equivalent information as the second harmonic in ac lock- in type measurements [11] , i.e., it provides information about the thermally excited magnons. A current of I = 300 µA was applied to the injector. The experiments were carried out in a liquid- He cryostat at temperatures between 2.5 and 10 K. A magnetic field of H = 1 T was applied in the plane of the sample and rotated (defined by the angle 𝛼𝛼 ) while the resulting voltage VISHE,E was measured in one of the detectors . Fig. 1 (d) shows a representative measurement . The signal obtained is proportional to sin 𝛼𝛼 , which is indicative of the diffusive magnon spin current [12] . The magnitude of the signal is defined as 𝑉𝑉𝑁𝑁𝑁𝑁,𝐸𝐸 [see Fig. 1 (b)]. The magnon spin current decays exponentially with d [13] . Therefore, the VNL measured in our devices is given by 6 𝑉𝑉𝑁𝑁𝑁𝑁= 𝐴𝐴𝑜𝑜𝑒𝑒−𝜆𝜆𝑆𝑆∗ 𝑑𝑑, (1) where A0 is a pre- factor that is independent of d and 𝜆𝜆𝐼𝐼∗, is the effective magnon spin diffusion length . The experimental data obtained for both the opto- thermal and the electro -thermal magnon spin excitation are shown in Fig. 2 and analyzed using Eq. (1). At high temperatures, the data fits very well to a single exponential as expected. Surprisingly , at low temperatures, the fit analysis reveals that there must actually be two different decay lengths. For inst ance, for the opto- thermal case, it is observed that the quality of the fit rapidly decreases below a correlation coefficient of r 2=0.985 when the distances considered range from the smallest measured (5.5 𝜇𝜇m) to greater than 37.5 𝜇𝜇m. This indicates that the application of the spin decay model is only appropriate up to 37.5 𝜇𝜇 m. If distances greater than 37.5 𝜇𝜇 m are considered and the data is fit to Eq. (1) , a lower r2 factor is obtained, indicating a low quality fit . This observation inspires us to separate the 𝑉𝑉𝑁𝑁𝑁𝑁,𝑂𝑂 data into two distinct regions defined as the 𝜆𝜆1 and 𝜆𝜆2 region s [see Fig. 2( a)]. Equation (1) is fit to each individual region. Th e effective magnon spin diffusion length 𝜆𝜆𝐼𝐼∗ is extracted for each region separately and plotted in Fig. 3 . The same FIG 2. (a) 𝑉𝑉𝑁𝑁𝑁𝑁,𝑂𝑂 as a function of 𝑑𝑑 with the measurement shown at different temperatures. The measurement results are divided into two regions defined as 𝜆𝜆1 and 𝜆𝜆2. Dotted lines represent single exponential fits of the data to Eq. (1) in each region. The decay in 𝜆𝜆1 is shorter, while it appears to be much longer in 𝜆𝜆2. (b) 𝑉𝑉𝑁𝑁𝑁𝑁,𝐸𝐸 as a function of 𝑑𝑑 with the measurement shown at multiple temperatures. Dividing the data also into the 𝜆𝜆1 and 𝜆𝜆2 regions confirms the existence of the two different characteristic decay lengths. Dashed li nes are fits to Eq. (1) in each region. 7 analysis was performed for the electro - thermal measurements and the existence of two different decay lengths was confirmed (See Fig . 2(b)). Fig. 3 shows the extracted values of the magnon spin diffusion lengths in each of the two regions as a function of temperature for both the opto- thermal and electro -thermal measurements. At low temperature, both measurements indicate an effective spin diffusion length of about 10 𝜇𝜇m in the 𝜆𝜆 1 region, which is in excellent agreement with p reviously reported values and temperature dependence of the magnon spin diffusion length [7] . Note that in the earlier opto -thermal study [9] the data indicated only a single exponential decay, which was interpreted as the spin diffusion length. In the opto- thermal measurements reported here, the improved signal to noise ratio of the experiment reveals the double exponential character of the spin decay profile. The current data can still be fitted to a single exponential decay at 23 K of 47 µm, consistent with the earlier report, however the improved data set in the current study demonstrates that a double exponential decay fit is far better quality. A larger 𝜆𝜆 𝐼𝐼∗ in the 𝜆𝜆 2 region is observed in both the opto- thermal and electro -thermal measurements . At temperatures above 10 K in the electro -thermal measurement, the non- local signal magnitude strongly decreased and could not be measured at enough values of d in order to make a meaningful exponential fit to extract 𝜆𝜆𝐼𝐼∗ in the 𝜆𝜆 2 region . The effective magnon spin FIG 3. The extracted decay parameters 𝜆𝜆𝐼𝐼∗ from the 𝜆𝜆1 and 𝜆𝜆2 regions as a function of temperature and for both experiments. 𝜆𝜆𝐼𝐼∗ values reported in Ref. 7 are included for comparison. Inset: zoomed view of low temperature data. 8 diffusion length in the 𝜆𝜆2 region is approximately one order of magnitude larger than in the 𝜆𝜆 1 region at low temperatures and decreases monotonically with increasing temperature. The maximum value of 83.03 𝜇𝜇 m occurs at 9.72 K and the minimu m value of 14.05 𝜇𝜇 m at 247.5 K. A zoom of the data at low T is shown in the inset to Fig. 3. In the electro -thermal measurements, the maximum value of 𝜆𝜆2 is not at the lowest temperature, but at ~10 K in agreement with the optothermal measurements . This is consistent with the origin of 𝜆𝜆2 as from intrinsic SSE associated with the temperature profile in YIG since as T approaches 0 K, thermal conductivity becomes negligible . To justify the existence of the long range spin current persisting well beyond the intrinsic magnon spin diffusion length, t he measurements are compared to a simulation of the diffusive transport of thermally generated magnons , which is obtained using three dimensional (3D) finite element modeling (FEM). The simulation is solved using COMSOL Multiphysics and is based on the spin and heat transport formalism that is developed in [14,15] . In the simulation , the length scale of the inelastic phonon and magnon scattering is assumed to be small, implying that the phonon temperature , 𝑇𝑇𝑝𝑝, is equal to the magnon temperature 𝑇𝑇𝑚𝑚 over the length s of interest . In addition, the simulation neglects the spin Peltier effect . Thus, the spin and heat transport equations are only partially coupled. The simplified spin transport equation that is used to model the magnon spin current within YIG is 𝜎𝜎∇2𝜇𝜇+ 𝜍𝜍∇2𝑇𝑇=𝑔𝑔𝜇𝜇 (2) and t he Pt/ YIG interfacial boundary condition states 𝑗𝑗𝑚𝑚,𝑧𝑧=𝜎𝜎∇𝜇𝜇𝑧𝑧+𝜍𝜍∇𝑇𝑇𝑧𝑧= 𝐺𝐺𝐼𝐼𝜇𝜇 (3) 9 where 𝑗𝑗𝑚𝑚,𝑧𝑧 is the simulated spin current perpendicular to the Pt/YIG interface, 𝜎𝜎 is the spin conductivity in the YIG , 𝜇𝜇 is the magnon chemical potential, 𝜍𝜍 is the intrinsic spin Seebeck coefficient, 𝑔𝑔 describes the magnon relaxation, 𝑇𝑇=𝑇𝑇𝑝𝑝~𝑇𝑇𝑚𝑚 is the temperature in YIG , 𝐺𝐺𝐼𝐼 is the interfacial magnon spin conductance, and ∇𝜇𝜇𝑧𝑧 and ∇𝑇𝑇𝑧𝑧 represent the gradient of the magnon chemical potential and temperature along the direction perpendicular to the Pt/YIG interface , respectively. We first solve for the temperature profile in a simulated Pt/YIG system using the parameters listed in Table I. The geometry of the model is the same as the experimental geometry of the opto- thermal measurement including the Pt absorbers . As previously stated, d is defined as the distance from the edge of the Pt detector to the center of the (simulated) laser heat source at the center of the absorber . Table I – Parameters used in the 3D FEM modeling. 𝜎𝜎 and 𝐺𝐺𝐼𝐼 are calculated based on data reported in [15] . 𝜅𝜅𝑌𝑌𝐼𝐼𝑌𝑌 is taken from [19] and 𝜅𝜅𝑃𝑃𝑃𝑃 is from [20] . 𝑇𝑇(K) 𝜎𝜎(JmV⁄ ) 𝐺𝐺𝐼𝐼(Sm2⁄ ) 𝜅𝜅𝑌𝑌𝐼𝐼𝑌𝑌 (WmK)⁄ 𝜅𝜅𝑃𝑃𝑃𝑃 (WmK)⁄ 10 3.10×10−8 5.84×1010 60.00 1214.98 70 8.32×10−8 1.08×1012 37.59 91.82 175 1.32×10−7 4.27×1012 11.41 75.56 300 1.73×10−7 9.60×1012 6.92 73.01 The decay profile for the interfacial spin current 𝑗𝑗𝑚𝑚,𝑧𝑧 is obtained by using the calculated temperature profile as an input in Eq. (3) . We report the total interfacial spin current that reaches the detector 𝑗𝑗𝑚𝑚,𝑧𝑧 by evaluating the surface integral ∬𝑗𝑗𝑚𝑚,𝑧𝑧(𝑥𝑥,𝑦𝑦)𝑑𝑑𝐴𝐴 beneath the detector. The decay profile is calculated as a function of simulated laser position , at multiple different temperatures, 10 ranging from 5 – 300 K. The values of the physical parameters used in the model are recorded in Table I. From Eq. (3) one can see that 𝒋𝒋𝑚𝑚,𝑧𝑧 can be broken up into two components 𝑗𝑗𝑚𝑚,𝑧𝑧 ∇𝜇𝜇, which is a component that is proportional to the interfacial gradient of the magnon chemical potential , and 𝑗𝑗𝑚𝑚,𝑧𝑧 ∇𝑇𝑇, which is a component that is proportional to the interfacial gradient of the magnon temperature. The decomposition of the simulated spin current at the detector is shown in Fig. 4(a) , which depicts a representative plot of the total 𝒋𝒋𝑚𝑚,𝑧𝑧 as a funct ion of 𝑑𝑑 at 70 K , as well as the components 𝑗𝑗𝑚𝑚,𝑧𝑧 ∇𝜇𝜇 and 𝑗𝑗𝑚𝑚,𝑧𝑧 ∇𝑇𝑇 . By analyzing the decay lengths of these individual components of 𝒋𝒋𝑚𝑚,𝑧𝑧 separately , it is possible to qualitatively understand the existence of the experimentally observed short and long range decay lengths . As shown in Fig. 4(a), the component of 𝒋𝒋𝑚𝑚,𝑧𝑧 that is proportional to ∇ 𝜇𝜇 decays much more rapidly than the component of 𝒋𝒋𝑚𝑚,𝑧𝑧 that is proportional to ∇ 𝑇𝑇. This indicates that the total spin current that reaches the Pt det ector should consist of a short er decay component and a long er decay component. We hypothesize that the driving force of the short er range component is the gradient of the magnon chemical potential, ∇𝜇𝜇 and that the driving force of the long er range component is the gradient of the magnon temperature ∇𝑇𝑇. To verify this conjecture, the plot of the simulated 𝒋𝒋𝑚𝑚,𝑧𝑧 vs. 𝑑𝑑 is divided into the same 𝜆𝜆 1 and 𝜆𝜆2 regions as in the opto- thermal experimental measurement (where the 𝜆𝜆2 region is defined as 𝑑𝑑 > 37.5 𝜇𝜇m). Equation ( 1) is fit independently to the simulated 𝑗𝑗𝑚𝑚,𝑧𝑧 ∇𝜇𝜇 within the 𝜆𝜆 1 region , where the short er range driving force is expected to dominate , and to the simulated 𝑗𝑗𝑚𝑚,𝑧𝑧 ∇𝑇𝑇 with in the 𝜆𝜆 2 region where the long er range driving force will be most prevalent , as shown in the representative 70 K plot in Fig. 4(a). The decay parameters of these fits, 𝜆𝜆∇𝜇𝜇∗ and 𝜆𝜆∇𝑇𝑇∗, are extracted and plotted as a function of temperature 11 in Fig. 4(b). The intrinsic spin diffusion length, 𝜆𝜆∇𝜇𝜇∗, is relatively constant as a function of temperature, implying that ∇𝜇𝜇 is responsible for the short er range spin current observed in the 𝜆𝜆1 region (Fig. 3) . On the other hand, the bulk generated magnon current, characterized by 𝜆𝜆∇𝑇𝑇∗, decays monotonically with temperature, in agreement with the observed long er decay in the 𝜆𝜆2 region (Fig. 3) , thus implying that ∇𝑇𝑇 is the driving force for the long range spin current . Since it is the temperature profile within YIG that determines 𝜆𝜆∇𝑇𝑇∗, it will vary with the thermal boundary conditions. This explains why the long range spin current manifests in bulk YIG at low temperatu re [9], but not in YIG/GGG thin films [7]. It should be noted that while the monotonic decay with temperature of the simulated 𝜆𝜆 ∇𝑇𝑇∗ agrees with the measured opto- thermal and electro -thermal long range decay in the λ2 region , the simulated magnitude of 𝜆𝜆∇𝑇𝑇∗ is smaller than the one obtained experimentally . This is attributed to uncertainties in the temperature dependence of the inputs to the FEM modeling, particularly of the magnon scattering time 𝜏𝜏 , which is used to calcul ate 𝜎𝜎𝑚𝑚. At low temperatures magnon relaxation FIG 4. 3D FEM modeling simulation of the opto-thermal measurement. (a) Dashed lines represent the total spin current (black), the component of spin current proportional to ∇𝜇𝜇 (green) and the component of spin current proportional to ∇𝑇𝑇 (pink). Solid lines represent individual exponential fits to the corresponding component of the spin current in each of the distinct 𝜆𝜆1 and 𝜆𝜆2 regions (blue and red respectively). (b) The magnon spin diffusion lengths 𝜆𝜆∇𝜇𝜇∗ and 𝜆𝜆∇T∗ extracted for each region are plotted as a function of temperature. 12 is primarily governed by magnon- phonon interactions that create or annihilate spin waves by magnetic disorder and 𝜏𝜏 ~ ℏ𝛼𝛼𝑌𝑌𝑘𝑘𝐵𝐵𝑇𝑇⁄ where 𝛼𝛼𝑌𝑌= 10−4 [16] . This leads to calculated values of 𝜎𝜎𝑚𝑚 that vary with experimental measurements by orders of magnitude [15] . Such discrepancies may be explained by recent works that attribute the primary contributors to the SSE as low -energy subthermal magnons [5,17] , however an analysis of the complete temperature dependence of effective magnon scattering time based on the spectral dependence of the dominant magnons involved in SSE is outside the scope of this work. Another source of uncertainty in the simulations is the role of spin sinking into the Pt absorbers (present in the opto- thermal measurements) on the spin current decay profile . To test this, identical simulations, as described above, are carried out but with the Pt absorber pads removed. The absorbers cause a decrease in 𝜆𝜆∇𝜇𝜇∗ of 1-2 µm , while the 𝜆𝜆∇𝑇𝑇∗ shows no significant change within the uncertainty. During the review of this paper, we became aware of a related paper discussing the role of intrinsic spin Seebeck in the nonlocal spin currents decay profile [18] . In conclusion, opto- thermal and electro -thermal measurements independently demonstrate the existence of a longer range magnon spin current at low temperatures persisting well beyond the intrinsic spin diffusion length. By representing the total magnon spin current by its individual components , one of which is proportional to the gradient in magnon chemical potential and the other of which is proportional to the gradient in magnon temperature, the driving force of the longer range magnon spin diffusion can be attributed to the gradient in magnon temperature , i.e. the intrinsic spin Seebeck effect . The authors thank B art van Wees, Ludo Cornel issen, Yaroslav Tserkovnyak and Benedetta Flebus for valuable discussions. This work was primarily supported by the Army Research Office MURI 13 W911NF -14-1-0016. J.J. acknowledges the Center for Emergent Materials at The Ohio State University, an NSF MRSEC (Award Number DMR -1420451), for providing partial funding for this research. The work at CIC nanoGUNE was supported by the Spanish MINECO (Project No. MAT2015- 65159- R) and by the Regional Council of Gipuzkoa (Project No. 100/16). J.M.G.- P. thanks the Spanish MINECO for a Ph.D. fellowship (Grant No. BES -2016- 077301). [1] K. Uchida, M. Ishida, T. Kikkawa, A. Kirihara, T. Murakami, and E. Saitoh, J. Phys. Condens. Matter 26, 343202 (2014). [2] A. Prakash, J. Brangham, F. Yang, and J. P. Heremans, Phys. Rev. B 94, 014427 (2016). [3] A. Kehlberger, U. Ritzmann, D. Hinzke, E.- J. Guo, J. Cramer, G. Jakob, M. C. Onbasli, D. H. Kim, C. A. Ross, M. B. Jungfleisch, B. Hillebrands, U. Nowak, and M. Kläui, Phys. Rev. Lett. 115, 096602 ( 2015). [4] E.-J. Guo, J. Cramer, A. Kehlberger, C. A. Ferguson, D. A. MacLaren, G. Jakob, and M. Kläui, Phys. Rev. X 6, 031012 (2016). [5] T. Kikkawa, K. Uchida, S. Daimon, Z. Qiu, Y. Shiomi, and E. Saitoh, Phys. Rev. B 92, 064413 (2015). [6] L. J. Cornelissen, J. Liu, R. A. Duine, J. B. Youssef, and B. J. van Wees, Nat Phys 11, 1022 (2015). [7] L. J. Cornelissen, J. Shan, and B. J. van Wees, Phys. Rev. B 94, 180402 (2016). [8] J. Shan, L. J. Cornelissen, N . Vlietstra, J. Ben Youssef, T. Kuschel, R. A. Duine, and B. J. van Wees, Phys. Rev. B 94, 174437 (2016). [9] B. L. Giles, Z. Yang, J. S. Jamison, and R. C. Myers, Phys. Rev. B 92, 224415 (2015). [10] S. Vélez, A. Bedoya -Pinto, W. Yan, L. E. Hueso, and F. Casanova, Phys. Rev. B 94, 174405 (2016). [11] F. L. Bakker, A. Slachter, J.- P. Adam, and B. J. van Wees, Phys. Rev. Lett. 105, 136601 (2010). [12] S. R. Boona, R. C. Myers, and J. P. Heremans, Energy Environ. Sci. 7, 885 (2014). [13] T. Valet and A. Fert, Phys. Rev. B 48, 7099 (1993). [14] B. Flebus, S. A. Bender, Y. Tserkovnyak, and R. A. Duine, Phys. Rev. Lett. 116, 117201 (2016). [15] L. J. Cornelissen, K. J. H. Peters, G. E. W. Bauer, R. A. Duine, and B. J. van Wees, Phys. Rev. B 94, 014412 (2016). [16] S. Hoffman, K. Sato, and Y. Tserkovnyak, Phys. Rev. B 88, 064408 (2013). [17] I. Diniz and A. T. Costa, New J. Phys. 18, 052002 (2016). [18] J. Shan, L. J. Cornelissen, J. Liu, J. B. Youssef, L. Lian g, and B. J. van Wees, ArXiv170906321 Cond- Mat (2017). [19] S. R. Boona and J. P. Heremans, Phys. Rev. B 90, 064421 (2014). [20] J. E. Jensen, W. A. Tuttle, H. Brechnam, and A. G. Prodell, Brookhaven National Laboratory Selected Cryogenic Data Notebook (Brookhaven National Laboratory, New York, 1980).

2017-08-06

The longitudinal spin Seebeck effect refers to the generation of a spin current when heat flows across a normal metal/magnetic insulator interface. Until recently, most explanations of the spin Seebeck effect use the interfacial temperature difference as the conversion mechanism between heat and spin fluxes. However, recent theoretical and experimental works claim that a magnon spin current is generated in the bulk of a magnetic insulator even in the absence of an interface. This is the so-called intrinsic spin Seebeck effect. Here, by utilizing a non-local spin Seebeck geometry, we provide additional evidence that the total magnon spin current in the ferrimagnetic insulator yttrium iron garnet (YIG) actually contains two distinct terms: one proportional to the gradient in the magnon chemical potential (pure magnon spin diffusion), and a second proportional to the gradient in magnon temperature ($\nabla T_m$). We observe two characteristic decay lengths for magnon spin currents in YIG with distinct temperature dependences: a temperature independent decay length of ~ 10 ${\mu}$m consistent with earlier measurements of pure ($\nabla T_m = 0$) magnon spin diffusion, and a longer decay length ranging from about 20 ${\mu}$m around 250 K and exceeding 80 ${\mu}$m at 10 K. The coupled spin-heat transport processes are modeled using a finite element method revealing that the longer range magnon spin current is attributable to the intrinsic spin Seebeck effect ($\nabla T_m \neq 0$), whose length scale increases at lower temperatures in agreement with our experimental data.

Thermally Driven Long Range Magnon Spin Currents in Yttrium Iron Garnet due to Intrinsic Spin Seebeck Effect

1708.01941v3

arXiv:1603.03578v2 [cond-mat.str-el] 11 Apr 2016Investigation of anomalous-Hall and spin-Hall effects of an tiferromagnetic IrMn sandwiched by Pt and YIG layers T. Shang,1,a)H. L. Yang,1Q. F. Zhan,1,b)Z. H. Zuo,1Y. L. Xie,1L. P. Liu,1S. L. Zhang,1Y. Zhang,1H. H. Li,1B. M. Wang,1Y. H. Wu,2S. Zhang,3,c)and Run-Wei Li1,d) 1)Key Laboratory of Magnetic Materials and Devices &Zhejiang Province Key Laboratory of Magnetic Materials and Application Technology, Ningbo Institute of Material Tech nology and Engineering, Chinese Academy of Sciences, Ningbo, Zhejiang 315201, China 2)Department of Electrical and Computer Engineering, Nation al University of Singapore, 4 Engineering Drive 3 117583, Singapore 3)Department of Physics, University of Arizona, Tucson, Ariz ona 85721, USA (Dated: 14 June 2021) We report an investigation of temperature and IrMn layered thickn ess dependence of anomalous-Hall resistance (AHR), anisotropic magnetoresistance (AMR), and ma gnetization on Pt/Ir 20Mn80/Y3Fe5O12 (Pt/IrMn/YIG) heterostructures. The magnitude of AHR is dram atically enhanced compared with Pt/YIG bilayers. The enhancement is much more profound at higher temper atures and peaks at the IrMn thickness of 3 nm. The observed spin-Hall magnetoresistance (SMR) in the te mperature range of 10-300 K indicates that the spin current generated in the Pt layer can penetrate the entire thickness of the IrMn layer to interact with the YIG layer. The lack of conventional anisotropic magnetore sistance (CAMR) implies that the inser- tion of the IrMn layer between Pt and YIG efficiently suppresses the magnetic proximity effect (MPE) on induced Pt moments by YIG. Our results suggest that the dual role s of the IrMn insertion in Pt/IrMn/YIG heterostructures are to block the MPE and to transport the spin current between Pt and YIG layers. We discuss possible mechanisms for the enhanced AHR. I. INTRODUCTION Antiferromagnts (AFMs) are promising candidates for spintronic applications.1Compared to ferromagnetic (FM) materials, the AFMs exhibit unique advantages, e.g., zero net magnetization, insensitivity to the exter- nal magnetic perturbation, lack of stray field, and ac- cess to extremely high frequency. Recently, the gener- ation and transmission of spin current in AFMs have attracted great attention. The spin pumping studies on (Pt, Ta)/(NiO, CoO)/Y 3Fe5O12(YIG) heterostruc- turesdemonstratethatthespincurrentgeneratedinYIG layer can pass through the antiferromagnetic (AFM) in- sulator NiO or CoO layer and can be detected in Pt or Ta layer by inverse spin-Hall effect (ISHE).2–5Sim- ilar results were also revealed in (Pt, Ta)/IrMn/CoFeB or Pt/NiO/FeNi heterostructures by spin-torque ferro- magnetic resonance(ST-FMR) technique, where the spin current generated by spin-Hall effect (SHE) in Pt or Ta layer can propagate through IrMn or NiO layer and change the FMR linewidth.6–8The spin current gen- erated by spin pumping or spin Seebeck was also ob- served in IrMn/YIG, Cr/YIG, and XMn/Py ( X= Fe, Pd, Ir, and Pt) bilayers through ISHE.9–13Moreover, the IrMn/YIG, Pt/Cr 2O3, and Pt/MnF 2exhibit spin- a)Present address: Swiss light source & Laboratory for Scient ific Developments and Novel Materials, Paul Scherrer Institut, CH- 5232 Villigen PSI, Switzerland b)Electronic mail: zhanqf@nimte.ac.cn c)Electronic mail: zhangshu@email.arizona.edu d)Electronic mail: runweili@nimte.ac.cnHall magnetoresistance (SMR) and large ISHE voltage, respectively, implying that the AFMs can be both spin- current detector and generator.14–16These investigations open up new opportunities in developing the AFMs- based spin-current devices. The IrMn alloy, which have been widely used to pin an adjacent FM layer in spin valve devices via exchange bias,17demonstrates large ISHE voltage when in con- tacts with YIG.9Recently, a large SHE and anomalous- Hall effect (AHE) have been theoretically proposed in Cr, FeMn, and IrMn AFMs owing to their large spin- orbit coupling (SOC) or Berry phase of the non-collinear spintextures.18–20Thesetheoreticalpredictionswerealso found to be valid for other cubic non-collinear AFMs, e.g., SnMn 3and GeMn 3, where the calculations have beenrepeatedwithcomparableresults.21Theexperimen- tal investigation of AHE and SHE on the AFMs could be helpful from both fundamental and practical viewpoints for AFMs spintronics. As previously revealed in Cr/YIG bilayers, the large anomalous-Hall resistance (AHR) in thin unprotected Cr film is likely caused by the surface FM Cr oxides.11Similar situation is expected in unpro- tected IrMn/YIG bilayers. Since the Pt/YIG bilayer is well studied,22in this study, we choose the Pt as cap layer to protect the IrMn from oxidation to investigate the AHE and SHE of IrMn by measuring the spin trans- port properties in Pt/IrMn/YIG heterostructures. II. EXPERIMENTAL DETAILS The Pt/IrMn/YIG heterostructures were prepared in a combined ultra-high vacuum (10−9Torr) pulsed laser/s50/s48 /s51/s48 /s52/s48 /s53/s48 /s54/s48 /s55/s48 /s56/s48/s40/s98/s41/s40/s52/s52/s52/s41 /s32/s32/s73/s110/s116/s101/s110/s115/s105/s116/s121/s32/s40/s97/s46/s117/s46/s41 /s40/s100/s101/s103/s114/s101/s101 /s41/s40/s50/s50/s50/s41 /s50/s48/s54/s48 /s50/s48/s56/s48 /s50/s49/s48/s48 /s50/s49/s50/s48 /s50/s49/s52/s48 /s50/s49/s54/s48 /s32/s32/s100/s73 /s70/s77/s82/s47/s100/s72/s32/s40/s97/s46/s117/s46/s41 /s70/s105/s101/s108/s100/s32/s40/s79/s101/s41/s72/s126/s56/s79/s101/s40/s100/s41/s52/s57/s46/s53 /s53/s48/s46/s48 /s53/s48/s46/s53 /s53/s49/s46/s48 /s53/s49/s46/s53 /s53/s50/s46/s48 /s53/s50/s46/s53/s40/s97/s41 /s71/s71/s71 /s32/s32/s73/s110/s116/s101/s110/s115/s105/s116/s121/s32/s40/s97/s46/s117/s46/s41 /s50 /s32 /s40/s100/s101/s103/s114/s101/s101 /s41/s89/s73/s71 /s45/s48/s46/s54/s110/s109/s80/s116/s47/s73/s114/s77 /s110/s47/s89/s73/s71 /s40/s99/s41/s48/s46/s54/s110/s109 /s53/s46/s48 /s109/s48/s46/s48 FIG. 1. (Color online) (a) A representative 2 θ-ωXRD pat- terns for YIG/GGG film near the (444) peaks of GGG sub- strate and YIG film. (b) The full range of XRD patterns from 20 to 80 degree. (c) Atomic force microscope surface topogra - phy of Pt/IrMn(3)/YIG heterostructure over an area of 5 µm ×5µm. (d) A FMR derivative absorption spectrum of a 60 nm YIG film with an in-plane magnetic field; the line-width is estimated to be 8 Oe. deposition (PLD) and magnetron sputter system. The high quality epitaxial YIG films were deposited on (111)- orientated single crystalline Gd 3Ga5O12(GGG) sub- strate via PLD technique as described elsewhere.23The Ir20Mn80(IrMn) and Pt films were sputtered at room temperature in argon atmosphere in an in situprocess. The thickness and crystal structure of films were char- acterized by Bruker D8 Discover high-resolution x-ray diffractometer (HRXRD). The thickness was estimated by using the software package LEPTOS (Bruker AXS). The surface topography of the films was measured in a Bruker Icon atomic force microscope. The ferromagnetic resonance (FMR) was measured by Bruker electron spin resonance spectrometers. The measurements of trans- verse Hall resistance, longitudinal resistance, and mag- netization were carried out in a Quantum Design physi- cal properties measurement system (PPMS) with a rota- tionoptionandmagneticpropertiesmeasurementsystem (MPMS), respectively. III. RESULTS AND DISCUSSION Figure 1(a) plots a representative room-temperature 2θ-ωXRD pattern of epitaxial YIG/GGG film near the (444)reflections. ClearLaueoscillationsindicatetheflat- ness and uniformity of the epitaxial YIG film. As shown in the Fig. 1(b), only the (222) and (444) reflections can FIG. 2. (Color online) (a)-(c) Schematic plot of longitudin al resistance and transverse Hall resistance measurements. T he magnetic fields are applied in the xy,xz, andyzplanes with anglesθxy,θxz, andθyzrelative to the y-,z-, andz-axes. The electric current is applied along the x-axis. Anomalous- Hall resistance R AHRfor Pt/YIG (d) and Pt/IrMn(1)/YIG (e) as a function of magnetic field at different temperatures. (f) Temperature dependence of the ρAHRfor Pt/IrMn/YIG with various IrMn thicknesses. The ρAHRare replotted as a function of IrMn thickness at various temperatures in (g). A ll ρAHRare averaged by[ ρAHR(70 kOe)- ρAHR(-70 kOe)]/2. The error bars are the results of subtracting OHR in different fiel d ranges be observed, and no indication of impurities or misorien- tation was detected in the full range of 2 θ-ωscan. In this study, the thicknesses of YIG and Pt films, determined by simulation of the x-rayreflectivity (XRR) spectra, are approximately 60 nm and 3 nm, respectively, while the IrMn thickness ranges from 0 nm to 8 nm. The atomic force microscope surface topography of Pt/IrMn(3)/YIG heterostructure over an area of 5 µm×5µm in Fig. 1(c) reveals a root-mean-squaresurface roughness of 0.18 nm, indicating atomical flat of prepared films. The other films show similar surface roughness. The number in the brackets represents the thickness of IrMn layer in nm unit. A representative FMR derivative absorption spec- trum of YIG film (60 nm) shown in Fig. 1(d) exhibits a line width ∆H = 8 Oe, which was measured at radio frequency 9.39 GHz and power 0.1 mW with an in-plane magnetic field at room temperature. The above proper- ties indicate excellent quality of our prepared films. 2/s48 /s49/s48/s48 /s50/s48/s48 /s51/s48/s48/s48/s50/s52/s54/s56 /s48 /s49/s48/s48 /s50/s48/s48 /s51/s48/s48/s48/s51/s48/s54/s48/s57/s48 /s40/s103/s41/s32/s80/s116/s47/s73/s114/s77/s110/s47/s89/s73/s71/s48/s32/s40/s49/s48/s45/s53 /s41 /s84/s32/s40/s75/s41 /s84/s32/s40/s75/s41 /s47 /s48/s40/s49/s48/s45/s53 /s41 /s120 /s121 /s120 /s122 /s121/s122 /s40/s104/s41/s32/s80/s116/s47/s89/s73/s71/s52/s50/s48/s45/s50/s45/s52/s48/s51/s54/s57 /s48 /s57/s48 /s49/s56/s48 /s50/s55/s48 /s51/s54/s48/s45/s57/s45/s54/s45/s51/s48/s48/s50/s48/s52/s48/s54/s48 /s45/s50/s48/s45/s49/s48/s48 /s48 /s57/s48 /s49/s56/s48 /s50/s55/s48 /s51/s54/s48/s45/s54/s48/s45/s51/s48/s48/s40/s98/s41/s48/s32/s40/s49/s48/s45/s53 /s41/s80/s116/s47/s73/s114/s77/s110/s47/s89/s73/s71 /s32/s49/s48/s75 /s32/s53/s48 /s32/s49/s48/s48 /s32/s49/s53/s48 /s32/s50/s48/s48 /s32/s50/s53/s48 /s32/s51/s48/s48/s40/s97/s41 /s40/s99/s41 /s40/s100/s101/s103/s114/s101/s101/s41/s80/s116/s47/s89/s73/s71/s40/s100/s41 /s40/s101/s41 /s47 /s48/s40/s49/s48/s45/s53 /s41 /s40/s102/s41 /s32/s40/s100/s101/s103/s114/s101/s101/s41 FIG. 3. (Color online) Anisotropic magnetoresistance for Pt/IrMn(1)/YIG at various temperatures down to 10 K with the magnetic field varied within xy(a),xz(b), and yz(c) planes. The results of Pt/YIG are shown in (d)-(f). Temper- ature dependence of AMR amplitudes for Pt/IrMn(1)/YIG (g) and Pt/YIG (h) heterostructures. The cubic, circle and triangle symbols standfor the θxy,θxz,θyzscans, respectively. A. Anomalous-Hall resistance As shown in the top panel of Fig. 2, in order to measure the transverse Hall resistance and longitudinal resistance, all the Pt/IrMn/YIG heterostructures were patterned into Hall-bar configuration (central area: 0.3 mm×10 mm; electrode 0.3 mm ×1 mm). The trans- verse Hall resistance R xyof Pt/IrMn/YIG was measured in the temperature range of 10 K to 300 K with per- pendicular magnetic field ranging from -70 to 70 kOe. In metal thin film, the ordinary-Hall resistance (OHR) ROHRis subtracted from the measured R xy, i.e., R AHR = Rxy- ROHR×µ0H, where R AHRis AHR. As shown in Figs. 2(d)-(e), the resulting R AHRas a function of magnetic field for Pt/YIG and Pt/IrMn(1)/YIG are pre- sented. It is noted that the Pt becomesmagnetic when in contacts with the YIG due to its proximity to the stoner ferromagnetic instability, i.e., magnetic proximity effect (MPE), as previously shown experimentally by x-ray magnetic circular dichroism (XMCD) and theoretically by first-principles calculation.24,25The magnetized Pt shares some common features as magnetic YIG film, i.e., strong anisotropy.23Thus, when the magnetic field ap- proaches zero, the magnetized Pt moments are randomlydistributed, the R AHRexhibits irregular M-shaped be- haviorclosetozerofield. However,forPt/IrMn/YIG,the RAHRcontinuously decreases as approaching zero field, implying that the Pt/IrMn and IrMn/YIG interfaces are freeofMPE,beingconsistentwith theabsenceofconven- tional anisotropic magnetoresistance (CAMR) (see be- low). We summarize the derived anomalous-Hall resis- tivityρAHRofPt/IrMn/YIGheterostruturesasfunctions of temperature ( T) and IrMn thickness ( tIrMn) in Figs. 2(f)-(g). The ρAHR(T) for all the Pt/IrMn/YIG exhibits rich characteristics whose magnitude and sign are highly non-trivial, which were also found in Pt/LaCoO 3bilay- ers.26. As shown in Fig. 2(h), the magnitude of ρAHR decrease with temperature and then it increases again below 100 K. Simultaneously, the ρAHRchange its sign at the temperature which is independent of IrMn thick- ness. We also replotted all the ρAHRas a function of IrMn thickness in Fig. 2(g). In the studied temperature range, as increasing the tIrMn, theρAHRalso increases and reaches a maximum around tIrMn= 3 nm, which excludes the interfacial origin of the observed AHR. B. Spin-Hall magnetoresistance The anisotropic magnetoresistance (AMR) for Pt/IrMn/YIG was also measured down to low temper- atures. As an example, the AMR of Pt/IrMn(1)/YIG and Pt/YIG for three different field scans are presented in top panel of Fig. 3. When the magnetic field scans within the xyplane [Fig. 3(a)(d)], both the CAMR and SMR contribute to the total AMR; for the xzplane [Fig. 3(b)(e)], the resistance changes are attributed to the MPE-induced CAMR; for the yzplane [Fig. 3(c)(f)], the CAMR is zero, and only the SMR are expected.29,30 As shown in Fig. 3(b), the θxzscan shows negligible AMR and the resistance is almost independent of θxz, indicating the extremely weak MPE at the interface even down to low temperatures. However, the MPE is significant at Pt/YIG interface [see Fig. 3(e)]: the max- imum amplitude of CAMR is around 2.2 ×10−4, which is comparable to the SMR. Thus, the IrMn can be used as clean spin current detector and generator, similar to the normal Rh or AFM Cr metals.11,23Since the CAMR is negligible in Pt/IrMn(1)/YIG, the SMR dominates the AMR when the magnetic field is varied within the xyplane, the amplitudes of θxyscan are almost identical toθyzscan. While for Pt/YIG, due to the MPE-induced CAMR, none of the amplitudes is identical to each other. The temperature dependence of the AMR amplitudes for allθxy,θxz, andθyzscans are summarized in Fig. 3(g) and Fig. 3(h) for Pt/IrMn(1)/YIG and Pt/YIG, respectively. Upon decreasing the temperature, the SMR persists down to 10 K, with the amplitudes monotonically decreasing from 7.5 ×10−5(300 K) to 3.0 ×10−5(10 K) in Pt/IrMn(1)/YIG. For Pt/IrMn/YIG, the amplitudes of SMR are almost an order smaller than that of the Pt/YIG due to the smaller spin-Hall angle, 3/s45/s49/s50/s48 /s45/s54/s48 /s48 /s54/s48 /s49/s50/s48/s45/s49/s46/s48/s45/s48/s46/s53/s48/s46/s48/s48/s46/s53/s49/s46/s48 /s48 /s49/s48/s48 /s50/s48/s48 /s51/s48/s48/s48/s45/s50/s48/s45/s52/s48/s45/s54/s48 /s48 /s49/s48/s48 /s50/s48/s48 /s51/s48/s48/s48/s49/s48/s50/s48/s51/s48/s52/s48/s32/s50/s75 /s32/s49/s48 /s32/s50/s48 /s32/s51/s48 /s32/s52/s48 /s32/s53/s48 /s32/s54/s48 /s32/s55/s48 /s32/s56/s48 /s32/s57/s48 /s32/s49/s48/s48 /s32/s49/s50/s48 /s32/s49/s53/s48 /s32/s49/s56/s48 /s32/s51/s48/s48/s77/s47/s77 /s115 /s70/s105/s101/s108/s100/s32/s40/s79/s101/s41/s40/s97/s41 /s32/s80/s116/s47/s73/s114/s77/s110/s40/s49/s32/s110/s109/s41/s47/s89/s73/s71 /s72 /s69/s32/s40/s79/s101/s41/s32/s49/s110/s109 /s32/s51 /s32/s53/s40/s98/s41 /s32 /s84/s32/s40/s75/s41/s72 /s67/s32/s40/s79/s101/s41 /s84/s32/s40/s75/s41 FIG. 4. (Color online) (a) Field dependence of normalized magnetization M/Msfor Pt/IrMn(1)/YIG at various temper- atures down to 2 K. The magnetic field is applied parallel to the film surface. The paramagnetic background of the GGG substrate has been subtracted. (b) The in-plane exchange bias field HEversus temperature. The arrows indicate the AFM block temperatures Tb. The inset plots the coercivity fieldHCas a function of temperature for Pt/IrMn(1)/YIG. shorter spin diffusion length, and larger electrical re- sistivity of IrMn.9,12,29The temperature characteristics of SMR amplitudes in Pt/IrMn/YIG are significantly different from the Pt/YIG or Pd/YIG bilayers, where the SMR amplitudes exhibit nonmonotonic tempera- ture dependence and acquire a maximum around 100 K.31,32For Pt/YIG, the temperature dependence of SMR amplitude can be described by a single spin- relaxation mechanism.31The spin diffusion length is defined as λ=/radicalbigDτsf, whereDandτsfare diffusion constant and spin-flip relaxation time, respectively. Within the Elliot-Yafet spin-orbit scattering model, bothDandτsfare proportional to the reciprocal of temperature dependence of the resistivity 1/ ρ(T).33,34 In Pt metal, the electrical resistivity mainly comes from phonon-electron scattering at high temperature, thenλ∝1/T. However, the extra magnetic electron scattering need to be considered in Pt/IrMn/YIG heterostructures, the assumption of λ∝1/Tis invalid. It is noted that the heterostructures with different IrMn thicknesses exhibit similar temperature dependent characteristics with different numerical values compared to the Pt/IrMn(1)/YIG heterostructure shown here. For example, the Pt/IrMn(3)/YIG exhibits the SMR amplitude of 6.8 ×10−5at room temperature. The sizable SMR observed in Pt/IrMn/YIG heterostructures indicates that the spin current can transport through IrMn layer.C. Magnetization Since the magnetic transitions of very thin AFMs are expected to be well below the ordering temper- ature of bulk forms, we measured the field depen- dence of magnetization down to low temperatures, from which we can track the AFM blocking temperature Tbfor Pt/IrMn/YIG heterostructures. As an exam- ple, the normalized magnetic hysteresis loops M/Msfor Pt/IrMn(1)/YIGatvarioustemperaturesafterfieldcool- ing from 300 K are presented in Fig. 4(a). The derived exchange bias field HEversus temperature are summa- rized in Fig. 4(b), from which the Tbare approximately estimated to be 150 K, 180 K, and 220 K for 1 nm, 3 nm and 5 nm IrMn, respectively, as the arrow indicated. Similar blocking temperatures were previously reported in IrMn/MgO/Ta tunnel junctions and IrMn/NiFe bi- layer.35,36Moreover, the coercivity HCalso exhibits a step-like increase near the blocking temperature, as the arrow shown in the inset of Fig. 4(b), indicating the strongly enhanced exchange coupling between IrMn and YIG layer below Tb. D. Discussion Based on the above experimental results, we discuss the origins of the significant AHR in Pt/IrMn/YIG het- erostructures and the effect of AFM order on spin trans- port properties. There are at least four contributions to the observed AHR in Pt/IrMn/YIG: MPE, spin-Hall based SMR, spin-dependent interface scattering, and in- trinsic properties of IrMn metal. In contrast to the Pt/YIG, the negligible CAMR in Pt/IrMn/YIG indi- cates the extremely weak MPE at Pt/IrMn or IrMn/YIG interfaces, which is different from the previous studied of IrMn/YIG bilayer.14The SMR model based on SHE also predicts an anomalous-Hall-like resistance,29whose magnitude and sign are determined by the spin diffu- sion length and spin-Hall angle of the metal and the imaginary part of the spin mixing conductance, respec- tively. Though the thickness dependence of the AHR in Pt/IrMn/YIG can be described by the SMR model, it fails to explain the AHR by the following reasons: (i) An arbitrary temperature dependence of the imag- inary part of the spin mixing conductance parameter is required to qualitatively describe the temperature- dependent AHR data, i.e., signreversal; (ii) Accordingto the spin pumping studies, both the spin-Hall angle and the spin diffusion length of IrMn are smaller than Pt, which cannot explain the enhancement of AHR by in- creasing the IrMn thickness.9,12,13Spin-dependent scat- tering at the interface, combined with the conventional skew-scatteringand side-jump mechanisms, can also give rise to AHR.37Again, the enhancement of AHR by in- creasing the IrMn thickness excludes the interfacial ori- gin. Finally, the theoretical calculations predict a large AHE and SHE in IrMn metal not only attributed to the 4large SOC of heavy Ir atoms which is transferred to the magnetic Mn atoms by hybridization effect but also the Berry phase of the non-collinear spin structures.18–20We conclude that the large AHR observed in Pt/IrMn/YIG is likely associated with SOC and non-collinear magnetic structure of IrMn. However, the non-trivial temperature dependence of AHR demands further theoretical and ex- perimental investigations. Now we discuss the possible interplay between AFM order and spin transport properties. As shown in Fig. 2 and Fig. 3, there is no clear anomalous in AHR or SMR near the blocking temperatures of IrMn, imply- ing weak correlations between the AHE or SHE and the AFM order in IrMn. Similar results were also observed in Cr/YIG bilayers, where the ISHE voltage and AHR is also independent of AFM ordering temperature.11Ac- cording to our magnetization results (Fig. 4), the AFM ordering temperatures of our IrMn films are well below room temperature. However, the enhancement of AHR in Pt/IrMn/YIG happens in the whole studied tempera- ture range [see Fig. 2(g)]. There are two possible reasons for this phenomenon, one is that the AHE and SHE at- tributed to non-collinear magnetism is generated on a length scale of nanometer and is a local property not relying on long range magnetic order, i.e., regardless of how IrMn grains are orientated, as reported previously in Mn 5Si3film.38The second one is that the strength of SOC is independent of AFM order in IrMn metal, which is mainly determined by the Ir atoms. IV. CONCLUSIONS In summary, we report an investigation of AHE and SHE by measuring the AHR and SMR in Pt/IrMn/YIG heterostrucutres. The significant AHR in Pt/IrMn/YIG islikelyassociatedwiththestrongSOCandnon-collinear magnetic structure of IrMn, and the sizable SMR in- dicates that the spin current can transport through IrMn. The observed non-trivial temperature dependence of AHR cannot be consistently explained by the existing theories, further investigations are needed to clarify this issue. Moreover, both the AHR and SMR are uncoupled to the AFM order of IrMn metal. The negligible MPE at Pt/IrMn or IrMn/YIG interface and large ISHE indi- cate that IrMn can be another model system to explore physics and devices associated with antiferromagnetism and pure spin current. ACKNOWLEDGMENTS We thank the high magnetic field laboratory of Chi- nese Academy of Sciences for the FMR measurements. This work is financially supported by the National Nat- ural Science foundation of China (Grants No. 11274321, No. 11404349, No. 51502314, No. 51522105) and the Key Research Program of the Chinese Academy of Sci-ences (Grant No. KJZD-EW-M05). S. Zhang was par- tiallysupportedbytheU.S.NationalScienceFoundation (Grant No. ECCS-1404542). 1A. H. MacDonald and M. Tsoi, Phil. Trans. R. Soc. A 369, 3098 (2011). 2C. Hahn, G. de. Loubens, O. Klein, M. Viret, V. V. Naletov, and J. B. Youssef, Europhys. Lett. 108, 57005 (2014). 3H. L. Wang, C. H. Du, P. C. Hammel, and F. Y. Yang, Phys. Rev. Lett. 113, 097202 (2014). 4Z. Qiu, J. Li, D. Hou, E. Arenholz, A. T. NDiaye, A. Tan, K. Uchida, K. Sato, Y. Tserkovnyak, Z. Q. Qiu, and E. Saitoh, arXiv: 1505.03926. 5W. Lin, K. Chen, S. Zhang, and C. L. Chien, arXiv: 1603.00931. 6T. Moriyama, M. Nagata, K. Tanaka, K-J. Kim, H. Almasi, W. Wang, T. Ono, arXiv: 1411. 4100. 7H. Reichlov´ a, D. Kriegner, V. Hol´ y, K. Olejn´ ık, V. Nov´ ak , M. Yamada, K. 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Loehneysen, arXiv : 1601.01840. 6

2016-03-11

We report an investigation of temperature and IrMn layered thickness dependence of anomalous-Hall resistance (AHR), anisotropic magnetoresistance (AMR), and magnetization on Pt/Ir20Mn80/Y3Fe5O12 (Pt/IrMn/YIG) heterostructures. The magnitude of AHR is dramatically enhanced compared with Pt/YIG bilayers. The enhancement is much more profound at higher temperatures and peaks at the IrMn thickness of 3 nm. The observed spin-Hall magnetoresistance (SMR) in the temperature range of 10-300 K indicates that the spin current generated in the Pt layer can penetrate the entire thickness of the IrMn layer to interact with the YIG layer. The lack of conventional anisotropic magnetoresistance (CAMR) implies that the insertion of the IrMn layer between Pt and YIG efficiently suppresses the magnetic proximity effect (MPE) on induced Pt moments by YIG. Our results suggest that the dual roles of the InMn insertion in Pt/IrMn/YIG heterostructures are to block the MPE and to transport the spin current between Pt and YIG layers. We discuss possible mechanisms for the enhanced AHR.

Investigation of anomalous-Hall and spin-Hall effects of antiferromagnetic IrMn sandwiched by Pt and YIG layers

1603.03578v2

EPJ manuscript No. (will be inserted by the editor) Building instructions for a ferromagnetic axion haloscope Nicol o Crescini1 Univ. Grenoble Alpes, CNRS, Grenoble INP, Institut Nel, 38000 Grenoble, France Received: date / Revised version: date Abstract. A ferromagnetic haloscope is a rf spin-magnetometer used for searching Dark Matter in the form of axions. A magnetic material is monitored searching for anomalous magnetization oscillations which can be induced by dark matter axions. To properly devise such instrument one rst needs to understand the features of the searched-for signal, namely the eective rf eld of dark matter axions Baacting on electronic spins. Once the properties of Baare dened, the design and test of the apparatus may start. The optimal sample is a narrow linewidth and high spin-density material such as Yttrium Iron Garnet (YIG), coupled to a microwave cavity with almost matched linewidth to collect the signal. The power in the resonator is collected with an antenna and amplied with a Josephson Parametric amplier, a quantum-limited device which, however, adds most of the setup noise. The signal is further amplied with low noise HEMT and down-converted for storage with an heterodyne receiver. This work describes how to build such apparatus, with all the experimental details, the main issues one might face, and some solutions. 1 Introduction The axion is an hypothetical beyond the Standard Model particle, rst introduced in the seventies as a consequence of the strong CP problem of QCD. Axions can be the main constituents of the galactic Dark Matter halos. Their experimental search can be carried out with Earth-based instruments immersed in the Milky Way's halo, which are therefore called \haloscopes". Nowadays haloscopes rely on the inverse Primako eect to detect axion-induced excesses of photons in a microwave cavity under a static magnetic eld. This work describes the process leading to the successful operation of a ferromagnetic axion haloscope, which does not exploit the axion-to-photon conversion but its interaction with the electron spin. The study of the axion-spin interaction and of the Dark Matter halo properties yields the features of the axionic signal, and is fundamental to devise a proper detector. A scheme of a realistic ferromagnetic haloscope is drawn to realize the challenges of its development. It emerges that there are a number of requirements for a this setup to get to the sensitivity needed for a QCD-axion search. These are kept in mind when designing the prototypes, to overcome the problems without compromising other requirements. A state-of-the-art sensitivity to rf signals allows for the detection of extremely weak signals as the axionic one. The number of monitored spins is necessarily large to increase the exposure of the setup, thus its scalability is a key part of the design process. A ferromagnetic haloscope consists in a transducer of the axionic signal, which is then measured by a suitable detector. The transducer is a hybrid system formed by a magnetic material coupled to a microwave cavity through a static magnetic eld. Its two parts are separately studied to nd the materials which match the detection conditions imposed by the axion-signal. The detector is an amplier, an HEMT or a JPA, reading out the power from the hybrid system collected by an antenna coupled to the cavity. A particular attention is given to the measurement of the noise temperature of the amplier. As it measures variation in the magnetization of the sample, the ferromagnetic haloscope is congured as a spin-magnetometer. The present haloscope prototype [1] works at 90 mK and reaches the sensitivity limit imposed by quantum mechan- ics, the Standard Quantum Limit, and can be improved only by quantum technologies like single photon counters. The haloscope embodies a large quantity of magnetic material, i. e. ten 2 mm YIG spheres, and is designed to be further up-scaled. This experimental apparatus meets the expected performances, and, to present knowledge, is the most sensitive rf spin-magnetometer existing. The minimum detectable eld at 10.3 GHz results in 5 :51019T for 8 h integration, and corresponds to a limit on the axion-electron coupling constant gaee1:71011. This result is the best limit on the DM-axions coupling to electron spins in a frequency span of about 150 MHz, corresponding to an axion mass range from 42 :4eV to 43:1eV. The eorts to enhance the haloscope sensitivity include improvements in both the hybrid system and the detector. The deposited axion power can be increased by means of a larger material volume, possibly with a narrower linewidth.arXiv:2201.04081v1 [hep-ex] 11 Jan 2022#### Page 2 of 18 Eur. Phys. J. Plus ##################### To overcome the standard quantum limit of linear ampliers one must rely on quantum counters. Novel studies on microwave photon counters, together with some preliminary results, are reported. Other possible usages of the spin- magnetometer are eventually discussed. 2 Overview on axions A long-standing puzzle of beyond the Standard Model physics consists in the dark matter (DM) problem. In 1933 Fritz Zwicky used two dierent techniques to estimate the mass of the Coma and Virgo clusters, one was based on the luminosity of the galaxies in the clusters, while the other used the velocity dispersion of individual galaxies. These two independent estimations did not agree by orders of magnitude [2]. It is only in the seventies that this discrepancy started to be studied systematically. In particular, Vera Rubin studied the rotation curves of spiral galaxies and observed a violation of the second Kepler's law which can be explained assuming that the mass prole does not vanish beyond the stars [3,4]. This was an early indication that spiral galaxies could be surrounded by an halo of DM. Despite these evidences, the nature of DM is still unknown. Its possible composition could be baryonic or non-baryonic. The former considers matter similar to the one already known, while the latter comprehends hypothetical particles of beyond the Standard Model (BSM) physics. The case of a non-baryonic DM is where cosmology meets particle physics. Approaching this problem, physicists glimpse the possibility of merging dierent questions which are apparently uncorrelated. New theories, remarkably su- persymmetric DM [5], triggered experimental searches in dierent forms and with various techniques. Low-background laboratory experiments aim at a direct detection [6], accelerators could produce such particles and observe their miss- ing energy and momentum [7], while indirect evidences are based on their decay or annihilation [8]. The detection of BSM particles would shed light on fundamental question like DM or the unication of all forces. Up to now the results of the LHC and experiments therein showed no evidence of new physics up to the 10 TeV scale. On the other hand, there are signicant hints for physics at the sub-eV scale, like neutrino oscillation or the vacuum energy density of the Universe [9]. The physics case of weakly interacting sub-eV particles (WISPs) is motivated by the fact that any theory introducing a high-energy global symmetry breaking implies a light particle by the Nambu-Goldstone theorem [10]. Among other WISPs, the axion appears as a well-motivated BSM particle. Originally introduced to account for a ne-tuning issue in the SM known as the \strong CP problem" of quantum chromodynamics (QCD), it quickly became a prominent DM candidate. The existence of axions is a very attractive perspective, since its addition to the Standard Model would solve two major problems of modern physics in a single shot [11]. QCD is a non-Abelian SU(3) cgauge theory which describes the strong interactions. Its Lagrangian LQCD contains a CP-violating term which is compatible with all symmetries of the SM gauge group. However, since there is no experimental sign of CP violation in strong interactions, one needs to unnaturally suppress of this term. The rst SU(3) ctheory was proposed as CP-conserving to agree with experimental observations, but had an issue at low energy known as Weinberg's U(1) Amissing meson problem. The strong CP problem arises following the solution of the missing meson problem proposed by t'Hooft [12, 13]. This solution brings on CP violation in QCD, parametrized by =QCD+2Y, two angles relative to QCD which, according to the theory, are independent. However, to conserve CP either should be zero or one of the quarks should be massless. Among the measurable observables containing and thus CP violation, the neutron electric dipole moment results [14,15,16]. Recent results by the nEDM collaboration constrain the parameter even more 1010[17,18,19]. Since it is unlikely that nature chose very small YandQCD or a ne tuning among them, a naturalness problem arises. It is normally denoted as the strong CP problem, which is an important hint of BSM physics. A solution to this problem is a scenario where is promoted from a parameter to an actual particle. This was realized, albeit in a dierent way, by Peccei and Quinn [20], who introduced a new U(1) PQsymmetry to the SM to dynamically interpret . The idea was further developed by, among others, Weinberg and Wilczek [21,22,23,24,25], who realized that it implies the existence of a new light pseudo-Nambu Goldstone boson which was called axion. The minimization of the meson potential adjusts the axion vacuum expectation value to cancel any eect of CP violation, addressing the strong CP problem. The axion mass eigenstate can then be computed from the masses of the pion m, of the up and down quarks, mu andmd, and from the decay constants of the pion and of the axion itself, fandfa, resulting in the axion mass m2 a'=mumd (mu+md)2m2 f2  f2a: (1) The energy scale fais the PQ-symmetry breaking scale, and as the axion is the pseudo-Goldstone boson arising from this process, its mass and couplings are proportional to f1 amaking it very light and weakly-interacting. The so-called \invisible-axion models" consider fa'1012GeV, and evade current experimental limits [26]. There are two main classes of invisible axions whose archetype are the Kim-Shifman-Vainshtein-Zakharov (KSVZ) and Dine-Fischler- Srednicki-Zhitnitsky (DFSZ) models [27,28,25,23,29], where the main dierence is the coupling to SM particles.Eur. Phys. J. Plus ##################### #### Page 3 of 18 It is now possible to analyze the axion as a constituent of DM. Cold DM particles must be present in the Universe in a sucient quantity to account for the observed DM abundance and they have to be eectively collisionless, i. e. to have only signicant long-range gravitational interactions. The axion satises both these criteria. Even if it is very light, the axion population is non-relativistic since it is produced out of equilibrium by vacuum realignment, string decay or domain wall decay [30,31,29,32,33,34,35,36]. Being the main cold axions production mechanism, vacuum realignment is basically explained hereafter. In a time when the Universe cools down to a temperature lower than the axion mass, the axion eld is sitting in a random point of its potential, and not necessarily at its minimum. As a consequence the eld starts to oscillate and, since the axion has extremely weak couplings, it has no way to dissipate its energy. This relic energy density is a form of cold Dark Matter [37,38]. Lattice QCD calculation can be used together with the present Dark Matter density to give an estimation of the QCD axion mass [39,40,41,42,43,44] The axion has to be framed in the context of present physical theories, since one can wonder if the presence of light scalars may in uence the behavior of already studied physical systems. Several constraints come from tting the axion theory into astrophysical and cosmological observations. As other weakly interacting low-mass particles, they can contribute to the cooling of stars and be produced in astrophysical plasmas and in the Sun, contribute to stellar evolution and even aect supernovae [45,46,47,48,49,50,51,52,53,54,55,56,57]. To sum up, these observations suggest that ma10 meV. Cosmology provides both upped and lower limits for the axion mass, but being the upper ones weaker than the ones already described, the focus will be on lower bounds. These limits on the axion mass come from the production of DM-axions in the early Universe [30,58,59,60,61,62,63,64,65,66]. In particular the axion mass must be higher than 6 eV to avoid the overclosure problem, i. e. an axion density exceeding the observed DM density. Lighter masses are still possible within the so-called \anthropic axion window". A general case of BSM particles are the so-called \axion-like particles" (ALPs). The interest in ALPs relies on the fact that its mass and coupling constants can be unrelated (other than for axions), thus they do not necessarily solve the strong-CP problem but still can account for the whole DM density of the Universe [67]. Any experimental search not reaching the axion sensitivity is still a probe of ALPs. 2.1 Experimental searches In the last decades several experimental techniques have been proposed to detect axions and ALPs [9,68]. Most experiments do not reach the axion-required sensitivity, but the physics result of these measurements is to limit the ALPs parameter space. The most tested eects of axions are related to their coupling to photons, being this one the strongest and thus most accessible parameter. These limits mostly rely on the inverse Primako eect: in a strong static magnetic eld it is possible to convert an itinerant axion into a photon that can be detected. Amongst all the experiment proposed or realised to detect axions, only haloscopes are treated in some details. As already discussed axions may constitute DM, and if existing at least a fraction of DM have to be composed of axions. DM is an interesting source of axions and triggered multiple experimental searches. Instruments searching for DM-axions composing the Milky Way's halo are called haloscopes. Haloscopes are particularly interesting in the scope of this work, which is devoted to the study of a ferromagnetic one. In 1983, Sikivie proposed new ways to detect the axion by resonantly converting them into microwave photons inside a high quality factor ( Q) cavity under a static magnetic eld [69]. The resonance condition implies that the apparatus is sensitive to axions in a very narrow frequency range. The frequency of the axion signal is related to its mass and its width depends on the virial DM velocities in the galaxy. These kinds of experiments need to change resonant frequency to scan for dierent masses. The Axion Dark Matter eXperiment (ADMX) reached the sensitivity of KSVZ axions in the range 1 :9eV3:3eV [70,71] assuming virialized axions composing the whole DM density %DM= 0:45 GeV=cm3. The setup was improved by using SQUID ampliers [72], and then reached the line of the DFSZ model [73,74]. The HAYSTAC experiment searched for heavier axions by operating a setup similar to the ADMX one but using a Josephson Parametric Amplier (JPA), and achieving quantum limited sensitivity [75] and beyond [76]. The collaborations UF and RBF also reached remarkable limits, and the ORGAN experiment operated a pathnding haloscope at 110 eV [77,78]. Several new concepts have been proposed to search for DM axions with next-generation haloscopes based not only on the Primako eect but also on axion-induced electric dipole moments or on the axion-spin interaction [79,80,81,82,83,84,85,86]. 3 The eective magnetic eld of DM axions The coupling between axions and electron spins can be used for axion detection as an alternative to the coupling to photons [87,88,89,90]. Being it weaker than the axion-photon coupling it was not immediately exploited, but recently new experimental schemes were presented. Besides the axion discovery, the axion-electron coupling is interesting for distinguishing between dierent axion models. The possibility of detecting galactic axions by means of converting them into collective excitations of the magnetization (magnons) was considered by Barbieri et al. in [87], laying the founda- tions to the Barbieri Cerdonio Fiorentini Vitale (BCFV) scheme and to the following experimental proposal [84]. The#### Page 4 of 18 Eur. Phys. J. Plus ##################### original idea is to use the large de Broglie wavelength of the galactic axions to detect the coherent interaction between the axion DM cloud and the homogeneous magnetization of a macroscopic sample. To couple a single magnetization mode to the axion eld, the sample is inserted in a static magnetic eld. The interaction yields a conversion rate of axions to magnons which can be measured by monitoring the power spectrum of the magnetization. The form of the interaction is calculated hereafter in terms of an eective magnetic eld. Such a eld is the searched-for signal. Its features are derived and characterized as follows. The axion derivative interaction with fermions is invariant under a shift of the axion eld a!a+a0and reads L =C 2fa  5 @a (2) where is the spinor eld of a fermion of mass m , andC is a model-dependent coecient. The dimensionless couplings can be dened as ga =C m =fa (3) and play the role of Yukawa couplings, while the ne structure constant of the interaction is a =g2 a =4. The tree-level coupling coecient to the electrons of the DFSZ model is [27,28] Ce= cos20=3, where tan 0=vd=vu, the ratio of the vacuum expectation values of the Higgs eld. The axion-electron derivative part of the interaction can be expressed as Le=gaee 2me@a(x) e(x)  5e(x)
'igaeea(x)e(x) 5e(x); (4) where the last term is an equivalent Lagrangian obtained by using Dirac equation and neglecting quadridivergences. The Feynman diagram of this interaction is reported in Fig. 1 and suggests how the process happens: an axion is absorbed and causes the fermion to ip its spin, and the macroscopic eect is a change in the magnetization of the sample containing the spin. aigaeeγ5e− e− Fig. 1. Feynman diagrams of the axion-fermion interaction, showing how the eect of the axion is to be absorbed and cause the spin ip of the fermion. The corresponding interaction Lagrangian is reported in Eq. (4). By taking the non-relativistic limit of the Euler{Lagrange equation, the time evolution of a spin 1/2 particle can be described by the usual Schroedinger equation i~@' @t= ~2 2mer2gaee~ 2mee
ra '; (5) whereeis the Pauli matrices spin vector. The rst term on the right side of Eq. (5) is the usual kinetic energy of the particle, while the second one is analogous to the interaction between a spin and a magnetic eld. One can notice that gaee~ 2mee
ra=2e~ 2mee
gaee 2e ra=2ee
gaee 2e ra; (6) sinceeis the magnetic moment of the particle, it can be both Bohr magneton or a nuclear magneton, depending on the considered fermion. From Eq. (6) it is clear that the eect of the axion is the one of a magnetic eld, but since it does not respect Maxwell's equations, calling it an eective magnetic eld is more appropriate Bagaee 2e ra: (7) This denition is useful to quantify the performances of a ferromagnetic haloscope in terms of usual magnetometers sensitivity. In the BCFV case, the signal is given by the electrons' magnetization. An intuitive connection between theEur. Phys. J. Plus ##################### #### Page 5 of 18 macroscopic magnetization and the spin is given by M/BNSwhereBis Bohr magneton and NSis the number of spins which take part to the magnetic mode [91]. In such a way, the spin- ip can be classically considered a variation of the magnetization at a frequency given by the axion eld. It is now necessary to understand which are the features of Bato design a proper detector. The isothermal model of the Milky Way's DM halo predict a local density of %DM'0:45 GeV=cm3[92]. An Earth-based laboratory is thus subjected to an axion-wind with a speed va'300 km/s, that is the relative speed of Earth through the Milky Way. Using the vector notation, the value of vi acan be calculated from the speed of the galactic rest frame. The speed on Earthvi Eis given by the sum of vi S,vi Oandvi R, which are respectively the Sun velocity in the galactic rest frame (magnitude 230 km/s), the Earth's orbital velocity around the Sun (magnitude 29.8 km/s), and the Earth's rotational velocity (magnitude 0.46 km/s). The observed axion velocity is then vi a=vi E, which follow a Maxwell-Boltzmann distribution. As will be shown hereafter, the eect of this motion is a non-zero value of the axion gradient, and a modulation of the signal with a periodicity of one sidereal day and one sidereal year [47,93,94]. The numeric axion density in the DM halo depends on the axion mass and results na'31012(104eV=ma) cm3. The coherence length of the axion eld is related to the de Broglie wavelength of the particles, which is given by a=h mava'14104eV ma m: (8) Such wavelength allows for the use of macroscopic samples to detect the variation of the magnetization. The large occupation number na, coherence length a, anda=va=c'103permit to treat Baas a classical eld1. The coherent interaction of a(x) with fermions has a mean value a(x) =a0eip ax=a0ei(p0 atpi axi); (9) wherepi a=mavi Eandp0 a=p m2a+jpiaj2'ma+jpi aj2=(2ma). The production of DM axions is discussed in Section 2, where it is shown that they are indeed cold DM since their momentum is orders of magnitude smaller than the mass. The axion kinetic energy is expected to be distributed according to a Maxwell-Boltzmann distribution, with a mean relative to the rest mass of 7 107and a dispersion about the mean of MB'5107[47,93,95]. The eect of the mean is a negligible shift of the resonance frequency with respect to the axion mass. The consequence of the dispersion on the eective magnetic eld is a natural gure of merit Qa=1 MB'ma hpiai2 =1 2a'2106: (10) To calculate the eld amplitude a0, the momentum density of the axion eld is equated to the mean DM momentum density yielding a2 0p0 1pi a=nahpi ai=namava)a0=p na=ma: (11) For calculation purposes natural units are dropped and the Planck constant ~and speed of light care restored. The eective magnetic eld associated to the mean axion eld reads Bi a=gaee 2ena~ mac1=2 pi asinp0 act+pi axi ~ : (12) From Eq. (12), the frequency and amplitude of the axionic eld interacting with electrons result Ba=gaee 2ena~ mac1=2 mava'51023ma 50eV T; !a 2'cp0 a ~=mac2 ~'12ma 50eV GHz:(13) As the equivalent magnetic eld is not directly associated to the axion eld but to its gradient, the corresponding correlation length and coherence time must be corrected to [84] ra'0:74a= 0:74~ mava'2050eV ma m; ra'0:68a= 0:682~ mav2a'4650eV maQa 1:9106 s:(14) The nature of the DM axion signal is now well-dened: an eective magnetic eld of amplitude Ba, frequency fa= !a=2, and quality factor Qawith values dened by Eq.s (13) and (14). 1The average speed vacalso justies the approximation of Eq. (5), i. e. the use of the non-relativistic limit of Euler- Lagrange equations.#### Page 6 of 18 Eur. Phys. J. Plus ##################### 4 The axion-to-electromagnetic eld transducer A viable experimental scheme must be designed to detect the eld Bi a, whose features are dened by Eq.s (13) and (14). A magnetic sample with a high spin density nSand a narrow linewidth m= 2=T 2(i. e. long spin-spin relaxation timeT2) can be used as a detector. The magnetic eld Badrives a coherent oscillation of the magnetization over a maximum volume of scale ra. The sensitivity increases with the sample volume Vsup to (ra)3. For electrons e'(2)28 GHz/T, so the corresponding magnetic eld B0is of order 1 T and experimentally readily obtainable. The electrons' spins of a magnetic sample under a uniform and constant magnetic eld result in a magnetization M(x;t) that can be divided in magnetostatic modes. The space-independent mode of uniform precession is called Kittel mode. The axionic eld couples to the components of Mtransverse to the external eld, depositing power in the material. More power is deposited if the axion eld is coherent with the Kittel mode for a longer time. The best-case scenario is a material with a quality factor Qm= eB0= mwhich matches Qa, so that the coherent interaction between spins and DM-axions lasts for ra. For this reason the magnetic eld uniformity over the sample must be1=Qmto avoid inhomogeneous broadening of the ESR. According to these considerations, it is possible to detect an axion-induced oscillation of the magnetisation by monitoring a large sample with an precise magnetometer. However, the limit of this scheme lies in the short coherence time of the magnetic sample. In fact at high frequency, i. e. above 1 GHz, the rate of dipole emission becomes higher than the intrinsic material dissipation, this eect is know as radiation damping [96]. Since radiation damping is related to the sample dipole emission, a possible way to reduce its contribution is to limit the phase-space of the radiated light by working in a controlled environment like a resonant cavity [96,97,98,99]. By housing the sample in a mw cavity and tuning the static magnetic eld such that !m= eB0'!c, where!cis the resonance frequency of a cavity mode with linewidth c, one obtains a photon-magnon hybrid system (PMHS). An exact description of the system is given by the Tavis-Cummings model [99]. It discusses the interaction of NStwo-level systems with a single mw mode, and predicts a scaling of the cavity-material coupling strength gcm/pNS. The single-spin coupling is gs= e 2s 0~!m Vc; (15) whereVcis the cavity volume, 0is the vacuum magnetic permeability and a mode-dependent form factor [100], with this relation gcm=gspNS. Such scaling has been veried experimentally down to mK temperatures for an increasing number of spins NS[100,101,102]. For a quantity of material such that gcm c, the single cavity mode splits into two hybrid modes with frequencies !+,!and 2gcm=!+!. For!m=!cthe linewidths of the hybrid modes are the average of the cavity mode linewidth cand of the material one m, namely h= ( c+ m)=2. The coupling gcmis in fact a conversion rate of the material magnetization quanta (magnons) to cavity photons and viceversa. If gcm> h the system is in the strong-coupling regime, meaning that for a magnon (photon) it is more likely to be converted than to be dissipated. In this way, magnetisation uctuations, which might be induced by axions, are continuously converted to electromagnetic radiation that be collected with an antenna coupled to the cavity mode, as schematically shown in Fig. 2. a ωam ωmc ωcgam≪1 gcm∼1 Fig. 2. The coupled harmonic oscillators are reported in orange, green and blue for cavity c, materialmand axionarespectively. The uncoupled normal-modes frequencies of the HOs are !c,!mand!aand the couplings are gamandgcm, represented by springs. The aim of a PMHS devised for an axion haloscope is to maximise the axionic signal, and it eectively works as an axion-to-photon transducer. A rendering of the resulting device is reported in Fig. 3. An important result for designing the transducer is that multiple spheres can be coherently coupled to a single cavity mode [103,102]. The measurements demonstrate that all the spins participate in the interaction, thus the samples act as a single oscillator. This is guaranteed by the fact that the static eld is uniform over the spheres and that the rf eld is degenerate over the axis of the cavity where they are placed. Several tests were performed to understand dierent features of the system in the light of the two properties mentioned before. To understand the results of the dierent measurements one can use a simple oscillators model as is done in [102]. The PMHS can beEur. Phys. J. Plus ##################### #### Page 7 of 18 described by introducing two magnon modes and two cavity modes, hereafter the photon modes are labeled as cand dwhile the magnon modes are mandn. In the matrix form, the system can be modeled by the hamiltonian Hcdmn =0 B@!ci c=2gcdgcm gcn gcd!di d=2gdm gdn gcmgdm!mi m=2gmn gcngdngmn!ni n=21 CA; (16) where!, andgare the frequencies, linewidth and coupling of the dierent modes. The autofunction of the system can be calculated as the determinant of !I4H cdmn thus the function used to show the anticrossing curve reads fcdmn(!) = det !I4H cdmn
: (17) To ideally have a coherent coupling, one needs to let the spins of dierent spheres cooperate and make them indistinguishable, so they need to be uncoupled and their resonant frequencies must be the same. These conditions translate to gmn= 0 and!m=!n, which clearly can be extended to an arbitrary number of oscillators (in this case, the ten spheres). The interaction between two spheres yields non-zero value of gmn, and its eect is to introduce other resonances besides the two main ones of the PMHS. This eect needs to be avoided to have control over the system and couple all the spins of the samples to the cavity mode, avoiding magnons bouncing between dierent magnetic modes and eventually being dissipated before their photon conversion. Fig. 3. Rendering of the whole system, constituted by the cavity and the pipe with ten YIG spheres, ready to be tested at milli-Kelvin tempera- tures. The external part shows the superconducting magnet (in brown) which surrounds the cavity and the spheres to provide a magnetic eld with uniformity better than 7 ppm. The magnet is immersed in the liquid helium bath outside the vacuum chamber of the dilution unit. The cavity is at the centre of the magnet, is anchored to the mixing chamber of the dilution refrigerator with two copper bars and is equipped with two an- tennas, one is xed and weakly coupled, while the second one is movable and is used to extract the signal. The YIG spheres are inside the cavity, held by a fused silica pipe lled with helium and separated by thin PTFE spacers. The cap used to seal the pipe is made of copper and is anchored to the cavity body to ensure the thermalisation of the exchange helium and therefore of the YIG spheres. YIG sphere were produced on site with a technique described in [102]. This open the possibility of studying spheres of dierent diameters coupled to the same mode. One of the ndings is that, trying to couple spheres with dierent diameter to the same mode, the volume of the sample is linearly related to the oset eld [102]. The axion- to-electromagnetic eld transducer of a ferromagnetic haloscope. The constraints to remember for its design are in the following, and were tested with a room temperature setup consisting in a 10.7 GHz cavity with conical endcaps and a fused silica pipe holding the YIG spheres. The magnetic eld is given by a SC magnet which, to perform quick tests, it is equipped with a room temperature bore allowing the magnet to be in a liquid helium bath during operation. First the minimum separation between two spheres is tested by gradually increasing the distance between them and verifying that a usual anticrossing curve is reproduced. The minimum distance between 2 mm spheres results in 3 mm. A YIG sample then occupies 5 mm of space, and since the cylindrical part of the cavity is 6 cm it can house a maximum of twelve samples. Ten spheres are inserted in the pipe for them not to be too close to the conical part of the cavity. Multiple spheres of dierent diameters were fabricated and rened to verify that they hybridize with the cavity for the same value of the magnetic eld.#### Page 8 of 18 Eur. Phys. J. Plus ##################### The setup must ensure a proper thermalization of the cavity and of the YIG spheres, the preparation of the fused silica pipe is as follows. A vacuum system is designed in such a way to empty the pipe from air which is then immersed in a 1 bar helium controlled atmosphere. This way the pipe is lled with helium, and can be sealed by using a copper plug and Stycast. First the sealing is tested without the samples by measuring the shift of the TM110 mode of the cavity-pipe system with and without helium. The frequency is measured with the helium-lled pipe, which is then immersed in liquid nitrogen and again placed in the cavity. Re-measuring the same frequency excludes the presence of leaks. The used cavity is made of oxygen-free high-conductivity copper, and features a cylindrical body with two conical endcaps, as shown in Fig. 3. The central body is not a perfect cylinder but it has two at surfaces used to remove the angular degeneration of the mode. This creates two modes rotated of =2 with dierent frequencies, which is the second cavity mode in Eq. (16). The function fcdmn(!) is tted to the measured PMHS dispersion relation to extract the parameter of our setup, and in particular the hybridization results 638 MHz which is compatible with the single 1 mm sphere since 638 MHz =p810 = 71 MHz. This value of the single sphere coupling is compatible with what previously obtained in simpler PMHS, indicating that the measured spin density of YIG is consistent both with the previous results and with the values reported in the literature. Remarkably, the lower frequency resonance is almost unaected by the behaviour of the rest of the PMHS, in the sense that its frequency does not dier from the one of a usual anticrossing curve, thus it can possibly be safely used for a measurement [104]. Since haloscopes need to scan multiple frequencies to search for axions, the resonant frequency of the PMHS mode used for the measurement need to be changed. The tuning is made extremely easy by the fact that it is controlled only by means of the external magnetic eld. A high stability of B0is necessary to perform long measurements over a single frequency band. This is set by the linewidth of the hybrid mode, which in this case is 2 MHz, and is tuned to cover a range close to 100 MHz [1,104]. Thanks to the anticrossing curve it is easy to identify the frequency of the correct mode to study. The hybrid mode is not aected by disturbances caused by other modes in a range that largely exceeds ten times its linewidth. These clean frequencies are selected for the measurements whenever it is possible to match them with the working frequencies of the amplier described in the next Section. 5 Quantum-limited amplication chain The PMHS described previously in this Chapter acts as a transducer of the axionic signal. The power coming from the PMHS must be measured and acquired with a suitable detection chain, and, as it is extremely weak, needs to be amplied. The intrinsic noise of an haloscope is essentially related to the temperature of the setup, and since axionic and Johnson power have the same origin it is the ultimate limit of the SNR. The amplication process inevitably introduces a technical noise which, for these setups, is useful to quantify in terms of noise temperature Tnto compare to the Johnson noise. This stage of the measurement is setting the overall sensitivity of the apparatus since, as shown hereafter, for very low working temperatures the noise temperature is higher than the thermodynamic one. Minimizing Tnis a key part of the development of an haloscope, and is complementary to the maximization of the axion deposited power. The mw ampliers used for precision measurement are mostly high electron mobility transistors (HEMT), since they have high gain and low noise, of the order of 4 K. The most sensitive amplier available is the Josephson parametric amplier (JPA), which reaches the quantum noise limit [105,106,107,108,109,110,111]. This type of amplier is used in the present haloscope, and its performances can be overcome only by using a photon counter. HEMT are eld eect transistors based on an heterojunction, i. e. a PN junction of two materials with dierent band gaps [112]. The proper doping prole and band alignment gives rise to extremely high electron mobilities, and thus to ampliers which can have high gain, very low noise temperature, and working frequency in the microwave domain. Even if their noise temperature is low, HEMTs are not the most sensitive ampliers available. At a frequency 10 GHz the SQL of linear ampliers is close to 0.5 K, which is about one order of magnitude lower than theTnof HEMTs. Such remarkably low Tnis achieved by JPAs, resonant ampliers with a narrow bandwidth but with quantum-limited noise. This feature makes them the ideal tool to measure faint rf signals, and thus to be implemented in ferromagnetic or Primako haloscopes. The non-linear mixing is given by a Josephson-RLC circuit with a quadratic time-dependent Hamiltonian, which can be degenerate or non-degenerate depending on whether the signal and idler waves are at the same frequency or not [113]. A non-degenerate device consists in a three-modes, three-input circuit made of four Josephson junctions forming a Josephson ring modulator. It eectively is a three-wave purely dispersive mixer which can be used for parametric amplication [114]. It can be computed that the non-linear mixing process appears as a linear scattering, conguring the JPA is a linear amplier. As such, it is quantum limited and its noise temperature depends on the working frequency. Being based on resonant phenomena, the JPA has a narrow working band of tens of MHz. To use the amplier in a wide frequency range, a bias eld is applied to the ring and the resonance frequencies of the signal, idler and pump mode are tuned. This is achieved with a small SC coil placed below the ring, biased with a current Ib. The implementation of a JPA in a ferromagnetic haloscope is shown and described in Fig. 4.Eur. Phys. J. Plus ##################### #### Page 9 of 18 Fig. 4. Rendering of the implementation of this JPA in a ferromagnetic haloscope. The golden pipe is connected to the mixing chamber of the dilution refrigerator used to cool down the setup, the circulators are only in thermal contact with this last stage, as is the shielding cage of the JPA (also drawn in gold). The blue component is a switch, present in one of the possible congurations of a ferromagnetic haloscope; attenuators are drawn in blue as well. The JPA is inside two concentric cans, the external one is made of Amuneal and the external is of aluminum. The rst is useful to reduce the Earth magnetic eld in which the superconducting parts (shields and junctions) undergo the transition, while the second screens from external disturbances. Everything is attached to the mixing chamber plate of a dilution refrigerator with a base temperature of about 90 mK. This image corresponds to the conguration reported in Fig. 5b. The characterisation the rf chain used for the measurements is described hereafter. It will focus on the setup described by Fig. 5a, as is the one used in [1] as it was found to be more reproducible and in general more reliable than 5b. The conguration of the electronics allows the testing of both the JPA and the PMHS. Transmission measurements of the PMHS can be performed by turning o the JPA (i.e. no bias eld and no pump) to re ect the signal on it, the input is the SO line and the output is the readout line. The JPA can be tested with the help of the Aux line, by uncoupling the antenna from the cavity and re ecting the incoming signal. Some rf is still absorbed at the cavity modes frequency but this does not compromise the measurement. The external static eld of the PMHS does not aect the resonances of the Josephson ring modulator as no dierence has been detected between the measurements with and without eld. Runs are performed with bias currents Ib'170A and 460A at frequencies ranging from 10.26 GHz to 10.42 GHz. Using the SO line and critically coupling the antenna to the hybrid mode, a signal is injected in the system and read with the whole amplication chain. The rst test is to verify the linearity of the JPA (and of the whole chain) using signals of growing intensity until the system saturates. These measurements show the linear and saturate behavior of the amplier. It is possible to calibrate the gain of the JPA by using a signal large enough to be measured with the JPA o but also not to saturate it once it is turned on. This is useful to know the gain of the amplier at the dierent working points to have a preliminary calibration of the system and to understand whether an output noise with higher amplitude is due to the JPA or to something else. Since the electronics above the 4 K line was already characterized for the previous prototype, the baseline noise with JPA o is roughly the amplier noise temperature T(hemt) n'10 K. The measured noise spectra with the JPA turned o is white in a bandwidth of several hundreds of MHz, when the parametric amplier is turned on its resonance exceeds this noise of roughly 10 dB. Since it is possible to calibrate the gain of the JPA GJPA'20 dB, the amplied noise level can be extracted as T(JPA) n =T(hemt) n=10(GJPA10 dB)=10'1 K, which is the noise temperature amplied by the JPA. The value of 1 K is reasonable, since a single quantum at this frequency is 0.5 K such noise corresponds to two quanta. Even if this procedure is somewhat correct, it is not a proper calibration of the setup and something better is explained hereafter. Since some problems were encountered in the noise calibration with hot load (see Fig. 5b), the rf setup of Fig. 5a is designed to calibrate all the dierent lines with the help of the variable antenna coupling. By moving the antenna one can arbitrarily choose the coupling to a mode, if it is weakly coupled a test signal from the Aux port gets re ected and goes to the JPA, while if the antenna is critically coupled to the mode, a signal from SO is transmitted through the cavity and than to the JPA. Almost the same result can be obtained by slightly changing the frequency of the test signal to be within the JPA band but out of the cavity resonance. The critical coupling can be reached by doubling the linewidth of the mode or equivalently by minimizing the re ected signal from the Aux line to the Readout line. The procedure to calibrate all the lines is: 1. with the weakly coupled antenna or by detuning the mode the losses of the Aux-Readout line LARare measured;#### Page 10 of 18 Eur. Phys. J. Plus ##################### (a) (b) Fig. 5. Two possible electronics layout for a ferromagnetic haloscope. The blue lines show the temperature ranges, the crossed rectangles are the magnet, and the orange rectangle is the cavity with black YIG circles inside. The boxed numbers are attenuators and the red circled Ts are the thermometers. At the top of the cavity are located the weakly coupled antenna (empty dot) and the variably-coupled antenna (full dot). The weakly coupled antenna is connected to an attenuator and then to the source oscillator SO. In conguration (a) the the variable antenna is connected to the JPA through a circulator, whose other input is used for auxiliary measurements. The output of the JPA is further amplied by two HEMTs A1 and A2. Conguration (b) is basically the same as (a), where the input can be switched from the cavity antenna to a matched load with variable temperature regulated by a current Ih, and used for calibration. In both (a) and (b) the A2 output is down-converted and acquired. 2. the antenna is critically coupled to the mode and a signal is sent through the Aux-SO line to get LAS; 3. with the same critical coupling the transmission of the SO-Readot line LSRis acquired. At this point a signal of power Ainis injected in the SO line, the fraction of this power getting into the cavity through the weakly coupled antenna is Acal=AinLSO. The attenuation of the line can be calculated as LSO'p LSRLAS=LAR, which gives the power collected by the critically coupled antenna. Since Acalis eectively a calibrated signal, it can be used to measure gain and noise temperature of the Readout line. Dierent Ainare used to get increasingly large signals to be detected by the JPA-based chain. This calibration has some minor biases, the rst is given by the cable from the cavity to the rst circulator which is accounted for two times in the Aux-Readout line. This contribution can be safely neglected as the cable is superconducting, making its losses negligible. Another bias is related to the antenna coupling, which is not perfect. With a proper antenna coupling the re ected signal is reduced of 10 dB, so there is a bias of a factor 10% intrinsic to the measurement which will be accounted for when calculating the error. As the calibration procedure is long it is not repeated for every run, however no important dierences are expected when changing the JPA frequency. As an example a run at 10.409 GHz is considered. The gain of the JPA at this frequency results GJPA'18 dB, and its bandwidth is 8 MHz, the hybrid mode is tuned the its central frequency and the calibration procedure is carried out,resulting in a noise temperature of Tn= 1:0 K and the total gain of the whole amplication chain is Gtot'120:4 dB. The value of Tnis compatible with the one estimated previously, and corresponds to two quanta. The coupling of the antenna with the hybrid mode is checked for every run. It is controlled by moving the dipole antenna in and out the cavity volume, the critical coupling is reached when the uncoupled linewidth of the mode is doubled. To verify the proper antenna positioning one may rely on the fact that depending on the temperature dierence between the cavity and a 50 , some power may be absorbed or added to the load thermal noise. The load under consideration is the hottest between the rst JPC isolator and the 20 dB attenuator of the Aux line. The hybrid resonance has a critical linewidth of about 2 MHz, so the depth will not be as narrow as the one of the cavity. In that case the temperature dierence is about 10%, which is about 10 mK, and if the temperature of the load and cavity are precisely measured the spectra can be used to get a two-points calibration. The selected calibration procedure was notEur. Phys. J. Plus ##################### #### Page 11 of 18 this one because the temperatures of loads and HS are not easily accessible. With two dedicated thermometers the temperatures of the loads could be measured, but it is not trivial to measure the temperature of the cavity and of the spheres with the needed precision. Since a small temperature dierence is expected, the measurement with the antenna coupled to the hybrid mode should be dierent from the uncoupled one. As reported in [1], there is a dierence between the two measurements and it is compatible with the thermal noise of the hybrid mode at a temperature slightly higher then the loads one. 6 Data acquisition and analysis A ferromagnetic haloscope's scientic run consists in several measurements in the common bands between the frequen- cies of the lower hybrid mode unaected by disturbances, and the JPA working range. The low temperature electronics is described in the previous section, and is completed by its following part hereafter. The room temperature electronics consists of a HEMT amplier (A2) followed by an IQ mixer used to down-convert the signal with a local oscillator (LO). In principle, it is possible to acquire the signal coming from both hybrid modes using two mixers working at f+andf. In this case it is chosen to work only with f+, thus setting the LO frequency tofLO=f+0:5 MHz. The amplied antenna output at the hybrid mode frequency is down-converted in the 0 - 1 MHz band, allowing to eciently digitize the signal. The phase and quadrature outputs are fed to two low frequency ampliers (A3 I;Q), with a gain of G3'50 dB each, and are acquired by a 16 bit ADC sampling at 2 MS/s (see [103]). A dedicated DAQ software is used to control the oscillators and the ADC, and veries the correct positioning of the LO with an automated measurement of the hybrid mode transmission spectrum. Some other online checks include a threshold monitor of the average amplitude, as well as of the peak amplitude, which ags the le if some unexpected large signal is present. The ADC digitizes the time-amplitude down-converted signal coming from A3 Iand A3Qand the DAQ software stores collected data binary les of 5 s each. The software also provides a simple online diagnostic, extracting 1 ms of data every 5 s, and showing its 512 bin FFT together with the moving average of all FFTs. The signal is down-converted in its in-phase and quadrature components fngandfqng, with respect to the local oscillator, that are sampled separately. Fig. 6. Second amplication stage of the setup, and rst room temperature amplier. The image shows the top part of the vacuum vessel containing the dilution fridge stages, the cavity and the electronics. Just outside it, the rst amplier is A1 (reported in yellow), while the second one is already at room temperature. The blue box corresponds to the variable temperature load of conguration (b). The stability of the measurement is tested by injecting a signal in the SO line slightly o resonance with the PMHS peak, and with an amplitude guaranteeing a large SNR. Monitoring its amplitude is a way to continuously check the peak position. In this setup the stability results well below the percent level, which is more than enough for the purpose of the experiment, thanks to the lower and more stable working temperature and to an extremely stable current generator produced in the Padua University electronic workshop. The signal is analysed using a complex FFT on the combination of phase and quadrature fsng=fng+ifqngto get its power spectrum s2 !with positive frequencies for f >f LOand negative frequencies for f <f LO, in a total bandwidth#### Page 12 of 18 Eur. Phys. J. Plus ##################### of 2 MHz which contains the whole hybrid linewidth. Some bins are found to be aected by disturbances resulting in systematic noise, but they can be easily removed by analyzing the data with longer FFT. Using 32768 points the single frequency resolution is 61 Hz  aand thus single bins which do not respect the uctuation dissipation theorem can be substituted with the average of the ten nearest neighbors. This procedure articially reduces the variance of the data, but the number of corrupted bins is negligible and so is the eect on the variance. This process does not cut the signal since it aects only known or single bins, while the axion signal is expected to be distributed over many. For debugging purposes a simulated signal is injected into the analysis code. It is created by generating a high- frequency noise to which are added a simulated axion signal and some disturbances. For creating the in-phase and in-quadrature component it is multiplied by a sine or to a cosine wave, to then extract only one point every 104and simulate the mixing and down-conversion processes. The FFT of this signal produces a white noise with some peaks (the axion signal plus disturbances), and is eventually integrated. The analysis procedure is veried to remove bad bins and to preserve the signal and SNR. Thanks to the stability and to the tests on the acquisition and analysis, all the les of a single run can be safely RMS averaged together. The calibration gives a noise temperature Tn= 1:0 K, which includes all the amplier noises, the cavity thermodynamic noise, and the losses. This can be used to calibrate the setup by setting the mean value of the FFT to kBTn
f. As for the calibration errors, the uctuations of the cavity temperature are of order 10 mK, thus they can change Tnof a fraction close to 1%. A larger contribution is intrinsic to the procedure, which requires the coupling of the antenna to be changed from weak to critical. This is believed to be the larger contribution to the uncertainty of the measurement, and even if in principle this is not a uctuation it will be used to estimate an error. The fact that the coupling is close to critical is also supported by a thermal noise measurement [1]. The control over this parameter can be estimated by injecting a signal through the Aux port in the weak-coupling position, and then by reducing the re ected amplitude by increasing the coupling. Typically the signal power can be decreased of more than 8 dB, this value can be used to estimate the larger uncertainty on the coupling resulting in a 16% error. To estimate the sensitivity to the axion eld the resolution bandwidth is set to 5 kHz, which at this frequency is the value producing the best SNR. The spectrum is tted with a degree ve polynomial to extract the residuals, whose standard deviation is the sensitivity of the apparatus in terms of power. For every run, by using the run length, the RMS power is compared with the estimated sensitivity obtained with Dicke radiometer equation. The measured uctuations are close to the expectations for almost every run, and are always compatible when considering the 16% error which was previously calculated. This shows that the measured power is compatible thermal noise and that the integration is eective over the whole measurement time, since it follows the 1 =p time trend. The best sensitivity reached isP= 5:11024W for an integration time of 9.3 h, corresponding to an excess rate .10 photons/s per bin. As is discussed in Section 1 this setup is a spin-magnetometer, as it is sensitive to variations of the sample magnetization. In this sense, it is interesting to determine the performances of this setup to get some physical intuition on its sensitivity and thus on all the possible phenomena that could be measured. The eld sensitivity can be estimated for a eld whose coherence length stretches on all the YIG spheres, and whose linewidth is narrower than the PMHS one. In the case of the present apparatus the threshold coherence length and time are then 10 cm and 100 ns, respectively. Iffandare the frequency and coherence time of the hybrid mode and NSis the number of spins, the magnetic eld sensitivity of the setup is B=P 4 eBfNS1=2 = 5:51019h10:4 GHz f83 ns 1:01021spins NSi1=2 T; (18) which is the minimum eective magnetic eld detectable by the spin magnetometer for a unitary SNR. A 0.5 aT sensitivity is remarkable by itself, and shows the high potential of HS-based magnetometers. The results of the previous section are used to extract a limit on the axion-electron coupling, which is a possible eect that modulates the sample magnetization and produces an excess of photons. The expected power deposited by DM-axions in the PMHS is Pout= 71033ma 43eV3NS 1:01021spinsh 83 ns W: (19) The 10.4 GHz photon rate corresponding to this power is ra= 109Hz. By comparing the rate rawith(3) P=(~!) one obtains that there are ten orders of magnitude to get the sensitivity required to detect the axion, which corresponds to ve in terms of eld (and thus of coupling constant). As discussed in Section 1, even if an instrument is not capable of limiting the QCD-axion parameter space it can still probe the presence of ALPs. These can also constitute the totality of DM and their mass and coupling are unrelated. The present measurements are used to extract a limit on the coupling of ALPs with electrons reported in [1]. No evidence of a signal due to axions has been detected, and theEur. Phys. J. Plus ##################### #### Page 13 of 18 measured spectra are compatible with noise. The 2 (95% C. L.) upper limit on the coupling reads gaee>e mavas 2tac(3) P 2B naNS; (20) wheretacis a frequency dependent coecient that takes into account that the axion-deposited power is not uniform in the haloscope operation range, as is discussed in [115]. All the experimental parameters used to extract the limits are measured within every run, making the measurement highly self-consistent. The limit on the ALP-electron coupling described by Eq. 20 is calculated for every run using the corresponding measured parameters. This result is compared with other techniques used for testing the axion-electron couling constant Fig. 7. Overview of axion searches based on their coupling with electrons [116]. The result obtained with the ferromagnetic haloscpe described in this work is labelled as \QUAX". The analysis is repeated by shifting the bins of half the RBW to exclude the possibility of a signal divided into two bins. The best limit obtained, and corresponding to P, isgaee<1:71011. The improvement of a longer integration time is not much, since the limit on the coupling scales as the fourth root of time, and to improve the current best limit of a factor 2 the needed integration time is six days. 7 Conclusions Low-energy measurements, precisely testing known physical laws, are a powerful probe of BSM physics, mainly com- plementary to accelerator physics. As shown in Fig. 8, thanks to Nambu-Goldstone theorem, extremely high energy scales can be explored by measuring faint eects at the limit of present technology. Eventually, new instruments and devices can be built to push the current technological limits to new levels and hopefully help not only fundamental physics but many other elds. Among these, haloscopes play a pivotal role while searching for Dark Matter axions. The scope of this work is to illustrate the construction and outline the operation of the rst ferromagnetic axion haloscopes. Such instruments can be used to measure the DM-axion wind which blows on Earth, as this last one is moving through the halo of the Milky Way. The axions interact with the spin of electrons causing spin ips that are, macroscopically, oscillations of a sample magnetization. The model of the isothermal galactic halo and of the axion yield the features of the searched for signal, namely its linewidth, frequency and amplitude. A frequency of 10 GHz, a linewidth of 5 kHz and an amplitude of about 1023T are expected for axion masses of order 40 eV. A proper haloscope has a transducer that converts the axion ux into rf power, followed by a sensitive detector to measure it. For a ferromagnetic haloscope, the transducer consists of a magnetic material containing the electron spins with which the axions interact. In order to maximize the axion- deposited power, the sample should have a large spin density and a narrow linewidth. The detector is a rf amplication chain based on a JPA. The described instrument features a power sensitivity limited by quantum uctuations, in this sense no linear amplier is or will be able to improve the haloscope. In future setups only bolometers or quantum#### Page 14 of 18 Eur. Phys. J. Plus ##################### 10−6eV 104GeVLHC 1012GeVNambu-Goldstonetheorem Precision tests Symmetry breaking scale Fig. 8. The usage of Nambu-Goldstone theorem to infer on physics at energy scales inaccessible to accelerators. counters can yield better results. For example, recent developments on quantum technologies [117,118] demonstrated the detection of uorescence photons emitted by an electron spin ensemble, and could be adopted for axion searches. Thermodynamic uctuations are already negligible due to the extremely low working temperature, so it is not necessary to decrease them by orders of magnitude. As the rate of thermal photons of a cavity mode is exponentially decreasing with temperature, the present dilution refrigeration technology is enough to reach the axion-required noise level. As discussed in Section 1, to get a rate of axion-induced photons which can be measured in a reasonable amount of time, a much increased quantity of material and a narrower linewidth are required. This setup features 0.05 cc of YIG, such volume must be increased by three orders of magnitude to get the required rate. This large quantity can be achieved by increasing the quantity of material in a single cavity and the number of cavities. In conclusion, the successful operation of an ultra cryogenic quantum-limited prototype demonstrates the possibility of scaling up the setup of orders of magnitude without compromising its sensitivity. To further increase the axionic signal there are two parameters to work on: the hybrid mode linewidth and the sample spin-density and volume. To nally achieve the sensitivity required by a QCD axion search, it is necessary to use a photon counter. The upgrades planned until now are implemented and result eective, as the apparatus behaves as expected. No showstoppers were identied so far. 8 Acknowledgment N.C. is thankful INFN and the Laboratori Nazionali di Legnaro for hosting and encouraging the experiment. The help and support of Giovanni Carugno and Giuseppe Ruoso is deeply acknowledged. References 1. N. Crescini, D. Alesini, C. Braggio, et al. Axion search with a quantum-limited ferromagnetic haloscope. Phys. Rev. Lett. , 124:171801, May 2020. 2. F. Zwicky. Die Rotverschiebung von extragalaktischen Nebeln. Helvetica Physica Acta , 6:110{127, 1933. 3. V. C. Rubin, N. Thonnard, and W. K. Ford, Jr. Extended rotation curves of high-luminosity spiral galaxies. 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2022-01-11

A ferromagnetic haloscope is a rf spin-magnetometer used for searching Dark Matter in the form of axions. A magnetic material is monitored searching for anomalous magnetization oscillations which can be induced by dark matter axions. To properly devise such instrument one first needs to understand the features of the searched-for signal, namely the effective rf field of dark matter axions $B_a$ acting on electronic spins. Once the properties of $B_a$ are defined, the design and test of the apparatus may start. The optimal sample is a narrow linewidth and high spin-density material such as Yttrium Iron Garnet (YIG), coupled to a microwave cavity with almost matched linewidth to collect the signal. The power in the resonator is collected with an antenna and amplified with a Josephson Parametric amplifier, a quantum-limited device which, however, adds most of the setup noise. The signal is further amplified with low noise HEMT and down-converted for storage with an heterodyne receiver. This work describes how to build such apparatus, with all the experimental details, the main issues one might face, and some solutions.

Building instructions for a ferromagnetic axion haloscope

2201.04081v1

Evidence for exchange Dirac gap in magneto-transport of topological insulator-magnetic insulator heterostructures S. R. Yang1#, Y . T. Fanchiang2#, C. C. Chen1, C. C. Tseng1, Y . C. Liu1, M. X. Guo1, M. Hong2, S. F. Lee3*, and J. Kwo1* 1Department of Physics, National Tsing Hua University, Hsinchu 30013, Taiwan 2Department of Physics, National Taiwan University, Taipei 10617, Taiwan 3Institute of Physics, Academia Sinica, Taipei 11529, Taiwan # Authors who have equal contributions to this work * Corresponding authors Abstract: Transport signatures of exchange gap opening because of magnetic proximity effect (MPE) are reported fo r bilayer structures of Bi 2Se3 thin films on yttrium iron garnet (YIG) and thulium iron garnet (TmI G) of perpendicular magnetic anisotropy (PMA). Pronounced negative magnetoresistanc e (MR) was detected, and attributed to an emergent weak localization (WL) effect superimposing on a weak antilocalization (WAL). Thickness-dependent study shows that the WL originates from the time-reversal-symmetry breaking of topological surface states by interfacial exchange coupling. The weight of WL declined when the interfacial magnetization was aligned toward the in-plane direction, which is understood as the effect of tuning the exchange gap size by varying the perpendicular magnetization component. Importantly, magnetotransport study revealed anomalous Hall effect (AHE) of square loops and anisotropic magnetoresistance (AMR) char acteristic, typifyi ng a ferromagnetic conductor in Bi 2Se3/TmIG, and the presence of an interfacial ferromagnetism driven by MPE. Coexistence of MPE-induced ferromagnetism and the finite exchange gap provides an opportunity of realizing zero magnetic-field dissipation-less transport in topological insulator/ferromagne tic insulator heterostructures. Breaking time-reversal symmetry (TRS) in topological insulators (TIs) leads to several exotic phenomenon such as quantum anomalous Hall effect (QAHE), topological magnetoelectric effect, and magnetic monopole [1,2] . A prerequisite of these novel quantum state is an energy gap opened at the Dirac surface state induced by exchange interaction with magnetic elements [3]. Magnetic doping is a prevalent way of introducing ferromagnetism in TIs [4-7] . Study of TRS breaking in magnetically doped TIs was ignited by the direct observation of an exchange gap opening of topological surface states (TSS s) via angle-resolved photoemission spectroscopy (ARPES) [5], and culminated with the real ization of QAHE in Cr-doped (Bi,Sb) 2Te3 [8]. Although magnetic doping is proven to be effective in breaking TRS, the observation temperature of QAHE reported so far was less than 2 K [8-12] , order-of-magnitude lower than the ferromagnetic Curie temperature ( ܶC). It is suggested that the disorder created by dopants, as well as the small exchange gap size induced by low doping concentration, poses a limit of raising the QAHE temperature [12,13] . Recently, magnetic proximity effect (M PE) of TI/ferromagnetic insulator (FI) heterostructures was demonstrated as another promising route of breaking TRS [14-17] . Besides the benefit of much higher ܶC, the induced interfacial magnetization is uniform, free of crystal defects. A room -temperature ferromagnetism by MPE is directly observed in epitaxial EuS/Bi 2Se3 by polarized neutron reflectometry [16] . Moreover, robust anomalous Hall (AH) resistances up to 400 K has been detected in (Bi,Sb) 2Te3 films on TmIG with perpendicular magnetic anisotropy (PMA) [17] . Despite the clear observations of ferromagnetism and presumably pronounced TRS breaking, the experimental indications of exchange gap opening following MPE in these cases are still vague. Unlike magnet ically doped-TIs where the gapped surface can be exposed to the probe of ARPES technique [5], the gapped surface state caused by MPE is buried at the interface, making it difficult to investigate using typical ARPES. Attempts to detect MPE-induced exchange gap by transport measurements have been made by various groups [18-20] . One signature of exchange gap opening is an emerging weak localization (WL) taking the form of negative magnetoresistance (MR) accompanied by a suppressed weak antilocalization (WAL) [21]. However, negative MR in TI/FI was hitherto observed in samples comparable or beyond the Ioffe-Regel limit (sheet resistance ܴୱ݁/݄ ଶ) [18-20] . Above the limit, the Anderson (strong) localization due to disorder c ould also give rise to a negative MR [22,23] , which cannot be easily distinguished from the one due to an exchange gap. Therefore, a definite transport signature of MPE-induced exchange gap remains elusive. Despite the observation of negative MR and suppressed WAL [24-26], robust MPE-induced ferromagnetism in TIs, whic h should be best manifested as large remnant magnetization ܯ୰ pointing out-of-plane and hysteric AHE, has not been observed in those systems. Conversely, although clear ܯ୰ was detected in (Bi,Sb) 2Te3/TmIG, the negative MR and transport signature of gap opening were not simultaneously presented [17]. To achieve QAHE in TI/FI, both ingredients, robust ferromagnetism of TSS and exchange gap opening, must be fulfilled concurrently, which is a condition yet to be demonstrated for TI/FI. In pursuing high-temperature QAHE through MPE, it is important to identify the transport signature of the exchange gap associated with MPE-induced ferromagnetism, before one moves on to the ultimate goal of QAHE. Moreover, although AHE is often taken as an evidence of interfacial ferromagnetism, the possible spin current effect in TI/FI, such as spin Hall MR [27] , can also lead to similar AH resistances. Additional transport study is demanded to elucidate possible modulations of magneto-transport by spin current effects. In this work we report a pronounced WL that competes with the WAL in Bi2Se3/YIG and Bi 2Se3/TmIG bilayers. The ferrimagnetic garnet films were chosen because of their high ܶC above 500 K [28] , good thermal stability in conjunction with TIs, and their technological importance [29] . The emergent WL effect is strong enough to manifest a negative magnetoresistance (MR), showing systematic changes with the perpendicular magnetization controlled by tilted external fields. This strongly suggests that the observed WL arose from a finite MPE-induced exchange gap, whose size could be further tuned by the external field directions. Most importantly, the breaking of TRS by MPE is corroborated by the observation of AHE up to 180 K, together with clear anisotropic magnetoresistance (AMR), confirming ferromagnetic TSS. Our study thus presents a coherent picture of the long sought MPE-induced ferromagnetism and exchange gap in electron transport. Our YIG and TmIG thin films are fabricated using off-axis sputtering [30,31] . YIG is a ferrimagnetic insulator with ultralow magnetic damping ideal for spin wave and spin current transport study [29] , while TmIG films, when under a tensile strain, exhibit robust and tunable perpendicular magnetic anisotropy (PMA) [31-33] which is essential for exchange gap of TSSs. Bi 2Se3 thin films of 6 – 40 quintuple layer (QL) were deposited on the garnet substrates by molecular beam epitaxy (MBE). High quality Bi 2Se3 thin films and sharp Bi 2Se3/garnet interface were obtained by the invention of a novel growth procedure, a key factor for conveying strong exchange interaction of the localized magnetic moments of garnets layers and TSSs [34] . For transport measurements, Bi 2Se3/garnet bilayer samples were made into Hall bars (650 µm ൈ 50 µm ) by standard photolithography. Four points measurement was carried out in a 9 T Quantum Design physical property measurement system with a 10 µA DC current. Figure 1(a) shows the temperature dependence of sheet resistance ( ܴୱ) of Bi2Se3/YIG bilayers with Bi 2Se3 thickness of 6, 10, 16, and 40 QL. All samples exhibit metallic behavior of decreasing ܴୱ when the samples were cooled down from room temperature. The sheet carrier density of these samples is in the range of ሺ1.5 െ 3 ሻൈ1 0ଵଷcmିଶ, indicating that the bulk carriers of Bi 2Se3 participate in the electron transport. Due to an increasing surface scattering, the ܴୱ tends to be larger in thinner Bi 2Se3 [35]. Note that the maximum ܴୱ of these sample were well below h/݁ଶሺൎ 25.8 kΩ ሻ satisfying the condition of transport regime, thus quantum interference effects of 2D electron systems in TI thin films, such as WAL and possible WL, can be described by well-developed theories [22] . Figure 1(b) displays the MR data taken at 2 K under a perpendicular applied field for the four Bi 2Se3/YIG bilayers and, for comparison, a 9 QL Bi 2Se3 grown on Al 2O3. For Bi 2Se3/Al 2O3, a sharp cusp feature at low fields, characteristic of WAL effect, was observed, and the MR stayed positive up to 9 T. WAL in thin Bi 2Se3 is generally attributed to destructive interference because of difference of Berry’s phase 2ߨ accumulated by the Dirac fermion travelling in two time-reversed paths [1]. In contrast, notable negative MR were observed in 6 and 10 QL Bi 2Se3/YIG. Specifically, at low fields ( ൏1 T ), a weakened positive MR or suppressed WAL was observed in all the four Bi 2Se3/YIG bilayers. At intermediate fields ( 1െ4 T ), MR becomes negative for the thinner two samples. While the MR of 16 and 40 QL Bi 2Se3/YIG remained positive, that from 16 QL Bi 2Se3/YIG did show the much weaker positive MR compared to that of Bi2Se3/sapphire. When the external field exceeded 4 T, all the samples exhibited positive MR that results from the Lorentz force on moving electrons. The suppressed WAL effect of Bi 2Se3/YIG suggests that an additional transport mechanism shows up that contributes to a negative MR and competes with the WAL. Since the film quality of Bi 2Se3 grown on YIG is comparable to that on sapphire, the distinct MR behavior of Bi 2Se3/YIG is most likely originated from the interaction between the bottom surface of Bi 2Se3 and the YIG layer. As the YIG is ferrimagnetic at 2 K [36] , a sizable interfacial exchange coupling should exist in Bi 2Se3/YIG [15,25,30] , which is otherwise negligible for the non-magnetic sapphire substrate. Therefore, the stark contrast in the MR behaviors suggest that the suppressed WAL and negative MR are the indication on transport properties of TRS-breaking in TIs. Other possible mechanisms for the negative MR of TIs are defect-induced hopping transport [37] , hybridization gap of TSSs [21], and bulk subbands in thin TIs [38] . Each scenario can be excluded straightfo rwardly, as discussed in details in Supplemental Materials [39] . When the TSS is subjected to an exchange field, the Dirac fermion becomes massive, as expressed by an effective Hamiltonian ܪൌെ i ݒ ிሺෝൈܢොሻڄસෝ ଶۻڄ [21], where ݒி is the Fermi velocity, ෝ is Pauli matrix, ܬ is the exchange coupling constant and ۻ is the magnetization unit factor. The resulting energy dispersion is ܧൌേ ඨ൬ݒ ி݇௫ܬ 2ܯ௬൰ଶ ൬ ݒ ி݇௬െܬ 2ܯ௫൰ଶ ൬ܬ 2ܯ௭൰ଶ (1) with an exchange gap size of ܯܬ . It has been shown that electrons travelling a closed path would acquire a Berry’s phase as ߨሺ1െܯܬ ௭/2ܧ Fሻ, where E F is Fermi energy measured from the Dirac point [21] . The modulated Berry’s phase weakens the associated destructive interference, or even induces a crossover from WAL to WL. In the case of Bi 2Se3/YIG, interfacial exchange coupling can induce a finite gap through MPE, which further leads to competing WL. To quantitatively describe the negative MR, we calculated the longitudinal conductivity ܩ୶୶ from the tensor relation ܴ୶୶/ ሺܴ ୶୶ଶܴ ୷୶ଶሻ. The competition between WAL and WL can be described using the modified Hikami-Larkin-Nagaoka (HLN) equation [21,40] , Δܩ ൌ ܩ ୶୶ሺܤሻെܩ ୶୶ሺ0ሻൌ ߙ ୧ቆ݁ଶ ݄ߨቇቈ ߰ቆ 4݈݁୧ଶܤ1 2ቇെl nቆ 4݈݁୧ଶܤቇଵ ୧ୀܤߚ ଶ (2) , where ߰ is the digamma function, ߙ represents the weights of WL ( iൌ0 ) or WAL ( iൌ1 ), ݈୧ is the corresponding effective phase coherence length. The ܤߚଶ term primarily results from the Lorentz deflection of carriers [39]. To clearly reveal the presence of the WL component, the MC curves subtracted by the ܤߚଶ background are plotted in Fig. 1(c) . Positive MC was observed for 6, 10, and 16 QL Bi2Se3/YIG. Figure 1(d) shows the Bi 2Se3 thickness dependence of ߙ and ߙଵ. For 6 and 10 QL Bi 2Se3/YIG bilayers, large ߙ values of 0.7 and 2.7 were extracted, respectively. A crossing of the magnitudes of ߙ and ߙଵ was observed in thicker Bi2Se3 as ߙ decreased substantially for 16 QL Bi 2Se3 and became vanishing for 40 QL Bi 2Se3. Meanwhile, the ߙଵ value in general remains lower than -0.5, showing a slight decrease toward thinner Bi 2Se3. The smaller ߙଵ value suggests suppressed WAL channels in the bulk-surface-coupled Bi 2Se3 [23,35] . The extracted ߙ’s and ߙଵ’s thus reveal the competitive behavior between WL and WAL, whose interfacial origin are indicated by the stronger WL and weaker WAL in thinner Bi 2Se3. It is noteworthy that Eq. 2 is derived from a model considering an effective Hamiltonian of a single Dirac surface state. In reality, the Bi 2Se3 films have two conducting surfaces interacting through the bulk carriers in transport [23]. The complexity may give rise to the unexpectedly large ߙ in thin Bi 2Se3/YIG. Nevertheless, Eq. 2 and Fig. 1(d) do capture the concept of emergent WL from TRS breaking at the interfaces [13]. TmIG films with PMA are more desirable for exchange gap opening at zero applied field due to their robust ܯ ୰. To further verify the relation between WL and exchange gap, we tilted the applied field from the z direction. Based on Eq. (1), tilted field should effectively vary the size of ܯ and thus tune the exchange gap size ܯܬ . In Bi 2Se3/TmIG, ܯ stands for the z component of (i) magnetization of TmIG near the interface or (ii) MPE-driven magnetization on the TI side ଵ. Figure 2(c) -(f) shows the MR results of Bi 2Se3/Al 2O3 and Bi 2Se3/YIG under applied fields of different angles ߠ୷ൌ 90°, 60°, and 30° . For Bi 2Se3/Al 2O3, although the MR curves change with ߠ୷ in Fig. 2(c) , when plotted as a function of perpendicular field ܤ൫ൌ ܤsinߠ ୷൯ in Fig. 2(d) , the curves collapse into one. The observation implies that MR here is sensitive to ܤ only and can be well explained by an ordinary WAL effect in the Bi 2Se3/Al 2O3 where ܬൎ0 . In sharp contrast, unusual MR behaviors were seen for Bi 2Se3/TmIG. Firstly, Bi2Se3/TmIG also shows clear negative MR in the intermediate fields as Bi 2Se3/YIG (Fig. 2(e) ). Note that it is difficult to directly measure the ܤ-dependent magnetization of TmIG films because of large low- ܶ paramagnetic background from GGG [30,41] . The total anisotropy field of TmIG is ~0.07 T at room temperature [31], which should not increase dramatically at low ܶ as it is compromised by increasing saturation magnetization of TmIG. At fields ܤ 1.2 ܶ where negative MR starts to appear, we expect that the magnetization of TmIG has been saturated by ܤ .Secondly, as shown in Fig. 2(f) , the ܤ dependence of MR systematically changes with ߠ୷. At low ܤ, the MR curves for different ߠ୷’s coincide well because WAL is governed primarily by ܤ. The MR curves split when ܤ1 . 2 ܶ and possess weaker negative MR for smaller ߠ୷. The correspondence between negative MR and ߠ୷ can be best explained by a tunable exchange gap. As illustrated in Fig. 2(a) , when the applied field was sufficient to align ۻ at interface, the exchange gap size is tuned by re-orienting ۻ .Since the exchange gap ܯܬ ܯן ןs i n ߠ ୷, it follows that the negative MR of Bi 2Se3/TmIG, or the weight of WL, is in positive correlation with the exchange gap size. Because the exchange gap size determines the deviations of the Berry’s phase from ߨ ,the effect of gap tuning is manifested as the variable negative MR with ߠ୷. In principle, an exchange gap can be induced locally by individual magnetic impurities [42] . Although magnetic impurities deposited on a TI surface can acquire ferromagnetism via Ruderman–Kittel–Kasuya–Yosidas (RKKY) type interaction mediated by Dirac fermions [42], this may not be the leading mechanism for ferromagnetism in a TI/FI system, where interlayer exchange coupling plays the major role. To realize QAHE, ferromagnetism needs be establishe d for an exchange gap opened macroscopically without an applied field [13]. In the following, we show that the Bi 2Se3/TmIG does meet the criterion. Figure 3(a) shows a representative curve of AHE at 100 K. A square hysteresis loop of Hall resistance was observed after the contribution from the ordinary Hall effect was subtracted, based on the empirical formula ܴ୷୶ൌܴ Hሺܤሻܴ AHሺܤሻ, where ܴH is the ordinary Hall resistance, and ܴAH represents the AH resistance. Since TmIG layer is insulating, the AH resistance dominantly comes from the TI layer. The hysteresis loop resembles that of TmIG magnetization [31] . As displayed in Fig. 3(b) , the switching field of the hysteresis loops ܤୡ increases rapidly as temperature was lowered. The enhanced ܪୡ is likely associated with the larger strain in TmIG at low temperatures [33] . Moreover, the effect of the stray field on ܴAH was negligible as we did not observe an AH resistance in Bi 2Se3/Al 2O3/TmIG, in which the interfacial exchange coupling is greatly suppressed by nonmagnetic Al 2O3 (see Supplemental materials, Fig. S2(a), (b) [39]). The above observations indicate that a spontaneous magnetization, ଵ, has developed at Bi 2Se3/TmIG interface because of MPE, with the magnetized Bi 2Se3 bottom surface effectively acting as a magnetic conductor. The AH resistance can be further transformed to AH conductance using ߪ AHߩ؆ AH/ߩ୶୶. Figure 3(b) shows the temperature dependence of AH conductance amplitude ߪAH. ߪAH decays moderately with increasing ܶ below 50 K, and persists up to 180 K. In the weakly disorder limit, the exchange gap size can be estimated from total ߪAH using ߪAHൎమ ቀ ாFቁଷ , taking into account of extrinsic AH conductivity [43]. With ܧF0 . 1 5 eV for bulk-conductive Bi 2Se3 [23] , a lower bound of exchange gap size ~7.7 meV at 10 K is determined. The gap size is in good agreement with 9 meV obtained from density-functional theory calculations for EuS/Bi 2Se3 [44]. The gap is one order of magnitude smaller than that observed in magnetically doped TIs of ~100 meV [13,45] . However, the very large surface state gap in doped TIs is likely nonmagnetic and caused by resonant states induced by impurities near the Dirac point [45]. To clarify the role of MPE-induced magn etization, the MR measurements were conducted at 100 K to preclude low-fiel d quantum interference effects of TSS. Figure 4(a) shows the field-dependent resistance of our samples with longitudinal ( צܴ ,) transverse ( ܴT), and perpendicular ( ܴୄ) fields. Distinct turnings of צܴ and ܴT were observed at ~ 0.5 T, which were attributed to the field needed to fully saturate the perpendicular magnetization in Bi 2Se3/TmIG toward in-plane direction. Below the saturation field, צܴ and ܴT progressively increase with the increasing in-plane field, and צܴܴ T in particular. Meanwhile, ܴୄ is parabolic because of ܤ-induced Lorentz deflection (see also Figs. 4(c) and (d)). We recognize the MR behaviors in Fig. 4(a) as features of AMR caused by MPE. Regardless of the domain configuration, an in-plane field promotes (diminishes) the average of in-plane (perpendicular) MPE-induced magnetization ݉ۃଵ୶,ଵ୷ ۄ( ݉ۃଵۄ )until the saturation field was reached. The subsequent increase of צܴ and ܴT implies that ݉ۃଵ୶ۄ and ݉ۃଵ୷ۄ contribute larger resistances than ݉ۃଵۄ does, i.e. ܴ,צܴTܴ ୄ. Indeed, in the regime |ܤ|൏0 . 7 ܶ where ܴୄ was not overwhelmingly enhanced by ܤ ,the AMR relation of magnetic thin films, צܴܴ Tܴ ୄ [46], has already appeared. Furthermore, Figure 4(a) also rules out SMR to be the dominant source of the AH resistances because SMR features צܴൎܴ ୄܴ T [27] . The AMR amplitude צܴെܴ T continues to build up as ܤ went larger, further justified by the ሺcosሻଶ dependence on ߶୶୷ shown in Fig. 4(b) . Corresponding planar Hall effect characteristic of fe rromagnetic conductors was also detected (Supplemental materials, Fig. S4(a), (b) [39] ). Alternatively, צܴെܴ T can be extracted from the resistance difference of ߠ୷ and ߙ୶ scans displayed in Fig. 4(c) , despite the large contributions of Lorentz deflection in ܴୄ. As a comparison, nearly no ߶୶୷ dependence of resistances was detected in Bi 2Se3/Al 2O3 (Fig. 4(d) ). We notice that the field-enhanced צܴെܴ T was also reported in Pt/YIG, where the authors stated that the large-field צܴെܴ T mostly came from the "hybrid MR" of MPE [47] , which exhibits the same relation צܴൎܴ ୄܴ T as that of SMR. Notwithstanding the similarity between the two systems, we point out that Bi 2Se3 cannot be simply treated as a heavy metal with strong spin-orbit-coupling even at an elevated temperature. The MPE in Bi 2Se3/TmIG involves hybridization between Fe d-orbitals and the paramagnetic TSS arising from the Bi and Se p-orbitals, as opposed to Pt/YIG or other Pt/FM structures where d-d interaction is responsible for MPE. The distinct MR behaviors, AMR in Bi 2Se3/TmIG contrary to SMR/hybrid MR in Pt/YIG, may be an important clue to the microsc opic transport property of MPE in TI/FI. To summarize, a competing WL along with a suppressed WAL has been observed in Bi 2Se3/YIG and Bi 2Se3/TmIG. Bi 2Se3 thickness dependence study suggests that the WL comes from an exchange gap of TSSs opened at interface. In addition, the weight of WL evolves with tilted MPE-induced magnetization. Such angular dependence consolidates the exchange gap as the origin of WL, and the variable WL strength signifies tunability of the gap. Moreover, the well-defined square ܴ AH loops in Bi 2Se3/TmIG unambiguously point to a long-range ferromagnetic order at the interface, and thus ensure a macroscopic and uniform exchange gap at zero field. The MPE-induced ferromagnetism in Bi 2Se3/TmIG is doubly evidenced by typical AMR characteristics, alleviating the concern of spin current effects on the magneto-transport. The simultaneous presence of the MPE-induced long-range ferromagnetic order at the surface and the exchange gap is thus realized in the prototypical TI Bi 2Se3, pending the Fermi-level tuning to deplete the bulk conduction. Lastly, by circumventing the inhomogeneous magnetic doping and the impurity-induced resonance state problems that have been encountered in magnetically doped Bi 2Se3 [13,45] , our study demonstrates that MPE could be a more viable way of introducing ferromagnetism in various TI systems. 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Transport properties of Bi 2Se3/YIG of various Bi 2Se3 thickness and one Bi2Se3/Al 2O3 bilayer. (a) Sheet resistance ܴୱ vs temperature ܶ( .b) Magnetoresistance (MR) measured at 2 K. (c) The magnetoconductance (MC) obtained by subtracting the contribution from the ܤߚଶ term of Eq. (2). Inset: decomposition of the MC curve into the WL, the WAL, and the B2 components for the 10 nm Bi 2Se3/YIG samples. The MC curve can be we ll-fitted to Eq. (2). (d) Thickness dependence of ߙ and ߙଵ extracted by curve fitting to Eq. (2). Figure 2. (a) Illustration of exchange gap opening and its size dependence on the direction of ( .b) Configurations of three different angular dependent resistance with field rotating in the xy-plane ( ߶ ୶୷-scan), yz-plane ( ߠ୷-scan), and xz-plane (ߙ୶-scan). (e)-(f) MR measurement at 2 K with magnetic field applied at ߠ୷ൌ 90°, 60° and 30°. In (c) and (e), resistances are plotted as a function of the magnetic field strength for Bi 2Se3/Al 2O3 and Bi 2Se3/TmIG, respectively. The field data are further transformed by ܤsinߠ ௬ൌܤ to show the ܤ dependence of MR in (d) and (f). Figure 3. (a) A representative ܴAHെܤ hysteresis loop of Bi 2Se3/TmIG at 100 K. (b) Amplitude of AH conductance ΔߪAH and ܤୡ as a function of temperature. Figure 4. Field- and angular dependent resistances of Bi 2Se3/TmIG and Bi 2Se3/Al 2O3 at 100 K. (a) Field-dependent צܴ ,ܴT, and ܴୄ of Bi 2Se3/TmIG. (b) ߶୶୷-, (c) ߠ୷-, and ߙ୶-dependent resistances of Bi 2Se3/TmIG. Here MR is defined as ൫ܴୱሺ݅ሻെ ܴୱሺ90°ሻ൯/ܴୱሺ90°ሻ with ݅ൌ߶ ୶୷,ߠ୷, and ߙ୶. (d) ߶୶୷-, ߠ୷-, and ߙ୶-dependent resistances of Bi 2Se3/Al 2O3. Fig. S1. Dependence of the fitted parameters on the curve fitting range . The figures in each column display data belonging to a specific sample indicated by the column headlines . The top row displays the MC curve s of each sample , where the insets show the MC curves at 𝐵<0.5 T. The 2nd, 3rd, and 4th rows show the extracted 𝛼0 and 𝛼1, 𝑙0 and 𝑙1, and the electron mobility obtained by curve fitting ( 𝜇L) and Hall measurement s (𝜇H), respectively. Error bars represent the standard errors of the fitted parameters. Fig. S2. Magneto -transport data of 6 nm Bi 2Se3/3 nm Al 2O3/TmIG . (a) The MR curve measured at 2 K. (b) The anomalous Hall resistance 𝑅AH taken at 100 K. Black and redy symbols represent magnetic field sweeping up and down, respectively. Fig. S3 . In-plane MR curve of (a) 9 nm Bi 2Se3/Al 2O3 and (b) 9 nm Bi 2Se3/TmIG bilayers measured at 2 K. Fig. S4 . Planar Hall effect data of 9 nm Bi 2Se3/TmIG. (a) 𝜙xy-dependent 𝑅yx for various 𝐵. Solid lines are fitted curves using Eq. (S2) . (b) Comparison of the extracted AMR and PHE amplitude 𝑅∥−𝑅T as a function of 𝐵. Fig. S5 . 𝑇-dependent MR curves of (a) 7 nm Bi 2Se3/YIG and (b) 9 nm Bi 2Se3/TmIG. 1. Discussion s of other possible origins of the negative MR at intermediate field Negative MR in TI thin films could also result from physical mechanisms other than TRS -breaking of surface states . Below we describe these mechanisms and discuss their possibilities in our Bi 2Se3/YIG and Bi 2Se3/TmIG samples. (i) Defect -induced hopping transport It has been shown that when crystal defects are deliberately introduced into TI thin films, a negative MR shows up as the applied fi eld increases. As described in R ef. [37], the samples that underwent ion milling treatme nts show considerably modified MR characteristics. At low fields, positive MR still dominates indicating the robustness of TSSs and the WAL ef fect against defects. However, a negative MR starts to take over when 𝐵>3 T and shows a quadratic dependence up to 𝐵=9 T. The large - 𝐵 negative MR was attributed to field -dependent hopping probabilities among defect states in transport. In disordered semiconductors, localized spins hos ted by defects lead to spin-dependent scattering of electrons. Such a process is suppressed when a large 𝐵 is applied that effectively align s the localized spins, giving rise to a negative MR. Here we note a key difference of the negative MR of Bi 2Se3/YIG and Bi2Se3/TmIG from that reported in R ef. [37]. The negative MR presented in our work occur s at smaller fields ( 3 T>𝐵>0.3 T), and at large field s the classical positive MR from electron cyclotron motion dominates. This is in sharp contrast to the defect -induced negative MR sho wing quadratic behaviors, who se magnitude is large enough to overcome the contribution of Lorentz deflection . Hence, the defect -induced ho pping transport is unlikely to be the origin of negative MR in Bi 2Se3/YIG and Bi 2Se3/TmIG. (ii) Formation of hybridization gap of top and bottom TSSs It is well -known that in a 3D TI , when its thickness approach 2D limits, the overlap of wave functions of the top and bottom TSSs causes a hybridization gap opened at Dirac point [48]. The 2D limit is 6 QL for Bi2Se3. The negative MR due to the hybridization gap -induced WL has been detected in bulk-insulating (Bi 0.57Sb0.43)2Te3 thin films [49]. The negative MR due to hybridization gap observed in Ref. [49] is actually similar to that in Bi 2Se3/YIG and Bi 2Se3/TmIG and primarily locate at even small er 𝐵< 0.3 T. The emergence of the WL effect upon hybridization gap opening can also be understood as a result of Berry phase 𝜋(1−∆H/2𝐸F) deviating from 𝜋, where ∆H denotes the size of the hybridization gap. In this aspect, both hybrid ization - and TRS -breaking -induced gaps give rise to WL effect . In our experiments, the WL was also detected in Bi 2Se3 thicker than 6 QL grown on YIG and TmIG . The positive MC component can be extracted for Bi2Se3 as thick as 16 QL as shown in Fig. 3(c) . As a comparison, the 9 QL Bi2Se3/Al 2O3 sample show s a very sharp negative MC cusp of WAL. Hence, we can conclude that for Bi2Se3 films at 9 QL or thicker, hybridization between top and bottom surfaces should not be a concern in interpreting the WL in Bi 2Se3/YIG and Bi 2Se3/TmIG. Nevertheless, the similarities between the negative MR reported in Ref. [49] and our work suggests that a Dirac gap is indeed opened, despite of a completely different physical origin. (iii) Quantized 2D bulk bands in thin TIs As studied in Ref. [38], a WL could also arise from quantized 2D subbands in ultrathin TI films. We again compare the MC of 9 QL Bi2Se3/Al 2O3 and 10 QL Bi 2Se3/YIG in Fig. 3(c) . Since the two samples exhibit comparable carrier concentration 𝑛2D, 𝑅s and thickness , the contribution of quantized bulk bands participating in transport in one sample should not differ significantly from the other. Obviously, the negative MR is absent in Bi 2Se3 9 QL/Al 2O3. Hence , we rule out such subbands as the main source of the WL in Bi 2Se3/YIG and Bi 2Se3/TmIG. 2. Curve fitting detail s of Fig. 1 The cusp-like feature at small fields resulted from WAL or WL is usually described by the one -component HLN equation. For our sample s, the negative MR at intermediate fields implies a much larger dephasing field of WL than that of WAL. To characterize the WL component, the curve fitting range is extended , and anoth er component of HLN equation including 𝛽𝐵2 (Eq. (2) ) is introduced into the fitting function. It has been shown that a 𝛽𝐵2 term is necessary to perform the curve fitting in wide ranges of field and tempera ture [50]. The first two terms of Eq. (2) present the competition effects of WL and WAL. The 𝛽𝐵2 term account for the classical cyclotronic moti on and other terms of quantum correction s in the conventional HLN equation [50]. In the following , we discuss the analyses of the MC data and the curve fitting at small and large fiel ds separately. (i) Small -field regime ( 𝐵<1 𝑇): suppressed WAL In this regime the weight of the 𝐵2 term is negligible, so the fitting involves four independent parameters, 𝛼0, 𝛼1, 𝑙0, and 𝑙1 in the beginning . We found that four -parameter fittings do not render reliable results. Instead, it is possible to obtain several sets of fitted parameters that all give reasonably good fits by manually adjust the parameter values. Since WL and WAL terms share the sam e mat hematical form, the curve fitting is valid only when the effective phase coherence lengths 𝑙0 and 𝑙1 differ to some extent s and the fitting range exceeds dephasing fields , otherwise an unique set of fitted parameters cannot be found . In this regime, we thus set 𝛼0 and 𝑙0 of WL to be zero. The results are shown in the 2nd and 3rd rows of Fig. S1. Therefore, only a suppressed WAL term can be concluded in this regime for our sample s. (ii) Larger field regime (𝐵>1 𝑇): negative MR and WL The presence of WL is seen in negative MR located at intermediate -field regime (Fig. 1(b) ). To disclose the characteristics of the WL effect, the curve fittin g range is extended to several T eslas. The MC curves within 4 T are shown in the first row of Fig. S1, and they exhibit a parabolic 𝐵 dependence toward 9 T. In this regime, five -parameter fitting, including the 𝛽𝐵2 term, has been performed . 𝛽 is composed of the classical cyclotronic part 𝛽c and the quantum correction one 𝛽q from the other two term s of the original HLN equation : 𝛽q𝐵2≈−𝑒2 24𝜋ℎ[𝐵 𝐵SO+𝐵𝑒]2 + 3𝑒2 48𝜋ℎ[𝐵 (4/3)𝐵SO+𝐵𝜙]2 , where 𝐵SO and 𝐵𝜙 are characteristic fields of the spin-orbit scattering length 𝑙SO and phase -coherence length 𝑙𝜙. Here, 𝛽<0 due to the negative MC at l arge fields. The five -parameters fitting results are shown in the 2nd to 4th rows of Fig. S1. The MC curves of all samples can be well fitted to Eq. (2), except the one of 40 nm Bi 2Se3/YIG which deviates the most from a 2D electron system . Since fitted param eters depend on the data range selected for the curve fitting , we di splay them as a functio n of the curve fitting range. The fitted pa rameters show a moderate ~10 % variations with respect to the fitting ranges. Throughout th e fitting ranges, the magnitudes of 𝛼0, 𝛼1, 𝑙0, and 𝑙1 can be compared without ambiguity. The reliability of the five - parameter fit s of the data is further justi fied by the following three observations . First ly, the 𝛼1’s and 𝑙1’s of the WAL component obtained from two -parameter fitting s at small fields are in good agreement with those from five -parameter fitting s. This implies the suitability of Eq. (2) for the MC behavior of Bi 2Se3/YIG and Bi 2Se3/TmIG in a wide range of 𝐵. Secondly, from the dephasing field 𝐵i calculated from ℏ/(4𝑒𝑙i2), we note that 𝐵0 is much larger than 𝐵1, which agrees with the observation that the negative MR shows up at larger fields . The notable difference of 𝑙0 and 𝑙1 causes the WL and WAL to manifest themselves in different regimes of 𝐵. Thirdly , if we set 𝛽q≈0, the electron mobility calculated by 𝜇L= √−𝛽c𝑅s overlap well with that calculated by our Hall measurement data 𝜇H=1/(𝑒𝑛2D𝑅xx), indicated by the blue dashed line. 𝛽q≈0 corresponds to a very large 𝐵SO or small 𝑙SO. 3. Data of the controlled sample Bi2Se3/Al 2O3/TmIG A trilayer sample 6 nm Bi 2Se3/3 nm Al 2O3/TmIG has been fabricated to test the effect of stray fields of TmIG. Here, the 3 nm Al 2O3 layer was deposited using atomic layer deposition (ALD). The nonmagnetic Al 2O3 insertion layer ought to suppress the interlayer exchange coupling of Bi 2Se3 and TmIG, while allow s the stray field to penetrate . Fig. S2(a) show s the MR of Bi 2Se3/Al 2O3/TmIG at 2 K. A cusp-like positive MR of the WAL effect was observed, and no negative MR was detected. Fig. S2(b) displays the 𝑅AH data as a function of 𝐵. No hysteresis loop was detected. Therefore, the data in Fig. S2 implies that stray field s are not the root cause of the negative MR and hysteric 𝑅AH loops observed in the Bi2Se3/TmIG. 4. In-plane MR data at 2 K Fig. S3(a) and (b) show the MR data under an in-plane applied (𝜃yz=0) field taken at 2 K for 9 nm Bi 2Se3/Al 2O3 and 9 nm Bi 2Se3/TmIG, respectively. For the Bi2Se3/Al 2O3 bilayer, a positive MR is detected . The MR induced by an in -plane field in Bi 2Se3 has been studied extensively in R ef. [51]. In short, the applica tion of in-plane fields force s the electron to scatter between top and bottom surface s, and the presence of bulk state is essential in understanding the in-plane MR. It was demonstrated that no existing theory can well -describe the distinct transport prope rties o f TIs under an in -plane field, thus highlighting the important role of bulk-surface coupling of TIs . The qualitatively difference between perpendicular and in -plane MR of Bi 2Se3 can already been seen by comparing Fig. 2(d) and S3(a) . While the MR with tilted field angles can be very well explained by WAL governed by 𝐵z, it can be inferred that when 𝑩 is rotated across a critical angle of 𝜃yz (< 30°), another physical picture of magneto -transport that dictates the in -plane MR comes into play in this regime . For Bi 2Se3/TmIG, the in -plane MR exhibit distinct feature s from those of Bi2Se3/Al 2O3: the MR is positive at 𝐵<3 T and becomes negative when 𝐵 goes larger. We may differentiate the physical origin of the in -plane negative MR from that of perpendicular one. From Eq. (1), we see that a gap in the TSS can only be opened by a perpendicular magnetization 𝑀z, while 𝑀x and 𝑀y shift the gapless Dirac cone in the momentum space. Although it is argued that an in -plane magnetic field can also break TRS of TIs when the field is aligned with a certain cry stal axis [52], this should not be of importance since our Bi 2Se3 films grown on gar net substrates contain randomly oriented in-plane domains. It is beyond the scope of this work to clarify the in -plane negative MR in Bi 2Se3/TmIG, especially when the magnetic scatt ering due to MPE adds to the complexity of the syste m. However, we emphasize that the observation of in -plane negative MR does not pose a major problem of our interpretations of the negative MR under tilted fields. As in the case of Bi 2Se3/Al 2O3, a different scheme of physical model is needed for the in -plane MR. 5. Planar Hall effect (PHE) in Bi 2Se3/TmIG For a measurement configuration defined Fig. 2(b), the anisotropic resistivity tensor induced by an in -plane field can be reduced to two elements , 𝑅s (or 𝑅xx) and 𝑅yx, with respect to the sample coordinate. Phenomenologically , the field - angle -dependent 𝑅s and 𝑅yx are identified as AMR and planar Hall resistances, respectively when they are expressed as, AMR : 𝑅s(𝜙xy)=𝑅T+(𝑅∥−𝑅T)cos2𝜙xy (S1) PHE : 𝑅yx(𝜙xy)=(𝑅∥−𝑅T)sin𝜙xycos𝜙xy (S2) Fig. S4(a) shows the 𝑅yx(𝜙xy) data of 9 QL Bi 2Se3/TmIG . The 𝑅yx satisfies the angular dependence sin𝜙xycos𝜙xy of PHE . By fitting the data in Fig. 4(b) and Fig. S4(a) to Eq. (S1) and (S2) respectively, we extract the c oefficient of the angular terms 𝑅∥−𝑅T for 𝑅s and 𝑅yx data. Fig. S 4(b) compares the 𝑅∥−𝑅T obtained from 𝑅s and 𝑅xy data at various fields . One immediately sees a good agreement between the two sets of data . PHE in TI has been previously observed in non-magnetic (Bi,Sb) 2Te3 films [53] and EuS/(Bi,Sb) 2Te3 [54]. In Ref. [53], the PHE results from anisotropic scattering of Dirac fermions due to T RS broken by an i n-plane field. T he PHE amplitude can be altered dramati cally by dual -gating, showing a unique two -peak profile as the Fermi level moves across the Dirac point. We are not able to completely preclude such a scenario in Bi 2Se3/TmIG bilayer, where a similar effect could also be caused by the interfacial e xchange effective field. Fermi -level dependent measurements enabled by top -gating will be performed to investigate this kind of PHE . In Ref. [54], an unconventional PHE was detected, whose angular dependence cannot be described by Eq. (S2). The autho rs argue that a non -linear Hall response defined a s 𝑗y=𝜎yxx𝐸x2 should be considered. The proposed possible origins of the non-linea r Hall response includes current -induced spin -orbit torques from TSSs, asymmetric scattering of electrons by magnons in magnetic TIs, and interband transitions between the two branches of the Dirac surface states. These scenario play minor roles , if any, in Bi 2Se3/TmIG because such an unconventional PHE was not observed in this work . Reference: [48] Y . Zhang, K. He, C. -Z. Chang, C. -L. Song, L. -L. Wang, X. Chen, J. -F. Jia, Z. Fang, X. Dai, W. -Y . Shan, S. -Q. Shen, Q. Niu, X. -L. Qi, S. -C. Zhang, X. -C. Ma, and Q. -K. Xue, Nat. Phys. 6, 584 (2010). [49] Z. Tang, E. Shikoh, H. Ago, K. Kawahara, Y . Ando, T. Shinjo, and M. Shiraishi, Physical Review B 87 (2013). [50] B. A. Assaf, T. Cardinal, P. Wei, F. Katmis, J. S. Moodera, and D. Heiman, Appl. Phys. Lett. 102, 012102 (2013). [51] C. J. Lin, X. Y . He, J. L iao, X. X. Wang, V . S. Iv, W. M. Yang, T. Guan, Q. M. Zhang, L. Gu, G. Y . Zhang, C. G. Zeng, X. Dai, K. H. Wu, and Y . Q. Li, Phys. Rev. B 88, 041307(R) (2013). [52] C. K. Chiu, J. C. Y . Teo, A. P. Schnyder, and S. Ryu, Rev. Mod. Phys. 88 (2016). [53] A. A. Taskin, H. F. Legg, F. Yang, S. Sasaki, Y . Kanai, K. Matsumoto, A. Rosch, and Y . Ando, Nat. Commun. 8, 1340 (2017). [54] D. Rakhmilevich, F. Wang, W. Zhao, M. H. W. Chan, J. S. Moodera, C. Liu, and C. Z. Chang, Phys. Rev. B 98, 094404 (2018).

2018-11-02

Transport signatures of exchange gap opening because of magnetic proximity effect (MPE) are reported for bilayer structures of Bi2Se3 thin films on yttrium iron garnet (YIG) and thulium iron garnet (TmIG) of perpendicular magnetic anisotropy (PMA). Pronounced negative magnetoresistance (MR) was detected, and attributed to an emergent weak localization (WL) effect superimposing on a weak antilocalization (WAL). Thickness-dependent study shows that the WL originates from the time-reversal-symmetry breaking of topological surface states by interfacial exchange coupling. The weight of WL declined when the interfacial magnetization was aligned toward the in-plane direction, which is understood as the effect of tuning the exchange gap size by varying the perpendicular magnetization component. Importantly, magnetotransport study revealed anomalous Hall effect (AHE) of square loops and anisotropic magnetoresistance (AMR) characteristic, typifying a ferromagnetic conductor in Bi2Se3/TmIG, and the presence of an interfacial ferromagnetism driven by MPE. Coexistence of MPE-induced ferromagnetism and the finite exchange gap provides an opportunity of realizing zero magnetic-field dissipation-less transport in topological insulator/ferromagnetic insulator heterostructures.

Evidence for exchange Dirac gap in magneto-transport of topological insulator-magnetic insulator heterostructures

1811.00689v1

Magnon contribution to unidirectional spin Hall magnetoresistance in ferromagnetic-insulator/heavy-metal bilayers W.P. Sterk and D. Peerlings Institute for Theoretical Physics, Utrecht University, Princetonplein 5, 3584 CC Utrecht, The Netherlands R.A. Duine Institute for Theoretical Physics, Utrecht University, Princetonplein 5, 3584 CC Utrecht, The Netherlands and Department of Applied Physics, Eindhoven University of Technology, PO Box 513, 5600 MB Eindhoven, The Netherlands (Dated: March 1, 2019) We develop a model for the magnonic contribution to the unidirectional spin Hall magnetore- sistance (USMR) of heavy metal/ferromagnetic insulator bilayer films. We show that diffusive transport of Holstein-Primakoff magnons leads to an accumulation of spin near the bilayer interface, giving rise to a magnoresistance which is not invariant under inversion of the current direction. Un- like the electronic contribution described by Zhang and Vignale [Phys. Rev. B 94, 140411 (2016)], which requires an electrically conductive ferromagnet, the magnonic contribution can occur in fer- romagnetic insulators such as yttrium iron garnet. We show that the magnonic USMR is, to leading order, cubic in the spin Hall angle of the heavy metal, as opposed to the linear relation found for the electronic contribution. We estimate that the maximal magnonic USMR in Pt|YIG bilayers is on the order of 108, but may reach values of up to 105if the magnon gap is suppressed, and can thus become comparable to the electronic contribution in e.g. Pt|Co. We show that the magnonic USMR at a finite magnon gap may be enhanced by an order of magnitude if the magnon diffusion length is decreased to a specific optimal value that depends on various system parameters. PACS numbers: 73.43.Qt, 75.76.+j I. INTRODUCTION Thetotalmagnetoresistanceofmetal/ferromagnethet- erostructures is known to comprise several independent contributions, including but not limited to anisotropic magnetoresistance (AMR) [1], giant magnetoresistance (GMR, in stacked magnetic multilayers) [2] and spin Hall magnetoresistance (SMR) [3]. A common characteristic of these effects is that they are linear; in particular, this means the measured magnetoresistance is invariant un- der reversal of the polarity of the current. In 2015, however, Avci et al.[4] measured a small but distinct asymmetry in the magnetoresistance of Ta|Pt and Co|Pt bilayer films. Due to its striking similarity to the current-in-plane spin Hall effect (SHE) and GMR, save for its nonlinear resistance/current characteristic, this effect was dubbed unidirectional spin Hall magne- toresistance (USMR). In the years following its discovery, USMR has been detected in bilayers consisting of magnetic and nonmag- netictopologicalinsulators[5], andthedependenceofthe USMR on layer thickness has been investigated experi- mentally for Co|Pt bilayers [6]. Additionally, Avci et al. [7] have shown that USMR may be used to distinguish between the four distinct magnetic states of a ferromag- net|normal metal|ferromagnet trilayer stack, highlighting itspotentialapplicationinmultibitelectricallycontrolled memory cells. Although USMR is ostensibly caused by spin accumu- lationattheferromagnet|metalinterface, acompletethe- oretical understanding of this effect is lacking. In bilayerfilms consisting of ferromagnetic metal (FM) and heavy metal (HM) layers, electronic spin accumulation in the ferromagnet caused by spin-dependent electron mobility provides a close match to the observed results [8]. It re- mains unknown, however, whether this is the full story; indeed, this model’s underestimation of the USMR by a factor of two lends plausibility to the idea that there may be additional, as-yet unknown contributions provid- ing the same experimental signature. Additionally, the electronic spin accumulation model cannot be applied to bilayers consisting of a ferromagnetic insulator (FI) and a HM, as there will be no electric current in the ferro- magnet to drive accumulation of spin. Kim et al.[9] have measured the USMR of Py|Pt (where Py denotes for permalloy) bilayer and claim, us- ing qualitative arguments, that a magnonic process is in- volved. Likewise, for Co|Pt and CoCr|Pt, more recent re- sults by Avci et al.[10] argue in favor of the presence of a magnon-scatteringcontributionconsistingoftermslinear and cubic in the applied current, and having a magnitude comparable to the electronic contribution of Zhang and Vignale [8]. Although these experimental results provide a great deal of insight into the underlying processes, a theoretical framework against which they can be tested is presently lacking. In this work, we aim to take first steps to developing such a framework, by considering an accumulation of magnonic spin near the FI|HM bilayer interface, which we describe by means of a drift-diffusion model. The remainder of this article is structured as follows: in Sec. II, we present our analytical model as genericallyarXiv:1810.02610v2 [cond-mat.mes-hall] 28 Feb 20192 as possible. In Sec. III we analyze the behavior of our model using parameters corresponding to a Pt|YIG (YIG being yttrium iron garnet) bilayer as a basis. In par- ticular, in Sec. IIIA we give quantitative predictions of the magnonic USMR in terms of the applied current and layer thicknesses, and in Sec. IIIB we take into account the effect of Joule heating. In the remainder of Sec. III, we investigate the influence of various material param- eters. Finally, in Sec. IV we summarize our key results and present some open questions. II. MAGNONIC SPIN ACCUMULATION To develop a model of the magnonic contribution to the USMR, we focus on the simplest FI|HM heterostruc- ture: a homogeneous bilayer. We treat the transport of magnonic and electronic spin as diffusive, and solve the resulting diffusion equations subject to a quadratic boundary condition at the interface. In this approach, valid in the opaque interface limit, current-dependent spin accumulations—electronic in the HM and magnonic in the FI—form near the interface. In particular, the use of a nonlinear boundary condition breaks the invariance of the SMR under reversal of the current direction, i.e. it produces USMR. We consider a sample consisting of a FI layer of thick- nessLFIdirectlycontactingaHMlayerofthickness LHM. We take the interface to be the xyplane, such that the FI layer extends from z= 0toLFIand the HM layer from z=LHMto 0. The magnetisation is chosen to lie in the positive y-direction, and an electric field E=E^ xis applied in the x-direction. The set-up is shown in Fig. 1. The extents of the system parallel to the interface are taken to be infinite, and the individual layers com- pletely homogeneous. This allows us to treat the system as quasi-one-dimensional, in the sense that we will only consider spin currents that flow in the z-direction. We account for magnetic anisotropy only indirectly through the existence of a magnon gap. We further assume that our system is adequately described by the Drude model (suitably extended to include spin effects[11]), and that the interface between layers is not fully transparent to spin current, i.e., has a finite spin-mixing conductance [12]. For simplicity, we assume electronic spin and charge transport may be neglected in the ferromagnet, as is the case for ferromagnetic insulators. Wedescribethetransferofspinacrosstheinterfacemi- croscopically by the continuum-limit interaction Hamil- tonian Hint=Z d3rd3r0J(r;r0)h by(r0)cy #(r)c"(r) +b(r0)cy "(r)c#(r)i ; wherecy (r)[c(r)] are fermionic creation [annihilation] operators of electrons with spin 2f";#gat position rin the HM, and by(r0)[b(r0)] is the bosonic creation FIG. 1. Schematic depiction of our system. The magnetiza- tionMof the FI layer lies in the +ydirection, an electric field of magnitude Eis applied to the heavy metal layer (HM) in thexdirection, and the interface between the layers lies in thexyplane. [annihilation] operator of a circularly polarized Holstein- Primakoff magnon [13] at position r0inside the ferro- magnet. We leave J(r;r0)to be some unknown coupling between the electrons and magnons, which is ultimately fixed by taking the classical limit [14, 15]. Transforming to momentum space and using Fermi’s golden rule, we obtain the interfacial spin current jint s, which can be expressed in terms of the real part of the spin mixing conductance per unit area g"# ras [14, 16] jint s=g"# r sZ d"g(")("
)  nB"
 kBTe nB"m kBTm :(1) (Similar expressions were derived by Takahashi et al.[17] and Zhang and Zhang [18], although these are not given in terms of the spin-mixing conductance.) Here,sis the saturated spin density in the FI layer, g(")is the magnon density of states, nB(x) = [ex1]1 is the Bose-Einstein distribution function, kBis Boltz- mann’s constant, and TmandTeare the temperatures of the magnon and electron distributions, respectively, which we do not assume a priori to be equal (al- though the equal-temperature special case will be our primary interest). Of crucial importance in Eq. (1) are the magnon effective chemical potential m—which we shall henceforth primarily refer to as the magnon spin accumulation—and the electron spin accumulation
"#, which we define as the difference in chem- ical potentials for the spin-up and spin-down electrons. (In both cases, a positive accumulation means the major- ity of spin magnetic moments point in the +ydirection.) We employ the magnon density of states g(") =p "
42J3 2s("
): Here,Jsis the spin wave stiffness constant, (x)is the Heavisidestepfunction, and
isthemagnongap, caused3 by a combination of external magnetic fields and in- ternal anisotropy fields in ferromagnetic materials [19]. In our primary analysis of a Pt|YIG bilayer, we take
B1 TkB0:67 KwithBthe Bohr mag- neton, in good agreement with e.g. Cherepanov et al. [20], and in Sec. IIIE we specifically consider the limit of a vanishing magnon gap. To treat the accumulations on equal footing, we now redefinem!mand
!
, expand Eq. (1) to second order in , and set= 1to obtain jint s'" kBTmI0+Ie
+Imm+Iee kBTe(
)2 +Imm kBTm2 m+Ime kBTmm
# g"# r(kBTm)3 2 43J3 2ss:(2) Here, the Iiare dimensionless integrals given by Eqs. (A.1) in the Appendix. All Iiare functions of Tm and
, andI0,IeandIeeadditionally depend on Te. In the special case where Tm=Te,I0vanishes,Im=Ie, andIee=(Imm+Ime). Inadditionto jint s, thespinaccumulationsandtheelec- tric driving field Egive rise to the following spin currents in thezdirection: je s=h 2e  2e@
 @zSHE ;(3a) jm s=m h@m @z: (3b) Hereje sandjm sare the electron and magnon spin cur- rents, respectively. is the electrical conductivity in the HM,mis the magnon conductivity in the ferromagnet, eis the elementary charge, and SHis the spin Hall angle. In line with Cornelissen et al.[21] and Zhang and Zhang [22], we assume the spin accumulations mand
obey diffusion equations along the z-axis: d2m dz2=m l2m;d2
 dz2=
 l2e; wherelmandleare the magnon and electron diffusion lengths, respectively. We solve these equations analyti- cally subject to boundary conditions that demand con- tinuity of the spin current across the interface and con- finement of the currents to the sample: jm s(0) =je s(0) =jint s(0); jm s(LFI) =je s(LHM) = 0: This system of equations now fully specifies the magnonic and electronic spin accumulations mand
, the latter of which enters the charge current jcvia the spin Hall effect: jc(z) =E+SH 2e@
(z) @z: (4)The measured resistivity at some electric field strength Eis then given by the ratio of the electric field and the averaged charge current: (E) =E 1 LHMR0 LHMdzjc(z): (5) Finally, we define the USMR Uas the fractional differ- ence in resistivity on inverting the electric field: U(E)(E) (E)=1 +R0 LHMdzjc(z;E) R0 LHMdzjc(z;E): It should be noted that the even-ordered terms in the expansion of the interface current are vital to the appear- anceofunidrectionalSMR.Supposeoursystemhasequal magnon and electron temperature, such that the interfa- cial spin Seebeck term I0vanishes (see Section IIIB), and we ignore the quadratic terms in Eq. (2). Then be- cause the only term in the spin current equations (3) that is independent of the accumulations is hSH 2eEin Eq. (3a), we have that
/m/E. Then by Eqs. (4) and (5),jc/Eand(E)/E E, such thatU= 0. Con- versely, with quadratic terms in the interfacial spin cur- rent,(E)E E+E2, and likewise if I0does not vanish, (E)E 1+E. BothcasesgivenonvanishingUSMR.Phys- ically, one can say that the spin-dependent electron and magnon populations couple together in a nonlinear fash- ion (namely, through the Bose-Einstein distributions in Eq. (1)), leading to a nonlinear dependence on the elec- tric field. III. RESULTS A. Equal-temperature, finite gap case Although our model can be solved analytically (up to evaluation of the integrals Ii), the full expression ofUis unwieldy and therefore hardly insightful. To get an idea of the behavior of a real system, we use a set of parameters—listed in Table I—corresponding to a Pt|YIG bilayer as a starting point. (Unless otherwise specified, all parameters used henceforth are to be taken from this table.) Fig. 2 shows the magnonic USMR of a Pt|YIG bilayer versus applied driving current ( E) whenTm=Te=T, at the temperature of liquid nitrogen ( 77 K, blue), room temperature ( 293 K, green) and the Curie temperature of YIG ( 560 K[20], red). FI and HM layer thicknesses used are90 nmand3 nm, respectively, in line with experimen- tal measurements by Avci et al.[23]. In all cases the magnonic USMR is proportional to the applied electric current—that is, the cubic term found by Avci et al.[10] is absent—and at room temperature has a value on the order of 109at typical measurement currents [4]. This is roughly four orders of magnitude weaker than the USMR obtained—both experimentally4 0.0 0.2 0.4 0.6 0.8 1.0 σE(A m−2)×10120246U×10−9 T(K) 77 293 5600 600T(K)0.00.5×10−8 FIG. 2. USMRUversus driving current Efor a Pt|YIG bilayer at liquid nitrogen temperature ( 77 K, blue), room temperature ( 293 K, green) and the YIG Curie temperature (560 K, red). Inset: USMR versus system temperature Tat fixed current E= 11012A m2. and theoretically—for FM|HM hybrids [4, 6, 8, 23], and is consistent with the experimental null results obtained for this system by Avci et al.[23]. Note, however, that the thickness of the FI layer used by these authors is significantly lower than the magnon spin diffusion length lm= 326 nm , which results in a suppressed USMR. Furthermore, it can be seen in the inset of Fig. 2 that the magnonic USMR is, to good approximation, linear in the system temperature, in agreement with observations by Kim et al.[9] and Avci et al.[10]. 1 2 3 4 5 LFI(µm)1020304050LHM(nm) 0.00.51.01.52.02.53.03.54.0 U×10−8 FIG. 3. Pt|YIG USMR UatTm=Te= 293 K versus FI layer thickness LFIand HM layer thickness LHM. A driving currentE= 11012A m2is used. A maximal USMR of 4:2108is reached at LHM= 4:5 nm,LFI= 5µm. In Fig. 3 we compute the USMR at E = 11012A m2as a function of both LFIandLHM. A maximum is reached around LHM4:5 nm, while in terms ofLFI, a plateau is approached within a few spindiffusion lengths. By varying the layer thicknesses, a maximal USMR of 4:2108can be achieved, an im- provement of one order of magnitude compared to the thicknesses used by Avci et al.[23]. B. Thermal effects We take into account a difference between the electron and magnon temperatures TeandTmby assuming these parameters are equal to the temperatures of the HM and FI layers, respectively, which we take to be homogeneous. We assume that the HM undergoes ohmic heating and dissipates this heat into the ferromagnet, which we take to be an infinite heat bath at temperature Tm. We only take into account the interfacial (Kapitza) thermal resis- tanceRthbetween the HM and FI layers, leading to a simple expression for the HM temperature Te: Te=Tm+RthE2LHM: Using this model, we still find a linear dependence in the electric field, U 'uE(Tm)E, but the coefficient uE(Tm)increases by three orders of magnitude compared to the case where the electron and magnon temperatures are set to be equal. The overwhelming majority of this increase can be attributed to an interfacial spin Seebeck effect (SSE) [21, 24]: it is caused by the accumulation- independent contribution I0(Eq. (A.1a)) in the interface current. When I0is artificially set to 0, uE(Tm)changes less than 1% from its equal-temperature value. Furthermore, the overall magnitude of the interfacial SSE in our system can be attributed to the fact that we have a conductor|insulator interface: the current runs through the HM only, resulting in inhomogeneous Joule heating of the sample and a large temperature disconti- nuity across the interface. C. Spin Hall angle The electronic spin accumulation
at the interface in the standard spin Hall effect is linear in the electric fieldEand spin Hall angle SH[3]. From the linearity inE, we may conclude that the terms in Eq. (2) that are linear in
have a suppressed contribution to the USMR. Thus, the contribution of the interface current is of order2 SH. Furthermore,
enters the charge cur- rent (Eq. (4)) with a prefactor SH, leaving the magnonic USMR predominantly cubic in the spin Hall angle. In- deed, in the special case Tm=Te, expanding the full expression forU(which spans several pages and is there- fore not reproduced within this work) in SHreveals that the first nonzero coefficient is that of 3 SH. This suggests a small change in SHpotentially has a large effect on the USMR. In Fig. 4 we plot the USMR for a Pt|YIG bilayer— once again using Tm=Te= 293 K—consisting of 4:5 nm5 0.0 0.1 0.2 0.3 0.4 0.5 θSH01234U×10−6 Computed Cubic fit FIG. 4. USMRUatTm=Te= 293 K versus spin Hall angle SH. A driving current E= 11012A m2and FI and HM layer thicknesses LFI= 5µmandLHM= 4:5 nmare used. Blue curve: computed value. Dashed green curve: fit of the formU=u3 SH, withu'3:1104. of Pt and 5µmof YIG, in which we sweep the spin Hall angle. Included is a cubic fit U=u3 SH, where we findu'3:1104. Here it can be seen that the magnonic USMR in HM|FI bilayers can, as expected, po- tentiallyacquiremagnitudesroughlycomparabletothose in HM|FM systems, provided one can find or engineer a metal with a spin Hall angle several times greater than that of Pt. This suggests that very strong spin-orbit coupling (SOC) is liable to produce significant magnon- mediated USMR in FI|HM heterostructures, although we expect our model to break down in this regime. D. A note on the magnon spin diffusion length Although we use the analytic expression for the magnon spin diffusion length[18, 21, 22], lm=vthr 2 3mr —wherevthis the magnon thermal velocity, is the com- binedrelaxationtime, and mristhemagnonicrelaxation time (see Table I)—this is known to correspond poorly to reality, being at least an order of magnitude too low in the case of YIG [21]. Artificially setting the magnon spin-diffusion length to the experimental value of 10µm (while otherwise continuing to use the parameters from Table I) results in a drop in USMR of some 4 orders of magnitude. It follows directly that there exists some optimal value oflm(which we shall label lm;opt) that maximizes the USMR, which we plot as a function of the FI layer thicknessLFIin Fig. 5, at LHM= 4:5 nmandE= 11012A m2, and for various values of the magnon- phonon relaxation time mp, which is the shortest and therefore most important timescale we take into account.For the physically realistic value of mp= 1 ps(blue curve), the optimal magnon spin diffusion length is just 24 nm. Although lm;optitself depends on mp, the con- ditionlm=lm;optacts to cancel the dependence of the USMR on the magnon-phonon relaxation time. Curi- ously, the USMR additionally loses its dependence on LFI, reaching a fixed value of 4:14107for our param- eters. 0 5 10 15 LFI(µm)0.00.51.01.52.02.53.0lm,opt (µm)τmp(s) 10−12 10−11 10−10 10−9 10−8 ∞ FIG. 5. Value of the magnon spin diffusion length lmthat maximizes the USMR, as a function of FI layer thickness LFI, at various values of the magnon-phonon relaxation time mp. We further find that lm;optis independent of the spin Hall angle and driving current, and shows a weak de- creasewithincreasingtemperatureprovidedthemagnon- phonon scattering time is sufficiently short. A significant increase in the optimal spin diffusion length is only found at low temperatures and large mp. Similarly, a weak de- pendence on the Gilbert damping constant is found, becoming more significant at large mp, with lower val- ues ofcorresponding to larger lm;opt. Whenis swept, again the USMR at lm=lm;optacquires a universal value of4:14107for our system parameters. E. Effect of the magnon gap We have thus far utilised a fixed magnon gap with a valueof
=B= 1 TforYIG.Althoughthisisreasonable for typical systems, it is possible to significantly reduce the gap size by minimizing the anisotropy fields within the sample, e.g. using a combination of external fields [25], optimized sample shapes [19, 26] and temperature [27, 28]. This leads us to consider the effect a decreased or even vanishing gap may have on our results. Fig. 6 shows the USMR Ufor a Pt|YIG system ( 4:5 nm of Pt and 5µmof YIG) at room temperature, plotted against the driving current E, now for different values of the magnon gap
. Here it can be seen that while U is linear in Efor large gap sizes and realistic currents,6 0.00 0.25 0.50 0.75 1.00 σE(A m−2)×10120.00.51.01.5U×10−5 ∆/µB(T) 10−9 10−6 10−5 10−4 10−3 100 FIG. 6. USMRUof a Pt( 4:5 nm)|YIG( 5µm) bilayer at room temperatureversusappliedcurrent Eatvariousvaluesofthe magnon gap
. For large gaps, linear behavior is recovered at realistic currents, while for smaller gap sizes, the USMR saturates as the current is increased. it shows limiting behavior at smaller gaps, becoming in- dependent of the electric current above some threshold (provided one neglects the effect of Joule heating). At low current and intermediate magnon gap, the current dependenceisnonlinearat O(I2)asopposedtothe O(I3) behavior found by Avci et al.[10]. Note also that the saturation value of the USMR is two to three orders of magnitude greater than the values found previously in our work, and of the same magni- tude as the electronic contribution found by Zhang and Vignale [8]. The maximal value of the USMR that can be achieved may be found by considering the full analytic expression forUin terms of the generic coefficients Iirepresenting the dimensionless integrals given by Eqs. (A.1) in the Appendix. In the gapless limit
!0and at equal magnonandelectrontemperature( Tm=Te), thesecond- order coefficients ImmandImediverge, while their sum takes the constant value Imm+Ime'0:323551at room temperature. Ieedoes not diverge, and obtains the value. Now working in the thick-ferromagnet limit ( LFI! 1), we substitute Ime!Imm+and take the limits E!1andImm!1. By application of l’Hôpital’s rule in the latter, all coefficients Iidrop out of the ex- pression forU. This leaves only the asymptotic value, which, after expanding in SH, reads Umax=4e2l2 s2 SHmtanh2 LHM 2ls h2lmLHM+4lse2LHMmcoth LHM ls+O(4 SH): (6) Whereas the linear-in- Eregime of the magnonic USMR growsas3 SH,wethusfindthattheleading-orderbehavior 10−1710−1410−1110−810−510−2101 ∆/µB(T)10−810−710−610−510−4UσE(A m−2) 107 108 109 1010 1011 1012FIG. 7. USMRUof a Pt( 4:5 nm)|YIG( 5µm) bilayer at room temperature as function of the magnon gap size
, for various valuesofthebasechargecurrent E. Notethelog-logscaling. Solid colored lines: computed USMR. Dashed colored lines: continuationsofthehigh-gaptailsofthecorrespondingcurves accordingtotheone-parameterfit U=u0=p
. Dashedblack line: asymptotic value of the USMR as given by Eq. (6). of the asymptotic value is only 2 SH, and the third-order term vanishes completely. Physically, this can be ex- plained by the fact that the asymptotic magnonic USMR is purely a bulk effect: all details about the interface van- ish, while parameters originating from the bulk spin- and charge currents remain. The appearance of lmin the de- nominator and its absence in the numerator of Eq. (6) once again highlights that a large magnon spin diffusion length acts to suppress the USMR. Fig. 7 is a log-log plot of the USMR versus gap size
at various values of the driving current E. Here the valueUmaxis shown as a dashed black line, indicating that this is indeed the value to which Uconverges in the gapless limit or at high current. Moreover, it shows that for given E, one can find a turning point at which the USMR switches relatively abruptly from being nearly constant to decreasing as 1=p
. A (backwards) continuation of the decreasing tails is included in Fig. 7 as dashed lines following the one- parameter fitU=u0=p
, and we define the threshold gap
thas the value of
where this continuation in- tersectsUmax. We then find that
thscales asE2, or conversely, that the driving current required to saturate the USMR scales as the square root of the magnon gap. We note that although the small-gap regime is math- ematically valid (even in the limit
!0, as
may be brought arbitrarily close to 0 in a continuous manner), it does not necessarily correspond to a physical situa- tion: when the anisotropy vanishes, the magnetization of the FI layer may be reoriented freely, which will break our initial assumptions. Nevertheless, in taking the gap- less limit, we are able to predict an upper limit on the magnonic USMR.7 IV. CONCLUSIONS Using a simple drift-diffusion model, we have shown that magnonic spin accumulation near the interface be- tween a ferromagnetic insulator and a heavy metal leads to a small but nonvanishing contribution to the unidirec- tional spin Hall magnetoresistance of FI|HM heterostruc- tures. Central to our model is an interfacial spin current originating from a spin-flip scattering process whereby electronsintheheavymetalcreateorannihilatemagnons in the ferromagnet. This current is markedly nonlinear in the electronic and magnonic spin accumulations at the interface, and it is exactly this nonlinearity which gives rise to the magnonic USMR. For Pt|YIG bilayers, we predict that the magnonic USMRUis at most on the order of 108, roughly three orders of magnitude weaker than the measured USMR in FM|HM hybrids (where electronic spin accumulation is thought to form the largest contribution). This is fully consistent with experiments that fail to detect USMR in Pt|YIG systems, as the tiny signal is drowned out by the interfacial spin Seebeck effect, which has a similar ex- perimental signature and is enhanced compared to the FM|HM case due to inhomogeneous Joule heating. We have shown that the magnon-mediated USMR is approximately cubic in the spin Hall angle of the metal, suggesting that metals with extremely large spin Hall angles may provide a significantly larger USMR than Pt. It is therefore plausible that a large magnonic USMR can exist in systems with very strong spin-orbit coupling, even though our model would break down in this regime. The magnonic USMR depends strongly on the magnon spin diffusion length lmin the ferromagnet. Motivated by a large discrepancy between experimental values and theoretical predictions of lm, we have shown that a sig- nificant increase in USMR can be realized if a method is found to engineer this parameter to specific, optimal values that, for realistic values of the magnon-phonon relaxation time mp(on the order of 1 psfor YIG), are significantly shorter than those measured experimentally or computed theoretically. We further find that when the magnon spin diffusion length has its optimal value, the USMR becomes independent of the ferromagnet’s thick- ness and Gilbert damping constant. Although in physically reasonable regimes, themagnonic USMR is to very good approximation linear in the applied driving current E, it saturates to a fixed valuegivenextremelylargecurrentsorastronglyreduced magnon gap
. The transition from linear to constant behavior in the driving current is heralded by a turn- ing point which is proportional to the square root of the magnon gap. The asymptotic behavior of the USMR be- yond the turning point is governed by the bulk spin- and charge currents, and is completely independent of the de- tails of the interface. While a vast reduction in
is required to bring the saturation current of a Pt|YIG bilayer within experimen- tally reasonable regimes, the magnonic USMR scales as 1=p
at currents below the turning point, suggesting that highly isotropic FI|HM samples are most likely to produce a measurable magnonic USMR. The increase in magnonic USMR at low gaps (and large currents) is in good qualitative agreement with the recent experimental work of Avci et al.[10], as is the linear dependence on system temperature. A notable disagreement with the experimental data of Avci et al.[10] is found in the scaling of the current de- pendence, which in our results lacks an O(I3)term at large magnon gaps and contains an O(I2)term at inter- mediate gaps. It is still unclear whether this discrepancy can be explained by system differences, such as the fi- nite electrical resistance of Co or the presence of Joule heating. Finally, we note that while our results apply to fer- romagnetic insulators, it is reasonable to assume a magnoniccontributionalsoexistsinHM|FMheterostruc- tures, although the possibility of coupled transport of magnons and electrons makes such systems more diffi- cult to model. Additionally, various extensions of our model may be considered, such as the incorporation of spin-momentum locking [5], ellipticity of magnons, heat transport and nonuniform temperature profiles [21], di- rectional dependence of the magnetization, etc. V. ACKNOWLEDGEMENTS R.A.D. is member of the D-ITP consortium, a pro- gram of the Dutch Organisation for Scientific Research (NWO) that is funded by the Dutch Ministry of Educa- tion, Culture and Science (OCW). This work is funded by the European Research Council (ERC). [1] T. McGuire and R. Potter, IEEE Transactions on Mag- netics11, 1018 (1975). [2] G. Binasch, P. Grünberg, F. Saurenbach, and W. Zinn, Physical Review B 39, 4828 (1989). [3] Y.-T. Chen, S. Takahashi, H. Nakayama, M. Althammer, S. T. B. Goennenwein, E. Saitoh, and G. E. W. Bauer, Physical Review B 87, 144411 (2013), arXiv:1302.1352 [cond-mat.mes-hall].[4] C. O. Avci, K. Garello, A. Ghosh, M. Gabureac, S. F. Alvarado, and P. Gambardella, Nature Physics 11, 570 (2015), arXiv:1502.06898 [cond-mat.mes-hall]. [5] K. Yasuda, A. Tsukazaki, R. Yoshimi, K. S. Takahashi, M. Kawasaki, and Y. Tokura, Physical Review Letters 117, 127202 (2016), arXiv:1609.05906 [cond-mat.mtrl- sci].8 [6] Y. Yin, D.-S. Han, M. C. H. de Jong, R. Lavrijsen, R. A. Duine, H. J. M. Swagten, and B. Koopmans, Applied Physics Letters 111, 232405 (2017), arXiv:1711.06488 [cond-mat.mtrl-sci]. [7] C. O. Avci, M. Mann, A. J. Tan, P. Gambardella, and G. S. D. Beach, Applied Physics Letters 110, 203506 (2017). [8] S. S.-L. Zhang and G. Vignale, Physical Review B 94, 140411 (2016), arXiv:1608.02124 [cond-mat.mes-hall]. [9] K. J. Kim, T. Moriyama, T. Koyama, D. Chiba, S. W. Lee, S. J. Lee, K. J. Lee, H. W. Lee, and T. Ono, ArXiv e-prints (2016), arXiv:1603.08746 [cond-mat.mtrl-sci]. [10] C. O. Avci, J. Mendil, G. S. D. Beach, and P. Gam- bardella, Physical Review Letters 121, 087207 (2018). [11] E. M. Chudnovsky, Physical Review Letters 99, 206601 (2007), arXiv:0709.0725 [cond-mat.mes-hall]. [12] W. Zhang, W. Han, X. Jiang, S.-H. Yang, and S. S. P. Parkin, Nature Physics 11, 496 (2015), arXiv:1504.07929 [cond-mat.mes-hall]. [13] T. Holstein and H. Primakoff, Physical Review 58, 1098 (1940). [14] S. A. Bender and Y. Tserkovnyak, Physical Review B 91, 140402 (2015), arXiv:1409.7128 [cond-mat.mes-hall]. [15] Y. Tserkovnyak, A. Brataas, and G. E. Bauer, Physical Review Letters 88, 117601 (2002), cond-mat/0110247. [16] S. A. Bender, R. A. Duine, and Y. Tserkovnyak, Physi- cal Review Letters 108, 246601 (2012), arXiv:1111.2382 [cond-mat.mes-hall]. [17] S. Takahashi, E. Saitoh, and S. Maekawa, in Journal of Physics Conference Series ,JournalofPhysicsConference Series, Vol. 200 (IOP Publishing, 2010) p. 062030.[18] S. S.-L. Zhang and S. Zhang, Physical Review B 86, 214424 (2012), arXiv:1210.2735 [cond-mat.mes-hall]. [19] C. Tang, M. Aldosary, Z. Jiang, H. Chang, B. Madon, K. Chan, M. Wu, J. E. Garay, and J. Shi, Applied Physics Letters 108, 102403 (2016). [20] V. Cherepanov, I. Kolokolov, and V. L’vov, Physics Re- ports229, 81 (1993). [21] L. J. Cornelissen, K. J. H. Peters, G. E. W. Bauer, R. A. Duine, and B. J. van Wees, Physical Review B 94, 014412 (2016), arXiv:1604.03706 [cond-mat.mes-hall]. [22] S. S.-L. Zhang and S. Zhang, Physical Review Let- ters109, 096603 (2012), arXiv:1208.5812 [cond-mat.mes- hall]. [23] C. O. Avci, K. Garello, J. Mendil, A. Ghosh, N. Blasakis, M. Gabureac, M. Trassin, M. Fiebig, and P. Gam- bardella, Applied Physics Letters 107, 192405 (2015). [24] M. Schreier, A. Kamra, M. Weiler, J. Xiao, G. E. W. Bauer, R. Gross, and S. T. B. Goennenwein, Physi- cal Review B 88, 094410 (2013), arXiv:1306.4292 [cond- mat.mes-hall]. [25] M. G. Pini, P. Politi, and R. L. Stamps, Physical Review B72, 014454 (2005), cond-mat/0503538. [26] H. Skarsvåg, C. Holmqvist, and A. Brataas, Physical Re- viewLetters 115,237201(2015),arXiv:1506.06029[cond- mat.mes-hall]. [27] J. F. Dillon, Physical Review 105, 759 (1957). [28] G. P. Rodrigue, H. Meyer, and R. V. Jones, Journal of Applied Physics 31, S376 (1960). [29] ASM Handbook Committee, ASM handbook , Vol. 2 (ASM International, Materials Park, Ohio, 1990). Appendix: Interfacial spin current integrals The following dimensionless integrals appear in the second-order expansion of the interfacial spin current to the spin accumulations, Eq. (2): I0=Z1
kBTmdxr x
kBTmx nB(x)nBTm Tex ; (A.1a) Ie=Z1
kBTmdxr x
kBTm nBTm Tex nB(x)Tm TexeTm Tex nBTm Tex2! ; (A.1b) Im=Z1
kBTmdxr x
kBTmxex[nB(x)]2; (A.1c) Iee=Z1
kBTmdxr x
kBTm eTm Tex nBTm Tex3 eTm Tex1Tmx 2Te eTm Tex+ 1! ; (A.1d) Imm=Z1
kBTmdxr x
kBTmx 2ex[ex+ 1] [nB(x)]3; (A.1e) Ime=Z1
kBTmdxr x
kBTmex[nB(x)]2: (A.1f)9 Description Symbol Expression Value at T= 293 K Ref. YIG spin-wave stiffness constant Js 8:4581040J m2[21] YIG spin quantum number per unit cell S 10 [21] YIG lattice constant a 1:2376 nm [21] YIG Gilbert damping constant  1104[21] YIG spin number density s Sa35:27541027m3[21] YIG magnon gap
9 :31024J[20] YIG magnon-phonon scattering time mp 1 ps[21] YIG magnon relaxation time mrh 2kBTm130 ps[21] Combined magnon relaxation time  1 mr+1 mp1 1 ps[21] Magnon thermal de Broglie wavelength q 4Js kBTm1:62 nm[21] Magnon thermal velocity vth2p JskBT h35:1 km s1[21] Magnon spin diffusion length lmvthq 2 3mr 326 nm [21] Magnon spin conductivity m3 2
2Js 3 1:351024J s m1[21] Real part of spin-mixing conductance g"# r 51018m2[16] Pt electrical conductivity  1107S m1[29]a Pt spin Hall angle SH 0.11 [21] Pt electron diffusion length ls 1:5 nm[21] Pt|YIG Kapitza resistance Rth 3:58109m2K W1[24] aThe conductivity of Pt is approximately inverse-linear in temperature over the regime we are considering. However, as we are not interested in detailed thermodynamic behavior, we use the fixed value = 1107S m1throughout this work. TABLE I. System parameters for a Pt|YIG bilayer film.

2018-10-05

We develop a model for the magnonic contribution to the unidirectional spin Hall magnetoresistance (USMR) of heavy metal/ferromagnetic insulator bilayer films. We show that diffusive transport of Holstein-Primakoff magnons leads to an accumulation of spin near the bilayer interface, giving rise to a magnoresistance which is not invariant under inversion of the current direction. Unlike the electronic contribution described by Zhang and Vignale [Phys. Rev. B 94, 140411 (2016)], which requires an electrically conductive ferromagnet, the magnonic contribution can occur in ferromagnetic insulators such as yttrium iron garnet. We show that the magnonic USMR is, to leading order, cubic in the spin Hall angle of the heavy metal, as opposed to the linear relation found for the electronic contribution. We estimate that the maximal magnonic USMR in Pt|YIG bilayers is on the order of $10^{-8}$, but may reach values of up to $10^{-5}$ if the magnon gap is suppressed, and can thus become comparable to the electronic contribution in, e.g., Pt|Co. We show that the magnonic USMR at a finite magnon gap may be enhanced by an order of magnitude if the magnon diffusion length is decreased to a specific optimal value that depends on various system parameters.

Magnon contribution to unidirectional spin Hall magnetoresistance

1810.02610v2

Magnons at low excitations: Observation of incoherent coupling to a bath of two-level systems Marco Prrmann,1,Isabella Boventer,1, 2Andre Schneider,1Tim Wolz,1Mathias Kl aui,2Alexey V. Ustinov,1, 3and Martin Weides1, 4,y 1Institute of Physics, Karlsruhe Institute of Technology, 76131 Karlsruhe, Germany 2Institute of Physics, Johannes Gutenberg-University Mainz, 55099 Mainz, Germany 3Russian Quantum Center, National University of Science and Technology MISIS, 119049 Moscow, Russia 4James Watt School of Engineering, University of Glasgow, Glasgow G12 8LT, United Kingdom (Dated: November 26, 2019) Collective magnetic excitation modes, magnons, can be coherently coupled to microwave photons in the single excitation limit. This allows for access to quantum properties of magnons and opens up a range of applications in quantum information processing, with the intrinsic magnon linewidth rep- resenting the coherence time of a quantum resonator. Our measurement system consists of a yttrium iron garnet sphere and a three-dimensional microwave cavity at temperatures and excitation powers typical for superconducting quantum circuit experiments. We perform spectroscopic measurements to determine the limiting factor of magnon coherence at these experimental conditions. Using the input-output formalism, we extract the magnon linewidth m. We attribute the limitations of the coherence time at lowest temperatures and excitation powers to incoherent losses into a bath of near-resonance two-level systems (TLSs), a generic loss mechanism known from superconducting circuits under these experimental conditions. We nd that the TLSs saturate when increasing the excitation power from quantum excitation to multiphoton excitation and their contribution to the linewidth vanishes. At higher temperatures, the TLSs saturate thermally and the magnon linewidth decreases as well. Strongly coupled light-spin hybrid systems allow for coherent exchange of quantum information. Such sys- tems are usually studied either classically at room tem- perature [1] or at millikelvin temperatures approaching the quantum limit of excitation [2{4]. The eld of cav- ity magnonics [5{9] harnesses the coherent exchange of excitation due to the strong coupling within the system and is used to access a new range of applications such as quantum transducers and memories [10]. Nonlinear- ity in the system is needed to gain access to the control and detection of single magnons. Because of experimen- tal constraints regarding required light intensities in a purely optomagnonic system [11], hybridized systems of magnon excitations and non-linear macroscopic quantum systems such as superconducting qubits [12, 13] are used instead, which opens up new possibilities in the emerging eld of quantum magnonics [14, 15]. An ecient interac- tion of magnonic systems and qubits requires their life- times to exceed the exchange time. Magnon excitation losses, expressed by the magnon linewidth m, translate into a lifetime of the spin excitation. Identifying its lim- iting factors is an important step toward more sophisti- cated implementations of hybrid quantum systems using magnons. Studies in literature show the losses in magnon excitations from room temperature down to about liquid helium temperatures [9]. The main contribution changes with temperature from scattering at rare-earth impu- rities [16, 17] to multi-magnon scattering at imperfect sample surfaces [18, 19]. For a typical environment of superconducting quantum circuit experiments, temper- atures below 100 mK and microwave probe powers com- parable to single-photon excitations, temperature sweepsshow losses into TLSs [3]. In this paper, we present both temperature- and power-dependent measurements of the magnon linewidth in a spherical yttrium iron gar- net (YIG) sample in the quantum limit of magnon ex- citations. We extract the critical saturation power and present on- and o-resonant linewidth that is mapped to the ratio of magnon excitation in the hybrid system. For large detuning, the fundamental linewidth can be extracted, thereby avoiding unwanted saturation eects from the residual cavity photon population. This renders the o-resonant linewidth a valuable information on the limiting factors of spin lifetimes. The magnetization dynamics inside a magnetic crys- tal is described by bosonic quasiparticles of collective spin excitation, called magnons. These magnons man- ifest as the collective precessional motion of the partic- ipating spins out of their equilibrium positions. Their energies and spatial distribution can be calculated ana- lytically using the Walker modes for spherical samples [20, 21]. We focus on the uniform in-phase precession mode corresponding to the wave vector k= 0, called the Kittel mode [22], treating it equivalently to one sin- gle large macro spin. The precession frequency (magnon frequency) of the Kittel mode in a sphere changes lin- early with a uniform external magnetic bias eld. The precessional motion is excited by a magnetic eld os- cillating at the magnon frequency perpendicular to the bias eld. We use the conned magnetic eld of a cav- ity photon resonance to create magnetic excitations in a macroscopic sample, biased by a static external magnetic eld. Tuning them into resonance, the magnon and pho- ton degree of freedom mix due to their strong interaction.arXiv:1903.03981v3 [quant-ph] 25 Nov 20192 185.5 186.25 187 187.75 188.5 Ext. magnetic field (mT) 5.205.225.245.265.28Probe frequency / (GHz) 5.20 5.22 5.24 5.26 5.28 Probe frequency / (GHz) 50 48 46 44 42 40 38 36 Reflection amplitude || (dB) / = . 1.8 2.0 2.2 2.4Coil current (A) a b Refl. amplitude || (arb. units) FIG. 1. (a) Color coded absolute value of the re ection spectrum plotted against probe frequency and applied current at T= 55 mK and P=140 dBm. The resonance dips show the dressed photon-magnon states forming an avoided level crossing with the degeneracy point at I0= 2:09 A, corresponding to an applied eld of B0= 186:98 mT (dashed vertical line). The inset displays the squared gradient of the zoomed-in amplitude data. The kink in the data represents a weakly coupled magnetostatic mode. This was also seen in Ref. [9]. (b) Raw data of the cross section at the center of the avoided level crossing and t to the input-output formalism. The t gives a magnon linewidth of m=2= 1:820:18 MHz. The data are normalized by the eld independent background before tting and is multiplied to the t to display it over the raw data. This creates hybridized states described as repulsive cav- ity magnon polaritons, which are visible as an avoided level crossing in the spectroscopic data with two reso- nance dips at frequencies !(see Supplemental Material [23]) appearing in the data cross section. The interaction is described by the macroscopic magnon-photon coupling strengthg. The system is probed in re ection with mi- crowave frequencies using standard ferromagnetic reso- nance techniques [24]. We use the input-output formal- ism [25] to describe the re ection spectrum. The complex re ection parameter S11, the ratio of re ected to input energy with respect to the probe frequency !p, reads as S11(!p) =1 +2c i (!r!p) +l+g2 i(!m!p)+m;(1) with the cavity's coupling and loaded linewidths cand l, and the internal magnon linewidth m(HWHM). For our hybrid system we mount a commercially avail- able YIG (Y 3Fe5O12) sphere with a diameter d= 0:5 mm [26] inside a three-dimensional (3D) rectangular cavity made of oxygen-free copper and cool the device in a dilu- tion refrigerator down to millikelvin temperatures (see gure in Supplemental Material [23]). YIG as a ma- terial is particularly apt for microwave applications, as it is a ferrimagnetic insulator with a very low Gilbert damping factor of 103to 105[27{29] and a high net spin density of 2 :11022B=cm3[30]. The single crys- tal sphere comes pre-mounted to a beryllium oxide rod along the [110] crystal direction. The 3D cavity has a TE102mode resonance frequency of !bare r=2= 5:24 GHzand is equipped with one SMA connector for re ection spectroscopy measurements. For low temperatures and excitation powers, we nd the internal and coupling quality factors to be Qi=!r=2i= 712597 and Qc=!r=2c= 543929, combining to a loaded quality factorQl= (1=Qi+ 1=Qc)1= 308424 (see Supple- mental Material [23]). We mount the YIG at a magnetic anti-node of the cavity resonance and apply a static mag- netic eld of about 187 mT perpendicular to the cavity eld to tune the magnetic excitation into resonance with the cavity photon. The magnetic eld is created by an iron yoke holding a superconducting niobium-titanium coil. Additional permanent samarium-cobalt magnets are used to create a zero-current oset magnetic eld of about 178 mT. The probing microwave signal is provided by a vector network analyzer (VNA). Microwave attenu- ators and cable losses account for 75 dB of cable atten- uation to the sample. We apply probe powers between 140 dBm and65 dBm at the sample's SMA port. To- gether with the cavity parameters, this corresponds to an average magnon population number hmifrom 0:3 up to the order of 107[23] in the hybridized case. The probe signal is coupled capacitively to the cavity photon using the bare inner conductor of a coaxial cable positioned in parallel to the electric eld component. The temperature of the sample is swept between 55 mK and 1 :8 K using a proportional-integral-derivative (PID) controlled heater. After a change in temperature, we wait at least one hour for the sample to thermalize before measuring. All data acquisition and analysis are done via qkit [31].3 0.1 1 Temperature (K) 0.81.01.21.41.61.82.02.2Magnon linewidth / (MHz) = = 140 120 100 80 60 Probe power (dBm) b a = = 1 10Temperature / (GHz) 110102103104105106107Average magnon number FIG. 2. (a) Temperature dependence of the magnon linewidth mat the degeneracy point. For low probe powers, m follows a tanh (1 =T) behavior (crosses), while for high probe powers (circles) the linewidth does not show any temperature dependence. (b) Power dependence of the magnon linewidth mforT= 55 mK and 200 mK at the degeneracy point. Both temperature curves show a similar behavior. At probe powers of about 90 dB mmdrops for both temperatures, following the (1 +P=P c)1=2trend of the TLS model. All linewidth data shown here are extracted from the t at matching frequencies. A typical measurement is shown in Fig. 1(a), mea- sured atT= 55 mK with an input power level of P= 140 dBm. Figure 1(b) shows the raw data and the t of the cavity-magnon polariton at matching resonance fre- quencies for an applied external eld of B0= 186:98 mT. We correct the raw data from background resonances and extract the parameters of the hybridized system by t- ting to Eq. (1). The coupling strength g=2= 10:4 MHz of the system exceeds both the total resonator linewidth l=2=!r 2Ql=2= 0:85 MHz and the internal magnon linewidthm=2= 1:82 MHz, thus being well in the strong coupling regime ( gl; m) for all temperatures and probe powers. The measured coupling strength is in good agreement with the expected value gth= e 2r 0~!r 2Vap 2Nss; (2) with the gyromagnetic ratio of the electron e, the mode volumeVa= 5:406106m3, the Fe3+spin number s= 5=2, the spatial overlap between microwave eld and magnon eld , and the total number of spins Ns[9]. The overlap factor is given by the ratio of mode volumes in the cavity volume and the sample volume [1]. We nd for our setup the overlap factor to be = 0:536. For a sphere diameter of d= 0:5 mm we expect a total number ofNs= 1:371018spins. We nd the expected coupling strengthgth=2= 12:48 MHz to be in good agreement with our measured value. Even for measurements at high powers, the number of participating spins of the order of 1018is much larger than the estimated number of magnon excitations (107). We therefore do not expect to seethe intrinsic magnon nonlinearity as observed at excita- tion powers comparable to the number of participating spins [32]. The internal magnon linewidth decreases at higher temperature and powers (Fig. 2) while the coupling strength remains geometrically determined and does not change with either temperature or power. This behav- ior can be explained by an incoherent coupling to a bath of two-level systems (TLSs) as the main source of loss in our measurements. In the TLS model [33{36], a quantum state is conned in a double-well potential with dierent ground-state energies and a barrier in-between. TLSs be- come thermally saturated at temperatures higher than their frequency ( T&~!TLS=kB). Dynamics at low tem- peratures are dominated by quantum tunneling through the barrier that can be stimulated by excitations at sim- ilar energies. This resonant energy absorption shifts the equilibrium between the excitation rate and lifetime of the TLSs and their in uence to the overall excitation loss vanishes. Loss into an ensemble of near-resonant TLSs is a widely known generic model for excitation losses in solids, glasses, and superconducting circuits at these ex- perimental conditions [37]. We t the magnon linewidth to the generic TLS model loss tangent m(T; P ) =0tanh ( ~!r=2kBT)p 1 +P=P c+o: (3) Directly in the avoided level crossing we nd 0=2= 1:050:15 MHz as the low temperature limit of the linewidth describing the TLS spectrum within the sam- ple ando=2= 0:910:11 MHz as an oset linewidth4 186.25 186.75 187.25 187.75 Ext. magnetic field (mT) 0.81.21.62.0Magnon linewidth (MHz) 186.25 186.75 187.25 187.75 Ext. magnetic field (mT) 01 Ratiophoton magnon 01 Ratio5.22 5.23 5.24 5.25 5.26Magnon frequency / (GHz) 5.22 5.23 5.24 5.25 5.26Magnon frequency / (GHz) a b = = FIG. 3. Magnetic eld (magnon frequency) dependence of the the magnon linewidth mfor dierent probe powers at T= 55 mK (a) and T= 200 mK (b). The shown probe powers correspond to the ones at the transition in Fig. 2 (b). The number of excited magnons depends on the detuning of magnon and photon frequency. At matching frequencies (dashed line) the magnon linewidth has a minimum, corresponding to the highest excited magnon numbers and therefore the highest saturation of TLSs. A second minimum at about 187 :25 mT corresponds to the coupling to an additional magnetostatic mode within the sample [inset of Fig. 1 (a)]. The insets show the ratio of excitation power within each component of the hybrid system. At matching frequencies, both components are excited equally. The magnon share drops at the plot boundaries to about 20 %. The coupling to the magnetostatic mode is visible as a local maximum in the magnon excitation ratio. The xaxes are scaled as in the main plots. The legends are valid for both temperatures. added as a lower boundary without TLS contribution. The critical power Pc=816:5 dBm at the SMA port describes the saturation of the TLS due to res- onant power absorption, corresponding to an average critical magnon number of hmci= 2:4
105. Using nite-element simulations, we map the critical excita- tion power to a critical AC magnetic eld on the or- der ofBc3
1010T at the position of the YIG sam- ple. Looking at the linewidths outside the anti-crossing at constant input power, we nd a minimum at match- ing magnon and photon frequencies (dashed lines in Fig. 3). Here, the excitation is equally distributed between photons and magnons, reaching the maximum in both magnon excitation power and TLS saturation, respec- tively. At detuned frequencies the ratio between magnon and photon excitation power changes, less energy excites the magnons (insets in Fig. 3), and therefore less TLSs get saturated. The magnon linewidth increases with de- tuning, matching the low power data for large detunings. This eect is most visible at highest excitation powers. We calculate the energy ratios by tting the resonances in each polariton branch individually and weight the stored energy with the eigenvalues of the coupling Hamiltonian [38]. For higher powers a second minimum at about 187:25 mT can be seen at both studied temperatures. We attribute this to the coupling to a magnetostatic mode within the YIG sample and therefore again an increased number of excited magnons [see inset of Fig. 1 (a)]. Thiscan also be seen in the inset gures as a local magnon excitation maximum. We attribute the TLS-independent losseso=2= 0:910:11 MHz to multi-magnon scat- tering processes on the imperfect sphere surface [18, 19]. As described in Ref. [9], we model the surface of the YIG with spherical pits with radii of2 3of the size of the polish- ing material (2 =30:05µm) and estimate a contribution of about 2 
1 MHz that matches our data. We attribute the slight increase in the linewidth visible in the high- power data (circles) in Fig. 2 (a) to the rst in uence of rare-earth impurity scattering, dominating the linewidth behavior of the TLS-saturated system at higher temper- atures [9, 16, 39]. In principle, loss due to TLS can also be determined indirectly by weak changes of the reso- nance frequency [40{43] while keeping the eld constant. Our system, however, operates at xed frequency and magnetic remanence within the magnetic yoke leads to uncertainties in absolute magnetic eld value beyond the required accuracy. In this work, we studied losses in a spherical YIG sample at temperatures below 2 K and excitation powers down from 107photons below a single photon. We iden- tify incoherent coupling to a bath of two-level systems as the main source of excitation loss in our measurements. The magnon linewidth m=2at the degeneracy point ts well to the generic loss tangent of the TLS model with re- spect to temperature and power. It decreases from about 1:8 MHz in uenced by TLSs to about 1 MHz with satu-5 rated TLSs. The magnon linewidth shows a minimum at maximum magnon excitation numbers, again corre- sponding with TLS saturation with increasing excitation power. While TLSs are a common source of loss in super- conducting circuits, their microscopic nature is still not fully understood. Possible models for TLS origin include magnetic TLSs in spin glasses [44{47] that manifest in crystalline samples in lower concentration, surface spins [48, 49] that in uence the eective number of spins or magnon-phonon and subsequent phonon losses into TLSs [50] (see also Supplemental Material [23]. Improving the surface roughness and quality of the YIG crystal can lead to lower overall losses and lower TLS in uence which can lead to longer coherence lifetimes for application in quan- tum magnonic devices. Note added in proof - Recently, a manuscript study- ing losses in thin lm YIG that independently observed comparable results and reached similar conclusions was published by Kosen et al. [51]. This work was supported by the European Research Council (ERC) under the Grant Agreement 648011 and the Deutsche Forschungsgemeinschaft (DFG) within Project INST 121384/138-1 FUGG and SFB TRR 173. We acknowledge nancial support by the Helmholtz In- ternational Research School for Teratronics (M.P. and T.W.) and the Carl-Zeiss-Foundation (A.S.). A.V.U ac- knowledges partial support from the Ministry of Educa- tion and Science of Russian Federation in the framework of the Increase Competitiveness Program of the National University of Science and Technology MISIS (Grant No. K2-2017-081). marco.prrmann@kit.edu ymartin.weides@glasgow.ac.uk [1] X. Zhang, C.-L. Zou, L. Jiang, and H. X. Tang, Phys. Rev. Lett. 113, 156401 (2014). [2] H. Huebl, C. W. Zollitsch, J. Lotze, F. Hocke, M. Greifen- stein, A. Marx, R. Gross, and S. T. B. Goennenwein, Phys. Rev. Lett. 111, 127003 (2013). [3] Y. Tabuchi, S. Ishino, T. Ishikawa, R. Yamazaki, K. Us- ami, and Y. Nakamura, Phys. Rev. Lett. 113, 083603 (2014). [4] M. Goryachev, W. G. Farr, D. L. Creedon, Y. Fan, M. Kostylev, and M. E. Tobar, Phys. Rev. Appl. 2, 054002 (2014). [5] D. Zhang, X.-M. Wang, T.-F. Li, X.-Q. Luo, W. Wu, F. Nori, and J. You, npj Quantum Inf. 1, 15014 EP (2015). [6] L. Bai, M. Harder, Y. Chen, X. Fan, J. Xiao, and C.-M. Hu, Phys. Rev. Lett. 114, 227201 (2015). [7] Y. 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Lisenfeld, Reports on Progress in Physics 82, 124501 (2019). [38] M. Harder and C.-M. Hu, Cavity Spintronics: An Early Review of Recent Progress in the Study of Magnon-6 Photon Level Repulsion , edited by R. E. Camley and R. L. Stamps, Solid State Physics, Vol. 69 (Academic Press, 2018) pp. 47 { 121. [39] E. G. Spencer, R. C. LeCraw, and A. M. Clogston, Phys. Rev. Lett. 3, 32 (1959). [40] D. P. Pappas, M. R. Vissers, D. S. Wisbey, J. S. Kline, and J. Gao, Appl. Phys. Lett. 21, 871 (2011). [41] J. Gao, M. Daal, A. Vayonakis, S. Kumar, J. Zmuidzinas, B. Sadoulet, B. A. Mazin, P. K. Day, and H. G. Leduc, Appl. Phys. Lett. 92, 152505 (2008). [42] S. Kumar, J. Gao, J. Zmuidzinas, B. A. Mazin, H. G. LeDuc, and P. K. Day, Appl. Phys. Lett. 92, 123503 (2008). [43] J. Burnett, L. Faoro, I. Wisby, V. L. Gurtovoi, A. V. Chernykh, G. M. Mikhailov, V. A. Tulin, R. Shaikhaidarov, V. Antonov, P. J. Meeson, A. Y. Tza- lenchuk, and T. Lindstr om, Nat. Commun. 5, 4119 EP (2014). [44] M. Continentino, Solid State Communications 38, 981 (1981). [45] M. A. Continentino, Journal of Physics C: Solid State Physics 14, 3527 (1981). [46] E. F. Wassermann and D. M. Herlach, Journal of Applied Physics 55, 1709 (1984). [47] D. Wesenberg, T. Liu, D. Balzar, M. Wu, and B. L. Zink, Nature Physics 13, 987 EP (2017). [48] S. E. de Graaf, A. A. Adamyan, T. Lindstr om, D. Erts, S. E. Kubatkin, A. Y. Tzalenchuk, and A. V. Danilov, Phys. Rev. Lett. 118, 057703 (2017). [49] S. E. de Graaf, L. Faoro, J. Burnett, A. A. Adamyan, A. Y. Tzalenchuk, S. E. Kubatkin, T. Lindstr om, and A. V. Danilov, Nature Communications 9, 1143 (2018). [50] S. Streib, N. Vidal-Silva, K. Shen, and G. E. W. Bauer, Phys. Rev. B 99, 184442 (2019). [51] S. Kosen, A. F. van Loo, D. A. Bozhko, L. Mihalceanu, and A. D. Karenowska, APL Materials 7, 101120 (2019). [52] S. Probst, F. B. Song, P. A. Bushev, A. V. Ustinov, and M. Weides, Review of Scientic Instruments 86, 024706 (2015). [53] M. Newville, T. Stensitzki, D. B. Allen, and A. Ingar- giola, \LMFIT: Non-Linear Least-Square Minimization and Curve-Fitting for Python," (2014). [54] M. A. Continentino, Phys. Rev. B 27, 4351 (1983). [55] M. A. Continentino, Journal of Physics C: Solid State Physics 16, L71 (1983). [56] C. Arzoumanian, A. De Go er, B. Salce, and F. Holtzberg, J. Physique Lett. 44, 39 (1983). [57] R. N. Kleiman, G. Agnolet, and D. J. Bishop, Phys. Rev. Lett. 59, 2079 (1987). Cavity-Magnon coupling The frequencies of both arms of the avoided level cross- ing!are tted to the energy eigenvalues of a 2 2 matrix describing two coupled harmonic oscillators, one with constant frequency and one with a linearly changing frequency, !=!bare r+!I=0 m 2s!barer!I=0m 22 +g2:(S1)We use the current dependent data taken at T= 55 mK andP=140 dBm to obtain the bare cavity fre- quency!bare r, the zero-current magnetic excitation fre- quency!I=0 m, and the coupling strength g. The fre- quencies of the anticrossing !were obtained by track- ing the minima in the amplitude data. From the t we obtain the bare cavity frequency !bare r=2= 5:239 020:000 02 GHz and the zero-current magnetic excitation frequency !I=0 m=2= 4:98170:0002 GHz due to the oset magnetic eld by the permanent mag- nets. The magnon-cavity coupling strength stays nearly constant for all temperatures and excitation powers at g=2= 10:390:17 MHz. Magnon number estimation Using the cavity's resonance frequency, quality factors, and the input power Pinwe estimate the total number of magnon and photon excitation within the cavity hNeiin units of ~!r, hNei= 4Q2 l Qc1 ~!2r
Pin: (S2) Note that the input power Pinis in units of watts and not to be confused with the probe power (level) Pin units of dBm. For the strongly coupled system the exci- tation energy at matching frequencies is stored in equal parts in photons and magnons, hni=hmi=hNei 2. We measure the re ection signal of the cavity reso- nance at 55 mK at probe powers between 140 dBm and 65 dBm at zero current applied to the magnetic coil. The complex data is then tted using a circle t al- gorithm [52] to determine the power dependent qual- ity factors (coupling quality factor Qc=!r 2c, inter- nal quality factor Qi=!r 2i, and loaded quality factor Ql= (1=Qi+ 1=Qc)1) and resonance frequencies as shown in Fig. S1 (a-d). Besides an initial shift of the quality factors of less than 1 % going from 140 dBm to 130 dBm of input power the quality factors show no power dependence, varying only in a range below 0 :15 %. The tted (dressed) frequencies are shifted compared to the bare cavity frequency due to the residual magnetic eld. From Eq. (S1) we expect a zero-current dressed frequency of !I=0 r= 5:239 4520:000 002 GHz. The cir- cle t gives a resonance frequency at the lowest power of !CF r= 5:239 4740:000 002 GHz. We calculate the av- erage total excitation for all probe powers using Eq. (S2) and t a line to the logarithmic data (Fig. S1 (e)), hNei= 62:046
P1:0003 in fW1: (S3) The t agrees well with the data and is used throughout all data evaluation to map probe powers Pto average excitation numbers. This results in average magnon ex- citation numbers at matching frequencies for our experi- ments between 0 :31 and 9:85
106.7 140 120 100 80 60 Probe power (dBm) 110102103104105106107108Average total excitation number 1 0123/ (kHz) 140 120 100 80 60 Probe power (dBm) 70807120716072007240Internal Q factor 307830823086309030943098Loaded Q factor 542054255430543554405445Coupling Q factor a b cd e FIG. S1. (a-c) Loaded, coupling, and internal quality factors of the cavity resonance against probe power. The data was taken atT= 55 mK with zero current applied to the magnetic coil and does not show a power dependent behavior. (d) Shift of the tted cavity frequencies
!=2= [!r(P)!r(P=140 dB m)]=2with compared to the measurement at lowest probe power at zero current. Similar as with the quality factors, the cavity frequency does not show a power dependence. (e) Calculated average photon number in cavity against probe power. The t shows a linear dependence of the photon number calculated with Eq. (S2) to the input power. Note that this plot features a log-log scale, making the t linear again. The errors on the average photon number are estimated to be smaller than 0 :35 % and are not visible on this plot. Extracting the internal magnon linewidth We extract the internal magnon linewidth mby tting the re ection amplitude jS11(!)jusing the input-output formalism [25]. jS11(!p;I)j=1 +2c i (!r!p) +l+g2 i(!m(I)!p)+m;(S4) with the probe frequency !p, the magnon frequency !m, and the loaded, coupling and magnon linewidths l,c, andm(HWHM). Before tting, we normalize the databy the current independent baseline similar to Ref. [9]. We estimate the background value for each probe fre- quency by calculating a weighted average over all entries along the current axis, neglecting the areas around the dressed cavity resonances. The amplitude data is divided by this baseline to account for losses in the measurement setup. The normalized data together with the t results of Eq. (S1) and the circle tted cavity resonance at zero current are then tted to Eq. (S4) using the Python pack- age lmt [53].8 VNA3K 0.75K 55mK -20dB -20dB -20dB40dB32dBb aYIG coupler 3 mm FIG. S2. (a) Photograph of the sample in the cavity. The top half of the cavity resonator was removed and can be seen in the background. (b) Schematic diagram of the experimen- tal setup. The cavity holding the YIG sphere and the magnet providing the static eld are mounted at the mixing cham- ber plate of a dry dilution refrigerator. The microwave input signal is attenuated to minimize thermal noise at the sample. The attenuation of the complete input line to the input port of the cavity is 75 dB at the cavity resonance frequency. The output signal is amplied by a cryogenic amplier operating at 3 K and an amplier at room temperature. Two magneti- cally shielded microwave circulators protect the sample from amplier noise. Possible TLS origin The microscopic origins of TLSs is still unclear and part of ongoing research. Possible models include mag- netic TLSs proposed with analog behavior to the electricdipolar coupled TLSs [44, 45, 54, 55] and measured in spin glasses by thermal conductivity, susceptibility and magnetization measuements at low temperatures [46, 56]. With amorphous YIG showing spin glass behavior [47] it seems plausible to observe these eects in our crystalline YIG sample where in addition to the observed rare earth impurities [9] we can assume structural crystal defects. This is based on materials with electric dipolar coupled TLSs, where TLSs appear largely in disorderd crystals but also in single crystals with smaller density [57]. Another possibility could be surface spins leading to strong damping that were observed as an important loss mechanism in cQED experiments [48, 49]. We evaluated the coupling strength to nd a power or temperature de- pendence on the participating number of spins, see Fig. S3. We nd an increase in the coupling strength of about 1 % at the saturation conditions for the TLSs. With g/p Nthis translates to an increase in the number of participating spins of the order of 2 %, e.g. due to the in- creased participation of now environmentally decoupled surface spins. This should not be enough to explain the decrease in mby a factor of 2. A loss mechanism by magnon-phonon coupling and subsequent phonon losses due to TLS coupling can be neglected since for k= 0 magnons in YIG these magnon losses are proposed to be much smaller than the Gilbert damping [50].9 0.1 1 Temperature (K) 10.2510.3010.3510.4010.4510.50coupling strength / (MHz) = = 140 120 100 80 60 probe power (dBm) = = FIG. S3. (a) Temperature and (b) power dependence of the coupling strength evaluated at the same conditions as Fig. (2) in the main text. We nd a increase of the coupling strength of about 1 % going to higher powers that decreases at higher temperatures. This indicates an increase in participating spins on the order of 2 %.

2019-03-10

Collective magnetic excitation modes, magnons, can be coherently coupled to microwave photons in the single excitation limit. This allows for access to quantum properties of magnons and opens up a range of applications in quantum information processing, with the intrinsic magnon linewidth representing the coherence time of a quantum resonator. Our measurement system consists of a yttrium iron garnet (YIG) sphere and a three-dimensional (3D) microwave cavity at temperatures and excitation powers typical for superconducting quantum circuit experiments. We perform spectroscopic measurements to determine the limiting factor of magnon coherence at these experimental conditions. Using the input-output formalism, we extract the magnon linewidth $\kappa_\mathrm{m}$. We attribute the limitations of the coherence time at lowest temperatures and excitation powers to incoherent losses into a bath of near-resonance two-level systems (TLSs), a generic loss mechanism known from superconducting circuits under these experimental conditions. We find that the TLSs saturate when increasing the excitation power from quantum excitation to multi-photon excitation and their contribution to the linewidth vanishes. At higher temperatures, the TLSs saturate thermally and the magnon linewidth decreases as well.

Magnons at low excitations: Observation of incoherent coupling to a bath of two-level-systems

1903.03981v3

Control of spin current by a magnetic YIG substrate in NiFe/Al nonlocal spin valves F. K. Dejene,1,N. Vlietstra,1D. Luc,2X. Waintal,2J. Ben Youssef,3and B. J. van Wees1 1Physics of Nanodevices, Zernike Institute for Advanced Materials, University of Groningen, 9747AG, Groningen, The Netherlands 2CEA-INAC/UJF Grenoble 1, SPSMS UMR-E 9001, Grenoble F-38054, France 3Universit de Bretagne Occidentale, Laboratoire de Magnetisme de Bretagne CNRS, 6 Avenue Le Gorgeu, 29285 Brest, France (Dated: June 9, 2021) We study the eect of a magnetic insulator (Yttrium Iron Garnet - YIG ) substrate on the spin transport properties of Ni 80Fe20/Al nonlocal spin valve (NLSV) devices. The NLSV signal on the YIG substrate is about 2 to 3 times lower than that on a non magnetic SiO 2substrate, indicating that a signicant fraction of the spin-current is absorbed at the Al/YIG interface. By measuring the NLSV signal for varying injector-to-detector distance and using a three dimensional spin-transport model that takes spin current absorption at the Al/YIG interface into account we obtain an eective spin-mixing conductance G"#'581013 1m2. We also observe a small but clear modulation of the NLSV signal when rotating the YIG magnetization direction with respect to the xed spin polarization of the spin accumulation in the Al. Spin relaxation due to thermal magnons or roughness of the YIG surface may be responsible for the observed small modulation of the NLSV signal. The coupled transport of spin, charge and heat in non- magnetic (N) metals deposited on the magnetic insulator Y3Fe5O12(YIG) has led to new spin caloritronic device concepts such as thermally driven spin currents, the gen- eration of spin angular momentum via the spin Seebeck eect (SSE) [1], spin pumping from YIG to metals [2], spin-orbit coupling (SOC) induced magnetoresistance ef- fects [3, 4] and the spin Peltier eect, i.e., the inverse of the SSE that describes cooling/heating by spin cur- rents [5]. In these spin caloritronic phenomena, the spin- mixing conductance G"#of the N/YIG interface controls the transfer of spins from the conduction electrons in N to the magnetic excitations (magnons) in the YIG, or vice versa [6{10]. The interconversion of spin current to a voltage employs the (inverse) spin Hall eect in heavy- metals such as Pt or Pd. The possible presence of prox- imity induced magnetism in these metals is reported to introduce spurious magnetothermoelectric eects [11, 12] or enhance G"#[7]. Owing to the short spin-diusion lengthin these large SOC metals, the applicability of the diusive spin-transport model is also questionable. Experimental measurements that alleviate these concerns are however scarce and hence are highly required. In this article, we investigate the interaction of spin current (in the absence of a charge current) with the YIG magnetization using the NLSV geometry [13{15]. Using a metal with low SOC and long spin-diusion length allows to treat our experiment using the diu- sive spin-transport model. We nd that the NLSV signal on the YIG substrate is two to three times lower than that on the SiO 2substrate, indicating signicant spin- current absorption at the Al/YIG interface. By vary- ing the angle between the induced spin accumulation and the YIG magnetization direction we observe a small but clear modulation of the NLSV signal. We also nd that modifying the quality of the Al/YIG interface, us-ing dierent thin-lm deposition methods [4], in uences G"#and hence the size of the spin current owing at the Al/YIG interface. Recently, a low-temperature mea- surements of a similar eect was reported by Villamor et al.[16] in Co/Cu devices where G"#1011 1m2was estimated, two orders of magnitude lower than in the lit- erature [4, 8]. Here, we present a room-temperature spin- transport study in transparent Ni 80Fe20(Py)/Al NLSV devices. Figure 1 depicts the concept of our experiment. A non- magnetic metal (green) deposited on the YIG connects the two in-plane polarized ferromagnetic metals F1and F2, which are used for injecting and detecting spin cur- rents, respectively. A charge current through the F1/Al interface induces a spin accumulation s(~ r) = (0;s;0)T that is polarized along the ^ ydirection, parallel to the magnetization direction of F1. This non-equilibrium s, the dierence between the electrochemical potentials for spin up and spin down electrons, diuses to both +^ x and^xdirections of F1/Al interface with an exponen- tial decay characterized by the spin diusion length N. Spins arriving at the detecting F2/Al interface give rise to a nonlocal voltage Vnlthat is a function of the rela- tive magnetic conguration of F1andF2, being minimum (maximum) when F1andF2are parallel (antiparallel) to each other. For NLSV devices on a SiO 2substrate, spin relaxation proceeds via electron scattering with phonons, impuri- ties or defects present in the spin transport channel, also known as the Elliot-Yafet (EY) mechanism. The situa- tion is dierent for a NLSV on the magnetic YIG sub- strate where additional spin relaxation due to thermal magnons in the YIG and/or interfacial spin orbit cou- pling can be mediated by direct spin- ip scattering or spin-precession. Depending on the magnetization direc- tion ^mof the YIG with respect to sspins incident atarXiv:1503.06108v1 [cond-mat.mes-hall] 20 Mar 20152 on SiO 2 on YIG sµ 0ˆmAl YIGspin waves ˆm FIG. 1. (Color online) Concept of the experiment for ^ mks. (a) A charge current through the F 1/Al interface creates a spin accumulation sin the Al. The diusion of sto the F 2/Al interface is aected by spin- ip relaxation at the Al/YIG interface. Scattering of a spin up electron ( s=~=2) into spin down electron ( s=~=2) is accompanied by magnon emission (s=~) creating a spin current that is minimum (max- imum) when ^ sis parallel (perpendicular) to the magnetiza- tion of the YIG. (b) Prole of salong the Al strip on a SiO 2 (red) and YIG (blue) substrate. The spin accumulation at the F 2/Al is lower for the YIG substrate compared to that on SiO2. the Al/YIG surface are absorbed ( ^ m?s) or re ected ( ^mks) thereby causing a spin current density js(~ r) through the Al/YIG interface [9] js( ^m)jz=0=Gr^m( ^ms)+Gi( ^ms)+Gss:(1) Here ^m= (mx;my;0)Tis a unit vector parallel to the in-plane magnetization of the YIG, Gr(Gi) is the real (imaginary) part of the spin-mixing conductance per unit area andGsis a spin-sink conductance that can be in- terpreted as an eective spin-mixing conductance that quanties spin-absorption ( ip) eects that is indepen- dent of the angle between ^ mands. When ^mkssome of the spins incident on the YIG are re ected back into the Al while some fraction is ab- sorbed by the YIG. The absorption of the spin-current in this collinear case is governed by a spin-sinking eect either due to (i) the thermal excitation of the YIG mag- netization (thermal magnons) or (ii) spin- ip processes due to interface spin orbit eects or magnetic impurities present at the interface. This process can be character- ized by an eective spin-mixing interface conductance Gs which, at room temperature, is about 20% of Gr[5]. Be- cause of this additional spin- ip scattering, the maximum NLSV signal on the YIG substrate should also be smaller than that on the SiO 2. When ^m?sspins arriving at the Al/YIG interface are absorbed. In this case all threeterms in Eq. (1) contribute to a maximum ow of spin current through the interface. The nonlocal voltage mea- sured at F 2is hence a function of the angle between ^ m andsand should re ect the symmetry of Eq. 1. Fig. 2(a) shows the scanning electron microscope im- age of the studied NLSV device that was prepared on a 200-nm thick single-crystal YIG, having very low coercive eld [2, 4, 17], grown by liquid phase epitaxy on a 500 m thick (111) Gd 3Ga5O12(GGG) substrate. It consists of two 20-nm thick Ni 80Fe20(Py) wires connected by a 130-nm thick Al cross. A 5 nm-thick Ti buer layer was inserted underneath the Py to suppress direct exchange coupling between the Py and YIG. We studied two types of devices, hereafter named Type-A and Type-B devices. In Type-A devices (4 devices), prior to the deposition of the Al (by electron beam evaporation), Ar ion milling of the Py surface was performed to ensure a transparent Py/Al interface. This process, however, introduces un- avoidable milling of the YIG surface thereby introducing disordered Al/YIG interface with lower G"#[18]. To cir- cumvent this problem, in Type-B devices (2 devices), we rst deposit a 20 nm-thick Al strip (by DC sputtering) between the injector and detector Py wires. Sputtering is reported to yield a better interface [4]. Next, after Ar ion milling of the Py and sputtered-Al surfaces, a 130 nm-thick Al layer was deposited using e-beam evapora- tion. Similar devices prepared on SiO 2substrate were also investigated. All measurements were performed at room temperature using standard low frequency lock-in measurements. The NLSV resistance Rnl=Vnl=Ias a function of the applied in-plane magnetic eld (along ^ y) is shown in Fig. 2(b), both for SiO 2(red and orange) and YIG (blue) samples. Note that the magnetizations of the injector, detector and YIG are all collinear and hence no initial transverse spin component is present. The spin valve sig- nal, dened as the dierence between the parallel RPand anti-parallel RAPresistance values, RSV=RPRAP on the YIG substrate is about two to three times smaller than that on the SiO 2substrate. This reduction in the NLSV signal indicates the presence of an additional spin- relaxation process even for ^ mks. Assuming an iden- tical spin injection eciency in both devices, this means that spin relaxation in the Al on the YIG substrate occurs on an eectively shorter spin relaxation length N. To properly extract Nwe performed several measurements for varying distance between the Py wires, as shown in Figure 2(c) both on SiO 2(red diamond) and YIG (blue square) substrates. Also shown are dashed-line ts us- ing the expression for the nonlocal spin valve signal RSV obtained from a one-dimensional spin transport theory given by [14] RSV=2 FRNed=2N (RF RN+ 1)[RF RNsinh(d=2N) + cosh(d=2N)]:(2)3 (a) AlYIG xy1 23 4 V(b) (c) 300 nm FIG. 2. (Color online) (a) Scanning electron microscopy image of the measured Type-A device. Two Py wires (indicated by green arrows) are connected by an Al cross. A charge current Ifrom contact 1 to 2 creates a spin accumulation at the F1/Al interface that is detected as a nonlocal spin voltage Vnlusing contacts 3 and 4. (b) The NLSV resistance Rnl=Vnl=I for representative YIG (blue) and SiO 2(red and orange) NLSV samples. For comparison, a constant background resistance has been subtracted from each measurement. (c) Dependence of the NLSV signal on the spacing dbetween the injecting and detecting ferromagnetic wires together with calculated spin signal values using a 1D (dashed lines) and 3D (solid lines) spin-transport model. For each distance dbetween the injector and detector several devices were measured, with the error bars indicating the spread in the measured signal. HereRF= (12 F)F FandRN=N Nare spin area re- sistance of the ferromagnetic (F) and non-magnetic (N) metals, respectively. NandFare the corresponding spin diusion lengths, F(N) is the electrical conduc- tivity of the F (N), Fis the spin polarization of Fandd is the distance between the injecting and detecting ferro- magnetic electrodes. Fitting the SiO 2data using Eq. (2), we extract F=0.32 and N;SiO 2=320 nm, which are both in good agreement with reported values [13{15]. A similar tting procedure for the YIG data, assuming an identical spin injection eciency, yields an eectively shorter spin-diusion length N;YIG=190 nm due to the additional spin- ip scattering at the Al/YIG interface. This value of N;YIGtherefore contains important infor- mation regarding an eective spin-mixing conductance Gsthat can be attributed to the interaction of spins with thermal magnons in the YIG. When spin precession, due to the applied external eld as well as the eective eld due toGiis disregarded, we can now estimate Gsby relatingN;YIGtoN;SiO 2viaGsas (see Supplemental Material [19], Sec. I): 1 2 N;YIG=1 2 N;SiO 2+1 2r; (3) with2 r= 2Gs=tAlN[19]. Using the extracted values from the t, N=2107S/m andtAl=130 nm, we extract Gs'2:51013 1m2, which is about 25% of the maximumGr1014 1m2reported for Pt/YIG [4, 7] and Au/YIG [8] interfaces. To quantify our results we performed three- dimensional nite element simulations using COMSOL Multiphysics (3D-FEM) [19, 20] that uses a set ofequations that are equivalent to the continuous random matrix theory in 3 dimensions (CRMT3D) [21]. The charge current j c(~ r) and spin current j s(~ r), (where 2x;y;z ), are linked to their corresponding driving forces via the electrical conductivity as j c(~ r) j s(~ r) = F F  ~rc ~rs (4) wherec= ("+#)=2 ands= ("#)=2 are the charge and spin accumulation chemical potentials, re- spectively. We supplement Eq. (4) by the conservation laws for charge (r
j c(~ r) = 0) and spin current ( r
js= (12 F) s=2+~ !Ls ) where~ !L=gB~B=~with g= 2 is the Larmor precession frequency due to spin precession in an in-plane magnetic eld ~B= (Bx;By;0)T andBis the Bohr magneton (see Supplemental Mate- rial [19], Sec. II). To include spin-mixing at the Al/YIG interface we impose continuity of the spin current jsat the interface using Eq. (1). The input material param- eters such as ,andFare taken from Refs. 22 and 23. The calculated spin signals obtained from our 3D-FEM are shown in Fig. 2(c) for samples on SiO 2(red solid line) and YIG (blue solid line) substrates. By matching the experimentally measured NLSV signal on the SiO 2sub- strate with the calculated values in the model we obtain F= 0:3 andN=350 nm. Using these two values and settingGs'51013 1m2well reproduces the mea- sured spin signal on the YIG substrate. This value of Gs obtained here is consistent with that extracted from our 1D analysis based on Eq. 2. Hence, the interaction of spins with the YIG magnetization, as modeled here, can4 α ˆysµˆm SV S FIG. 3. (Color online) (a) Nonlocal spin valve resistance R nlof a Type-B device with d=500 nm between injecting and detecting Py wires and tAl=130 nm. A constant background resistance of 117 m was subtracted from the original data. (b) Angular dependence of the NLSV signal in the parallel and antiparallel congurations. The AP curve is average of 10 measurements and that of the P state is a single scan. Both resistance states exhibit a cos(2 ) dependence on the angle between ^ mands. The black solid lines are calculated using the 3D-FEM model for Gr= 11013 1m2that show a percentage modulation of only 12% corresponding to the green curve in (c) RSV=RSVis plotted. The angular dependent measurement in (b) is from a device for which complete set of measruements were peformed. A spin valves measurement as in (a) was also performed for another device with d= 300 nm. capture the concept of spin-mixing conductance being responsible for the observed reduction in the spin signal. In the following we investigate the dependence of Rnl on the angle betweensand ^m. We rotate the sam- ple under the application of a very low in-plane mag- netic eld B5 mT, enough to saturate the low-coercive (0:5 mT) YIG magnetization [4, 5] but smaller than the coercive elds of F1andF2(20 mT). This con- dition is important to maintain xed polarization axes ofs, along the magnetization direction of the injecting ferromagnet, and also have a well dened . The re- sult of such measurement in a Type-B device is shown in Fig. 3(b) for d= 400 nm between F 1and F 2. Al- though the measured NLSV signal [Fig.3(a)] is smaller than in Type-A devices, possibly due to a better Al/YIG interface,Rnlexhibits a cos(2 ) behavior with a maxi- mum (minimum) for = 0 (==2), consistent with Eq. (1). However, the maximum change (modulation) of the signalRs=Rnl(= 0)Rnl(==2)) is only 12% of the total spin signal RSV, which is at odds with the large spin-mixing conductance estimated from Fig. 2(b). From anistropic magnetoresistance measurements we ex- clude the possibility of any rotation of the magnetization of the injector and detector as the cause for the observed modulation in the NLSV signal (see Supplemental Mate- rial [19], Sec. III-B). Using the 3D-FEM we calculated the angular depen- dence ofRSVfor various values of Grwhere the percent- age modulation Rs=RSVis plotted as a function of , as shown in Fig. 3(c). The Grvalue of 11013 1m2 extracted from the NLSV signal modulation experiment is one order of magnitude less than reported elsewhere [4]. This can be possibly caused by the presence of disor- dered Al/YIG interface with r.m.s. roughness of 0.8 nm (as measured by AFM), which is close to the magnetic co-herence volume3pVc'1:3 nm [6] of the YIG. This length scale determines the eective width of the Al/YIG inter- face and also the extent to which spin current from the Al is felt by the YIG magnetization [6, 24]. Furthermore, the fact that there exists a nite spin-mixing when = 0, as discussed above, can also explain the observed small modulation. It is important to note that in our experi- ments the non-equilibrium spin accumulation induced by electrical spin injection into Al has a spin-polarization strictly along the direction of the magnetization of F1, which lies along the ^ yaxis. In the measurement results shown in Figs. 1(b) and 2(b) the magnetization of the F2is always kept either parallel or antiparallel to the de- tectorF1. This ensures that it is only the ^ ycomponent of the spin accumulation that is measured in our exper- iments as it is insensitive to other two spin-polarization /s48 /s57/s48 /s49/s56/s48 /s50/s55/s48 /s51/s54/s48/s45/s52/s48/s45/s50/s48/s48/s50/s48/s52/s48 /s48 /s57/s48 /s49/s56/s48 /s50/s55/s48 /s51/s54/s48/s45/s54/s45/s51/s48/s51/s54/s32/s82 /s110/s108/s32/s40 /s41 /s32/s40/s100/s101/s103/s114/s101/s101/s115/s41/s120/s45/s99/s111/s109/s112/s111/s110/s101/s110/s116 /s40/s97/s41 /s32/s32/s82 /s110/s108/s32/s40 /s41 /s32/s40/s100/s101/s103/s114/s101/s101/s115/s41/s32/s80 /s32/s65/s80/s32 /s122/s45/s99/s111/s109/s112/s111/s110/s101/s110/s116 /s40/s98/s41 FIG. 4. (Color online) Calculated NLSV signals showing the (a)x-component and (b) z-component of the NLSV signal Rnlin the parallel (red) and antiparallel (blue) magnetiza- tion congurations of the injector and detector ferromagnetic contacts for Gr= 11013 1m2andGi= 0:1Gr. Even if the injected spin accumulation is polarized along the magne- tization direction of the injecting electrode F 1, its interaction with the magnons via the spin-mixing conductance induces these spin accumulation components.5 directions. It is however possible that the interaction of the initially injected spin accumulation with the YIG magnetization, via G"#, to induce a nite NLSV signal with components polarized along the ^ x- and ^z-directions. Figure 4 shows the angular dependence of the ^ xand ^zcomponent of the NLSV signal as calculated using our 3D-FEM. While the ^ zcomponent exhibits a sin( ) de- pendence, the ^ xcomponent shows a sin(2 ) dependence which is consistent with Eq. (1). The size of the mod- ulation is determined by Grfor the ^xcomponent and byGifor the ^zcomponent. In a collinear measure- ment conguration these transverse spin accumulation components can induce local magnetization dynamics by exerting a spin transfer torque to the YIG. Separately measuring these spin accumulation using ferromagnetic contacts magnetized along the ^ xand ^zdirections can be an alternative way to extract G"#. In summary, we studied spin injection and relaxation at the Al/YIG interface in Ni 80Fe20/Al lateral spin valves fabricated on YIG. The samples on the YIG substrate yield NLSV signals that are two to three times lower than those grown on standard SiO 2substrates, indicating spin- current absorption by the magnetic YIG substrate. We also observed a small but clear modulation of the mea- sured NLSV signal as a function of the angle between the spin accumulation and magnetization of the YIG. The presence of a disordered Al/YIG interface combined with a spin- ip (sink) process due to thermal magnons or interface spin-orbit eects can be accounted for this small modulation. Using nite element magnetoelectronic cir- cuit theory as well as additional control experiments, we establish the concept of collinear (eective) spin mixing conductance due to the thermal magnons in the YIG. Our result therefore calls for the inclusion of this term in the analysis of spintronic and spin caloritronic phenom- ena observed in metal/YIG bilayer systems. The authors thank M. de Roosz and J.G. Holstein for technical assistance. This work is part of the research program of the Foundation for Fundamental Research on Matter (FOM) and is supported by NanoLab NL, EU- FET Grant InSpin 612759 and the Zernike Institute for Advanced Materials.6 SUPPLEMENTAL MATERIAL I. Derivation for the eective spin relaxation length in the collinear case The spin accumulation s, with polarization parallel to the magnetization direction of F1(see Fig. S5), injected in the Al is governed by the Valet-Fert spin diusion equation [25] @2 x+@2 y+@2 z s=s=2 N, which can be re-arranged to give @2 xs=s=2 N@2 zs: (5) Here we assume that, for a homogeneous system, the spin current along the ^ y-direction is zero. As discussed in the main text, when the YIG magnetization direction ^mksthe spin current jz=0 sat the Al/YIG interface, in the ^z-direction, is governed by the spin sink term Gsin Eq. 1 of the main text. Applying spin current continuity condition at the Al/YIG interface we nd that N 2@zs=Gss (6) whereNis the conductivity of the normal metal. Now after re-arranging Eq. (6) to obtain @zs, dierentiat- ing it once and using @zs=s=tAl, wheretAlis the thickness of the Al, we obtain @2 zs=2Gss NtAl: (7) Substituting Eq. (7) into Eq. (5) we obtain a modi- ed VF-spin diusion equation that contains two length scales @2 xs=s=2 N+ 2Grs=NtAl; (8a) =s 2 N+s 2r; (8b) where we dened a new length scale 2 r= 2Gs=NtAl that, together with the N, re-denes an eective spin relaxation length 2 e=2 N+2 r. This eective spin relaxation length in the Al channel is weighted by the spin-mixing conductance Gsof the Al/YIG interface. The modulation of the NLSV signal observed in our mea- surements is hence determined by the interplay between these two length scales, Nandr. While the rst quan- ties the eective spin-conductance of the Al channel (GN=NAN=Al) over the spin relaxation length, the second is a measure of the quality of the Al/YIG inter- face and is set by Gs. For the devices investigated in this work, using AN=tAlwAlwith the width of the Al channelwAl= 100nm and Al= 2107S/m, we ob- tainA1 NGN'61013 1m2, which is close to the Gs obtained in our experiments. This highlights the impor- tance of spin-relaxation induced by the thermal motion of the YIG magnetization, as discussed in the main text.Geometrical enhancement of the modulation can be ob- tained by reducing tAl, as shown in Fig. S5(d), thereby maximizing spin-absorption at the Al/YIG interface [16]. II. Three dimensional (3D) spin transport model Here we describe the our 3D spin transport model used to analyze our data. It is similar to that described in Ref. 20 for collinear spin transport with the possibility of studying spin-relaxation eects (i) due to the spin-mixing conductance at the Al/YIG interface as well as (ii) Hanle spin-precession due to the in-plane magnetic eld [see Sec. III below for detail]. The charge current j c(~ r) and spin currentj s(~ r), forj c(~ r) (where2x;y;z ), are re- lated to the charge c(~ r) and spin potentials sas j c(~ r) j s(~ r) = F F  ~rc ~rs (9) whereis the bulk conductivity and Fis the bulk spin polarization of the conductivity. The device geometry we model is shown in Fig. S5(a), showing schematic source- drain congurations as well as voltage contacts. We im- pose charge ux at contact 1 and drain it at 2. The nonlocal voltage, due to spin diusion, is obtained by taking the dierence between the surface integrated c at contacts 3 and 4 , both for the parallel (P) or antipar- allel (AP) magnetization congurations. To solve Eq. (9), we use conservation laws for charge ( r
j c(~ r) = 0) and spin current (r
js= (12 F)s=2) with spin pre- cession due to the in-plane applied eld also included in the model. By dening an angle betweensand the YIG magnetization ^ mand allowing for a boundary spin current at the Al/YIG interface using Eq. 1 of the main text, we can study the transport of spins in NLSV de- vices and their interaction with the YIG magnetization. The material parameters for the model, ,Fandsare taken from Ref. [23]. Our modeling procedure involves, rst, tting of the measured NLSV signal on a SiO 2sub- strate by varying Fand usingN= 350nm. Next, we aim to nd Gsof the Al/YIG interface that properly quanties spin transport properties of the YIG sample. Figure S5(b) shows the dependence of the NLSV signal onGs. As expected, when Gsvery low, the NLSV signal is not aected by the presence of the YIG as spins are not lost to the substrate. For Gs'51013 1m2we obtain the experimentally measured NLSV signal (shown in red dashed line). For even larger Gsvalues, the eect is maximum with the NLSV signal falling by almost one order of magnitude. It is important to remember that the value of Gsthat is extracted here is a simple mea- sure of spin- ip processes at the Al/YIG interface due to thermal uctuation of the YIG magnetization or disorder induced eects. At the temperatures of our experiment it is dicult to distinguish which one of the two processes is dominant.7 13 2 13 2G 8 10 m and G 5 10 mr s− −= × Ω = × Ω(c)Gs(b) 12 4 3cj/arrowrightnosp xz y(a) (d) SV S FIG. 5. (a) Geometry of the modeled device showing the measurement conguration with a 3D prole and the y-component of the spin accumulation. (b) The dependence of the NLSV signal on the eective (collinear) spin mixing conductance Gs. To reproduce the experimentally observed decrease in the spin signal from SiO 2to the YIG substrate, an eective spin mixing conductance of Gs= 51013 1m2is required. (c) The dependence of the NLSV signal on the angle between ^ mands forGs= 51013 1m2. (d) The dependence of the spin signal modulation amplitude on the thickness of the Al channel signifying the interplay between the spin-mixing conductance and the spin-conductance in the Al channel. For the angular dependent simulation we only vary the anglebetweensand ^mwhile keeping all other param- eters constant (such as F,NGs= 51013 1m1 andGr= 81013 1m1). As shown in Fig. S5(b) our simulation as described above reproduces the cos2() de- pendence observed in our experiments as well as by Vil- lamor et al. [16]. For the extracted values of Grfrom our analysis, the experimentally observed modulation of the NLSV signal by the rotating magnetization direction of the YIG is small. Possible ways to enhance the modulation are to 1) maximize the spin-mixing conductance via controlled interface engineering of the Al/YIG interface or 2) reduce the thickness of the spin transport channel. In the latter, for a xed Gr, the eect of decreasing the thickness of the spin transport channel is to eectively reduce the spin conductance GNalong the channel thereby maximizing the spin current through the Al/YIG interface. Fig- ure S5(c) shows the thickness dependence of the modula- tion of the spin signal Rs=Rs(= 00)Rs(= 900)normalized by Rsas a function of the thickness tAl, with the inset showing that for the P and AP congurations. As the thickness of the Al channel increases the spin cur- rent absorption at the Al/YIG interface decreases or vice versa. III. Investigation of possible alternative explanations for the observed modulation It can be argued that the experimentally observed modulation of the NLSV signal can be fully explained by (i) the Hanle spin-precession and/or (ii) the rotation of the magnetizations of the injector/detector electrodes due to the 5 mT in-plane magnetic eld. Below, we show that even the combined eect of both mechanisms is too small to explain the experimentally observed modulation of the NLSV signal.8 (b) (c)(a) AntiparallelParallel Injector Detector FIG. 6. (a) Modulation of the NLSV when only considering the Hanle eect due to the in-plane magnetic eld in the P (dashed lines) and AP (solid lines) at 5 mT (red), 50 mT (blue) and 100 mT (black). see text for more details. (b) Anisotropic magnetoresistance (AMR) measurement for the injector (left) and detector (right) ferromagnets at two dierent magnetic elds. The insets show the full-scale plot of the measurements at 5mT. A. Hanle spin-precession induced modulation of the NLSV signal Spins precessing around an in-plane magnetic eld ~Bwould acquire an average spin precession angle of =!LD, where!L=gB~B=~is the Larmor precession frequency,D=L2=2Dc= 25 ps is the average diusion time an electron takes to traverse the distance Lbetween the injector and the detector and Dc= 0:005m2/s is the diusion coecient [26]. For an applied eld of 5 mT andL=500 nm, we obtain = 1:25o, giving us a max- imum contribution of 1 cos=0.02% [see Eq. (10)] to the experimentally observed signal (compared to the 12% in Fig. 3(b) of the main text). This is expected be- cause the spin-precession frequency !1 L(8 ns) at such magnetic elds is three orders of magnitude slower than D. This simple estimate is further supported by our 3D - nite element model as we show next. Figure S6(a) shows the angle dependence of the nonlocal signal due to an in-plane magnetic eld when we only consider the Hanleeect both for the AP (solid lines) and P (dashed lines) congurations at three dierent magnetic eld values of 5 mT (red), 50 mT (blue) and 100 mT (black). The maximum modulation of the NLSV signal that the Hanle eect presents is only 0.001% at the measurement eld of 5 mT and only become relevant at high elds. There- fore, the Hanle eect alone can not explain the results presented in the main text. B. Magnetization rotation induced modulation of the NLSV signal The in-plane rotation of the sample under an applied magnetic eld of 5 mT might induce rotations in the ma- gentization of the injector/detector electrodes. In such a case, a relative angle rbetween the magnetization direc- tion of the injector and detector electrodes would result in a modulation of the NLSV signal given by Rnl Rnl(r= 0)=Rnl(r= 0)Rnl(r) Rnl(r= 0)=j1cosrj; (10)9 with +() corresponding to the P (AP) conguration. Using Eq. (10), we nd that a relative angle r'28o between the magnetization directions of the injector and detector is required in order to explain the experimentally observed modulation. To determine the eld induced in- plane rotation of the magnetization by the applied mag- netic eld, we carried out angle dependent anisotropic magnetoresistance (AMR) measurements both for the in- jector and detector electrodes, using a new set of devices with identical dimensions. The AMR measurements were repeated for dierent magnetic eld strengths, at 5 mT and at higher magnetic elds of 100 mT and 300 mT. Figure S6(b) and (c) show the two-probe AMR mea- surement of the injector and detector electrodes, respec- tively, at two dierent magnetic elds. For the injector electrode in Fig. S6(b), at an applied eld of 100 mT (red line), an AMR response
R=RkR?= 0:6 is observed, where Rk(R?) is the resistance of the ferro- magnet when the angle between the applied eld and the easy axis is = 0o(= 90o). For the same electrode, at an applied eld of 5 mT (blue line, see also the inset), the AMR response is only 0.025 . Now, by comparing these two measurements we conclude that the eect of the 5 mT eld would be to rotate the magnetization of this electrode by a maximum angle 1= 15ofrom the easy axis. A similar analysis for the detector electrode, using the AMR responses of 2 (at 300 mT) and 0.025 (at 5 mT) in Fig. S6(c), yields a maximum magneti- zation rotation 2= 10o. Relevant here is the net rel- ative magnetization rotation between the two electrodes r=12= 5oand, using Eq. (10), we conclude that it would only cause a modulation of 0.4 %, which is much smaller than the 12% observed in our experiments. Our analysis based on the AMR eect is equivalent to that in Ref. 16 where magneto-optical Kerr eect measurements were used to exclude a possible in-plane magnetization rotation as the origin for the observed modulation in the nonlocal spin valve signal [16]. To summarize this section, the Hanle eect and the magnetization rotation induced by the in-plane magnetic eld neither separately nor when combined are sucient to explain the experimentally observed modulation. Only after including the eect of the spin-mixing interaction viaG"#that it is possible to reproduce the modulation observed in the experiments. e-mail:f.k.dejene@gmail.com [1] K. Uchida, J. Xiao, H. Adachi, J. Ohe, S. Takahashi, J. Ieda, T. Ota, Y. Kajiwara, H. Umezawa, H. Kawai, et al. , Nature materials 9, 894 (2010). [2] V. Castel, N. Vlietstra, J. Ben Youssef, and B. J. van Wees, Applied Physics Letters 101, 132414 (2012). [3] M. Althammer, S. Meyer, H. Nakayama, M. Schreier,S. Altmannshofer, M. Weiler, H. Huebl, S. Gepr ags, M. Opel, R. Gross, D. Meier, C. Klewe, T. Kuschel, J.-M. Schmalhorst, G. Reiss, L. Shen, A. Gupta, Y.-T. Chen, G. Bauer, E. Saitoh, and S. Goennenwein, Phys. Rev. B 87, 224401 (2013). [4] N. Vlietstra, J. Shan, V. Castel, B. J. van Wees, and J. Ben Youssef, Physical Review B 87, 184421 (2013). [5] J. Flipse, F. K. Dejene, D. Wagenaar, G. E. W. Bauer, J. B. Youssef, and B. J. van Wees, Phys. Rev. Lett. 113, 027601 (2014). [6] J. Xiao, G. E. W. Bauer, K.-c. Uchida, E. Saitoh, and S. Maekawa, Physical Review B 81, 214418 (2010). [7] X. Jia, K. Liu, K. Xia, and G. E. W. Bauer, EPL (Eu- rophysics Letters) 96, 17005 (2011). [8] B. Heinrich, C. Burrowes, E. Montoya, B. Kardasz, E. Girt, Y.-Y. Song, Y. Sun, and M. Wu, Physical Re- view Letters 107, 066604 (2011). [9] Y.-T. Chen, S. Takahashi, H. Nakayama, M. Althammer, S. T. B. Goennenwein, E. Saitoh, and G. E. W. Bauer, Physical Review B 87, 144411 (2013). [10] M. Weiler, M. Althammer, M. Schreier, J. Lotze, M. Pernpeintner, S. Meyer, H. Huebl, R. Gross, A. Kamra, J. Xiao, Y.-T. Chen, H. Jiao, G. Bauer, and S. Goennenwein, Phys. Rev. Lett. 111, 176601 (2013). [11] S. Huang, X. Fan, D. Qu, Y. Chen, W. Wang, J. Wu, T. Chen, J. Xiao, and C. Chien, Phys. Rev. Lett. 109, 107204 (2012). [12] T. Kikkawa, K. Uchida, S. Daimon, Y. Shiomi, H. Adachi, Z. Qiu, D. Hou, X.-F. Jin, S. Maekawa, and E. Saitoh, Physical Review B 88, 214403 (2013). [13] F. J. Jedema, M. V. Costache, H. B. Heersche, J. J. A. Baselmans, and B. J. van Wees, Applied Physics Letters 81(2002). [14] F. J. Jedema, M. S. Nijboer, A. T. Flip, and B. J. van Wees, Physical Review B 67, 085319 (2003). [15] T. Kimura, T. Sato, and Y. Otani, Physical Review Let- ters100, 066602 (2008). [16] E. Villamor, M. Isasa, S. V elez, A. Bedoya-Pinto, P. Vavassori, L. E. Hueso, F. S. Bergeret, and F. Casanova, Phys. Rev. B 91, 020403 (2015). [17] V. Castel, N. Vlietstra, B. J. van Wees, and J. B. Youssef, Physical Review B 86, 134419 (2012). [18] Z. Qiu, K. Ando, K. Uchida, Y. Kajiwara, R. Takahashi, H. Nakayama, T. An, Y. Fujikawa, and E. Saitoh, Ap- plied Physics Letters 103, 092404 (2013). [19] See supplemental material for the derivation of Eq. 3, detailed procedure of the nite element simulation and additional control experiments. [20] A. Slachter, F. L. Bakker, and B. J. van Wees, Phys. Rev. B 84, 174408 (2011). [21] V. S. Rychkov, S. Borlenghi, H. Jares, A. Fert, and X. Waintal, Phys. Rev. Lett. 103, 066602 (2009). [22] F. L. Bakker, A. Slachter, J.-P. Adam, and B. J. van Wees, Phys. Rev. Lett. 105, 136601 (2010). [23] F. K. Dejene, J. Flipse, G. E. W. Bauer, and B. J. van Wees, Nature Physics 9, 636 (2013). [24] M. Schreier, A. Kamra, M. Weiler, J. Xiao, G. E. W. Bauer, R. Gross, and S. T. B. Goennenwein, Physical Review B 88, 094410 (2013). [25] T. Valet and A. Fert, Physical Review B 48, 7099 (1993). [26] F. J. Jedema, M. V. Costache, H. B. Heersche, J. J. A. Baselmans, and B. J. van Wees, Applied Physics Letters 81(2002).

2015-03-20

We study the effect of a magnetic insulator (Yttrium Iron Garnet - YIG) substrate on the spin transport properties of Ni$_{80}$Fe$_{20}$/Al nonlocal spin valve (NLSV) devices. The NLSV signal on the YIG substrate is about 2 to 3 times lower than that on a non magnetic SiO$_2$ substrate, indicating that a significant fraction of the spin-current is absorbed at the Al/YIG interface. By measuring the NLSV signal for varying injector-to-detector distance and using a three dimensional spin-transport model that takes spin current absorption at the Al/YIG interface into account we obtain an effective spin-mixing conductance $G_{\uparrow\downarrow}\simeq 5 - 8\times 10^{13}~\Omega^{-1}$m$^{-2}$. We also observe a small but clear modulation of the NLSV signal when rotating the YIG magnetization direction with respect to the fixed spin polarization of the spin accumulation in the Al. Spin relaxation due to thermal magnons or roughness of the YIG surface may be responsible for the observed small modulation of the NLSV signal.

Control of spin current by a magnetic YIG substrate in NiFe/Al nonlocal spin valves

1503.06108v1

arXiv:2202.03774v1 [cond-mat.mes-hall] 8 Feb 2022AIP/123-QED Modulation of Spin Seebeck Effect by Hydrogenation K. Ogata,1T. Kikkawa,2E. Saitoh,2, 3and Y. Shiomi1, 4 1)Department of Integrated Science, University of Tokyo, Meg uro, Tokyo 153-8902, Japan 2)Department of Applied Physics, University of Tokyo, Bunkyo , Tokyo 113-8656, Japan 3)Institute for AI and Beyond, University of Tokyo, Bunkyo, To kyo 113-8656, Japan 4)Department of Basic Science, University of Tokyo, Meguro, T okyo 153-8902, Japan (*Electronic mail: yukishiomi@g.ecc.u-tokyo.ac.jp) (Dated: 9 February 2022) We demonstrate the modulation of spin Seebeck effect (SSE) b y hydrogenation in Pd/YIG bilayers. In the presence of 3% hydrogen gas, SSE voltage dec reases by more than 50 % from the magnitude observed in pure Ar gas. The modulation o f the SSE voltage is reversible, but the recovery of the SSE voltage to the prehyd rogenation value takes a few days because of a long time constant of hydrogen desorption. We also demonstrate that the spin Hall magnetoresistance of the identical sample reduce s significantly with hydrogen exposure, supporting that the observed modulation of spin c urrent signals originates from hydrogenation of Pd/YIG. 1Hydrogen is an energy carrier which can be produced by an envi ronmentally clean process and therefore has a positive impact on decarbonization1. To utilize hydrogen as a clean and renewable alternative to carbon-based fuels, hydrogen safety sensor s are also critical to assure the develop- ment of hydrogen systems2. Metal-hydride systems have been widely studied for the pot ential of solid-state hydrogen storage and sensing. In particular, P d is frequently used as a catalyst for hy- drogen dissociation and adsorption. A hydrogen molecule de composes into independent hydrogen atoms when the molecule approaches a Pd surface due to the str ong interaction between Pd and H atoms. As the smallest single atom, a H atom can easily diffus e into the interstices of the Pd lattice and cause lattice expansion. As a result, hydrogen adsorpti on changes the density of states of Pd near the Fermi energy3,4, significantly modulating its electrical and optical prope rties. The hydrogenation of Pd films also impacts spintronic effect s. Pd is known to exhibit a strong spin-orbit coupling and has been used in many spintro nic experiments. Magnetic multi- layers and super-lattices which include Pd layers have been of particular interest. It was reported that in Co/Pd bilayers which possess a strong interfacial pe rpendicular magnetic anisotropy, the magnetic anisotropy and ferromagnetic resonance are re versibly modulated by hydrogen exposure5–11. Efficient hydrogen sensing based on magnetization dynamic s was also reported in similar materials5,12,13. Moreover, the inverse spin Hall effect (ISHE) induced by sp in pumping was successfully modulated by hydrogen exposure in Co/Pd bi layers14,15. Absorption of hydrogen gas at 3% concentration in the Pd layer reduces the ISHE volta ge by 20%15. This decrease in ISHE signals in the presence of hydrogen gas was attributed to the decrease in spin diffusion length due to enhanced scatterings to hydrogen atoms in the Pd layer. Af ter the hydrogen gas is flushed out of the setup, the ISHE voltage returns to the prehydrogenati on value; hence the observed effect is reversible. The studies of hydrogen effects on spintronic materials hav e been carried out for the combi- nation of Pd layers with itinerant magnetic films. For the mea surement of ISHE in ferromag- netic/nonmagnetic metallic bilayers, however, it is known that the precise estimation of ISHE voltage is difficult because of spurious spin rectification e ffects such as anisotropic magnetoresis- tance and anomalous Hall effect which couple the dynamic mag netization to microwave currents in the ferromagnetic layer16. Hence magnetic insulators such as Y 3Fe5O12(YIG) should be more suitable to investigate hydrogen effects on pure ISHE signa ls17. In this letter, we demonstrate the reversible manipulation of the spin Seebeck effect (SSE) by hydrogen exposure in Pd/YIG bilayers. The SSE is the generat ion of a spin current as a result of 2a temperature gradient applied across a junction consistin g of a magnet and a metal18,19. The spin current injected into a metal can be converted into a voltage by ISHE. Since the ISHE induced by SSE is a transverse thermoelectric effect, it can be employe d to realize transverse thermoelectric devices, which could potentially overcome the inherent lim itations of conventional thermoelectric devices20,21. Moreover, SSE is expected to be utilized for flexible heat-fl ow sensors22. The ma- nipulation of SSE by hydrogenation demonstrated below may o pen up new device potentials in spin caloritronics. We used epitaxial YIG films with 2 micron thickness grown by li quid phase epitaxy on Gd3Ga5O12(111) substrates. The YIG surfaces were mechanically polis hed, and then 5-nm-thick Pd films were sputtered at room temperature. The Pd layer is 5 m m long and 0.5 mm wide. For the Pd/YIG bilayers, the SSE measurements in the longitudin al configuration18were performed at room temperature using an electromagnet (3470 Electroma gnet System, GMW Associates). The bilayer sample was placed between sapphire and copper pl ates. A 1-k Ωresistive heater was attached to the upper sapphire plate and the lower copper pla te is a heat sink. To facilitate hydro- genation of the Pd layer, a breathable tape (TBAT-252, TRUSC O) was inserted between the upper plate and the sample [Fig. 1(a)]. The temperature gradient i s generated by applying an electric current to the heater. Two-pairs of leads were attached to th e Pd layer to measure not only the SSE but also the spin Hall magnetoresistance (SMR)23in the same setup. The distance between the voltage terminals is 2.5 mm. The thermoelectric voltage due to the ISHE induced by SSE was monitored with a Keithley 2182A nanovoltmeter. SMR was m easured by lockin detection using Anfatec USB Lockin Amplifier 250; the frequency and amp litude of ac electric current are 111 Hz and 0.8 mA. The sample was loaded into a small chamber to control the atmosphere. For hydrogenation measurements, the samples were first measure d in pure Ar gas ( >99.9999 vol.%) at atmospheric pressure followed by a 3%/97% H 2/Ar gas mixture. Before the measurements in Ar-H 2gas, we waited 20-40 minutes for the Pd layer to be completely hydrogenated15,24after the chamber was filled with 1 atm Ar-H 2gas. First, we measured SSE voltage VSSEof Pd/YIG in Ar atmosphere. Figure 1(b) shows the magnetic-field ( H) dependence of VSSEmeasured at several heater power levels. Here symmetric components of the output voltage with respect to Hare subtracted and antisymmetric components are plotted; note that the symmetric components which are no t to be attributed to the effect under study are almost independent of Hin our measurements. When the heater is off, VSSEis almost zero in the entire Hrange in Fig. 1(b). As the heater power Pincreases from zero, the clear 3FIG. 1. (a) Measurement setup of the SSE. A breathable tape wa s inserted between Pd/YIG and the heater part to facilitate hydrogen absorption and desorption in th e Pd film. (b) Magnetic field ( H) dependence of the SSE voltage ( VSSE) measured in 1 atm of Ar. The heater power ( P) was changed from 0 mW to 100 mW. (c) Heater power ( P) dependence of the SSE voltage ( VSSE) at 200 mT. The raw data is shown in (b). SSE signals appear and their magnitudes increase with P. The sign of VSSEis the same as that reported for Pt/YIG18. The saturated magnitude of VSSEis plotted against Pin Fig. 1(c). The VSSEmagnitude increases linearly with the heater power, which i ndicates that VSSEis proportional to temperature gradient generated across the Pd/YIG juncti on. The temperature difference ∆T generated at P=100 mW is estimated to be ∼1.5 K (see Fig. S1 in Supplementary Material). Next, the effect of exposing the 3% H 2mixture on the Pd/YIG sample is investigated in Fig. 2 (see also Fig. S2 in Supplementary Material for additional e xperimental results). Here the heater power Pis kept constant at 100 mW during the series of measurements. After the initial SSE measurement in pure Ar gas at atmospheric pressure already s hown in Figs. 1(b) and 1(c), the sample chamber was filled with H 23% H 2/Ar mixture and the SSE measurement was performed. As shown in Fig. 2, the magnitude of VSSEis found to be reduced by more than 50% in the presence of H 2gas. After completing the SSE measurement in Ar-H 2atmosphere, the sample was then remeasured in pure Ar. The VSSEmagnitude returned to the pristine value as shown in Fig. 2. Note that this data was taken 2.5 days after the cham ber was refilled with Ar. The observed decrease in the SSE signal is safely ascribed to the presence of hydrogen in Pd/YIG, and 4FIG. 2. Magnetic field ( H) dependence of the SSE voltage ( VSSE) measured before hydrogenation, during exposure to hydrogen gas, and after the hydrogen gas is flushe d out of the setup. The heater power is kept at 100 mW. importantly, the change is reversible. It is well known15,25that upon hydrogenation, Pd thin films undergo two stages of l attice ex- pansion depending on the hydrogen gas concentration. For li ght concentration levels up to 2-3 %, the lattice constant grows by approximately 1% in the out-of -plane direction only. This expansion is reversible. In the second stage, the lattice constant gro ws by up to 4% in both out-of-plane and in-plane directions. These changes are irreversible, caus ing structural changes to the Pd lattice. In our SSE measurements under 3% hydrogen gas, the sample sho uld undergo the first stage of lattice expansion and the SSE is thereby reversible. Note th at we confirmed by x-ray diffraction that the Pd films are (111) oriented as in the literature25(see Supplementary Material). Though the modulation of SSE by hydrogen absorption/desorp tion is reversible, the recovery 5FIG. 3. (a) Magnetic field ( H) dependence of the SSE voltage ( VSSE) measured 1.7-66 hours after Ar gas is refilled in the measurement chamber. The heater power is kept at 100 mW. (b) Time dependence of the SSE voltage ( VSSE) at 210 mT measured after Ar gas is refilled in the measurement chamber. The selected raw data is shown in (a). The black curve is a fit to the experimenta l data (see text). time of the SSE signal due to hydrogen desorption is as long as 2.5 days. It was reported that the hydrogen desorption takes a long time in contrast to the q uick hydrogen absorption9, and the response time depends significantly on materials. The time f or hydrogen desorption is typically at most several tens of minutes for Co/Pd5–11, while the completion of the entire desorption requires at least a few days at 10−3mbar for FePd alloys26. Our Pd/YIG also includes Fe and Pd, and the situation looks similar to FePd alloys. We then take a closer look on the dehydrogenation process by t he time dependent measurement of SSE in Fig. 3. Figure 3(a) shows VSSEcurves measured at different times after the measurement chamber is refilled with pure Ar gas. The VSSEmagnitude is approximately 0.5 µV just after the gas is replaced with Ar, and increases monotonically with ti me. After 50 hours, the VSSEmagnitude is almost saturated at ∼1µV. The time dependence of VSSEat 210 mT is plotted in Fig. 3(b). The VSSEmagnitude increases monotonically with time, as already shown in Fig. 3(a). We fit the experimental data by a standard relaxation function: VSSE∝1−e−t/τ, where tis the measurement time and τis a time constant of hydrogen desorption. The fitting curve matches the experi mental data very well, meaning that the hydrogen desorption follows an exponential function. T he same function was adopted for the hydrogenation effect on magneto-optical effects in Pd/Co/ Pd films9. The fit in Fig. 3(b) yields τ≈25 hour. Such a long time constant was not observed in the spin pumping measurement for 6FIG. 4. Magnetic field ( H) dependence of the magnetoresistance (MR) ratio ( ρ(H)/ρ(H=0)−1) measured before hydrogenation (a), during exposure to hydrogen gas ( b), and after the hydrogen gas is flushed out of the setup (c). Pd/Co bilayers15. In contrast to the spin pumping measurements, the attachmen t of the heater to the Pd surface is required in the SSE measurements, which may adversely affec t the absorption/desorption of hy- drogen because of small numbers of exposed surface atoms. To confirm that spin current signals in the Pd layer is indeed modulated by hydrogenation, we also perform the measurement of SMR (spin Hall magnetoresistance) for the same sample in the sam e setup. The SMR is a magnetoresis- tance effect related to a nonequilibrium proximity effect c aused by the simultaneous action of the SHE and ISHE23,27; the absorption/reflection of spin current at the ferromagn et/metal interface re- sults in magnetoresistance, since the spin-dependent scat tering at the metal/ferromagnet interface depends on the angle between the polarization of spin Hall cu rrent and the magnetization of the attached magnetic layer. The experimental setup is illustr ated in the inset to Fig. 4(a). Magnetic field is applied perpendicular to the electric-current dire ction in the film plane. Figure 4 shows the hydrogen effects on SMR in the Pd/YIG bilay er. Here, since the size of SMR is very small, the magnetoresistance measurements were repeated several times and aver- aged. The error bars stand for the standard errors. Before hy drogenation [Fig. 4(a)], a negative magnetoresistance effect is observed. The magnetic-field d ependence of resistance change follows the magnetization process of the YIG layer, consistent with the SMR23. The size of SMR is about 1×10−3%. This magnitude is about ten times smaller than that in Pt/Y IG23. A small SMR of about 10% compared to Pt/YIG was also reported in the literat ure28. During the exposure to 3% hydrogen gas, the SMR magnitude dec reases significantly as shown 7in Fig. 4(b). Although quantitative analysis is difficult be cause of the large error bars, the suppres- sion of SMR ratio by hydrogenation looks more than 50%, consi stent with the modulation in SSE voltages (Figs. 2 and 3). After the hydrogen gas is flushed out of the chamber and pure Ar gas is refilled, we confirmed that the size of SMR returns to the initi al value [Fig. 4(c)]. An important finding in the SMR measurement is that the SMR rat io has already returned to its original value 30 minutes after refilling Ar gas. Namely, the time constant of hydrogen desorption in the SMR measurement is much shorter than that in the SSE mea surement. Since both the measurements were performed for the same sample in the same s etup, the long time constant of hydrogen desorption in the SSE measurement cannot be attrib uted to impurities/defects in the Pd layer, surface oxidation, surface morphology9, or moisture which may trap hydrogen atoms and hinder the hydrogen desorption29,30. In our measurements of SSE and SMR, spin current signals are s ignificantly suppressed by hydrogen exposure as shown in Figs. 2-4. The decrease in the s pin Hall signals with hydrogen exposure is consistent with the previous spin pumping measu rements for Co/Pd14,15. Scatterings of conduction electrons to hydrogen atoms in the Pd layer dec rease the spin diffusion length due to the enhanced Elliot-Yafet relaxation mechanism, and res ult in the decrease in spin-pumping signals15. This mechanism should be also applicable to SSE and SMR. Sin ce the theory has shown that both effects depend on spin diffusion length and s pin Hall angle of the Pd layer19,27, the signal variation by hydrogenation can be attributed to the d ecrease in the spin diffusion length15. On the other hand, it is notable that magneto-optical Kerr si gnals are enhanced by hydrogenation in Co/Pd bilayers9,10, in contrast to the decrease in the spin current signals14,15. In transport measurements such as (inverse) spin Hall effects, enhanced electron scatterings due to interstitial hydrogen impurities are likely to play a dominant role in the hydrogenation effect. The significant scattering effect due to hydrogen atoms is also evidenced by the reduction of the anomalous Hall signal in hydrogenated Co xPd1−xfilms12. The decrease in the VSSEmagnitude ( >50%) by hydrogen exposure is noticeably greater than the change in ISHE signals reported in the spin pumping measu rements15; the decrease in the spin- pumping voltage in H 2/Ar mixture with 3% of hydrogen was only 20%. The larger signa l change in our results suggests that there may be other factors for the r eduction of VSSEbesides hydrogenation of the Pd layer. The first possibility is imperfect separatio n of ISHE signals from spin rectification effects in metallic Co/Pd bilayers14,15. Another possible origin is different interfacial stresse s to the Pd layer between YIG and Co. It is known that electrical re sistivity of single-layer Pd grown on 8Si substrates increases upon hydorogenation, while it tend s to decrease for bilayer cases15because of the interfacial compressive stress from the underlying l ayer. The interfacial stress can also affect the interface spin mixing conductance, modulating the inje ction efficiency of spin currents. Note that the resistivity of the Pd film on YIG decreases by hydroge nation, but the change in resistivity is as small as 1% (Fig. S3 in Supplementary Material), which c annot explain the large variation (>50%) of SSE voltage by hydrogen exposure. Moreover, since the SSE also depends on bulk spin transport i n the YIG layer31,32in contrast to the spin pumping, hydrogen effects on YIG may contribute t o the significant reduction in the VSSEmagnitude. Hydrogen diffusion in YIG was indeed reported fo r annealed samples in H 2 atmosphere33–35. The hydrogen diffusion in the YIG layer can suppress the mag non and phonon transport, which should reduce the VSSE. Also the interface spin-exchange coupling can be weak- ened by hydrogen around the interface, leading to the decrea se in the interface spin-injection efficiency. The presence of hydrogen effects on the YIG layer is also sugg ested by the different recovery time constants between SSE and SMR. Our measurements in Figs . 3 and 4 showed that the time constant for the signal recovery of SSE is much longer than th at of SMR. The different recovery time constants are attributable to different bulk sensitiv ity of these effects. In SMR, spin-dependent scattering at the Pd/YIG interface is essential. In contras t, bulk thermal spin current also plays an important role in the SSE voltage31,32in addition to the interfacial spin coupling. Bulk properti es of magnetic materials such as bulk magnetization, thermal c onductivity, and magnon transport coefficient contribute to the SSE signals, but not to SMR or sp in pumping. Also in the case of magneto-optical effects of Pd/Co frequently studied befor e for hydrogenation effects, the variation of perpendicular magnetic anisotropy originates from inte rface effects5,9. Hence the SSE is a rare spintronic phenomenon that depends not only on interface pr operties but also on bulk properties of magnons and phonons in the magnetic layer. Hydrogen effec ts on YIG could be related to the reduction of VSSEand also the long time constant for hydrogen desorption in SS E. In conclusion, we experimentally demonstrated the reversi ble modulation of SSE and SMR by hydrogenation in Pd/YIG bilayers. Absorption of hydrogen r esults in the decrease in both SSE and SMR signals by more than 50%. Enhanced scatterings of con duction electrons to hydrogen atoms in the Pd layer are partly responsible for the decrease in the spin-current signals, as reported in the previous spin pumping experiments. The modulation of SSE voltage is reversible, but the time constant for the signal recovery is longer than 2 days. T he long time constant for hydrogen 9desorption in the SSE measurement is in contrast to the case o f SMR, in which the SMR ratio already returned to the prehydrogenation value 30 minutes a fter the chamber was refilled with pure Ar. We speculate that the significant decrease in the SSE magnitude by hydrogen exposure and the long time constant for hydrogen desorption in SSE are related to the hydrogen modulation of bulk properties of the YIG layer, since the SSE depends not only on interfacial spin couplings but also on bulk properties of the magnetic layer. We hope tha t the present results will stimulate further research on hydrogen effects on Pd films grown on insu lating magnetic oxides. See the supplementary material for additional SSE data, x-r ay diffraction data, and resistivity change by hydrogen exposure. We thank Y . Miyazaki for the experimental help of sample prep aration and Dr. T. Yok- ouchi for the fruitful discussion. This research was suppor ted by JST CREST (JPMJCR20C1 and JPMJCR20T2), Institute for AI and Beyond of the Universi ty of Tokyo, and JSPS KAK- ENHI Grant Numbers JP20H05153, JP20H02599, JP20H04631, JP 21K18890, JP19H05600, and JP19H02424. The data that support the findings of this study are available from the corresponding author upon reasonable request. REFERENCES 1Etienne Rivard, Michel Trudeau, and Karim Zaghib. Hydrogen storage for mobility: A review. Materials , 12(12), 2019. 2William J. Buttner, Matthew B. Post, Robert Burgess, and Car l Rivkin. An overview of hydrogen safety sensors and requirements. 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2022-02-08

We demonstrate the modulation of spin Seebeck effect (SSE) by hydrogenation in Pd/YIG bilayers. In the presence of 3% hydrogen gas, SSE voltage decreases by more than 50% from the magnitude observed in pure Ar gas. The modulation of the SSE voltage is reversible, but the recovery of the SSE voltage to the prehydrogenation value takes a few days because of a long time constant of hydrogen desorption. We also demonstrate that the spin Hall magnetoresistance of the identical sample reduces significantly with hydrogen exposure, supporting that the observed modulation of spin current signals originates from hydrogenation of Pd/YIG.

Modulation of Spin Seebeck Effect by Hydrogenation

2202.03774v1

Eect of dipolar interactions on cavity magnon-polaritons Antoine Morin, Christian Lacroix, and David M enard Department of Engineering Physics, Polytechnique Montr eal, Montr eal, Qc (Dated: September 11, 2020) The strong photon-magnon coupling between an electromagnetic cavity and two yttrium iron garnet (YIG) spheres has been investigated in the context of a strong mutual dipolar interaction between the spheres. A decrease in the coupling strength between the YIG spheres and the elec- tromagnetic cavity is observed, along with an increase of the total magnetic losses, as the distance between the spheres is decreased. A model of inhomogeneous broadening of the ferromagnetic res- onance linewidth, partly mitigated by the dipolar narrowing eect, reproduces the reduction in the coupling strength observed experimentally. These ndings have important implications for the un- derstanding of strongly coupled photon-magnon system involving densely packed magnetic objects, such as ferromagnetic nanowires arrays, in which the total coupling strength with an electromagnetic cavity might become limited due to mutual dipolar interactions. I. INTRODUCTION Following recent works on the strong coupling between a magnonic mode of a ferromagnetic sample and a pho- tonic mode of a microwave cavity1,2, also called cavity magnon-polaritons3,4, much interest emerged in the sci- entic community to exploit the phenomenon as a mean to develop novel information transfer technologies5{10. Some interesting propositions involve multiple yttrium iron garnet (YIG) spheres placed inside an electromag- netic cavity, such as the magnon gradient memory11, the long distance modication of spin currents12, and the de- velopment of ultrahigh sensitivity magnetometers13. As these new ideas are being elaborated14, it is impor- tant to correctly predict the behavior of photon-magnon systems consisting of several ferromagnetic elements cou- pled to an electromagnetic resonator. In this context, the eect of dipolar interactions between the ferromagnetic objects on the strong photon-magnon coupling is crucial and remains relatively unexplored. The strong coupling of photon-magnon systems is well understood and has been recently reviewed15. Its ex- tension to multiple independent magnons is relatively straightforward16. For an ensemble of Nindependent and identical ferromagnetic objects, the ideal coupling is expected to be enhanced by a factor ofp Nas com- pared to the coupling strength of a single object to the cavity. However, due to dipolar interactions between the magnetic elements, some detuning along with inhomoge- neous broadening are expected for coupled magnon sys- tems. In this paper, we investigate the coupling strength of a simplied system consisting of two YIG spheres cou- pled to an adjustable microwave cavity. We show exper- imentally that the coupling constant gdecreases as the spheres are brought closer. The results are explained us- ing a model based on the Landau-Lifshitz equation and the Fourier expansion of the magnetization in order to include the coupling of the photons with the uniform fer- romagnetic mode as well as with the long wavelength spin wave modes, which are excited in presence of a non- uniform magnetic eld. Y I G s p h e r e s D e p t h c o n t r o l L e n g t h c o n t r o l M e t a l l i c r o d S h o r tH0hxYIG spheres Depth control Length control Metallic rodShortFIG. 1. Schematic representation of the tunable cavity used experimentally. The metallic rod allows the tuning of the resonance frequency and the losses of the cavity. The direction of the eld H0and the RF magnetic eld hxare also shown. II. EXPERIMENTAL PROCEDURE A tunable waveguide cavity consisting of a shorted X- band waveguide in which a metallic rod of 1 :36 mm of diameter is inserted in a slit located on one side of the waveguide17was used, as shown in Fig. 1. Varying the position of the rod along the slit and its length inside the waveguide allowed the tuning of the volume, resonance frequency and electromagnetic losses of the cavity. For the experiments, the TE 109mode with !c=2= 11:69 GHz (volume Vc= 32:37 cm3) was chosen18. The cavity and the YIG spheres were excited by a vector network analyzer, which was also used as a detector to obtain the resonance spectra for dierent applied elds through the S11re ection coecient. In order to observe the strong coupling regime, the spheres were placed on the shorted end of the waveguide resonator where the amplitude of the RF magnetic eld is maximum. The two spheres, which will be called YIG 1and YIG 2hereafter, have a radiusR1= 0:620:01 mm and R2= 0:610:01 mm, respectively. They were placed so that the center line (or axis) generated by the two spheres was parallel to the direction of the external DC magnetic eld H0. The center-to-center distance dof the spheres was varied from 1:41 mm to 3 :58 mm. The strong coupling regime is observed when the cou- pling constant gexceeds both the cavity losses cand the magnetic losses m6. This is illustrated in Fig. 2arXiv:2009.04557v1 [cond-mat.mtrl-sci] 9 Sep 20202 for YIG 1, where the hybridization of the cavity photonic mode and the ferromagnetic uniform mode of resonance is observed. In this work, the rod insertion was adjusted to haveccomparable to min order to facilitate the observation of the coupling. The coupling gis obtained by subtracting the resonance frequency of both modes for the whole range of magnetic elds, whereas the min- imum value is equal to 2 g. The value of the magnetic eld corresponding to this minimum will be called Hc. III. RESULTS A coupling constant of g1=2= 29:2 MHz and g2=2= 28:5 MHz was independently extracted for YIG 1and YIG 2, respectively. This agrees well with the theoreti- cal value given by5 g=r Vs Vc!M!c 21=2 ; (1) whereVsis the volume of the sphere, !M=0j jMs withMsthe saturation magnetization, the gyromag- netic ratio, and represents the spatial overlap between the cavity photonic mode and the magnonic mode. The factoris given by19 =1 hmaxmmaxVsZ sphere(h
m)dV(2) where his the dynamic magnetic eld of the cavity with hmaxbeing its maximum magnitude and mis the dy- namic magnetization of the sphere with mmaxbeing its maximum magnitude. The value of is usually equal to 1 when handmare both uniform, which is the case for a small sample placed at the maximum of the cavity eld. The input-output formalism6was used to extract the losses of each component. The losses of the cavity c=2 were8:65 MHz, similar to the losses m1=2= 8:44 MHz (YIG 1) andm2=2= 12:63 MHz (YIG 2) of the YIG spheres. The mean magnetic losses of both spheres, equal to 10:54 MHz, will be referred to as  m1hereafter. With two spheres in the cavity, the hybridization of the modes is still exhibited, but accompanied with a shift in the value of Hcand a change in the coupling strength as the spheres are brought closer. This is shown in Fig. 3 for two values of d. The eld shift, due to the dipo- lar interaction, can be calculated by solving the coupled Landau-Lifshitz equations of motion of the two spheres treated as macrospins: @ @t M1 M2 =0j j M1 M2  H+N M2 M1 (3) where N=1 3R d32 41 0 0 01 0 0 0 23 5; (4) 0.41 0.415 0.4211.511.611.711.811.9 Magnetic Field (T)Frequency (GHz) Cavity FMR g/2π = 29.2 MHz Hc = 0.4137 TFIG. 2. Strong coupling spectra obtained for the sphere YIG 1with the setup described in Sect. II. The extracted cou- pling constant is g1=2= 29:2 MHz. The hybridization of the modes occurs at a eld Hc= 0:4137 T. When there is no coupling, the resonance frequency of the cavity and the YIG sphere is represented by the red dashed line and the blue dot- ted line, respectively. 0.41 0.415 0.4211.511.611.711.811.9 Magnetic Field (T)Frequency (GHz)g/2π = 40.5 MHz Hc = 0.4133 T(a) 0.395 0.4 0.40511.511.611.711.811.9 Magnetic Field (T)Frequency (GHz)g/2π = 20 MHz Hc = 0.3983 T(b) FIG. 3. Strong coupling between the microwave cavity and two YIG spheres placed at a mutual distance of (a) d= 3:58 mm and (b) d= 1:41 mm. The dipolar interaction between the spheres shifts the value of Hcand decreases the total coupling constant g. H=H0^z+h,Ris the mean radius of the spheres, considered identical, and dis the distance between the macrospins. Using a small signal approximation, the cou- pled equations yield the resonance condition !res=!0+R d3 !M (5) where!0=0j jH0. Because the hybridization of the modes occurs for !res=!c, Eq. (5) shows that smaller distancesdlead to smaller values of Hc. This shift of Hcwas used to corroborate and correct the distances between the spheres, which were initially measured man- ually with a digital micrometer. A good agreement has been found between the two methods. The reduction of the coupling constant g, exhibited in Figure 3 as the spheres are brought closer, is reported in greater details in Fig. 4 (closed circles). For large dis- tances between the spheres, one expects from the input- output formalism20a total coupling strength of approxi- matelyp g2 1+g2 2(dotted line), which is indeed observed. However, for smaller distances d, the coupling constant3 11.522.533.5 d/2R2025303540g (MHz) ind. spin dip. narrowing(g12+g22) FIG. 4. Eect of dipolar interactions on the total coupling strengthgobtained experimentally (closed circles). Light gray curve: Expected decrease in the case of independent spins calculated with Eq. (8). Dark gray curve: Expected decrease in the case of dipolar narrowing calculated with Eq. (12). Dotted line: Coupling constant when d! 1 . 11.5 22.5 3 d/2R6810121416Mean losses (MHz)ind. spin dip. narrowingm FIG. 5. Magnetic losses  mobtained experimentally as the spheres get closer (closed circles). Light gray curve: Expected increase in linewidth when considering independent spins ex- tracted from the susceptibility calculated with Eq. (8). Dark gray curve:  m1+
!where
!is calculated using Eq. (11). Dotted line: Magnetic losses when d! 1 . is observed to decrease sharply from 40 :5 MHz down to 20 MHz. Considering the two YIG spheres as a whole, the usual expression of the S 11re ection coecient, calculated from the input-output formalism, was used to extract the magnetic losses of the two spheres as a function of the distance between the spheres  m(d) (closed circles in Fig. 5). In contrast with g, the magnetic losses increase sharply as the spheres get closer. For a distance d= 1:41 mm (d=2R= 1:13), the magnetic losses are just above 16 MHz, which is near the coupling strength of 20 MHz. For shorter distances, the magnetic losses would continue to increase while the coupling constant would decrease, causing the system to exit the strong coupling regime.IV. DISCUSSION In order to explain the reduction of the coupling con- stant, let us consider the impact of the dipolar interaction on. The dipolar eld can be separated into two com- ponents. A dominant non-uniform static component is added to the applied static eld and tend to spread the local eld on the spheres. A weaker non-uniform dynamic eld is further added to the cavity pumping eld, which could result in the excitation of non-uniform resonance modes. Assuming that the RF magnetic eld of the un- perturbed cavity is uniform, we can rewrite in terms of the uniform mode susceptibility using m=h. Since the real part of the susceptibility 00 near resonance, we keep only the imaginary part and rewrite Eq. (2) as =1 00maxVsZ sphere00dV=h00i 00max; (6) where the brackets h
irepresents the mean value over the volume of the sphere. Further insights are provided by examining two lim- iting cases. Case 1 corresponds to the macrospin ap- proximation, in which all spins in a sphere are strongly coupled and locked parallel to each other's, which was as- sumed earlier in Eq. (3). Our calculations indicate that the dynamic part of both spheres will be in phase, re- sulting in a constant factor = 1 for any distance d. In Fig. 4, the macrospin approximation corresponds to the dotted line and a value of g=p g2 1+g2 2. Likewise, the macrospin approximation does not lead to an increase in the linewidth observed in Fig. 5 but rather gives a con- stant linewidth of  m1(dotted line). In contrast, Case 2 assumes fully independent spins, that is, no long-range dynamic dipolar interaction and each spin constituent of the spheres is resonating at its own frequency depending on the value of its lo- cal static magnetic eld. This non-uniform magnetic eld, assumed to be along the ^ z-direction, is given by Hz=H0+Hdip., where Hdip.=R3(r2(3 cos2#1) + 4drcos#+ 2d2) 3(r2+ 2drcos#+d2)5=2Ms:(7) Here,Hdip.is the static dipolar magnetic eld and the variablesrand#determine the position in a spherical coordinates system centered on a sphere placed at a dis- tancedfrom the source dipole. One can then numerically compute the probability density function f(Hdip.) over the volume of the sphere as a function of dto calculate the value of the mean susceptibility of the independent spins ensemble at resonance ( !=!c). Assuming no magnetic anisotropy, we have h00i=Z Hzm1!M (0j jHz!c)2+ 2m1f(Hz)dHz;(8) which can be substituted in (6) and then (1) to calcu- late the coupling. In this limiting case, a strong decrease4 inis predicted, even for spheres separated by a rela- tively large distance d, as shown by the light gray curve in Fig. 4. Furthermore, the inhomogeneously broad- ened linewidth in the independent spins approximation is given by the light gray line in Fig. 5, which predicts a much broader linewidth than observed experimentally. In our two spheres experiment, we thus fall somewhere between these two limits: macrospin and independent spins. A more rigorous approach should include long- range dynamic dipolar interactions which are known to produce a phenomenon called \dipolar narrowing" in the literature21. We consider the original approach used by Clogston22in which the Landau-Lifshitz equation of mo- tion is solved for a non-uniform magnetic eld expanded in Fourier components as Hz=X kHkeik
r: (9) Assuming the eld inhomogeneity is low with respect to the sample dimensions, we can neglect the terms related to the exchange interaction in the equation of motion, but consider the terms associated with dynamic dipolar elds. Further expanding the magnetization in Fourier series and by following a procedure similar to Ref. 22, we can derive an analytical expression for the imaginary part of the susceptibility of the uniform mode of resonance, ac- counting for the coupling between the uniform mode and the long wavelength spin wave modes, a process called two-magnon scattering23. With some simplications, it can be written in the form h00i=(m1+
!)!M (!!c)2+ (m1+
!)2(10) where
!= 2Var(!dip.) !M 1 +1 2!M 3!c!M2  2 3!c 3!c!M1=2 (11) is an additional loss term directly related to the variance of the static dipolar magnetic eld through the quantity !dip.=0j jHdip., which can be calculated analytically (Appendix A). In the expression of
!, the division by !Mrepresents the dipolar narrowing eect. This addi- tional loss term is added to  m1, which yields a total loss term that can be compared with the measured mean losses of the magnetic system. As shown by the dark gray curve in Fig. 5, the general trend of the data is re- produced relatively well. Regarding the coupling constant g, the denition of  in Eq. (6) is extended to account for the fact that spins, whose resonance frequency 0j jHzis detuned from the resonance frequency of the cavity !c, can contribute to the coupling with the cavity. This can be achieved by introducing a weight function in the denition of sothat the spins whose resonance frequency is contained in- side the coupling range ( garound!c), have a stronger contribution (high energy exchange) to the total cou- pling than those whose resonance frequency falls outside the coupling range (low energy exchange). In contrast, in Eq. (6), only the spins resonating at frequency !c contribute, whereas the remaining spins (detuned from the cavity) do not contribute to the coupling. To in- clude this phenomenon, we use a weight function con- sisting in a Lorentz distribution L(!c;gmax) centered at !=!cand having a half-width at half maximum of gmax=p g2 1+g2 2. We thus have =Z1 0(m1+
!)!ML(!c;gmax) (!!c)2+ (m1+
!)2d! Z1 0m1!ML(!c;gmax) (!!c)2+ 2m1d!; (12) which equals unity if
!= 0, in absence of dipolar broadening. Equation (12) may be used with Eq. (1) to generate the dark gray curve in Fig. 4. The excellent agreement with the experimental data supports that the observed decrease in the coupling rate between the sys- tem of magnetic spheres and the cavity, as the spheres are brought closer together, originates from the increasingly non-uniform dipolar static magnetic eld on each sphere. It also shows that the long-range dynamic dipolar inter- action within each sphere, which gives rise to the dipolar narrowing eect, somewhat limits the adverse eect of the non-uniform eld distribution. Similarly, the expression of given in (12) implies that a larger coupling gmaxtends to smooth out the adverse eect of a given dipolar broadening
!in reducing the total coupling strength. V. CONCLUSION We have demonstrated that the dipolar interaction be- tween two ferromagnetic objects can strongly aect their coupling with a microwave cavity. As the distance be- tween the spheres is gradually reduced, dipolar interac- tions force the spins to resonate at increasingly dierent frequencies. This results in increased magnetic losses and decreased coupling strength gof the system. A model based on inhomogeneous broadening with dipolar nar- rowing reproduces the main features observed on a sys- tem consisting of two YIG spheres in a tunable microwave cavity. While the reduction in the coupling strength can be linked with the variance of applied eld caused by the dipolar interaction, this eect is attenuated by dipo- lar narrowing and by strong coupling of each individual sphere with the cavity. Our results suggest that a number of Nindividual fer- romagnetic objects inserted in an electromagnetic cavity will eventually exhibit a reduced coupling as compared to the expected g/p Nbehavior as the density is in- creased. Yet the dipolar broadening will be mitigated5 by a compensating dipolar narrowing eect. A trade-o must be found to determine the optimal density of fer- romagnetic objects to be placed in the cavity to reach a maximum coupling strength while reducing the impact of dipolar interaction.Appendix A: Analytical expression for Var (!dip.). Integrating by parts Eq. (7), we have h!dip.i=a3 12!M (A1) wherea= 2R=d (0a1). The integration by parts also leads to an analytical expression for h!2 dip.i. The denition of the variance, Var( !dip.) =h!2 dip.ih!dip.i2, then gives Var(!dip.) !2 M=a3 4a 3(4a2)3 5 +a2 2 1 +a2 8 +tanh1(a=2) 24a 43 32+a2 9 3(4a2) 512ln2 +a 2a :(A2) 1O. O. Soykal and M. Flatt e, Physical review letters 104, 077202 (2010). 2H. Huebl, C. W. Zollitsch, J. Lotze, F. Hocke, M. Greifen- stein, A. Marx, R. Gross, and S. T. Goennenwein, Physical review letters 111, 127003 (2013). 3Y. Cao, P. Yan, H. Huebl, S. T. Goennenwein, and G. E. Bauer, Physical Review B 91, 094423 (2015). 4B. Yao, Y. Gui, Y. Xiao, H. Guo, X. Chen, W. Lu, C. Chien, and C.-M. Hu, Physical Review B 92, 184407 (2015). 5Y. Tabuchi, S. Ishino, T. Ishikawa, R. Yamazaki, K. Usami, and Y. Nakamura, Physical review letters 113, 083603 (2014). 6X. Zhang, C.-L. Zou, L. Jiang, and H. X. Tang, Physical review letters 113, 156401 (2014). 7P. Hyde, L. Bai, M. Harder, C. Dyck, and C.-M. Hu, Physical Review B 95, 094416 (2017). 8M. Goryachev, W. G. Farr, D. L. Creedon, Y. Fan, M. Kostylev, and M. E. Tobar, Physical Review Applied 2, 054002 (2014). 9Y. Tabuchi, S. Ishino, A. Noguchi, T. Ishikawa, R. Ya- mazaki, K. Usami, and Y. Nakamura, Science 349, 405 (2015). 10N. Lambert, J. Haigh, S. Langenfeld, A. Doherty, and A. Ferguson, Physical Review A 93, 021803 (2016). 11X. Zhang, C.-L. Zou, N. Zhu, F. Marquardt, L. Jiang, andH. X. Tang, Nature communications 6(2015). 12L. Bai, M. Harder, P. Hyde, Z. Zhang, C.-M. Hu, Y. Chen, and J. Q. Xiao, Physical Review Letters 118, 217201 (2017). 13Y. Cao and P. Yan, Phys. Rev. B 99, 214415 (2019). 14J. T. Hou and L. Liu, Physical Review Letters 123, 107702 (2019). 15M. Harder and C.-M. Hu, in Solid State Physics , Vol. 69 (Elsevier, 2018) pp. 47{121. 16D. Zhang, X.-M. Wang, T.-F. Li, X.-Q. Luo, W. Wu, F. Nori, and J. You, npj Quantum Information 1, 1 (2015). 17A. Morin, C. Lacroix, and D. M enard, in 2016 17th Inter- national Symposium on Antenna Technology and Applied Electromagnetics (ANTEM) (IEEE, 2016) pp. 1{2. 18D. M. Pozar, Microwave engineering (John Wiley & Sons, 2009). 19N. Lambert, J. Haigh, and A. Ferguson, Journal of Applied Physics 117, 053910 (2015). 20D. Schuster, A. Sears, E. Ginossar, L. DiCarlo, L. Frunzio, J. Morton, H. Wu, G. Briggs, B. Buckley, D. Awschalom, et al. , Physical review letters 105, 140501 (2010). 21S. M. Rezende and A. Azevedo, Physical Review B 44, 7062 (1991). 22A. Clogston, Journal of Applied Physics 29, 334 (1958). 23R. D. McMichael, D. Twisselmann, and A. Kunz, Physical review letters 90, 227601 (2003).

2020-09-09

The strong photon-magnon coupling between an electromagnetic cavity and two yttrium iron garnet (YIG) spheres has been investigated in the context of a strong mutual dipolar interaction between the spheres. A decrease in the coupling strength between the YIG spheres and the electromagnetic cavity is observed, along with an increase of the total magnetic losses, as the distance between the spheres is decreased. A model of inhomogeneous broadening of the ferromagnetic resonance linewidth, partly mitigated by the dipolar narrowing effect, reproduces the reduction in the coupling strength observed experimentally. These findings have important implications for the understanding of strongly coupled photon-magnon system involving densely packed magnetic objects, such as ferromagnetic nanowires arrays, in which the total coupling strength with an electromagnetic cavity might become limited due to mutual dipolar interactions.

Effect of dipolar interactions on cavity magnon-polaritons

2009.04557v1

arXiv:2102.12181v1 [quant-ph] 24 Feb 2021Phase-controlled pathway interferences and switchable fa st-slow light in a cavity-magnon polariton system Jie Zhao,1, 2, 3, 4, ∗Longhao Wu,1, 2, 3,∗Tiefu Li,5, 6Yu-xi Liu,5 Franco Nori,7, 8Yulong Liu,9, 10,†and Jiangfeng Du1, 2, 3,‡ 1Hefei National Laboratory for Physical Sciences at the Microscale and Department of Modern Physics, University of Science and Technology of China, Hefei 230026 , China 2CAS Key Laboratory of Microscale Magnetic Resonance, University of Science and Technology of China, Hefei 230026 , China 3Synergetic Innovation Center of Quantum Information and Qu antum Physics, University of Science and Technology of China, Hefei 230026 , China 4National Laboratory of Solid State Microstructures, School of Physics, Nanjing University, Nanjing 210093, Chi na 5Institute of Microelectronics, Tsinghua University, Beij ing 100084, China 6Quantum states of matter, Beijing Academy of Quantum Information Sciences, Beijing 100193, China 7Theoretical Quantum Physics Laboratory, RIKEN, Saitama, 3 51-0198, Japan 8Department of Physics, The University of Michigan, Ann Arbor, Michigan 48109-1040, USA 9Beijing Academy of Quantum Information Sciences, Beijing 1 00193, China 10Department of Applied Physics, Aalto University, P .O. Box 15100, FI-00076 Aalto, Finland (Dated: February 25, 2021) 1Abstract We study the phase controlled transmission properties in a c ompound system consisting of a 3D copper cavity and an yttrium iron garnet (YIG) sphere. By tuning the relative phase of the magnon pumping and cavity probe tones, constructive and destructive interfer ences occur periodically, which strongly modify both the cavity field transmission spectra and the group dela y of light. Moreover, the tunable amplitude ratio between pump-probe tones allows us to further improve the si gnal absorption or amplification, accompanied by either significantly enhanced optical advance or delay. B oth the phase and amplitude-ratio can be used to realize in-situ tunable and switchable fast-slow light. The tunable phase and amplitude-ratio lead to the zero reflection of the transmitted light and an abrupt fast-s low light transition. Our results confirm that direct magnon pumping through the coupling loops provides a versatile route to achieve controllable signal transmission, storage, and communication, which can be fur ther expanded to the quantum regime, realizing coherent-state processing or quantum-limited precise mea surements. I. INTRODUCTION Interference, due to superposed waves, plays a considerabl e role in explaining many classical and quantum physical phenomena. Based on the phase-differe nce-induced interference patterns, ultraprecise interferometers have been created, impactin g the development of modern physics and industry [1]. In addition to the phases, waves or particles p ropagating through different path- ways can also introduce interference patterns. Among vario us types of multiple-path-induced interference, the Fano resonance [2] and its typical manife stations, the electromagnetically in- duced transparency (EIT) and electromagnetically induced absorption (EIABS) [3, 4], are the most well-known ones. The Fano resonance and EIT-like (or EI ABS-like) line shapes are not only experimentally observed in quantum systems but also in vari ous classical harmonic-resonator sys- tems. Quantum examples include quantum dots [5], quantum we lls [6], superconducting qubits [7– 10], as well as Bose-Einstein condensates [11]. Classical e xamples [12] include coupled optical cavities [13–16], terahertz resonators [17, 18], microwav e resonators [19, 20], mechanical res- onators [21, 22], optomechanical systems [23]. However, wh ether in quantum or in classical systems, the Fano resonance, EIT- or EIABS-like spectra are normally experimentally realized ∗These authors contributed equally to this work †liuyl@baqis.ac.cn ‡djf@ustc.edu.cn 2separately. The switchable electromagnetically induced t ransparency and absorption, as well as fast and slow light, have been proposed using dressed superc onducting qubits [8], hybrid optome- chanical system [24, 25], dark-mode breaking [26–28], and s o on. Particularly, there appears growing interest to control the EIT and EIABS by introducing exceptional points [29–31]. Photon stops [32, 33], chiral EIT [34], and infinite slow light [33] h ave recently been realized around exceptional points. Motivated by their potential applicat ions in rapid transitions between fast and slow light, which facilitate coherent state storage and retrieval, it is highly desirable to have experimental realizations of in situ tunable and switchable absorption, transparency, and even am- plification. Meanwhile, cavity magnon polaritons in an yttrium-iron-ga rnet (YIG) sphere-cavity coupled system has attracted much attention due to its strong [35–42 ] and even ultrastrong couplings [43– 45]. The compatibility and scalability with microwave and o ptical light enable magnons to be a versatile interface for different quantum devices [46–51] . At low temperatures, strong coupling between magnons, superconducting resonators and qubits ha ve been demonstrated [52–56]. Sub- sequently, the EIT-like magnon-induced transparency (MIT ) or the EIABS-like magnon-induced absorption (MIABS) of the transmitted cavity field were obse rved for different external coupling conditions [57]. The underlying mechanism is attributed to interferences between two transition pathways, i.e., the direct cavity pathway and the cavity-ma gnon-cavity pathway, to transmit the probe field. In addition to the coupling strength [57] and frequency detu ning [58–60] between coupled modes, phases play a vital role in wave interference control . We thus focus on the controllabil- ity of pathway interferences through the phase difference b etween the cavity-probe tone and the magnon-pump tone, which is introduced by the coupling loops ’ technology [61–64]. The direct magnon pump is becoming useful in realizing the light-wave i nterface [46–48], enhancing the Kerr nonlinearity [65–67], and has also been adopted to observe t he magnetostriction-induced quantum entanglement [68–72], among other applications. Together with the cavity-probe tone, a magnon-pump tone int roduces a controllable relative phase to the system, and thus the path interference can be rea l-time controlled. Changing the two- tone phase difference, we can switch the cavity-probe spectra from the original magnon-i nduced transparency instantly to the magnon-induced absorption, or even the Fano line shape . Further- more, the tunable pump-probe amplitude ratio allows us to fu rther improve the signal absorption, transparency, or amplification, accompanied by a significan t enhancement by nearly 2 orders of 3magnitude of the optical advance or delay time compared to the case with out magnon pump [57]. In particular, the tunable phase and amplitude ratio also le ad to the zero reflection of the trans- mitted light, which is accompanied by an abrupt transition o f delay time. Our results confirm that direct magnon pumping provides a versatile route to con trol signal transmission, storage, and communication, and can be further expanded to coherent stat e processing in the quantum regime. -150 -75 0 75 150 p(MHz)-4-20S11(dB) -150 -75 0 75 150 p(MHz)-8-6-4-2S11(dB)Port 1 Port 2 1 32Q IL R , AB Splitter IQ Mixer CirculatorVNAAWG CavityMagnon gCavity gMagnon High Low(a) (b) (c) (d)xy z Destructive C t ti 4FIG. 1. Measurement setup and phase-induced interference m echanism diagrams. (a) The system consisting of a three-dimensional (3D) copper cavity and a YIG sphere, w hich is coherently pumped by the coupling loops shown as a black coil surrounding the YIG sphere. The re d arrows and colors indicate the magnetic field directions and amplitudes of the TE 101mode distribution, respectively. The YIG sphere is placed a t the area with maximum magnetic field distribution inside a 3D copper cavity box to obtain a strong cavity- magnon coupling. A small hole at the cavity sidewall is assem bled with a standard SubMiniature version A connector (SMA connector), allowing us to do the reflection m easurement S11of the probe field, i.e., such a SMA connector works as both the signal input and readout port . A beam of coherent microwave comes out from port 1 of the vector network analyzer (VNA) and splits in to two beams, working as the magnon-pump tone and the cavity-probe tone. Here, we use an in-phase and q uadrature mixer (I-Q mixer) and an arbitrary waveform generator (AWG) to control and tune the phase diffe renceϕand pump-probe amplitude ratio δ=εm/εcbetween the pump and probe tones. The interfering results ar e extracted by the circulator and finally transferred to port 2 of the VNA. (b) Diagram showing t he relative phase between the magnon pump and cavity probe in the cavity-magnon coupled system. (c) Th e corresponding energy-level diagram. Two transition pathways to the higher energy level: 1/circleco√yrtprobe-tone-induced direct excitation, and 2/circleco√yrtpump-tone excites magnons and then coherently transfers there to cavi ty photons. (d) Measurements of the reflection spectraS11versus the detuning ∆p=ωc−ωp=ωm−ωp. The relative phase difference between pump and probe tones can be developed to realize an in situ switchable constructive and destructive interference, presented as MIABS with ϕ= 0.35π,δ= 1.2and MIT with ϕ= 1.35π,δ= 1.2. II. EXPERIMENTAL SETUP As shown in Fig. 1(a), our system consists of a 3D copper (Cu) c avity with an inner dimension of40×20×8mm3and an YIG sphere with a 0.3 mm diameter. A static magnetic fiel dHstatic applied in the x-yplane tunes the magnon frequency. The simulated cavity-mod e magnetic field distribution is shown at the bottom of Fig. 1(a), where the ar rows and colors indicate the cavity mode magnetic field directions and amplitudes. The YIG spher e is placed near the magnetic field antinode of the cavity TE101mode. The magnetic components (along the zaxis) of the microwave field at this antinode is perpendicular to the static magneti c bias field. Here, we are only interested in the low excited states of the K ittel mode, in which all the spins precess in phase. Under the Holstein-Primakoff transforma tion, such collective spin mode can be 5simplified to a harmonic resonator, which introduces the mag non mode. In our setup, the cavity mode couples to the magnon mode with coupling strength g= 7.6 MHz , which is larger than the magnon decay rate κm= 1.2MHz , but smaller than the cavity decay rate κc= 113.9MHz . In our experiment, a beam of coherent microwave is emitted fr om port 1 of a VNA and then divided through a splitter into two beams, one of which is use d to probe the cavity (probe tone) and another beam is used to pump the magnon (pump tone) by inco rporating the coupling loop technique, which is schematically shown in the dashed recta ngle of Fig. 1(a). The probe tone is injected into the cavity through antenna 1, which induces th e cavity external decay rate κc1= 21.8 MHz . The pump tone is injected through antenna 2, which introduc es the magnon external decay rate κm1= 0.6 MHz . Note that the phase ϕc= 0 and amplitude εcof the probe tone are fixed (i.e., working as a reference), and the phase ϕand amplitude εmof the magnon-pump tone are tunable and controlled by an arbitrary wave generator wi th an in-phase and quadrature mixer (I-Q mixer). III. MODEL By considering the cavity-magnon coupling, as well as the pu mp and probe tones [model in Fig. 1(b)], the system Hamiltonian becomes H=ωca†a+ωmm†m+g(a†m+m†a) +i/radicalbig 2ηcκcεc/parenleftbig a†e−iωpt−aeiωpt/parenrightbig +i/radicalbig 2ηmκmεm/parenleftbig m†e−iωpt−iϕ−meiωpt+iϕ/parenrightbig . (1) Here,a†(a) andm†(m) are the creation (annihilation) operators for the microwa ve photon and the magnon at frequencies ωcandωm, respectively, and we choose units with /planckover2pi1= 1. The magnon frequency ωmlinearly depends on the static bias field Hstaticand is tunable within the range of a few hundred MHz to about 45 GHz; εc(εm) is the microwave amplitude applied to drive the cavity (magnon). Here, we introduce the coupling parameter ηc=κc1/κc, (2) ηm=κm1/κm (3) 6to classify the working regime of the cavity (the magnon). Th e parameter ηc(ηm) classifies three working regimes for the cavity (magnon) into three types: ov ercoupling regime for ηc(ηm)>1/2; critical-coupling regime for ηc(ηm) = 1/2; and undercoupling regime for ηc(ηm)<1/2. In our experiment, the cavity works in the undercoupling regime ( ηc<1/2) and the magnon works in the critical coupling regime ( ηm= 1/2). Experimentally, the reflection signal from the cavity is cir culated and then transferred to port 2 of the VNA to carry out the spectroscopic measurement, whic h corresponds to the steady-state solution of the Hamiltonian Eq. (1). The transmission coeffi cienttpof the probe field is defined as the ratio of the output-field amplitude εoutto the input-field amplitude εcat the probe frequency ωp:tp=εout/εc. With the input-output boundary condition, εout=εc−/radicalbig 2ηcκc/angbracketlefta/angbracketright, (4) we can solve the transmission coefficient tpof the probe field as [73] tp=tprobe+tpump, (5) with tprobe= 1−2ηcκc(i∆p+κm) (i∆p+κc)(i∆p+κm)+g2, (6) tpump=ig√2ηcκc√2ηmκmδe−iϕ (i∆p+κc)(i∆p+κm)+g2. (7) Here∆pis the detuning between the probe frequency ωpand either the cavity resonant frequency ωcor the magnon frequency ωm. In our experiment, the cavity is resonant with the cavity, i .e., ∆p=ωc−ωp=ωm−ωp; (8) and δ=εm/εc (9) is the pump-probe amplitude ratio. Equation (5) clearly sho ws that the transmission coefficient can be divided into two parts: 1.tprobe in Eq. (6), the contribution from the cavity-probe tone, rep resents the traditional pathway-induced interference; 2.tpump in Eq. (7), the contribution from the magnon-pump field, affe cts the interference and modifies the transmission of the probe field. 7As shown in Fig. 1(c), there exist two transition pathways fo r the cavity: the probe-tone-induced direct excitation, and the photons transferred from magnon excitations. When the cavity decay rate (analog to broadband of states) is much larger than the m agnon decay rate (analog to a nar- row discrete quantum state in other quantum systems), Fano i nterference happens and has been successfully used to explain the MIT and MIABS phenomenon in cavity magnon-polariton sys- tems [57]. Besides pathway-induced interference, the stee red phase ϕof the wave provides another useful way to generate and especially control the interfere nces, as shown in Fig. 1(d). We emphasize that in this paper we focus on how the phase difference ϕand pump-probe ratio δ=εm/εcaffect the interference, and we explore its potential appli cations, such as controllable field transmission and in situ switchable slow-fast light . TheS11spectrum and group-time delay measurement are carried out on the VNA and then fitted by T=|tp| (10) and τ=−∂[arg(tp)] ∂∆p, (11) respectively. IV . PHASE INDUCED INTERFERENCE AND CONTROLLABLE MICROWA VE FIELD TRANS- PORT We first study how the phase of the magnon-pump tone affects th e transmission of the cavity- probe field. In Fig. 2 (a), we present experimental results of the transmission, when the pump- probe ratio is δ=εm/εc= 1.7. In this setup, the phase ϕis continuously increased from 0 to 2 π using an I/Q mixer, and is shown in the xaxis of Fig. 2 (a). Then we conduct the S11measurements and the recorded spectra are plotted versus the detuning fre quencies ∆p. The colors represent the relative steady-state output amplitude (in dB units) at dif ferent frequency and pump-probe ratios. Figure 2(a) shows that the interference mainly happens arou nd∆p= 0 and can be controlled in situby changing the phase ϕ. As shown in Fig. 2(b), where ϕis set to0.35π, destructive interference happens and an obvious dip appears around ∆p= 0. This behavior can be regarded as MIABS. However, if we set ϕ= 8Theory Experiment -150 75 0 75 150 (a) (b) 0 -6.5 -130 -4.5 -9 2 -1.5 -51 -2.5 -6 -150 150-150-75 15075 FIG. 2.S11spectrum versus relative phase difference ϕ. (a) Measured transmission spectrum S11versus phaseϕand detuning ∆p. The colors indicate the transmitted amplitudes in dB units . (b) Measured output spectrum S11with phases: 1/circleco√yrtϕ= 0.35π,2/circleco√yrtϕ= 0.85π,3/circleco√yrtϕ= 1.35π, and 4/circleco√yrtϕ= 1.85π. Here, the pump-probe amplitude ratio is fixed at δ= 1.7. Red-solid lines are the corresponding theoretical result s. 1.35π, constructive interference happens and an obvious amplific ation window appears around ∆p= 0. This behavior can be described as magnon-induced amplifica tion (MIAMP). When ϕ is set to0.85πor1.85π, sharp and Fano-interference-like asymmetry spectra are o bserved even when the cavity and magnon are exactly resonant. Although the interference originates from the coherent cav ity-magnon coupling, Fig. 2 clearly shows that the phase ϕplays a key role in realizing an in situ tunable and controlla ble interfer- ence (e.g., constructive or destructive interference) , which can be further engineered to control the probe-field transmission. Note that in previous studies [57 ] MIABS was only observed in the cav- 9ity overcoupling regime (i.e., ηa>1/2) and MIT was only observed in the cavity undercoupling regime (i.e., ηa<1/2). In contrast to this, here we realize a phase-dependent and switchable MI- ABS and MIT, as well as MIAMP in a fixed undercoupling regime ( ηc= 0.19in our experiment). We emphasize that the destructive interference-induced MI ABS is a unique result of phase mod- ulation. The observed asymmetric Fano line shapes could be u seful to realize Fano-interference sensors or precise measurements, using the magnon-pump met hod realized in our work. V . AMPLITUDE RATIO OPTIMIZED MAGNON-INDUCED-ABSORPTION Recall the magnon-pump transmission coefficient tpump in Eq. (7). There, the phase ϕdeter- mines the type of interference, e.g., constructive or destr uctive. However, the pump-probe ratio δ=εm/εcalso affects the degree of interference, and thus can be used to control the probe-field transmissions tp. As shown in Fig. 3(a), a color map is used to present the exper iment results. Along the xaxis, the amplitude ratio δis continuously increased from 0 to 6.5, by changing the overall voltage amplitude applied to the I and Q ports of an I- Q mixer. Then we conduct the S11 measurements and the steady-state output-field amplitudes are plotted versus the frequency de- tuning∆p. The colors in Fig. 3(a) represent the relative strength of t he steady-state output field (in dB units) at a different frequency. Here, the chosen phas eϕ= 0.35πresults in MITs when δ <0.32, while MIABSs dominate the output response in the regime δ >0.32. We then study how the pump-probe ratio δaffects the central absorption window of the S11spectra. Figure 3(a) shows that interference occurs around ∆p= 0and is in situ controlled by changing the pump-probe ratio δ. The center blue-colored area represents an ideal absorpti on (transmission T <0.01) of the probe field. Figure 3 (b) shows the extreme values of the transmission coe fficients around ∆p= 0 versus the pump-probe ratio δ. In the yellow area, we find the local maximum values of the MIT s, and the local minimum values are found for MIABSs in the blue area . An obvious dip appears around δ= 3 and the minimum transmission value is less than 1% (voltage a mplitude ratio), which corresponds to an optimized and ideal probe-field absorptio n. Figure 3(c) shows the evolution process from MIT to MIABS by g radually increasing the pump-probe ratio δ. Whenδ= 0, corresponding to case 1/circleco√yrtof Fig. 3(c), our scheme recovers the traditional MIT case when no magnon pump is applied. When the magnon pump is introduced and its strength is continuously increased, the transparen cy window disappears and is replaced by 10-150 -75 0 75 150Theory -150 -75 0 75 150Experiment(a)(
Extreme Amplitude (dB) -1.0 -2.5 -4.00 -20 -40 -60 0 2 4 6 -1.0 -2.5 -4.0 0 -20 -42 -150-75 1501.0 -4.0 -9.0 -150-75 150 FIG. 3. Measured transmission spectrum S11versus pump-probe amplitude ratio δwith phase fixed at ϕ= 0.35π. (a) Measured output spectrum versus amplitude ratio δand detuning ∆p. The colors indicate transmitted power in dBs. (b) The extreme values of the S11transmission spectra of the output field versus the amplitude ratio parameter δ. In the light-yellow (light-blue) regime, the extreme valu es represent the maximum (minimum) transmission amplitudes of the peaks (di ps) around ∆p= 0. (c) Measured transmis- sion spectrum S11with amplitude ratio: 1/circleco√yrtδ= 0,2/circleco√yrtδ= 0.3,3/circleco√yrtδ= 3.0, and 4/circleco√yrtδ= 5.7. Red-solid lines are the corresponding theoretical results. an obvious absorption dip, as shown in cases 2/circleco√yrtand 3/circleco√yrtof Fig. 3(c). With an even larger pump- probe ratio, the MIABS dips become asymmetry gradually, suc h as the spectrum in the case 4/circleco√yrtof Fig. 3(c). Comparing with other results in Fig. 3(c), we can fi nd that the experimental data do no fit so well with the theory in case 4/circleco√yrtof Fig. 3(c). This is induced by the additional cavity-anten na 112 coupling. Due to the existence of this tiny coupling, the ma gnon pump signal also pumps the cavity. With a modest magnon-pump strength, the additional cavity pump does not affect the sys- tem seriously, so that the theory fit the experiment data well . With a relatively strong magnon pump, the side effects of the additional cavity pump become l arger, though it does not change the line shape. Therefore, the experiment data and theory do not fit so well when the magnon pump is relatively strong [73]. Similar phenomena can also be obser ved in the case 4/circleco√yrtof Fig. 4(c). We emphasize one main result of this paper: the absorption dips appear with an under-coupling coefficient of ηa= 0.19in our experiment. However, absorptions only happen in the o vercoupling regime in traditional cases . Moreover, Figs. 3(a) and (c) show that δcan be used to switch the transmission behavior from the magnon-induced transparen cy to the magnon-induced absorption . Note that the type of interference, destructive interferen ce or constructive interference, depends on the value of the phase ϕ. However, the interference intensity is determined and opt imized by the pump-probe ratio δ. As shown in Fig. 3(c), the dip of S11is 42 dB lower than the baseline. The dip amplitude is quite close to zero, which indicates tha t a zero reflection is generated by the destructive interference. VI. AMPLITUDE RATIO OPTIMIZED MAGNON-INDUCED-AMPLIFICAT ION We now study how the amplitude ratio of δ=εm/εcaffects the MIAMP. In this case, the phase is fixed at ϕ= 1.35π, where constructive interference dominates the transmiss ion of the output field. As shown in Fig. 4(a), a color map is used to present the m easurement results. Along the xaxis, the pump-probe ratio δis continuously increased from 0 to 6.5. Then we conduct the S11 measurement, and the steady-state transmission spectra ar e plotted versus the frequency detuning parameter ∆p. The colors in Fig. 4(a) represent the transmission amplitu des of the steady-state output field (in dB units) at different frequencies. We then s tudy how the amplitude δaffects the center amplification window of the S11spectra. Figure 4(a) clearly shows that constructive interference h appens around ∆p= 0and are in situ controlled by changing the pump-probe ratio δ. Magnon-pump-induced constructive interference happens when the probe field is nearly resonant with the cavit y (also the magnon), and amplifi- cation windows appear. Around ∆p= 0, the color changes from light blue to orange when the pump-probe ratio δincreases from 0 to 6.5. This indicates that the higher ampli fication can be obtained with a larger pump-probe ratio δ. 120 1 2 3 4 5 6-30369Extreme Amplitude (dB) Theory Experiment (c)Experiment Theory (b)(a) -150 -75 0 75 150-4.0-2.00 -150 -75 0 75 150-4.004.0-150 -50 0 75 150-4-2.5-1 -150 -75 0 75 150-4.0-2.00 FIG. 4. Measured transmission spectrum S11versus pump-probe amplitude ratio δ=εm/εcwith phase fixed atϕ= 1.35π. (a) Measured output spectra S11versus amplitude ratio δand frequency detuning ∆p. The colors indicate the transmitted amplitude in dB units. (b) The extreme values of the S11trans- mission spectra of the output field versus the amplitude-rat io parameter δ. The extreme values represent the maximum transmission amplitude of the peaks around ∆p= 0. (c) Measured transmission spectra S11 with amplitude ratios: 1/circleco√yrtδ= 0,2/circleco√yrtδ= 0.9,3/circleco√yrtδ= 1.2, and 4/circleco√yrtδ= 4.5. The red-solid lines are the corresponding theoretical results. Figure 4(b) shows how the peak values in the amplification win dow change versus the ampli- tude ratio δ. The amplification coefficient is monotonously dependent on the increment of the pump-probe ratio δ. Although the maximum pump-probe ratio is δ= 6.5in our experiment, we emphasize that a higher transmission gain can be obtained us ing a larger pump power. 13Figure 4(c) clearly shows the evolution of the transmission spectrum from MIT to MIAMP when we gradually increase the pump-probe ratio δ. Whenδ <1.2, an obvious transparency window appears. When δ= 1.2, the peak value of the transparency window equals the value o f the baseline, showing the ideal MIT phenomenon. Further inc reasing the pump strength, we can observe MIAMP. When δ= 4.5, an obvious amplification window appears, producing MIAMP. Note that the phase is fixed at ϕ= 1.35πto produce constructive interference. When the amplitude ratio is set to δ= 0, i.e., no magnon pump, our scheme also recovers the traditio nal case without a magnon pump and only MIT is observed. This resu lt is, of course, the same as case 1/circleco√yrtin Fig. 3(c). We point out another main result that the pump-probe ratio δcan be used to realize and control the magnon-induced amplifications . Figures 4(a) and 4(c) show that δcan be used to switch the system response from MIT to MIAMP. Note t hat the interference type, such as constructive interference discussed here, depends on th e value of the phase ϕ; however, the interference intensity is determined and optimized by the p ump-probe ratio δ. VII. SWITCHABLE FAST- AND SLOW-LIGHT BASED ON THE PHASE AND A MPLITUDE RATIO The group delay or advance of light always accompanies EIT or EIABS. In this experiment, we show that the group delay (slow light) and group advance (f ast light) can also be realized in our cavity magnon-polariton system. Similar to the discuss ions above, the phase ϕis the key parameter that determines the interference type, e.g., des tructive or constructive. Therefore, the phaseϕprovides a tunable and in situ switched group advance or delay of the probe field. The extreme values of the delay time are measured and presented i n Fig. 5, choosing the same phases ϕ= 0.35πandϕ= 1.35π, which are also used in Figs. 3 and 4, respectively. In Fig. 5(a), the phase is set to ϕ= 0.35π. When we increase the pump-probe ratio δ, a longer advance time is achieved, but immediately changes to time de lay when δ >3.0. Further increasing δreduces the delay time. In Fig. 5(c), we present the phase of t ransmission signals at different probe frequencies with δ= 2.7(case 1/circleco√yrt) andδ= 3.3(case 2/circleco√yrt). The phase changes drastically around∆p= 0with opposite directions. The drastic changes of the phase r esult in a long advance or delay time, while the phase-change direction reversal re sults in the sharp transition from time advance to time delay. Accompanying the sharp transition in Fig. 5(a), we observe the longest either delay or advance times. Therefore, the pump-probe ra tioδallows to optimize and switch the 140 2 4 6-1000-50005001000 0 2 4 6205080Extreme Delay Time (ns) Phase(c) -50 500 -50 50 0-101 Phase (rad) -44 0 FIG. 5. Measured time delay versus pump-probe ratio δfor the phase ϕ= 0.35π(a); andϕ= 1.35π(b). Light-yellow area indicates the group-delay regime, and th e light-blue area indicates the group-advance regime. (c) Measured unwrapped phase versus frequency detu ning∆pwithδ= 2.7[point 1/circleco√yrtin (a)] and δ= 3.3[point 2/circleco√yrtin (a)] for ϕ= 0.35π. probe microwave from fast to slow light, or inversely . Comparing the abrupt transition in Fig. 5(a) with the zero reflection discussed in Sec. V , we find that the de lay time abrupt transition and the zero reflection occur at the same parameter setup. It is notab le that the discontinuity and abrupt transition are always accompanied by the zero reflection in c oupled resonator systems. In Fig. 5(b), we set the phase to ϕ= 1.35πand mainly observe constructive interference. In this case , the time delay monotonously increases with the pump-probe ratio δ. Note that the pump-probe ratio used in Fig. 5(b) is not its limitation, therefore longer delay ti mes can be achieved by further increasing δ. Figure 5 also shows that when the amplitude ratio δ≤3.0, the delay time is a negative number which corresponds to fast light with ϕ= 0.35π, and the positive delay time corresponds to slow 15TABLE I. Summary of MIT, MIABS, MIAMP and Fano resonance obse rved experimentally for different values in parameter space. Amplitude Ratio δ 0 - 0.3 0.3 0.3 - 1.2 1.2 1.2 - 3.0 >3.0 Phaseϕ0.35πMIT NULL MIABS MIABS MIABS Fano 1.35πMIT MIT MIT MIT (perfect) MIAMP MIAMP light with ϕ= 1.35π. Thus the phase parameter ϕcan also be used to switch fast and slow light. When δ= 0, i.e., no magnon pump, our scheme recovers the traditional M IT and only a 16-ns delay time is achieved. By applying the magnon pump and optimizing ϕandδ,the time delay, as well as advance, can be enhanced by nearly 2 orders o f magnitude compared with the case without magnon pump . For our scheme, the pump-probe amplitude ratio and phase di fference mediated path interference can result in the zero reflection , which is accompanied with a delay time abrupt transition. In our experiment, Fig. 5(a) clearly sho ws such an abrupt transition and greatly enhanced fast-slow light around this point. We can find that t he experimental data deviates from the theoretical result around the abrupt transition. This i s mainly induced by the imperfect system setups, such as limited output precision of AWG, imperfectn ess of the I-Q mixer and unstable magnon frequency [73]. VIII. CONCLUSION We experimentally study how the magnon pump affects the prob e-field transmission, and the observed results are summarized in Table. I. Two parameters , the relative phase ϕand the pump- probe ratio δbetween pump and probe tones, are studied in detail. The main results of this work are as follows: • the unconventional MIABS of the transmitted microwave fiel d is observed with the cavity in the undercoupling condition; • MIAMP phenomena is realized in our experiment; • asymmetric Fano-resonance-like spectra are observed eve n when the cavity is resonant with the magnon; 16• by tuning the phase of the magnon pump, we can easily switch b etween MIT, MIABS and MIAMP; • by tuning the pump and probe ratio, the MIABS and MIAMP can be further optimized, accompanied by greatly enhanced advanced or slow light by ne arly 2 orders of magnitude; • the tunable phase and amplitude ratio can lead to the zero re flection of the transmitted light and abrupt fast-slow light transitions.; • both the ϕandδcan be used to carry out the in situ switch of fast and slow light. Our results confirm that direct magnon pumping through the co upling loops provides a versa- tile route to achieve controllable signal transmission, st orage, and communication, which can be further expanded to coherent state processing in the quantu m regime. Furthermore, by exploiting multi-YIG spheres or multimagnon modes systems, the amplifi cation or absorption bandwidth can be increased, resulting in a broadband coherent signal stor e device. The sharp peak and asymmet- ric Fano line shape indicate that our platform has great pote ntial in the application of high-precision measurement of weak microwave fields [74, 75]. Our two-tone p ump scheme and phase-tunable interference can also be accomplished in other coupled-res onator systems, such as optomechanical resonators, which explores effects of mechanical pump on li ght transmission [76–84], and even in circuit-QED systems, in which photon transmission can be controlled through a circuit-QED system [85–88]. ACKNOWLEDGMENTS This work is supported by the National Key R&D Program of Chin a (Grant No. 2018YFA0306600), the CAS (Grants No. GJJSTD20170001 and No. QYZDY-SSW-SLH00 4), Anhui Initiative in Quantum Information Technologies (Grant No. AHY050000), a nd the Natural Science Foun- dation of China (NSFC) (Grant No. 12004044). F.N. is support ed in part by: NTT Research, Army Research Office (ARO) (Grant No. W911NF-18-1-0358), Ja pan Science and Technol- ogy Agency (JST) (via the CREST Grant No. JPMJCR1676), Japan Society for the Promotion of Science (JSPS) (via the KAKENHI Grant No. JP20H00134 and t he JSPS-RFBR Grant No. JPJSBP120194828), the Asian Office of Aerospace Research an d Development (AOARD), and the Foundational Questions Institute Fund (FQXi) via Grant No. FQXi-IAF19-06. 17Note added – Recently, we become aware of a study presenting a n infinite group delay and abrupt transition in a magnonic non-Hermitian system [33]. [1] M. Born, E. Wolf, A. B. Bhatia, P. C. Clemmow, D. 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2021-02-24

We study the phase controlled transmission properties in a compound system consisting of a 3D copper cavity and an yttrium iron garnet (YIG) sphere. By tuning the relative phase of the magnon pumping and cavity probe tones, constructive and destructive interferences occur periodically, which strongly modify both the cavity field transmission spectra and the group delay of light. Moreover, the tunable amplitude ratio between pump-probe tones allows us to further improve the signal absorption or amplification, accompanied by either significantly enhanced optical advance or delay. Both the phase and amplitude-ratio can be used to realize in-situ tunable and switchable fast-slow light. The tunable phase and amplitude-ratio lead to the zero reflection of the transmitted light and an abrupt fast-slow light transition. Our results confirm that direct magnon pumping through the coupling loops provides a versatile route to achieve controllable signal transmission, storage, and communication, which can be further expanded to the quantum regime, realizing coherent-state processing or quantum-limited precise measurements.

Phase-controlled pathway interferences and switchable fast-slow light in a cavity-magnon polariton system

2102.12181v1

arXiv:1606.03469v1 [cond-mat.mes-hall] 10 Jun 2016Indirect Coupling between Two Cavity Photon Systems via Fer romagnetic Resonance Paul Hyde,a)Lihui Bai,b)Michael Harder, Christophe Match, and Can-Ming Hu Department of Physics and Astronomy, University of Manitob a, Winnipeg, Canada R3T 2N2 (Dated: 14 September 2018) We experimentally realize indirect coupling between two cavity modes v ia strong coupling with the ferromag- netic resonance in Yttrium Iron Garnet (YIG). We find that some ind irectly coupled modes of our system can have a higher microwave transmission than the individual uncoup led modes. Using a coupled harmonic oscillator model, the influence of the oscillation phase difference betw een the two cavity modes on the nature of the indirect coupling is revealed. These indirectly coupled microwav e modes can be controlled using an external magnetic field or by tuning the cavity height. This work has potential for use in controllable optical devices and information processing technologies. The indirect coupling of cavity modes via a waveguide has been studied theoretically and experimentally for use inopticalinformationprocessing1. Thisindirectcoupling dramatically modifies the transmission spectra, and is widely used for optical filtering, buffering, switching, and sensinginphotoniccrystalstructures2–5. Formicro/nano disk optical cavities, coupling properties are determined by the spatial distance between the disk and the waveg- uide during the fabrication process. Therefore, a tunable coupling between indirectly coupled cavity modes is re- quired for potential applications. Recently, strong coupling between a microwave cavity mode and ferromagnetic resonance (FMR) has been re- alized at room temperature6–17. Exchange interactions lock the high density of spins in YIG into a macro-spin state, leading to strong coupling with a cavity mode which can be adjusted using an external magnetic field. Potential applications of this form of strong coupling are currently being explored. For example, indirect coupling between the FMR in two YIG spheres has produced dark magnon modes with potential uses in information stor- age technologies18, and the FMR of YIG has been indi- rectly coupled with a qubit through a microwave cavity mode19. Instead of using a microwave cavity mode to build a bridge between two oscillators, we have used the FMR in YIG to produce indirect coupling between two cavity modes. In this work, we present two cavity modes which in- directly couple via their strong coupling with the FMR in YIG at room temperature. The two cavity modes are labelled hω1(ω) andhω2(ω) respectively, and are inde- pendent of each other when there is no direct coupling between them. Here ω1andω2are the uncoupled reso- nance frequencies of each cavity mode and ωis the in- put microwave frequency. The two cavity modes can be indirectly coupled with each other when they both in- dividually interact with the FMR in YIG and this indi- rect coupling can be controlled using an external mag- netic field. We found that the microwave transmission a)Electronic mail: umhydep@myumanitoba.ca b)Electronic mail: bai@physics.umanitoba.ca FIG.1. (Colour online) (a)Inanuncoupledsystemindividual elements do not interact with each other. In our experimenta l system a YIG sphere simultaneously couples to two separate cavity modes, indirectly coupling the modes together. (b) The frequencies of the two cavity modes are functions of the height of the cylindrical microwave cavity and cross near a height of 36 mm. The inset shows a sketch of the microwave cavity.(c)Transmission spectrum S21of our indirectly cou- pled system, as a function of the external magnetic field at a microwave cavity height of 36.5 mm [dashed line in (b)]. (d) Transmission spectrum S21ofourcavitysystem, withaheight of 36.5 mm at an external field µ0H= 0.412 T, in an uncou- pled state (NO YIG in cavity) and (e)an indirectly coupled state (YIG in cavity), showing the influence of coupling on the resonant modes. properties change dramatically for the coupled modes as the external field is tuned. Our experimental results,2 together with an extended coupled harmonic oscillator model, demonstrate the nature of indirect coupling and coherent information transfer. This tunable interaction between orthogonal cavity modes could potentially be used to build controllable optical and microwave devices. The microwave cavity used in our experiment was made of oxygen-free copper with a height tunable cylin- drical structure. The diameter of the cavity is 25 mm and the height is tunable in a range between 24 mm and 45 mm. Although multiple modes can exist inside of the cavity, the TM 012mode (with a cavity frequency ofω1) and the TE 211mode (with a cavity frequency of ω2) were chosen to demonstrate indirect coupling in this work. With no YIG inside of the cavity, the microwave transmission, S21, was measured using a Vector Network Analyser (VNA) as a function of frequency. The out- put microwave power of the VNA is 1 mW. The am- plitude of the transmission is proportional to the res- onance amplitudes of both cavity modes at a given mi- crowavefrequency, |S21(ω)|2∝ |hω1(ω)+hω2(ω)|2. Here, hω1=Γ1ω2 ω2−ω2 1+2iβ1ω1ωh0andhω2=Γ2ω2 ω2−ω2 2+2iβ2ω2ωh0are the response functions of each cavity mode near the res- onance conditions. ω1,ω2,β1, andβ2are the cavity mode resonance frequencies and damping. Γ 1and Γ2de- note the impedance matching parameters for each cavity mode.h0(ω) is the microwave field used to drive reso- nance in the cavity and is eliminated by normalization in the microwave transmission. The microwave trans- mission spectra with no YIG in the cavity allows the individual cavity mode frequencies and damping to be evaluated. Fig. 1(b) plots the resonant frequencies of ω1andω2as a function of the height of the microwave cavity, both agree well with the solutions for Maxwell’s equations (solid lines) in a cylindrical microwave cavity. Thatthetwocavitymodescrosseachotherindicatesthat there is no direct coupling between them. The different microwave magnetic field distributions of the two modes inside the cavity leads to them having different coupling strengthswiththe FMRinYIG.Foragivencavityheight of 36.5 mm, the parameters of the two cavity modes were determined to be: ω1/2π= 12.357GHz, β1= 1.9×10−4, Γ1= 6.1×10−5,ω2/2π= 12.382 GHz, β2= 0.91×10−4, and Γ2= 3.7×10−5. A YIG sphere20placed inside the cavity allows for in- direct coupling between the two cavity modes. The YIG sphere has a diameter of 1 mm, saturation magnetiza- tionµM0= 0.178 T, gyromagnetic ratio γ= 28×2πµ0 GHz/T, and Gilbert damping α= 1.15×10−4. The YIG sphere was placed at the bottom of the cavity near the wall as shown in the inset of Fig. 1(b). An external mag- netic field, H, was applied to the YIG as shown in the inset. This magnetic field allows us to tune the FMR fre- quency of the YIG, ωFMR, following an ω-H dispersion ωFMR=γ(H+HAni). Here, the anisotropy field of the sphere is µ0HAni= 0.0294 T. Transmission measurements of our coupled system are plotted in Fig. 1(c), which shows the amplitude |S21|2 as a function of the input microwave frequency ( ω) and FIG. 2. (Colour online) (a)and(b)display the ω-H disper- sion and damping evolution (symbols) of each of the Normal Modes in our system. They are compared to calculations from Eq. 1 (solid curves). (c)The amplitudes of the Nor- mal Modes, |S21|2, are dramatically enhanced or suppressed during coupling. (d)The relative phase between the two cav- ity modes, φ1−φ2, was calculated during indirect coupling. The in-phase point of Mode B corresponds to its maximum amplitude in (c). the external magnetic field H. By increasing the Hfield, the FMR frequency ( ωFMR) first increases to the lower cavity mode frequency ω1, then reaches the higher cavity mode frequency ω2as indicated by the dashed lines. By doing this, the two cavity modes are indirectly coupled together via their direct coupling with the FMR in YIG, producing three coupled modes. We observed a maxi- mum in the microwave transmission amplitude when the middle mode (later labelled Mode B) crosses the disper- sion of the YIG FMR (dashed line) due to the resonances ofthe twocavitymodesbeing in-phase. Fig. 1(d) and (e) show how the addition of the YIG sphere into the cavity affects the observed resonant modes at an external field of 0.412 T; with both the number and position of the ob- served modes changing once the sphere is placed in the cavity. To further understand the nature of this indirect cou- pling between the two cavity modes, an expanded cou- pled harmonic oscillator system is used to calculate cou- plingfeaturesincludingthe ω-Hdispersion,dampingevo- lution, and amplitudes. Coupled harmonic oscillators have previously been used to accurately model strong coupling between a cavity mode and FMR in YIG21. The coupling strengths between each cavity mode and the FMR in YIG, κ1= 0.070 and κ2= 0.043, were eval- uated using the two coupled harmonic oscillator model when the two cavity mode frequencies were well sepa- rated (not shown here). The local microwave magnetic field distribution of each mode, with respect to the exter- nal field orientation, lead to different coupling strengths between each cavity mode and the FMR in YIG22. A3 slight change of the cavity height does not change the coupling strength of each mode. However, the coupled system observed in this work can no longer be modelled by the two coupled harmonic oscillator model. To take into account the second cavity mode, a three oscillatorsystemis consideredratherthan the twoin Ref.[21]. Two of the oscillators describe the cavity modes with ampli- tudes of hω1andhω2, each separately coupled with the third representing the FMR in YIG with amplitude m. Therefore, the indirect coupling model can be written in the form; ω2−ω2 1+i2β1ω1ω 0 −κ2 1ω2 1 0 ω2−ω2 2+i2β2ω2ω κ2 2ω2 2 −κ2 1ω2 1 κ2 2ω2 2 ω2−ω2 FMR+i2αωFMRω hω1 hω2 m =ω2 Γ1 Γ2 0 h0(1) Here the diagonal terms are the uncoupled resonance conditions of the two cavity modes and the FMR in YIG. The off-diagonal terms are the coupling strengths. The two zeros indicate that there is no direct coupling be- tween the two cavity modes. To explain our experimen- talobservationswemustinclude a π-phasedelaybetween the resonance frequencies of neighbouring cavity modes, although the physical source of this phase shift is still an open question. This is the source of the additional minus sign in the κ1terms. Eq. 1 allows us to predict the char- acteristics of indirect coupling between cavity modes via FMR in YIG. By finding the complex eigen-frequencies ωn(n = A, B, C, denoting the Modes labelled in Figure 2) of the coupling matrix at a given Hfield, we can plot the cal- culated resonance frequency Re(ωn), and the normalized line width |Im(ωn)|/Re(ωn) in Fig. 2(a) and (b) using solid curves. This matches the observed ω-H dispersion and damping evolution seen in the measurements (sym- bols). Furthermore, we are able to calculate the amplitude and relative phase of the microwave transmission hω1, FIG. 3. (Colour online) (a),(b), and(c)show the transmis- sion spectrum of the three Normal Modes for different ω2−ω1 values.(d)Amplitude of the in-phase point of Mode B as a function of ω2−ω1.hω2, andmusing Eq. 1. The calculated transmission amplitude |S21|2was plotted in Fig. 2(c) (solid curves) and compared with that from our experimental results (symbols). An amplitude peak is seen in both the experi- mental results and the theoretical calculation. The phase difference( φ1−φ2)between hω1andhω2iscalculatedand plotted in Fig. 2(d). The applied field strength at the maximum amplitude corresponds to an in-phase point, highlighted in Fig. 2(d), where the phases of the two cavitymodes hω1andhω2areequal ( φ1−φ2= 0). Mean- while the amplitude decrease of the other Normal Modes is due to the relative phase difference between the two cavity modes approaching π. Hence, coherent phase con- trol between two indirectly coupled cavity modes is de- tected through amplitude enhancement of the microwave transmissionand explainedby our three oscillatormodel. The in-phase point observed occurs when Mode B crosses the uncoupled dispersion of the YIG FMR. The amplitude of this in-phase point also depends on the dif- ference between the resonant frequencies of the two cav- ity modes ( ω2−ω1). By tuning the cavity height, the in- phasepointcanbemeasuredfordifferentvaluesof ω2−ω1 as shown in Fig. 3(a), (b), and (c). The amplitude of these in-phase points, highlighted in red, increases when the two cavity mode frequencies are near to each other. As summarized in Fig. 3(d), the transmission amplitude |S21|2decreases as the two cavity mode frequencies are separated. Therefore, the microwave transmission of the in-phasepointcanalsobecontrolledbythe cavityheight. In summary, we experimentally demonstrate control- lable indirect coupling between two microwave cavity modes through a YIG sphere. The coupling features are analysed and explained using a three coupled harmonic oscillator model. Microwave modes produced due to in- direct coupling were observed to have a higher transmis- sion rate than the two uncoupled cavity modes. We also demonstrated that these indirectly coupled modes can be controlled with an external field and by changing the cavity’s height. Therefore, due to the controllable nature of our findings, our work can be useful for designing new optical devices for information processing. The authors would like to thank B. Yao for useful discussions. P.H. is supported by the UMGF program. M.H. is supported by an NSERC CGSD Scholarship.4 This work has been funded by NSERC, CFI, and NSFC (No. 11429401) grants (C.-M. Hu). 1S. Fan, ’Sharp asymmetric line shapes in side-coupled waveguide- cavity systems ’, Appl. Phys. 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Ferguson, ’Dispersive readout of ferromagnetic resonance for strongl y cou- pled magnons and microwave photons ’, Phys. Rev. B 91, 104410 (2015) 13B.M. Yao, Y.S. Gui, M. Worden, T. Hegmann, M. Xing, X.S. Chen, W. Lu, Y. Wroczynskyj, J. van Lierop, and C.-M. Hu, ’Quantifying the complex permittivity and permeability of m ag- netic nanoparticles ’, Appl. Phys. Lett. 106, 142406 (2015) 14L. Bai, M. Harder, Y.P. Chen, X. Fan, J.Q. Xiao, and C.-M. Hu, ’Spin Pumping in Electrodynamically Coupled Magnon-Photon Systems ’, Phys. Rev. Lett. 114, 227201 (2015) 15L.V. Abdurakhimov, Y.M. Bunkov, and D. Konstantinov, ’Normal-Mode Splitting in the Coupled System of of Hybridize d Nuclear Magnons and Microwave Photons ’, Phys. Rev. Lett. 114, 226402 (2015) 16B.M. Yao, Y.S. Gui, Y. Xiao, H. Guo, X.S. Chen, W. Lu, C.L. Chien, and C.-M. Hu, ’ Theory and experiment on cavity magnon polariton in the 1D configuration ’, Phys. Rev. B 92, 184407 (2015) 17A Osada, R. Hisatomi, A. Noguchi, Y. Tabuchi, R. Yamazaki, K. Usami, M. Sadgrove, R. Yalla, M. Nomura, and Y. Naka- mura, ’Cavity optomagnonics with spin orbit coupled photons ’, arXiv:1510.01837 (2015) 18X. Zhang, C.-L. Zou, N. Zhu, F. Marquardt, L. Jiang, and H.X. Tang, ’ Magnon dark modes and gradient memory ’, Nature Comm.6, 8914 (2015) 19Y. Tabuchi, S. Ishino, A. Noguchi, T. Ishikawa, R. Yamazaki, K. Usami, and Y. Nakamura, ’ Coherent coupling between a ferro- magnetic magnon and a superconducting qubit ’, Science 24, 405 (2015) 20http://www.ferrisphere.com 21M. Harder, L. Bai, C. Match, and C.-M. Hu, ’ Study of the cavity- magnon-polariton transmission line shape ’, arXiv:1601.06049 22L. Bai, K. Blanchette, M. Harder, Y. Chen, X. Fan, J. Xiao, and C.-M. Hu, Control of the Magnon-Photon Coupling ’, IEEE Transactions on Magnetics, PP, 2527691 (2016)

2016-06-10

We experimentally realize indirect coupling between two cavity modes via strong coupling with the ferromagnetic resonance in Yttrium Iron Garnet (YIG). We find that some indirectly coupled modes of our system can have a higher microwave transmission than the individual uncoupled modes. Using a coupled harmonic oscillator model, the influence of the oscillation phase difference between the two cavity modes on the nature of the indirect coupling is revealed. These indirectly coupled microwave modes can be controlled using an external magnetic field or by tuning the cavity height. This work has potential for use in controllable optical devices and information processing technologies.

Indirect Coupling between Two Cavity Photon Systems via Ferromagnetic Resonance

1606.03469v1

Thermally controlled connement of spin wave eld in a magnonic YIG waveguide Pablo Borysa,, O. Kolokoltseva, Iv an G omez-Aristab, V. Zavislyakc, G. A. Melkovc, N. Qureshia, Csar L. Ordez-Romerod aInstituto de Ciencias Aplicadas y Tecnologa, Universidad Nacional Autnoma de Mxico (UNAM), Ciudad Universitaria, 04510, Mxico bCtedras Conacyt Instituto Nacional de Astrofsica, ptica y Electrnica, 72840, Mxico cFaculty of Radiophysics, Electronics and Computer Systems, Taras Shevchenko National University of Kiev, Ukraine dInstituto de Fsica, Universidad Nacional Autnoma de Mxico, Ciudad Universitaria, 04510, Mxico. Abstract Methods for detecting spin waves rely on electrodynamical coupling between the spin wave dipolar eld and an inductive probe. While this coupling is usually treated as constant, in this work, we experimentally and theoretically show that it is indeed temperature dependent. By measuring the spin wave magnetic eld as a function of temperature of, and distance to the sample, we demonstrate that there is both a longitudinal and transversal connement of the eld near the YIG-Air interface. Our results are relevant for spin wave detection, in particular in the eld of spin wave caloritronics. Keywords: YIG, spin waves, thermal connement, spin wave- waveguide, electrodynamic coupling, inductive probe, dipolar eld. 1. Introduction It is expected that the emergence of thin lm logic elements based on spin waves in thin-lm ferromagnetic solids can lead to a new generation of Boolean and analogue processors [1, 2, 3]. One of the important points here is the tech- nique of spin wave excitation and modulation of their parameters. Traditionally, 5 spin waves have been excited and detected using the inductive coupling of micro- electrodes to the dipolar magnetic eld of the spin wave system. Usually, this electrodynamical coupling is considered to be constant, however, as shown in this work, it can suer signicant variations, depending on the temperature of the ferromagnetic material. The temperature of the sample can change due to 10 spin wave dissipation, from 1 to 10oC [4, 5] or up to 100-300oC because of Corresponding author Email address: pabloborys@ciencias.unam.mx (Pablo Borys) Preprint submitted to Journal of Magnetism and Magnetic Materials April 30, 2022arXiv:1910.04304v1 [cond-mat.mes-hall] 9 Oct 2019Figure 1: A schematic view of experimental set-up. The inductive probe is attached to YX motorized translation stages external heating used to control the spin wave propagation [6, 7]. Recently, the typical electrodynamical and magneto-optical methods for spin wave detec- tion/excitation were enriched with the spin transfer torque (STT) [8, 9, 10] in Pt/magnet thin lm structures caused by electrical or thermal spin currents 15 [11, 12, 13, 14, 15]. STT has been recognized to be a much promising tool to detect exchange SW, to control dipole SW, and to generate thermo-electricity on the basis of spin Seebeck eect. The discovery of the later has stimulated a number of ideas involving magnetocaloritronics [16]. For example, thermally assisted STT has been used for enhancement of spin oscillations in resonators, 20 spin wave amplication and spin auto-oscillations [17, 18, 19, 20]. The aim of this work is to reveal lateral eects of sample heating in experimental con- gurations on the inductive coupling between micro-antennas and the dipolar magnetic eld of a spin wave system [21, 22, 23, 24, 25, 26, 27, 28, 29]. 2. Experiment 25 A schematic diagram of the experimental set-up, designed to investigate the coupling between spin waves propagating on a YIG/GGG sample and an 2inductive micro-transducer, is shown in Fig. 1. The sample is 1 mm wide in the Zdirection and 28 mm long in the Ydirection. The thickness of the YIG lm is 7m. The sample was biased by a tangential magnetic eld ( H0) applied 30 along theZaxis to provide the propagation of Magneto-Static Surface Waves (MSSW) in the Ydirection. MSSW were excited at one end of the sample (at Y=Y0), in a pulse regime, by dc electric current pulse owing through a 0.25 mm-wide microstrip line terminated to a 50 Ohm resistive load. This method provides very short spin wave packets, with duration of 10 ns. In the time 35 domain the shortest period of the magnetization precession in the wave packet is limited by the rise time of the electric current pulse, and in the k-space the largest wavenumber (k) is limited by the microstrip line width [3]. The MSSW pulse propagation characteristics were registered by an inductive frame-shaped probe [30] (Fig. 1) sensitive to the Ymagnetic component of microwave eld 40 (hy) induced by the spin wave in the vicinity of the YIG lm. The probe was scanned over the sample plane along the Ycoordinate (Fig. 1) by a motorized translation stage. The distance between the probe and the sample surface was also controlled by a motorized translation stage. It should be noted that we used a frame probe with reduced X-dimension to have high spatial resolution 45 of hy along the lm normal, as the probe is displaced in the X direction. The probe electrode was fabricated with a 50 m micro-wire. The sample was heated with a solid state green laser with variable output power ( Popt), from 40 to 300 mW. The laser spot on YIG was of 0.5 mm in diameter and was located at the distance of YLfrom the excitation port. 50 Fig 2 compares the time-space evolution of the amplitude of the hy pulse at room temperature (RT sample) in 2(a), with a sample heated at optical power Popt= 180 mW in 2(b). The pulse waveform was recorded by a real time Tektronix oscilloscope with 6 GHz- bandwidth, at dierent Y- positions of the probe, and at xed distance
X= 50m between the probe and the YIG lm 55 plane. The measurements were done with a uniform bias eld H0= 120 Oe, and the laser spot at the position YL= 15 mm. As seen in Fig.2, the wave packet in the optically heated sample acquires an additional group delay, compared to the sample at room temperature. This phenomenon has been discussed in ref. [31], and is caused by a reduction on the saturation magnetization Msthat in 60 turn decreases the slope in the MSSW dispersion relation. Fig. 3 shows the signal detected by the probe, as the probe moves along the Y axis, at dierent Popt. The value of each point in the curves in Fig. 3 represents the energy of the pulse envelope. As clearly seen in the gure, the signal induced in the probe increases in the vicinity of the laser spot, and this 65 increment is proportional to temperature of the hot zone. On the other hand, in the sample at room temperature (Curve 1) the MSSW pulse propagates and attenuates exponentially, in the usual way. The data presented in Fig.3 are proportional to the overlap integral between a small eective area of the probe frame and an evanescent function hy(x) [30]. Hence, displacing the probe along 70 the lm normal one can obtain the prole jhy(x)j2, shown in Fig.4. In this experiment the probe was located in the center of the hot zone Y=YL Fig.4 presents the principal result of this study: the density of hynear 3Figure 2: Propagation of MSSW pulse along the magnonic waveguide YIG/GGG. The inset in Fig. 2a) shows details the pulse waveform, with a duration of 8 ns, at three adjacent positions along the Y axis. The data represent the pulse waveform (amplitude) a) in the sample at room temperature, and b) in the sample heated at 380 K in its center. 4Figure 3: Fig. 3 The energy of the MSSW pulse at dierent distances from excitation port. The curves were recorded at dierent Popt, which induce dierent temperatures in the region Y=YL: 1) T = T ROOM ; 2) T = T ROOM + 50 K ; 3) T = T ROOM + 70 K ; 4) T = TROOM + 90 K. 5Figure 4: Energy of hycomponent as a function of the distance between the probe and YIG lm surface, at the xed Y-position of the probe Y=YL. Red and black experimental points show the energy density in evanescent MSSW eld in the heated sample (at T = 380 K) and the RT sample, respectively. 6the lm interface increases as the sample temperature increases, i.e. the heat modies the eld connement. 75 3. Theoretical Background The eect of the thermally dependent eld connement is caused by the decrease ofMsin the ferrite lm, as its temperature increases. It can be analyzed analytically by a full set of Maxwell equations. In our case, considering that the sample is innite in YZ plane, the solutions for the magnetic and electric elds 80 of MSSW are h= (hx;hy;0) and e= (0;0;ez), respectively. Let us compare transversal proles of monochromatic magnetic eld components hx; hyin hot and RT samples, taking into account that the elds have to be normalized to transmit a given power ow Pthrough the sample. It is clear that in both hot and RT samples a value of Pshould be the same, supposing equal excitation 85 eciency of MSSW. It can be shown that the Pointing vector for MSSW is calculated as: P=c 8 ezh yi+ezh xj (1) or P2=c 8k0? ke2 z+a ez@e z @x j (2) in the YIG lm, and P1;3=c 8k0ke2 zj (3) in air and substrate. 90 Here:kis the MSSW wavenumber, k0=!=c,cis the speed of light in the vacuum,!is the MSSW frequency, = (!2!2 1)=(!2!2 H),a=!!M=(!2 !2 H),!H= H0, and!M= 4MS,!1=!H(!+!M), and is the electron gyromagnetic ratio. 95 Then, taking into account that h,ein Eq. 1 are proportional to a certain constant,A, the value of Afor both hot and RT sample can be calculated using the conditionP i=1;2;3Pi(Hot sample ) =P i=1;2;3Pi(RT sample ) =Const . The explicit expressions for MSSW eld components in Eq. 1 are given in Appendix A. The calculated magnetic eld proles are shown in Fig. 5. 100 The results were obtained by using the experimental approximation for temperature dependence of the saturation magnetization in YIG: Ms= 140 
T(G),0:3 G/K [31]. The eld proles in Fig.5 correlate well with the experimental proles in Fig.4. 4. Discussion and Conclusions 105 The peculiarity of the results for the pulse group delay shown in Fig. 2 is that the local heating increases the pulse delay, however, it does not change the group velocity dispersion. As seen in Fig.2, the pulse width (the pulse duration) in the 7Figure 5: Fig.5. The values of the tangential (hy) an the normal (hx) eld components calculated for hot (red curves) and RT (blue curves) samples. 8hot region remains unchanged, with respect to the pulse width in the RT sample. This means that spatial width of the pulse along the Y coordinate decreases, 110 i.e. there is spatial, longitudinal compression of the pulse along the propagation direction. This leads to the increase of a peak and average amplitude of the pulse envelope for pulse power to be conserved. The eect has been analyzed in [32], where we used a large diameter loop antenna that was not sensitive to the eect of the transversal connement of the evanescent eld shown in Fig.4, 115 and 5. On the other hand, the results presented in Fig.3, 4, and 5 indicate that increasing the sample temperature increases the coupling between MSSW eld and the micro-antenna. The experimental, Fig.4, and theoretical, Fig.5, data demonstrate that this eect takes place due to an increasing concentration of magnetic elds near YIG-Air interface, the so-called transversal connement. 120 In conclusion, it is shown that the increase of the sample temperature leads to the increase of both longitudinal and transversal connement of MSSW in the vecinity of YIG lm. This eect, in turn, is revealed as the increase of the signal induced in a micro-antenna, that has to be taken into account in the experiments on spin-wave caloritronics. 125 5. Appendix A Full system of Maxwell equations for electromagnetic waves in the sample saturated in the Z direction describes two kind of waves. The subsystem @hz @y=i"k0ex @hz @y=i"k0ey @ey @x@ex @y=ik0hz(4) describes fast waves, which neglects magnetism, and the subsystem @hy @x@hx @y=i"k0ez ik0(hxiahy) =@ez @y ik0(iahx+hy) =@ez @y(5) that is used to describe MSSW. It can be reduced to 130 @2ez @x2+@2ez @y2+"?k2 0ez= 0 (6) MSSW elds, where ?= (22 a)=, which satisfy Eq. 2b, and Eq.3 are ez=Aeax+i(!tky);hx=k k0Aeax+i(!tky);hy=ia k0Aeax+i(!tky) | {z } Air(7) 9ez= (Bcosh (mx) +Csinh (mx))ei(!tky) hx=1 k0(22a)[k(Bcosh (mx) +Csinh (mx)) +am(Bsinh (mx) +Ccosh (mx))]ei(!tky) hy=1 k0(22a)[ak(Bcosh (mx) +Csinh (mx)) +m(Bsinh (mx) +Ccosh (mx))]ei(!tky) | {z } YIG (8) ez=Deax+i(!tky); hx=k k0Deax+i(!tky); hy=ia k0Deax+i(!tky); | {z } Substrate (9) witha=p k2k2 0, andm=p k2?k2 0. The standard electrody- namic boundary conditions at the structure interfaces determine the following relations between the coecients A,B,C,D A=B; Deas=Bcosh (ms) +Ccosh (ms); a 22 a
A=akB+mC; a 22 a
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Hansen, Temperature dependence of the saturation magneti- 235 zation of ion-implanted yig lms, Applied Physics A 29 (2) (1982) 83{86. [32] O. Kolokoltsev, I. G omez-Arista, N. Qureshi, A. Acevedo, C. L. Ord o~ nez- Romero, A. Grishin, Compression gain of spin wave signals in a magnonic yig waveguide with thermal non-uniformity, Journal of Magnetism and Magnetic Materials 377 (2015) 1{5. 240 13

2019-10-09

Methods for detecting spin waves rely on electrodynamical coupling between the spin wave dipolar field and an inductive probe. While this coupling is usually treated as constant, in this work, we experimentally and theoretically show that it is indeed temperature dependent. By measuring the spin wave magnetic field as a function of temperature of, and distance to the sample, we demonstrate that there is both a longitudinal and transversal confinement of the field near the YIG-Air interface. Our results are relevant for spin wave detection, in particular in the field of spin wave caloritronics

Thermally controlled confinement of spin wave field in a magnonic YIG waveguide

1910.04304v1

All-optical cryogenic thermometry based on NV centers in nanodiamon ds M. Fukami1, C. G. Yale1,†, P. Andrich1,‡, X. Liu1, F. J. Heremans1,2, P. F. Nealey1,2, D. D. Awschalom1,2,* 1. Institute for Molecular Engineering, University of Chicago, Chicago, IL 60637 2. Institute for Molecular Engineering and Materials Science Division, Argonne National Lab, Argonne, IL 60439 †Present address: Sandia National Laboratories, Albuquerque, NM, 87185 ‡Present address: University of Cambridge, Cavendish Laboratory, JJ Thomson Ave, Camb ridge CB3 0HE *Email: awsch@uchicago.edu ABSTRACT The nitrogen -vacancy (NV) center in diamond has been recognized as a high -sensitivity nanometer -scale metrology platform . Thermometry has been a recent focus, with attention largely confined to room temperature applications. Temperature sensing at low temperatures , however, remains challenging as the sensitivity decreases for many commonly used technique s, which rely on a temperature dependent frequency shift of NV center’s spin resonance and its control with microwaves . Here w e use an alternative approach that does not require microwaves , ratiometric all -optical thermometry , and demonstrate that it may be utilized to liquid nitrogen temperatures without deterioration of the sensitivity . The use of an array of nanodiamonds embedded within a portable polydimethylsiloxane (PDMS) sheet provides a versatile temperature sensing platform that can probe a wide variety of systems without the configurational restrictions needed for applying microwaves . With this device, w e observe a temperature gradient over tens of microns in a ferromagnetic -insulator substrate (yttrium iron garnet, YIG) under local heating by a resistive heater . This thermometry technique provides a cryogenically compatible, microwave - free, minimally invasive approach capable of probing local temperatures with few restriction s on the substrate materials . I. INTRODUCTION Local temperature variation plays a central role in many -body physics governed by hydrodynamic description s [1,2] , in biomolecular science [3], as well as in thermal engineering of integrated circuit s. Among the existing high -sensitivity nanometer -scale thermometers, nitrogen vacancy (NV) centers in nanodiamond s (NDs) have emerged as promising temperature -sensitive fluorescent probes . The negatively -charged NV -center (NV-) consists of a ground state spin triplet manifold with a zero-field splitting 𝒟⋍2.87 GHz that sensitively responds to temperature s, where the shift can be measured by reading out the spin optically [3–6]. By vi rtue of diamond’s high thermal conductivity an d NV- centers’ long spin coherence time, ND-based thermometry has been demonstrated in a variety of systems , such as within a living cell at room temperature [3]. The temperature response of 𝒟 is significantly smaller at low temperatures, however, which reduces sensitivity and hinders the conventional thermometry technique [7,8] . Ratiometric all -optical thermometry has been proposed as an alternative to the convent ional microwave spin-resonance thermometry technique with compatible sensitivity at room temperature [9–12]. It also enables temperature sensing without the application of microwave s, which removes concern s of microwave heating . Interestingly, the temperature sensitivity of the all -optical thermometer is estimated to improve at lower temperatures (see Supplementary Material, Sec. A [13]), and indicates that this tech nique can offer a path forward towards ND-based cryogenic thermometry . The use of an array of NDs on a polydimethylsiloxane ( PDMS ) sheet [13] combined with all -optical thermometry completely removes configurational restrictions needed for microwave application s, offering a versatile device capable of probing a wide variety of solid -state systems over tens of microns with an adjustable spatial resolution on the order of a few microns. This makes all-optical thermome try suitable for probing and imaging a variety of condensed matter systems , and may have advantages over conventional NV-center thermometry technique s depending on the required thermal or spatial resolutions as well as the potential microwave response of the target system . Here we extend the all -optical thermometry technique based on the NV- centers in NDs from room temperature to liquid nitrogen temperatures , 85 K T 300 K , and demonstrate its application on a ferromagnetic insulato r (yttrium iron garnet, YIG) substrate . In particular, we focus on YIG as a platform to demonstrate our sensing approach both because the microwave s used to manipulate NV centers in conventional thermometry would impact the magnetic spins in the YIG [14–19], and the low temperature thermal response of YIG is of interest in the study of the spin -Seebec k effect [20–24]. We initially demonstrate that a laser -pulse sequence to control the NV centers’ charge states improve s the sensitivity of the all -optical thermometer by approximately a factor of 3 . Next , we systematically study the temperatu re dependence of the sensitivity , demonstrating that it improves at cryogenic temperatures . Finally , we apply this all-optical cryogenic thermometry technique at T 170 K to measure the surface temperature profile of a YIG slab in contact with a resistive heater, with the array of NDs embedded on the surface of a flexible PDMS sheet . The observed temperature gradient over a range of tens of micrometers confirms the applicability of the technique on the YIG substrate , indicating that it provides a tool for study ing local thermal properties of a wide variety of substrates over a broad range of temperatures. II. DEMONSTRATION OF CRYOGENIC ALL -OPTICAL THERMOMETRY We focus on the temperature dependence of the NV- centers’ zero phonon line (ZPL) amplitude ratio ( A), which is defined as the ratio of the ZPL intensity with respect to an average photoluminescence (PL) intensity in a spectral range around the ZPL. The ratio A strongly responds to temperature change due to the presence of a coupling between the orbital state of NV- and vibrational modes in diamond [25] (see Supplementary Material, Sec. B [13]), which leads to a high temperature sensitivity . The experiment was conducted on an array of NDs containing ensemble s of NV- centers measured with a confocal microscope using a high numerical aperture objective (NA =0.9) as shown in Fig. 1(a). An array of NDs embedded into the flexible PDMS sheet was placed on the surface of a 3.05- m-thick YIG film grown on a 500-m-thick gadolinium gallium garnet ( GGG ) substrate (MTI Corp.) . A Ti/Au (thickness: 8nm/200nm) resistive heater , for local heating , was patterned on the YIG film using a lithographic process . The bottom of the GGG substrate was affixed to a copper thermal sink within a flow cryostat . Both characterization (section II) and application (section III) of the thermometry were conducted on the same device with a YIG substrate for consistency (for data without a PDMS sheet on a quartz substrate, see Supplementary Material, Sec. I [13]). Figure 1(b) shows a two-dimensional PL scan of a n individual spot in the array of NDs under continuous 594-nm excitation measured by an avalanche photodiode (APD). The 594 -nm light does not excite the neutrally - charged NV -center (NV0) [26,27] and removes the noisy NV0 phonon -sideband spectral emission from the NV-‘s ZPL spectrum . The diameter of the spot is 1000 nm which is defined by our microfabrication technique [28], and contains tens of NDs, where each ND contains hundreds of NV centers [28]. Figure 1(c) shows a horizontal cut through the maximum of Fig. 1(b) . Interestingly, when we applied pulse sequence s of the 594-nm and 532-nm laser s as shown in F ig. 1(d) , which is in contrast to the previous studies with a continuous -wave excitation [11,12] , the PL count rate was enhanced by approxi mately a factor of three (see Supplementary Material, Sec. D [13]). The enhancement is due to the charge -state conversion between NV- and NV0 [13,29 –31]. While charge -state conversions of NV centers in NDs have not been comprehensively studied to our knowledge, we simply assume the results reported in bulk diamonds are applicable and attribute the PL enhancement to the charge -state convers ion. Since the sensitivity of the all-optical thermometer is limited by shot noise , improving the PL count rate by a factor of 3 increases sensitivity by a factor of 3 (see Supplementary Material, Sec. E [13]). In the following spectral measurements , we send the PL to a spectrometer and gate the intensifier of a single -photon sensitive CCD camera in the spectrometer (iStar 334T, Andor) triggered by the pulse sequence s. Every spectra l measurement was followed by a background measurement taken off the ND and the background counts were subtracted . (see Supplemental Material, Sec. F [13]). Figure 2(a) shows the PL spectra ()Lh of NV- centers in the temperature range 85 K T 100 K . Monotonic change in the spectra is observed except near T ≃230 K and T ≃150 K , which are due to the melting point and the glass transition point of the PDMS , respectively . We note that the presence of the PDMS sheet does not change the thermometry property of NV centers except PL count rate s, which is verified by the measurements done on NDs without a PDMS sheet (See Supplementary Material, Sec. I [13]). To maximize the PL count rat e, we widely opened the slit in the spectrometer, which results in a wavelength resolution =3.5 nm. For the temperature sensing, we focus on the ZPL emission peak at h≃1.94 eV (637 nm) . Importantly, the ZPL becomes sharper and more prominent at lower temperatures . In this experiment , we focused on the PL in the wavelength ranging from 605 nm to 660 nm , which we define as the spectral range (ℛ) (for the choice of this range, see Supplemental Materia l, Sec. G [13]). As shown in the inset of F ig. 2(b), we fit the relative spectrum LLR by a sum of a squared -Lorentzian function and an exponential function ZPL 2 2 2 B ZPL() 1( ) exp[ ( ) ]hL h L A Bk w h h R (1) where Bk is the Boltzmann constant , ℎ is the Plank constant, LR is the average PL intensity in the spectral window ℛ and ZPL { , , , , }A B w are fitting parameters. A squared -Lorentzian function instead of a Lorentzian function is used as suggested in Ref. [32] for better fit s at cryogenic temperatures. Temperature dependence of the ratio A is shown as solid marker s in Fig. 2(b) , where the solid and dot ted curves are derived from the two fits of the reduced Debye -Waller factor and the ZPL linewidth shown in Figs. 2(c) and 2(d). We note that the reduced Debye - Waller factor is defined in this work as the ratio of the integra ted ZPL emission , which corresponds to the area under the squared -Lorentzian fit , to the total PL in the range ℛ. Importantly , we find a maximum in the slope of the ratio 𝐴 around T 150 K, which coincidentally corresponds to the glass transition temperature of the PDMS , though does not appear to be related to it (See Supplementary Material, Sec. H and Sec. I [13]). While the stronger temperature response dA dT at lower temperatures observed in this study is desirable for the improved temperature sensing, t he presence of the maximum cannot be explained by a currently existing model , since it predicts a monotonic increase of the temperature response at lower temperatures . This can be resolved by taking into account a constant term ( a) in the linewidth 2w a bT , modifying t he analytical expression of the ZPL amplitude ratio to be (see Supplemental Material, Sec. J [13]) 2 22 exp( ) ()TAa bT R (2) where and are fitting parameters of the reduced Debye -Waller factor and R is the size of the spectral window ℛ. The constant contribution is due both to a resolution of the spectrometer and an inhomogeneous broadening. W avelength resolution can be improved by narrowing down the slit in the spectrometer with a trade -off of the PL count rate. The inhomogeneous broadening is not negligible at lower temperatures due to crystal strain variations both between different NDs and within the individual commercial NDs used in this study . These limitations could be overcome by introducing engineered nanoparticles [33,34] , leading to an enhanced temperature response at cryogenic temperatures . The temperature sensitivity of a thermometer, which is sometimes referred to as the noise floor, is not only quantified by the temperature response dA dT but also by the uncertainty A in the measurement of A . They are related by 1 At dA dT , where t is the measurement time . While the temperature response increases at lower temperatures , A grow s along with the temperature response. To fully characterize the sensitivity of the thermometry technique, we studied the uncertainty A as a function of temperature T . At each temperature, PL spectrum measurements with an integration time of t 2.5 s were repeated one hundred times (F ig. 3(a)). We then calculate the standard deviation A for each data set and show its temperature dependence in F ig. 3(b). Note that the standard deviation A is rescaled by a factor ZPLCt to quantitatively compare the results at different temperatures, where ZPLC is the ZPL count rate shown in the inse t of F ig. 3(c) that corresponds to the area under the squared -Lorentzian fit (see Supplemental Material, Sec. L [13]). The dashed curve shows the lower bound when the noise is coming only from photon shot noise, while the dotted curve shows the lower bound when the CCD camera’s dark-current shot noise also contributes to the noise in the measurement of the ratio A (see Supplementary Material, Sec. M [13]). The experimental observation is well explained by the dotted curve, demonstrating that the standard deviation A is limited both by the ZPL photon shot noise and the CCD’s dark current shot noise. Comb ining the temperature dependencies of A and dA dT as shown in F igs. 3(c) and 2(b), we plot the temperature dependence of the sensitivity in Fig. 3(d) . The lower bounds shown are derived from the same model s as in F ig. 3(c). Importantly, the sensitivity improves at cryogenic temperatures in contrast to the conventional thermometry technique based on the temperature dependent shift in the zero -field splitting . We note that the sensitivity calculated in this study at T 300 K does not reach the level of the sensitivity provided in the previous report on all -optical thermometry at room temperature [11]; however, taking into account detection efficiency differences, our result is found to be fully consistent with the one in Ref. [11]. This can be confirmed by introducing a projected sensitivity proj as shown in F ig. 3(d), wh ich assumes as high ZPL counts rate s as in Ref. [11] and shows an anticipated sensitivity compatible with their result (for detail, see Supplemental Material, Sec. O [13]). The highest temperature sensitivity is achieved near T 200 K, which can be understood through the simplified analytical model that only considers the temperature evolution of the DWF (for detail on the necessary assumptions , see Supplemental Material, Sec. P [13]) 12 tot 011exp2 2 (DWF)TTT C (3) resulting in a minimum at 1T =218 K, where totC is the total PL counts rate of NV- and 0(DWF)T is the (non-reduced) Debye -Waller factor at absolute zero (For the discussion of the effect of the PDMS sheet, see Supplemental Material, Sec. Q [13]). While there is a quantitative mismatch due to oversimplification in the model , this model captures the existence of the minimum well. To further improve the sensitivity at low temperature s, one could, for instance, increase the ZPL count rate by improving the detection efficiency and utilize brighter NDs that contain more NV- centers . III. SURFACE TEMPERATURE IMAGING OF A YIG FILM To demonstrate the applicability of the all -optical thermometer , we apply an 80-mA current to the resistive heater to generate a temperature gradient in the YIG and measure the spatial temperature variation of the YIG surface using an array of NDs, as illustrated in F ig. 1(a) . Since the YIG has spin -wave resonances at microwave frequencies near 𝒟 [14–19], this measurement confirms that the all -optical thermometry technique can be used independently of substrate materials where microwave control is problematic . In the se experiments , the base temperature of the copper heat sink is stabilized at T =170 K (see Supplemental Material , Sec. R [13]). Figure 4(a) shows a two -dimensional spatial scan of the PL from the array of NDs used in this study. To construct the temperature profile, we repeat temperature measurements at multiple spots in the array . The accuracy of the measured temperature is ensured by calibrating NDs individually (see Supplemental Material , Sec. S [13]) and the temperature dependencies of ZPL { , , , }Bw in addition to A are utilized for calculating the local temperature (see Supplemental Material , Sec. T [13]). For each measurement, the PL is collected in total for 500 s. Figure 4(b) shows the resulting temperature profile of the YIG surface, where we observe a temperature decay on the order of tens -of-microns from the heat source. The temperature of each spot as a function of the distance from the heater is shown in F ig. 4(c), where the error bars include both the uncertainty of the sensing and the err or in the calibration. The data is fit well by the Green’s function to the two -dimensional Poisson equation, showing that the temperature field in the YIG approximately follows the steady state diffusion equation with a single heat carrier. We note that the Poisson equation is not accurate in YIG because there are two kinds of heat carriers, phonons and magnons. A deviation from the Poisson equation is expected near the heat source within a length scale of a magnon -phonon thermalization, which is much small er than a few micrometers [35]. In our experiment, NDs directly measure temperatures of the YIG lattice, or phonon s, and we do not observe any perturbation to the qualitative feature of the steady -state phononic temperature profile by the presence of magnons in YIG , which is expected due to our thermal and spatial resolutions . (see Supplementary Material, Sec. U [13]) [20,24] . IV. CONCLUSION We demonstrate and characterize an all-optical thermometry technique based on NV- center ensembles in ND that can be deployed from room temperature to liquid nitrogen temperatures , with a sensitivity that increases with decreasing temperature . Furthermore , the PL intensity of NV- centers is enhanced by implem enting pulse sequences to convert NV0 into NV-, leading to a higher temperature sensitivity by approximately a factor of 3 . Systematic noise analysis reveal s that the sensitivity is limited by the shot noise and the inhomogeneous broadening of the ZPL linewidth , suggesting a pathway for further sensitivity improvements by optimizing the spectral resolution, improving the PL detection efficiency , and introducing engineered NDs with high brightness and homogeneous crystal strain s. Taking advantage of an array of NDs embedded i n a flexible PDMS sheet, w e show the utility of the all -optical thermometer at T =170 K by measuring the surface temperature profile of a YIG slab thermally driven by a resistive heater . This all-optical thermome try technique along with the versatility of the ND membrane array provides a microwave -free, minimally invasive , and cryogenic ally compatible way of measuring local temperatures within a variety of substrate materials . ACKNOWLEDGMENTS This work was supported by the Air Force Office of Scientific Research and the Army Research Office through the MURI program, grant no. W911NF -14-1-0016. The fabrication of the diamond nanoparticle arrays was supported by the US Department of Energy, Office of Science, Basic Energy Sciences, Materials Sciences and Engineering Division. FJH, PFN , and DDA were supported by the US Department of Energy, Office of Science, Basic Energy Sciences, Materials Sciences and Engineering Division. 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Phys. 15, (2013). [32] G. Davies, Reports Prog. Phys. 44, (1981). [33] P. Maletinsky, S. Hong, M. S. Grinolds, B. Hausmann, M. D. Lukin, R. L. Walsworth, M. Loncar, and A. Yacoby, Nat. Nanotechnol. 7, 320 (2012). [34] and F. J. H. S. O. Hruszkewycz, W. Cha, P. Andrich, C. P. Anderson, A. Ulvestad, R. Harder, P. H. Fuoss, D. D. Awschalom, 026105 , (2017). [35] A. Prakash, B. Flebus, J. Brangham, F. Yang, Y. Tserkovnyak, and J. P. Heremans, Phys. Rev. B 020408 , 1 (2018). FIG. 1. (a) Schematic of an array of nanodiamonds (NDs) on a 3.05 -m YIG film grown on a GGG substrate. NDs are embedded on the surface of a flexible PDMS sheet and the YIG film was patterned with a resistive heater (central wire has a width of 5 m and a length of 200 m). (b) Two -dimensional photoluminescence (PL) image of NV centers in NDs collected under continuous 594 -nm excitation. PL intensity is measured by an ava lanche photodiode (APD). The measurement was conducted at T =170 K. Scale bar, 0.5 m. (c) Line cuts of PL intensity profiles of NV centers under two different excitation pulse sequences. (d) Schematic of the pulse sequences of a 532-nm laser ( NV- charge state initialization), a 594 -nm laser ( NV- detection) and a detector (APD/CCD camera). FIG. 2. (a) Evolution of NV centers’ PL spectrum ()Lh between temperatures T =85 K and T =300 K. The areas under the spectra are normalized to one. Discontinuities at T 230 K and T 150 K are associated with the PDMS’s phase transitions and not related to NV centers. Top (bottom) graph shows the spectrum at 300 K (85 K). (b) Temperature dependence of the ZPL amplitude ratio A (left axis) and its temperature respo nse dA dT (right axis). The solid blue curve is calculated from two fits: (i) temperature dependence of the reduced Debye -Waller factor (DWF) (shown in (c)), and (ii) temperature dependence of the ZPL linewidth (shown in (d)). The dotted red curve is the derivative of the solid (blue) curve with respect to temperature T . Inset shows the fit of the ZPL at T =170 K with a sum of an exponential function and a squared -Lorentzian function (black curve). The exponential - function part only is shown with a gray curve. LR is the mean PL intensity in the range ℛ from 605 nm to 660 nm. (c) Reduced DWF as a function of temperature T . A Gaussian -functional fit is shown. (d) ZPL linew idth as a function of temperature T . The solid blue fit is the second -order polynomial 2a bT and the dotted orange curve shows 2bT . FIG. 3. (a) ZPL amplitude ratio scanned over 100 times. A single scan consists of a total PL accumulation time of 2.5 s. The measurements were conducted at temperature T =170 K. (b) Histogram built from the measurements in (a). The standard deviation A is depicted. (c) Rescaled standard deviation ZPL AA Ct as a function of temperature T , where ZPLC is the PL counts rate under the squared -Lorentzian fit of ZPL and t is the total PL accumulation time. The dashed red curve shows the lower bound determined by photon shot noise and the dotted blue curve shows the lower bound dete rmined by photon and dark current shot noise. Inset shows ZPL counts rat e ZPLC as a function of temperature T . Solid black curve shows a one -parameter ( 1a ) fit of the ZPL counts rate () ZPL 1( ) DWFTC T aR , where ()DWFT R is the curve shown in F ig. 2 (c). (d) Temperature sensitivity as a function of temperature T . The dashed red and the dotted blue curves identify the lower bounds for the sensitivity as defined in (c). The spike near 160 K arises from the dip in the experimental data of ZPLC as shown in the inset of (c). Right axis shows a projected sensitivity proj under the assumption of a higher detection rate of the PL as explained in the main text. FIG. 4. (a) Spatial PL scan of NV centers in NDs in the array. (b) Two -dimensional temperature imaging of the YIG surface using NV centers in the array of NDs embedded on the surface of the PDMS sheet measured by the all - optical thermometry technique. An 8 0-mA current is applied to the resistive heater. The base temperature was set to T =170 K. (c) YIG surface temperature as a function of the distance from the resistive heater. Fit with a logarithmic function is shown. Supplemental Information for All-optical cryogenic thermometry based on NV centers in nanodiamonds M. Fukami1, C. G. Yale1,†, P. Andrich1,‡, X. Liu1, F. J. Heremans1,2, P. F. Nealey1,2, D. D. Awschalom1,2,* 1. Institute for Molecular Engineering, University of Chicago, Chicago, IL 60637 2. Institute for Molecular Engineering and Materials Science Division, Argonne National Lab, Argonne, IL 60439 †Present address: Sandia National Laboratories, Albuquerque, NM, 87185 ‡Present address: University of Cam bridge, Cavendish Laboratory, JJ Thomson Ave, Cambridge CB3 0HE *Email: awsch@uchicago.edu A. Temperature Dependence of the Sensitivity in a Range 𝟑𝟎𝟎 𝐊≲𝑻≲𝟒𝟎𝟎 𝐊 Reference [S1] provides a model that explains the temperature dependence of the zero-phonon line ( ZPL) amplitude ratio 𝐴 under temperature 𝑇 in the range 300 K≲𝑇≲400 K. The authors fit the ZPL with a sum of a Lorentzian function and an exponential function, with the coefficient s, 𝐴 and 𝐵, respectively . In the model, t he ratio 𝐴 is proportional to the Debye -Waller factor (DWF) divided by a ZPL linewidth 𝑤. Then t he temperature dependence of the ratio 𝐴 is given by 𝐴=𝛼𝑇−2exp(−𝛾𝑇2), (𝑆1) resulting in the temperature response |𝑑𝐴 𝑑𝑇|=2𝑇(𝑇−2+𝛾)𝐴, (𝑆2) where 𝛼 and 𝛾 are temperature independent con stants which are related to the electron -phonon coupling 𝑆, the Debye temperature 𝑇𝐷 and reference values . We n ote that the DWF is defined as the ratio of the integral ZPL intensity to the total PL. From this expression, the temperature response is expected to be larger at lower temperatures, which potentially give s rise to a higher temperature sensitivity at lower temperatures though it also depends on the uncertainty of the measurement of the ratio 𝐴. The uncertainty 𝜎𝐴 is given by [S2] 𝜎𝐴=𝑓(𝑟)𝐴 √𝐶ZPL 𝛥𝑡(𝑆3) 𝑓(𝑟)=√𝑐1+𝑐2𝑟+𝑐3√𝑟2+𝑟, 𝑤𝑖𝑡ℎ [𝑐1,𝑐2,𝑐3]=[3,3,1], (𝑆4) where 𝑟=𝐵/𝐴, 𝐶ZPL is the ZPL counts rate and 𝛥𝑡 is the measurement time. From the equation (S3) , the temperature sensitivity, or the noise floor, can be written as 𝜂≡𝜎𝐴√𝛥𝑡|𝑑𝑇 𝑑𝐴⁄ |=𝑇𝑓(𝑟) 2(1+𝛾𝑇2 )√𝐶ZPL. (𝑆5) Assuming , for simplicity , that the temperature dependence of the total PL is negligible, we can write 𝐶𝑍𝑃𝐿=𝐶𝑡𝑜𝑡(DWF )|𝑇=0exp(−𝛾𝑇2), (𝑆6) where 𝐶tot is the total PL counts rate and (DWF )|𝑇=0 is the DWF at absolute zero . From equations (S5) and (S6), we get 𝜂=𝑇𝑓(𝑟) 2(1+𝛾𝑇2 )√𝐶tot(DWF )|𝑇=0exp (1 2𝛾𝑇2). (𝑆7) As the temperature decreases , the factor 𝑟=𝐵/𝐴 decreases, which is not shown in the Ref. [S1] but is confirmed in a regime 85 K≤𝑇≤300 K as shown in the F ig. S6(b). Then we get 𝑑𝜂/𝑑𝑇>0, demonstrating a higher sensitivity at lower temperatures , at least in a regime 300 K≲𝑇≲400 K where the model is confirmed . B. Temperature response of the ratio 𝑨 The model described in Sec. A assumes that the temperature response of the ratio 𝐴 is dominated by that of the DWF and the ZPL linewidth. Another possible contribution is the temperature response of the amount of phonon -sideband emission in the range of i nterest (in our case, the spectral range ℛ) with regard to the total PL , though its temperature dependence is negligible as shown in Sec . I. C. Temperature stability of the flow cryostat In the experiment, the base temperature of the sample was stabilized with PID control. Temperature deviation was within ±0.3 K for all measurements. Though the thermocouple was positioned a few centimeters away from the sample position, temperature accuracy within ±0.5 K was ensured in a calibration of the setup which has a thermocouple right next to the sample position. D. Enhancement of the PL at Different Spot s in the Array Figure. 1(c) in the main text shows the enhancement of the PL at 𝑇=170 K with the pulse sequences shown in F ig. 1(d) in the main text. In the Fig. S1, we show the enhancement of the PL at different spots in the array. The figure S1(a) is identical to the Figure. 1(c) in the main text , while figures S1(b) and S1(c) show the PL scans at other spots. Each PL peak was fit by a sum of a Gaussian function and a constant, where the amplitude s of the Gaussian function s were extracted from the fits. The enhancement in the amplitudes due to the pulse sequences was observed, where the enhancement factors were approximately 3.1, 2.4, and 3.4 for F igs. S1(a), S1(b) and S1(c), respectively . Though the factor depends on the spots, enhancement s by approximately a factor of three were observed. The enhancement is due to the charge -state conversion between NV− and NV0. While the 594 -nm excitation preferentially converts NV− into NV0, the 532 -nm excitation preferentially converts NV0 back into NV− [S3–S5]. The time scale of the char ge-state conversion depends on the laser power [S4]. To m inimize the heating while keeping the PL counts high enough in our experiments, the powers of the two lasers were both set to 200 𝜇W, leading to an estimate that the relaxation time of the charge -state conversion is larger than 1 𝜇s. FIG. S1. Enhancement of the PL due to the pulse sequences as shown F ig. 1(d) in the main text under three different spots in the array. (a) shows the same figure as the F ig. 1(c) in the main text. E. Discussion of the practical sensitivity under the pulse sequence Since the sensitivity of the all -optical thermometer is limited by a shot noise, a higher PL count rate by roughly a factor of three results in a higher sensitivity of temperatures by approximately a factor of √3. Note that the pulse sequences also reduce the fraction of the measurement time in the total scanning time. While it improves the physical sensitivity 𝜂 of temperatures, it may result in worse practical sensitivity 𝜂practical if the enhancement factor is less than two. We observed, however , improved sensitivity with the pulse sequences not only because the enhancement is larger than two but also because it reduces the noise due to the CCD dark counts and the background counts . Here we note that physical sensitivity 𝜂 is defined as the minimum temperature difference that can be resolved by a given amount of NV -center -measurement time, while the practical sensitivity 𝜂practical is the minimum temperature difference that can be resolved by a given total time including the time necessary for charge state preparation, background measurement, control of the equipment, and feedback control to focus on the target spot. F. Background Measurement Each spectral measurement was followed by an off -spot background measurement with the same measurement duration. This not only deteriorates the practical sensitivity 𝜂practical , but also adds additional noise to the physical sensitivity 𝜂. The factor is considered in the calculation of the noise model where CCD camera’s dark - current s hot noise also contributes in addition to the ZPL photon shot noise, as explained in Sec. L . G. Choice of the Range 𝓡={𝒉𝝂| 𝒉𝒄(𝟔𝟔𝟎 𝐧𝐦)−𝟏≤𝒉𝝂≤𝒉𝒄(𝟔𝟎𝟓 𝐧𝐦)−𝟏 } The spectral range ℛ is chosen such that it is consistent with previous report s [S1,S6,S7]. With a choice of the grating in our spectrometer (Acton SP -2750, Princeton Instrument , 300 gr/mm with 750 nm blaze ; iStar 334T, Andor ), the range can be measured in the CCD with a single scan . This allowed us to take measurement s without stitching different spectral scans under different angles of the grating in the spectrometer. In contrast, the spectra shown in the F ig. 2(a) in the main text are stitched over multiple scans under different angles of the grating. H. Fitting the reduced Debye -Waller factor and the ZPL linewidth To get a curve for the ratio 𝐴 in the main text , we fit temperature dependencies of a reduced Debye -Waller factor (DWF )ℛ and a ZPL linewidth 𝑤. We fit the temperature dependence of (DWF )ℛ by a Gaussian function (DWF )ℛ=𝛼exp(−𝛾𝑇2), where 𝛼 and 𝛾 are fitting parameters but 𝛾 is related to the electron -phonon coupling 𝑆 and the Debye temperature 𝑇D by a relation 𝛾=2𝜋2𝑆3𝑇D2⁄ [S8]. In our experiment, we measured 𝛾=(218 K)−2 which corresponds to 𝑇D/√𝑆=560 K. The value of 𝛾 was consistent with a measurement conducted on NDs without the PDMS sheet as shown in Sec. I. The temperature dependence of the ZPL linewidth 𝑤 is fit by a second - order polynomial 𝑤=𝑎+𝑏𝑇2. As shown in Figure 2(d) in the main text , the constant contribution 𝑎 is not negligible at lower temperatures in our experiment due to the inhomogeneous broadening. Based on the two fits, we obtained the curve in Fig. 2(b). Based on the model under a simplifying assumption 𝑎≫𝑏𝑇2, one can easily find that the temperature response |𝑑𝐴 𝑑𝑇⁄ | takes maximum at 𝑇≃1√2𝛾 ⁄ =154 K, which is consistent with the exper imental observation. I. Temperature Dependencies of the Parameters without the PDMS Sheet The dependency of the ratio 𝐴 on the temperature shown in F ig. 2(b) in the main text is not largely affected by the presence of the PDMS sheet. To support this statement, we show the PL spectra of NV centers without the PDMS sheet in the F ig. S2 where the spectra were measured at 𝑇=85 K,110 K,150 K,200 K,250 K and 300 K. The measurement was conducted on NDs scattered on a quartz substrate, where hundreds of NDs existed under a laser -focused spot. The same analysis as in the main text is conducted. From the fit of the (DWF )ℛ as shown in F ig. S3(a) we got 𝛾=(283 K)−2, which stays within 25% from the value 𝛾=(218 𝐾)−2 of the NDs embedded in the PDMS array, showing that the presence of the PDMS sheet does not change the main result in this report. When we fit the reduced Debye -Waller factor with a Gaussian function in the main text , there was an implicit assumption that the temperature dependence of (DWF )ℛ is approximately that of (DWF ) since the temperature dependence of the DWF is known to be (DWF )=exp (−𝑆(1+2𝜋2𝑇2 3𝑇D2)). (𝑆8) Figure S3 (a) compares (DWF ) and (DWF )ℛ scanned on NDs without the PDMS sheet. Figure S3(b) shows almost constant ratio (DWF )/(DWF )ℛ over the temperature range 85 K≤𝑇≤300 K, confirming the assumption . FIG. S2. PL spectra and ZPL amplitude ratio of NV centers without PDMS sheet scanned under multiple temperatures 𝑇=85 K,110 K,150 K,200 K,250 K and 300 K. (b) inset shows the spectrum at 𝑇=150 K. FIG. S3. (a) Temperature dependencies of the Debye -Waller factor (DWF ) (left axis) and the reduced Debye -Waller factor (DWF )ℛ (right axis) measured on NDs without the PDMS sheet. (b) Temperature dependence of the ratio (DWF )(DWF )ℛ ⁄ , which equals to the fraction of the integrated PL in the range ℛ to the total PL. J. Temperature Dependence of the ZPL Amplitude Ratio In the fit of the spectrum as shown in the inset of the F ig. 2(b) in the main text, the PL intensity 𝐿(ℎ𝜈) was firstly divided by the mean PL intensity in the range ℛ={ℎ𝜈| ℎ𝑐(660 nm)−1≤ℎ𝜈≤ℎ𝑐(605 nm)−1 }. The mean PL intensity ⟨𝐿⟩ℛ can be explicitly written as ⟨𝐿⟩ℛ=1 𝛥ℛ∫𝐿(𝑛𝜈)ℎ𝑑𝜈𝜈f 𝜈i, (𝑆9) where 𝜈i=𝑐(660 nm)−1, 𝜈f=𝑐(605 nm)−1 and 𝛥ℛ=ℎ(𝜈f−𝜈i). Therefore, the reduced Debye -Waller factor (DWF )ℛ, the ZPL amplitude ratio 𝐴, and the linewidth 𝑤 are related by (DWF )ℛ=𝜋 2𝐴𝑤 𝛥ℛ. (𝑆10) We note that the reduced Debye -Waller factor is defined as the ratio of the integra ted ZPL emission intensity to the total PL in the range ℛ. With the use of the coefficients from the fits of (DWF )ℛ and 𝑤, the ZPL amplitude ratio 𝐴 can be written as 𝐴=2𝛼exp(−𝛾𝑇2)𝛥ℛ 𝜋(𝑎+𝑏𝑇2). (S11) From the equation (S11), we get |𝑑𝐴 𝑑𝑇|=2𝑇(𝑏 𝑎+𝑏𝑇2+𝛾)𝐴. (𝑆12) K. Discussion of the value 𝑻𝐃/√𝑺 From the fit of the reduced Debye -Waller factor in the main text, we obtained 𝑇𝐷/√𝑆=560 K. Though this is relatively small considering the bulk Debye temperature 𝑇Dbulk≃2200 K of diamond, Debye temperatures in nanodiamonds are know n to be around 30% smaller [S6]. Mismatch from the literature value of nanodiamonds 𝑇D/√𝑆|literature=1.0(1)×103 K given in Ref. [S1] would be due to the different ensemble of NVs used in our experiment. While tens of commercial NDs with 100 -nm diameter were used in this study, Ref. [S1] reports meas urements on a single ND with diameter smaller than 50 nm prepared from synthetic sub -micron diamond powder. In the measurement without the PDMS sheet shown in F ig. S3(a), we got similar value 𝛾=(283 K)−2 which corresponds to 𝑇D/√𝑆=725 K. This supports that the smaller value of 𝑇D/√𝑆 measured in our experiment compared to the value in Ref. [S1] is due to the different ensembles of NVs . L. Temperature Dependence of the ZPL Counts Rate There is a subtlety in modeling the temperature dependence of the ZPL counts rate 𝐶ZPL because it is not only determined by the temperature dependence of the NV center’s optical lifetime, but also affected by the temperature dependencies of the steady sta te population of NV− and the PDMS sheet’s optical transparency in our experiment. For simplicity, we conducted a one -parameter ( 𝑎1) fit of the ZPL counts rate 𝐶ZPL(𝑇)=𝑎1(DWF )ℛ(𝑇) where (DWF )ℛ(𝑇) is the curve we got in F ig. 2(c) in the main text. The underlying assumption is that the temperature dependence of 𝐶ZPL is dominated by that of the reduced DWF, which is valid when the integra ted PL intensity in the range ℛ is not significantly temperature dependent compared to th e temperature dependence of (DWF )ℛ. A detailed study of the temperature dependence of the ZPL counts rate is beyond the scope of this report. M. Two Models for the Rescaled Standard Deviation The dotted and dashed curves in the F ig. 3(c) in the main text show lower bounds for the rescaled standard deviation under different models. The dashed curve is the lower bound when the noise is coming only from the photon shot noise of the ZPL and the phonon sideband under the ZPL, while the dotted curve shows the lower bound when the CCD camera’s dark -current shot noise also contributes to the uncertainty 𝜎𝐴. In a model where photon counts under the ZPL peak add noise to the fit of the ZPL, the rescaled standard deviation 𝜎𝐴√𝐶ZPL𝛥𝑡 is given by 𝜎𝐴√𝐶ZPL𝛥𝑡=𝑔(𝑦)𝐴 (𝑆13) 𝑔(𝑦)=√𝑐1+𝑐2𝑦+𝑐3√𝑦2+𝑦, with [𝑐1,𝑐2,𝑐3]≃[2.00,1.98,0.763 ]. (𝑆14) The function 𝑔(𝑦) differs from the case when u sing a Lorentzian function [S2]. The dashed curve is drawn by setting 𝑦=𝐵/𝐴, while the dotted curve is drawn by setting 𝑦=𝐵/𝐴+2𝑐dark/⟨𝐿⟩ℛ̅̅̅̅̅̅𝐴ℎ𝛿𝜈, where the temperature dependence of 𝐵/𝐴 was fit by an exponential function as shown in F ig. S5, 𝛿𝜈 is the frequency range corresponds to one line of vertically binned pixel s in the CCD camera, ⟨𝐿⟩ℛ ̅̅̅̅̅̅≡(1𝑁⁄)∑ ⟨𝐿⟩ℛ(𝑇𝑖) 𝑁 𝑖=1 represents the average of ⟨𝐿⟩ℛ over temperatures 𝑇𝑖={85,90,⋯,300 K},, and 𝑐dark is the counts due to the CCD’s dark current whose average value is cancelled by the background measurement while it adds noise to the spectrum. The dotted curve explains the experimentally observed standard deviation 𝜎𝐴 and the residual would be associa ted with the background counts from the surroundings of NDs such as the PDMS sheet. The noise due to 𝑐dark is non -negligible because the PL is spread over thousands of pixels in the CCD camera in the spectrometer. N. Derivation of the Function 𝒈(𝒚) Applicatio n of the theory given in Ref. [S2] to the case with squared -Lorentzian function gives 𝑔(𝑦)=√𝑓2(𝑦) 𝑓1(𝑦)𝑓2(𝑦)−(𝑓3(𝑦))2√𝜋𝛤(𝛽−1 2) 𝛤(𝛽)(𝑆15) 𝑓1(𝑦)=∫𝑑𝑥∞ −∞((𝑥2+1)𝛽 𝑦(𝑥2+1)𝛽+1)(1 (𝑥2+1)𝛽)2 (𝑆16) 𝑓2(𝑦)=∫𝑑𝑥∞ −∞((𝑥2+1)𝛽 𝑦(𝑥2+1)𝛽+1)(𝑥2 (𝑥2+1)𝛽+1)2 (𝑆17) 𝑓3(𝑦)=∫𝑑𝑥∞ −∞((𝑥2+1)𝛽 𝑦(𝑥2+1)𝛽+1)(𝑥2 (𝑥2+1)𝛽+1), (𝑆18) where 𝛽=2 and 𝛤(𝑥) is the Gamma function. Instead of evaluating them analytically, we computed them numerically and fit the function 𝑔(𝑦) by a form √𝑐1+𝑐2𝑦+𝑐3√𝑦2+𝑦 as shown in the F ig. S4, where {𝑐1,𝑐2,𝑐3} are fitting parameters. The function was well fit by [𝑐1,𝑐2,𝑐3]≃[2.00,1.98,0.763 ]. FIG. S4. Numerical evaluation of the function 𝑔(𝑦) and the fit. Inset shows the residuals. O. Calculation of the projected sensitivity In our experiment, the ZPL counts rates were orders of magnitude smaller than those measured in the former study, where the ZPL counts rate from a single ND was observed to be 𝐶ZPL ,1(295 K)=900 kcps at 𝑇= 295 K [S1], in contrast to our measurement of 𝐶ZPL ,2(295 K)=760 kcps at 𝑇=295 K. High sensitivity all -optical thermometry with 𝜂=300 mK Hz−1/2 was demonstrated with this high ZPL detection rate 𝐶ZPL ,1(295 K). To compare our result with the previous study, we define a projected sensitivity 𝜂proj =√𝐶ZPL ,2(295 K)𝐶ZPL ,1(295 K)⁄ 𝜂 (𝑆19) and it is shown in the right axis of the F ig. 3(d) in the main text. Though the projected sensitivity only gives a rough estimate of a sensitivity given a higher detection efficiency of the PL, it shows our result is consistent with the previous report. P. Temperature Dependence of Sensitivity From the equations (S12 ) and (S13), we get the rescaled sensitivity 𝜂√𝐶ZPL≡𝜎𝐴√𝐶ZPL𝛥𝑡|𝑑𝑇 𝑑𝐴|=𝑔(𝑦) 2𝑇(𝑏 𝑎+𝑏𝑇2+𝛾). (𝑆19) The rescaled sensitivity represents the minimum temperature difference that can be resolved by a single ZPL photon detection. We show the temperature dependence of the rescaled sensitivity in F ig. S5. Two lower bounds due to the models ex plained in the previous section are shown. The equation (S19) gives a low temperature behavior 𝜂√𝐶ZPL∼1/𝑇, which is consistent with the experimental data shown in F ig. S5. FIG. S5. Temperature dependence of the rescaled sensitivity 𝜂√𝐶ZPL. Two curves showing the lower bound due to the two limitations as explained in the F ig. 3 in the main text are shown. The rescaled sensitivity represents the minimum possible temperature difference that can be measured by a single ZPL photon detection. Temperature dependence of the sensitivity can be derived from the equations (S4) and (S19), resulting in 𝜂=𝑔(𝑦) 2𝑇(𝑏 𝑎+𝑏𝑇2+𝛾)√𝐶tot(DWF )|𝑇=0exp (1 2𝛾𝑇2). (𝑆20) Under simplifying approximations 𝑎≫𝑇𝑏2, 𝛾≫𝑏/𝑎 and 𝑔(𝑦)≃𝑔(0)≃√2, we get 𝜂≃1 𝑇𝛾√2𝐶tot(DWF )|𝑇=0exp (1 2𝛾𝑇2), (𝑆21) which gives a minimum at 𝑇=1√𝛾⁄. Q. Discussion of the Effect of the PDMS Sheet on the Sensitivity Measurement Figure. 3(d) in the main text shows non -negligible effects of the PDMS sheet. This is mainly due to the temperature dependence of the absolute ZPL counts rate shown in the inset of F ig.3(c) , which is largely affected by the optical transparency of the PDMS sheet that modifies the PL collection efficiency of our setup. Temperature dependence of the rescaled sensitivity shown in F ig. S5 support s this statement, since there are no observable dips/peaks in the figure. While the inset of F ig. 3(c) and F ig. 3(d) are affected by the existence of the PDMS sheet, the general tendency of these figures are expected to be due to the NV-center’s intrinsic pr operties , since the temperature dependence of the total PL of NV centers below room temperatures are reported to be negligible [S9– S11], leading to the decrease of the ZPL counts rate with temperature increase, due to the Debye -Waller factor . R. Choice of the Base Temperature 𝑻=𝟏𝟕𝟎 𝐊 for the Temperature Imaging of YIG We chose the base temperature of 𝑇=170 K in the measurement of F ig. 4 in the main text . This is because there is a glass transition of PDMS at 𝑇≃150 K. Below the glass transition of PDMS, the proximity of the NDs on the YIG surface is not ensured and the local temperature measurement s become untrustworthy. Above the transition temperature, the NDs are in goo d contact with the YIG surface and they measure the local temperatures of the YIG. Note that the glass transitio n does not affect the main results in other parts of this report since the temperature gradient was not applied. S. Calibration of the temperature sensor For the calibration of the temperature sensors, we conducted multiple scans of the spectrum at 𝑇=170 K and 𝑇=180 K by changing the base temperatures of the copper thermal sink. The average value and the variance of the fitting coefficients {𝐴,𝐵,𝛩,𝑤,𝜈𝑍𝑃𝐿} were extracted. Then we calculated the linear dependence to convert the value of {𝐴,𝐵,𝛩,𝑤,𝜈𝑍𝑃𝐿} into temperatures. The calibration was conducted for each spot in the array. T. Temperature Imaging of YIG As mentioned in the main text, the temperature dependencies of the parameters {𝐵,𝛩,𝑤,𝜈𝑍𝑃𝐿} in addition to 𝐴 were used for temperature sensing by taking the weighted average of the temperatures measured by fitting coefficients . In the Figure S 6, the temperature dependencies of 𝛩,𝐵/𝐴 and 𝜈ZPL are shown, where 𝐵/𝐴 was fit by an exponential function which is empirical but the specific functional form does not matter in this report . The parameter 𝛩 represents the slope of the exponential function in the fit of the phonon sideband. The value is different from the true temperature by a factor of order one, which is called Urbach’s rule and similar dependencies are observ ed in many other materials [S12]. We note that t he temperature dependence of this exponential tail can potentially be used as a temperature sensor below the liquid nitrogen temperatures for future applications. FIG. S 6. Temperature dependencies of (a) 𝛩, (b) 𝐵/𝐴 and (c) 𝜈ZPL. The ratio 𝐵/𝐴 was fit by an exponential function in (b) with a solid (red) curve. Before the temperature measurements, YIG was magnetized to one direction by applying a DC magnetic field. After taking temper ature measurements on multiple spots in the ND array , the temperatures around the scanned spots are smoothly interpolated or extrapo lated and shown in the F ig. 4(b) in the main text . The spots used in the temperature measurement is shown in Fig. S7, where the white circles represent the spots that were used . FIG. S7. Two -dimensional scan of PL and temperatures as shown in the F ig. 4 in the main text with circles representing the spots in the array that were used for the temperature measurement. Temperatures around the scanned spots were smoothly interpolated or extrapolated in (b). U. Discussion of the temperature profile In the F ig. 4(c) in the main text, the YIG surface temperature 𝑇(𝑥) was fit by a logarithmic function 𝑇(𝑥)=−𝜉log(𝑥−𝜁), (𝑆19) where 𝑥 is the distance from the resistive heater and {𝜉,𝜁} are fitting parameters. A logarithmic function is used because the Green’s function to the steady state two -dimensional diffusion equat ion with a single hear carrier is logarithmic. Since the w ire has the length of 200 𝜇𝑚 and the center of the PDMS sheet was displaced to the left by approximately 45 𝜇m due to experimental imperfection , there would be a deviation from the logarithm ic function due to the imperfection of the two -dimensionality , i.e., the resistive heater is not infinitely long . A finite YIG thickness and the existence of an interface between YIG and GGG can also be a potential cause of the deviation from the logarithmic function. The deviation is, however, not clearly observed. In addition, both phonons and magnons are the heat carriers in YIG . According to the coupled magnon - phonon heat transport theory [S13], the steady state phonon temperature profile 𝑇𝑝(𝐱) does not obey a simple Poisson equation, but it obeys (𝜅𝑚𝜅𝑝𝛻4−𝑔(𝜅𝑚+𝜅𝑝)𝛻2)𝑇𝑝=−(𝜅𝑚𝛻2−𝑔)𝑄𝑝+𝑔𝑄𝑚. (𝑆20) Here we ignored , for simplicity, the spatial derivatives of the thermal conductivities, 𝜅𝑚 and 𝜅𝑝. Parameters 𝑄𝑝 and 𝑄𝑚 are the power densities of external heating absorbed by phonons and magnons, respectively [S14]. It is shown in the reference [S14], however, that the equation (S20) can be approximated to the Poisson equation in a regime where the phononic temperature gradient is dominant over the gradient of the magnon -phonon temperature difference . The observed logarithmic behavior of the phononic temperature profile supports this approximation and that the steady -state phononic temperature profile is not largely disturbed by magnons in YIG . For further study of the temperature profile of the YIG film, higher temperature sensitivity is required . References: [S1] T. Plakhotnik, H. Aman, and H. C. Chang, Nanotechnology 26, (2015). [S2] E. A. Donley and T. Plakhotnik, Single Mol. 2, 23 (2001). [S3] X. D. Chen, S. Li, A. Shen, Y. Dong, C. H. Dong, G. C. Guo, and F. W. Sun, Phys. Rev. Appl. 7, 1 (2017). [S4] X. D. Chen, L. M. Zhou, C. L. Zou, C. C. Li, Y. Dong, F. W. Sun, and G. C. Guo, Phys. Rev. B - Condens. Matter Mater. Phys. 92, 1 (2015). [S5] N. Aslam, G. Waldherr, P. Neumann, F. Jelezko, and J. Wrachtrup, New J. Phys. 15, (2013). [S6] T. Plakhotnik, M. W. Doherty, J. H. Cole, R. Chapman, and N. B. Manson, Nano Lett. 14, 4989 (2014). [S7] P. C. Tsai, C. P. Epperla, J. S. Huang, O. Y. Chen, C. C. Wu, and H. C. Chang, Angew. Chemie - Int. Ed. 56, 3025 (2017). [S8] D. B. Fitchen, R. H. Silsbee, T. A. Fulton, and E. L. Wolf, Phys. Rev. Lett. 11, 275 (1963). [S9] A. T. Collins, M. F. Thomaz, M. I. B. Jorge, A. T. Collins, A. T. Collins, A. T. Collins, P. M. Spear, A. T. Collins, A. T. Collins, M. Stanley, A. T. Collins, S. C. Lawson, and J. Walker, 2177 , (1983) . [S10] T. Plakhotnik and D. Gruber, Phys. Chem. Chem. Phys. 12, 9751 (2010). [S11] D. M. Toyli, D. J. Christle, A. Alkauskas, B. B. Buckley, C. G. Van de Walle, and D. D. Awschalom, Phys. Rev. X 2, 1 (2012). [S12] J. D. Dow and D. Redfield, Phys. Rev. B 5, 594 (1972). [S13] M. Schreier, A. Kamra, M. Weiler, J. Xiao, G. E. W. Bauer, R. Gross, and S. T. B. Goennenwein, Phys. Rev. B - Condens. Matter Mater. Phys. 88, 1 (2013). [S14] K. An, K. S. Olsson, A. Weathers, S. Sullivan, X. Chen, X. Li, L. G. Marshall , X. Ma, N. Klimovich, J. Zhou, L. Shi, and X. Li, Phys. Rev. Lett. 117, 1 (2016).

2019-03-05

The nitrogen-vacancy (NV) center in diamond has been recognized as a high-sensitivity nanometer-scale metrology platform. Thermometry has been a recent focus, with attention largely confined to room temperature applications. Thermometry has been a recent focus, with attention largely confined to room temperature applications. Temperature sensing at low temperatures, however, remains challenging as the sensitivity decreases for many commonly used techniques which rely on a temperature dependent frequency shift of the NV centers spin resonance and its control with microwaves. Here we use an alternative approach that does not require microwaves, ratiometric all-optical thermometry, and demonstrate that it may be utilized to liquid nitrogen temperatures without deterioration of the sensitivity. The use of an array of nanodiamonds embedded within a portably polydimethylsiloxane (PDMS) sheet provides a versatile temperature sensing platform that can probe a wide variety of systems without the configurational restrictions needed for applying microwaves. With this device, we observe a temperature gradient over tens of microns in a ferromagnetic-insulator substrate (YIG) under local heating by a resistive heater. This thermometry technique provides a cryogenically compatible, microwave-free, minimally invasive approach capable of probing local temperatures with few restrictions on the substrate materials.

All-optical cryogenic thermometry based on NV centers in nanodiamonds

1903.01605v1

0018-9464 (c) 2020 IEEE. Personal use is permitted, but republication/redistribution requires IEEE permission. See http://www.ieee.org/publications_standards/publications/rights/index.html for more information.This article has been accepted for publication in a future issue of this journal, but has not been fully edited. Content may change prior to final publication. Citation information: DOI 10.1109/TMAG.2020.3021646, IEEE Transactions on Magnetics 1 Origin of Perpendicular Magnetic Anisotropy in Yttrium Iron Garnet Thin Films Grown on Si (100) Zurbiye Capku,1,2 Caner Deger,3 Perihan Aksu,4 Fikret Yildiz 1 1Department of Physics, Gebze Technical University, Gebze, Kocaeli, 41400, Turkey 2Department of Physics, Bo ğaziçi University, Beşiktaş, Istanbul, 34342, Turkey 3Department of Physics, Marmara University, Kadikoy, Istanbul, 34722, Turkey 4Institute of Nanotechnology, Gebze Technical University, Gebze, Kocaeli, 41400, Turkey We report the magnetic properties of yttrium iron garnet (YIG) thin films grown by pulsed laser deposition technique. The films were deposited on Si (100) substrates in the range of 15-50 nm thickness. Magnetic characterizations were investigated by ferromagnetic resonance spectra. Perpendicular magnetic easy axis was achieved up to 50 nm thickness. We observed that the perpendicular anisotropy values decreased by increasing the film thickness. The origin of the perpendicular magnetic anisotropy (PMA) was attributed to the texture and the lattice distortion in the YIG thin films. We anticipate that perpendicularly magnetized YIG thin films on Si substrates pave the way for a cheaper and compatible fabrication process. Index Terms —Ferromagnetic Resonance (FMR); Perpendicular Magnetic Anisotropy (PMA); Yttrium Iron Garnet (YIG). I. INTRODUCTION Magnetic garnet films have recently begun to take the place of conducting ferromagnetic materials in spintronic applications. The insulating features of the garnets eliminates the disadvantages of Eddy currents, which causes loss of information in relevant applications [1]. They have attracted great attention for high frequency and fast switching of magnetic properties [2]. In particular, perpendicular magnetization in garnet films is very crucial for the field of spintronics i.e. spin-orbit switching, spin transfer torque and, a reliable and rapid response [3, 4]. Yttrium iron garnet (YIG) is considered to be one of the most important magnetic insulators. Static and dynamic magnetic properties of bulk crystal or YIG films in the micrometer thickness range have been investigated in great detail and widely used in microwave applications (filtering, tunabling, isolators, phase shifters, etc.) [5, 6]. However, the process of thin/ultrathin YIG films plays a key role for spintronic [7-11] and magneto-optical applications [12- 14]. Many spintronic applications require a fine tuning of the orientation and magnitude of the magnetic anisotropy [15, 16] . Perpendicular magnetic anisotropy (PMA) has led to a revolutionary breakthrough in the technology such as the invention of high-density Magnetoresistive Random-Access Memory devices (MRAM). The effective control over the magnetic anisotropy leads to highly remarkable features such as increased data storage capacity in the magnetic recording media, magnon transistor [17] , and advancement in the logic devices [16]. Despite the fact that enhancement of PMA in metal thin films is a well-established phenomenon [18, 19] , generating PMA in insulating materials such as YIG remains a challenge. For such reasons, ferromagnetic insulators with PMA have been of particular importance for both fundamental scientific research and technological applications. Recent developments in the magnonic field have attracted great attention to ultrathin/thin YIG films perpendicularly magnetized. For example; YIG with the possess of PMA, has a unique feature in spin-orbit torque (SOT) applications [20] . The typical anisotropy in YIG films is in-plane anisotropy (IPA) which mainly originates from the strong shape anisotropy. When the magnetocrystalline anisotropy overcomes the shape anisotropy, the direction of the magnetic easy axis switches to the out of the film plane, resulting in PMA. In the literature, the control on magnetic anisotropy in YIG thin films has been studied by various substrate, temperature, and thicknesses [21, 22]. PMA in YIG films were achieved by using a buffer layer [23] and/or doping with rare earth elements [1, 24, 25]. In these studies, garnet substrates such as Yttrium aluminium garnet (YAG) [26] and Gadolinium gallium garnet (GGG) were used to grow epitaxial YIG thin films due to their similar crystalline structure [22, 27]. Lattice constants of YIG film and GGG substrate are aYIG= 12.376 Å and aGGG= 12.383 Å, respectively [28]. This lattice match between YIG and GGG provides high quality crystallized YIG films [29]. However, the use of the GGG substrate in certain areas is limited and also costs much for large area applications. The use of Silicon (Si) as a substrate has many advantages; cost-effectiveness and widespread use in electronic devices and integrated circuits. Si has an fcc diamond cubic crystal structure with a lattice constant of 5.43 Å. The nearest neighbor distance between two Si atoms is 2.35 Å [30]. On the other hand, YIG has a cubic structure consisting of Y3+ ions in dodecahedral (c) sites, Fe3+ ions in tetrahedral (d) and octahedral (a) sites in polyhedron of oxygen ions [31]. The nearest interionic distance in YIG is reported as (Y3+ - O2-) at 2.37 Å [31]. The atomic distances are comparable; thus, one can achieve crystalline / texture YIG on Si (100). In this study, we report the PMA enhancement in YIG films grown on Si substrates by pulsed laser deposition (PLD) technique. Several parameters such as oxygen pressure, substrate temperature, post-annealing treatment, and laser power play an important role in the stoichiometry and Authorized licensed use limited to: University of Edinburgh. Downloaded on September 05,2020 at 02:30:35 UTC from IEEE Xplore. Restrictions apply. 0018-9464 (c) 2020 IEEE. Personal use is permitted, but republication/redistribution requires IEEE permission. See http://www.ieee.org/publications_standards/publications/rights/index.html for more information.This article has been accepted for publication in a future issue of this journal, but has not been fully edited. Content may change prior to final publication. Citation information: DOI 10.1109/TMAG.2020.3021646, IEEE Transactions on Magnetics 2 crystallinity of the YIG films fabricated by PLD. In this report, YIG films with different thicknesses were grown on Si (100) substrates. A post-annealing process was carried out for all films to improve the crystallization and substrate-film lattice mismatch. The effect of the thickness on the magnetic anisotropy values was studied. In some reports, PMA in YIG films was obtained in the thickness range of 10-20 nm [22, 23, 32]. However, in this study, the lattice distortion/texture in YIG films gave rise to PMA in 15-50 nm thickness. We anticipate that our study not only offers a basis for fundamental understanding but also will inspire the integration of perpendicularly magnetized YIG thin films with technological applications. II. EXPERIMENTAL STUDIES Before the thin film deposition, the oxide layer was first etched from the surface of Si (100) substrates with diluted Hydrofluoric (HF) acid for a few minutes. The substrates were further cleaned in acetone, methanol and Isopropyl alcohol for 15 minutes by using an ultrasonic bath. Subsequently, the surfaces were spray dried with Nitrogen gas. Following the chemical cleaning, the substrates were introduced into high vacuum chamber and annealed at 500 oC for an hour. PLD with a KrF excimer laser, a Coherent COMPex Pro 205F operating at λ = 248 nm (20 ns pulse duration) was used to obtain the desired YIG film stoichiometry by adjusting the oxygen pressure and deposition temperature. The base pressure of the deposition chamber was 1.0 x 10-9 mbar. The commercial polycrystalline sintered YIG was used as the deposition target. The distance between the target and substrate was about 60 mm. The films were fabricated using laser energy of 220 mJ at a pulse repetition rate of 10 Hz in an oxygen atmosphere of 1.0 x 10-5 mbar. The substrate temperature was 400 °C during growth. The deposition rate was 0.96 nm/min. The films were cooled within a rate of 9.6 oC/min inside the chamber. Thereafter, the films were annealed at 850 °C for 2 hours in an air atmosphere and cooled down to room temperature by a ratio of 1.2 oC/min. The thicknesses of the annealed films were defined as 15 nm, 20 nm, 35 nm and 50 nm using X-ray reflectivity (XRR) method. Ⅲ. RESULTS Atomic Force Microscopy (AFM) was performed for the surface morphology and roughness of the films. A representative AFM image of the annealed YIG film at 850 o C with a root mean square (RMS) roughness value of 0.8 nm was given at the inset of Fig.1. Structural properties of the films were characterized by X-ray Diffraction (XRD) measurement using a Rigaku 2000 DMAX diffractometer with a Cu (alpha) wavelength of 1.54 nm. The θ -2θ scan XRD pattern was demonstrated in Fig.1. A typical (420) peak of YIG was observed for the annealed films. At each measurement, a signal from the sample plate was detected at 44o. The additional peaks around the (400) plane of the Si substrate correspond to the Kβ, Lα1, Lα2, Kα1 and Kα2 li nes of the incident x-ray. Fig. 1. XRD pattern of YIG thin film on a Si substrate. θ -2θ scan which shows the (420) characteristic peak of 20 nm YIG. (Inset: AFM image of a 20 nm YIG film.) The chemical analysis was performed by X-ray photoelectron spectroscopy (XPS) measurement. The survey scan XPS spectrum is represented in Fig. 2(a). The spectrum confirmed the presence of Y, Fe and O elements on the surface of YIG film grown on Si. The fittings of the spectral ranges related to the elements which are used to determine the composition ratio are given in Figs. 2(b)-2(d). XPS spectrum of the Fe 2p region in Fig. 2(c) shows the valance state of the Fe ions. Y/Fe compositional ratio was found to be 0.59 and Fe/O ratio was 0.44. These values are very close to the bulk YIG compositional ratios within the experimental error (Y/Fe = 0.6 and Fe/O = 0.42 in bulk YIG) [11]. Fig. 2. XPS spectrum of YIG thin film grown on Si substrate. (a) XPS survey scan. Fitted XPS spectrum of (b) O 1s. (c) Y 3p. (d) Fe 2p. Authorized licensed use limited to: University of Edinburgh. Downloaded on September 05,2020 at 02:30:35 UTC from IEEE Xplore. Restrictions apply. 0018-9464 (c) 2020 IEEE. Personal use is permitted, but republication/redistribution requires IEEE permission. See http://www.ieee.org/publications_standards/publications/rights/index.html for more information.This article has been accepted for publication in a future issue of this journal, but has not been fully edited. Content may change prior to final publication. Citation information: DOI 10.1109/TMAG.2020.3021646, IEEE Transactions on Magnetics 3 Ferromagnetic resonance (FMR) measurements were performed to the annealed YIG films by an X-Band (9.1 GHz) JEOL series ESR spectrometer at room temperature. FMR is a powerful technique, which the analysis of the spectra also provides the values of anisotropy constants [33- 35]. The resonance profıle is determined by the field (H) derivative of the absorbed RF power (P) dP/dH curve as a function of the applied magnetic field. The sample dimension for the FMR measurement was around 3 mm × 3 mm. The FMR spectra were registered by sweeping the applied field angle around the sample plane (SP) and sample normal (SN). In SP measurement, the magnetic component of the microwave field is al ways in the film plane, whereas the external magnetic field is rotated from the film plane towards the film normal. In SN measurement, the magnetic component of the microwave field is perpendicular to the film plane and the external magnetic field is rotat ed in the film plane for each spectrum. Representative FMR spectra of the YIG films in SP configuration were given in Fig. 3(c) for the applied field direction along with the film normal and in the film plane (Figs. 3(a) and 3(b)). When the applied field was parallel to the film normal, the spectrum was at low field (red spectra), whereas it shifted to higher field (black spectra) when the applied field was parallel to the film plane, for all samples. This behavior refers that the easy axis of the magnetic anisotropy is perpendicular to the film plane. In SN geometry measurements, there was no any anisotropic behavior, which is not surprising. Thin films having PMA do not represent any anisotropic behavior in the film plane [18, 23, 36] . Further analysis on intrinsic magnetic properties of the system is performed by angular FMR measurements and numerical calculations. To reveal the micromagnetic parameters of YIG/Si (100) structure, the energy Hamiltonian presented in Eq.1 is employed and numerically solved. (1) The Hamiltonian consists of two energy terms used to represent the magnetic behavior of the systems [33, 37]. Here, (θ, θH) and (φ, φH) are, respectively, the polar and azimuth angles for magnetization vector M and external DC magnetic field vector H with respect to the film plane. External DC magnetic field is represented by the first term of the Hamiltonian, i.e., Zeeman energy. Effective magnetic anisotropy energy consists of the demagnetization energy, the interface energy and the first-order term of magnetocrystalline energy of the system. And, the last term represents the second- order magnetocrystalline energy. In Eq. (1), Meff, Keff, and Keff_q are the effective magnetization, effective magnetic anisotropy energy density, second order term of magnetocrystalline energy density, respectively. We scan the DC magnetic field from 0 to 1 T to determine the field corresponding to the maximum value of the dynamic susceptibility, which is called as the resonance field (Hres). Dynamic susceptibility spectra are recorded by using the Soohoo formulation for ferromagnetic resonance in multilayer thin films [38- 40]. Fig. 3. FMR spectra of the YIG films in SP measurement geometry. (a) The applied field direction is along the film normal (H // [001] of Si substrate) and (b) along the film plane (H // [100] of Si substrate). (c) The black and red lines indicate the FMR spectrum when the magnetic field is parallel (H // [100]) and perpendicular (H // [001]) to the film plane. The resonance fields are extracted from the recorded spectrum for SP geometry. By performing the aforementione d procedure for different angles of the magnetic field with respect to the film plane, we are able to reproduce the experimental data. All calculations were performed at room temperature. Meff, Keff, and Keff_q were obtained by the simulation model for all samples. Here, the total energy was minimized with the values of the magnetic parameters given in Table Ⅰ. The angular dependence of the resonance field for different thicknesses in SP geometry is shown in Fig. 4. Table Ⅰ represents the result of the nu merical calculations. The positive effective anisotropy energy density confirms that the easy axis is perpendicular to the film plane. The effective magnetization is lower than the bulk YIG value, which may be Authorized licensed use limited to: University of Edinburgh. Downloaded on September 05,2020 at 02:30:35 UTC from IEEE Xplore. Restrictions apply. 0018-9464 (c) 2020 IEEE. Personal use is permitted, but republication/redistribution requires IEEE permission. See http://www.ieee.org/publications_standards/publications/rights/index.html for more information.This article has been accepted for publication in a future issue of this journal, but has not been fully edited. Content may change prior to final publication. Citation information: DOI 10.1109/TMAG.2020.3021646, IEEE Transactions on Magnetics 4 caused by possible crystal vacancies / deficiencies, inter diffusion between the substrate and the film, and Fe ion variation in the film [10]. In general, shape anisotropy is in the film plane . The increase in thickness strengthens the contribution of shape anisotropy to in -plane magnetic anisotropy, while the strain between th e substrate -film tends to relax and, therefore, t he effective perpendicular magnetic anisotropy reduces by increasing thickness as seen in Table Ⅰ. Fig. 4. A plot of angular variation of FMR resonance fields. Symbols and solid lines indicate the experimental and theoretical results; respectively. Table Ⅰ. Magnetic parameters obtained from the simulation for YIG thin films wit h PMA. Thickness(nm) Keff (J/m3) Keff_q(J/m3) Meff (kA/m) 15 nm 1773 320 105 20 nm 1456 250 105 35 nm 1260 150 105 50 nm 1180 120 105 Ⅳ. DISCUSSIONS In this section, the structural and magnetic characterization of YIG films grown on Si (100) will be discussed. The crystallization of the films was analyzed by XRD measurements. Since we did not get the characteristic XRD peaks in as-grown films, an annealing process was required to generate the YIG phase [28]. After annealing at the temperature of 850 °C for 2 hours, we were able to observe (420) peak of the YIG from θ -2θ scan of the XRD measurement as seen in Fig. 1, which indicates the formation of the YIG phase. Some studies report the polycrystalline YIG film grown on quartz with three characteristic peaks [27, 32] . However, it seems that there is a preferentia l crystalline ordering or texturing in our films. When the film was annealed, the lattice of YIG locates on Si by making an angle of 26.6o between (400) plane of Si and (420) plane. There are two different crystallographic orientations / domain of the film lattice repeating each other on the substrate with respect to the symmetry axis of c, as shown in Fig. 5. The lattice mismatch is , where “a” is the lattice constant. Since the lattice constant of YIG is larger than that of the substrate, a small compressive strain occurred at the interface, which can lead to a tetragonal distortion of the crystalline structure of the film. Fig. 5. A two-dimensional configuration of the lattice orientation of the film on Si substrate. Two preferential crystalline orderings/texturing were present in YIG after the annealing procedure. The composition and electronic state of the Y- Fe-O elements on the surface of the YIG film were investigated by XPS analysis. The stoichiometry of the YIG film was determined by the percentage of O 1s, Fe 2p and Y 3p XPS peak areas shown in Figs. 2(b)-2(d) using relative sensitivity factors in CasaXPS software. The stoichiometry of our samples was found to be Y: 3.06, Fe: 5.17, O: 11.7 which is close to the expected one for Y:Fe:O of 3:5:12. The Fe percentage in our YIG film stoichiometry is 20.6 % which is similar to the ratio of 20% in bulk YIG as known from the literature [11]. Fig. 2d shows the core level Fe 2p spectra. Both Fe3+ and Fe2+ are present in the films [41]. 711.1 eV and 724.4 eV are the binding energy values for the 2p3/2 and 2p1/2 peaks of Fe3+and Fe2+. Two peaks at 710.9 eV and 725.8 eV correspond to Fe3+ 2p3/2 and Fe3+ 2p1/2, and the binding energies at 708.86 eV and 724.16 eV refer to Fe2+ 2p3/2 and Fe2+ 2p1/2, respectively. The satellite structure of Fe 2p 3/2 was located at 718.8 eV, binding energy higher than 710.9 eV. This shows that the Fe ions are in +3 valance states in the spectrum and located at tetrahedral sites of YIG lattice [28, 42, 43]. Representative FMR spectra of the samples are given in Fig. 3 for the external magnetic field parallel to the film normal and film plane. Resonance field is determined by taking the minimum value when applied along the easy axis. Authorized licensed use limited to: University of Edinburgh. Downloaded on September 05,2020 at 02:30:35 UTC from IEEE Xplore. Restrictions apply. 0018-9464 (c) 2020 IEEE. Personal use is permitted, but republication/redistribution requires IEEE permission. See http://www.ieee.org/publications_standards/publications/rights/index.html for more information.This article has been accepted for publication in a future issue of this journal, but has not been fully edited. Content may change prior to final publication. Citation information: DOI 10.1109/TMAG.2020.3021646, IEEE Transactions on Magnetics 5 The spectra clearly show PMA for all thicknesses. The FMR spectrum for H // [001] orientation of Si shifts towards higher field values as the thickness increases. In contrast, for H // [100] orientation of Si, the FMR spectrum shifts to lower values while increasing the thickness. This behavior indicates that the uniaxial perpendicular magnetic anisotropy decreases as the thickness increases. Magnetic properties of the films can be affected by many factors such as thickness, substrate, interfacial energy, and strain. The strain is compressive by 1.9% and tensile by 0.65% when the film is grown on Si (100) and GGG, respectively. In our case, the strain in the YIG films grown on the Si substrate is much greater than that grown on GGG. Due to the lattice mismatch between Si and YIG, available thickness with PMA, the value of magnetic anisotropies and FMR linewidth were different from studies using lattice-compatible substrates in the YIG thin film fabrication process. In this study, all the YIG thin films had a single uniform ferromagnetic resonance peak. In Fig.3, t he shape and intensity of the FMR spectra vary depending on the thickness, crystalline quality, and magnetic homogeneity of films. Inhomogeneous broadening of the linewidth of the FMR spectra is due to the imperfections in film such as defects, roughness, symmetry breaking in surface and interface, oxygen vacancies, and inter ion diffusion between the layers [27]. The thinnest film has a wider and lower intensity FMR profile while the spectra gets clearer with the increase in thickness. Meanwhile, surface and interfacial strain effects show tendency to decrease and the increase in the amount of spins which interact with microwave field increases the FMR intensity. The reason for the FMR shape and intensity which do not vary in a systematic manner might be due to some uncontrollable parameters during deposition. However, it is observed that the out -of-plane magnetic anisotropy behavior of the films still exist in the pronounced thicknesses. The FMR linewidth was determined as the distance between the minimum and maximum point of the dP/dH curve, so called peak to peak lin ewidth. 20 nm YIG film has a linewidth of 230 Oe when the applied field is parallel to the sample plane and linewidth of 160 Oe when the applied field is perpendicular to the sample plane. The linewidths of the spectra are relatively larger compared to the reported value on GGG substrate [44]. However, there are similar linewidth values of that grown on quartz in the literature, as well [22, 27] . For example, 12 nm YIG film was reported to have an FMR linewidth of 250 Oe. For the film thickness range between 100 nm and 290 nm, linewidth values were between 340 Oe and 70 Oe. It is thought that defects due to the surface roughness and Fe iron deficiency may lead to magnon scatterings and increase of the FMR linewidth [11]. It is known that the magnetic easy axis in most of thin films are in the film plane due to the shape/dipolar anisotropy. Additional factor is necessary to overcome the shape anisotropy and switch the orientation of the easy axis from the film plane to the film normal. The crystalline or surface anisotropy or textured structure can trigger a perpendicular magnetic anisotropy [36]. Here, the lattice mismatch between Si and YIG thin film induced a compressive strain at the interface which led to a distortion of the lattice structure [4] . The compressive strain in the film plane results in an expansion along the c-axis, which switches the easy axis from the film plane to the film normal [36, 45]. In previous studies, PMA was realized in YIG films grown on different substrates in the thickness range of 10-20 nm [23, 32]. However, we achieved PMA up to 50 nm thickness as a result of texture and the lattice distortion of YIG. In the literature, PMA in YIG films was observed in those grown on the buffer layer except GGG [23, 46]. This study indicates that PMA was attained successfully in YIG films on a non-garnet substrate without using any additional buffer layer or doping. Ⅴ. CONCLUSION YIG thin films with perpendicular magnetic anisotropy can pave the way for cutting-edge magnonic and spin-related technologies, i.e. for the fast response in microwave devices, logic devices, spin-transfer torque and magneto-optical device applications. Existence of the PMA in an insulator material is a rare magnetic phenomenon. In this work, we have achieved perpendicular magnetization in YIG thin films grown on Si substrate which is a common and base material of the present- day electronic industry. The effect of post annealing- temperature on the crystal structure and magnetic anisotropy was explored. XRD analysis revealed that the crystallization of YIG films improved after annealing. The compressive strain due to the lattice mismatch between Si and YIG led to a distortion in the YIG films, resulting in PMA in the thickness range of 15-50 nm. As far as our best knowledge, we report PMA in pure YIG thin films grown on Si substrate for the first time. We anticipate that perpendicular magnetized YIG thin films will allow the YIG magnetic insulator to be widely used in many areas. ACKNOWLEDGEMENTS The authors are grateful to Dr. Ilhan Yavuz for his fruitful discussions on the results. REFERENCES [1] H. Wang, C. Du, P. C. Ham mel, and F. 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2021-07-12

We report the magnetic properties of yttrium iron garnet (YIG) thin films grown by pulsed laser deposition technique. The films were deposited on Si (100) substrates in the range of 15-50 nm thickness. Magnetic characterizations were investigated by ferromagnetic resonance spectra. Perpendicular magnetic easy axis was achieved up to 50 nm thickness. We observed that the perpendicular anisotropy values decreased by increasing the film thickness. The origin of the perpendicular magnetic anisotropy (PMA) was attributed to the texture and the lattice distortion in the YIG thin films. We anticipate that perpendicularly magnetized YIG thin films on Si substrates pave the way for a cheaper and compatible fabrication process.

Origin of Perpendicular Magnetic Anisotropy in Yttrium Iron Garnet Thin Films Grown on Si (100)

2107.05591v1

Spin wave based tunable switch between superconducting flux qubits Shaojie Yuan1*, Chuanpu Liu2*, Jilei Chen2*,Song Liu1, Jin Lan5, Haiming Yu2†, Jiansheng Wu†1, Fei Yan1, Man-Hong Yung1, Jiang Xiao3†, Liang Jiang4†, Dapeng Yu1† 1Institute for Quantum Science and Engineering, and Department of Physics, Southern University of Science and Technology, Shenzhen 518055, China 2Fert Beijing Research Institute, School of Electronic and Information Engineering, BDBC, Beihang University, 100191 Beijing, China 3Department of Physics and State Key Laboratory of Surface Physics, Fudan University, Shanghai 200433, China 4Pritzker School of Molecular Engineering, The University of Chicago, Chicago, Illinois 60637, USA 5Center of Joint Quantum Studies and Department of Physics, School of Science, Tianjin University, Tianjin 300350, China * Equally contributed authors. † Corresponding authors Quantum computing hardware has received world-wide attention and made considerable progress recently. YIG thin film have spin wave (magnon) modes with low dissipation and reliable control for quantum information processing. However, the coherent coupling between a quantum device and YIG thin film has yet been demonstrated. Here, we propose a scheme to achieve strong coupling between superconducting (SC) flux qubits and magnon modes in YIG thin film. Unlike the direct √𝑵 enhancement factor in coupling to the Kittel mode or other spin ensembles, with N the total number of spins, an additional spatial-dependent phase factor needs to be considered when the qubits are magnetically coupled with the magnon modes of finite-wavelength. To avoid undesirable cancelation of coupling caused by the symmetrical boundary condition, a CoFeB thin layer is added to one side of the YIG thin film to break the symmetry. Our numerical simulation demonstrates avoided crossing and coherent transfer of quantum information between the flux qubit and the standing spin waves in YIG thin films. We show that the YIG thin film can be used as a tunable switch between two flux qubits, which have modified shape with small direct inductive coupling between them. Our results manifest that it is possible to couple flux qubits while suppressing undesirable cross-talk. Quantum computing and simulation based on superconducting qubits have achieved significant progress in recent years (1-3). Many efforts were devoted to hybridizing the solid-state qubits with other physical systems, such as mechanical or magnetic systems (4-9). For instance, the Kittel mode of a macroscopic YIG sphere was coherently coupled to a transmon qubit in a 3D cavity with the microwave photons manipulated inside the cavity (8). Besides, the superconducting flux qubit was successfully hybridized with spin ensembles, i.e., nitrogen-vacancy (NV) centers in diamond via magnetic interaction (4-6). On the other hand, because of the zero Joule heating, the wave nature with microwave working frequency, spin wave (whose quanta is called magnon) has become a promising candidate for conventional information transmission and processing and acquired the potential to establish a spin-wave based computing technology, far beyond its CMOS counterpart (10-16). Due to its favorably low damping, ferrimagnetic insulator yttrium iron garnet (YIG) is particularly promising for these applications (17-19). In this work, we propose a novel hybrid system, consisting of superconducting flux qubits and the standing spin waves (20) in ferrimagnetic YIG thin film. The latter system has been widely used in spintronics and magnonics (17-19), while, its magnetic coupling to superconducting qubits and the corresponding application in quantum information processing has not been extensively investigated. As shown in the following, unlike the coupling to spin ensembles or Kittel mode of spin waves (4, 5, 7), the enhancement factor for the coupling strength does not follow the √𝑁 law, but carries a modulation associated with the finite spin-wave wavelength. In our proposal, an additional thin pinning layer of CoFeB is deposited on one side of the YIG thin film to break the symmetry at the boundary conditions (21-30). Avoided crossing of the energy spectrum can be numerically simulated by solving Heisenberg equation based on the full Hamiltonian of the flux qubit, the spin waves in the YIG thin film and their coupling. We find that it is possible to transfer quantum information coherently between the flux qubit and the spin wave mode in the YIG thin film. Moreover, we propose an experimentally feasible design to switch “on” and “off” the coupling between two shape-modified flux qubits or to entangle them via the perpendicular standing spin waves (PSSWS) of the YIG thin film. Hybridizing one flux qubit with and further “tuning” the inductive coupling, which causes cross-talk (31), between multiple flux qubits or entangling them through PSSWs highlights the application of spin wave bus in quantum computing, further expanding the application of spin wave-based computation technology (32-35). A superconducting (SC) loop with three Josephson junctions compose of a flux qubit with the superposition of the clockwise and counter clockwise persistent currents state as the qubit ground state: |g>=|↺>−|↻> and first excited state, |e>=|↺>+|↻> (36, 37), respectively. The net currents and the resulting the magnetic field threading the loop for the |g> and |e> states are distinct. Consequently, the Rabi oscillation between the two states of the flux qubit generates an alternating magnetic field perpendicular to the SC loop, which can be used to excite spin waves in YIG system. The basic setup of the hybrid system is shown schematically in Fig. 1, which consists of a 5x 5 µm! superconducting loop and a 3x 0.08x 3 µm" YIG thin film above. A much thinner CoFeB capping layer ~ 10 nm in thickness is deposited on the top side of the YIG thin film to pin the magnetization in YIG at the interface. The magnetization follows Dirichlet boundary condition at the pinned surface, and Neumann boundary condition at the other free surface (21-27). The resonant frequencies of the perpendicular standing spin wave (PSSW) modes are (20), 𝑓#$$%=&'!!(23𝐻)*++!,"#'!-$5.(/6!73𝐻)*++!,"#'!-$5.(/6!+𝑀07 (1) with gyromagnetic ratio &!(=28 GHz/T, vacuum permeability 𝜇1=1.256 ∗1023 𝑁/𝐴!, saturation magnetization 𝑀$=192 kA/m for YIG, thicknessδ=80 nm, the exchange constant 𝐴)*=3.1 pJ/m, external field 𝐻456 and mode number 𝑛=1,2,3,…, . Values of 𝑀$ and 𝐴45 are obtained by fitting the resonance of a 295 nm YIG thin film from Ref (19) using equation (1) with mode number n=1,2,3,4,5,6. Experiment has measured the resonance value for the PSSW mode of 80 nm YIG thin film at near zero external field to be 4.57 GHz, which is different from theoretical prediction 3.39 GHz. The discrepancy may be due to choosing of order parameter to be integer for unsymmetrical pinning in the fitting process, as actually there are ¾ wavelength in thickness direction for n=1 as illustrated in Fig. 1 b. For our quantum control schemes, we will use the experimental resonance values and design a flux qubit with transition frequency close to 𝑓#$$%(.89) and sufficiently detuned from the CoFeB resonance. Using the geometric confinement, the proper boundary conditions and the suitable coupling strength (see Eq. 4 with later discussion), the PSSW of wavelength of λ=;"δ=;* =1" nm can be excited. An external field of 10 Gauss is applied to align spins in YIG and the field created by the YIG thin film and the CoFeB capping layer on the flux qubit is of the same order (see Fig 1. c), assuming the spin density 𝑛>?@4A=1.61x 10!B 𝑚2" for CoFeB and 𝑛CDE=2.14x10!= 𝑚2" for YIG. The distance between the flux qubit and the YIG thin film is chosen to be around 1-1.5 µm for later simulation in Fig.3. At these distances, the total magnetic field on the qubit is between 21.5 to 37 Gauss, which is less than the critical field of the aluminum superconductor (around 100 Gauss) and guarantees superconductivity of the flux qubit. In addition, other superconducting material such as Niobium can be used to fabricate the loop and junctions of the flux qubit, which has a much higher critical magnetic field for superconductivity, i.e., above 1000 Gauss. From Ref (19), the decay rates for YIG thin film and CoFeB pinning layer are estimated as ΓFGH,J8!~40 MHz and ΓKLM)N~300 MHz, where n is the PSSW mode number. Since the decay rate is proportional to the frequency and the frequency is approximately proportional to the square of mode number, intrinsic decay rate for n=1 PSSW mode is ~ 10 MHz. The resonance frequency for n=1 PSSW mode in YIG thin film and CoFeB pinning layer are 𝑓CDE~4.6 GHz and𝑓>?@4A~1.35 GHz and the exchange coupling strength is 𝑔KLM)N,FGH~500 MHz, which makes converted decay rate of CoFeB on n=1 PSSW mode as Γ>?@4A→CDE,.89=(P%&'()*+,-Q+,-2Q%&'())!∗300~7 MHz and total decay rate for n=1 PSSW mode being 17 to 20 MHz. In our proposal, we replace the microwave antenna in Ref (19) a flux qubit loop, which has a lower decay rate and a much smaller inductive coupling with the sample, and we expect the magnon decay rates will be further reduced. Therefore, it is reasonable to assume that the decay rate for n=1 PSSW mode is about 20-30 MHz. Fig. 1 Hybrid structure coupling a flux qubit and a YIG thin film (spin wave, or a magnon). (a) A YIG thin film with the dimension of 3 x 0.08 x 3 µm" is placed in the center of the 5 x 5 µm! flux qubit loop separated by a distance d. An external field of 10 Gauss is applied along the x axis to align the spins in the YIG thin film. The thickness of YIG thin film is δFGH=80 nm. A perpendicular standing spin wave (n = 1) with wavelength λ=;"δ is excited. The frequency of the flux qubit is chosen to match that of the spin wave, which is experimentally around 4.7 GHz. The alternating magnetic field in the flux qubit loop excites the spin wave in the YIG thin film. (b) A cartoon depicts the PSSW of mode number n=0,1,2,3 with bottom spins being pinned and top spins unpinned. In our proposal, n=1 mode is selected. (c) The magnetic field on the flux qubit created by the YIG thin film and CoFeB thin layer. The spins in both YIG and CoFeB are fully aligned along x. In the following, we consider the coupling strength between the flux qubit and YIG thin film. Hamiltonian for a ferromagnetic or ferrimagnetic material in a magnetic field takes the following general form (38) 𝐻o=−∑𝑆r𝐦𝐦,𝐧.𝐽𝐦,𝐧.𝑆r𝐧+ ∑𝑆r𝐦𝐦.𝐵 (2) with coupling matrix J𝐦,𝐧 between the spins, with the assumptions that deviations from the group state are small, we can perform the Holstein–Primakoff approximation and transforming into the momentum space, we obtain 𝐻o=−𝐽𝑁𝑆!+𝑁𝑆𝜇N𝐵T+∑∑(ℏ𝜔U."V89+𝜇N.𝐵T)𝑎𝐤†𝑎𝐤A.Y.𝐤+∑𝑎𝐤†A.Y.𝐤∑(2𝑆)/0𝜇Nx9√[𝑒\𝐤.𝐦5A#]\A1!6z+𝑎𝐤∑(2𝑆)/0𝜇Nx9√[𝑒2\𝐤.𝐦5A#2\A1!6z𝐦𝐦 (3) Where i=1,2,3, ℏωU.=2JS(1−coskV)=4JSsin!5U.!6, 𝑆 is total spin at each lattice site, 𝑘⃗ is the wavevector of the spin wave and 𝑁 is the number of lattice sites in each direction. From Equation (3), by replacing the summation over each site with integration over space and insert the spin density, the integral form of the coupling strength between the flux qubit and YIG thin film is obtained as following: 𝑔4QQ^_⃗~(!a)/0∫c').42344⃗.744⃗d)892):0ef;g(∫cf;g)/0 (4) where B(x,y,z) is the microwave excitation field created by the flux qubit and 𝜌 is the spin density in YIG. Unlike the simple √𝑁 enhancement associated with coupling to Kittel mode, there is an extra spatial-dependent phase factor 𝑒\^_⃗.g⃗ in Eq. (4). For long wavelength spin wave, |𝒌|≪1 𝜇𝑚29 and 𝑒\𝐤.𝐫 ~1and if 𝐵* or 𝐵i only vary slowly compared to 9|𝒌| in real space, 𝑔4QQ𝐤 will be proportional to √𝑁. However, for the short wavelength spin wave, 𝑔4QQ𝐤 is not necessarily proportional to √𝑁 and can even be zero if the integration region covers exactly integer times of the wavelength along wavevector direction. This is also the reason, to excite PSSW mode in YIG thin film by an almost homogenous field, an asymmetric boundary condition is required to avoid zero coupling strength caused by the phase factor. Given the dimension of the flux qubit square loop 5x 5 µm! and the persistent current I ~ 500 nA (39, 40), the magnetic field produced by the flux qubit can be evaluated using Ampere’s law 𝐵(r)='!;(∮𝐼fl____⃗×gn___⃗|gn___⃗|; and 𝐵o dominates while, 𝐵5, 𝐵p is close to zero in Fig .1. Given a net spin density 𝜌=2.14x 10!= 𝑚2" in YIG, we obtain the absolute value 𝑔4QQ^ as a function of the separation distance d between of the coupling strength between the flux qubits and the YIG thin film as in Fig. 2: Fig. 2, Coupling strength |𝒈𝐞𝐟𝐟𝐤| as a function of the separating spacing d. For small distance (d<2 µm), |𝒈𝒆𝒇𝒇𝒌| for 𝑘⃗o=!(u , where y is the direction in Fig. 1, decreases slowly with the distance and is above 30 MHz, which is larger than decay rate of magnon in YIG thin film. For large d, |𝒈𝐞𝐟𝐟𝐤| decreases as 𝑑2" indicated by the red curve. With the coupling strength estimated above, the full Hamiltonian with the flux qubit and YIG thin film can be written as 𝐻o=−𝐽𝑁𝑆!+𝑁𝑆𝜇N𝐵T+∑(ℏ𝜔𝐤+𝜇A.𝐵)𝑎𝐤]𝑎𝐤A.Y.𝐤+v!(∆𝜎5+𝜀𝜎p)−ℎ𝑔4QQU𝑎𝐤†+𝑔4QQ^∗𝑎𝐤𝜎p+ℎ𝜆cos𝜔𝑡.𝜎p (5) where the first and second terms are the ferromagnetic and Zeeman terms, the third term describes the spin wave excitation, the fourth term is the flux qubit with ∆ the tunneling energy splitting and 𝜀 being the energy bias between the two qubit states, the fifth term characterizes interaction between the two devices, and the last term is the external driving of the flux qubit. Here, 𝜎5,p are the Pauli matrices. The first two terms can be neglected for the reason that spin wave energy is a small perturbation compared to these two energies. By changing the basis of the flux-qubit, neglecting the Zeeman splitting and performing the rotating wave approximation, the Hamiltonian becomes, 𝐻o=v!(√∆!+𝜀!−𝜔)𝜎p+(ℏ𝜔^+𝜇A.𝐵)𝑎^]𝑎^+ℎ∆√∆0]y0𝑔4QQ^𝑎^]𝜎2+𝑔4QQ^∗𝑎^𝜎]+ℎ∆√∆0]y0u!(𝜎]+𝜎2) (6) Where 𝜎],𝜎2are rasing and lowering operator. 𝜎p=2∗(𝜎]𝜎2−9!) . Approximating the flux qubit as a harmonic oscillator and let 𝜎]→𝑐̂] and 𝜎2→𝑐̂, the Hamiltonian can be written in a different form. Employing the Heisenberg relation fẑf6=[𝑐̂,𝐻], sloving in Fourier space and transforming back to the lab frame, we obtain simulation of the energy spectrum 𝜎2,|~ 9|2√∆0]y0]\}<=2~P(<<3∆?∆09@0~0/(|2|AB]\}AB) (7) with 𝜔 being the driving pulse frequency and 𝜔$% being the resonance frequency of the standing spin wave of the YIG thin film. The expression of Eq. (7) describes the spectroscopic measurement of the flux qubit hybridized with spin waves in YIG thin film. Chossing parematers as ∆!(=4.52 GHz, }'C!(=2 MHz, |AB!(=4.57 GHz and }AB!(=20 MHz, which is a rsonable number since the decay rate for Kittel spin wave in a perfect sphere is around 1 MHz (7) and for finite wavelength spin wave in the YIG thin film is 6.8 MHz at 20 mk with GGG substrate and 1.4 MHz without substrate (41), let |P(<<3|!(=0 MHz and 30 MHz, we obtain a simulated spectrum for a bare qubit and a hybridized qubit-spin wave system, as shown in Fig.3. The avoided cross or gap shows the strong coupling between flux qubit and standing spin wave of YIG thin film with vacuum Rabi splitting 2g = 60 MHz, which supports coherent energy or information exchange between them. Before preceding further, let us have a brief discussion about the influence of the CoFeB thin layer on the flux qubit. Using Eq. 4, with long wavelength approximation (𝑘~0) and spin density of CoFeB being 1.61∗10!B 𝑚2" (Co), and 𝑑=1.2 𝜇𝑚 as the parameter chosen in Fig. 3, a rough coupling strength between flux qubit and Co thin layer is 200 MHz. Decay rate for CoFeB is Γ>?@4A ~ 300 MHz and the converted influence on the flux qubit from Co electrons would be Γ>? *$"∆%$~ 1.2 MHz, where 𝑔 is the coupling strength and ∆ is the off resonance between the flux qubit and CoFeB. We may introduce the the damping constant 𝛼=}Q , where Γ is the decay rate and 𝑓 is the resonace frequency. For YIG, 𝛼 is on the order of 102 to 102;, which makes decay rate as small as 3.3 MHz at a resonace of 4.57 GHz, most possibly by improving the thin film growing quality. In addition, a low ferromagntic alloy Co25Fe75 with damping constant as low as 5∗102; is reoprted. This material could substitute the CoFeB capping layer, which would have the decay rate ΓKL!M)<1 MHz instead of ΓKLM)𝐁~300 MHz and decrease the total decay rate of YIG-pinning layer to below 5 MHz. These further ensure the possibilities to implement thicknees mode of YIG thin film in quantum information processing. Fig. 3 Simulation of the energy spectrum of a flux qubit coupled to standing spin waves in the YIG thin film. (a) Spectrum of a bare flux qubit with ∆=4.52 GHz, Γ@=2 MHz and 𝑔4QQ^=0 in Eq. (7). (b) Spectrum of a flux qubit coupled to the standing spin wave of the flux qubit with |P(<<3|!(=30 MHz, |AB!(=4.57 GHz, }AB!(=20 MHz. Next, we propose a scheme to entangle and further switch the coupling “on” and “off” between two shape-modified flux qubits through PSSW mode in YIG thin film. Fig. 4 shows the schematic: two modified flux qubits with center-to-center distance of 20√2 𝜇𝑚, are placed on top of a YIG thin film with a vertical separation d. The left/right arc of a flux qubit is a quarter of a 10 𝜇𝑚 radius circle and the top/down arc is a quarter of a 13.2 𝜇𝑚 radius circle. Mutual inductance of the two loops is given by the Neumann formula 𝐿,.='!;(∮∮f𝑿Df𝑿E|𝑿D2𝑿E|. The designed orientations of those arcs are to decrease the mutual inductance between the two flux qubit loops from several tens MHz for comparable size square loops to 3.97 MHz for the current design with circulating current as much as 500 nA. YIG thin film is ~ 80 nm in thickness with left/right sides being a quarter of a 10 𝜇𝑚 circle and top/down sides having the length of 10√2 𝜇𝑚, which is also deposited with 10 𝑛𝑚 CoFeB on one side. As oscillation occurs between the two states of a flux qubit, alternating magnetic fields are created outside the loop and Fig. 4 (d) shows the coupling strength between each flux qubit and the YIG thin film as a function of the distance d in between. As shown in Fig. 4, stray magnetic field created by the YIG-CoFeB thin film is below the superconducting critical field of material of Niobium, i.e., 1000 Gauss, that is used to fabricate the flux qubit. Readout of a flux qubit can be realized via another shaped-modified squid loop as in Fig. 4 c. Mutual inductance between the squid loop and flux qubit is 3.8∗10299 𝐻, while the one between the squid loop and the neighboring flux qubit is 5.6∗1029; 𝐻. This guarantees that reading-out flux qubit will not be influenced much by the state of neighboring qubit, even operating simultaneously. Microwave line which is not shown, can quickly tune flux quit resonance frequency to the frequency of the (PSSW) spin wave mode of 4.57 GHz. At distance 𝑑=0.5 𝜇m, the absolute value of coupling strength is about 50 MHz. If both flux qubits are detuned simultaneously to 630 MHz below 4.57 GHz, effective coupling strength J between the two flux qubits can be J~"!""%!∆!&!∆"'$≈−3.97 MHz (8) This will cancel the mutual inductive coupling between (+ 3.97 MHz) the two flux qubits loops, thus switching off the coupling. On the other hand, if detuning both flux qubit to 400 MHz above 4.57 GHz, J would be 6.25 MHz, and plus additional mutual inductive 3.97 MHz, the total coupling strength would be about 10 MHz. Since the intrinsic life time for flux qubit can be about 1 µs, coupling strength of 10 MHz is strong enough to entangle the two qubits. In this way the coupling between two flux qubits is switched “on” and “off‘. In addition, the intrinsic decay rate of thickness mode spin wave in YIG thin film is about ΓCDE =10 MHz, which will introduce an extra broadening of 10*51;116!=0.15 MHz on the flux qubit. Similarly, the CoFeB thin layer gives rise to another 300*5!1""116!~0.01 MHz broadening on flux qubit. Fig. 4 Proposed setup for a tunable switch between two shape-modified flux qubits utilizing (with) YIG thin film. (a) two shape modified flux qubits are placed at a distance d above the 80 nm thick YIG thin film, which is capped with 10 nm CoFeB layer on one side. Special geometry of flux qubits is to decrease mutual inductance and detail dimensions of both flux qubits and YIG thin film are given in the context. (a) the sideview (b) the top view. (c) a special designed squid loop used for reading out the state of flux qubit. Mutual inductance between flux qubit and squid loop is given in the context and reading out one flux qubit will not be influenced much by the neighboring qubit. (d) the absolute value of effective coupling strength (left axis) between one flux qubit and YIG thin film and the total magnetic field (right axis) at point p as in (a) created by YIG thin film as a distance of d. As demonstrated above, different from coupling to spin ensembles or Kittel mode of spin waves, the coupling of the flux qubit with finite-wavelength (fundamental) spin wave mode has an extra phase term, which enables us to obtain the coupling strength and proposed a scheme to hybridize flux qubit with a perpendicularly standing spin wave in the YIG thin film. We further show the PSSW spin wave mode in an YIG thin film can switch “on” and “off” the coupling between two flux qubits and generate entanglement. Our results manifest that it is possible to couple flux qubits while suppressing cross-talk. This opens a possibility of utilizing YIG thin film for quantum information processing. The authors thank Huaiyang Yuan, Peihao Huang, Xiuhao Deng for fruitful discussions. This work is supported by Key-Area Research and Development Program of GuangDong Province (No. 2018B030326001), the National Key Research and Development Program of China (2016YFA0300802), the National Natural Science Foundation of China (Grants No. 11704022, No. U1801661), the Guangdong Innovative and Entrepreneurial Research Team Program (Grant No. 2016ZT06D348), the Science, Technology and Innovation Commission of Shenzhen Municipality (Grant No. KYTDPT20181011104202253). 1. A. D. Córcoles et al., Nature Communications 6, 6979 (2015). 2. R. Barends et al., Nature 508, 500 (2014). 3. M. Mirrahimi et al., New Journal of Physics 16, 045014 (2014). 4. D. Marcos et al.,Physical Review Letters 105, 210501 (2010). 5. X. Zhu et al., Nature 478, 221 (2011). 6. X. Zhu et al., Nature Communications 5, 3424 (2014). 7. Y. Tabuchi et al., Physical Review Letters 113, 083603 (2014). 8. Y. Tabuchi et al., Science 349, 405 (2015). 9. D. 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2020-04-05

Quantum computing hardware has received world-wide attention and made considerable progress recently. YIG thin film have spin wave (magnon) modes with low dissipation and reliable control for quantum information processing. However, the coherent coupling between a quantum device and YIG thin film has yet been demonstrated. Here, we propose a scheme to achieve strong coupling between superconducting flux qubits and magnon modes in YIG thin film. Unlike the direct $\sqrt{N}$ enhancement factor in coupling to the Kittel mode or other spin ensembles, with N the total number of spins, an additional spatial dependent phase factor needs to be considered when the qubits are magnetically coupled with the magnon modes of finite wavelength. To avoid undesirable cancelation of coupling caused by the symmetrical boundary condition, a CoFeB thin layer is added to one side of the YIG thin film to break the symmetry. Our numerical simulation demonstrates avoided crossing and coherent transfer of quantum information between the flux qubits and the standing spin waves in YIG thin films. We show that the YIG thin film can be used as a tunable switch between two flux qubits, which have modified shape with small direct inductive coupling between them. Our results manifest that it is possible to couple flux qubits while suppressing undesirable cross-talk.

Spin wave based tunable switch between superconducting flux qubits

2004.02156v1

Engineering Entangled Coherent States of Magnons and Phonons via a Transmon Qubit Marios Kounalakis,1, 2,∗Silvia Viola Kusminskiy,2and Yaroslav M. Blanter1 1Kavli Institute of Nanoscience, Delft University of Technology, 2628 CJ Delft, The Netherlands 2Institute for Theoretical Solid State Physics, RWTH Aachen University, 52074 Aachen, Germany (Dated: September 29, 2023) We propose a scheme for generating and controlling entangled coherent states (ECS) of magnons, i.e. the quanta of the collective spin excitations in magnetic systems, or phonons in mechani- cal resonators. The proposed hybrid circuit architecture comprises a superconducting transmon qubit coupled to a pair of magnonic Yttrium Iron Garnet (YIG) spherical resonators or mechanical beam resonators via flux-mediated interactions. Specifically, the coupling results from the mag- netic/mechanical quantum fluctuations modulating the qubit inductor, formed by a superconducting quantum interference device (SQUID). We show that the resulting radiation-pressure interaction of the qubit with each mode, can be employed to generate maximally-entangled states of magnons or phonons. In addition, we numerically demonstrate a protocol for the preparation of magnonic and mechanical Bell states with high fidelity including realistic dissipation mechanisms. Furthermore, we have devised a scheme for reading out the prepared states using standard qubit control and res- onator field displacements. Our work demonstrates an alternative platform for quantum information using ECS in hybrid magnonic and mechanical quantum networks. I. INTRODUCTION The development of quantum technologies aims to- wards disruptive practical applications in several fields such as computing, communication and sensing by ex- ploiting the effects of quantum mechanics [1, 2]. The success of this venture largely relies on the evolution of hybrid quantum systems that incorporate the advantages of different physical platforms in a constructive way [3, 4]. For example, circuit quantum electrodynamics (QED), where light-matter interactions in superconducting cir- cuits are used to manipulate quantum information, is one of the leading platforms in quantum computing, combin- ing strong nonlinearities with advanced quantum control and readout as well as high coherence times relative to qubit operations [5, 6]. However, superconducting cir- cuits do not directly couple to optical photons, hindering their integration with optical networks [3]. In this direc- tion, the development of hybrid circuit QED platforms based on mechanical and magnetic systems is an essen- tial requirement towards networked quantum computa- tion [4]. In addition, the evolution of high-quality me- chanical systems operating in the quantum regime pro- vides unique opportunities not only in transduction but also in building quantum memories and sensors [3, 4, 7]. Moreover, hybrid quantum systems based on magnons, i.e., the quanta of the collective spin excitations in mag- netic materials, offer distinctive advantages, such as uni- directional propagation and chiral coupling to phonons and photons [8, 9], making them prime candidates for technological applications in quantum information sci- ences [10, 11]. The ability to generate entanglement is at the heart of most protocols in quantum information. For macro- scopic mechanical and magnonic resonators, which carry bosonic degrees of freedom and typically operate in thelinear regime, the special class of entangled coherent states (ECS) [12, 13] is of particular interest. Such states exhibit continuous-variable entanglement between different bosonic modes and provide a valuable resource for quantum teleportation [14, 15], quantum computa- tion [16–18] and communication [19, 20]. In addition, ECS are useful for fundamental studies of quantum me- chanics with applications in quantum metrology [21, 22] and tests of collapse models [23, 24]. Macroscopic entanglement between mechanical modes has recently been achieved on aluminum drum res- onators [25, 26] and micromechanical photonic/phononic crystal cavities [27, 28], however, an experimental demon- stration of entanglement between metallic nanobeams such as the ones studied in Refs. [29–31] is currently lack- ing. Furthermore, while entanglement between atomic ensembles has been experimentally realised in an optical setup [32], entangling magnons in two distant magnets still remains a challenge. Recent theoretical proposals have investigated the possibility of entangling magnons in two Yttrium Iron Garnet (YIG) spheres interacting via photons in a microwave cavity. More specifically, in Ref. [33] the emerging Kerr nonlinearity in strongly driven magnons is used, relying on driving the magnon modes far from equilibrium in order to create entangle- ment. In Ref. [34] the nonlinearity stemming from the parametric magnetorestrictive interaction is employed to create magnon-magnon entanglement, although requir- ing a much larger magnetorestrictive coupling strength than experimentally attainable [35]. Alternatively, in Refs. [36, 37] it is shown that two YIG spheres can be en- tangled by driving the magnon-cavity system with strong squeezing fields. However, while the above schemes show promise for creating magnon-magnon entanglement in distant YIG spheres, the absence of a highly controllable nonlinear element, such as a qubit, hinders the gener- ation and control of more complex states and ECS, inarXiv:2309.16514v1 [quant-ph] 28 Sep 20232 particular. Here we propose a scheme for generating ECS of magnons/phonons in a hybrid circuit QED architecture comprising a superconducting transmon qubit and two magnonic/mechanical modes. Concerning magnonic sys- tems, without loss of generality, we consider two YIG sphere modes in a hybrid qubit-magnon setup similar to Ref. [38], where the qubit-magnon coupling is me- diated via a superconducting quantum interference de- vice (SQUID). We showcase a protocol for generating maximally-entangled states such as Bell and NOON states with high fidelity, by exploiting the parametric na- ture of the qubit-magnon radiation-pressure interaction and the transmon quantum control toolbox. Further- more, we analyze a readout scheme for verifying the en- tanglement in the system based on qubit measurements and displacements of the magnon field. Contrary to pre- vious proposals for generating magnon-magnon entangle- ment, there is no need for placing the YIG spheres inside a cavity, therefore, increasing scalability and modularity. Furthermore, we numerically demonstrate the validity of our proposal for entangling SQUID-embedded mechan- ical beam resonators [29–31, 39, 40], thereby extending the possibilities for quantum control using mechanical ECS. II. HYBRID SYSTEM DESCRIPTION The fundamental element in the proposed circuit ar- chitecture is a dc SQUID, i.e., a superconducting loop interrupted by two Josephson junctions, as schematically depicted in Fig. 1. When shunted by a capacitance C, with charging energy EC= 2e2/C, this nonlinear induc- tor can realize a flux-tunable transmon qubit described by the Hamiltonian, ˆHT= 4ECˆN2−EJcosˆδ, (1) where ˆN,ˆδare conjugate operators describing the tunnel- ing Cooper-pairs and the superconducting phase across the SQUID, respectively [41, 42]. In the case where the two junctions are the same (symmetric SQUID), an external flux bias Φ btunes the Josephson energy EJ=Emax J|cosϕb|, where ϕb˙ =πΦb/Φ0and Φ 0is the flux quantum. For magnetic systems, without loss of generality we focus our description on micro-sized YIG spheres sim- ilar to Refs. [38, 43]. Upon application of an in-plane magnetic field Bz, a YIG sphere acquires a magne- tization Msand its excitations can be approximated as a set of independent quantum harmonic oscillators with Hamiltonian ˆHM=ℏP mωmˆa† mˆam, where a(†) m are bosonic operators describing the annihilation (cre- ation) of single magnons [44, 45]. Note that this de- scription is valid in the limit ⟨ˆm†ˆm⟩ ≪ NS, where NSis the total number of spins in the sphere [44, 45]. The fundamental excitation, or Kittel mode, is a uni- EJC 1 EJ2Φb δm^ δm^(a) Bz (b) Bzδx δx^ ^EJ1 EJ2C ΦbFIG. 1. Proposed hybrid circuit architecture. A flux- tunable transmon qubit, formed by a C-shunted SQUID loop, is coupled to (a) two nearby YIG spheres or (b) two SQUID- embedded mechanical beams. The magnetization of both spheres in (a) is oriented by an in-plane field Bz. The mag- netic quantum fluctuations ˆδmmodulate the SQUID flux as well as the transmon inductive energy, thereby giving rise to a qubit-magnon coupling. In (b) the coupling stems from the mechanical quantum fluctuations ˆδxinducing a modulating flux in the SQUID in the presence of the in-plane field Bz. An additional flux bias Φ bcan be externally applied to tune the qubit frequency and modulate the coupling. formly polarized state of all the spins acting as a sin- gle “macrospin” precessing around z, with ferromagnetic resonance (FMR) frequency ω0=γ0(Bz+Bani), where Baniis the anisotropy field [46]. Higher mode frequencies are given by ωm=ω0+γ0Msl−1 3(2l+1)depending on the magnon angular momentum quantum number l[47]. The mechanical systems of interest in this work consist of SQUID-embedded aluminum beams [29, 30, 39, 48]. Such mechanical beams are realised by suspending part of the SQUID loop such that it can freely oscillate out of plane [29, 30]. Similar to the YIG sphere, its ex- citations can also be described by a set of indepen- dent quantum harmonic oscillators, with Hamiltonian ˆHX=ℏP xωxˆa† xˆax, where a(†) xare bosonic operators that annihilate (create) a phonon. The fundamental mode, which is the one considered in this work, oscil- lates with frequency ω0=ℏ/(2mx2 zpf), where mis the beam mass and xzpfthe magnitude of its zero-point mo- tion [29]. Upon application of an in-plane magnetic field Bz, the quantum fluctuations in the out-of-plane displace- ment of the beam ˆδx=xzpf(ˆax+ ˆa† x) induce a flux Φ(ˆδx) =β0Bzlˆδxthrough the loop, where lis the beam length and β0is a geometric factor that depends on the mode shape [29]. Similarly, quantum fluctuations of the magnetic moment in the magnetized YIG sphere, ˆδm=µzpf(ˆam+ ˆa† m), result in an additional flux Φ( ˆδm) through the SQUID loop. Let us assume that the sphere is placed at an in-plane and out-of-plane distance dfrom the closest point in the loop. Then in the far-field limit Φ(ˆδm) =µ0ˆδm/(4√ 2πd) [38]. The additional flux from each source of quantum fluc-3 tuation, Φ( ˆδj), modulates the SQUID flux and conse- quently its Josephson energy, E′ J(ϕb,ˆδj)≃EJ 1−tanϕbX jϕ(ˆδj) , (2) where we assume ϕ(ˆδj) ˙ =πΦ(ˆδj)/Φ0≪1 and a symmetric SQUID; for a full treatment including fi- nite junction asymmetry see Refs. [38, 39]. Re- placing EJwith E′ Jin Eq. (1) and expressing the transmon operators in terms of annihilation (cre- ation) operators ˆ c(†), i.e., ˆN=i[EJ/(32EC)]1/4(ˆc†−ˆc), ˆδ= [2EC/EJ]1/4(ˆc+ ˆc†) [42], yields the total system Hamiltonian ˆH=ˆHq+ℏX jh ωjˆa† jˆaj−gjˆc†ˆc(ˆaj+ ˆa† j)i ,(3) where ˆHq=ℏωqˆc†ˆc−EC 2ˆc†ˆc†ˆcˆc, is the bare transmon Hamiltonian (valid for EJ≫EC), with qubit frequency ωq= (√8EJEC−EC)/ℏ[41]. The last term in Eq. (3) describes the radiation- pressure interaction between the qubit and each bosonic mode, with coupling strength gj=∂ωq ∂ϕjϕzpf j, (4) where ϕzpf jis the magnitude of the flux fluctuations induced by either the beam or the magnet, given by ϕzpf x=πβ0Bzlxzpf/Φ0andϕzpf m=µ0µzpf/(4√ 2dΦ0), re- spectively. In the case of a symmetric SQUID, the trans- mon frequency sensitivity to flux changes is ∂ωq ∂ϕj=ωp 2sinϕb√cosϕb, (5) where ωp=p8Emax JEC/ℏis the Josephson plasma fre- quency at ϕb= 2πk(k∈Z). The behavior of the cou- pling strength as a function of the SQUID asymmetry andϕbis studied in detail in Ref. [38]. III. ENTANGLED COHERENT STATE GENERATION The system Hamiltonian in Eq. (3) describes a qubit interacting with a set of bosonic modes via bipartite radiation-pressure interactions. However, in the ab- sence of additional driving, these radiation-pressure cou- plings lead to interesting dynamics only in the ultra- strong coupling regime, gj≳ωj[39, 49, 50]. Typi- cally, mechanical beam resonators have frequencies of a few MHz [29, 30] and operating magnon frequencies lie above 100 MHz [44, 45], whereas gj≲10 MHz [38, 39]. Therefore, while the ultrastrong coupling condition seems promising for optomechanical setups [39], it is far fromrealistic for magnonic devices. On the other hand, when external driving is introduced to the system, the radiation-pressure interaction can be “activated” even for gj< ωj, e.g., by a stroboscopic application of short π qubit pulses [51] or by modulating the coupling [38, 52]. Here, without loss of generality, we consider the case gj≪ωjand assume that the radiation-pressure inter- action is activated by applying a weak flux modula- tion through the SQUID loop as in Ref. [38]. In this scheme the qubit operates around the transmon “sweetspot”, i.e., ϕb≃0, and an applied ac flux with amplitude ϕacat frequency ωac, modulates the flux, ϕb=ϕaccos (ωact−θ)≪1, resulting in a modulated coupling strength gj(t) =ωp 2ϕzpf jcos (ωact−θ), where θ is a constant phase. In the frame rotating at ωacthe transformed Hamiltonian reads, ˆeH=ˆHq+ℏX jh ∆jˆa† jˆaj−egjˆc†ˆc(ˆajeiθ+ ˆa† je−iθ)i ,(6) where egj=ωp 4ϕzpf j, ∆j=ωj−ωacand we have omit- ted fast-rotating terms ˆ c†ˆcˆa(†) je±i(ωj+ωac)twhich do not contribute to the dynamics since egi≪(ωj+ωac). We now describe a simple protocol for generating ECS that are maximally entangled using the Hamiltonian in Eq. (6). Let us assume there are Nbosonic modes, in- teracting with the qubit via bipartite radiation-pressure couplings. First, a microwave pulse, prepares the qubit in a superposition state |χ⟩q˙ =(|0q⟩+eiχ|1q⟩)/√ 2. The next step is to activate the bipartite interaction of the qubit with each mode. In the simple case where all the modes we want to entangle have the same frequency, ωj, then by turning on the flux modulation, i.e., setting ωac=ωj, for a variable duration, τj, the system evolves into a hybrid generalized Greenberger–Horne–Zeilinger state |ψ⟩GHZ=1 N |0q01···0N⟩+eiχ|1qα1···αN⟩
,(7) where |αj⟩denotes a coherent state with complex phase space amplitude αj=−iegjτj. For |αj|≳4 the normal- ization factor is N ≃√ 2 [53]. Note that if there are M modes with different frequencies, then the flux modula- tion should be activated Mtimes in order to prepare the state in Eq. (7). Applying a qubit pulse Rˆy,π 2followed by a strong pro- jective measurement collapses the qubit in its ground or excited state and projects the bosonic system into 1 N± |0102···0N⟩ ±eiχ|α1α2···αN⟩
, where the “+” or “−” state results from measuring the qubit in |0q⟩or |1q⟩, respectively. For the case of two bosonic modes with eg1,2, τ1,2chosen such that α1=α2=αandχ= 0 the prepared state corresponds to the maximally-entangled Bell state, |±ΨBell⟩=1 N±(|00⟩ ± |αα⟩), (8)4 where N±=p 2(1±e−|α|2)≃√ 2 for |α|≳4 [54]. Alternatively, in the case of different frequency modes, ω1̸=ω2, a maximally-entangled NOON state of the form |±ΦNOON⟩=1 N±(|0α⟩ ± |α0⟩), (9) can be obtained by performing a πpulse to flip the qubit state right after turning on the first interaction and be- fore the second one. The protocol would then require the following steps: (a) start modulating at ωac=ω1, (b) turn off the interaction after time τ1, (c) apply πqubit pulse, and (d) switch on the second flux modulation with ωac=ω2for time τ2=τ1eg1/eg2. Additionally, more general ECS of the form, |Ψ⟩ij=c00|0i0j⟩+c1α|0iαj⟩+cα0|αi0j⟩+cαα|αiαj⟩, (10) with cα0, c0α̸= 0, may also be generated using appro- priately adjusted protocols. For example, starting from |ψ⟩qij=|(0q+ 1q)0i0j⟩, then turning on the interaction with mode ifor time τisuch that |α| ≡ |egiτi|≳4, and applying a Rˆyπ 2qubit pulse, results in the state |ψ⟩qij=1 2[|0q0i0j⟩+|0qαi0j⟩+|1q0i0j⟩ − |1qαi0j⟩]. If we subsequently turn on the interaction with mode j(for time τj=α/egj) and apply another Rˆyπ 2qubit pulse, the resulting state is, |ψ⟩qij=1√ 2 |0⟩q|Ψ⟩+ ij+|1⟩q|Ψ⟩− ij , where |Ψ⟩± ij=1 2(|0i0j⟩+|0iαj⟩ ± |αi0j⟩ ∓ |αiαj⟩).(11) Finally, a strong measurement collapses the qubit in |0⟩q or|1⟩q, projecting the system in the maximally-entangled two-mode state |Ψ⟩+ ijor|Ψ⟩− ij, respectively. IV. NUMERICAL MODELING & BENCHMARKING We benchmark the protocol described above for gener- ating the Bell state |+ΨBell⟩against realistic experimen- tal conditions including dissipation using the quantum statistical Lindblad master equation [55] ˙ρ=i ℏ[ρ,ˆeH] +X jωj Qj nth jL[ˆa† j]ρ+ (nth+ 1)L[ˆaj]ρ +1 T1L[ˆc]ρ+1 T2L[ˆc†ˆc]ρ, (12) where Qjis the quality factor of each resonator, L[ˆo]ρ= (2ˆ oρˆo†−ˆo†ˆoρ−ρˆo†ˆo)/2 are superoper- ators describing each bare dissipation channel and nth j= 1/[exp(ℏωj/(kBT))−1] is the number of thermally excited magnons/phonons at temperature T.T1and T2are the qubit relaxation and dephasing times, re- spectively, for which we pick a realistic value of 50 µs throughout our simulations [6]. Of note, the in-plane (a) (b) (c) (d)FIG. 2. Bell state benchmarking for the case of two Kittel modes in two identical YIG spheres, as schematically shown in Fig. 1(a). (a) Magnon number in each magnonic mode, as a function of time during the protocol, shown for different resonator quality factors. (b) Wigner function of the individ- ual magnonic state in one mode, after tracing out the other mode, at the end of the protocol for Qm= 105. The fidelity of the prepared state to the ideal Bell state |+ΨBell⟩is shown as a function of time in (c) and as a function of the magnon number in (d). System parameters: ω1,2/(2π) = 1 GHz, eg1,2/(2π) = 2 MHz, T1=T2= 50 µs,T= 10 mK. magnetic field that is required to enable the qubit cou- pling to the magnonic or the mechanical resonator, Bz∼10−50 mT [38, 39], is not expected to limit the qubit performance [56]. In addition, while the transmon is effectively a qubit, it is more accurately described as a three-level system with negative anharmonicity given by ∼ −EC. We therefore model it as such choosing a typical value of EC/h= 300 MHz [6, 41]. We first study the case, schematically depicted in Fig. 1(a), of two YIG spheres placed diametrically op- posite with respect to the center of the SQUID. For simplicity, we assume two identical spheres and Kittel modes with the same frequency, ω1,2/(2π) = 1 GHz, as well as coupling to the qubit, eg1,2/(2π) = 2 MHz, and study the performance of the protocol proposed above as a function of the resonator quality factor, Qm, at T= 10 mK ( nth 1,2≃0.01). For typical val- ues of the Gilbert damping constant αGwe expect Qm= 1/αG∼103−105[45, 57, 58]. In Fig. 2(a) we plot the evolution of the magnon num- ber in either mode jand compare it to the ideal case, i.e., without dissipation, where ⟨ˆa† jˆaj⟩(t) =|egmt|2/2. In ad- dition, in Fig. 2(b) we plot the Wigner quasi-probability5 (a) (b) FIG. 3. (a) Logarithmic negativity and (b) conditional quan- tum entropy as a function of the magnon number for the magnon-magnon system described in Fig. 2. In the absence of dissipation (dashed-dotted curves) an ideal Bell state is created for magnon numbers ⟨ˆa† jˆaj⟩>2 with EN→1 and S(m1|m2)→ − log 2. distribution at t= 0.24µs for Qm= 105, which is de- fined as W(αj) = 2 /πTrn D†(αj)ρjD(αj)eiπˆa† jˆajo , where ρj≡Tri[ρij] is the reduced density matrix of mode j andD(αj) =eαˆa† j−α∗ˆajis the displacement operator act- ing on this mode. The two-mode density matrix, ρij, is obtained after projecting on |+q⟩, and tracing out the qubit, i.e., ρij≡Trq[ρ|+q⟩⟨+q|]. We note that since we have two identical modes, the magnon number evo- lution as well as the reduced-state Wigner functions are exactly the same for both. Furthermore, Figs. 2(c) and 2(d) show the fidelity F=p ⟨+ΨBell|ρ12|+ΨBell⟩[55, 59] of the prepared two-mode state to the ideal Bell state, as a function of time and magnon number, respectively. Evidently, for realistic values of the magnonic quality fac- torsQm≳104[40, 57, 58], the desired Bell state can be prepared with high fidelity F≲90%. To showcase the evolution of the bipartite entangle- ment during the protocol, in Fig. 3(a) we plot the loga- rithmic negativity EN= log2(2N(ρ12)+1), where N(ρ12) is the sum of negative eigenvalues of the partial transpose of the two-mode density matrix ρ12[60]. The dashed- dotted curve shows the logarithmic negativity evolution in the ideal case, EN(t) = log2h 2/(e−|egjt|2+ 1)i [61]. For |α| ≡ |egjt|≳2 it approaches the ideal value of Emax N= 1, where the two modes are maximally entangled, before magnon dissipation eventually takes over and the entan- glement gets lost. Furthermore, in Fig. 3(b) we plot the conditional quantum entropy S(m1|m2) =S(ρ12)−S(ρ2) [62, 63], where S(ρij) and S(ρj) are the Von Neumann en- tropies of the joint and reduced state, respectively, with S(ρ) =−Tr[ρlnρ]. Negative conditional quantum en- tropy serves as a sufficient criterion for the quantum state to be entangled and provides a measure of the degree of coherent quantum communication between the two en- tangled modes [62, 63]. For maximally-entangled Bell (a) (b) (c)FIG. 4. (a) Bell state fidelity, (b) logarithmic negativity and (c) conditional quantum entropy as a function of the phonon number for the case of two SQUID-embedded mechanical nanobeams interacting via the transmon. System parameters: ω1,2/(2π) = 10 MHz, eg1,2/(2π) = 100 kHz, T1=T2= 50 µs, T= 10 mK, initial nth 1,2= 0.1. states we have S(ρij) = 0 and S(ρj) = ln 2. There- fore, in the limit of large magnon numbers, we ex- pect S(m1|m2)→ − ln 2, as illustrated by the dashed- dotted curve plotting the ideal (dissipationless) case. However, as the entanglement starts decreasing due to magnon dissipation, the joint entropy of the system be- comes positive and both S(ρij) and S(ρi) start increas- ing. Therefore, as expected, the positive value threshold forS(m1|m2) is surpassed faster and at lower magnon numbers as the quality factors get smaller. Note that initially S(m1|m2)>0 due to the fact that the modes start in a thermal state with nth≃0.01. The protocol described above can also be applied to entangle mechanical beam resonators embedded in the SQUID loop, as depicted in Fig. 1(b). These can be realized using carbon nanotubes [40] or aluminum-based mechanical beams [29–31, 39] interacting via radiation- pressure couplings with the transmon. The former have operating frequencies and quality factors similar to the magnonic case studied above, therefore, the results in Figs. 2 and 3 are applicable as well. On the other hand, mechanical beam resonators made of aluminum typically operate in the range 1 −10 MHz, with quality factors Qx≳105[29–31]. Therefore, in conjunction with the magnonic case, we numerically test the same protocol for creating me- chanical Bell states between two SQUID-embedded alu- minum beam resonator modes [30], with the same fre- quency ω1,2/(2π) = 10 MHz and coupling to the qubiteg1,2/(2π) = 100 kHz. Typical temperatures of T∼10 mK, correspond to high thermal population at these frequencies, however, cooling schemes can reduce the number of thermal phonons to ≲0.1 [39, 40]. We therefore assume an attainable initial thermal population nth 1,2= 0.1 and an operating temperature of T= 10 mK. In Figs. 4(a) and 4(b) we plot the Bell-state fidelity and the logarithmic negativity, respectively, as a function of6 the phonon number during the protocol for quality fac- tors in the range Qx= 105−107. Note that initially the fidelity is less than 1, due to the finite thermal popula- tion in both resonators, however, as the protocol evolves it starts increasing before phonon dissipation takes over. We find that, for realistic quality factors Qx≳106, high phonon number Bell-states can be prepared with high fi- delity and sufficiently high entanglement as quantified by EN. However, as shown in Fig. 4(c), the effects of the initial thermal population seem to be detrimental to the conditional quantum entropy S(x1|x2) which remains far from the ideal limit during the whole protocol and only reaches negative values for Qj∼106. Experimental verification of the prepared states can be obtained by performing state tomography. For example, in the case of mechanical resonators, by sideband driving on the qubit one may engineer beam-splitter and two- mode squeezing interactions that can be used to detect correlations of the entangled state similar to Ref. [26]. This method may also be applied to the magnonic res- onators, for which independent state tomography tech- niques exist as well [64]. However, strong driving may severely impact the qubit state [65] limiting the suc- cess of such protocols. For this reason we have also analyzed an alternative scheme for reading out the en- tangled states, presented in the Appendix, which relies solely on switching on/off the interaction and performing magnon/phonon displacements and qubit measurements. V. CONCLUSION In summary, we have proposed a scheme for generat- ing ECS of magnons/phonons in a hybrid circuit QED architecture comprising a superconducting transmon qubit coupled to different magnonic/mechanical modes via bipartite flux-mediated interactions. In particu- lar, we have highlighted several schemes for creating maximally-entangled states and, as a proof-of-principle demonstration, we have numerically tested a simple protocol for generating magnonic and mechanical Bell states under realistic experimental conditions. We show that high-fidelity Bell states can be prepared in the presence of typical dissipation mechanisms in the system. Furthermore, in the Appendix we have analyzed a readout scheme, using standard circuit operations, that can be used as an alternative to existing tomography methods for verifying the prepared states. Our results pave the way towards creating controllable quantum networks of entangled magnons in a flexible and scalable platform without relying on microwave 3D cavities or strong driving. Although for simplicitywe have considered identical YIG spheres, our results are also applicable to nonidentical modes and other geometries such as micro-disk resonators [66]. Finally, as we demonstrate numerically, the proposed scheme for creating and controlling ECS is also applicable to SQUID-embedded mechanical beam resonators, opening up new opportunities for quantum information tasks in this platform and potentially giving rise to novel magnonic-mechanical hybrid devices. ACKNOWLEDGMENTS We thank Sanchar Sharma and Victor Bittencourt for helpful discussions. This research was supported by the Dutch Foundation for Scientific Research (NWO). M.K. and S.V.K. would like to acknowledge financial support by the German Federal Ministry of Education and Research (BMBF) project QECHQS (Grant No. 16KIS1590K). APPENDIX: READOUT SCHEME We now describe a method for reading out the two- mode ECS discussed in the main text, using only qubit measurements and displacement operations on the bosonic modes. We start with the assumption that the most general state one can prepare with the system Hamiltonian in Eq. (6) is of the following form, |Ψ⟩ij=c0eiθ0|0i0j⟩+c1eiθ1|0iαj⟩ +c2eiθ2|αi0j⟩+c3eiθ3|αiαj⟩, (A13) where cjare real positive numbers andP3 j=0c2 j= 1. Our assumption is based on the fact that the engineered radiation-pressure interaction in Eq. (6) can only lead to magnon/phonon displacements when the qubit is in the excited state, therefore, for the protocols described in the main text, where the interaction is activated at least once for each bosonic mode, Eq. (A13) describes the mode general state one can prepare. In addition, single-photon losses acting on coherent states result in a coherent state of smaller amplitude, therefore this decay channel does not alter the form of the state described in Eq. (A13). Assuming the state in Eq. (A13) has been prepared, we start the readout protocol by preparing the qubit in a general superposition state |ϕ⟩q= (|0⟩q+eiϕ|1⟩q)/√ 2. After switching on both interactions, the system wave- function evolves as7 U(i) intU(j) int|ϕ⟩q|Ψ⟩ij=1√ 2h |0⟩q c0eiθ0|0i0j⟩+c1eiθ1|0iαj⟩+c2eiθ2|αi0j⟩+c3eiθ3|αiαj⟩
+ |1⟩qei(ϕ+¯ϕ) c0eiθ0|βiβj⟩+c1eiθ1+γj|βi(α+β)j⟩+c2eiθ2+γi|(α+β)iβj⟩+c3eiθ3+γi+γj|(α+β)i(α+β)j⟩
i , (A14) where U(j) int= expn iegjˆc†ˆc(ˆajeiθ+ ˆa† je−iθ)to . The displacement amplitudes and corresponding ge- ometric phases, which arise from the radiation pres- sure interactions, are given by βi,j˙ =β(ti,j) = (gi,j/ωi,j) (eiωi,jti,j− 1) and ¯ϕ˙ =¯ϕ(ti,j) = (gi,j/ωi,j)2(ωi,jti,j−sin (ωi,jti,j)) [38, 67]. For simpli- fication purposes we have assumed that the latter are equal and, since ϕis arbitrarily determined at the qubit preparation stage, they can be absorbed into a redefini- tion of ϕ→¯ϕ+ϕ. The phases γi,j= Im( α∗βi,j) arise fromthe fact that in general two consecutive displacements do not commute. The above state can also be written as |ψ⟩qij=1 2 |+⟩qΨ+ ij+|−⟩qΨ− ij , (A15) where |±⟩= (|0⟩ ± |1⟩)/√ 2 are the eigenstates of the Pauli ˆ σxoperator and Ψ± ij=c0eiθ0|0i0j⟩+c1eiθ1|0iαj⟩+c2eiθ2|αi0j⟩+c3eiθ3|αiαj⟩ ± c0eiθ0+ϕ|βiβj⟩+c1eiθ1+ϕ+γ|βi(α+β)j⟩+c2eiθ2+ϕ+γ|(α+β)iβj⟩+c3eiθ3+ϕ+2γ|(α+β)i(α+β)j⟩ . (A16) The expectation value of the qubit in the |±⟩basis is then given by ⟨ˆσx⟩β,β=1 4 |⟨Ψ+ ij|Ψ+ ij⟩|2− |⟨Ψ− ij|Ψ− ij⟩|2
.(A17) We now consider several cases for each displacement: (I) First, assuming the coupling strength and inter- action times for both resonators are chosen such that βi,j=αi,j, we have ( γi,j= 0): Ψ± ij=c0eiθ0|0i0j⟩+c1eiθ1|0iαj⟩+c2eiθ2|αi0j⟩+(c3eiθ3±c0eiθ0+ϕ)|αiαj⟩ ±c1eiθ1+ϕ|αi(2α)j⟩ ±c2eiθ2+ϕ|(2α)iαj⟩ ±c3eiθ3+ϕ|(2α)i(2α)j⟩ (A18) From Eq. (A17) we obtain ⟨ˆσx⟩α,α=|c3eiθ3+c0eiθ0+ϕ|2− |c3eiθ3−c0eiθ0+ϕ|2 =c0c3cos (ϕ+θ0−θ3). (A19) Additionally, for βi,j=−αi,jit can be shown that ⟨ˆσx⟩−α,−α=c0c3cos (ϕ+θ3−θ0). (A20) (II) For the case βi=αi,βj=−αj, using Eq. (A16)and Eq. (A17), it follows that ⟨ˆσx⟩α,−α=c1c2cos (ϕ+θ1−θ2). (A21) Similarly for βi=−αi,βj=αjwe obtain ⟨ˆσx⟩−α,α=c1c2cos (ϕ+θ2−θ1) (A22) (III) For the cases βi=αi,βj= 0 and βi=−αi, βj= 0 we have ⟨ˆσx⟩α,0=c0c2cos (ϕ+θ0−θ2) +c1c3cos (ϕ+θ1−θ3) (A23)8 and ⟨ˆσx⟩−α,0=c0c2cos (ϕ+θ2−θ0)+c1c3cos (ϕ+θ3−θ1) (A24) respectively. (IV) For βi= 0,βj=αiandβi= 0,βj=−αiwe find two more equations, ⟨ˆσx⟩0,α=c0c1cos (ϕ+θ0−θ1) +c2c3cos (ϕ+θ2−θ3). (A25) and ⟨ˆσx⟩0,−α=c0c1cos (ϕ+θ1−θ0)+c2c3cos (ϕ+θ3−θ2) (A26) Finally for βi,j= 0 we obtain the following relation ⟨ˆσx⟩0,0= c2 0+c2 1+c2 2+c2 3
cosϕ, (A27) which is equivalent to the normalisation condition for |Ψ⟩ijwith the additional degree of freedom ϕ. The above equations are not yet in a form where they can be used to obtain all pairs of ci, θistraightforwardly. However, they can be combined and further simplified using basic trigonometric relations as shown below: (i) First, adding and subtracting equations (A19) and (A20) we obtain ⟨ˆσx⟩α,α+⟨ˆσx⟩−α,−α= 2c0c3cosϕcos (θ3−θ0),(A28) and ⟨ˆσx⟩α,α− ⟨ˆσx⟩−α,−α= 2c0c3sinϕsin (θ3−θ0).(A29) If the qubit is prepared such that ϕ=π/4 then by com- bining the above two equations we obtain a relation for c0, c3that does not depend on θ0, θ3: c0c3=q |⟨ˆσx⟩α,α|2+|⟨ˆσx⟩−α,−α|2. (A30)Ifc0c3̸= 0 we can also determine the phases. First, eiθ0 in Eq. (A13) can be absorbed into a global phase factor multiplying |Ψ⟩ijfollowed by a redefinition of θ1,2,3→ θ1,2,3/θ0(equivalent to defining θ0= 0 or 2 π). Then for ϕ=π/4 we have θ3= arctan⟨ˆσx⟩α,α− ⟨ˆσx⟩−α,−α ⟨ˆσx⟩α,α+⟨ˆσx⟩−α,−α . (A31) (ii) Following the same recipe we can obtain similar relations for c1, c2andθ1, θ2. In this case, by combining equations (A21) and (A22) for ϕ=π/4 we obtain the following equations c1c2=q |⟨ˆσx⟩α,−α|2+|⟨ˆσx⟩−α,α|2, (A32) and (assuming c1c2̸= 0) θ2−θ1= arctan⟨ˆσx⟩α,−α− ⟨ˆσx⟩−α,α ⟨ˆσx⟩α,−α+⟨ˆσx⟩−α,α . (A33) (iii) Furthermore, from equations (A23) and (A24) we obtain (for ϕ=π/4) (⟨ˆσx⟩α,0+⟨ˆσx⟩−α,0)2±(⟨ˆσx⟩α,0− ⟨ˆσx⟩−α,0)2 = 2 (c0c2)2+ (c1c3)2+ 2c0c1c2c3cos (θ2±θ1∓θ3) . (A34) Using equations (A30), (A31), (A32) and (A33) we can obtain a relation for c0, c1, c2, c3with no dependence on the phases: (c0c2)2+ (c1c3)2=f(⟨ˆσx⟩α,0,⟨ˆσx⟩−α,0,⟨ˆσx⟩α,α,⟨ˆσx⟩−α,−α,⟨ˆσx⟩α,−α,⟨ˆσx⟩−α,α) = 2⟨ˆσx⟩α,0⟨ˆσx⟩−α,0−2"q (|⟨ˆσx⟩α,α|2+|⟨ˆσx⟩−α,−α|2) (|⟨ˆσx⟩α,−α|2+|⟨ˆσx⟩−α,α|2) ×cos arctan⟨ˆσx⟩α,α− ⟨ˆσx⟩−α,−α ⟨ˆσx⟩α,α+⟨ˆσx⟩−α,−α + arctan⟨ˆσx⟩α,−α− ⟨ˆσx⟩−α,α ⟨ˆσx⟩α,−α+⟨ˆσx⟩−α,α# . (A35) (iv) Similarly, from equations (A25) and (A26) we ob- tain (for ϕ=π/4) (⟨ˆσx⟩0,α+⟨ˆσx⟩0,−α)2±(⟨ˆσx⟩0,α− ⟨ˆσx⟩0,−α)2 = 2 (c0c1)2+ (c2c3)2+ 2c0c1c2c3cos (θ1±θ2∓θ3) . (A36) Again, using equations (A30), (A31), (A32) and (A33) we can obtain another relation for c0, c1, c2, c3with no dependence on the phases:9 (c0c1)2+ (c2c3)2=g(⟨ˆσx⟩0,α,⟨ˆσx⟩0,−α,⟨ˆσx⟩α,α,⟨ˆσx⟩−α,−α,⟨ˆσx⟩α,−α,⟨ˆσx⟩−α,α) = 2⟨ˆσx⟩0,α⟨ˆσx⟩0,−α−2"q (|⟨ˆσx⟩α,α|2+|⟨ˆσx⟩−α,−α|2) (|⟨ˆσx⟩α,−α|2+|⟨ˆσx⟩−α,α|2) ×cos arctan⟨ˆσx⟩α,α− ⟨ˆσx⟩−α,−α ⟨ˆσx⟩α,α+⟨ˆσx⟩−α,−α −arctan⟨ˆσx⟩α,−α− ⟨ˆσx⟩−α,α ⟨ˆσx⟩α,−α+⟨ˆσx⟩−α,α# . (A37) In our case we are interested in reading out the Bell state |Ψ⟩ij=1√ N |0i0j⟩+eiθ|αiαj⟩
, (A38) i.e. the state in Eq. (A13) with θ3=θ,c0=c3=1√ Nand c1=c2= 0. Let us assume that we have prepared the general state in Eq. (A13). First, we can measure ⟨ˆσx⟩α,α and⟨ˆσx⟩−α,−αand from Eq. (A30) determine c0c3. If we have indeed prepared the target state shown in Eq. (A38) then this product should be nonzero. Then we proceedby measuring ⟨ˆσx⟩α,−αand⟨ˆσx⟩−α,αwhich should both be zero indicating that either c1= 0 or c2= 0 according to Eq. (A32). Additionally equations (A35) and (A37) should also equate to zero indicating c1=c2= 0. Finally, combining equations (A27) and (A30) we have (c0−c3)2=√ 2⟨ˆσx⟩0,0−2q |⟨ˆσx⟩α,α|2+|⟨ˆσx⟩−α,−α|2. (A39) If indeed the state in Eq. (A38) is prepared then we should find that ⟨ˆσx⟩0,0=p 2 (|⟨ˆσx⟩α,α|2+|⟨ˆσx⟩−α,−α|2), and therefore c0=c3= |⟨ˆσx⟩α,α|2+|⟨ˆσx⟩−α,−α|2
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2023-09-28

We propose a scheme for generating and controlling entangled coherent states (ECS) of magnons, i.e. the quanta of the collective spin excitations in magnetic systems, or phonons in mechanical resonators. The proposed hybrid circuit architecture comprises a superconducting transmon qubit coupled to a pair of magnonic Yttrium Iron Garnet (YIG) spherical resonators or mechanical beam resonators via flux-mediated interactions. Specifically, the coupling results from the magnetic/mechanical quantum fluctuations modulating the qubit inductor, formed by a superconducting quantum interference device (SQUID). We show that the resulting radiation-pressure interaction of the qubit with each mode, can be employed to generate maximally-entangled states of magnons or phonons. In addition, we numerically demonstrate a protocol for the preparation of magnonic and mechanical Bell states with high fidelity including realistic dissipation mechanisms. Furthermore, we have devised a scheme for reading out the prepared states using standard qubit control and resonator field displacements. Our work demonstrates an alternative platform for quantum information using ECS in hybrid magnonic and mechanical quantum networks.

Engineering Entangled Coherent States of Magnons and Phonons via a Transmon Qubit

2309.16514v1

arXiv:1407.4957v2 [cond-mat.mes-hall] 22 Jul 2014Microwave-induced spin currents in ferromagnetic-insula tor|normal-metal bilayer system Milan Agrawal,1,2,a)Alexander A. Serga,1Viktor Lauer,1Evangelos Th. Papaioannou,1Burkard Hillebrands,1 and Vitaliy I. Vasyuchka1 1)Fachbereich Physik and Landesforschungszentrum OPTIMAS, Technische Universit¨ at Kaiserslautern, 67663 Kaiserslautern, Germany 2)Graduate School Materials Science in Mainz, Gottlieb-Daim ler-Strasse 47, 67663 Kaiserslautern, Germany (Dated: 1 September 2018) A microwave technique is employed to simultaneously examine the spin p umping and the spin Seebeck effect processes in a YIG |Pt bilayer system. The experimental results show that for these t wo processes, the spin current flows in opposite directions. The temporal dynamics of the longitudinal spin Seebeck effect exhibits that the effect depends on the diffusion ofbulk thermal-magnonsin t he thermal gradientin the ferromagnetic- insulator |normal-metal system. Since its discovery in 2008, the spin Seebeck effect1—a route to generate a spin current by applying a heat cur- rent to ferromagnets—has given a new dimension to the field of spin-caloritronics2. In particular, the longitudi- nal spin Seebeck effect (LSSE)3, where the spin current flows along the thermal gradient in the magnetic mate- rial, drives the field due to its technologically promising applications in energy harvesting4, and in temperature, temperature gradient, and position sensing5. Having conceptual understanding and future applica- tions in the centre of attention, a comparative study of the spin current direction and its temporal evolution for different spin-current-generation processes like spin pumping (SP) and spin Seebeck effect (SSE) is very im- portant. In previous experiments6,7, these issues have not been explicitly addressed. In this letter, we demon- strate microwaves as a simple-and-controlled tool to in- vestigate both, SP and SSE, processes simultaneously in a single experiment. Such investigations are not pos- sible with other techniques including laser heating8or direct-current heating9employed to study the SSE. For example, in Ref. 10, the direction of the spin current in the SP and SSE processes has been determined by combining the FMR technique with additional Peltier or dc/ac based heating techniques. Our results reveal that inaferromagnet |normalmetal(paramagnet)system, the spin current flows from the ferromagnet (FM) to the nor- mal metal (NM) in the case of SP process, while the flow reverses for the LSSE provided that the NM is hot- ter than the FM. The time-resolved measurements show that the spin current dynamics of the LSSE is on sub- microsecond timescale compared to nanosecond fast spin pumping process11. The experiment was realized using a bilayer of a mag- netic insulator, Yttrium Iron Garnet (YIG), and a nor- mal metal, Pt. The sample structure consists a 6.7- µm- thick YIG film of dimensions 14 mm ×3 mm, grown by a)Electronic mail: magrawal@physik.uni-kl.deYIG Microwave Generator HF-Diode Pt12 3 OscilloscopeLow-pass FilterY CirculatorAmplifier50 Ω load Voltage Amplifierzxy Temperature (°C) 16.5 19.0 21.5 FIG. 1. A schematic sketch of the experimental setup. Mi- crowaves were employed to heat a 10-nm-thick Pt strip grown on a 6.7- µm-thick YIG film placed on top of the micro- stripline. The reflected microwaves were monitored on an oscilloscope. The inverse spin Hall voltage generated in th e Pt strip was amplified and measured by the oscilloscope. The inset is an infrared thermal image of the sample obtained by continuous microwave heating of the Pt strip situated in the middle of the film. liquid phase epitaxy on a 500- µm-thickGallium Gadolin- ium Garnet (GGG) substrate, and a10-nm-thick Ptstrip (3 mm×100µm), structured by photolithography and deposited by molecular beam epitaxy at a growth rate of 0.05˚A/s under a pressure of 5 ×10−11mbar. A 0.6-mm- wide and 17- µm-thick copper (Cu) microstrip antenna wasdesignedona dieletricsubstratetoapplymicrowaves to the sample. In order to obtain a maximum microwave heating efficiency by eddy currents in the metal (Pt), the Pt-covered surface of the sample was placed on top of the micro-stripline. An insulating layer was inserted in between the sample and the micro-stripline to avoid any direct electric contact. Gold wires were glued to the Pt strip with silver paste to connect with the external cir- cuit. A schematic diagram of the experimental setup is shown in Fig. 1. Microwaves from an Anritsu MG3692 C generator were amplified (+30 dB) by an amplifier and guided to the sample structure. The microwaves are2 absorbed by the thin Pt metal strip and start to heat it up. As a result, a thermal gradient along the + z- direction was established. The reflected microwavesfrom the micro-stripline were received by connecting a Y- circulator to the microwave circuit-line. The reflected microwaves were rectified using a high-frequency (HF) diode and monitored on an oscilloscope. This signal pro- vides the information about the shape of the microwave signal envelope and its duration. The YIG film was mag- netized in plane by an applied magnetic field µ0Halong thex-axis. The perpendicular thermal gradient in the YIG |Pt bi- layer generates a spin current in the system due to the LSSE. The generated spin current flows along the ther- mal gradient ( z-axis). In the normal metal Pt, the spin current converts into a charge current by the inverse spin Hall effect (ISHE)12asJISHE∝Js×σ,whereJsis the spin current, and σthe spin polarization. The gener- ated charge current along the y-axis in the Pt strip was passed through a low-pass filter to block alternating cur- rents generated directly by the electric field components of the microwaves. The filtered signal was amplified and observed on the oscilloscope. In the first experiment, continuous microwave mea- surements at a fixed frequency of 6 .8 GHz were carried out by varying the magnetic field. In Fig. 2, the in- verse spin Hall voltage VISHEis plotted versus the ap- plied magnetic field µ0H. Clearly, three features can be noticed here: (i) two peaks with opposite polarities at the magnetic fields of +168 .6 mT and −168.6 mT, (ii) their unequal magnitude, and (iii) an offset for all non- resonancemagnetic fields, which hasan opposite polarity to the peaks. The first feature originates from the spin pumping processby spin wavesexcited close to ferromag- netic resonance (FMR) in the YIG film13–15. The FMR conditions were achieved for both positive and negative magnetic fields. Corresponding to these fields, a spin current is injected into Pt by the spin pumping process. The inverse spin Hall voltage generated in Pt is given by VISHE∝θSHE(Js×σ), where θSHEdenotes the spin Hall angle. The direction of σdepends on the direction of the magnetic field. Therefore, on inverting the magnetic field direction, the polarity of VISHEreverses. In order to understand the second feature of the spec- trum that the signals have unequal amplitude, it is im- portant to discuss the spin-wave modes excited in the YIG film for our experimental geometry. It is clear from the sample orientation, shown in Fig. 1 and inset to Fig. 2, that the spin waves excited by the Oersted field of the micro-stripline propagate along the y-axis, perpen- dicular to the magnetic field applied along the x-axis. These kinds of spin waves with wave vector k⊥Hare known as magnetostatic surface spin waves (MSSW) or Damon-Eshbach (DE) spin-waves16. The DE spin-waves are nonreciprocal spin waves and travel along a direction given by k=H×n, wherenis the normal to the film surface. Therefore, the propagation of these spin waves on the surface of a film can be reversed by inverting the-150 -100 -50 0 50 100 150-6-4-20246 VISHV(µV) Magnetic field (mT) µ H0Spin pumping by DE spin-waves Longitudinal SSE signal H kn PtMW stripline σE JSPt-H-k nMW stripline -σ -EJS zxy FIG. 2. The inverse spin Hall voltage ( VISHE) generated in the Pt strip as a function of the applied magnetic field µ0H. An asymmetry in the amplitude of VISHEat FMR-magnetic field appears due to unequal efficiency of Damon-Eshbach spin- waves excitation (shaded area) for two opposite directions of the applied magnet field, shown in the insets. direction of the magnetic field. In our experiment, when a magnetic field is applied along the + x-direction, The DE-spin waves, in the YIG film surface close to the micro-stripline17, can only be ex- cited along the −y-direction ( ˆk=x×z) with respect to the micro-stripline as shown in the inset to Fig 2. On the other hand, when the magnetic field is applied along the −x-direction, the spin waves can propagate only along the +y-axis. If the YIG film is not positioned symmet- rically around the micro-stripline, as in our case, the ef- fective YIG film area, where spin waves can be excited, will be unequal for two opposite fields as shown in the inset to Fig 2. Since the strength of VISHEsignal is pro- portional to the spin-wave intensity in the system7, we observed an unequal amplitude of VISHEin our experi- ment. We performed alike measurements with displacing the YIG film and find that the amplitude of the spin pumping signals can be altered by varying the relative positions of the film with respect to the micro-stripline. Therefore, we conclude that the unidirectional nature of the DE spin-waves regulates the asymmetry of the ISHE signal18. The third feature seen in Fig. 2, i.e., an offset for non- resonant magnetic fields; is attributed to the LSSE. A similar signal could also be produced by the anomalous Nernst effect in Pt, magnetized due to the proximity ef- fect. However, recent observations19,20discard any such possibility in YIG |Pt systems. The polarity of the LSSE signal changes with the direction of the magnetic field; however, it is important to notice that the LSSE signal has an opposite polarity than that of the signal at FMR for a same direction of the magnetic field. This evidence excludes the possibility of non-resonant spin pumping in the system. When the Pt strip is heated by microwave absorption, a thermal gradient ( ∇Tz) from YIG to Pt develops normal to the interface. The thermal gradient generates a spin current flowing along the z-axis. Since the Pt strip is hot, the spin currentgeneratedvia the lon- gitudinal SSE ( Js∝ −∇T) flows from Pt to YIG21,22, in3 (a) VLSSE(µV) 0 0.4 0.8 1.204812 Microwave power (W) μ0H(mT)(b) VLSSE(µV) -20 -10 0 10 20-40418.6 mW 117.5 mW 295.1 mW 468 mW 741 mW 933 mW 1175 mW8 FIG. 3. (a) Plotted is VLSSEas a function of the applied magnetic field for various applied microwave powers. (b) The peak-to-peak amplitude of VLSSEis plotted versus the ap- plied microwave power. The peak-to-peak amplitude of VLSSE scales linearly with the microwave power. contrast to spin pumping where the spin current flows from YIG to Pt23. This argument explains the oppo- site polarities of the resonant (spin pumping) and the non-resonant (longitudinal SSE) inverse-spin-Hall volt- ages (VISHE) observed in Fig. 2. These results are con- sistent with previous experimental studies6,7,10. As dis- cussed above, the non-resonant VISHEis attributed to the longitudinalSSE;henceforth,wedenotethenon-resonant VISHEvalues as VLSSE. Magnetic field scans for various microwave input pow- ers were carried out. In Fig. 3, the VLSSEversusthe mag- netic field data is plotted for various applied microwave powers. With increasing microwave power, the tempera- ture increases in the Pt strip which enlarges the thermal gradient close to the YIG |Pt interface and, hence, in- jects a larger spin current ( Js∝ −∇T) into the YIG film3,21,22. Impact of the large spin current appears as a higherVLSSEsignal highlighted in Fig. 3(b). The peak- to-peak amplitude of VLSSEscales linearly with the ap- plied microwave power. The signature that the VLSSE signal scales linearly with the microwave power verifies that the signal originates from the heating produced in Pt shown in the inset to Fig. 1. The above experiment demonstrates that microwaves can be utilized to create a thermal gradient in ferromagnetic-insulator |normal-metal system, thereby, to study the LSSE along with the SP. The experimental setup shown in Fig. 1 can also be employed to investi- gate the temporal dynamics of the longitudinal SSE and to compare it with the SP dynamics11. In the second ex- periment, instead of continuous microwaves, 10- µs-long microwave pulses with rise-fall times of less than 10 ns were used to perform the time-resolved measurements of the longitudinal SSE. The frequency of the microwave pulses (6 .8 GHz) was chosen such that the magnetic sys- tem stayed at non-resonance condition of the magnetic field in the range of interest ( ±25 mT). The experiment was executed at various microwave powers. The mea- surementswere recordedfor both positive (+25mT) and negative (-25 mT) magnetic fields, and an average valueof the non-resonant VISHE, i.e.,VLSSEwas considered. In Fig. 4, VLSSEis plotted versus time. The longitu- dinal SSE signal takes around 1 µs to reach to the sat- uration level. The 10% −90% rise time of the signal is found to be around 530 ns. The longitudinal SSE sig- nal (VLSSE) shows similar features as reported in Ref. 8, where a pulsed laser is employed to create the vertical thermal gradient in the YIG |Pt system. The main dif- ference observed here is that the VLSSEsignal appears as soon as the microwave current runs; contrarily, in the laser heating experiment8a time lag (200 ns) exists due to the laser switching time. The model of thermal magnon diffusion8,22,24is em- ployed here to understand the timescale of the longitudi- nal SSE. The model states that the spin current from a FM injected into a NM depends on the diffusion of ther- malmagnonsinthe FM. Thedensityofthermalmagnons is proportional to the local phonon temperature25,26. Due to the thermal gradient in the FM, magnons diffuse from hotter regions (higher population) to colder regions (lower population) of the FM and create a magnon den- sity inequilibrium at the FM |NM interface which leads to the injection of a spin current into the NM21. The timescale of the effect depends on the temporal devel- opment of the magnon density inequilibrium, i.e., the thermal gradient in the system. According to the model, VLSSEis given by8 VLSSE(t)∝l/integraldisplay interface∇Tz(z,t)exp(−|z| L)dz,(1) where∇Tzis the phonon thermal gradient in the FM, perpendicular to the interface, lis the magnetic film thickness, and Lis the effective magnon diffusion length. Wefitted ourexperimentaldatashownin Fig4(a)with Eq. 1 using ∇Tz(z,t) calculated numerically by solving the heat equation for our system in accordance with a model described in Ref. 8. In Fig. 4(b), the normalized experimental VLSSE-signal was plotted together with the FIG. 4. (a) Plotted is the temporal evolution of VLSSEon the application of a 10 µs long microwave pulse which creates a vertical temperature gradient in the YIG |Pt structure by heatingthePtstrip. (b)Acomparison ofexperimentallymea - suredVLSSEdata with calculated values using Eq. (1) for vari- ous effective magnon diffusion lengths L= 300,500,700 nm.4 calculated ones for various magnon diffusion lengths of 300 nm, 500 nm, and 700 nm. The model resembles the experimental data well. The fitting shows that a typi- cal magnon diffusion length for thermal magnons in the YIG|Pt system is around 500 nm. An identical value for the magnon diffusion length was obtained in the laser heating experimental performed on the same sample, re- ported in Ref. 8. In summary, we presented microwavesas a perspective heating technique to generate a thermal gradient in fer- romagnetic insulator |normal metal systems to study the static and temporal dynamics of the longitudinal spin Seebeck effect. The static measurements provide cru- cial information about the direction of the spin current flow in the spin pumping and longitudinal SSE processes. The experiment demonstrates that in the longitudinal SSE a spin current flows from the normal metal (hot) towards the ferromagnet (cold) while in the spin pump- ing case, the flow is opposite. The temporal dynamics of the longitudinal SSE experiment manifests the sub- microsecond timescale of the effect which is slower than the spin pumpingprocess. Thethermal magnondiffusion model can explain the outcomes of the experiment and leadstoconcludethatthetimescaleoftheeffect reliesthe evolution of the vertical thermal gradient in the vicinity of the ferromagnet |normal metal interface. From our ex- periment, a typical magnon diffusion length of 500 nm is estimated for the YIG |Pt system. The authors thank A. V. Chumak, M. B. Jungfleisch, and P. Pirro for valuable discussions. M.A. was sup- ported by a fellowship of the Graduate School Material Sciences in Mainz (MAINZ) through DFG funding of the Excellence Initiative (GSC-266). We acknowledge financial support by Deutsche Forschungsgemeinschaft (SE 1771/4) within Priority Program 1538 “Spin Caloric Transport”, and technical support from the Nano Struc- turing Center, TU Kaiserslautern. 1K. Uchida, S. Takahashi, K. Harii, J. 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2014-07-18

A microwave technique is employed to simultaneously examine the spin pumping and the spin Seebeck effect processes in a YIG|Pt bilayer system. The experimental results show that for these two processes, the spin current flows in opposite directions. The temporal dynamics of the longitudinal spin Seebeck effect exhibits that the effect depends on the diffusion of bulk thermal-magnons in the thermal gradient in the ferromagnetic-insulator|normal-metal system.

Microwave-induced spin currents in ferromagnetic-insulator|normal-metal bilayer system

1407.4957v2

arXiv:2301.05820v1 [quant-ph] 14 Jan 2023Quantum entanglement generation on magnons assisted with m icrowave cavities coupled to a superconducting qubit Jiu-Ming Li and Shao-Ming Fei∗ School of Mathematical Sciences, Capital Normal Universit y, Beijing 100048, China We present protocols to generate quantum entanglement on no nlocal magnons in hybrid systems composed of yttrium iron garnet (YIG) spheres, microwave ca vities and a superconducting (SC) qubit. In the schemes, the YIGs are coupled to respective mic rowave cavities in resonant way, and the SC qubit is placed at the center of the cavities, which int eracts with the cavities simultaneously. By exchanging the virtual photon, the cavities can indirect ly interact in the far-detuning regime. Detailed protocols are presented to establish entanglemen t for two, three and arbitrary Nmagnons with reasonable fidelities. Keywords: magnon, superconducting qubit, quantum electro dynamics, quantum entanglement, indirect in- teraction I. INTRODUCTION Quantum entanglement is one of the most important features in quantum mechanics. The quantum entan- gled states [1–4] are significant ingredients in quantum information processing. Over past decades, various the- oretical and experimental proposals have been presented for processing quantum information by using varioussys- tems such as atoms [5–14], spins [15–21], ions [22–29], photons [5, 30–39], phonons [40–42], and so on. With the development of technologies, the quantum entanglement hasbeenestablishednotonlyin microscopicsystems, but also in the macroscopic systems such as superconducting circuits [43–48] and magnons system [49–54]. Hybrid systems exploit the advantages of different quantum systems in achieving certain quantum tasks, such as creating quantum entanglement and carryingout quantum logic gates. Many works have been presented so far for quantum information processing in the hybrid systems [55–58]. For instance, as an important quan- tum technology [59], the hybrid quantum circuits com- bine superconducting systems with other physical sys- tems which can be fabricated on a chip. The supercon- ducting (SC) qubit circuits [60, 61], based on the Joseph- son junctions, can exhibit quantum behaviors even at macroscopic scale. Generally, the interaction between the SC qubits and the environment, e.g., systems in strong or even ultrastrong coupling regime via quantized electromagnetic fields, would result in short coherence time. Thus many researches on circuit quantum electro- dynamics (QED) [62] have been presented with respect to the SC qubits, superconducting coplanar waveguide resonators, LCresonators and so on. This circuit QED focuses on studies of the light-matter interaction by us- ing the microwave photons, and has become a relative independent research field originated from cavity QED. The hybrid systems composed of collective spins (magnons) in ferrimagnetic systems and other systems ∗Electronic address: feishm@cnu.edu.cnare able to constitute the magnon-photon [63, 64], magnon-phonon[65–67], magnon-photon-phonon[49, 50, 68] systems and so on, giving rise to new interesting ap- plications. Ferrimagnetic systems such as yttrium iron garnet (YIG) sphere have attracted considerable atten- tion in recent years, which provide new platforms for in- vestigating the macroscopic quantum phenomena partic- ularly. Such systems are able to achieve strong and even ultrastrong couplings [69] between the magnons and the microwave photons, as a result of the high density of the collective spins in YIG and the lower dissipation. The YIG has the unique dielectric microwave properties with very lower microwave magnetic loss parameter. Mean- while, some important works have been presented on magnon Kerr effect [70, 71], quantum transduction [72], magnon squeezing [73, 74], magnon Fock state [75] and entanglement of magnons. For example, In 2018 Li et al. [49] proposed a system consisted of magnons, microwave photonsandphononsforestablishingtripartiteentangled states based on the magnetostrictiveinteraction and that the entangled state in magnon-photon-phonon system is robust. In 2019 Li et al.[50] constructed the entangled state of two magnon modes in a cavity magnomechani- cal system by applying a strong red-detuned microwave field on a magnon mode to activate the nonlinear mag- netostrictive interaction. In 2021 Kong et al.[52] used the indirect coherent interaction for accomplishing two magnonsentanglementand squeezingvia virtualphotons in the ferromagnetic-superconducting system. Inthiswork,wefirstpresentahybridsystemcomposed of two YIG spheres, two identical microwave cavities and a SC qubit to establish quantum entanglement on two nonlocal magnons. In this system, two YIGs are coupled to respective microwave cavities that cross each other. And a SC qubit is placed at the center of the crossing of twoidenticalcavities,namely,theSCqubitinteractswith the two cavities simultaneously. The magnons in YIGs can be coupled to the microwave cavities in the resonant way, owing to that the frequencies of two magnons can be tuned by biased magnetic fields, respectively. Com- pared with other works, the SC qubit is coupled to the two microwavecavities in the far-detuning regime, mean-2 FIG. 1: (Color online) Schematic of the hybrid system com- posed oftwoyttriumiron garnet spheres coupledtorespecti ve microwave cavities. Two cavities cross each other, and a su- perconducting qubit (black spot) is placed at the center of t he crossing. ing that the two identical cavities indirectly interact with each other by exchanging virtual photons. Then, we give the effective Hamiltonian of the subsystem composed of the SC qubit and two cavities, and present the protocol of entanglement establishment. In Sec. III, we consider the caseofthreemagnons. Inthe hybridsystemshownin Fig.3, the three identical microwave cavities could indi- rectly interact via the virtual photons, and each magnon is resonant with the respective cavity by tuning the fre- quency of the magnon. At last, we get the isoprobability entanglement on three nonlocal magnons. Moreover, the hybrid system composed of Nmagnons, Nidentical mi- crowave cavities and a SC qubit is derived in Sec. IV. We summarize in Sec. V. II. QUANTUM ENTANGLEMENT ON TWO NONLOCAL MAGNONS A. Hamiltonian of the hybrid system We consider a hybrid system, see Fig.1, in which two microwavecavitiescrosseachother,twoyttriumirongar- net (YIG) spheres are coupled to the microwave cavities, respectively. A superconducting (SC) qubit, represented by black spot in the Fig.1, is placed at the center of the crossing in order to interact with the two microwave cav- ities simultaneously. The YIG spheres are placed at the antinode of two microwave magnetic fields, respectively, and a static magnetic field is locally biased in each YIG sphere. In our model, the SC qubit is a two-level system with ground state |g/an}bracketri}htqand excited state |e/an}bracketri}htq. The magnetostatic modes in YIG can be excited when the magnetic component of the microwave cavity field is perpendicular to the biased magnetic field. We only con- sider the Kittel mode [76] in the hybrid system, namely,the magnon modes can be excited in YIG. The fre- quency of the magnon is in the gigahertz range. Thus the magnon generally interacts with the microwave pho- ton via the magnetic dipole interaction. The frequency of the magnon is given by ωm=γH, whereHis the biased magnetic field and γ/2π= 28 GHz/T is the gyro- magnetic ratio. In recent years, some experiments have already real- izedthestrongandultrastrongmagnon-magnoncoupling [77–79] as well as the magnon-qubit interaction [80, 81], which means that in the hybrid system shown in Fig.1 the magnon is both coupled to the SC qubit and an- other magnon. However, we mainly consider that the magnons which frequencies are tuned by the locally bi- ased static magnetic fields can be resonant with the cav- ities. In the meantime, the two cavities modes interact indirectly in the far-detuning regime for exchanging pho- tons. The entanglement of two nonlocal magnons can be constructed by using two cavities and the SC qubit. Given that there are magnon-magnon and magnon-qubit interactions, the magnon can be detuned with the qubit and another magnon in order to neglect their interac- tions. In the rotating wave approximation the Hamilto- nian of the hybrid system is ( /planckover2pi1= 1 hereafter) [82] H(S)=H0+Hint H0=ωm1m† 1m1+ωm2m† 2m2+1 2ωqσz +ωa1a† 1a1+ωa2a† 2a2 Hint=λm1(a1m† 1+a† 1m1)+λm2(a2m† 2+a† 2m2) +λq1(a1σ++a† 1σ)+λq2(a2σ++a† 2σ).(1) Here,H0is the free Hamiltonian of the two cavities, two magnonsandtheSCqubit. Hintisthe interactionHamil- tonian among the cavities, magnons and SC qubit. ωm1 andωm2are the frequencies of the two magnons, which are tunable under biased magnetic fields, respectively. ωa1andωa2are the frequencies of two cavities, and ωq is the state transition frequency between |g/an}bracketri}htq↔ |e/an}bracketri}htqof the SC qubit. In the Kittel mode, the collective spins in YIGs can be expressed by the boson operators. m1(m2) andm† 1(m† 2) are the annihilation and creation opera- tors of magnon mode 1 (2). a1(a2) anda† 1(a† 2) denote the annihilation and creation operators of cavity mode 1 (2), respectively. They satisfy commutation relations [O,O†] = 1 forO=a1,a2,m1,m2.σz=|e/an}bracketri}htq/an}bracketle{te|−|g/an}bracketri}htq/an}bracketle{tg|. σ=|g/an}bracketri}htq/an}bracketle{te|andσ+=|e/an}bracketri}htq/an}bracketle{tg|are the lowing and rais- ing operators of the SC qubit. λq1(λq2) is the coupling strengthbetweenthe SCqubit andthe cavitymode1(2). λm1(λm2) is the coupling between the magnon mode 1 (2) and the cavity mode 1 (2). As mentioned above, the two microwave cavities are identical ones with the same frequency ωa1=ωa2=ωa. Meanwhile, one can assume that λq1=λq2=λq. In the3 interaction picture with respect to e−iH0t, the Hamilto- nian is expressed as H(I)=λm1a1m† 1eiδ1t+λm2a2m† 2eiδ2t+λqa1σ+ei∆1t +λqa2σ+ei∆2t+H.c., (2) whereδ1=ωm1−ωa,δ2=ωm2−ωa, ∆1=ωq−ωa and ∆ 2=ωq−ωa. The SC qubit is coupled to the two cavities simultaneously. Owing to ∆ 1= ∆2= ∆0/ne}ationslash= 0 and ∆ 0≫λq, the two identical microwave cavities indi- rectlyinteractwitheachotherinthefar-detuningregime. Therefore, the effective Hamiltonian of the subsystem composedofthe twomicrowavecavitiesand the SC qubit in the far-detuning regime is given by [83] Heff=/tildewideλq/bracketleftBig σz(a† 1a1+a† 2a2+a† 1a2+a1a† 2)+2|e/an}bracketri}htq/an}bracketle{te|/bracketrightBig ,(3) where/tildewideλq=λ2 q/∆0. B. Entangled state generation on two nonlocal magnons We now give the protocol of quantum entanglement generation on two nonlocal magnons. Generally, the magnon can be excited by a drive magnetic field. For convenience the state of magnon 1 is prepared as |1/an}bracketri}htm1 via the magnetic field. The initial state of the hybrid system is |ϕ/an}bracketri}ht0=|1/an}bracketri}htm1|0/an}bracketri}htm2|0/an}bracketri}hta1|0/an}bracketri}hta2|g/an}bracketri}htq, in which the two cavities are all in the vacuum state, magnon 2 is in the state|0/an}bracketri}htm2, and the SC qubit is in state |g/an}bracketri}htqwhich is unal- tered all the time due to the indirect interaction between the two cavities. step 1: The frequency of magnon 1 is tuned to be ωm1=ωa1so that the cavity 1 could be resonated with it. Therefore, the magnon 1 and cavity 1 are in a super- posed state after time T1=π/4λm1. The local evolution is|1/an}bracketri}htm1|0/an}bracketri}hta1→1√ 2(|1/an}bracketri}htm1|0/an}bracketri}hta1−i|0/an}bracketri}htm1|1/an}bracketri}hta1), which means that the states of SC qubit, magnon 2 and cavity 2 are unchanged due to decoupling between the SC and two cavities, and the magnon 2 is far-detuned with cavity 2. The state evolves to |ϕ/an}bracketri}ht1=1√ 2(|1/an}bracketri}htm1|0/an}bracketri}hta1−i|0/an}bracketri}htm1|1/an}bracketri}hta1) ⊗|0/an}bracketri}htm2⊗|0/an}bracketri}hta2⊗|g/an}bracketri}htq. (4) step 2: The magnons are tuned to far detune with respective cavities. From Eq. (3), the evolution of sub- system composedof twomicrowavecavities and SC qubit is given by |χ(t)/an}bracketri}htsub=ei/tildewideλqt/bracketleftbig cos(/tildewideλqt)|1/an}bracketri}hta1|0/an}bracketri}hta2+isin(/tildewideλqt)|0/an}bracketri}hta1|1/an}bracketri}hta2/bracketrightbig ⊗|g/an}bracketri}htq (5) under the condition ∆ 0≫λq. After timeT2=π/2/tildewideλq, the evolution between two cav- ities is|1/an}bracketri}hta1|0/an}bracketri}hta2→ −|0/an}bracketri}hta1|1/an}bracketri}hta2, which indicates that thephoton can be indirectly transmitted between the two cavities, with the state of SC qubit unchanged. There- fore, the state after this step changes to |ϕ/an}bracketri}ht2=1√ 2(|1/an}bracketri}htm1|0/an}bracketri}hta1|0/an}bracketri}hta2+i|0/an}bracketri}htm1|0/an}bracketri}hta1|1/an}bracketri}hta2) ⊗|0/an}bracketri}htm2⊗|g/an}bracketri}htq. (6) step 3: The frequency of magnon 2 is tuned with ωm2=ωa2to resonate with the cavity 2. In the mean- time the cavities are decoupled to the SC qubit and the magnon 1 is far detuned with the cavity 1. After time T3=π/2λm2, the local evolution |0/an}bracketri}htm2|1/an}bracketri}hta2→ −i|1/an}bracketri}htm2|0/an}bracketri}hta2 is attained. The final state is |ϕ/an}bracketri}ht3=1√ 2(|1/an}bracketri}htm1|0/an}bracketri}htm2+|0/an}bracketri}htm1|1/an}bracketri}htm2) ⊗|0/an}bracketri}hta1⊗|0/an}bracketri}hta2⊗|g/an}bracketri}htq, (7) which is just the single-excitation Bell state on two non- local magnons. In the whole process, we mainly consider the interac- tions between the magnonsand the cavities, and between the cavities and the SC qubit. However, the SC qubit can be coupled to the magnons. In terms of Ref.[80], the interactions between the magnons and the SC qubit are described as Hqm,1=λqm,1(σ+m1+H.c.) and Hqm,2=λqm,2(σ+m2+H.c.) whereλqm,1=λqλm1/∆0 andλqm,2=λqλm2/∆0, while the conditions ωq=ωm1 andωq=ωm2are attained. In the meantime, the two magnons are interacts each other by using the SC qubit. Generally, the frequencies of two magnon modes are tuned by the locally biased magnetic fields. There- fore, the magnon can be detuned with the SC qubit and another magnon in order to neglect the interactions be- tween the magnons and the SC qubit. C. Numerical result We here simulate [84] the fidelity of the Bell state on two nonlocal magnons by considering the dissipations of all constituents of the hybrid system. The realistic evo- lution of the hybrid system composed of magnons, mi- crowave cavities and SC qubit is governed by the master equation ˙ρ=−i[H(I),ρ]+κm1D[m1]ρ+κm2D[m2]ρ +κa1D[a1]ρ+κa2D[a2]ρ+γqD[σ]ρ.(8) Here,ρis the density operator of the hybrid system, κm1 andκm2are the dissipation rates of magnon 1 and 2, κa1andκa2denote the dissipation rates for the two mi- crowave cavities 1 and 2, γqis the dissipation rate of the SC qubit, D[X]ρ=(2XρX†−X†Xρ−ρX†X)/2 for X=m1,m2,a1,a2,σ. The fidelity of the entangled state of two nonlocal magnons is defined by F=3/an}bracketle{tϕ|ρ|ϕ/an}bracketri}ht3. The related parameters are chosen as ωq/2π= 7.92 GHz,ωa/2π= 6.98 GHz,λq/2π= 83.2 MHz,λm1/2π=4 74 78 82 86 90 9488909294 λq/2π [MHz] Fidelity (%)(a) 0.6 0.8 1.0 1.2909294 (κa)−1 [µs] Fidelity (%)(b) 0.8 1.0 1.2 1.4 1.6909294 (κm)−1 [µs]Fidelity (%)(c) 0.6 0.8 1.0 1.29091929394 (γq)−1 [µs] Fidelity (%)(d) FIG. 2: (a) The fidelity of the Bell state of two nonlocal magno ns with respect to the coupling strength λq. Since/tildewideλq=λ2 q/∆0 in Eq.(3), the fidelity is similar to parabola. (b)-(d) The fid elity of the Bell state versus the dissipations of cavities, magnons, and SC qubit, respectively. 15.3 MHz,λm2/2π= 15.3 MHz [81], κm1/2π=κm2/2π= κm/2π= 1.06 MHz,κa1/2π=κa2/2π=κa/2π= 1.35 MHz [69], γq/2π= 1.2 MHz [80]. The fidelity of the entanglement between two nonlocal magnons can reach 92.9%. The influences of the imperfect relationship among pa- rameters is discussed next. The Fig.2(a) shows the fi- delity influenced by the coupling strength between the microwave cavities and the SC qubit. Since /tildewideλq=λ2 q/∆0 in Eq.(3), the fidelity is similar to parabola. In Fig.2(b)- (d), we give the fidelity varied by the dissipations of cav- ities, magnons, and SC qubit. As a result of the virtual photon, the fidelity is almost unaffected by the SC qubit, shown in Fig.2(d). III. ENTANGLEMENT GENERATION FOR THREE NONLOCAL MAGNONS A. Entangled state of three nonlocal magnons Similar to the protocol of entangled state generation for two nonlocal magnons in two microwave cavities, we consider the protocol for entanglement of three nonlocalmagnons. As shown in Fig.3, similar to the hybrid sys- tem composed of two magnons coupled to the respective microwave cavities and a SC qubit in Fig.1, there are three magnons in three YIGs coupled to respective mi- crowave cavities and a SC qubit placed at the center of the three identical cavities ( ωa1=ωa2=ωa3=ωa). Each magnon is in biased static magnetic field and is located at the antinode of the microwave magnetic field. In the interaction picture, the Hamiltonian of the hy- brid system depicted in Fig.3 is H(I) 3=λm1a1m† 1eiδ1t+λm2a2m† 2eiδ2t +λm3a3m† 3eiδ3t+λqa1σ+ei∆1t +λqa2σ+ei∆2t+λqa3σ+ei∆3t+H.c.,(9) whereλm3is the coupling strength between magnon 3 and microwavecavity3, a3andm† 3areannihilation oper- ator of the cavity 3 and creation operator of the magnon 3, respectively. λqis the coupling between the SC qubit and three cavities, δ3=ωm3−ωa. The frequency ωm3 can be tuned by the biased magnetic field in microwave cavity 3. ∆ 3=ωq−ωa= ∆0. At the beginning we have the initial state |ψ/an}bracketri}ht(3) 0= |ψ/an}bracketri}ht(3) m⊗|ψ/an}bracketri}ht(3) a⊗|g/an}bracketri}htqwith|ψ/an}bracketri}ht(3) m=|1/an}bracketri}htm1|0/an}bracketri}htm2|0/an}bracketri}htm3=|100/an}bracketri}htmand5 FIG. 3: (Color online) Schematic of the hybrid system com- posed of three yttrium iron garnet spheres coupled to respec - tivemicrowave cavities. Asuperconductingqubit(blacksp ot) is placed at the center of the three cavities. |ψ/an}bracketri}ht(3) a=|0/an}bracketri}hta1|0/an}bracketri}hta2|0/an}bracketri}hta3=|000/an}bracketri}hta. Thesingle-excitationissetin the magnon 1. The magnon 1 is resonant with the cavity 1 by tuning the frequency of magnon 1, and the SC qubit is decoupled to the cavities. After time T(3) 1=π/2λm1, the local evolution |1/an}bracketri}htm1|0/an}bracketri}hta1→ −i|0/an}bracketri}htm1|1/an}bracketri}hta1is attained. The state is evolved to |ψ/an}bracketri}ht(3) 1=−i|000/an}bracketri}htm|100/an}bracketri}hta|g/an}bracketri}htq. (10) The SC qubit is coupled to the three identical mi- crowave cavities at the same time in far-detuning regime ∆0≫λq. Therefore, the effective Hamiltonian of the subsystem composed of the SC qubit and the three iden- tical cavities is of the form [83] H(3) eff=/tildewideλq/bracketleftbigg σz(a† 1a1+a† 2a2+a† 3a3)+3|e/an}bracketri}htq/an}bracketle{te| +σz(a1a† 2+a1a† 3+a2a† 3+H.c.)/bracketrightbigg .(11) The magnons are then all detuned with the cavities. The local evolution e−iH(3) efft|100/an}bracketri}hta|g/an}bracketri}htof the subsystem is given by |χ(t)/an}bracketri}ht(3) sub=/bracketleftbigg C(3) 1,t|100/an}bracketri}hta+C(3) 2,t|010/an}bracketri}hta+C(3) 3,t|001/an}bracketri}hta/bracketrightbigg ⊗|g/an}bracketri}htq, (12) whereC(3) 1,t=ei3/tildewideλqt+2 3andC(3) 2,t=C(3) 3,t=ei3/tildewideλqt−1 3. It is easy to derive that |C(3) 1,t|2+|C(3) 2,t|2+|C(3) 3,t|2= 1. (13) Fig.4 shows the probability related to the states |100/an}bracketri}hta|000/an}bracketri}htm|g/an}bracketri}htq,|010/an}bracketri}hta|000/an}bracketri}htm|g/an}bracketri}htqand|001/an}bracketri}hta|000/an}bracketri}htm|g/an}bracketri}htq. In0 1 2 3 4 5 6 70.00.51.0 3λ2 qt/∆0Pn |C1,t(3)|2 |C2,t(3)|2=|C3,t(3)|2 FIG. 4: (Color online) Evolution probabilities of the state s: P1=|C(3) 1,t|2for|100/angbracketrighta|000/angbracketrightm|g/angbracketrightq(red),P2=|C(3) 2,t|2for |010/angbracketrighta|000/angbracketrightm|g/angbracketrightq,P3=|C(3) 3,t|2for|001/angbracketrighta|000/angbracketrightm|g/angbracketrightq, andP2= P3(blue). particular, one has |C(3) 1,t|2=|C(3) 2,t|2=|C(3) 3,t|2=1 3, with C(3) 1=√ 3+i 2√ 3,C(3) 2=C(3) 3=−√ 3+i 2√ 3(14) at timeT(3) 2= 2π/9/tildewideλq. Correspondingly, the state evolves to |ψ/an}bracketri}ht(3) 2=/bracketleftbigg√ 3+i 2√ 3|100/an}bracketri}hta+−√ 3+i 2√ 3|010/an}bracketri}hta+−√ 3+i 2√ 3|001/an}bracketri}hta/bracketrightbigg ⊗(−i)|000/an}bracketri}htm⊗|g/an}bracketri}htq. (15) Finally, the magnonscan be resonatedwith the respec- tive cavities under the condition {δ1,δ2,δ3}= 0. The local evolution and the time are |0/an}bracketri}htmk|1/an}bracketri}htak→ −i|1/an}bracketri}htmk|0/an}bracketri}htak andT(3) 3k=π/2λmk(k= 1,2,3), respectively. Thus the final state is |ψ/an}bracketri}ht(3) 3=−/bracketleftbigg√ 3+i 2√ 3|100/an}bracketri}htm+−√ 3+i 2√ 3|010/an}bracketri}htm+−√ 3+i 2√ 3|001/an}bracketri}htm/bracketrightbigg ⊗|000/an}bracketri}hta⊗|g/an}bracketri}htq. (16) In the whole process, the state of the SC qubit is kept unchanged. B. Numerical result The entanglement fidelity of three nonlocal magnons is given here by taking into account the dissipations of hy- brid system. Firstly, the master equation which governs the realistic evolution of the hybrid system composed of three magnons, three microwave cavities and a SC qubit can be expressed as ˙ρ(3)=−i[H(I) 3,ρ(3)]+κm1D[m1]ρ(3)+κm2D[m2]ρ(3) +κm3D[m3]ρ(3)+κa1D[a1]ρ(3)+κa2D[a2]ρ(3) +κa3D[a3]ρ(3)+γqD[σ]ρ(3), (17)6 0.6 0.8 1.0 1.281838587 (κa)−1 [µs]Fidelity (%)(a) 0.8 1.0 1.2 1.4 1.6838485 (κm)−1 [µs]Fidelity (%)(b) 0.6 0.8 1.0 1.28283848586 (γq)−1 [µs]Fidelity (%)(c) FIG. 5: (a)-(c) The fidelity of the entanglement on three nonlocal magnons versus the dissipations of cavities, magn ons and SC qubit. whereρ(3)is the density operator of realistic evolu- tion of the hybrid system, κm3is the dissipation rate of magnon 3 with κm3/2π=κm/2π= 1.06 MHz [69],κa3denotes the dissipation rate for the microwave cavities 3 with κa3/2π=κa/2π= 1.35 MHz [69], D[X]ρ(3)=(2Xρ(3)X†−X†Xρ(3)−ρ(3)X†X)/2 for any X=m1,m2,m3,a1,a2,a3,σ. The entanglement fidelity for three nonlocal magnons is defined by F(3)=(3) 3/an}bracketle{tψ|ρ(3)|ψ/an}bracketri}ht(3) 3, which can reach 84.9%. The fidelity with respect to the parameters is FIG. 6: (Color online) Schematic of the hybrid system com- posed of Nyttrium iron garnet spheres coupled to respective microwave cavities. A superconducting qubit is placed at th e center of the Nidentical microwave cavities. shown in Fig.5. IV.NMAGNONS SITUATION In Sec. II and Sec. III, the entanglement of two and three nonlocal magnons have been established. In this section we consider the case of Nmagnons. In the hy- brid system shown in Fig.6, the SC qubit is coupled to Ncavity modes that have the same frequencies ωa. A magnon is coupled to the cavity mode in each cavity. Each magnon is placed at the antinode of microwave magnetic field of the respective cavity and biased static magnetic field. In the interaction picture the Hamiltonian of whole system shown in Fig.6 can be expressed as H(I) N=/summationdisplay n/bracketleftbigg λmn(anm† neiδnt+H.c.) +λq(anσ+ei∆nt+H.c.)/bracketrightbigg ,(18) whereanandm† n(n= 1,2,3,···,N) are the annihi- lation operator of the nth cavity mode and the creation operatorofthe nth magnon, λmnis the couplingbetween thenth magnon and the nth cavity mode, λqdenotes the coupling strength between the SC qubit and the nth cav- ity mode,δn=ωmn−ωa,ωmnis the frequency of the nth magnon, ∆ n= ∆0=ωq−ωa. The initial state is prepared as |ψ/an}bracketri}ht(N) 0=|ψ/an}bracketri}ht(N) m⊗|ψ/an}bracketri}ht(N) a⊗|g/an}bracketri}htq, (19) |ψ/an}bracketri}ht(N) m=|1/an}bracketri}htm1|0/an}bracketri}htm2|0/an}bracketri}htm3···|0/an}bracketri}htmN=|100···0/an}bracketri}htm, |ψ/an}bracketri}ht(N) a=|0/an}bracketri}hta1|0/an}bracketri}hta2|0/an}bracketri}hta3···|0/an}bracketri}htaN=|000···0/an}bracketri}hta. At first, we tune the frequency of magnon 1 under the conditionδ1= 0. The magnon 1 is resonant with the7 cavity 1, which means that the single photon is trans- mitted to cavity 1, and the SC qubit is decoupled to all the cavities. The state evolves to |ψ/an}bracketri}ht(N) 1=−i|000···0/an}bracketri}htm|100···0/an}bracketri}hta|g/an}bracketri}htq(20) after timeT(N) 1=π/2λm1. Next the magnons are tuned to detune with respec- tive cavities. The SC qubit is coupled to the Nmi- crowave cavities at the same time in far-detuning regime ∆0≫λq. Under the condition ∆ n= ∆0, the effective Hamiltonian of the subsystem composed of the SC qubit andNmicrowave cavities is of the form [83] H(N) eff=/summationdisplay n/tildewideλq/bracketleftbigg σza† nan+|e/an}bracketri}htq/an}bracketle{te|/bracketrightbigg +/summationdisplay l<n/tildewideλq/bracketleftbigg σz(ala† n+H.c.)/bracketrightbigg .(21) Consequently, the evolution of the hybrid system is given by |ψ/an}bracketri}ht(N) 2=/bracketleftbigg C(N) 1,t|100···0/an}bracketri}hta+C(N) 2,t|010···0/an}bracketri}hta +C(N) 3,t|001···0/an}bracketri}hta+···+C(N) N,t|000···1/an}bracketri}hta/bracketrightbigg ⊗(−i)|000···0/an}bracketri}htm⊗|g/an}bracketri}htq, (22) where C(N) 1,t=eiN/tildewideλqt+(N−1) N, C(N) 2,t=C(N) 3,t=···=C(N) N,t=eiN/tildewideλqt−1 N.(23) In addition, we have the following relation /summationdisplay n|C(N) n,t|2=|C(N) 1,t|2+|C(N) 2,t|2+|C(N) 3,t|2+···+|C(N) N,t|2 = 1 (24) by straightforward calculation. At last, the SC qubit is decoupled to the cavities, and the magnons are resonant with the cavities, respectively. Thus, after the time T(N) 3n=π/2λmn, the final state is given by |ψ/an}bracketri}ht(N) 3=−/bracketleftbigg C(N) 1,t|100···0/an}bracketri}htm+C(N) 2,t|010···0/an}bracketri}htm +C(N) 3,t|001···0/an}bracketri}htm+···+C(N) N,t|000···1/an}bracketri}htm/bracketrightbigg ⊗|000···0/an}bracketri}hta⊗|g/an}bracketri}htq. (25) In the whole process, the state of SC qubit is unchanged all the time. [Remark] Concerning the coefficients Eq. (23), the probabilities with respect to the states|100···0/an}bracketri}htm|000···0/an}bracketri}hta|g/an}bracketri}htq,|010···0/an}bracketri}htm|000···0/an}bracketri}hta|g/an}bracketri}htq, |001···0/an}bracketri}htm|000···0/an}bracketri}hta|g/an}bracketri}htq,···,|000···1/an}bracketri}htm|000···0/an}bracketri}hta|g/an}bracketri}htq arep(N) 1=|C(N) 1,t|2,p(N) 2=|C(N) 2,t|2,p(N) 3=|C(N) 3,t|2,···, p(N) N=|C(N) N,t|2, respectively, and p(N) 2=p(N) 3=···= p(N) N. If the condition p(N) 1=p(N) 2can be attained, the isoprobability entanglement can be obtained. For instance, for N= 4, the entangled state of the four nonlocal magnons is given by |ψ/an}bracketri}ht(4) 3=−1 2/bracketleftbigg |1000/an}bracketri}htm−|0100/an}bracketri}htm−|0010/an}bracketri}htm−|0001/an}bracketri}htm/bracketrightbigg ⊗|0000/an}bracketri}hta⊗|g/an}bracketri}htq. (26) However, if N/greaterorequalslant5, the isoprobability entanglement does not exist as a result of p(N) 1/ne}ationslash=p(N) 2, see illustration in Fig.7(b)(c). V. SUMMARY AND DISCUSSION We have presented protocols of establishing entan- glement on magnons in hybrid systems composed of YIGs, microwave cavities and a SC qubit. By exploit- ing the virtual photon, the microwave cavities can indi- rectly interact in far-detuning regime, and the frequen- cies of magnons can be tuned by the biased magnetic field, which leads to the resonant interaction between the magnons and the respective microwavecavities. We have constructed single-excitation entangled state on two and three nonlocal magnons, respectively, and the entangle- ment for Nmagnons has been also derived in term of the protocol for three magnons. By analyzing the coefficients in Eq. (23), the isoprob- ability entanglement has been also constructed for cases N= 2,N= 3 andN= 4. In particular, such isoproba- bility entanglement no longer exists for N/greaterorequalslant5. In the protocol for the case of two magnons discussed in Sec. II, we have firstly constructed the superposition of magnon 1 and microwave cavity 1. Then the photon could be transmitted |1/an}bracketri}hta1|0/an}bracketri}hta2→ −|0/an}bracketri}hta1|1/an}bracketri}hta2between two cavities. At last, the single-excitation Bell state is finally constructedinresonantway. Asfor N/greaterorequalslant3,however,such method is no longer applicable because of |100···0/an}bracketri}hta/notarrowright α2|010···0/an}bracketri}hta+α3|001···0/an}bracketri}hta+···+αN|000···1/an}bracketri}hta, namely, p(N) 1/ne}ationslash= 0. Acknowledgements This work is supported by the National Natural Sci- ence Foundation of China (NSFC) under Grant Nos. 12075159and 12171044,Beijing Natural Science Founda- tion (Grant No. Z190005), the Academician Innovation Platform of Hainan Province.8 0 2 4 6 8 10 12 1400.51 4λq2t/∆0pn(4)(a) p1(4) p2(4) 0 2 4 6 8 10 12 1400.51 5λq2t/∆0pn(5)(b) p1(5) p2(5) 0 2 4 6 8 10 12 1400.51 6λq2t/∆0pn(6)(c) p1(6) p2(6) FIG. 7: (a)-(c) Evolution probabilities for N= 4 (left), N= 5 (middle), and N= 6 (right). If N/greaterorequalslant5,p(N) 1/negationslash=p(N) 2implies that the isoprobability entanglement does not exist. [1] A. Einstein, B. Podolsky, and N. Rosen, Can quantum- mechanical description of physical reality be considered complete, Phys. Rev. 47, 777 (1935) [2] D. Bouwmeester, J.-W. Pan, M. Daniell, H. Wein- furter, and A. Zeilinger, Observation of three-photon Greenberger-Horne-Zeilinger entanglement, Phys. Rev. Lett.82, 1345 (1999) [3] W. D¨ ur, G. Vidal, and J. I. 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2023-01-14

We present protocols to generate quantum entanglement on nonlocal magnons in hybrid systems composed of yttrium iron garnet (YIG) spheres, microwave cavities and a superconducting (SC) qubit. In the schemes, the YIGs are coupled to respective microwave cavities in resonant way, and the SC qubit is placed at the center of the cavities, which interacts with the cavities simultaneously. By exchanging the virtual photon, the cavities can indirectly interact in the far-detuning regime. Detailed protocols are presented to establish entanglement for two, three and arbitrary $N$ magnons with reasonable fidelities.

Quantum entanglement generation on magnons assisted with microwave cavities coupled to a superconducting qubit

2301.05820v1

1 Observation of nonlinear planar Hall effect in magnetic insulator/topological insulator heterostructures Yang Wang1, Sivakumar V. Mambakkam2, Yue-Xin Huang3, Yong Wang2, Yi Ji1, Cong Xiao4,5, Shengyuan A. Yang3, Stephanie A. Law1,2, and John Q. Xiao1,* 1 Department of Physics and Astronomy, University of Delaware, Newark, Delaware 19716, USA 2 Department of Materials Science and Engineering, University of Delaware, Newark, Delaware, 19716, USA 3 Research Laboratory for Quantum Materials, Singapore University of Technology and Design, Singapore 487372, Singapore 4 Department of Physics, The University of Hong Kong, Hong Kong, China 5 HKU -UCAS Joint Institute of Theoretical and Computational Physics, Hong Kong, China *jqx@udel.edu Abstract Interfacing topological insulators (TIs) with magnetic insulators (MIs) has been widely used to study the interaction between topological surface states and magnetism. Previous transport studies typic ally interpret the suppression of weak antilocalization or appearance of the anomalous Hall effect as signatures of magnetic proximity effect (MPE) imposed to TIs. Here, we report the observation of nonlinear planar Hall effect (NPHE) in Bi 2Se3 films grown on MI thulium and yttrium iron garnet (TmIG and YIG) substrates, which is an order of magnitude larger than that in Bi 2Se3 grown on nonmagnetic gadolinium gallium garnet (GGG) substrate. The nonlinear Hall resistance in TmIG/Bi 2Se3 depends linearly on the external magnetic field, while that in YIG/Bi 2Se3 exhibits an extra hysteresis loop around zero field. The magnitude of the NPHE is found to scale inversely with carrier density. We speculate the observed NPHE is related to the MPE -induced exchange gap opening and out -of-plane spin textures in the TI surface states, which may be used as an alternative transport signature of the MPE in MI/TI heterostructures. I. INTRODUCTION Magnetic topological insulators [1] have been intensively stu died in the past decade, because scientifically fundamental as well as technologically promising phenomena like quantum anomalous Hall effect (QAHE) [2] and topological magnetoelectric effect [ 3] can arise in the surface states of magnetic TIs. So far doping magnetic elements like Cr or V [4,5] into the TI (Bi,Sb) 2Te3 has been the most developed and robust method to observe QAHE [ 6,7]. Another approach to magnetize the topological surface states (TSS) is to proximately couple the TI to a magnetic material [8 -11], preferentially an insulator, so there is no doping or current shunting effect. Although robust anomalous Hall effect (AHE) has been reported in (Bi,Sb) 2Te3 coupled to MIs like TmIG [12] o r Cr 2Ge2Te6 [13], weak or absence of AHE is more commonly seen in Bi 2Se3 or Bi 2Te3 when grown or transferred on top of MI substrates, and the suppression of weak antilocalization (WAL) is often taken a s a qualitative signature of the MPE [14 -2 17]. Conventional surface -sensitive spectroscopic methods like a ngle-resolved photoemission spectroscopy (ARPES) cannot be applied to probe the band structure at the buried MI/TI interface, and x - ray- or neutron -based measurements have not been successful in detecting any induced magnetic moments in TIs grown on M Is [18 -20]. Therefore, searching an alternative and more sensitive transport probe of the MPE will be helpful to clarify and understand the interaction between magnetism and TSS in MI/TI heterostructures . In recent years, planar Hall effect (PHE) in both linear [21] and nonlinear [22] re gimes have been reported in nonmagnetic TIs. The linear PHE was interpreted as a result of magnetic -field-induced anisotropic backscattering [21] or magnetic -field-induced tilting of the Dirac cone with particle -hole (p - h) asymmetry [23,24]. The nonlinear PHE was attributed to the distortion or tilting of the Dirac dispersion with higher -order k terms like p -h asymmetry ( k2) or hexagonal warping ( k3) by the external magnetic field [22,24] . In this work, we report the observation of NPHE in Bi 2Se3 (BS) films grown on magnetic Tm 3Fe5O12 (TmIG) and Y 3Fe5O12 (YIG) substrates, which is an order of magnitude larger than that in Bi 2Se3 film grown on nonmagnetic Gd 3Ga5O12 (GGG) substrate. While the NPHE in TmIG/BS with out -of-plane (OP) easy axis shows a linear dependence on the in -plane (IP) magnetic field, the NPHE in YIG/BS with IP easy axis takes an extra hysteretic jump around zero field, reflecting the reversal of the IP YIG magnetization. The same temperature dependence of the linear -in-B and hysteretic components suggests they share the same origin. The carrier density dependence of the NPHE excludes hexagonal warping as the dominant mechanism. The enhancement of the NPHE in Bi 2Se3 grown on magnetic substrates as well as the sharp increase o f it below 30 K indicate MPE plays a critical role. Our results suggest the NPHE may work as a convenient and sensitive transport probe of the MPE in MI/TI heterostructures. II. METHODS The 8 quintuple -layer (QL) Bi 2Se3 films used in this study were grown on Tm 3Fe5O12, Y3Fe5O12, and Gd3Ga5O12 substrates with very close lattice constants and smooth surfaces in a molecular beam epitaxy (MBE) system with a base pressure of 1 ×10-9 Torr, following the two -step Se -buffer layer method reported in Ref. [25]. The TmIG (30 nm) and two YIG (2.5 𝜇m and 100 nm) films with OP and IP magnetic anisotropy respectively were deposited on GGG(111) substrates by magnetron sputtering or liquid phase epitaxy method . Prior Bi 2Se3 growth, the Tm IG and YIG substrates were soaked in Piranha solution (H2SO 4:H2O2=3:1) for 5 min to clean the surface [26]. After annealing the substrates at 650 °C for 30 min and cooling down to 50 °C in the growth chamber, ~2 nm thick amorphous Se and 1 nm Bi xSe1-x were deposited. Then the substrate temperature was slowly ramped to 325 °C at 10 °C/min to evaporate extra Se and crystalize the first QL Bi 2Se3. And the remaining 7 QL Bi 2Se3 was subsequently deposited by co - evaporating Bi and Se. After cooling to room t emperature, the Bi 2Se3 films were capped with 5 nm SiO 2 layer for protection in another magnetron sputtering chamber. As shown in Fig. 1(a), streaky reflection high-energy electron diffraction (RHEED) patterns were observed from the very first QL and x-ray diffraction (XRD) results of all the Bi 2Se3 films exhibit clear (0,0,3 n) peaks, indicating the c-axis growth orientation and the high and similar crystalline quality. 3 After the growth, the Bi 2Se3 films were fabricated into 200×100 𝜇m Hall bar devices by standard photolithography method and contacted with Ti/Au electrodes. The magneto -transport measurements were carried out in a home -built cryogenic system with base temperature 4.5 K. Low -frequency lock -in technique was used to detect the first - and second -harmonic longitudinal and transverse voltages. III. RESULTS AND DISCUSSION A. Linear transport properties Fig. 2(a) displays the temperature dependence of the longitudinal resistance of four Bi 2Se3 devices made on TmIG, YIG and GGG substrates. All of them show metallic behavior above 50 K . Below 50 K, the resistance upturn in TmIG and YIG/BS samples is more pronounced compared to that of the GGG/BS sample, especially in the YIG2/BS sample with a lower carrier density. Such insulating behavior may result from the suppression of WAL due to MPE [27] and is consistent with previous reports on iron garnet/Bi 2Se3 bilayers [14,16,17] . As shown in Fig. 2(b), from ordinary Hall effect (OHE) measurements we extracted the sheet carrier densities for these devices to be 2.45 -3.46 ×1013 cm-2. This means these Bi2Se3 samples are n-doped with a Fermi level of ~0.3 eV, so significant amount of the current is carried by the bulk states. We did not observe hysteretic or nonlinear AHE in the TmIG or YIG/BS samples after subtracting the linear OHE background. This suggests that the garnet/Bi 2Se3 interfaces formed here may not be as good as those in Ref. [17], where weak AHE was observed. Thi s does not rule out the existence of a small exchange -interaction -induced gap in the TSS in our samples, because when the gap ∆ is small, e.g. ~1 meV , and it is much smaller than the Fermi energy 𝜀F, the AH conductivity 𝜎𝑦𝑥AH∝8𝑒2 ℏ(∆ 𝜀F)3 [28] will give rise to a n AH resistance in the order of 0.1 mΩ, which can not be discerned from the large OHE background. Although AHE was not detected, we observed suppression of WAL in Bi 2Se3 films grown on TmIG and YIG substrates as compared with that on GGG [Fig. 2(c)]. By fitting to the Hikami -Larkin - Nagaoka equation, the electron phase coherence of length of the three Bi 2Se3 samples on GGG, TmIG, and YIG are 203, 119, and 90 nm, respectively. Given the similar crystalline quality [Fig. 1] and carrier densities [Fig. 2(b)], the suppressed WAL in TmIG and YIG/BS is most likely due to MPE -induced OP spin textures and correspondingly, the reduced Berry phase of TI surface electrons [30]. B. Observat ion of the NPHE The linear PHE has sin𝜙𝐵cos𝜙𝐵 dependence on the IP magnetic field direction [21,23] where 𝜙𝐵 is the angle between the current and magnetic field directions. Thus, the first -order Hall voltage is zero at 𝜙𝐵=𝑛90° with n being an integer. Differently, the NPHE depends on the IP magnetic field as 𝐵cos𝜙𝐵 [22,24], so it can be detected by sweepin g the magnetic field between 0 and 180 °. As illustrated in Fig. 3(a) inset, in our experiments we sent a sinusoidal a.c. current to the Hall channel in the x-direction, and measured the second harmonic Hall voltage 𝑉𝑦2𝜔 while sweeping the external magne tic field also in the x- direction. As shown in Fig. 3, the second harmonic Hall resistance 𝑅𝑦𝑥2𝜔=𝑉𝑦2𝜔/𝐼 as a function of B for three Bi 2Se3 devices fabricated on GGG, TmIG, and YIG substrates have dramatically different behaviors. Compared with the TmIG and YIG/BS samples, the magnitude of the NPHE in GGG/BS measured by the slope 𝑑𝑅 𝑦𝑥2𝜔/𝑑𝐵 is an order of magnitude smaller. In the TmIG/BS bilayer with OP magnetic anisotropy, 4 the NPHE is enhanced and exhibits a linear dependence on the IP magne tic field [Fig. 3(b)]. When Bi2Se3 is grown on YIG substrate with IP anisotropy, 𝑅𝑦𝑥2𝜔 not only exhibits a linear dependence on B, but also takes a hysteretic shape around zero field, corresponding to the switching of the IP magnetization of YIG. Moreover, the NPHE in both TmIG and YIG/BS samples is only observable below ~30 K and shows sharp increase when temperature is decreased from 30 K to 4.5 K. Such enhance d NPHE with similar characteristics was observed in different YIG/Bi 2Se3 samples grown at different times and in different TmIG/Bi 2Se3 devices , confirming the reproducibility of the results . The NPHE previously reported in Al 2O3/Bi2Se3 was attributed to the nonlinear, i.e., the hexagonal warping k3 and the p -h asymmetry k2 terms in the topological surface dispersion [22]. Although this contribution should exist in all three samples shown here, it cannot account the enhancement of the NP HE in TmIG and YIG/BS. When scaled by the coefficient 𝛾𝑦≡𝑅𝑦𝑥2𝜔 𝑅𝑥𝑥𝐼𝐵, the magnitude of the NPHE in TmIG and YIG/BS at 4.5 K is 0.025 and 0.031 respectively, which is one and two orders of magnitude larger than that in GGG/BS ( 𝛾𝑦=3×10-3) and Al2O3/BS (𝛾𝑦=1×10-4) [23] , respectively. Similarly, if Nernst effect was responsible for the measured second harmonic voltage, it should appear on the same order of magnitude in all three samples. The large difference shows that it should not be th e dominant contribution. Therefore, the observed NPHE is likely to have a magnetic origin. Ref. [31] reported a large hysteretic NPHE in magnetic TI Cr x(Bi,Sb) 2-xTe3/(Bi,Sb) 2Te3 heterostructures and explains it as a result of asymmetric scattering of surface electrons by magnons. This magnon scattering mechanism cannot be applied to the TmIG/BS sample, because with OP magnetization, the magnons are polarized in the z-direction an d cannot participate the scattering of surface electrons with IP - polarized spins. To see whether it is responsible for the hysteretic loop of the 𝑅𝑦𝑥2𝜔 𝑣𝑠 𝐵 curve in the YIG/BS sample, we parsed the NPHE into the linear -in-B (𝑑𝑅 𝑦𝑥2𝜔/𝑑𝐵) and hysteretic ( ∆𝑅𝑦𝑥2𝜔) part, as displayed in Fig. 4(a). Fig. 4(b) shows that these two components have almost the same temperature dependence, suggesting that they share the same origin. Therefore, in the YIG/BS sample, it is likely that the IP exchange field experienced by the Dirac electrons from the IP magnetic moments of YIG plays the same role with the external magnetic field, which gives rise to the hysteretic loop around zero field. We note that although the easy axis of YIG is IP, the Dirac -electron -mediated exchange interaction [8,32] tilts the magnetic moments to the OP direction at the interface [8,32 -34], which can also open an exchange gap for the TSS, just like that in TmIG/BS. As will be discussed in Section III.C, this gap o pening may play a decisive role in generating the large NPHE in MI/TI heterostructures. The other characteristics of the NPHE are shown in Fig. 5. First, the second -harmonic Hall resistance divided by field 𝑑𝑅𝑦𝑠2𝜔/𝑑𝐵 depends linear on the current de nsity [Fig. 5(a) top inset], demonstrating its nonlinear nature. The deviation from linear relationship at higher current densities is due to Joule heating. Second, as shown in Fig. 5(a), when scanned under a y-direction field, 𝑅𝑦𝑥2𝜔 becomes much small er, consistent with the cos𝜙𝐵 dependence of NPHE. Third, we also measured the longitudinal second - harmonic resistance 𝑅𝑥𝑥2𝜔 under a y-direction magnetic field scan and observed the longitudinal counterpart response , the so-called bilinear magnetoresistance (BMR) [35]. Lastly, as summarized in Fig. 5(b), the magnitude of the NPHE measure by 𝛾𝑦 is enhanced at low carrier densities. This also points out the insignificant role of the hexagonal warping effect in the NPHE observed here, because hexagonal warping is negligible at low Fermi levels and is enhanced when carrier density increases . 5 C. Discussion The nonlinear charge current under IP electric and magnetic fields can be expressed as 𝑗𝑎(2)= 𝜒𝑎𝑏𝑐𝑑 𝐸𝑏𝐸𝑐𝐵𝑑, where 𝜒𝑎𝑏𝑐𝑑 is the nonlinear conductivity tensor and 𝑎,𝑏,𝑐,𝑑=𝑥 or 𝑦. For an ideal linear Dirac dispersion, an IP magnetic field merely shift s the Dirac cone in k-space, leaving no effect on transport properties [22 -24]. However, in real TIs, higher -order k terms exist like the quadratic p -h asymmetry term and the cubic warping term. Here, for simplicity, we consider a linear TI dispersion with a parabolic 𝐷𝑘2 term. As sketched in Fig. 6(a) , when there is no IP magnetic field, under a driving electric field, there are equal number of electrons moving to the left and right with opposite spin polarizations. This generates a second -order spin current 𝑗s(2) but there is no net charge current. When an IP magnetic field 𝐵𝑥 is applied [Fig. 6(b)], it not only shifts the Dirac cone, but also tilts it [23,24] in the y-direction due to the existence of the 𝑘2 term. As a result, the transverse currents carried by the left - and right -moving electrons with opposite spin polarizations no longer cancel each other, and the nonlinear spin current is partially converted into a nonlinear charge current [22,35] , giving rise to a nonlinear Hall conductivity 𝜒𝑦𝑥𝑥𝑥. When the TI is further coupled to a magnetic material , the exchange interaction with OP moments opens a gap and introduces OP spin textures to the TSS [30]. Expanding around the shifted Dirac point, the Hamiltonian of TSS can be put in the form of 𝐻=ℏ𝑣F(𝑘𝑥𝜎𝑦−𝑘𝑦𝜎𝑥)+𝛼𝐵𝑥𝑘𝑦+∆𝜎𝑧 [23,24] , where ℏ is the reduced Planck constant, 𝑣F the Fermi velocity, and 𝝈 the Pauli matrices. The term with coefficient 𝛼𝐵𝑥 describes the magnetic -field-induced tilting in the y-direction , and 2∆ is the OP exchange - interaction -induced gap. The IP magnetic field Bx ~0.1 T used in this study is presumably smaller than the perpendicular magnetic anisotropy fields, so the size of the exchange gap is not affected by the small Bx. For similar tilted mas sive Dirac models, previous theoretical studies show that both intrinsic Berry -phase - related [36 -39] and extrinsic skew -scattering or side -jump mechanisms [40-42] can contribute to the NPHE with different 𝜏 scaling . In our experiment, the one to two orders of magnitude increase of 𝜒𝑦𝑥𝑥𝑥 (∝𝑑𝑅𝑦𝑥2𝜔/𝐼𝑑𝐵) [Fig. 3] from 30 K to 4.5 K and the relatively small change of the linear conductivity in this regime [Fig. 2(a)] suggest that some other factor other than t he relaxation time, governs the temperature dependence of the NPHE. As a result, we are not able to use 𝜏-scaling to narrow down the candidate NPHE mechanisms. More systematic future study is needed to clarify the dominant mechanism in the observed effect . Our experimental result s also suggest the importance of the OP spin textures formed in the TSS (due to MPE) for the NPHE , which was not considered in previous studies. As sketched in Fig. 6(c), with broken time-reversal symmetry, the spins of the electro ns with ±𝑘 momenta are no longer orthogonal with each other, and backscattering between these states by nonmagnetic impurities is allowed. This is reflected as the suppression of WAL in the linear transport regime [Fig. 2(c)] . Ref. [23] shows that the tilting -induced PHE on the surface of TIs can be enhanced by scattering off nonmagnetic impurities. Here, we speculate that similar effect also occur s in the nonlinear regime. When backscattering is allowed due to OP spin texture formation, the nonlinear s pin to charge current conversion may be increased , resulting in the large NPHE in MI/TI heterostructures. We note that in the above analysis, we only considered the TI surface states and did not consider the contribution from the bulk states . Because of th e short -range nature of the 6 MPE, the inversion symmetry of the bulk is presumably preserved. As a result, neither second -order spin nor charge current can be generated from the bulk [4 3]. IV. CONCLUSION In summary, we observed enhanced nonlinear planar Hall effect in Bi 2Se3 films grown on magnetic TmIG and YIG substrates as compared to that on nonmagnetic GGG substrate. This NPHE is only observable below 30 K and scales inversely with carrier density. Compared with the previously reported NPHE in Al 2O3/Bi2Se3 [22] arising from hexagonal warping or p -h asymmetry, the NPHE in MI/TI heterostructures is orders of magnitude larger, indicating a different origin. In YIG/Bi 2Se3 we find the IP exchange field plays the same role as the external magnetic field, giv ing rise to an extra hysteresis loop in the 𝑅𝑦𝑥2𝜔 𝑣𝑠 𝐵 scans. Actually, a large hysteretic NPHE was previously observed in EuS/(Bi,Sb) 2Te3 bilayers in Ref. [4 4], and a mechanism based on tilting of the p -h asymmetric Dirac cone by the IP exchange field was speculated. Our control experiments suggest the necessary role of the OP exchange - interaction -induced gap opening or modified spin textures in generating the large NPHE. Further experimental and theoretical work is needed to reveal the u nderlying mechanism and establish the NPHE as a convenient and sensitive transport probe of the MPE in MI/TI heterostructures. ACKNOWLEDGMENTS This work was supported by the U.S. DOE, Office of Basic Energy Sciences under Contract No. DE- SC0016380 and by NSF DMR Grant No. 1904076. Y.-X. H and S. A. Yang are supported by Singapore NRF CRP22 -2019 -0061. C. X. is supported by the UGC/ RGC of Hong Kon g SAR (AoE/P -701/20). The authors acknowledge the use of the Materials Growth Facility (MGF) at the University of Delaware, which is partially supported by the National Science Foundation Major Research Instrumentation under Grant No. 1828141 and UD -CHARM, a National Science Foundation MRSEC, under Award No. DMR - 2011824. 7 References [1] Y. Tokura, K. Yasuda, and A. Tsukazaki, Magnetic Topological Insulators , Nat. Rev. Phys. 1, 126 (2019). [2] R. Yu, W. Zhang, H. J. Zhang, S. C. Zhang, X. Dai, and Z. Fang, Quantized anomalous Hall effect in magnetic topological insulators , Science 329, 61 (2010). [3] X. L. Qi., T. L. Hughes, and S. C. Zhang, Topological field theory of time -reversal invariant insula tors, Phys. Rev. B 78, 195424 (2008). [4] C. Z. Chang et al., Thin Films of Magnetically Doped Topological Insulator with Carrier - Independent Long -Range Ferromagnetic Order , Adv. Mater. 25, 1065 (2013). [5] M. Mogi, R. Yoshimi, A. Tsukazaki , K. Yasuda, Y. Kozuka, K. S. Takahashi, M. Kawasaki, and Y. 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Fu, Quantum nonlinear Hall effect induced by Berry curvature dipole in time -reversal invariant materials , Phys. Rev. Lett. 115, 216806 (2015). [37] Y. Gao, S. A. Yang, and Q. Niu, Field induced positional shift of Bloch electrons and its dynamical implications , Phys. Rev. Lett. 112, 166601 (2014). [38] C. Wang, Y. Gao, and D. Xiao, Intrinsic Nonlinear Hall Effect in Ant iferromagnetic Tetragonal CuMnAs , Phys. Rev. Lett. 127, 277201 (2021). [39] H. Liu, J. Zhao, Y. -X. Huang, W. Wu, X. -L. Sheng, C. Xiao, and S. A. Yang, Intrinsic Second -Order Anomalous Hall Effect and Its Application in Compensated Antiferromagnets , Phys. Rev. Lett. 127, 277202 (2021) . [40] Z. Z. Du, C. M. Wang, H. -Z. Lu, and X. C. Xie, Disorder -induced nonlinear Hall effect with time -reversal symmetry , Nat. Commun. 10, 3047 (2019). [41] S. Nandy and I. Sodemann, Symmetry and quantum kinetics of the nonlinear Hall effect , Phys. Rev. B 100, 195117 (2019). [42] C. Xiao, Z. Z. Du, and Q. Niu, Theory of nonlinear Hall effects: Modified semiclassics from quantum kinetics , Phys. Rev. B 100, 165422 (2019) . [43] K. Hamamoto, M. Ezawa , K. W. Kim, T. Morimoto, and N. Nagaosa, Nonlinear spin current generation in noncentrosymmetric spin -orbit coupled systems , Phys. Rev. B 95, 224430 (2017). [44] D. Rakhmilevich, F. Wang, W. Zhao, M. H. W. Chan, J. S. Moodera, C. Liu, and C. -Z. Chang, Unconventional planar Hall effect in exchange -coupled topological insulator – ferromagnetic insulator heterostructures , Phys. Rev. B 98, 094404 (2018). 10 FIG. 1 XRD and RHEED results of Bi 2Se3(8 QL) films grown on TmIG, YIG and GGG substrates. FIG. 2 (a) Temperature dependence of the longitudinal resistance 𝑅𝑥𝑥 of four Garnet/Bi 2Se3(8 QL) samples. YIG and YIG2 are two films with thicknesses 2.5 𝜇m and 100 nm respectively. The result of YIG2/BS is plotted in the inset due to its large resistance. (b) Hall resistance as a function of OP magnetic field and extracted sheet carrier densities. Inset: Schematic of the measurement setup. (c) Change of sheet conductance under an OP magnetic field for three Garnet/Bi 2Se3(8 QL) devices at 4.5 K. 10 20 30 40 50 60 70Intensity (a.u.) 2q (degree)(006) (009)(012)(015) (018)(021) GGG/Bi2Se3 YIG/Bi2Se3 TmIG/Bi2Se3Garnet (444)Bi2Se3 1QL 8QL 0 100 200 3002.533.54 0 100 200 30010.511Rxx (kW) T (K)YIG2/Bi2Se3Rxx (kW) T (K)Bi2Se3(8 QL) on TmIG YIG GGG(a) (b) -0.5 0 0.5-10-50 DGs (mW) B (T)Bi2Se3 (8 QL) on TmIG YIG GGG 4.5 K (c) -0.1 0 0.1-4-2024Ryx (W) B (T)Garnet/Bi2Se3(8 QL) ns (´1013 cm-2) TmIG 3.46 YIG 3.02 YIG2 2.45 GGG 3.16 11 FIG. 3 The nonlinear Hall resistance 𝑅𝑦𝑥2𝜔=𝑉𝑦2𝜔/𝐼 as a function of x-direction magnetic field for three Bi 2Se3 samples grown on (a) GGG, (b) TmIG , and (c) YIG substrates at various temperatures. The bottom inset in (a) is an optical image of a device with illustrated measurement setup. Scale bar, 100 𝜇m. The top insets in (a) -(c) display the temperature dependence of the slope −𝑑𝑅 𝑦𝑥2𝜔/𝑑𝐵. (a) (b) (c) -2-1012 -2-1012 -0.1 0 0.1-505 4.5 K 6 K 7.5 K 10 K 15 K 20 K 30 K 50 K 150 KR2w yx (mW) 25 K 30 K 50 K 100 K 200 K 4.5 K 6 K 7.5 K 10 K 12.5 K 15 K 20 K 0 100 200010-dR2w yx/dB (mW/T) T (K) 0 100 200 30001020-dR2w yx/dB (mW/T) T (K) B (T) 25 K 30 K 50 K 100 K 200 K 4.5 K 6 K 7.5 K 10 K 12.5 K 15 K 20 K 0.2 mA 0 100 200010-dR2w yx/dB (mW/T) T (K) 12 FIG. 4 (a) The 𝑅𝑦𝑥2𝜔 𝑣𝑠 𝐵 curve (black) of the YIG/BS sample consists of a linear -in-B component (red) with slope 𝑑𝑅 𝑦𝑥2𝜔/𝑑𝐵 and a hysteretic component (blue) with magnitude ∆𝑅𝑦𝑥2𝜔. (b) Temperature dependence of −𝑑𝑅 𝑦𝑥2𝜔/𝑑𝐵 and −∆𝑅𝑦𝑥2𝜔. Inset shows the linear relationship between them. (a) -0.1 0.0 0.1-505R2w yx (mW) B (T) Total Linear-in- B HystereticdR2w yx/dB DR2w yx 0.2 mA (b) 10 10001020 0 10 20012-DR2w yx (mW) -dR2w yx/dB (mW/T) -dR2w yx/dB -DR2w yx T (K)-dR2w yx/dB (mW/T) 012 -DR2w yx (mW)13 FIG. 5 (a) Longitudinal and transverse second -harmonic resistance as a function of IP x- or y-direction magnetic field for the TmIG/Bi 2Se3 sample. Top inset: Current density dependence of the nonlinear Hall resistance −𝑑𝑅 𝑦𝑥2𝜔/𝑑𝐵. Bottom inset: Schematic of the measurement setup. (b) Carrier density dependence of the coefficient 𝛾𝑦=𝑅𝑦𝑥2𝜔 𝑅𝑥𝑥𝐼𝐵 in several devices. The solid l ine fit to 𝑛𝑠−3 is guide for the eye. FIG. 6 Illustration of the hypothetical backscattering enhanced NPHE mechanism. (a) For a TI surface dispersion containing a parabolic term, when there is no IP magnetic field, under a driving electric field, the number of electrons traveling to the right carrying up spins is equal to that of the electrons traveling to the left with down spins. This genera tes a second -order spin current 𝑗s(2) but there is no charge current. (b) An IP magnetic field tilts the upright Dirac cone and causes an imbalance between the left - and right -moving electrons, resulting in a second -order charge current, i.e., the NPHE. (c) When the TI is further coupled with a magnetic material, the exchange interaction opens an exchange gap 2∆ and introduces OP spin polarization components to the surface electrons. The restriction on backscattering is lifted, which may enhance the nonlinear spin to charge conversion efficien cy, giving rise to the large NPHE in MI/TI heterostructures. -0.1 0 0.1-4-2024 0 0.1 0.2 0.3010-dR2w yx/dBx (mW/T) I (mA)R2w y(x)x (mW) B (T)R2w xx vs By R2w yx vs Bx R2w yx vs By TmIG/Bi2Se3 4.5 K 2.5 3 3.5 40.020.030.040.05YIG/BS Device1 Device2 Device3 YIG2/BS gy (A-1T-1) ns (´1013 cm-2)TmIG/BS Device1 Device2 Device3(a) (b) (a) (b) (c) B=0

2022-03-12

Interfacing topological insulators (TIs) with magnetic insulators (MIs) has been widely used to study the interaction between topological surface states and magnetism. Previous transport studies typically interpret the suppression of weak antilocalization or appearance of the anomalous Hall effect as signatures of magnetic proximity effect (MPE) imposed to TIs. Here, we report the observation of nonlinear planar Hall effect (NPHE) in Bi2Se3 films grown on MI thulium and yttrium iron garnet (TmIG and YIG) substrates, which is an order of magnitude larger than that in Bi2Se3 grown on nonmagnetic gadolinium gallium garnet (GGG) substrate. The nonlinear Hall resistance in TmIG/Bi2Se3 depends linearly on the external magnetic field, while that in YIG/Bi2Se3 exhibits an extra hysteresis loop around zero field. The magnitude of the NPHE is found to scale inversely with carrier density. We speculate the observed NPHE is related to the MPE-induced exchange gap opening and out-of-plane spin textures in the TI surface states, which may be used as an alternative transport signature of the MPE in MI/TI heterostructures.

Observation of nonlinear planar Hall effect in magnetic insulator/topological insulator heterostructures

2203.06293v2

arXiv:1506.02902v1 [cond-mat.mtrl-sci] 9 Jun 2015Identification of spin wave modes strongly coupled to a co-ax ial cavity. N. J. Lambert,1J. A. Haigh,2and A. J. Ferguson1 1)Microelectronics Group, Cavendish Laboratory, Universit y of Cambridge, Cambridge, CB3 0HE, UK 2)Hitachi Cambridge Laboratory, Cavendish Laboratory, Univ ersity of Cambridge, Cambridge, CB3 0HE, UK (Dated: 16 October 2018) We demonstrate, at room temperature, the strong coupling of th e fundamental and non-uniform magnetostatic modes of an yttrium iron garnet (YIG) ferrimagnetic sphere to theelectromagnetic modesof aco-axial cavity. The well- defined field profile within the cavity yields a specific coupling strength for each magneto static mode. We experimentally measure the coupling strength for the different mag netostatic modes and, by calculating the expected coupling strengths, are able to ide ntify the modes themselves. 1A magnet may be excited in a uniform mode1,2, where all the constituent moments are precessing in phase, or in non-uniform modes3,4where there is a spatially varying phase difference between the moments. The uniform oscillating field that us ually drives ferromag- netic resonance excites only the uniform mode or higher order mode s with a net dynamic magnetisation. In contrast, if the oscillating field is spatially depende nt, perhaps due to the skin depthinthecaseofa metalferromagnet5,6or bydesign inanelectromagnetic waveguide or cavity7,8, then the modes are excited according to the spatial symmetry of the drive field. Such modes are the standing spin waves, and their propagating cou nterparts are central to the research field of magnonics which introduces the possibility to tr ansfer information over millimeter length scales9,10and perform specific information processing tasks11. Recently there has been a surge of interest in the coupling of magne ts to high quality fac- tor electromagnetic cavities12,13, motivated by the possibility of performing experiments in quantummagnonicswhichmightallowsinglelocalisedmagnonstatestob ecreatedandmea- sured. Sofar, thestrongcouplingregimeofquantumelectrodyna mics hasbeenreached8,14–16 along with demonstrations of magnetically induced transparency14. The strong coupling has been enabled by the high moment density and low magnetic damping17in YIG. Both uni- form and non-uniform modes have shown strong coupling8. The work reported in this Letter has been performed in such a context. We fabricate an easily made cavity (Fig. 1a) with a well-defined non-un iform field specif- ically so that we can couple into the non-uniform excited modes. It is m ade from a short (L= 28 mm) length of 3.5 mm diameter copper semi-rigid coaxial cable cut fl at at each end. These ends are brought into proximity with similarly flat ends in connec torised leads, with a small air gap forming the coupling capacitance. SMA screw connect ors provide mechan- ical stability and allow the size of the air gap, and hence the coupling ca pacitance, to be varied in a controlled way. At one extreme, the coaxial cables can be brought into contact with each other, transforming the cavity back into a transmission lin e. We find that the internal quality factor ( Q) of our cavity is 515, in close agreement with the theoretical value ofQ= 517 calculated from the specified attenuation in the co-axial cable . For the cavity experiments described in this Letter, we tuned the coupling streng ths to be κc/2π= 3.3 MHz, giving a loaded Qof 261, a fundamental frequency of ω0/2π= 3.535 GHz and a total cavity linewidth of (2 κc+κint)/2π= 13.5 MHz. A YIG sphere18of diameter 1 mm is inserted into the cable dielectric at the midpoint 23.50 3.55 3.60-6-4-20S22,S11 (dB) f (GHz)-40-30-20-10(a) (b) (c)xy zBx,yYIG sphere S21S11 S22SMA female thread SMA male nut Coupling capacitance gaps FIG. 1. The cavity and YIG sphere. (a) Diagram and longitudin al cross-section of the cavity. It is made from 3.5 mm diameter (UT141) semirigid coaxial cable, a nd the gap capacitances controlled with SMA coupling threads. (b) |S21|,|S11|and|S12|for the cavity configuration used in this experiment. (c) Non-uniform magnetic field around the YIG sp here due to the alternating cavity drive. The global field is applied in the zdirection. of the cavity (Fig. 1c). A key feature of our cavity is the well define d and non-uniform magnetic field profile in the dielectric gap, which has a 1 /rform in the radial direction. This non-uniform field allows the cavity to couple to both uniform and n on-uniform spin- wave modes. We measure the transmission, S21, of the system using a vector ne twork analyser. The incident power on the cavity is -10 dBm; the driven FMR in this regime is lin ear, as ob- served by the independence of S21 on power. We sweep the freque ncy from 2 GHz to 8 GHz, encompassing both the fundamental mode and the second ha rmonic of the cavity. A magnetic field is applied parallel to the cavity, and is varied between 50 and 330 mT. In this field range the magnetization of the YIG is fully saturated. The transmission of the system is shown in Fig. 2. In Fig. 2a we show d|S21|/dHfor the case in which the coupling capacitors are shorted; this is theref ore simply transmission line FMR. The magnetostatic band can be clearly seen, comprising a mu ltitude of modes. Unambiguous identification of each one is not trivial; the intensity of e ach line depends on 3|S21| (dB) 78 6 5 4 3 78 6 5 4 3 2Frequency (GHz) 50 100 150 200 250 300-20 -40 -60 -80 -100 -120 -140 d|S21| dH 01 0.75 0.5 0.25 Applied field (mT)(a) (b)(a.u.) (2,1) (1,1) FIG. 2. Transmission of the system. (a) Derivative of cavity transmission amplitude, d|S21|/dH, with both coupling capacitors shorted; it acts as a 50 Ω trans mission line. Many magnetostatic modes are visible. (b) Transmission amplitude |S21|of the cavity with both coupling capacitances setto≈28fF.Anticrossingsbetweencavity modesandmagnetostati cmodesareseen. Thecoupling depends strongly upon which magnetostatic mode is being exc ited. The anticrossing between the (2,1) mode and the second cavity harmonic is labelled. both the coupling of the magnetostatic mode to the transmission line , and the damping of that mode19, and the linewidth is also dependent on the measurement method20. In Fig 2b we revert to the gap coupled cavity as earlier described. An ticrossings between magnetostatic modes and the cavity resonances at both 3 .53 GHz and 7 .12 GHz are seen, with a maximum coupling strength of 130 MHz for the uniform FMR mode and the funda- mental cavity frequency. Coupling to the second harmonic of the c avity is in general much weaker, as the sphere is positioned at a magnetic field node of this ca vity mode. The spatial form and resonant frequencies of modes in magnetized spheres is well known3,4. Following Walker3we label them with indices nandm21. The radial form of the mode is characterized by n, andmdetermines the number of lobes in the mode pattern. 4The coupling of the ( n,m) mode to the cavity is given by16 gj=ηn,m 2γ/radicalbigg /planckover2pi1ωcµ0ǫr Vc√ 2Ns. Hereωris the resonance frequency, Vcis the volume of the cavity mode, Nis the total number of spins in the YIG sphere, s= 5/2 is the spin per site, µ0is the permeability of free space and ǫris the relative permitivity of the dielectric within the co-axial cable. Th e overlap between the cavity mode and the sphere mode ( n,m) is described by ηn,m, which is given by ηn,m=/vextendsingle/vextendsingle/vextendsingle/vextendsingle1 HmaxMmaxVs×/integraldisplay sphere(H·M)dV/vextendsingle/vextendsingle/vextendsingle/vextendsingle. His the r.f. driving field, and Mis the complex time-dependent off zaxis sphere mag- netization for mode ( n,m).HmaxandMmaxare the maximum magnitudes of these, and Vs is the sphere volume. The coupling strength is independent of magne tostatic damping. The coupling to a particular FMR mode is dependent on the relative sym metries of the mode and the r.f. drive field. It is forced to zero if the mode is antisym metric with respect to thedrive. Inparticular, forthecoupling tothefundamental cavit y modetobesignificant the FMR mode must be symmetric and low-order in z(as the cavity mode is also symmetric). This condition is only met by modes for which n=m. In contrast, in order to couple to the second harmonic cavity mode, the mode must be antisymmetric abou tz= 0. We tabulate calculated coupling constants larger than 1 MHz in Table I. In order to compare these values to our measurement we model th e transmission of strongly coupled cavity using the input-output formalism14–16,22. Close to the fundamental mode of the cavity S21 = κc i(ω−ωc)−1 2(2κc+κint)+/summationtext j|gj|2 −1 2γj+i(ω−ωj), wherejruns over the magnetostatic modes and γjare the FMR linewidths. In Fig. 3 we examine the region around the uniform mode’s anticrossing with th e cavity fundamental 5110 120 130 140 Applied field (m T)(a) (b) -20 -40 -60 -80 -100 -120 -140|S21| (dB)Frequency (GHz)3.23.43.63.84.0 3.23.43.63.84.0(1,1)(2,2) (3,3) FIG. 3. Strong coupling betwen cavity and FMR modes. (a) The r egion around the anticrossing of the uniform mode and the fundamental mode of the cavity. Th e most strongly coupled modes are labelled. (b) Simulation of the same region using the inp ut-output formalism. more closely. In Fig. 3a we show the measured transmission, and in Fig . 3b show the calcu- lated transmission over the same range. For m=nmodes the two are in good agreement. We attribute the appearance of additional weakly coupled modes to the YIG sphere being slightly off-center in the cavity, which lifts the symmetry conditions d escribed above. This also accounts for the weak coupling of the uniform mode to the seco nd harmonic of the cavity. In conclusion, we have described a simple tunable cavity-spin ensemb le system which can nevertheless achieve the strong coupling limit due to the high spin den sity in ferrimagnetic YIG. We show that the coupling to the uniform mode is 130 MHz, giving a cooperativity ofC=g2/κγ≈200. Furthermore, the asymmetric but well defined field profile in th e cavity permitsaquantitative understanding ofthecoupling tohighe r orderspinwave modes. Coupling between microwave cavities andhighlytunablemagnonicexcit ations isa candidate building block for hybrid quantum systems, and the ability to selective ly excite specific spin wave modes offers intriguing possibilities in the emerging field of quantu m magnonics. 6TABLE I. Calculated coupling strengths of selected FMR mode s to the fundamental and second harmonic cavity resonances. g/2π(MHz) n m Fundamental Second harmonic 1 1 130 0 2 1 0 2.9 2 2 27.1 0 3 3 8.1 0 4 4 2.8 0 5 5 1.1 0 We would like to acknowledge support from Hitachi Cambridge Labora tory, and EPSRC Grant No. EP/K027018/1. A.J.F. is supported by a Hitachi Researc h fellowship. REFERENCES 1C. Kittel, “Interpretation of Anomalous Larmor Frequencies in Fer romagnetic Resonance Experiment,” Physical Review 71, 270–271 (1947). 2C. Kittel, “On the Theory of Ferromagnetic Resonance Absorption ,” Physical Review 73, 155–161 (1948). 3L. R. Walker, “Magnetostatic Modes in Ferromagnetic Resonance,” Physical Review 105, 390–399 (1957). 4P. C. Fletcher and R. O. Bell, “Ferrimagnetic Resonance Modes in Sph eres,” Journal of Applied Physics 30, 687 (1959). 5C. Kittel, “Excitation of Spin Waves in a Ferromagnet by a Uniform rf F ield,” Physical Review110, 1295–1297 (1958). 6M. H. Seavey andP. E. Tannenwald, “Direct observationof spin-wa ve resonance,” Physical Review1, 168–169 (1958). 7Y. V. Khivintsev, L. Reisman, J. Lovejoy, R. Adam, C. M. Schneider , R. E. Camley, and Z. J. Celinski, “Spin wave resonance excitation in ferromagnetic films using planar waveguide structures,” Journal of Applied Physics 108, 023907 (2010). 78M. Goryachev, W. G. Farr, D. L. Creedon, Y. Fan, M. Kostylev, an d M. E. Tobar, “High-Cooperativity Cavity QED with Magnons at Microwave F requencies,” Physical Review Applied 2, 054002 (2014). 9J. R. Eshbach, “Spin-wave propogationandthe magnetoelastic int eraction inYttriumIron Garnet,” Physical Review Letters 8, 357–359 (1962). 10M. Tsoi, A. G. M. Jansen, J. Bass, W. Chiang, V. Tsoi, and P. Wyder, “Generation and detection of phase-coherent current-driven magnons in magetic multilayers,” Nature 406, 46–48 (2000). 11A. V. Chumak, A. A. Serga, and B. Hillebrands, “Magnon transistor for all-magnon data processing.” Nature Communications 5, 4700 (2014). 12O. O. Soykal and M. E. Flatt´ e, “Strong Field Interactions betwee n a Nanomagnet and a Photonic Cavity,” Physical Review Letters 104, 077202 (2010). 13O. O. Soykal and M. E. Flatt´ e, “Size dependence of strong couplin g between nanomagnets and photonic cavities,” Physical Review B 82, 104413 (2010). 14X. Zhang, C.-L. Zou, L. Jiang, and H. X. Tang, “Strongly Coupled Ma gnons and Cavity Microwave Photons,” Physical Review Letters 113, 156401 (2014). 15H. Huebl, C. W. Zollitsch, J. Lotze, F. Hocke, M. Greifenstein, A. Ma rx, R. Gross, and S. T. B. Goennenwein, “High Cooperativity in Coupled Microwave Reso nator Ferrimag- netic Insulator Hybrids,” Physical Review Letters 111, 127003 (2013). 16Y. Tabuchi, S. Ishino, T. Ishikawa, R. Yamazaki, K. Usami, and Y. Na kamura, “Hybridizing Ferromagnetic Magnons and Microwave Photons in the Q uantum Limit,” Physical Review Letters 113, 083603 (2014). 17E. G.Spencer, R.C. Lecraw, andA.M. Clogston, “Low-temperat ureline-width maximum in Yttrium Iron Garnet,” Physical Review Letters 3, 32–33 (1959). 18Ferrisphere, Inc. 19C. Bilzer, T. Devolder, P. Crozat, C. Chappert, S. Cardoso, and P . P. Freitas, “Vector net- work analyzer ferromagnetic resonance of thin films on coplanar wa veguides: Comparison of different evaluation methods,” Journal of Applied Physics 101, 074505 (2007). 20S. S. Kalarickal, P. Krivosik, M. Wu, C. E. Patton, M. L. Schneider, P . Kabos, T. J. Silva, and J. P. Nibarger, “Ferromagnetic resonance linewidth in metallic th in films: Comparison of measurement methods,” Journal of Applied Physics 99, 093909 (2006). 21For some modes, the resonance equation admits more than one solu tion, which is generally 8labelled with a third index. As none of the modes we explicitly discuss her e have multiple resonances, for simplicity we omit this index. 22A. A. Clerk, M. H. Devoret, S. M. Girvin, F. Marquardt, and R. J. Schoelkopf, “Introduction to quantum noise, measurement, and amplification,” Reviews of Modern Physics 82, 1155–1208 (2010). 9

2015-06-09

We demonstrate, at room temperature, the strong coupling of the fundamental and non-uniform magnetostatic modes of an yttrium iron garnet (YIG) ferrimagnetic sphere to the electromagnetic modes of a co-axial cavity. The well-defined field profile within the cavity yields a specific coupling strength for each magnetostatic mode. We experimentally measure the coupling strength for the different magnetostatic modes and, by calculating the expected coupling strengths, are able to identify the modes themselves.

Identification of spin wave modes strongly coupled to a co-axial cavity

1506.02902v1

Micro magnet location using spin waves Michael Balinskiy and Alexander Khitun* Electrical Engineering Department, University of California - Riverside, Riverside, CA, USA, 92521 Abstract. In this work, we present experimental data demonstrating the feasibility of magnetic object location using spin wave s. The test structure includes a Y3Fe2(FeO 4)3) (YIG) film with four micro -antennas placed on the edges . A constant in -plane bias magnetic field is provided by NdFeB permanent magnet . Two antennas are used for spin wave excitation while the other two are used for the inductive voltage measurement. There are nine selected places for the magnet on the film. The magnet was subsequentl y placed in all nine positions and spin wave transmission and reflection were measured. The obtained experimental data show the difference in the output signal amplitude depending on the magnet position. All nine locations can be identified by the frequenc y and the amplitude of the absolute minimum in the output power. All experiments are accomplished at room temperature. Potentially, spin waves can be utilized for remote magnetic bit read -out. The disadvantages and physical constraints of this approach are also discussed. I. Introduction The ability to locate objects using transmitted or reflected waves is widely used in d ifferent technologies 1-3. The position is related to the amplitude/frequency of the reflected/ transmitted wave. It provides a convenient tool for non -contact object location. It may be possible to apply s imilar techniques to locate magnetic objects using spin waves. Spin wave is a collective oscillation of spins in a spin lattice around the direct ion of magnetization. Spin waves appear in magnetically ordered structures, and a quantum of spin wave is called a “magnon”. The collective nature of spin wave phenomena manifests itself in relatively long coherence length, which may be order of the tens o f micrometers in conducting ferromagnetic materials (e.g. Ni 81Fe19) and exceed millimeters in non -conducting ferrites (e.g. YIG) at room temperature 4. The latter makes it possible t o build magnonic interferometers exploiting spin waves within the coherence length. For example, a Mach –Zehnder -type spin wave interferometer based on YIG structure was demonstrated in 2005 by M. Kostylev et al. 5. The phase difference among the interfering spin waves was controlled by the magnetic field produced by an electric current flowing through a conducting wire under one of the arms. In general, spin wave dispersion depends on the strength and direction of the bias magnetic field 6. Even a relatively weak (e.g. tens of Oersted) magnetic field produced by a micro -scale magnet placed on the top of magnonic waveguide may result in a prominent phase shift /amplitude change 7. This phenomenon was utilized for magnonic holographic imaging of magnetic microstructures 8. Here, we consider the feasibility of magnetic object location using spin waves. The rest of the paper is organized as follows. In the next Section II, we describe the experimental setup and present experimental data. The Discussion and Conclusions are given in Sections III and IV, respectively. II. Experimental data The schematics of the experimental setup are shown in Fig.1. The cross -section of the device under study is shown in Fig. 1(a). It consists of a permanent magnet made of NdFeB, a Printed Circuit Board (PCB) substrate with four short -circuited antennas, a ferrite film made of YIG, and a micro magnet that can be placed on different parts on top of the film. The permanent magnet is aimed to create a constant bias magnetic field. This bias magnetic field defines the frequency window as well as the type of spin waves that can propagate in the ferrite film. The strength of the field depends on the type of perman ent magnet and the substrate thickness. The bias field is about 195 Oe and directed in -plane on the film surface. The photo of the PCB substrate with four antennas is shown in Fig. 1(b). Each antenna is 6 mm long and 150 μm wide. The antennas are marked as 1,2,3, and 4 in the figure. The four antennas are placed on the side of a virtual square with a length of 11.4 mm. These antennas are aimed to excite and detect spin waves. The antennas are connected to the Programmable Network Analyzer (Keysight N5241A). The details of the spin -wave measurements with micro antennas can be found elsewhere 9,10. The ferrite film is made of YIG. YIG was chosen due to the low spin wave damping. The thickness of the film is 31.5 μm. The saturation magnetization is close to 1750 G, the dissipation parameter ΔH = 0.6 Oe. The planar dimensions of YIG -film are significantly larger than a virtual s quare of antennas providing the total coverage of all microstrip antennas. The schematics in Fig.1(c) show the top view of the YIF film. It is shown a frame of reference , where one division corresponds to 0.75 mm. The red circles depict the nine possible p ositions of the magnet. Hereafter, the position of the magnet will be referred to this reference frame. The magnet has a disk shape whose diameter is about 1 mm and the thickness is 0.3 mm . The magnet is made of magnetic steel . The first set of experiments is aimed to confirm spin -wave propagation through the film. These experiments are accomplished without a magnet on the top of the film. The collection of data showing S21 parameter measured with PNA is presented in Fig.2. In Fi g.2(a), there are shown data for the case when the signal is excited by antenna #1 and detected by antenna #3 (see Fig.1(c)). The data are taken in the frequency range from 1.3 GHz to 2 .9 GHz. The spectrum reveals Magnetostatic Surface Spin Wave (MSSW) propagation perpendicular to the direction of the bias magnetic field. In Fig. 2(b), there is shown S21 parameter measured at PNA for the frequency range from 1. 4 GHz to 2. 2 GHz. The signal is excited by antenna # 2 and detected by antenna # 4. The spectrum reveals Backward Volume Magnetostatic Spin Wave ( BVMSW) propagation directed along the bias magnetic field. In Fig.1(c), there are shown experimental data when two input antennas #1 and #3 operate simul taneously. The output signal is detected at antennas # 2 and #4. The detected inductive voltage is a superposition of the two signal s transmitted by MSSW and BVMSW. This configuration with two working input antennas is the most preferable for magnet location as it allows us to see the difference in spin wave p ropagation along the X and Y axes at a same time. Next, the experiments with two operating input antennas (i.e., #1 and #2) and two output antennas (i.e., #3 and #4) were repeated for the different position s of the magnet. The magnet was sequentially placed in the nine positions marked with the red circles in Fig.1(c). The measurements were accomplished in the frequency range from 1. 5 GHz to 3. 0 GHz. In Fig.3, there are shown data obtained after the subtraction (i.e., signal without magnet) and filtering. The graph shows the change of the output transmitted power ΔP (i.e., measured by the two output antennas) as a function of frequency. There are nine curves of different color s that correspond to the nine magnet locations. The data reveal a prominent variation in the output power depending on the magnet position. In Fig.4., there are shown data on spin wave signal transmission for the four selected cases corresponding to four selected positions of the magnet. The position of the magnet is shown in the inset . Fig.4(a) shows the normalized output power (i.e., after the subtraction) for the case when the magnet is placed in the center of the film. Figs. 4(b) and 4(c) show the output power for the magnet located on the corners of the film. In Fig.4(d), there are shown data for magnet shifted from the center towards the excitation antenna #1. The minimum of the signal transmission appears at different frequencies and reach es different amplitudes . That is the key result that allows us to conclude on the magnet position by the results of spin wave transport measurements. The summary of the experimental data obtained for different magnet locations is shown In Table I . The first two column s contain data on the magnet location while the last two columns show the frequency of the signal minimum and the normalized minimum amplitude. As one can see, there is a unique combination of frequency/minimum amplitude for each of the nine positions. The accuracy of the frequency measurements is 1 MHz . The accuracy of the output power measurements is 4.5 pW. All measurements are done at room temperature. It is also interesting to investigate the change in the signal reflection (i.e. , the S11 parameter) depending on the position of the magnet on the film. In Fig. 5, there are shown data on signal reflection. The reflected signal is measured by antenna #1 and antenna #2. The change in the reflected power is shown after the subtracti on (i.e., signal reflection without magnet) and filtering. The graph shows the change of the reflected power ΔP (i.e., measured by the two antennas) as a function of frequency. There are nine curves of different colors that correspond to the nine magnet lo cations. The re is a lso a difference in the reflection where the minimum of the reflected power occurs at different frequencies and reaches different minimum values. There is a less difference in the frequency compared to the transmission signal. The diffe rence in the amplitude of the reflected signal is prominent for the different locations of the magnet. In Fig. 6., there are shown data on spin wave signal reflection for the four selected cases corresponding to four selected positions of the magnet. The position of the magnet is shown in the inset. Fig. 6(a) shows the normalized reflected power (i.e., after the subtraction) for the case when the magnet is placed in the center of the film. Figs. 4(b) and 4(c) show the reflected power for the magnet located on the corners of the film. In Fig.4(d), there are shown data for magnet shifted from the center towards the excitation antenna #1. The summary of the experimental dat a obtained for the reflected signal is shown In Table II. The first two columns contain data on the magnet location while the last two columns show the frequency of the signal minimum and the normalized minimum amplitude. There are two prominent minima in the reflected power that occur for several magnet positions (e.g., (2,0), (2,1), (1,2), and (2,2). The minima in the reflected power appear on the same frequencies for the different magnet locations. Overall, the reflected spectra are less informative for the magnet location compared to the ones obtained for the transmitted signal. III. Discussion There several observations we want to make based on the obtained experimental data. (i) The output power spectra for transmitted signal are not symmetric for sym metric position of the magnet. For instance, one can see in Table I that the minimum of the output power differs in frequency and amplitude for magnet placed in different corners. The movement of the magnet along the X or Y axes results in the different ou tput power depending the direction of motion. This fact ca n be attributed to the asymmetry of the spin wave diffraction on the magnet and different dispersion of MSSW and BVMSW. The using of the same types of spin waves (e.g., only MSSW or only BVMSW) woul d smash the difference in the output characteristics. The calculation of the spin wave intensity profile over the film is a quite complicated computational task that goes beyond the scope of this work. The main focus of this work is the feasibility of magn et location via spin waves. (ii) The difference in the output power is quite prominent in the range of tens or a hundred of pW. It may be possible to recognize hundreds of magnet locations only by the amplitude of the output signal. The difference in the f requency of the output is also prominent, which provides an additional degree of freedom for magnet location. (iii) The exper iments were accomplished on a relatively large template (e.g., the area of the film with four antennas is about 1 cm2, the size of the magnet is about 1mm2). These millimeter -scale dimensions are mainly defined by the wavelength of the spin waves. It is estimated that the wavelength of MSSW is about 0.5 mm . The wavelength of BVMSW is about 0.5 mm as well . These large dim ensions are possible due to the long coherence length of spin waves in YIG. There is a lot of room for scaling down and increasing the number of possible magnet positions. The scaling down will require the reduction of the spin wave wavelength to microme ter range. In this work , the wavelength of the spin waves is mainly defined by the thickness of the YIG -film and the size of the micro -antennas. There should be a different mechanism for micrometer wavelength spin wave generation. For example, spin waves c an be excited and detected by synthetic multiferroics 11. However, this technique remains mainly unexplored. The ability to search fo r a number of possible magnet positions is the most appealing property of the described approach to magnet location using spin waves. In contrast to the existing technologies based on magnetoresistance measurements, i t does not require any physical contact between the magnet and the sensing element. Overall, it may provide a fundamental advantage over the existing practices for magnetic bit addressing and read -out. The spin wave location technique may be further extended by increasing the number of input/o utput ports 12 or/and exploiting spin wave interference 13. It would be of great interest to validate the possibility to identify multiple magnet configurations (i.e., configuration of several magnets on selected locations) . That would significantly enhance the read -out information capacity and lead to a new clas s of magnetic memory. There are several physical limitations and constraints inherent in the spin wave approach. The recognition of the magnet position requires the scan over a frequency range to find the location of the absolute minimum. It complicates th e whole search procedure and requires additional resources for input frequency modulation. The accuracy of output power measurements is another physical constraint that limits the number of possible magnet locations or magnets configuration to be recognize d. The physical origin of the prominent change in the signal transmission/reflection depending on the position of the magnet is not well understood. There are multiple factors that affect spin wave propagation (e.g., non - uniformity of the bias magnetic fi eld, non -uniformity of the magnetic field produced by the magnet, etc). The position of the magnet may also affect the generation of spin waves by the input antennas. One of the critical concerns is related to the scalability of the proposed approach. On t he one hand, quite a large propagation length of spin waves (i.e., up to 1 cm) at room temperature in YIG serves as the base for further device scaling. On the other hand, it is not clear if the difference in the signal transmission will be still recogniza ble for nanometer scale magnets. IV. Conclusions We present experimental data showing the change of spin wave transmitted and reflected signal depending on the magnet position on the film. Overall, the data show a prominent variation in the frequency an d the amplitude of the signal depending on the magnet position. It is possible to conclude on the location of the magnet (i.e., one of the nine pre -selected positions) based on the spin wave measurements. All experiments are accomplished at room temperatur e. It demonstrates the practical feasibility of using spin wave for magnetic object location. It may be utilized for magnetic bit addressing and read -out. The physical origin of the prominent signal modulation is not clear. There are multiple factors affe cting spin wave propagation/generation/detection which need further investigation. The experiments are accomplished on a relatively large template with millimeter -sized antennas. The main practical challenge toward nanometer magnet location is associated w ith a short -wavelength spin wave generation and detection. Author Contributions M.B. carried out the experiments. A.K. conceived the idea of magnet location using spin wave and wrote the manuscript . All authors discussed the data and the results and co ntributed to the manuscript preparation. Competing financial interests The authors declare no competing financial interests. Data availability All data generated or analyzed during this study are included in this published article . Acknowledgment This work was supported by the National Science Foundation (NSF) under Award # 2006290. Figure Captions Figure 1. (a) Schematics of the test structure. It consists of a permanent magnet made of NdFeB , a Printed Circuit Board (PCB) substrate with four short -circuited antennas, a ferrite film made of YIG, and a micro magnet that can be placed on different parts on top of the film. (b) The photo of the PCB substrate with four antennas. Each antenna is 6 mm long and 150 μm wide. The antennas are marked as 1,2,3, and 4 in the figure. The antennas are connected to the Programmable Network Analyzer (Keysight N5241A . (c) The top view of the YIF film. It is shown a frame of reference, where one division corres ponds to 0.75 mm. The red circles depict the nine possible positions of the magnet. The magnet has a disc oidal shape whose diameter is about 1 mm and the thickness is 0.3 mm. The magnet is made of magnetic steel. Figure 2. The collection of data showing S 21 parameter measured with PNA without a magnet. (a) Experimental data for the case when the signal was is excited by antenna #1 and detected by antenna #3. The data are taken in the frequency range from 1.3 GHz to 2.4 GHz. (b) Data obtained for the case w hen the signal is excited by antenna #2 and detected by antenna #4. Data are collected in the frequency range from 1.4 GHz to 2.2 GHz. (c) Experimental data for the case when two input antennas #1 and #3 operate simultaneously. The output signal is detecte d at antennas #2 and #4. Figure 3. Experimental data on spin wave transmission collected for nine positions of the magnet on the top of YIG film. There are nine curves of different colors that correspond to the nine magnet locations. The measurements were accomplished in the frequency range from 1.5 GHz to 3.0 GHz. The data are after the subtraction (i.e., signal without magnet) and filtering. Figure 4. Experimental data on spin wave signal transmission for the four selected positions of the magnet. The position of the magnet is shown in the inset. (a) data for magnet placed in the center of the film ; (b) data for magnet placed in the left top corner; (c) magnet is placed in the right down corner; (d) magnet is shifted from the center towards antenna #1. Table I. Summary of the experimental data showing the frequency and the amplitude of the absolute minimum of the transmitted signal for nine magnet locations. The first two columns contain data on the magnet location while the last two columns show the frequency of the signal minimum and the normalized minimum amplitude. Figure 5. Experimental data on spin wave reflection collected for nine positions of the magnet on the top of YIG film. There are nine curves of different colors that correspond to t he nine magnet locations. The measurements were accomplished in the frequency range from 1.5 GHz to 3.0 GHz. The data are after the subtraction (i.e., signal without magnet) and filtering. Figure 6. Experimental data on spin wave signal reflection for th e four selected positions of the magnet. The position of the magnet is shown in the inset. (a) data for magnet placed in the center of the film; (b) data for magnet placed in the left top corner; (c) magnet is placed in the right down corner; (d) magnet is shifted from the center towards antenna #1. Table I I. Summary of the experimental data showing the frequency and the amplitude of the absolute minimum of the reflected signal for nine magnet locations . The first two columns contain data on the magnet lo cation while the last two columns show the frequency of the signal minimum and the normalized minimum amplitude. Figure 1 Figure 2 Figure 3 1.6 1.8 2.0 2.2 2.4 2.6 2.8 3.0-0.1896-0.1580-0.1264-0.0948-0.0632-0.03160.00000.0316 DP [nW] (2,2) (1,2) (0,2) (2,1) (1,1) (0,1) (2,0) (1,0) (0,0) Frequency [GHz] Figure 4 Table I Figure 5 1.6 1.8 2.0 2.2 2.4 2.6 2.8 3.0-0.72-0.60-0.48-0.36-0.24-0.120.000.12 (2,2) (1,2) (0,2) (2,1) (1,1) (0,1) (2,0) (1,0) (0,0) DP [nW] Frequency [GHz] Figure 6 Table II References 1 Born, W. T. A REVIEW OF GEOPHYSICAL INSTRUMENTATION. GEOPHYSICS 25, 77-91, doi:10.1190/1.1438705 (1960). 2 Sarvazyan, A. P., Urban, M. W. & Greenleaf, J. F. Acoustic Waves in Medical Imaging and Diagnostics. Ultrasound in Med icine & Biology 39, 1133 -1146, doi:https://doi.org/10.1016/j.ultrasmedbio.2013.02.006 (2013). 3 Zhukov, V. Y. & Shchukin, G. G. Current problems of meteorological radiolocation. Journal of Communications Technology and Electronics 61, 1069 -1080, doi:10.1134/S1064226916100235 (2016). 4 Serga, A. A., Chumak, A. V. & Hillebrands, B. YIG MAgnonics. Journal of Physics D: Applied Physics 43 (2010). 5 Kostylev, M. P., Serga, A. A., Schneider, T., L even, B. & Hillebrands, B. Spin -wave logical gates. Applied Physics Letters 87, 153501 -153501 -153503 (2005). 6 Gurevich, A. G. & Melkov, G. A. (CRC press, 1996). 7 Gertz, F., Kozhevnikov, A., Filimonov Y., Nikonov, D. E. & Khitun, A. Magnonic Holograph ic Memory: From Proposal to Device. IEEE Journal on Exploratory Solid -State Computational Devices and Circuits 1, 67-75 (2015). 8 Gutierrez, D. et al. Magnonic holographic imaging of magnetic microstructures. Journal of Magnetism and Magnetic Materials 428, 348 -356, doi:10.1016/j.jmmm.2016.12.022 (2017). 9 Khivintsev, Y. et al. Prime factorization using magnonic holographic devices. Journal of Applied Physics 120, doi:10.1063/1.4962740 (2016). 10 Balinskiy, M. et al. Spin wave interference in YIG cross junc tion. Aip Advances 7, doi:10.1063/1.4974526 (2017). 11 Cherepov, S. et al. Electric -field -induced spin wave generation using multiferroic magnetoelectric cells. Applied Physics Letters 104, doi:10.1063/1.4865916 (2014). 12 Balynsky, M. et al. Quantum compu ting without quantum computers: Database search and data processing using classical wave superposition. Journal of Applied Physics 130, 164903, doi:10.1063/5.0068316 (2021). 13 Balinskiy, M., Chiang, H., Gutierrez, D. & Khitun, A. Spin wave interference detection via inverse spin Hall effect. Applied Physics Letters 118, 242402, doi:10.1063/5.0055402 (2021).

2022-04-14

In this work, we present experimental data demonstrating the feasibility of magnetic object location using spin waves. The test structure includes a Y$_3$Fe$_2$(FeO$_4$)$_3$) (YIG) film with four micro-antennas placed on the edges. A constant in-plane bias magnetic field is provided by NdFeB permanent magnet. Two antennas are used for spin wave excitation while the other two are used for the inductive voltage measurement. There are nine selected places for the magnet on the film. The magnet was subsequently placed in all nine positions and spin wave transmission and reflection were measured. The obtained experimental data show the difference in the output signal amplitude depending on the magnet position. All nine locations can be identified by the frequency and the amplitude of the absolute minimum in the output power. All experiments are accomplished at room temperature. Potentially, spin waves can be utilized for remote magnetic bit read-out. The disadvantages and physical constraints of this approach are also discussed.

Micro magnet location using spin waves

2204.07238v1

arXiv:1502.05244v1 [cond-mat.mtrl-sci] 13 Feb 2015Applied Physics Letters Spin-current injection and detection in strongly correlat ed organic conductor Z. Qiu∗,1,2M. Uruichi,3D. Hou,1,2K. Uchida,2,4,5H. M. Yamamoto,6,7and E. Saitoh1,2,4,8 1WPI Advanced Institute for Materials Research, Tohoku University, Sendai 980-8577, Japan 2Spin Quantum Rectification Project, ERATO, Japan Science and Technology Agency, Aoba-ku, Sendai 980-8 577, Japan 3Institute for Molecular Science, Okazaki 444-8585, Japan. 4Institute for Materials Research, Tohoku University, Send ai 980-8577, Japan 5PRESTO, Japan Science and Technology Agency, Saitama 332-0 012, Japan 6Research Center of Integrative Molecular Systems (CIMoS), Institute for Molecular Science, 38 Nishigounaka, Myodaiji, Okazaki 444-8585, Japan. 7RIKEN , 2-1 Hirosawa, Wako 351-0198, Japan. 8Advanced Science Research Center, Japan Atomic Energy Agency, Tokai 319-1195, Japan ∗Author to whom correspondence should be addressed; electronic mail: qiuzy@imr.tohoku.ac.jp 1Abstract Spin-current injection into an organic semiconductor κ-(BEDT-TTF) 2Cu[N(CN) 2]Br film in- duced by the spin pumping from an yttrium iron garnet (YIG) fil m. When magnetiza- tion dynamics in the YIG film is excited by ferromagnetic or sp in-wave resonance, a volt- age signal was found to appear in the κ-(BEDT-TTF) 2Cu[N(CN) 2]Br film. Magnetic-field- angle dependence measurements indicate that the voltage si gnal is governed by the inverse spin Hall effect in κ-(BEDT-TTF) 2Cu[N(CN) 2]Br. We found that the voltage signal in the κ-(BEDT-TTF) 2Cu[N(CN) 2]Br/YIG system is critically suppressed around 80 K, around which magnetic and/or glass transitions occur, implying that the efficiency of the spin-current injection is suppressed by fluctuations which critically enhanced nea r the transitions. PACS numbers: 72.80.Le, 85.75.-d, 72.25.Pn Keywords: Organic semiconductor, spintronics, inverse spin Hall e ffect, spin pumping 2Thefieldofspintronics hasattractedgreatinterest inthelastdec adebecauseofanimpact on the next generation magnetic memories and computing devices, w here the carrier spins play a key role in transmitting, processing, and storing information [ 1]. Here, a method for direct conversion of a spin current into an electric signal is indispens able. The spin-charge conversion has mainly relied on the spin-orbit interaction, which caus es a spin current to induce an electric field EISHEperpendicular to both the spin polarization σand the flow directionofthespin current Js:EISHE/ba∇dblJs×σ. This phenomenon isknown astheinverse spin Hall effect (ISHE) [2, 3]. To investigate spin-current physics and r ealize large spin-charge conversion, the ISHE has been measured in various materials rangin g from metals and semiconductors to an organic conjugated polymer [4–11]. A recent study has suggested that conjugated polymers can work as a spin-charge converter [10, 11 ], and further investigation of the ISHE in different organic materials is now necessary. In the present study, we report the observation of the ISHE in an organic molecular semi- conductor κ-(BEDT-TTF) 2Cu[N(CN) 2]Br (called κ-Br). The κ-Br consists of alternating layers of conducting sheets (composed of BEDT-TTF dimers) and in sulating sheets (com- posed of Cu[N(CN) 2]Br anions), which is recognized as an ideal system with anisotropic strongly correlated electrons (Fig. 1(a)). The ground state of b ulkκ-Br is known to be superconducting with the transition temperature Tc≈12 K[12], which becomes antiferro- magnetic and insulating by replacing Cu[N(CN) 2]Br with Cu[N(CN) 2]Cl (Fig. 1(d))[13]. To inject a spin current into the κ-Br, we employed a spin-pumping method by using κ-Br/yttrium iron garnet (YIG) bilayer devices. In the κ-Br/YIG bilayer devices, mag- netization precession motions driven by ferromagnetic resonance (FMR) and/or spin-wave resonance (SWR) in the YIG layer inject a spin current across the in terface into the con- ductingκ-Br layer in the direction perpendicular to the interface[14]. This inje cted spin 3current is converted into an electric field along the κ-Br film plane if κ-Br exhibits ISHE. The preparation process for the κ-Br/YIG bilayer devices is as follows. A single- crystalline YIG film with the thickness of 5 µm was put on a gadolinium gallium garnet wafer by a liquid phase epitaxy method. The YIG film on the substrate was cut into a rect- angular shape with the size of 3 ×1 mm2. Two separated Cu electrodes with the thickness of 50 nm were then deposited near the ends of the YIG film. The dista nce between the two electrodes was 0.4 mm. Here, to completely avoid the spin injection int o the Cu electrodes, 20-nm-thick SiO 2films were inserted between the electrodes and the YIG film, as show n in Fig. 1(b). Finally, a laminated κ-Br single crystal, grown by an electrochemical method [15], was placed on the top of the YIG film between the two Cu electrod es (Fig. 1(b)). We prepared two κ-Br/YIG samples A and B to check reproducibility. The thicknesses o f theκ-Br films for the samples A and B are around 100 nm but a little different from each other, resulting in the difference of the resistance of the κ-Br film (Fig. 1(b)). To observe the ISHE in κ-Br induced by the spin pumping, we measured the Hdependence of the mi- crowave absorption and DC electric voltage between the electrode s at various temperatures with applying a static magnetic field Hand a microwave magnetic field with the frequency of 5 GHz to the device. Figure 1(c) shows the temperature dependence of resistance of theκ-Br films on YIG for the samples A and B. The κ-Br films in the present system exhibit no superconducting transition [12, 16], but do insulator-like behavior similar to a bulk κ-Cl [13, 16]. This result can be ascribed to tensile strain induced by the substrate du e to the different thermal expansion coefficients of κ-Br and YIG. The similar phenomenon was reported in κ-Br on a SrTiO 3substrate[15], of which the thermal expansion coefficient is close to that of YIG (∼10 ppm/K at room temperature[17, 18]). Thus, the ground state o f theκ-Br film on 4YIG is expected to be slightly on the insulator side of the Mott transit ion. The red arrow in Fig. 1(d) schematically indicates a state trajectory of our κ-Br films with decreasing the temperature [15, 16, 19–25]. Figure 2(a) shows the FMR/SWR spectrum dI/dHfor theκ-Br/YIG sample A at 300 K. Here, Idenotes the microwave absorption intensity. The spectrum shows that the mag- netization in the YIG film resonates with the applied microwave around the FMR field HFMR≈1110 Oe. As shown in Fig. 2(b), under the FMR/SWR condition, electr ic voltage with peak structure was observed between the ends of the κ-Br film at θ=±90◦, where θdenotes the angle between the Hdirection and the direction across the electrodes (Fig. 2(b)). The voltage signal disappears when θ= 0◦. Thisθdependence of the peak voltage is consistent with the characteristic of the ISHE induced by the spin pumping. Because the SiO 2film between the Cu electrode and the YIG film blocks the spin-curren t injection across the Cu/YIG interfaces, the observed voltage signal is irre levant to the ISHE in the Cu electrodes. The magnitude of the electric voltage is one or two or ders of magnitude smaller than that in conventional Pt/YIG devices [26–29], but is close to that observed in polymer/YIG devices [10]. To establish the ISHE in the κ-Br/YIG sample exclusively, it is important to separate the spin-pumping-induced signal from thermoelectric voltage induced b y temperature gradients generated by nonreciprocal surface-spin-wave excitation [30], s ince thermoelectric voltage in conductors whose carrier density is low, such as κ-Br, may not be negligibly small. In order to estimate temperature gradient under the FMR/SWR condition, w e excited surface spin waves ina 3-mm-lengthYIG sample by using a microwave of which the po wer is much higher than that used in the present voltage measurements, and measur ed temperature images of the YIG surface with an infrared camera (Figs. 3(a) and (b)). We f ound that a temperature 5gradient is created in the direction perpendicular to the Hdirection around the FMR field and its direction is reversed by reversing H, consistent with the behavior of the spin-wave heat conveyer effect (Fig. 3(c) and (d)) [30]. Figure 3(e) shows th at the magnitude of the temperature gradient is proportional to the absorbed microwave power. This temperature gradient might induce an electric voltage due to the Seebeck effect in κ-Br with the similar symmetry as the ISHE voltage. However, the thermoelectric volta ge is expected to be much smaller than the signal shown in Fig. 2(b); since all the measurement s in this work were carried out with a low microwave-absorption power (marked with a gr een line in Fig. 3(e)), the magnitude of the temperature gradient in the κ-Br/YIG film is less than 0.015 K/mm. Even when we use the Seebeck coefficient of κ-Brat the maximum valuereported inprevious literatures [31–33], the electric voltage due to the Seebeck effect in theκ-Br film is estimated to be less than 0 .01µVat 300 K, where the effective length of κ-Br is 0.4 mm. This is at least one order of magnitude less than the signals observed in our κ-Br/YIG sample. Therefore, we can conclude that the observed electric voltage with the peak st ructure is governed by ISHE. Figure 4 shows the electric voltage spectra in the κ-Br/YIG sample A for various values of temperature, T. Clear voltage signals were observed to appear at the FMR fields whe n T >80 K. We found that the sign of the voltage signals is also reversed wh enHis reversed, which is consistent with the ISHE as discussed above. Surprisingly, h owever, the peak voltage signals at the FMR fields decrease steeply with decreasing Tand merge into noise around 80 K. This anomalous suppression of the voltage signals cann ot be explained by the resistance Rchange of the κ-Br film because no remarkable Rchange was observed in the same temperatures (Fig. 1(c)). At temperatures lower than 60 K, large voltage signals appear around the FMR fields as shown in Fig. 4, but its origin is not con firmed because 6of the big noise and poor reproducibility. Therefore, hereafter we focus on the temperature dependence of the voltage signals above 80 K. In Fig. 5, we plot the Tdependence of V∗/Rfor theκ-Br/YIG samples A and B, where V∗=/parenleftbig VFMR(−H)−VFMR(+H)/parenrightbig /2 withVFMR(±H)being the electric voltage at the FMR fields, to take into account the resistance difference of the κ-Br films. V∗/Rfor both the samples exhibit almost same Tdependence, indicating that the observed voltage suppression is a n intrinsic phenomenon in the κ-Br/YIG samples. Here we discuss a possible origin of the observed temperature depe ndence of the voltage in theκ-Br/YIG systems. ISHE voltage is determined by two factors. One is spin-to- charge conversion efficiency, i.e. the spin-Hall angle, in the κ-Br film. The mechanism of ISHE consists of intrinsic contribution due to spin-orbit coupling in the band structure and extrinsic contribution due to the impurity scattering [34]. In or ganic systems such asκ-Br, the extrinsic contribution seem to govern the ISHE since intrin sic contribution is expected to be weak because of their carbon-based light-element composition. Judging from the predicted rather weak temperature dependence of impurity s cattering, the temperature dependenceofthespin-Hallanglecannotbetheoriginofthesharp suppressionofthevoltage signal in the κ-Br/YIG systems (Fig. 5). The other factor is the spin-current in jection efficiency across the κ-Br/YIG interface, which can be affected by spin susceptibility [35] in κ-Br. Importantly, the temperature dependence of the spin susc eptibility for the κ-X family was shown to exhibit a minimum at temperatures similar to those at whic h the anomalous suppression of the spin-pumping-induced ISHE voltage was observ ed [16], suggesting an importance of the temperature dependence of the spin-current injection efficiency in the κ- Br/YIG systems. We also mention that the temperature at which th e ISHE suppression was observed coincides with a glass transition temperature of κ-Br films [36, 37]. However, at the 7present stage, thereisnoframeworktodiscusstherelationbetw eenthespin-current injection efficiency and such lattice fluctuations. To obtain the full understa nding of the temperature dependence of the spin-pumping-induced ISHE voltage in the κ-Br/YIG systems, more detailed experimental and theoretical studies are necessary. In summary, we have investigated the spin pumping into organic semic onductor κ-(BEDT-TTF) 2Cu[N(CN) 2]Br (κ-Br) films from adjacent yttrium iron garnet (YIG) films. The experimental results show that an electric voltage is generate d in the κ-Br film when ferromagnetic or spin-wave resonance is excited in the YIG film. Sinc e this voltage signal was confirmed to be irrelevant to extrinsic temperature gradients generated by spin-wave excitation and the resultant thermoelectric effects, we attribute it to the inverse spin Hall effect in the κ-Br film. The temperature-dependent measurements reveal tha t the voltage signal in the κ-Br/YIG systems is critically suppressed around 80 K, implying that t his suppression relates with the spin and/or lattice fluctuations in κ-Br. This work was supported by PRESTO “Phase Interfaces for Highly E fficient Energy Utilization”, Strategic International Cooperative Program ASPIM ATT from JST, Japan, Grant-in-Aid for Young Scientists (A) (25707029), Grant-in-Aid f or Young Scientists (B) (26790038), Grant-in-Aid for Challenging Exploratory Research ( 26600067), Grant-in-Aid for Scientific Research (A) (24244051), Grant-in-Aid for Scientifi c Research on Innovative Areas “Nano Spin Conversion Science” (26103005) from MEXT, Jap an, NEC Corporation, and NSFC. 8[1] I. 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Sasaki, Physical Review B 65, 144521 (2002). 11T (K) Cu[N(CN)2]Br-1 BEDT-TTF+0.5 Cu[N(CN)2]Br-1 BEDT-TTF+0.5 Cu[N(CN)2]Br-1 0.4 mm Tensile Compressive StrainSC PM PI AFIMixed phase10 100κ-Clκ-BrR/R300 K 10 2κ-Cl (bulk) κ-Br (bulk)sample Bsample A 10 100 T (K)SiO2 (c) (d)Cu YIGV κ-Br(a) (b) SS SSS S SSBEDT-TTF H microwave FIG. 1: (a) Structural formula of the BEDT-TTF molecule (upp er panel) and schematic cross- section of the (BEDT-TTF) 2Cu[N(CN) 2]Br (κ-Br) crystal, where cationic BEDT-TTF and anionic Cu[N(CN) 2]Br layers alternate each other (lower panel). (b) Schemati c illustration of the sample structure and experimental setup. Hdenotes the static external maganetic field applied along th e film plane. (c) Temperature dependence of R/R300Kof the two κ-Br/YIG samples A and B, a bulk κ-Br crystal, and a bulk κ-Cl crystal. Here, R(R300K) denotes the resistance between the ends of theκ-Br film at each temperature (at 300 K). (d) Conceptual phase d iagram of κ-X systems. PI, PM, AFI, and SC denote paramagnetic insulator, paramagneti c metal, antiferromagnetic insulator, and superconductor, respectively. The red arrow indicates the trajectory that the κ-Br crystal on the YIG substrate experiences upon cooling. 12800 1200 1600-1 01 -90°θ=90° 0° H (Oe)V (µV) 0dI/dH (a.u.)(a) (b)V H θH FIG. 2: (a) The FMR/SWR spectrum dI/dHof theκ-Br/YIG sample A at 300 K. Here, Iand denotes the microwave absorption intensity. The dashed lin e shows the magnetic field HFMRat which the FMR is excited. (b) The electric voltage Vbetween the ends of the κ-Br film as a function of H. 13-1 0 1-0.20.00.2 Position from center (mm)T-Tcenter (K) -1 0 1 0.050.100.15 0 5 10T (K/mm) Pab (mW)exp. fitting H H 1.5 mm 1.5 mm -0.4 0.0 (K) 0.4 (a) (b) (c) (d) (e) FIG. 3: (a),(b) Temperature distributions of the YIG surfac e near the FMR fields (5 GHz) for the opposite orientations of H, measured with an infrared camera. (c),(d) Temperature pro files of the YIG surface. (e) The microwave-power absorption Pabdependence of the temperature gradient ∇Tof the YIG surface. The voltage measurements were carried ou t with a low Pabvalue (marked with a green line). 141 μV FIG. 4:Hdependence of Vin theκ-Br/YIG sample A for various values of the temperature T. The scales of the longitudinal axis for the data at T≤70 K are shrinked by a factor of 0.1. 100 200 3000.010.101 T (K)Normalized V*/Rsample A sample B FIG. 5: Tdependence of V∗/Rfor the κ-Br/YIG samples A and B. Here, V∗=/parenleftbig VFMR(−H)−VFMR(+H)/parenrightbig /2 withVFMR(±H)being the electric voltage at HFMR. 15

2015-02-13

Spin-current injection into an organic semiconductor $\rm{\kappa\text{-}(BEDT\text{-}TTF)_2Cu[N(CN)_2]Br}$ film induced by the spin pumping from an yttrium iron garnet (YIG) film. When magnetization dynamics in the YIG film is excited by ferromagnetic or spin-wave resonance, a voltage signal was found to appear in the $\rm{\kappa\text{-}(BEDT\text{-}TTF)_2Cu[N(CN)_2]Br}$ film. Magnetic-field-angle dependence measurements indicate that the voltage signal is governed by the inverse spin Hall effect in $\rm{\kappa\text{-}(BEDT\text{-}TTF)_2Cu[N(CN)_2]Br}$. We found that the voltage signal in the $\rm{\kappa\text{-}(BEDT\text{-}TTF)_2Cu[N(CN)_2]Br}$/YIG system is critically suppressed around 80 K, around which magnetic and/or glass transitions occur, implying that the efficiency of the spin-current injection is suppressed by fluctuations which critically enhanced near the transitions.

Spin-current injection and detection in strongly correlated organic conductor

1502.05244v1

Spatial Control of Hybridization-Induced Spin-Wave Transmission Stop Band Franz Vilsmeier∗1, Christian Riedel1, and Christian H. Back1 1Fakult¨ at f¨ ur Physik, Technische Universit¨ at M¨ unchen, Garching, Germany March 23, 2024 Abstract Spin-wave (SW) propagation close to the hybridization-induced transmission stop band is investigated within a trapezoid-shaped 200 nm thick yttrium iron garnet (YIG) film using time-resolved magneto-optic Kerr effect (TR-MOKE) microscopy and broadband spin wave spectroscopy, supported by micromagnetic simulations. The gradual reduc- tion of the effective field within the structure leads to local variations of the SW dispersion relation and results in a SW hybridization at a fixed position in the trapezoid where the propagation vanishes since the SW group velocity approaches zero. By tuning external field or frequency, spatial control of the spatial stop band position and spin-wave propagation is demonstrated and utilized to gain transmission control over several microstrip lines. I Introduction Driven by potential spin-wave-based applications in computing and data processing, the field of magnonics has garnered growing interest in recent years [1–11]. To perform logic operations encoded within magnon currents various approaches were suggested and realized, such as interference-based logic gates [1,10,12,13] or magnonic crystals that ex- ploit the periodicity-induced formation of bandgaps in the spin-wave spectrum [14–21]. These devices rely on precise control and manipulation of spin- waves with wave vector kwithin a material with magnetization M. Recently, a hybridization-induced ∗franz.vilsmeier@tum.despin-wave-transmission stop band was demonstrated in 200 nm yttrium iron garnet (YIG) [22], adding to the list of options for engineering spin-wave propa- gation. It was shown that the hybridization of two different Damon Eshbach-like (DE) ( k⊥M) SW modes causes a frequency- and field-dependent sup- pression of SW propagation in a film with in-plane magnetization. Furthermore, it is well known that at the edges of thin magnetic films, depending on the magnetization direction, the effective field is locally reduced in order to avoid the generation of magnetic surface charges [16, 23, 24]. This allows for shape- modulated local variations of SW propagation. Com- bining this effect with the transmission stop band may provide enhanced control over spin-wave prop- agation dynamics and facilitate the implementation of magnonic devices. In this report, we investigate the effect of the geometry-induced variation of the effective field in a 200 nm YIG film on the hybridization-induced stop band. We demonstrate that the spin-wave propa- gation distance can be actively controlled within a trapezoid-shaped magnetic film as the reduced ef- fective field locally enforces the hybridization con- dition. Experimental dispersion measurements and micromagnetic simulations using TetraX [25] are conducted to determine the full film stop band con- dition. From further micromagnetic simulations us- ingMuMax3 [26], the effective internal field of a trapezoid geometry is determined. An inhomoge- neous field distribution with a gradual decrease along the trapezoid’s length is observed. We experimen- tally investigate the corresponding spin-wave prop- agation within the trapezoid in a DE-like geometry using time-resolved magneto-optic Kerr effect (TR- MOKE) microscopy [27–31] and broadband spin- wave spectroscopy. Bending of wavefronts, the for- 1arXiv:2403.15840v1 [physics.app-ph] 23 Mar 2024(a) (b)Figure 1 Sketch of the experimental setup. (a) Schematic for TR-MOKE measurement. Spin-waves are excited in the dipolar regime by the CPW and propagate through the trapezoid structure. The trapezoid was chosen to have a maximum width of 30 µm, a minimum width of 5 µm and a length of 80 µm. A static external field along the x-direction was applied throughout the experiment. (b) Schematic of all-electrical VNA spin-wave spectroscopy measurement. Spin-waves are excited from the first microstrip and detected via two more microstrips at different positions along the trapezoid geometry. Each microstrip is connected to a separate port of a four-port VNA. mation of edge channels, and a gradual decrease of wavelength along the propagation direction are ob- served. At a distinct position in the trapezoid, the propagation ceases. We show that this stop posi- tion is locally induced by the reduced effective field, which grants access to the spin-wave transmission stop band. Based on these findings, we demonstrate spatial control of spin-wave propagation within a trapezoid-shaped device by tuning the static external field close to the stop band. We utilize this effect for the active transmission control between microstrip lines. II Experimental Results The first set of experiments was carried out us- ing time-resolved magneto-optic Kerr effect (TR- MOKE) microscopy. Here, the dynamic out-of-plane magnetization component δmzis spatially mapped in the xy-plane, and a direct observation of spin- wave propagation in the sample is obtained. Simul- taneously, the reflectivity is detected, providing a to- pographic map of the sample. The measurements were conducted on a 200 nm thick yttrium iron gar- net (YIG) film grown by liquid phase epitaxy on a gadolinium gallium garnet (GGG) substrate. Thetrapezoid shape, with a gradual continuation back to the full film, was patterned by means of opti- cal lithography and subsequent Argon sputtering of the YIG film. For the excitation of spin-waves, a coplanar waveguide (CPW) was fabricated on top of the YIG film by optical lithography and electron beam evaporation of Ti(5 nm)/Au(210 nm). During the measurements, the external bias field was fixed along the CPW, so spin-waves in a DE-like geometry were excited [32]. A schematic of the measurement geometry can be found in Fig. 1(a). As a preliminary step, the spin-wave stop band in the unpatterned plane YIG film was identified by examining SW propagation far away from the patterned trapezoid structure. In this context, line scans of the Kerr signal along the y-direction were recorded as a function of the applied external field at a constant microwave frequency of f= 2.8 GHz. The result is depicted in Fig. 2(a). Here, a clear suppres- sion of spin-wave propagation around 32 mT can be observed. Previous work [22,33] has shown that hy- bridization between the DE-mode and the first-order perpendicular standing spin-wave (PSSW) mode can create a spin-wave stop band in 200 nm YIG. This is further illustrated in Fig. 2(b) by micromag- netic simulations with TetraX [25], an open-source 2Figure 2 (a) Measurement of SW propagation ex- cited by the CPW (gold) in full film YIG as a func- tion of the external field. A suppression of propaga- tion is visible around 32 mT. The grey-scale repre- sents the measured Kerr amplitude. (b) Micromag- netic simulations with TetraX forf= 2.8 GHz. The DE-mode (blue dash-dotted line) and the n=1 mode (blue dashed line) hybridize and form an anti- crossing in the micromagnetic simulations (red line). This results in an attenuation of SW propagation since the group velocity approaches zero. For all the simulations, the following material parameters were used: saturation magnetization Ms= 1.4·105A m, ex- change stiffness Aex= 3.7·10−12J m, gyromagnetic ratio γ= 176GHz T, film thickness L= 200 nm. Python package for finite-element-method micro- magnetic modelling [25]. In zeroth-order pertur- bation theory, according to Kalinikos and Slavin (KS) [34], the n=0 mode (blue dash-dotted line) and n=1 mode (blue dashed line) cross each other. This degeneracy is lifted by the formation of an avoided crossing in the micromagnetic simulations (red line). This leads to a flattening of the disper- sion relation and, in turn, to a decrease in group ve- locity [22]. For the given experimental parameters, the stop band is predicted at approximately 32 mT, consistent with the observed suppression of propa- (a) 313233µ0Hx,eff(mT) 0 20 40 60 80 y (µm)313233µ0Hx,eff (mT)(c)03060 µ0Hx,eff (mT)010203040 x (µm)(b)Figure 3 Micromagnetic simulations of the x- component of the effective field inside the trapezoid structure. The effective magnetic field varies locally across the geometry in (a). In the vicinity of edges, it is significantly reduced. The grey contour depicts the spatial boundaries of the simulated YIG struc- ture. Across the width of the structure shown in (b), a strong dip of the field at the edges of the ge- ometry is visible. Here, the grey-shaded rectangles indicate the areas outside of the magnetic structure. The effective field along the length of the trapezoid gradually decreases, as shown in (c). The full film hybridization condition at 2.8 GHz is marked with a green dot. gation in the full film line scans. We thus conclude that the pronounced attenuation can be attributed to the hybridization-induced stop band. To understand the influence of the trapezoid ge- ometry on SW propagation, and in turn, on the hy- bridization condition, further micromagnetic simu- lations were performed [26] to determine the effec- tive field of the tapered SW waveguide. Fig. 3(a) shows the spatial distribution of the x-component µ0Hx,effof the effective field at an externally ap- plied field µ0Hx= 33 .5 mT. The simulations re- vealed that the effective field varies locally inside the trapezoid and is strongly reduced at the YIG edges. The iso-field lines (black lines), which dis- play rounded triangular-like features, further illus- trate the inhomogeneous spatial distribution. Along the width of the trapezoid (Fig. 3(b)), we observe sharp edge pockets of low internal field, and a grad- ual decrease of field along the axis of spin-wave prop- agation (Fig. 3(c)). The origin of this inhomogeneity of the effective field lies in the geometry-induced de- magnetizing field, which aims to avoid the formation of magnetic surface charges [23,24]. Another important consideration regarding the ef- 3fect of the modified trapezoid waveguide is the emer- gence of additional width modes in the SW disper- sion relation due to the finite waveguide width [35– 37]. However, this width quantization does not af- fect the PSSWs, and the intersection of modes is still present. Thus, we argue that the key influence of the trapezoid geometry on the stop band is the reduction in effective field which results in a local variation of the dispersion relation. A more detailed discussion concerning the width quantization can be found in the supplementary material. Next, we experimentally investigate the effect of the geometry-induced field distribution on spin-wave propagation. Fig. 4(a) displays TR-MOKE measure- ments in which plane spin-waves are launched from the CPW into the trapezoid in the y-direction, with an excitation frequency f= 2.8 GHz and a static ex- ternal field µ0Hx= 33.5 mT. Changes in the prop- agation characteristics are observed upon entering the trapezoid. Apart from a prominent mode with slightly bent wavefronts in the trapezoid center, a localized mode with strongly bent wavefronts close to the edges appears. We also note that a magnetic contrast right at the edges of the patterned struc- ture was observed in some Kerr images. We argue that this artifact is due to imperfections in the fab- rication process and discuss it in more detail in the supplementary material. The observed bending of wavefronts can be at- tributed to the inhomogeneous internal field pro- file [24], where a local reduction in the effective field causes a shift towards lower fields and lower wave- lengths in the spin-wave dispersion relation. Addi- tionally, the edge localization of modes is a direct consequence of the low-field pockets in the effec- tive field distribution (Fig. 3(b)) as reported previ- ously [30,38,39]. Furthermore, the center mode changes wavelength as it travels through the trapezoid structure. No- tably, it comes to a halt at a specific position in space, beyond which the spin-wave propagation is almost entirely suppressed. This behaviour is illus- trated in more detail in Fig. 4(b), where a y-line pro- file (red curve) along the red dashed line in Fig. 4(a) is plotted. A line scan on plane YIG (blue curve), far away from any patterned structure, is also presented for comparison. As the spin-wave enters the trape- zoid, its wavelength gradually decreases up to more than 50%, consistent with the simulated decrease in the effective field (Fig. 3(c)). However, the propa- gation abruptly ceases at a specific position in space Figure 4 Kerr images at 2.8 GHz. (a) Propaga- tion of the main mode stops at a distinct position in space. The golden area depicts the excitation source. Light red and blue indicate saturation of the grey-scale. (b) Gradual decrease in wavelength and suppression of propagation along the trapezoid (red line in (a)) is observed. Propagation in the non- patterned plane YIG (blue curve) is also shown. (c) Below the hybridization field, propagation along the full trapezoid is observed. (d)-(e) By slightly tuning the field above the stop band, spin-wave propagation vanishes at different positions in space. 4(y≈74µm). From Fig. 3(c), we observe that the stop position of the center mode within the wedge corresponds to an estimated effective field of about 32 mT which aligns well with the measured full film hybridization condition (Fig. 2). Thus, we conclude that the reduction in effective field at different po- sitions in space leads to the local dispersion enter- ing the hybridization regime at a specific position in space, resulting in a sharp local attenuation of spin- wave propagation. Now, we aim to apply our findings towards the active manipulation of spin-wave propagation. To this end, additional Kerr images as a function of the external field were taken and are depicted in Figs. 4(c)-(e). Below the hybridization field, at 31.5 mT (Fig. 4(c)), propagation along the full length of the trapezoid without any sharp attenuation is present. No spatial suppression of propagation is observable since the effective field is only further reduced inside the trapezoid, and thus, the stop band regime is never reached. We also note a com- plex spatial beating profile with a prominent node aty≈45µm, and several less prominent ones. This self-focusing effect results from interference of the width modes induced by the tapered waveguide geometry and has been reported in magnonic mi- crostripes before [35, 36]. Furthermore, caustic-like beams induced by the corners where the full film transitions into the trapezoid may emerge [22, 40]. These caustic-like beams are reflected back and forth at the edges, resulting in non-equidistant areas of higher and lower amplitude. On tuning the external field slightly above 32 mT (Figs. 4(d)-(e)), however, the spin-wave propagation ceases at different positions in space. Furthermore, the boundaries of the spin-wave pattern display a shape reminiscent of the iso-field lines in the effective field. As the external field increases, the positions where the dispersion relation locally gains access to the transmission stop band also shift further outward along the y-direction. As a result, the geometry- induced hybridization allows to actively control the spin-wave propagation distance merely by tuning the external field within a reasonable range. Further all-electrical Vector Network Analyzer (VNA) spin-wave spectroscopy measurements were conducted with the intention to demonstrate the control of spin-wave propagation within a potential magnonic device. For this purpose, three 800 nm wide Au microstrips were patterned at different posi- tions along the trapezoid structure. One microstripserved as a source of spin-wave excitation, while the other two served for detection. The microstrips were connected to separate ports of a four-port VNA, and broadband spin-wave spectroscopy was performed. Note that the choice to employ microstrips instead of CPWs was made in order to obtain a more contin- uous range of wavenumbers for both the excitation and detection processes. A sketch of the measure- ment geometry is depicted in Fig. 1(b). Fig. 5(a) displays the detected spin-wave transmis- sion spectra showcasing the amplitudes of the scat- tering parameters S21,S31in terms of |∆S21|and |∆S31|. We point out that, in the following discus- sion,|Sij|denotes the absolute values of the detected scattering parameters whereas |∆Sij|refers to ab- solute values where a high-field subtraction method was applied. Also note that for better visibility, only data close to the stop band condition is de- picted. Full transmission spectra, along with more detailed information about the data processing pro- cedure, can be found in the supplementary material. Both spectra exhibit amplitude oscillations with the |∆S31|spectrum displaying shorter spacing between these oscillations. This is due to the change of the lateral spin wave profile, where the positions of high- amplitude and caustic-like nodes shift due to changes in the external magnetic field and applied frequency. This effect leads to a smaller node spacing in the field domain at the location of the third microstrip due to the gradual decrease in trapezoid width [35]. Moreover, distinct wide regions with low to no transmission in the spectra (highlighted by red dot- ted lines) occur at conditions in accordance with the spin wave stop band. For the transmission |∆S31|, this band is noticeably broader compared to the |∆S21|spectrum and is reached at lower frequencies at a given field (compare white dashed lines). This is a direct consequence of the spatially varying oc- currence of the hybridization condition suppressing the propagation of spin waves over a broader range of fields and frequencies the further they advance along the trapezoid. To put it differently, distinct external field and frequency conditions exist where transmission is absent in both |∆S21|and|∆S31|, transmission is observed only in |∆S21|, and trans- mission occurs in both |∆S21|and|∆S31|. As a re- sult, selective control over the transmission between the microstrips can be achieved by slightly tuning the frequency or the applied bias field. This is further illustrated in the continuous wave (CW) mode measurements at fixed frequen- 515 25 35 45 field (mT)2.22.32.42.52.62.72.82.9f (GHz)|∆S21| 15 25 35 45 field (mT)|∆S31| min max Magnitude (arb. u.) 1001012.45 GHz(a) (b) 18 19 20 21 22 23 24 25 field (mT)100101|S21||S31| 1001012.7 GHz 24 25 26 27 28 29 30 31 field (mT)100101Magnitude (arb. u.)Figure 5 Selective control of transmission between microstrips along trapezoid-like structure. (a) Spin-wave transmission spectra (amplitudes |∆S21|and|∆S31|). A region of low to no transmission consistent with the expected hybridization conditions occurs in both spectra (red dotted lines serve as guides to the eye). This region appears broader in the transmission spectrum from port 1 to port 3. Moreover, at a given external field, the stop band starts at lower frequencies in |∆S31|compared to |∆S21|as highlighted by white dashed lines. (b) Transmission signals in CW mode at fixed frequencies. The hybridization-induced stop band (roughly marked by gray-shaded areas) spans to higher applied fields in the |S31|transmission trace. cies shown in Fig. 5(b). The regions of suppressed transmission shift with applied frequency and span over a broader field range in the |S31|parameter. The hybridization-induced stop band (highlighted by gray-shaded regions) extends to higher fields due to localized effective field reduction. For instance, at 2.45 GHz and with a field of 23 mT, we observe trans- mission in the |S21|channel but minimal transmis- sion in the |S31|trace, similar to 2.7 GHz at 29 mT. Interestingly, the transmission in |S31|also appears to be suppressed for fields slightly below the hy- bridization field. Additional TR-MOKE data in the supplementary material reveals that this behavior can be attributed to the formation of caustic-like beams that are significantly attenuated upon propa- gation along the geometry. To conclude this section, we suggest employing multiple microstrips along the trapezoid geometry for potential logic operation. Moreover, in the sup- plementary material, we provide further discussion on properties of the hybridization, such as its thick- ness dependence.III Conclusion In conclusion, we demonstrated the feasibility of ac- tively controlling the spin-wave propagation distance by combining the hybridization-induced stop band and a geometry-induced variation of the effective field in 200 nm YIG within a trapezoid-shaped mag- netic film. Experiments and micromagnetic simula- tions were performed to gain insight into the effect of the trapezoid geometry on the effective field. 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Back, “Observation of a goos-h¨ anchen-like phase shift for magnetostatic spin waves,” Phys. Rev. Lett. , vol. 121, Sept. 2018. [32] R. Damon and J. Eshbach, “Magnetostatic modes of a ferromagnet slab,” J. Phys. Chem. Solids , vol. 19, pp. 308–320, May 1961. [33] M. Vaˇ natka, K. Szulc, O. Wojewoda, C. Dubs, A. V. Chumak, M. Krawczyk, O. V. Dobrovol- skiy, J. W. K/suppress los, and M. Urb´ anek, “Spin-wave dispersion measurement by variable-gap propa- gating spin-wave spectroscopy,” Phys. Rev. Ap- plied, vol. 16, Nov. 2021. [34] B. A. Kalinikos and A. N. Slavin, “Theory of dipole-exchange spin wave spectrum for ferro- magnetic films with mixed exchange bound- ary conditions,” J. Phys. C: Solid State Phys. , vol. 19, pp. 7013–7033, Dec. 1986. [35] V. E. Demidov, S. O. Demokritov, K. Rott, P. Krzysteczko, and G. Reiss, “Mode interfer- ence and periodic self-focusing of spin waves in permalloy microstripes,” Physical Review B , vol. 77, Feb. 2008. [36] X. Xing, Y. Yu, S. Li, and X. Huang, “How do spin waves pass through a bend?,” Scientific Reports , vol. 3, Oct. 2013. [37] A. V. Chumak, “Fundamentals of magnon- based computing,” 2019. [38] J. Topp, J. Podbielski, D. Heitmann, and D. Grundler, “Internal spin-wave confinement in magnetic nanowires due to zig-zag shaped mag- netization,” Phys. Rev. B , vol. 78, July 2008. [39] J. Jorzick, S. O. Demokritov, B. Hillebrands, M. Bailleul, C. Fermon, K. Y. Guslienko, A. N. Slavin, D. V. Berkov, and N. L. Gorn, “Spin wave wells in nonellipsoidal micrometer size magnetic elements,” Phys. Rev. Lett. , vol. 88, Jan. 2002. [40] A. Wartelle, F. Vilsmeier, T. Taniguchi, and C. H. Back, “Caustic spin wave beams in soft thin films: Properties and classification,” Phys. Rev. B , vol. 107, Apr. 2023. 9Supplementary Material: Spatial Control of Hybridization Induced Spin-Wave Transmission Stop Band Franz Vilsmeier∗1, Christian Riedel1, and Christian H. Back1 1Fakult¨ at f¨ ur Physik, Technische Universit¨ at M¨ unchen, Garching, Germany March 23, 2024 I Time-Resolved Magneto-Optic Kerr Effect Microscopy As a source of illumination, a mode-locked Ti:Sa laser with a centre wavelength of 800 nm and a pulse width of around 150 fs is used. The pulse trains are applied at a fixed repetition rate of 80 MHz. Subsequently, we fix the polar- ization plane of the laser and focus it onto the sample through an objective lens with a numerical aperture of 0.7, giving a maximum resolution of ∼0.6µm. Upon reflection at a magnetic surface, the polarization changes due to the po- lar magneto-optical Kerr effect. Here, the change of polarization rotation is directly proportional to the change in the dynamic out-of-plane magnetization component. A Wollaston prism splits the reflected signal into two beams with orthogonal polarization components, which are detected by two photodiodes. The difference between the two photodiode signals then gives a direct represen- tation of the change in magnetization - the Kerr signal -, and the sum of the two is proportional to the sample’s reflectivity. Since the sample is mounted onto a piezo stage, the relative laser focus position can be spatially scanned in the sample plane. Hence, Kerr image and topography are obtained. In addition, during the acquisition of a Kerr image, the relative phase relation between the applied rf-frequency in the GHz regime and the laser repetition rate is fixed. This requires the driving field frequency to always be a multiple of the laser repetition rate. As a result of the constant phase and the short laser pulses (much shorter than one period of the excitation), we can directly access the dynamic out-of-plane magnetization component and observe the propagation of spin-waves excited by our antenna structure. Fig. 1 shows some spatial Kerr maps approaching the full film hybridization condition from lower fields. Caustic-like beams emerge, which are damped along the trapezoid with increasing field. ∗franz.vilsmeier@tum.de 1arXiv:2403.15840v1 [physics.app-ph] 23 Mar 2024020406080 y (µm)02040x (µm)32.2 mT 020406080 y (µm)32.3 mT 020406080 y (µm)32.4 mT 020406080 y (µm)32.5 mT min maxδmzFigure 1 Kerr images recorded at f= 2.8 GHz at different fields coming from below the stop band. Caustic-like beams are reflected back and forth at the edges and fade away along the trapezoid with increasing field. II Effect of Waveguide Width on Hybridiza- tion Following the analytical model by Kalinikos and Slavin [1], the full film disper- sion relation in the presence of an in-plane external field with totally unpinned surface states is given by ω2 n=/parenleftbig ωH+l2 exk2 nωM/parenrightbig/parenleftbig ωH+l2 exk2 nωM+ωMFnn/parenrightbig , (1) where ωH=γµ0H, (2) ωM=γµ0MS, (3) Fnn=Pnn+/parenleftbigg 1−Pnn/parenleftbig 1 + cos2φ/parenrightbig +ωMPnn(1−Pnn) sin2φ ωH+l2exk2nωM/parenrightbigg , (4) and Pnn=k2 k2n−k4 k4nFn1 1 +δ0n, Fn=2 kL/parenleftbig 1−(−1)ne−kL/parenrightbig .(5) Furthermore, n= 0,1,2, ...denote the eigenmode orders across the film thick- nessL,kn=/radicalig k2+/parenleftbignπ L/parenrightbig2, and φdescribes the angle between kandM(so for k⊥M,φ=π 2). Considering a spin wave waveguide of finite width w, an additional quanti- zation across the waveguide width is introduced and the dispersion relation can be represented using equ. (1) by letting k→/radicalig k2+/parenleftbigmπ w/parenrightbig2andφ→ φ−arctan/parenleftbigmπ kw/parenrightbig [2–4]. Here, m= 0,1,2, ...denote the eigenmode orders across the width, and kdenotes the wavenumber along the waveguide. In the specific case of a tangentially magnetized waveguide in the DE-geometry ( k⊥M), de- magnetization also has to be taken into account and the non-uniform effective field µ0Heffneeds to be considered in the dispersion relation in place of the externally applied field. In the case of spin wave propagation in the center, a uniform field is assumed, but an effective waveguide width weffis introduced to account for the strong reduction in the effective field at the edges. The effective width can be defined in different ways. Here, we follow the definition by Chu- mak [4], where weffis given by the distance of points across the width where the effective field is reduced by 10%. i.e., to the value 0 .9·µ0Hmax eff. 2From micromagnetic simulations [5], the effective field within the trapezoid geometry at an externally applied field of 32 mT (close to the full film hybridiza- tion field at 2.8 GHz) was determined. From the field distribution, the effective field and effective width for the center mode were extracted as a function of trapezoid width w. The respective results are depicted in Figs 2(a)-(b). At the smallest width, the effective field is reduced by almost 3 mT with respect to the applied field. The effective width is maximally reduced to about 65 % of the actual waveguide width, allowing for a rather wide region of uniform field and mode propagation across the width. 29303132µ0Hx,eff(mT)(a) 5 10 15 20 25 30 w(µm)0.70.80.9weff/w(b)2.12.42.73.03.3f (GHz)(c) wg, m=0 wg, m=1 wg, m=2wg, n=1 ff, n=0 ff, n=1 0 1 2 3 k (µm−1)2.72.9f (GHz)(d) n=1, m=0 n=1, m=1 Figure 2 Effect of width modes on dispersion relation. (a) Effective field from micromagnetic simulations for the center mode as a function of trapezoid width w. (b) Ratio of extracted effective width weffand actual trapezoid width w. (c) Dispersion relations of the waveguide (wg) modes considering effective field and effective width for w= 6µm. Modes with n=0 and m=0 (blue dash-dotted line), n=0 and m=1 (red dash-dotted line), n=0 and m=2 (purple dash-dotted line), and n=1 and m=0 (red dashed line) are shown. The full film (ff) modes with n=0 and n=1 are also depicted (grey lines) for comparison. (d) Waveguide modes with n=1 and m=0, and n=1 and m=1. The width modification does not significantly affect the first-order PSSW. Forw= 6µm, the resulting dispersion relations for a waveguide with several width modes (m=0, m=1, m=2) and thickness modes (n=0, n=1) are displayed in Fig. 2(c). Note that for the waveguide case, only the m=0 mode of the first- order (n=1) PSSW mode is shown, as the width quantization doesn’t notably affect the higher-order PSSWs (see Fig. 2(d)). The reduced effective field gener- ally shifts the dispersion relation towards lower frequencies compared to the full film case (grey lines). The higher-order width modes (m=1, m=2) also display lower frequencies in the dipolar regime than the m=0 mode. More importantly, however, the n=0 and n=1 thickness modes still intersect in the dipolar regime, facilitating a hybridization and corresponding stop band. From this, we con- clude that the main effect of the width modulation on the hybridization is the reduced effective field and the resulting shift in the hybridization condition. This is especially the case for the m=0 width mode, which should be dominant in the trapezoid structure due to the transmission of spin waves from the full film into the tapered waveguide. 3III Broadband Spin-Wave Spectroscopy A four-port vector network analyzer (Agilent N5222A) was used for the broad- band spin-wave spectroscopy. All measurements were conducted at a microwave power of 3 dBm, and the real and imaginary parts of the complex scattering pa- rameters S21andS31were recorded. A frequency sweep method was applied at different external magnetic field values for the transmission spectra. The mag- netic field’s strength was changed stepwise (5 mT steps) from high to low field. To improve contrast, a high-field subtraction method was applied. Reference data S21,refandS31,reftaken at 200 mT was recorded. The presented spectra were then obtained by subtracting the absolute value of the reference data from the absolute value of the scattering parameters, i.e., |∆S21|=|S21|−|S21,ref|and |∆S31|=|S31|−|S31,ref|. Exemplary recorded spectra are shown in Fig. 3 where modes close or at the FMR are prominent. In the CW mode measurements, no reference was taken. −60−40−200204060 field (mT)0.51.01.52.02.53.03.5f (GHz)|∆S21| minmax Magnitude (arb. u.) −60−40−200204060 field (mT)|∆S31| Figure 3 Broadband spin-wave spectroscopy spectra. Modes close to FMR are very prominent in the transmission spectra |∆S21|and|∆S31|. IV Edge Mode Due to Imperfect Fabrication Fig. 4 shows TR-MOKE measurements at different frequencies and relative phases between the microwave excitation and laser pulses at an applied field of 33.5 mT. Apart from the propagation inside the trapezoid, Kerr images where intense caustic-like beams are detected (2.4 GHz, 2.48 GHz, 2.56 GHz) also ex- hibit a magnetic contrast right at the edges of the patterned YIG. In close vicinity to the points where the beams are reflected from the trapezoid edges, a localized mode profile outside the previous propagation region in x-direction is visible. However, no such distinct feature seems to occur when DE-like modes become dominant in the profile (2.72 GHz). Fig. 5(a) depicts an AFM image of the patterned YIG trapezoid. Linescans across the edges (Fig. 5(b)) reveal that the transition from YIG to GGG sub- strate is not sharp, but rather gradual along a distance of about 1-2 µm. This is attributed to imperfections in the fabrication process. The boundary regions, from which the caustic-like beams scatter, serve as secondary point-like excitation sources with a finite size of the order of the 402040x (µm)2.4 GHz 0 deg 2.48 GHz 0 deg 2.56 GHz 0 deg 2.72 GHz 0 deg 020 40 60 y (µm)02040x (µm)2.4 GHz 90 deg 020 40 60 y (µm)2.48 GHz 90 deg 020 40 60 y (µm)2.56 GHz 90 deg 020 40 60 y (µm)2.72 GHz 90 deg min maxδmzFigure 4 TR-MOKE measurements for different frequencies and phases be- tween microwave excitation and laser pulses at an external field of 33.5 mT. In the vicinity where the caustic-like beams scatter from the edges, an additional mode profile is observed. We note that the antisymmetric beam directions stem from a slight mismatch of the external field angle with respect to the DE- geometry. 051015202530 y (µm)05101520253035x (µm)(a) 0 100 200profile (nm) −2−1012 ∆x (µm)050100150200250profile (nm)(b) upper lower −6−4−20246 ky(µm−1)−8−6−4−202468kx(µm−1)(c) 50 nm 100 nm150 nm 200 nm Figure 5 (a) AFM profile of patterned YIG structure. (b) Linescans taken across the edges highlighted in blue and red in (a). A gradual transition from the YIG to the GGG is observed. (c) Iso-frequency curves for several film thicknesses at 2.48 GHz and 33.5 mT. beam’s width, as noted in previous works [4,6]. Consequently, spin wave modes may be excited within the transitional region where the thickness of YIG de- creases. Examining the iso-frequency curves at 2.48 GHz and 33.5 mT (see Fig. 5 (c)) reveals that the modes potentially excited fall within our resolution limits across a considerable range of film thicknesses. 5V Some Properties of Hybridization-Induced Stop Band This section offers a brief overview of some characteristics of the anticrossing in full YIG films. All micromagnetic simulations were executed utilizing the TetraX [7] software package. To provide a qualitative understanding of the hybridization’s coupling strength, we introduce the quantity ∆ fas the minimal gap between the upper and lower band determined by micromagnetic simulations. We note that here, we only consider the strength of hybridization in the frequency domain, as this needed significantly less computing time. Furthermore, we inferred the wave vector of hybridization khybby the intercept of the n=0 and n=1 modes according to the model by Kalinikos and Slavin [1]. 150 250 350 450 YIG thickness (nm)01234khyb(µm−1)(a)khyb ∆f 0.000.020.040.060.080.10 field (mT)12345khyb(µm−1)(b)170 nm 200 nm 230 nm 020406080100120 ∆f(MHz) Figure 6 Some properties of hybridization. (a) Trend of hybridization wave number khyband hybridization strength with increasing film thickness at ex- ternal field of 32 mT. Both parameters decrease with increasing thickness. (b) Dependence of khybon the external field for different film thicknesses. khybcan be increased to some extent by the external field strength. In Fig. 6(a), the relationship between film thickness and both ∆ fandkhyb is illustrated at an external field strength of 32 mT. As film thickness increases, both these parameters exhibit a decreasing trend. Notably, below a thickness of 170 nm, the formation of an anticrossing appears to be absent. Here, the increased separation between n=0 and n=1 prevents hybridization from occur- ring. We also note that potential crossing is only possible in the dipolar regime as modes follow k2-dependence in the exchange regime. Fig. 6(b) depicts the influence of the external field on khyb. Higher external field strengths result in higher wave numbers. Furthermore, the external field strength also affects the existence of frequency degeneracy. At higher fields, the n=0 mode becomes flatter and no longer intersects with the n=1 mode. In summary, the manipu- lation of external field strength and sample thickness allows for the adjustment of hybridization properties, facilitating higher wave number values and stronger coupling within a specific range. 6References [1] B. A. Kalinikos and A. N. Slavin, “Theory of dipole-exchange spin wave spec- trum for ferromagnetic films with mixed exchange boundary conditions,” J. Phys. C: Solid State Phys. , vol. 19, pp. 7013–7033, Dec. 1986. [2] V. E. Demidov and S. O. Demokritov, “Magnonic waveguides studied by microfocus brillouin light scattering,” IEEE Transactions on Magnetics , vol. 51, p. 1–15, Apr. 2015. [3] T. Br¨ acher, O. Boulle, G. Gaudin, and P. Pirro, “Creation of unidirectional spin-wave emitters by utilizing interfacial dzyaloshinskii-moriya interaction,” Physical Review B , vol. 95, Feb. 2017. [4] A. V. Chumak, “Fundamentals of magnon-based computing,” 2019. [5] A. Vansteenkiste, J. Leliaert, M. Dvornik, M. Helsen, F. Garcia-Sanchez, and B. V. Waeyenberge, “The design and verification of MuMax3,” AIP Adv., vol. 4, p. 107133, Oct. 2014. [6] T. Schneider, A. A. Serga, A. V. Chumak, C. W. Sandweg, S. Trudel, S. Wolff, M. P. Kostylev, V. S. Tiberkevich, A. N. Slavin, and B. Hille- brands, “Nondiffractive subwavelength wave beams in a medium with exter- nally controlled anisotropy,” Phys. Rev. Lett. , vol. 104, May 2010. [7] L. K¨ orber, A. Hempel, A. Otto, R. A. Gallardo, Y. Henry, J. Lindner, and A. K´ akay, “Finite-element dynamic-matrix approach for propagating spin waves: Extension to mono- and multi-layers of arbitrary spacing and thickness,” AIP Adv. , vol. 12, p. 115206, Nov. 2022. 7

2024-03-23

Spin-wave (SW) propagation close to the hybridization-induced transmission stop band is investigated within a trapezoid-shaped 200\,nm thick yttrium iron garnet (YIG) film using time-resolved magneto-optic Kerr effect (TR-MOKE) microscopy and broadband spin wave spectroscopy, supported by micromagnetic simulations. The gradual reduction of the effective field within the structure leads to local variations of the SW dispersion relation and results in a SW hybridization at a fixed position in the trapezoid where the propagation vanishes since the SW group velocity approaches zero. By tuning external field or frequency, spatial control of the spatial stop band position and spin-wave propagation is demonstrated and utilized to gain transmission control over several microstrip lines.

Spatial Control of Hybridization-Induced Spin-Wave Transmission Stop Band

2403.15840v1

Detection sensitivity enhancement of magnon Kerr nonlinearity in cavity magnonics induced by coherent perfect absorption Guo-Qiang Zhang,1,Yimin Wang,2,yand Wei Xiong3,z 1School of Physics, Hangzhou Normal University, Hangzhou, Zhejiang 311121, China 2Communications Engineering College, Army Engineering University of PLA, Nanjing 210007, China 3Department of Physics, Wenzhou University, Zhejiang 325035, China (Dated: February 23, 2023) We show how to enhance the detection sensitivity of magnon Kerr nonlinearity (MKN) in cavity magnonics. The considered cavity-magnon system consists of a three-dimensional microwave cavity containing two yttrium iron garnet (YIG) spheres, where the two magnon modes (one has the MKN, while the other is linear) in YIG spheres are simultaneously coupled to microwave photons. To obtain the e ective gain of the cavity mode, we feed two input fields into the cavity. By choosing appropriate parameters, the coherent perfect absorption of the two input fields occurs, and the cavity-magnon system can be described by an e ective non-Hermitian Hamil- tonian. Under the pseudo-Hermitian conditions, the e ective Hamiltonian can host the third-order exceptional point (EP3), where the three eigenvalues of the Hamiltonian coalesce into one. When the magnon frequency shift
Kinduced by the MKN is much smaller than the linewidths of the peaks in the transmission spectrum of the cavity (i.e.,
K), the magnon frequency shift can be amplified by the EP3, which can be probed via the output spectrum of the cavity. The scheme we present provides an alternative approach to measure the MKN in the region
Kand has potential applications in designing low-power nonlinear devices based on the MKN. I. INTRODUCTION In the past decade, the progress in cavity-magnon systems has been impressive, where magnons (i.e., collective spin ex- citations) in ferrimagnetic materials are strongly coupled to photons in microwave cavities via the collective magnetic- dipole interaction [1–3]. Experimentally, the most widely used cavity-magnon system is composed of the millimeter- scale yttrium iron garnet (Y 3Fe5O12or YIG) crystal and the three-dimensional (3D) microwave cavity [4–7]. Up to now, various exotic phenomena have been extensively in- vestigated in cavity-magnon systems, such as magnon dark modes [8], manipulating spin currents [9, 10], steady-state magnon-photon entanglement [11], magnon blockade [12– 14], non-Hermitian physics [15–17], cooperative polariton dynamics [18], enhancing spin-photon coupling [19], quan- tum states of magnons [20–23], microwave-to-optical trans- duction [24, 25], and dissipative coupling [26, 27]. Based on the coherent perfect absorption (CPA), the second-order exceptional point (EP2) was observed [28] and the third-order EP (EP3) was subsequently predicted [29] in cavity-magnon systems. The CPA refers to a phenomenon that when two (or more) coherent electromagnetic waves are fed into a medium, the waves are completely absorbed by the medium due to both destructive interference between them and medium dissipation, and there are no output waves from the medium [30, 31]. Intriguing applications of CPA include, e.g., engineering EPs [28, 29, 32, 33], antilasing [34, 35], optical switches [36, 37], and coherent polarization con- trol [38, 39]. The nth-order EP (EP n) refers to the degen- erate point in non-Hermitian systems, where neigenvalues zhangguoqiang@hznu.edu.cn yvivhappyrom@163.com zxiongweiphys@wzu.edu.cnas well as corresponding neigenvectors coalesce simultane- ously [40]. Owing to its fundamental importance and po- tential applications, the EPs have been explored in various physical systems (see, e.g., Refs. [41–49]). Contrary to the degenerate point in Hermitian systems, the EPs have some unique features. For example, the energy splitting follows a1=ndependence around the EP nwhen the non-Hermitian systems are subjected to a weak perturbation with strength  (1) [50, 51], which makes it possible to enhance the detec- tion sensitivity [52–54]. It is worth noting that the cavity-magnon system also has reached the nonlinear regime [55], where the magnon Kerr nonlinearity (MKN) stems from the magnetocrystalline anisotropy in the YIG [56]. The MKN not only results in cavity-magnon bistability [57–59] and tristability [60–62], nonreciprocal microwave transmission [63], and strong long- distance spin-spin coupling [64], but it also leads to magnon- photon entanglement [65, 66] as well as dynamical quantum phase transition [67, 68]. In experiments, many phenomena induced by MKN can be detected by measuring the transmis- sion spectrum of the microwave cavity, where the MKN is equivalent to the magnon frequency shift
Kdependent on the magnon population [55–63]. This probe method works well only when the magnon frequency shift
Kis comparable to (or larger than) the linewidths of the peaks in the transmis- sion spectrum of the cavity (i.e.,
K), while it is not valid in the region
K[18, 69]. In this paper, we propose a scheme to enhance the detection sensitivity of MKN around an EP in cavity magnonics when
K. Here, the considered hybrid system consists of a 3D microwave cavity with two YIG spheres (YIG 1 and YIG 2) embedded (cf. Fig. 1), where the magnon mode in YIG 1 has the MKN, while the auxiliary magnon mode in YIG 2 is linear. By feeding two input fields with the same frequency into the 3D microwave cavity via its two ports, an e ective pseudo- Hermitian Hamiltonian of the cavity-magnon system can be obtained, where the e ective gain of the cavity mode resultsarXiv:2211.08922v2 [quant-ph] 22 Feb 20232 from the CPA of the two input fields. In the absence of the MKN (corresponding to
K=0), we analyze the eigenvalues of the pseudo-Hermitian Hamiltonian and find the EP3 in the parameter space. Further, we show that the magnon frequency shift
K() induced by the MKN can be amplified by the EP3. Finally, we derive the output spectrum of the 3D cavity and display how the amplification e ect can be probed via the output spectrum. Recently, Ref. [70] has proposed to enhance the sensitivity of the magnon-population response to the coe cient of MKN via the anti-parity-time-symmetric phase transition, where the strength of the drive field on the system is fixed. In contrast to Ref. [70], we show that the EP3 can enhance the sensitivity of the eigenvalue response to the small magnon frequency shift induced by MKN in the present work. Our study provides a possibility to detect the MKN in the region
K, which is a complement to the existing approach (i.e., measuring the transmission spectrum of the microwave cavity) [55–63] and may find promising applications in designing low-power non- linear devices in cavity magnonics. In addition to MKN, other weak signals (such as a weak magnetic field), which can re- sult in the changes of system parameters, can also be detected using our scheme. II. THE MODEL As shown in Fig. 1, the considered cavity-magnon system consists of two YIG spheres (YIG 1 and YIG 2) and a 3D mi- crowave cavity, where YIG 1 and YIG 2 are uniformly mag- netized to saturation by the bias magnetic fields B1andB2, re- spectively. Here, to enhance the detection sensitivity of MKN in YIG 1, the YIG 2 provides a magnon mode serving as an ancilla. Now the entire cavity-magnon system is described by the Hamiltonian [56, 57] H=!caya+X j=1;2h !jby jbj+Kjby jbjby jbj+gj(aybj+aby j)i + d(by 1ei!dt+b1ei!dt); (1) where aanday(bjandby jwith j=1;2) are the annihilation and creation operators of the cavity mode (magnon mode in YIG j) at frequency !c(!j),gjis the coupling strength be- tween the cavity mode aand the magnon mode bj, and d (!d) is the strength (frequency) of the drive field on YIG 1. In the two YIG spheres, the magnetocrystalline anisotropy re- sults in the MKN term Kjby jbjby jbj, where the nonlinear coef- ficient Kjcan be continuously tuned from negative values to positive values by adjusting the angle between the crystallo- graphic axis of YIG jand the bias magnetic field Bj[71, 72]. Without loss of generality, we assume K1>0 and K2=0 in our scheme. When macroscopic magnons are excited in YIG 1 (i.e.,hby 1b1i1), the system Hamiltonian in Eq. (1) can be linearized as H=!caya+X j=1;2h !jby jbj+gj(aybj+aby j)i +
Kby 1b1 + d(by 1ei!dt+b1ei!dt); (2) FIG. 1. Schematic of the proposed setup for enhancing the detection sensitivity of MKN in YIG 1. The cavity magnonic system is com- posed of two YIG spheres coupled to a 3D microwave cavity, where YIG 1 (YIG 2) is magnetized by a static magnetic field B1(B2). To measure the weak MKN in YIG 1, one microwave field with Rabi frequency dis used to drive YIG 1. In addition, two input fields a(in) 1anda(in) 2are fed into the microwave cavity via ports 1 and 2, re- spectively, and a(out) 1anda(out) 2denote the corresponding output fields. with the frequency shift
K=2K1hby 1b1iof the magnon mode b1, where the mean-field approximation by 1b1by 1b1 2hby 1b1iby 1b1has been used [56, 57]. When the magnon frequency shift
Kis comparable to (or larger than) the linewidths of the peaks in the transmission spectrum of the cavity (i.e.,
K), the MKN can be probed by measuring the transmission spectrum of the cavity [55– 63], where the linewidths are comparable to the decay rates of cavity mode and magnon modes. However, in the case of
K, it is di cult to probe the MKN in this way [18, 69]. For measuring the magnon frequency shift
Kin this circumstance, we feed two weak input fields a(in) 1anda(in) 2with same frequency !pinto the microwave cavity via ports 1 and 2, respectively. Using the input-output formalism [73], we get the equations of motion of the cavity-magnon system as follows: ˙a=i(!cic)aX j=1;2 igjbjq 2ja(in) jei!pt +p 2cf(in) a; ˙b1=i(!1+
Ki 1)b1ig1ai dei!dt+p 2 1f(in) b1; ˙b2=i(!2i 2)b2ig2a+p 2 2f(in) b2; (3) where 1( 2) is the decay rate of the magnon mode b1(b2), the total decay rate c=int+1+2of the cavity mode is composed of the intrinsic decay rate intand the decay rates 1 and2induced by the ports 1 and 2, and f(in) a(f(in) bj) with zero mean valuehf(in) ai=0 (hf(in) bji=0) describes the quantum3 noise from the environment related to the cavity mode (the magnon mode bj). Following the above equations of motion, the expected values haiandhbjisatisfy h˙ai=i(!cic)haiX j=1;2 igjhbjiq 2jha(in) jiei!pt ; h˙b1i=i(!1+
Ki 1)hb1iig1haii dei!dt; h˙b2i=i(!2i 2)hb2iig2hai: (4) In the absence of the two input fields (corresponding to ha(in) 1i=ha(in) 2i=0), we denotehai=Aei!dtandhbji= Bjei!dt. When the input fields are considered, we assume that the changes of haiandhbjican be expressed as Aei!pt andBjei!pt, i.e., hai=Aei!dt+Aei!pt; hbji=Bjei!dt+Bjei!pt; (5) wherejAj  j AjandjBjj  j Bjj[56]. This assumption is reasonable, because compared with the drive field, the input fields are very weak and can be treated as a perturbation. Now the magnon frequency shift becomes
K=2K1jB1j2. Substi- tuting Eq. (5) into Eq. (4), we have ˙A=i(cdic)A ig1B1ig2B2; ˙B1=i(1d+
Ki 1)B1ig1A i d; ˙B2=i(2di 2)B2ig2A; (6) and ˙A=i(cpic)AX j=1;2 igjBjq 2jha(in) ji ; ˙B1=i(1p+
Ki 1)B1ig1A; ˙B2=i(2pi 2)B2ig2A; (7) wherecd=!c!d(jd=!j!d) is the frequency detuning between the cavity mode (magnon mode j) and the drive field, andcp=!c!p(jp=!j!p) is the frequency detuning between the cavity mode (magnon mode j) and the two in- put fields. Eq. (6) determines the magnon frequency shift
K, while Eq. (7) determines the output spectrum of the cavity. According to the input-output theory [73], the output field ha(out) jifrom the port jof the cavity is given by ha(out) ji=q 2jAha(in) ji: (8) Under the pseudo-Hermitian conditions [cf. Eq. (12) in Sec. III], the CPA may occur by carefully choosing appropri- ate parameters of the two input fields [cf. Eqs. (16) and (17) in Sec. III] [29]. The CPA means that the two input fields are nonzero but there are no output fields, i.e., ha(in) 1i,0 and ha(in) 2i,0 butha(out) 1i=ha(out) 2i=0 [28, 32, 33]. When ha(out) 1i=ha(out) 2i=0, ha(in) ji=q 2jA: (9)Inserting the above relation into Eq. (7) to eliminate ha(in) ji, Eq. (7) can be rewritten as 0BBBBBBB@˙A ˙B1 ˙B21CCCCCCCA=iHe0BBBBBBB@A B1 B21CCCCCCCA; (10) where He=0BBBBBBB@cp+ig g1 g2 g11p+
Ki 1 0 g2 02pi 21CCCCCCCA(11) is the e ective non-Hermitian Hamiltonian of the cavity- magnon system. Due to the occurrence of CPA, the cavity mode has an e ective gaing=1+2int(>0) [28, 29]. III. ENHANCING THE DETECTION SENSITIVITY OF MKN A. The EP3 in the cavity-magnon system In this section, we study the EP3 in the cavity-magnon system when
K=0. Usually, the eigenvalues of a non- Hermitian Hamiltonian are complex. However, when the sys- tem parameters satisfy the pseudo-Hermitian conditions [29], g=(1+) 2;
2=
1;
2 1=1+k2 (1+)g2 1 2 2;g1gmin; (12) the eective non-Hermitian Hamiltonian Hein Eq. (11) has the pseudo-Hermiticity and thus can also own either three real eigenvalues or one real and two complex-conjugate eigenval- ues [74–76]. The parameter = 1= 2(k=g2=g1) de- notes the ratio between the decay rates 1and 2(coupling strengths g1andg2),
j=!j!cis the frequency detun- ing of the magnon mode jrelative to the cavity mode, and gmin=[(1+)=(1+k2)]1=2 2is the allowed minimal value of the coupling strength g1for ensuring
2 10. For engineering the EP3 under the pseudo-Hermitian con- ditions in Eq. (12), the parameters andkmust satisfy the following constraint [29]: k= 1+2 2+2!3=2 : (13) In the symmetric case of =k=1, the non-Hermitian Hamiltonian Hehas three eigenvalues, 0=cpand = cpq 3g2 14 2 2[29]. Obviously, 0is real and indepen- dent of the coupling strength g1and the decay rate 2, while are functions of g1and 2. To have three real eigenvalues, the coupling strength g1should be in the region g1>gEP3, where gEP3=2 2=p 3. For g1=gEP3in particular, the three eigenvalues and 0coalesce to = 0= EP3=cp, and the corresponding three eigenvectors of Healso coalesce4 toji=ji0=jiEP3=1p 3 1;1+p 3i 2;1p 3i 2T . This co- alescent point at g1=gEP3is referred to as the EP3. While gming1<gEP3, become complex. For the asymmet- ric case with ,1 and k,1, the expressions of and 0are cumbersome and not shown here, and we only give the coalesced eigenvalues = 0= EP3atg1=gEP3= [2(2+2)1=2=(1+2)] 2, where [29] EP3=cpp 3(1) 22+5+2 2: (14) At the EP3, the three eigenvectors of Hecoalesce to jiEP3=1p N0BBBBB@1;2p 2+2p 3i(1+2);2p 2+1p 3+i(2+)1CCCCCAT ;(15) with the normalization factor N=(22+5+2)=(2++1), i.e.,ji=ji0=jiEP3. Note that the results in Eqs. (14) and (15) are also valid for the symmetric case of =k=1. As stated in Sec. II, the e ective non-Hermitian Hamil- tonian Hein Eq. (11) is obtained in the presence of CPA. For engineering the CPA in the pseudo-Hermitian conditions in Eq. (12), the strengths of the two input fields should sat- isfy [29] ha(in) 2i ha(in) 1i=r2 1: (16) In addition, the same frequency of the two input fields need to be equal to the real eigenvalues of He[29], i.e., !(CPA) p= ;0when Im[ ;0]=0: (17) Therefore, the eigenvalues and the EP3 of the pseudo- Hermitian cavity-magnon system can be probed by measur- ing the CPA via the output spectrum of the cavity in experi- ments [28, 32, 33]. B. Eigenvalue response to the MKN near the EP3 Here we investigate the eigenvalue response to the MKN in YIG 1 near the EP3. Considering the magnon frequency shift
K(,0), the three eigenvalues of the cavity-magnon system can be obtained by solving the corresponding characteristic equation jHe Ij=0; (18) with an identity matrix I. Because the magnon frequency shift
Kis much smaller than other parameters of the cavity- magnon system, we can perturbatively expand the eigenvalue near the EP3 as = EP3+11=3 2+22=3 2 (19) using a Newton-Puiseux series [77–79], where only the first two terms are considered, and EP3is given in Eq. (14). The 0.0 0.1 0.2 0.3-1.0-0.50.00.51.01.5 0.0 0.1 0.2 0.3-1.0-0.50.00.51.01.50.0 0.1 0.2 0.3-1.0-0.50.00.51.01.5 0.0 0.1 0.2 0.3-1.0-0.50.00.51.01.5(a) (b)(c) (d) FIG. 2. The changes of the real and imaginary parts of the eigen- values and 0, (Re[ ;0] EP3)= 2and Im[ ;0]= 2, versus the magnon frequency shift
K= 2near the EP3, where =1 in (a,b), while=2 in (c,d). In (a)–(d), the thick curves correspond to the numerical results obtained by numerically solving the characteristic equation in Eq. (18), and the thin curves correspond to the analytical results in Eq. (23). Note that the thick curves almost overlap the thin curves in (b,d). coecients1and2are complex, while =
K= 2(1) is real. With Eq. (19), the characteristic equation of the cavity- magnon system in Eq. (18) can be expressed as f1+f4=34=3+f5=35=3+f22+f7=37=3=0; (20) where the coe cients are f1=3 142(1p 3i) 1+2; f4=3=32 122[p 3i(1+2)] 1+21; f5=3=312 222 12[p 3i(1+2)] 1+22; f2=3 2412; f7=3=22 2: (21) Since4=35=327=3, we can ignore the contri- butions from the last three terms in Eq. (20), and Eq. (20) is reduced to f1+f4=34=3=0. To ensure the relation f1+f4=34=3=0 is valid for any , the coe cients f1and f4=3must be zero, i.e., f1=f4=3=0. Solving f1=f4=3=0, we obtain three sets of solutions for the coe cients1and2, (l) 1= 82 1+2!1=3 eil; (l) 2=2[p 3i(1+2)] 3(1+2)(l) 1; (22) with l=;0, where+=17=9,=11=9, and0=5=9. Now the three complex eigenvalues of the cavity-magnon sys-5 tem read += EP3+(+) 11=3 2+(+) 22=3 2; 0= EP3+(0) 11=3 2+(0) 22=3 2; = EP3+() 11=3 2+() 22=3 2: (23) Clearly, the changes of the eigenvalues, ;0 EP3, are pro- portional to 1=3in the case of 1, i.e., ;0 EP3 (;0) 11=3 2. By numerically solving the characteristic equation in Eq. (18), we further study the eigenvalue response to the MKN near the EP3 when
K= 2<0:3. In the symmetric case of =1, we plot the changes of the real and imaginary parts of and 0, (Re[ ;0] EP3)= 2and Im[ ;0]= 2, as func- tions of magnon frequency shift
K= 2(i.e.,) in Figs. 2(a) and 2(b), where the thick curves correspond to the numerical results, and the thin curves correspond to the analytical results in Eq. (23). The analytical results and the numerical results are almost consistent for
K= 2<0:1, while the analytical results deviate from the numerical results when
K= 2>0:1 because the condition
K= 21 has been used in deriving Eq. (23). Obviously, (Re[ ;0] EP3)= 2and Im[ ;0]= 2 versus
K= 2sharply change. This is because the small fre- quency shift
Kis amplified by the EP3 [50, 51]. In the re- gion1, (Re[ ;0] EP3)= 2and Im[ ;0]= 2follow the cube-root of , i.e., (Re[ ;0] EP3)= 2Re[(;0) 1]1=3 and Im[ ;0]= 2Im[(;0) 1]1=3. It is very di erent from the existing approach of measuring MKN, where the energy split- ting follows a dependence [55–57]. Further, we find that the amplification e ect is more significant for a larger value of [cf. Figs. 2(a) and 2(c); Figs. 2(b) and 2(d)], which results from the monotonous increase of j(l) 1j=[82=(1+2)]1=3ver- sus. Considering the experimentally accessible parameters, we choose 13 in our study [1, 28, 29]. This amplifica- tion e ect of the EP3 can be used to measure the MKN in the case of
K= 2<1 (cf. Sec. IV). IV . MEASURING THE MKN VIA THE OUTPUT SPECTRUM OF THE CA VITY In the cavity-magnon system, we can measure the eigen- value response to the MKN via the output spectrum of the cavity [28, 29]. In the theory, the output spectrum can be de- rived using Eqs. (7) and (8). At the steady state, we solve Eq. (7) with ˙A=˙B1=˙B2=0 and obtain the change Aof the cavity fieldhaidue to the two input fields, A=p21ha(in) 1i+p22ha(in) 2i c+icp+P(!p); (24) where X (!p)=g2 1 1+i(1p+
K)+g2 2 2+i2p(25) -2 -1 0 1 2-200-150-100-500 (a) -2 -1 0 1 2-50-40-30-20-10 (b)(dB) (dB)FIG. 3. (a) The output spectrum jS(!p)j2of the cavity at the EP3, where
K=0. (b) The output spectrum jS(!p)j2of the cavity near the EP3 when
K,0 (e.g.,
K= 2=0:01). The (red) dashed ver- tical lines in (b) highlight the locations of the two dips in the output spectrum. Other parameters are chosen to be 1= 2=1,int= 2=1, and1= 2=2= 2=1:5. is the self-energy. Correspondingly, the two output fields ha(out) 1iandha(out) 2iin Eq. (8) can be expressed as ha(out) 1i=21ha(in) 1i+2p12ha(in) 2i c+icp+P(!p)ha(in) 1i; ha(out) 2i=2p12ha(in) 1i+22ha(in) 2i c+icp+P(!p)ha(in) 2i: (26) It follows from Eq. (26) that ha(out) 1i=S(!p)ha(in) 1iand ha(out) 2i=S(!p)ha(in) 2iunder the constraint in Eq. (16), where S(!p)=21+22 c+icp+P(!p)1 (27) is the output spectrum of the microwave cavity. It can be easily verified that in the case of
K=0, the output spec- trum S(!p) is zero [i.e., S(!p)=0] when the system pa- rameters satisfy the pseudo-Hermitian conditions in Eq. (12) and the same frequency of the two input fields is given in Eq. (17) [29]. At the EP3, the three eigenvalues and 0of the cavity- magnon system coalesce to EP3, and the CPA occurs at !(CPA) p= EP3, i.e., there is only one CPA point with jS(!p)j= 0 in the output spectrum [see Fig. 3(a)]. In the presence of the MKN (i.e.,
K,0), the CPA disappears, and there are6 0.00 0.05 0.10 0.15 0.20 0.25 0.300.00.51.01.52.0(a) 0.00 0.05 0.10 0.15 0.20 0.25 0.301101001000 (b) 10-510-410-310-210-110-210-1100 FIG. 4. (a) The distance !p= 2between the two dips in the output spectrum of the cavity versus the magnon frequency shift
K= 2for dierent. The inset displays the logarithmic relationship between !p= 2and
K= 2for di erent, where the three (violet) thin curves with a same slope of 1 =3 serve as guides to the eyes. (b) Detection sensitivity enhancement factor !p=
Kversus the magnon frequency shift
K= 2for di erent. Here=1 for the (black) solid curve, =2 for the (red) dashed curve, and =3 for the (blue) dotted curve. Other parameters are chosen to be 1= 2=,int= 2=1, and 1= 2=2= 2=1+0:5. two dips in the output spectrum highlighted by the two (red) dashed vertical lines in Fig. 3(b). The locations and linewidths of the dips in the output spectrum are determined by the real and imaginary parts of the complex eigenvalues of the cavity- magnon system given in Eq. (23). The left dip at !(dip1) p Re[ ] (right dip at !(dip2) pRe[ +]) corresponds to the eigenvalue ( +). Note that because jIm[ 0]j>jIm[ ]j [cf. Figs. 2(b) and 2(d)], there is no dip in the output spec- trum corresponding to the eigenvalue 0. Therefore, we can measure the MKN by the output spectrum of the cavity. To characterize the detection sensitivity enhancement of MKN near the EP3, we introduce an experimentally measur- able quantity !p=!(dip2) p!(dip1) p; (28) which presents the distance between the two dips in the out- put spectrum of the cavity. By numerically solving the output spectrum S(!p) in Eq. (27), we plot the frequency di erence !p= 2as a function of the magnon frequency shift
K= 2 for di erent values of in Fig. 4(a), where !p= 2increases monotonically with
K= 2. Obviously, for a given value of
K= 2, the corresponding frequency di erence!p= 2be-tween the two dips is far larger than the magnon frequency shift
K= 2, i.e.,!p
K. In contrast, the frequency dif- ference induced by
Kis approximately equal to
Kin the existing approach of measuring MKN [55–57]. This means that the magnon frequency shift
Kis amplified by the EP3. For su ciently small
K= 2,!pfollows a (
K= 2)1=3depen- dence [see the inset in Fig. 4(a)]. Especially, for a larger value of, the amplification e ect of the EP3 is more significant. Moreover, we also display the detection sensitivity enhance- ment factor !p=
Kversus the magnon frequency shift
K= 2 in Fig. 4(b), where !p=
Kmonotonically decreases for dif- ferent. In the region
K= 21,!p=
Kis proportional to (
K= 2)2=3. When
K= 2tends to 0, the sensitivity enhance- ment factor !p=
Ktends to infinity, i.e., !p=
Kdiverges at
K= 2=0. V . DISCUSSIONS AND CONCLUSIONS In our study, all results are based on the equations of motion in Eq. (4), which describes the average behavior of the cavity- magnon system in the mean-field approximation by neglecting the impacts of noises [including classical noise related to fluc- tuations of system parameters and quantum noise related to termsp2cf(in) aandp2 jf(in) bjin Eq. (3)] and quantum fluc- tuations (related to a=ahaiandbj=bjhbji). Using Eq. (4), we investigate the detection sensitivity enhancement of MKN by deriving the e ective non-Hermitian Hamiltonian Heof the cavity-magnon system in Eq. (11) and the output spectrum S(!p) of the microwave cavity in Eq. (27). This pro- cedure is widely applied in studying EP-based sensors [50– 54], and the related theoretical predictions have been demon- strated experimentally in various physical systems [80]. For example, the detection sensitivity enhancement factor of 23 has been realized experimentally in a ternary micro-ring sys- tem [77]. However, in the region with the signal being comparable to the noises and quantum fluctuations, the impacts of noises and quantum fluctuations on the EP-based sensor should be considered [80]. The classical noise caused by the techni- cal limitation can reduce the resolvability of frequency di er- ence!pby broadening the linewidth of the output spectrum S(!p) [81, 82]. In principle, the classical noise can be made arbitrarily small in the cavity-magnon system. Di erent from the classical noise, the quantum noise cannot be made arbi- trarily small owing to the vacuum noise. Due to the quantum noise and quantum fluctuations, the diverging sensitivity en- hancement factor [cf. Fig. 4(b) and related discussions] does not necessarily lead to arbitrary high measurement precision, where the measurement precision refers to the smallest mea- surable change of signal [83–86]. This is because the EP- based sensor is sensitive to not only the signal but also the quantum noise, and thus the quantum-limited signal-to-noise ratio cannot be improved [80]. Following the procedures in Refs. [84–86], one can derive the upper bound of the signal- to-noise ratio by calculating the quantum Fisher information based on Heisenberg-Langevin equations in Eq. (3). For the MKN term K1by 1b1by 1b1, the corresponding e ective Hamil-7 tonian for quantum fluctuations can be expressed as Hflu= 2
Kby 1b1+by 1by 1+b1b1with=K1hb1i2[64–66]. The two-magnon terms by 1by 1andb1b1can squeeze the quantum fluctuations of magnon mode b1, which can be transferred to cavity mode aand magnon mode b2via their interactions and leads to the squeezing of cavity mode aand magnon mode b2[66]. The squeezing of quantum fluctuations induced by MKN may be helpful for improving the measure- ment precision [87, 88]. Before concluding, we briefly analyze the experimental fea- sibility of the present scheme. In cavity magnonics, both the intrinsic decay rate of the 3D microwave cavity as well as the decay rate of the magnon mode are of the order 1 MHz (i.e., int=21 MHz and 1;2=21 MHz) [1], while the decay rates1;2due to the two ports of the cavity can be tuned from 0 to 8 MHz [28]. Since the frequency of the magnon mode in the YIG is proportional to the bias magnetic field, the fre- quencies!1;2can be easily controlled [8, 60]. In Ref. [28], the EP2 based on CPA has been observed, where the cavity- magnon coupling can be adjusted (ranging from 0 to 9 MHz) via moving the YIG sphere, and the relative amplitudes (rela- tive phases) of the two input fields, ha(in) 1iandha(in) 2i, are also tunable via a variable attenuator (a phase shifter). In addi- tion, the magnon frequency shift
Kcaused by the MKN isdependent on the strength of the drive field on the magnon mode [55, 57, 58]. These available conditions ensure that our scheme in the present work is experimentally accessible. In conclusion, we have presented a feasible scheme to en- hance the detection sensitivity of MKN via the CPA around an EP3. In the proposed scheme, the cavity-magnon system con- sists of a 3D microwave cavity and two YIG spheres. With the assistance of the CPA, an e ective pseudo-Hermitian Hamil- tonian of the cavity-magnon system can be obtained, which makes it possible to engineer the EP3 in the parameter space. Considering the magnon frequency shift caused by the MKN, we find that it can be amplified by the EP3. Moreover, we show that this amplification e ect can be measured using the output spectrum of the 3D cavity. Our proposal paves a way to measure the MKN in the case of
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2022-11-16

We show how to enhance the detection sensitivity of magnon Kerr nonlinearity (MKN) in cavity magnonics. The considered cavity-magnon system consists of a three-dimensional microwave cavity containing two yttrium iron garnet (YIG) spheres, where the two magnon modes (one has the MKN, while the other is linear) in YIG spheres are simultaneously coupled to microwave photons. To obtain the effective gain of the cavity mode, we feed two input fields into the cavity. By choosing appropriate parameters, the coherent perfect absorption of the two input fields occurs, and the cavity-magnon system can be described by an effective non-Hermitian Hamiltonian. Under the pseudo-Hermitian conditions, the effective Hamiltonian can host the third-order exceptional point (EP3), where the three eigenvalues of the Hamiltonian coalesce into one. When the magnon frequency shift $\Delta_K$ induced by the MKN is much smaller than the linewidths $\Gamma$ of the peaks in the transmission spectrum of the cavity (i.e., $\Delta_K\ll \Gamma$), the magnon frequency shift can be amplified by the EP3, which can be probed via the output spectrum of the cavity. The scheme we present provides an alternative approach to measure the MKN in the region $\Delta_K\ll \Gamma$ and has potential applications in designing low-power nonlinear devices based on the MKN.

Detection sensitivity enhancement of magnon Kerr nonlinearity in cavity magnonics induced by coherent perfect absorption

2211.08922v2

Electronic control of the spin-wave damping in a magnetic insulator A. Hamadeh,1O. d'Allivy Kelly,2C. Hahn,1H. Meley,1R. Bernard,2A.H. Molpeceres,2V. V. Naletov,1, 2, 3M. Viret,1A. Anane,2V. Cros,2S. O. Demokritov,4J. L. Prieto,5M. Mu~ noz,6G. de Loubens,1and O. Klein1, 1Service de Physique de l' Etat Condens e (CNRS URA 2464), CEA Saclay, 91191 Gif-sur-Yvette, France 2Unit e Mixte de Physique CNRS/Thales and Universit e Paris Sud 11, 1 av. Fresnel, 91767 Palaiseau, France 3Institute of Physics, Kazan Federal University, Kazan 420008, Russian Federation 4Department of Physics, University of Muenster, 48149 Muenster, Germany 5Instituto de Sistemas Optoelectr onicos y Microtecnolog a (UPM), Madrid 28040, Spain 6Instituto de Microelectr onica de Madrid (CNM, CSIC), Madrid 28760, Spain (Dated: November 28, 2021) It is demonstrated that the decay time of spin-wave modes existing in a magnetic insulator can be reduced or enhanced by injecting an in-plane dc current, Idc, in an adjacent normal metal with strong spin-orbit interaction. The demonstration rests upon the measurement of the ferromagnetic resonance linewidth as a function of Idcin a 5m diameter YIG(20nm) jPt(7nm) disk using a magnetic resonance force microscope (MRFM). Complete compensation of the damping of the fun- damental mode is obtained for a current density of 3
1011A.m2, in agreement with theoretical predictions. At this critical threshold the MRFM detects a small change of static magnetization, a behavior consistent with the onset of an auto-oscillation regime. The spin-orbit interaction (SOI) [1{3] has been re- cently shown to be an interesting and useful addition in the eld of spintronics. This subject capitalizes on adjoining a strong SOI normal metal next to a thin mag- netic layer [4]. The SOI converts a charge current, Jc, to a spin current, Js, with an eciency parametrized by SH, the spin Hall angle [5, 6]. Recently, it was demon- strated experimentally that the spin current produced in this way can switch the magnetization in a dot [7, 8] or can partially compensate the damping [9{11], allow- ing the lifetime of propagating spin-waves [12] to be in- creased beyond their natural decay time, . These two eects open potential applications in storage devices and in microwave signal processing. The eect is based on the fact that the spin current Js exerts a torque on the magnetization, corresponding to an eective damping s= Js=(tFMMs), wheretFMis the thickness of the magnetic layer, Msits spontaneous magnetization, and the gyromagnetic ratio. In the case of metallic ferromagnets [13{15], it was established that scan fully compensate the natural damping 1 =at a critical spin current J s, which determines the onset of auto-oscillation of the magnetization: J s=1 tFMMs : (1) An important benet of the SOI is that JcandJsare linked through a cross-product, allowing a charge current owing in-plane to produce a spin current owing out- of-plane. Hence it enables the transfer of spin angular momentum to non-metallic materials and in particular to insulating oxides, which oer improved performance compared to their metallic counterparts. Among all ox- ides, Yttrium Iron Garnet (YIG) holds a special place for having the lowest known spin-wave (SW) damping factor. In 2010, Kajiwara et al. reported on the e-cient transmission of spin current through the YIG jPt interface [16]. It was shown that Jsproduced by the excitation of ferromagnetic resonance (FMR) in YIG can cross the YIGjPt interface and be converted into Jcin Pt through the inverse spin Hall eect (ISHE). This nding was reproduced in numerous experimental works [17{23]. In the same paper, the reciprocal eect was also reported asJsproduced in Pt by the direct spin Hall eect (SHE) could be transferred to the 1.3 m thick YIG, resulting in damping compensation. However, attempts to directly measure the expected change of the resonance linewidth of YIG as a function of the dc current have so far failed [21, 22] [24]. This is raising fundamental questions about the reciprocity of the spin transparency, T, of the in- terface between a metal and a magnetic insulator. This coecient enters in the ratio between Jcin Pt andJsin YIG through: Js=TSHh 2eJc; (2) whereeis the electron charge and  hthe reduced Planck constant.Tdepends on the transport characteristics of the normal metal as well as on the spin-mixing conduc- tanceG"#, which parametrizes the scattering of the spin angular momentum at the YIG jPt interface [25]. At the heart of this debate lies the exact value of the threshold current. The lack of visible eects reported in Refs.[21, 22], although inconsistent with [16], is co- herent with the estimation of the threshold current of 101112A.m2using Eqs.(1) and (2) and typical pa- rameters for the materials [26]. This theoretical cur- rent density is at least one order of magnitude larger than the maximum Jcthat could be injected in the Pt so far. Importantly, the previous reported experiments were performed on large (millimeter sized) structures, where many nearly degenerate SW modes compete forarXiv:1405.7415v1 [cond-mat.mes-hall] 28 May 20142 TABLE I. Transport and magnetic properties of the Pt and bare YIG layers, respectively from Ref.[31] and Ref.[22]. PttPt(nm)( 1.m1)SD(nm)  SH 7 5:8
1063.5 0.056 YIGtYIG(nm) 4Ms(G) (rad.s1.G1)0 20 2 :1
1031:79
1072:3
104 feeding from the same dc source of angular momentum, a phenomenon that could become self-limiting and pre- vent the onset of auto-oscillations [11]. To isolate a single candidate mode, we have recently reduced the lateral di- mensions of the YIG pattern, as quantization results in increased frequency gaps between the dynamical modes [27]. This requires to grow very thin lms of high qual- ity YIG [23, 28{30]. Beneting from our progress in the epitaxial growth of YIG lms by pulsed laser deposition (PLD) [22], we propose to study the FMR linewidth as a function of the dc current in a micron-size YIG jPt disk. FIG.1 shows a schematic of the experimental setup. A YIGjPt disk of 5 m in diameter is connected to two Au contact electrodes (see the microscopy image) across which a positive voltage generates a current ow Jcalong the +^x-direction. The microdisk is patterned out of a 20 nm thick epitaxial YIG lm with a 7 nm thick Pt layer sputtered on top. The YIG and Pt layers have been fully characterized in previous studies [22, 31]. Their characteristics are reported in Table I. The sample is mounted inside a room temperature magnetic resonance force microscope (MRFM) which de- tects the SW absorption spectrum mechanically [32{34]. The excitation is provided by a stripline (not shown in the sketches of FIG.1) generating a linearly polarized microwave eld h1along the ^x-direction. The detec- tion is based on monitoring the de ection of a mechan- ical cantilever with a magnetic Fe particle axed to its tip, coupled dipolarly to the sample. The FMR spec- trum is obtained by recording the vibration amplitude of the cantilever while scanning the external bias magnetic eld,H0, at constant microwave excitation frequency, f=!=(2) [35]. The MRFM is placed between the poles of an electromagnet, generating a uniform magnetic eld, H0, which can be set along ^ yor ^z(i.e., perpendicularly to bothh1andJc). We start by measuring the eect of a dc current, Idc, on the FMR spectra when the disk is magnetized in- plane by a magnetic eld along the +^ y-direction (positive eld). The spectra recorded at f= 6:33 GHz are shown in FIG.1a in red tones. The middle row shows the ab- sorption at zero current. The MRFM signal corresponds to a variation of the static magnetization of about 2 G, i.e., a precession cone of 2.5. As the the electrical cur- rent is varied, we observe very clearly a change of the linewidth. At negative current, the linewidth decreases, FIG. 1. (Color online) MRFM spectra of the YIG jPt mi- crodisk as a function of current for dierent eld orientations: a)H0k+^yatf= 6:33 GHz (red tone); b) H0k+^zat f= 10:33 GHz (black); c) H0k^yatf= 6:33 GHz (blue tone). The highest amplitude mode is used for linewidth anal- ysis (shaded area). Field axes are shifted so as to align the peaks vertically. In-plane and out-of-plane eld orientations are sketched above. The top right frame is a microscopy im- age of the sample. to reach about half the initial value at Idc=8 mA. This decrease is strong enough so that the individual modes can be resolved spectroscopically within the main peak. Concomitantly the amplitude of the MRFM sig- nal increases. The opposite behavior is observed when the current polarity is reversed. At positive current, the linewidth increases to reach about twice the initial value atIdc= +8 mA, and the amplitude of the signal de- creases. Idc=12 mA is the maximum current that we have injected in our sample to avoid irreversible eects. We estimate from the Pt resistance, the sample temperature to be 90C at the maximum current. This Joule heating reduces 4M sat a rate of 4 :8 G/K, which results in an even shift of the resonance eld towards higher eld [36]. In FIG.1b, we show the FMR spectra at f= 10:33 GHz in the perpendicular geometry, i.e.,H0is along ^z. In con- trast to the previous case, the linewidth does not change with current. This is expected as no net spin transfer torque is exerted by the spin current on the precessing magnetization in this conguration. Note that due to Joule heating, the spectrum now shifts towards lower eld due to the decrease of Msas the current increases. We now come back to the in-plane geometry, but this time, the magnetic eld is reversed compared to FIG.1a,3 FIG. 2. (Color online) Variation of the full linewidth
Hk measured at 6.33 GHz as a function of IdcforH0k+^y(red) andH0k^y(blue). Inset: detection of VISHE as a function ofH0atf= 6:33 GHz and Idc= 0. i.e., applied along^y(negative eld). The correspond- ing spectra are presented in FIG.1c using blue tones. As expected for the symmetry of the SHE, the observed be- havior is inverted with respect to FIG.1a: a positive (neg- ative) current now reduces (broadens) the linewidth. We report in FIG.2 the values of
Hk, the full linewidth measured in the in-plane geometry, as a func- tion of current. The data points follow approximately a straight line, whose slope 0:5 Oe/mA reverses with the direction of H0along^yand whose intercept with the abscissa axis occurs at I 6.33 GHz =14 mA. More- over, we emphasize that the variation of linewidth covers about a factor ve on the full range of current explored. The inset of FIG.2 shows the inverse spin Hall voltage VISHE measured at Idc= 0 mA and f= 6:33 GHz. This voltage results from the spin current produced by spin pumping from YIG to Pt and its subsequent conversion into charge current by ISHE [16]. Its sign changes with the direction of the bias magnetic eld, as shown by the blue and red VISHE spectra. This observation conrms that a spin current can ow from YIG to Pt and that damping reduction occurs for a current polarity corre- sponding to a negative product of VISHE andIdc. To gain more insight into these results, we now an- alyze the frequency dependence of the full linewidth at half maximum for three values of dc current (0, 6 mA) for both the out-of-plane and in-plane geometries. We start with the out-of-plane data, plotted in FIG.3a. The dispersion relation displayed in the inset follows the Kit- tel law,!= (H04NeMs), whereNeis an eective demagnetizing factor close to 1 [37, 38]. The linewidth
H?increases linearly with frequency along a line that intercepts the origin, a signature that the resonance is homogeneously broadened [27]. In this geometry, the Gilbert damping coecient is simply =
H?=(2!) = 1:1
103and the reaxation time = 1=(!). We also report on this gure the fact that at 10.33 GHz,
H?= 7 Oe is independent of the current (see FIG.1b). FIG. 3. (Color online) Frequency dependence of the linewidth for three values of the dc current (0, 6 mA) a) in the per- pendicular geometry and b) in the parallel geometry. Insets show the corresponding dispersion relations f(H0). The damping found in our YIG jPt microdisk is signif- icantly larger than the one measured in the bare YIG lm0= 2:3
104(cf. Table I). This dierence is due to the spin pumping eect, and enables to determine the spin-mixing conductance of our YIG jPt interface through [39, 40]: =0+ h 4M stYIGG"# G0; (3) whereG0= 2e2=his the quantum of conductance. The measured increase of almost 9
104for the damping corresponds to G"#= 1:51014 1m2, in agree- ment with a previous determination made on similar YIGjPt nanodisks [27]. This value allows us to esti- mate the spin transparency of our interface [25], T= G"#=(G"#coth (tPt=sd) +=(2sd))'0:15, whereis the Pt conductivity and sdits spin-diusion length. Moreover, the spin-mixing conductance can be used to analyze quantitatively the dc ISHE voltage produced at resonance [21, 41, 42]. Using the parameters of Table I and the value of G"#, we nd that the 50 nV voltage measured in the inset of FIG.2 is produced by an angle of precession '3:5, which lies in the expected range. We now turn to the in-plane data, presented in FIG.3b. The dispersion relation plotted in the inset follows the Kittel law != p H0(H0+ 4NeMs). In this case, 1==(@!=@H 0) (!= ). ForIdc= 0 mA the slope of the linewidth vs. frequency is exactly the same as that in the perpendicular direction = 1:1
103. For this geometry, however, the line does not intercept the origin, indicating a nite amount of inhomogeneous broadening
H0= 2:5 Oe, i.e, the presence of several modes within the resonance line. Setting Idcto6 mA shifts
Hkby 3 Oe independently of the frequency, which is consistent with the rate of 0.5 Oe/mA reported at 6.33 GHz in FIG.2. In fact, in the presence of the eective damping4 FIG. 4. (Color online) a) Density plot of the MRFM spectra at 4.33 GHz vs. eld and current Idc2[12;+12] mA. The color scale represents 4 
Mz(white: 0 G, black: 1.5 G). b) Evolution of integrated power vs. Idc. c) Dependence of linewidth on Idc. d) Dierential measurements of Mz(Idc modulated by 0.15 mA pp, no rf excitation) vs. Idcat six dierent values of the in-plane magnetic eld. s, the linewidth of the resonance line varies as
Hk=
H0+ 2! + 2Js MstYIG: (4) This expression is valid when ( @!=@H 0)' ,i.e., at large enough eld or frequency (see inset of FIG.3b). It de- scribes appropriately the experimental data on the whole frequency range measured. In order to investigate the autonomous dynamics of the YIG layer and exceed the compensation current, I, we now perform measurements at lower excitation fre- quency, where the threshold current is estimated below 12 mA. In FIG.4a, we present a density plot of the MRFM spectra acquired at 4.33 GHz as a function of the in-plane magnetic eld and Idcthrough the Pt. The measured signal is clearly asymmetric in Idc. At positive current, it broadens and its amplitude decreases, almost disappearing above +8 mA, whereas at negative current, it becomes narrower and the amplitude is maximal at Idc<10 mA. The power integrated over the full eld range normal- ized by its value at 0 mA and the linewidth variation vs. Idcare plotted in FIGs.4b and 4c, respectively. The nor- malized integrated power varies by a factor of ve from +12 mA to12 mA following an inverse law on Idc(see continuous line), which is consistent with the spin trans- fer eect [11, 43]. The linewidth varies roughly linearly withIdc: it increases from 6 Oe at 0 mA up to 14 Oe at +12 mA and it reaches a minimum value close to 2 Oe between8 and11 mA. It is interesting to note that this happens in a region of the density plot where the evolution of the signal displays some kind of discontinu- ity, with the appearance of several high amplitude peaksin the spectrum (see arrow in FIG.4a). We tentatively ascribe this feature to the onset of auto-oscillations in the YIG layer, namely, one or several dynamical modes have their relaxation compensated by the injected spin current and are destabilized [16]. To conrm this hypothesis, we present in FIG.4d re- sults of an experiment where no rf excitation is applied to the system. Here, the dc current is modulated at the MRFM cantilever frequency by I= 0:15 mA ppand the inducedMzis probed as a function of Idc. This experi- ment thus provides a dierential measurement @M z=@Idc of the magnetization (in analogy with dV=dI measure- ments in transport experiments). At H0= 0:92 kOe, a peak in@M z=@Idcis measured around 9 mA. It corre- sponds to a variation of 4 M z'0:5 G, i.e., a change of the angle of precession by 1 :3induced by the modula- tion of current. Moreover, this narrow peak observed in @M z=@Idcshifts linearly in dc current with the applied magnetic eld, from 8 mA at 0.81 kOe to 10 mA at 1.1 kOe (see the continuous straight line in FIG.4d), in agreement with the expected behavior of the threshold current Eq.(1). Hence, FIG.4 presents a set of data consistent with the determination of a critical current of I=9 mA at H0= 0:92 kOe, corresponding to J c'3
1011A.m2, in agreement with the value of 2
1011A.m2expected from Eqs.(1) and (2) and the parameters of our system. Nevertheless, the destabilization of dynamical modes is rather small, as the jump of resonance eld at I(due to reduction of the magnetization) does not exceed the linewidth. We suspect that in our YIG jPt microdisk, the splitting of modes is not sucient to prevent nonlinear interactions that limit the amplitude of auto-oscillations [11]. In order to favor larger auto-oscillation amplitudes, YIG structures that are even more conned laterally (be- low 1m) should be used [27], or one should excite a bullet mode [13]. In conclusion, we have demonstrated that it is possi- ble to control electronically the SW damping in a YIG microdisk. Extending this result to one-dimensional SW guide [44] will oer great prospect in the emerging eld of magnonics [45, 46], whose aim is to investigate the ma- nipulation of SW and their quanta { magnons { with the benece of combining ultra-low energy consumption and compactness. To improve the magnonic paradigm, a so- lution will be to actively compensate damping in the YIG magnetic insulator by SW amplication through stimu- lated emission generated by a charge current in the ad- jacent metallic layer with strong SOI. This research was supported by the French Grants Trinidad (ASTRID 2012 program), by the RTRA Trian- gle de la Physique grant Spinoscopy, and by the Deutsche Forschungsgemeinschaft. We acknowledge C. Deranlot, E. Jacquet, and R. Lebourgeois for their contribution to the growth of the sample and A. Fert for fruitful discus- sion.5 Emails: oklein@cea.fr & gregoire.deloubens@cea.fr [1] S. O. Valenzuela and M. Tinkham, Nature (London) 442, 176 (2006) [2] T. Jungwirth, J. Wunderlich, and K. Olejnik, Nat Mater 11, 382 (May 2012), ISSN 1476-1122 [3] A. Manchon and S. Zhang, Phys. Rev. B 78, 212405 (2008) [4] A. Thiaville, S. Rohart, Emilie Ju e, V. Cros, and A. Fert, EPL (Europhysics Letters) 100, 57002 (2012) [5] M. I. Dyakonov and V. I. 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2014-05-28

It is demonstrated that the decay time of spin-wave modes existing in a magnetic insulator can be reduced or enhanced by injecting an in-plane dc current, $I_\text{dc}$, in an adjacent normal metal with strong spin-orbit interaction. The demonstration rests upon the measurement of the ferromagnetic resonance linewidth as a function of $I_\text{dc}$ in a 5~$\mu$m diameter YIG(20nm){\textbar}Pt(7nm) disk using a magnetic resonance force microscope (MRFM). Complete compensation of the damping of the fundamental mode is obtained for a current density of $\sim 3 \cdot 10^{11}\text{A.m}^{-2}$, in agreement with theoretical predictions. At this critical threshold the MRFM detects a small change of static magnetization, a behavior consistent with the onset of an auto-oscillation regime.

Electronic control of the spin-wave damping in a magnetic insulator

1405.7415v1

Electrical spectroscopy of the spin-wave dispersion and bistability in gallium-doped yttrium iron garnet Joris J. Carmiggelt1,Olaf C. Dreijer1,Carsten Dubs2,Oleksii Surzhenko2,Toeno van der Sar1; 1Department of Quantum Nanoscience, Kavli Institute of Nanoscience, Delft University of Technology, 2628 CJ Delft, The Netherlands 2INNOVENT e.V. Technologieentwicklung, D-07745 Jena, Germany Corresponding author. Email: t.vandersar@tudelft.nl Abstract Yttrium iron garnet (YIG) is a magnetic insulator with record-low damping, allowing spin- wave transport over macroscopic distances. Doping YIG with gallium ions greatly reduces the demagnetizing eld and introduces a perpendicular magnetic anisotropy, which leads to an isotropic spin-wave dispersion that facilitates spin-wave optics and spin-wave steering. Here, we characterize the dispersion of a gallium-doped YIG (Ga:YIG) thin lm using electrical spec- troscopy. We determine the magnetic anisotropy parameters from the ferromagnetic resonance frequency and use propagating spin wave spectroscopy in the Damon-Eshbach conguration to detect the small spin-wave magnetic elds of this ultrathin weak magnet over a wide range of wavevectors, enabling the extraction of the exchange constant = 1:3(2)1012J/m. The frequencies of the spin waves shift with increasing drive power, which eventually leads to the foldover of the spin-wave modes. Our results shed light on isotropic spin-wave transport in Ga:YIG and highlight the potential of electrical spectroscopy to map out the dispersion and bistability of propagating spin waves in magnets with a low saturation magnetization. 1arXiv:2109.05045v1 [cond-mat.mes-hall] 10 Sep 2021Yttrium iron garnet (YIG) is a magnetic insulator that is famous for its low Gilbert damping and long-range spin-wave propagation [1]. At low bias elds the YIG magnetization is typically pushed in the plane by the demagnetizing eld [2], leading to an anisotropic spin-wave dispersion at microwave frequencies. For applications that rely on spin-wave optics and spin-wave steering an isotropic spin- wave dispersion is desirable [3], which can be achieved by introducing gallium dopants in the YIG: The presence of the dopants reduces the saturation magnetization and thereby the demagnetizing eld [4], and induces a perpendicular magnetic anisotropy (PMA) [5, 6], such that the magnetization points out-of-plane. Isotropic transport of forward-volume spin waves has been observed even at zero bias eld [7], opening the door for spin-wave logic devices [8{10]. To harness isotropic spin waves it is essential to know the spin-wave dispersion, which is dominated by the exchange interaction for magnets with a low saturation magnetization [11]. Here, we use all-electrical spectroscopy of propagating spin waves to characterize the spin-wave dispersion of a 45- nm-thick lm of gallium-doped YIG (Ga:YIG). Rather than looking at the discrete mode numbers of perpendicular standing spin waves [12], this method enables extracting the exchange constant by monitoring the spin-wave transmission for a continuously-tunable range of wavevectors. We show that this technique has sucient sensitivity to characterize spin waves in nanometer-thick Ga:YIG lms, where perpendicular modes may be challenging to detect due to their high frequencies and small mode overlap with the stripline drive eld. We extract the anisotropy parameters from the eld dependence of the ferromagnetic resonance (FMR) frequency at dierent bias eld orientations and nd that the PMA is strong enough to lift the magnetization out of the plane. Next, we characterize the spin-wave dispersion from electrically- detected spin-wave spectra. We measure in the Damon-Eshbach conguration to boost the inductive coupling of the spin waves to the striplines [13], allowing the extraction of the spin-wave group ve- locity over a wide range of wavevectors from which we determine the exchange constant. When increasing the microwave excitation power, we observe clear frequency shifts of the spin-wave modes. The shifts result in the foldover of spin waves, which we verify by comparing upward and downward frequency sweeps. These results benchmark propagating spin wave spectroscopy as an accessible tool to characterize the exchange constant and spin-wave foldover in technologically attractive thin-lm magnetic insulators with low saturation magnetization and PMA. We use liquid phase epitaxy to grow a 45-nm-thick lm of Ga:YIG on an (111)-oriented gadolin- ium gallium garnet (GGG) substrate (supplementary material section 1). Using vibrating sample magnetometry (VSM) we determine the saturation magnetization Ms= 1:52(6)104A/m (Fig. 1a, 2the number in parentheses denotes the 95% condence interval), which is approximately an order of magnitude smaller than undoped YIG lms of similar thicknesses [14]. Out-of-plane B0 (mT)02 0 -20Magnetization (emu/cm3) 0 -2020 Magnetic field B0 (mT)FMR frequency (GHz)ab 0Out-of-plane In-plane 80 16048 0 Figure 1: The saturation magnetization and magnetic anisotropies of Ga:YIG. (a) Hysteresis loop of the magnetization of a 45-nm-thick Ga:YIG lm as a function of out-of-plane magnetic eld B0measured using vibrating sample magnetometry and corrected for magnetic background. The arrows denote the sweep direction of the magnetic eld. (b) FMR measurements using an out-of-plane (green) and in-plane (red) magnetic eld B0. From the ts of the FMR frequencies (solid lines) we determine the perpendicular and cubic anisotropy elds (see text). In addition to PMA, Ga:YIG lms also have a cubic magnetic anisotropy due to a cubic unit cell. We start by determining the cubic and perpendicular anisotropy elds from the ferromagnetic resonance (FMR) frequencies !FMR=2using an out-of-plane ( ?) and in-plane (jj) magnetic bias eld B0. For (111)-oriented lms the out-of-plane and in-plane Kittel relations are given by [14, 15] !FMR(?)= ?(B00Ms+2K2? Ms4 3K4 Ms); (1) !FMR(jj)= jjr B0(B0+0Ms2K2? MsK4 Ms): (2) Here ?;jj=g?;jjB=his the gyromagnetic ratio with g?;jjthe anisotropic g-factor, Bthe Bohr mag- neton and hthe reduced Planck constant, 0is the magnetic permeability of free space, K2?is the uniaxial out-of-plane anisotropy (e.g. PMA) constant and K4the cubic anisotropy constant. During the in-plane FMR measurement we apply the magnetic eld along the [1 10] crystallographic axis to minimize the out-of-plane component of the magnetization (supplementary material section 2). We neglect any uniaxial in-plane anisotropy as it is known to be small in YIG samples [14]. 3By substituting the value of Msthat we obtained with VSM into equations 1 and 2, we can deter- mineK2?andK4from the FMR frequencies (Fig. 1b) [16]. From the ts (solid lines) we extract the uniaxial out-of-plane anisotropy eld 2 K2?=Ms= 104:7(8) mT and the cubic anisotropy eld 2K4=Ms=8:2(5) mT (supplementary material section 3). Undoped YIG lms of similar thicknesses have comparable cubic anisotropy elds [14], which agrees with previous work on micrometer-scale lms showing that the cubic anisotropy of YIG does not depend on gallium concentration [17]. We determine the in-plane and out-of-plane g-factors to be gjj= 2:041(4) and g?= 2:101(3) [18]. Ga:YIGa +20 dBVNAPort 2 Port 1 w s 10 μmB0xy Frequency (GHz)b Magnetic field B0 (mT)100 7511.522.5 125d|S21|/dB0 (dB/mT) -0.5 0.5 Figure 2: All-electrical propagating spin wave spectroscopy. (a) Optical micrograph of a Ga:YIG lm with two gold striplines that are connected to the ports of a vector network analyser (VNA). Port 1 applies a microwave current (typical excitation power: 35dBm) that induces a radio-frequency magnetic eldBRFat the injector stripline. This eld excites propagating spin waves that couple inductively to the detector stripline at a distance s. The generated microwave current is amplied and detected at port 2. A static magnetic eld B0is applied in the Damon-Eshbach conguration and is oriented such that the chirality ofBRFfavours the excitation of spin waves propagating towards the detector stripline [19]. (b) Field-derivative of the microwave transmission jS21jbetween two striplines ( w= 1m,s= 6m) as a function ofB0and microwave frequency. The colormap is squeezed, such that also fringes corresponding to low-amplitude spin waves are visible. A masked background was subtracted to highlight the signal attributed to spin waves (supplementary material section 4). We now use propagating spin wave spectroscopy to characterize the spin-wave dispersion in Ga:YIG. We measure the microwave transmission jS21jbetween two microstrips fabricated directly on the 4Ga:YIG as a function of static magnetic eld B0and frequency f(Fig. 2a). The magnetic eld is applied in the Damon-Eshbach geometry to maximize the inductive coupling between the spin waves and the striplines [13]. We measure a clear Damon-Eshbach spin-wave signal in the microwave transmission spectrum when B0overcomes the PMA and pushes the spins in the plane (Fig. 2b, supplementary material section 4). The signal appears at a nite frequency, because the bias eld B0 is applied along the [11 2] crystallographic axis with a nite out-of-plane angle of 1(supplementary material section 2). The fringes in the transmission spectra result from the interference between the spin waves and the microwave excitation eld [20, 21]. Each fringe indicates an extra spin-wavelength that ts between the striplines. We can thus use the fringes to determine the group velocity vgof the spin waves via [22] vg=@!SW @k2
f 2=s=
fs: (3) Here!SW= 2fandk= 2= are the spin wave's angular frequency and wavevector,
fis the frequency dierence between two consecutive maxima or minima of the fringes (Fig. 3a) and sis the center-to-center distance between both microstrips. We extract the exchange constant of our Ga:YIG lm by tting the measured group velocity to an analytical expression derived from the spin-wave dispersion. The Damon-Eshbach spin-wave disper- sion for magnetic thin lms with cubic and perpendicular anisotropy is given by [15] (supplementary material section 5) !SW(k) =r !B(!B+!M!K) +!Mt 2(!M!K)k+ jjD(2!B+!M!K)k2+ 2 jjD2k4:(4) Here we dened for notational convenience !B= jjB0,!M= jj0Ms,!D= jjD Ms, and!K= jj(2K2?=Ms+K4=Ms),tis the thickness of the lm and D= 2=M sis the spin stiness, with  the exchange constant. Dierentiating with respect to kgives an analytical expression for the group velocity vg(k) =1 2p !SW(k)!Mt 2(!M!K) + 2 jjD(2!B+!M!K)k+ 4 2 jjD2k3
: (5) Since we determined Msand the anisotropy constants from the VSM and FMR measurements, the exchange constant is the only unknown variable in the dispersion. We determine the exchange constant from spin-wave spectra measured using two sets of striplines with dierent widths and line- to-line distances ( w= 1m,s= 6m andw= 2:5m,s= 12:5m) at the same static eld 5(Fig. 3a,b). First we extract vgas a function of frequency from the extrema in the spin-wave spectra using equation 3 (Fig. 3c). By then tting the measured vg(f) using equations 4 and 5 (solid line in Fig. 3c), we nd = 1:3(2)1012J/m andB0= 117:5(3) mT (supplementary material sec- tion 3). The determined exchange constant is about 3 times smaller compared to undoped YIG [12], which is in line with earlier observations of a decreasing exchange constant with increasing gallium concentration in micrometer-thick YIG lms [23]. Simultaneously the spin stiness is increased by about 3 times compared to undoped YIG [12] due to the strong reduction of the saturation mag- netization. For large wavelengths the group velocity is negative as a result of the PMA in the sample. The spin-wave excitation and detection eciency depends on the absolute value of the Fourier ampli- tude of the radio-frequency magnetic eld BRFgenerated by a stripline, which oscillates in kwith a period given by
k= 2=w (Fig. 3e) [20, 21]. To verify that the spin waves we observe are eciently excited and detected by our striplines, we substitute the extracted exchange constant into equation 4 and plot the spin-wave dispersion (Fig. 3f). The shaded areas correspond to the frequencies of the spin-wave fringes (Fig. 3a,b) and the dashed lines indicate the nodes in jBRF(k)jof both striplines (Fig. 3d,e). We conclude that the fringes in Fig. 3a correspond to spin waves excited by the rst maximum ofjBRF(k)jand that the fringes in Fig. 3b correspond to spin waves excited by the second maximum. Surprisingly, we do not observe fringes in Fig. 3b corresponding to the rst maximum of jBRF(k)j, but rather see a dip in this frequency range (arrows in Fig. 3b,f). This can be understood by noting that the average frequency dierence between the fringes would be smaller than the spin-wave linewidth (supplementary material section 6). Low-amplitude fringes corresponding to small-wavelength spin waves excited by the second k-space maximum of the 1- m-wide stripline are also visible (Fig. 2b, supplementary material section 7). These results demonstrate that the spin-wave dispersion in weak magnets can be reliably extracted using propagating spin wave spectroscopy by combining measure- ments on striplines with dierent widths and spacings. When strongly driven to large amplitudes, the FMR behaves like a Dung oscillator with a bistable response. Such bistability could potentially be harnessed for microwave switching [24]. Foldover of the FMR and standing spin-wave modes has been studied for several decades [24{26], but foldover of propagating spin waves was only observed before in active feedback rings [27], spin-pumped sys- tems [28] and magnonic ring resonators [29]. We show that we can characterize the foldover of propagating spin waves in Ga:YIG thin lms using our spectroscopy technique. 62.1 2 1.9 1.8-0.10-0.200.2 Frequency (GHz) |S21| (dB) 1.9 2-0.020 Freq. (GHz) Frequency (GHz) 1.81.922.1 k (1/μm)024 6a f s = 12.5 μm, w = 2.5 μm |S21| (dB)bΔfvg (m/s) Frequency (GHz)1.9 1.85 1.95 20200400c Fit a)Data from: b)1 0|BRF y(k)| |BRF z(k)| s = 6 μm, w = 1 μm d k (1/μm)1 00 10|BRF y(k)| |BRF z(k)|Fourier amp. (norm.) Fourier amp. (norm.)e Δk Figure 3: Extracting the exchange constant from spin-wave transmission spectra. (a,b) Background-subtracted linetraces of jS21jfor two sets of striplines (a: w= 1m,s= 6m, b:w= 2:5 m,s= 12:5m, excitation power: 35dBm). The red circles (a) and green squares (inset of b) mark the extrema of the spin-wave fringes. (c) From the frequency dierence between the extrema
fwe determine the group velocity vgof the spin waves at the center frequency between the extrema. The blue line ts the data with an analytical expression for vg, extracting the exchange constant = 1:3(2)1012 J/m. (d,e) Normalized Fourier amplitude of the yandzcomponents of the microwave excitation eld BRFfor striplines with widths w= 1m (d) andw= 2:5m (e). (f) Reconstructed spin-wave dispersion based on the t in (c). The shaded areas correspond to the frequencies of the extrema in (a,b). The dashed lines are the same as in (d,e) and indicate the nodes in jBRF(k)jof the striplines. Only spin waves that are eciently excited and detected by the striplines are observed in (a,b). 7When increasing the drive power we observe frequency shifts of the spin waves (Fig. 4a,c). These non-linear shifts result from the four-magnon self-interaction term in the spin-wave Hamiltonian. For an in-plane magnetized thin lm, the shifts are given by [30] ~!k=!k+Wkk;kkjakj2: (6) Here ~!k(!k) is the non-linear (linear) spin-wave angular frequency, Wkk;kkis the four-wave frequency- shift parameter and akis the spin-wave amplitude. In our case Wkk;kk is positive as a result of the PMA in the sample, leading to positive frequency shifts of the spin-wave modes (supplementary material section 8). The low-frequency spin waves start shifting rst, because the stripline is the most ecient in exciting spin waves with small wavenumbers (Fig. 3d,e). The spin-wave modes start shifting at a surprisingly low drive power of 30 dBm, potentially caused by reduced spin-wave scattering [26] due to the low density of states associated with the increased spin stiness and reduced saturation magnetization of our sample. In the high-power microwave spectra we observe an abrupt transition at which the spin waves fall back to their unshifted low-power frequencies, indicating the foldover of the spin waves. As the spin-wave amplitude increases the spin-wave modes shift to higher frequencies, until the maximum amplitude is reached and the spin waves fall back to their low-amplitude dispersion (Fig. 4b). To demonstrate the foldover behaviour, we compare upward and downward frequency sweeps (Fig. 4a,c). As expected the spin waves fall back to their unshifted dispersion earlier when sweeping against the frequency shift direction than when the sweep is in the same direction. The spin-wave amplitude and wavevector is thus bistable for the frequencies at which the foldover occurs. For these frequencies the stripline can excite two dierent wavelengths of spin waves at the same excitation power depending on the sweep direction that was used in the past. The observed frequency shifts provide an extra knob for tuning the dispersion of spin waves. They give rise to strong non-linear microwave transmission between the striplines as a function of excita- tion power, which may provide opportunities for neuromorphic computing devices that simulate the spiking of articial neurons above a certain input threshold [29, 31]. In summary, we used propagating spin wave spectroscopy to characterize the spin-wave dispersion in a 45-nm-thick lm of Ga:YIG. The gallium doping reduces the saturation magnetization of the YIG and introduces a small PMA that lifts the magnetization out of the plane and causes the dispersion 8-30 -20 -10 0 -30 -20 -10 01.822.22.4 Frequency (GHz) Power (dBm) 00.3 -0.3 f-f0 0 0 0 Increasing power P = PcP < PcP > PcSW amp. (norm.)01 |S21| (dB)a bc Figure 4: Observation of spin-wave frequency shifts and foldover. (a) Spin-wave spectra at dierent excitation powers ( w= 1m,s= 6m). Low-frequency spin waves shift to higher frequencies when the microwave excitation power is increased. The markers indicate the sharp transition at which the spin waves fall back to their unshifted frequencies and serve as a guide to the eye. (b) Sketch of the normalized spin-wave amplitude vs frequency for increasing drive power P, showing the upward frequency shift away from the low-power resonance frequency f0. Above a critical power Pcthe frequency shift results in the foldover of the spin-wave mode. As a result, the spin waves fall back to their unshifted dispersion at higher frequencies for upward frequency sweeps (red arrows, a) than downward sweeps (pink arrows, c). to be dominated by the exchange constant. We extract the exchange constant by tting the group velocity at dierent frequencies and demonstrate that the detected spin waves are eciently excited by the excitation elds of the striplines. Finally, we observe pronounced power-dependent frequency shifts of the spin waves that lead to foldover and mode bistability. Our results highlight the potential of all-electrical spectroscopy to shed light on the dispersion and nonlinear response of propagating spin waves in weakly-magnetic thin lms. Supplementary material: See the supplementary material for methods, details on the data analy- sis and error estimations, additional measurements and calculations of the FMR frequency, spin-wave dispersion and non-linear frequency-shift parameter. Author contributions: J.J.C. and T.v.d.S. conceived the experiment. J.J.C. and O.D. built the 9experimental setup, performed the experiments and analyzed the data. C.D. grew the Ga:YIG lm and O.S. performed the VSM measurement. J.J.C. fabricated the striplines. J.J.C. and T.v.d.S. wrote the manuscript with contributions from all coauthors. T.v.d.S. supervised the project. Acknowledgements: This work was supported by the Netherlands Organisation for Scientic Research (NWO/OCW), as part of the Frontiers of Nanoscience program and by the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) -271741898. The authors thank A.V. Chumak for reviewing the manuscript, A. Katan, E. Lesne for useful discussions and C.C. Pothoven for fabricating the magnet holders used in the experimental setup. We also thank the sta of the TU Delft electronic support division and the Kavli Nanolab Delft for their support in soldering the printed circuit board and fabricating the microwave striplines. Competing interests: The authors declare that they have no competing interests. Data availability: All data contained in the gures are available in Zenodo.org at http://doi. org/10.5281/zenodo.5494466 , reference number [32]. Additional data related to this paper are available from the corresponding author upon reasonable request. References (1) Serga, A. 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All-optical spiking neurosynaptic networks with self-learning capabilities. Nature 2019 ,569, 208{214. (32) Carmiggelt, J. J.; Dreijer, O. C.; Dubs, C.; Surzhenko, O.; van der Sar, T. Electrical spec- troscopy of the spin-wave dispersion and bistability in gallium-doped yttrium iron garnet, Zenodo: 2021. 12Supplementary material Electrical spectroscopy of the spin-wave dispersion and bistability in gallium-doped yttrium iron garnet Joris J. Carmiggelt1, Olaf C. Dreijer1, Carsten Dubs2, Oleksii Surzhenko2,Toeno van der Sar1; 1Department of Quantum Nanoscience, Kavli Institute of Nanoscience, Delft University of Technology, 2628 CJ Delft, The Netherlands 2INNOVENT e.V. Technologieentwicklung, D-07745 Jena, Germany Corresponding author. Email: t.vandersar@tudelft.nl 1 Ga:YIG sample and experimental setup . . . . . . . . . . . . . . . . . . . . . . . . . . 14 2 Eect of the magnetic eld alignment on the FMR frequency . . . . . . . . . . . . . . 14 2.1 FMR frequency and magnetization direction at B= 0. . . . . . . . . . . . . 16 2.2 FMR frequency and magnetization direction at B= 90. . . . . . . . . . . . 18 3 Systematic error in the applied bias eld . . . . . . . . . . . . . . . . . . . . . . . . . 19 4 Background-subtraction procedures of the spin-wave spectra . . . . . . . . . . . . . . 21 5 The spin-wave dispersion of a magnetic thin lm with perpendicular and cubic mag- netic anisotropy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22 6 Comparing the frequency dierence between fringes to the spin-wave linewidth . . . . 23 7 Zoomed-in spin-wave spectra displaying low-amplitude fringes . . . . . . . . . . . . . 24 8 Calculation of the non-linear frequency-shift coecient . . . . . . . . . . . . . . . . . 24 131 Ga:YIG sample and experimental setup A 45-nm-thick lm of gallium-doped yttrium iron garnet (Ga:YIG) was grown using liquid phase epitaxy on a one-inch (111) gadolinium gallium garnet (GGG) substrate and cut into chips of 5x5x0.5 mm3. Striplines were fabricated on top of the Ga:YIG by e-beam lithography using PMMA(A8 495)/PMMA(A3 950) bilayer resist with an Elektra92 coating to avoid charging, and subsequent evaporation of Ti/Au (10 nm/190 nm). We wirebond the striplines to a printed circuit board and connect them to our vector network analyser (VNA, Keysight, P9372A) via small, non-magnetic SMPM connectors (Amphenol RF, 925-169J-51PT) to minimize spurious magnetic-eld dependent signals and maximize the dynamic range of the bias eld. Before reaching the VNA, the signals are amplied by a low noise +20 dB amplier (Minicircuits, ZX60-83LN-S+) to avoid detection noise on the order of our signals. We place the sample between two large cylindrical permanent magnets (Supermagnete, S35-20-N) to apply a strong and homogeneous bias eld. The magnets sit in home-built magnet holders that are mounted on computer-controlled translation stages (Thorlabs, MTS25-Z8, 25 mm range), which allow sweeping the eld. We calibrate the magnetic eld using a Hall probe (Hirst Magnetic Instruments, GM08). All measurements were performed at room temperature. 2 Eect of the magnetic eld alignment on the FMR frequency In this section we show that for (111)-oriented lattices with cubic anisotropy the in-plane Kittel relation holds when a strong magnetic eld B0is applied along the in-plane [1 10] crystallographic axis. We also investigate the eect of a 1out-of-plane angle of B0on the FMR frequency and the magnetization direction. Such a small angle may be present due to the manual placement of the sample in our setup (section 1). The FMR frequency is calculated according to [1] !2 FMR= 2 sin(M)2
@2F @2 M@2F @2 M@2F @M@M
2 : (S1) 14HereMis the angle of the magnetization with respect to the lm's normal, Mis the in-plane angle of the magnetization with respect to the [1 10] crystallographic axis and F=F0 Ms, withF0the free energy density and Msthe saturation magnetization (Fig. S1). =gB his the gyromagnetic ratio, withBthe Bohr magneton and  hthe reduced Planck constant. The anisotropic g-factor is given by g=q g2 ?cos(M)2+g2 jjsin(M)2, withgjjandg?respectively the in-plane and out-of-plane g-factors [2]. For (111)-oriented lms with cubic and uniaxial out-of-plane magnetic anisotropies the normalized free energy density is given by [3, 4] F=B0 sin(M) sin(B) cos(MB) + cos(M) cos(B) +1 2 0Ms2K2? Ms
cos2(M) +1 2
2K4 Ms1 3cos4(M) +1 4sin4(M)p 2 3sin3(M) cos(M) sin(3M) ;(S2) withBandBthe angles of B0with respect to respectively the lm's normal and the in-plane [110] crystallographic axis (Fig. S1) and 0the vacuum permeability.2K2? Msand2K4 Msare respectively the uniaxial out-of-plane and cubic anisotropy elds, with K2?andK4the perpendicular and cubic anisotropy constants. Note that to calculate the FMR frequency using equation S1 at a certain B0, BandB, we rst need to nd MandMthat minimize the free energy by numerically solving @F @M(M;M) = 0 and@F @M(M;M) = 0. Using equations S1 and S2 we can calculate the FMR frequency for an out-of-plane magnetic eld and magnetization ( B=M= 0), which gives !FMR(?)= ?(B00Ms+2K2? Ms4K4 3Ms): (S3) For an in-plane magnetic eld and magnetization ( B=M= 90), we nd !FMR(jj)= jjr B0
B0+0Ms2K2? MsK4 Ms
2K4 Mscos(3M)
2: (S4) 15The factor 3 in the cosine arises from the triangular in-plane symmetry of a cubic unit cell with its normal along the [111] direction (Fig. S1). In our measurements a large in-plane magnetic eld is needed to overcome the perpendicular anisotropy and push the magnetization in the plane, such that generallyB0j2K4 Msj= 8:2 mT and we can ignore the last term [3] !FMR(jj)= jjr B0
(B0+0Ms2K2? MsK4 Ms): (S5) Equations S3 and S5 are the same as equations 1 and 2 in the main text. [1,1,1] [1,1,2][1,1,0]Ga:YIGUnit cellM B0θB θMφMφB Figure S1: Coordinate frame and crystallographic axes in Ga:YIG. The [110] axis is slightly displaced to highlight the triangular symmetry plane (light blue) of the (111) -oriented cubic unit cell. After [3]. 2.1 FMR frequency and magnetization direction at B= 0 Fig. S2a shows a ipchip FMR measurement with the magnetic eld applied along the [1 10] direction (B= 90,B= 0). The solid white line shows a t to equation S5 for magnetic elds at which the FMR frequency is increasing. Together with the tted out-of-plane FMR, we extract2K2? Ms= 104:7 mT,2K4 Ms=8:2 mT and jj 2= 28:56 MHz/mT. The same t and data are presented in Fig. 1b of the main text. We can calculate the FMR frequency also for low bias elds by substituting the extracted param- eters into equations S1 and S2. We obtain the black dashed line, which ts reasonably well to the 16Magnetic field B0 (mT)100 50 150100 50 150 Magnetic field B0 (mT)100 5012 150100 50 150Frequency (GHz) 3 0123 0θM (degrees) 04590 04590φB = 0˚ φB = 90˚ d|S21|/dB0 (mdB/mT) 5 -5 d|S21|/dB0 (mdB/mT) 5 -5a bc d θB = 89˚θB = 90˚Figure S2: Dependence of the FMR frequency on the direction of the external magnetic eld B0.(a) Flipchip FMR measurement with B0applied parallel to a 180- m-wide excitation stripline and along the in-plane [110]crystallographic direction ( B= 0). The FMR is extracted from the eld- derivative of the microwave transmission jS21j. The solid white line shows a t to the Kittel relation (equation S5). Using the extracted anisotropy elds and gyromagnetic ratio, the FMR frequency for the entireB0-range was numerically calculated assuming B= 90(black dashed line) and B= 89(red dashed line). (b) The minimum FMR frequency is raised at B= 89because the magnetization does not abruptly turn into the plane. (c) Similar FMR measurement, but with B0applied along the [112]direction (B= 90). The white line is the same as in (a). The black and red dashed lines are the calculated FMR frequencies for B= 90at respectively B= 90andB= 89using the parameters extracted in (a). (d) The magnetization maintains a nite out-of-plane component even when B= 90. The blue dashed line indicates the eld at which the exchange constant was determined from the spin-wave spectra. In (a) and (c) a similar background subtraction was performed as in Fig. S4. 17measured FMR, even when the FMR frequency is decreasing with eld. The red dashed line shows the calculated FMR frequency when B0has an 1out-of-plane angle ( B= 89), which dramatically increases the minimum FMR frequency. This is because the magnetization turns only asymptotically into the plane when the angle is oset, instead of abruptly (Fig. S2b, black line: B= 90, red line: B= 89). We note that in Fig. S2a at large bias elds both the black and red dashed lines overlap with the white t. Therefore, we conclude that the in-plane FMR at B= 0is quite robust to any small out-of-plane component of the static eld that might be present in our experimental setup, validating the white t using equation S5 [3]. 2.2 FMR frequency and magnetization direction at B= 90 Fig. S2c shows a similar ipchip FMR measurement as in Fig. S2a, but now with the eld applied along the [11 2] direction ( B= 90,B= 90, the white line is the same as in Fig. S2a and is added as a reference). The FMR reaches a minimum frequency of about 1 GHz, which is signicantly larger than the minimum in the B= 0geometry. We reproduce this enhanced frequency minimum by calculating the expected FMR frequency using the parameters extracted in section 2.1 (black dashed line, we ignore any potential in-plane anisotropy of the g-factor). The calculated FMR frequency matches the measured FMR remarkably well for all magnetic eld values, demonstrating the accuracy of the white t. Again we attribute the enhanced FMR minumum to the fact that the magnetization only slowly turns into the plane, even for a perfect in-plane magnetic eld B= 90(Fig. S2d, black line). As a result the FMR frequency asymptotically approaches the in-plane Kittel relation (equation S5, white line). Similar to before, a change of 1inBlifts the minimum FMR frequency, explaining the minimum FMR frequency of about 1.25 GHz observed in Fig. 2b in the main text. Variations on the order of 1inBare expected in our measurement setup since we manually place the sample between two permanent magnets (section 1). 18Fig. S2d shows that the magnetization does not point exactly in the plane during our propagating spin wave spectroscopy measurements, even though this is assumed in the data analysis. We derived the exchange constant from spin-wave spectra taken at approximately B0= 117:5 mT, at which the magnetization points 3-6 degrees out of the plane (blue dashed line in Fig. S2d). We neglect this small out-of-plane angle, because we expect the induced error to be negligible compared to the 15% error obtained from the t in Fig. 3c in the main text. 3 Systematic error in the applied bias eld In this section we calculate how a systematic error in the applied bias eld aects the error of the anisotropy elds, which we extracted from the FMR frequency (Fig. 1b of the main text). From the ts of the FMR frequency we obtain ?= 29:40(3) MHz/mT and =0Ms+2K2? Ms2 32K4 Ms= 91:1(2) mT (out-of-plane bias eld), jj= 28:56(4) MHz/mT and =0Ms2K2? Ms1 22K4 Ms=81:5(1) mT (in- plane bias eld). Since we know from vibrating sample magnetometry (VSM) that Ms= 1:52(6)
104 A/m, we can calculate the magnetic anisotropy elds 2K4 Ms=6 7(+) = 8:2(2) mT; 2K2? Ms=0Ms+3 74 7= 104:7(8) mT:(S6) Since we manually place our sample between the magnets (section 1), it may have a small oset of 1 mm with respect to the center position. Such an oset would cause a systematic error in the applied magnetic eld B0, which enhances the error of the anisotropy elds. To obtain a conservative estimate of these errors, we determine the systematic error in the applied magnetic eld via
B0(x) =B0(x+ 1) +B0(x1)2
B0(x): (S7) 19B0(x) is the magnetic eld of a cylindrical magnet at a distance of xmm along its symmetry axis B0(x) =Br 2x+Lp r2+ (x+L)2xp r2+x2 : (S8) HereBr= 1320 mT is the remanence, L= 20 mm and r= 17:5 mm are the length and radius of the magnet. Fig. S3 shows the calculated error
B0(x) for a 1-mm-oset against the magnetic eld B0at the center position between the magnets. Including this error in the t of Fig. 1b in the main text gives= 91:1(3) mT and =81:5(5) mT, resulting in a slight increase in the error of the cubic anisotropy eld2K4 Ms= 8:2(5) mT. The errors in the gyromagnetic ratios and perpendicular anisotropy eld do not change signicantly. At the magnetic eld B0= 117:5 mT at which we took the spin-wave spectra the error in the eld
B0is0:3 mT. Error Δ B0 (mT) 0.1 0 Magnetic field B0 (mT)05 00.20.30.40.5 100 150 Figure S3: Error in the static magnetic eld as a result of a 1 mm oset of the sample with respect to the center position between the magnets. For magnetic elds between 100 mT and 150 mT an error of0.3-0.5 mT is expected. 204 Background-subtraction procedures of the spin-wave spectra For the spin-wave spectra in Fig. 3a,b and Fig. 4a,c in the main text a background spectrum was subtracted consisting of the mean jS21jtransmission at 100 mT and 138 mT, for which there are no spin waves in the frequency range of interest. In Fig. 2b in the main text a background was subtracted using Gwyddion (Fig. S4). Magnetic field B0 (mT)100 75 12511.522.5Frequency (GHz) Magnetic field B0 (mT)100 75 12511.522.5Frequency (GHz) Magnetic field B0 (mT)100 75 12511.522.5Frequency (GHz) - = d|S21|/dB0 (dB/mT) -0.5 0.5 d|S21|/dB0 (dB/mT) -0.5 0.5 d|S21|/dB0 (dB/mT) -0.5 0.5 Figure S4: Background-subtraction procedure of the microwave spectrum in Fig. 2b of the main text. The measured data (left gure) contains spurious signals attributed to small changes in the microwave transmission of the cables and connectors that attach the VNA to the striplines as a function of magnetic eld. We lter these signals by rst masking the high-curvature part of measured data that contains the spin-wave fringes. Then we t a fth-order polynomial through each horizontal line, excluding the masked data, and subtract it as a background (middle gure). The resulting spectrum only contains the spin-wave fringes (right gure, same as Fig. 2b in the main text). The image processing was performed using Gwyddion (version 2.58). 215 The spin-wave dispersion of a magnetic thin lm with perpendicular and cubic magnetic anisotropy The spin-wave dispersion for magnetic thin lms with perpendicular magnetic anisotropy (PMA) and cubic anisotropy was derived in reference [5]. Equation 30 of this work states the dispersion for an (111)-oriented lm with in-plane magnetization, similar as in our experiment !SW(k) = jjr B0+Dk2+0Ms(1f)2K2? MsK4 Ms

B0+Dk2+0Msfsin2()
2K4 Mscos(3M)
2: (S9) Here!SWis the angular frequency of a spin wave with wavevector kthat propagates at an angle  with respect to the magnetization. D= 2=M sis the spin stiness, with the exchange constant, andf= 1(1ekt)=ktwithtthe thickness of the lm and Mis the angle of the magnetization with respect to [1 10] crystallographic direction. We note that if we set k= 0 in equation S9, we obtain the in-plane FMR frequency derived before (equation S4). In our experiment we measure spin waves in the Damon-Eshbach conguration ( ==2), we apply the external eld B0along [11 2] (M==2) and the wavelengths of the detected spin waves are much smaller than the thickness of the lm ( kt1), such that we can approximate fkt=2. This gives !SW(k) =q !B+ jjDk2!K+!M(1kt=2)
!B+ jjDk2+!Mkt=2
; (S10) where we dened !B= jjB0,!M= jj0Ms, and!K= jj(2K2? Ms+K4 Ms) for convenience of notation. Working out the brackets and rearranging the terms in orders of kgives !SW=r !B !B+!M!K
+!Mt 2 !M!K
k+ jjD 2!B+!M!K(!Mt 2)2
k2+ 2 jjD2k4: (S11) 22For the spin-wave spectra taken at B0= 117:5 mT we nd (!Mt 2)22!B+!M!Kdue to the low saturation magnetisation and thickness of our lm, such that we can further approximate !SW=r !B(!B+!M!K) +!Mt 2(!M!K)k+ jjD(2!B+!M!K)k2+ 2 jjD2k4;(S12) which is equation 4 in the main text. We derive the group velocity vgby dierentiating with respect to k vg=@!SW @k=1 2p!SW!Mt 2(!M!K) + 2 jjD(2!B+!M!K)k+ 4 2 jjD2k3
; (S13) which is equation 5 in the main text. 6 Comparing the frequency dierence between fringes to the spin-wave linewidth In this section we calculate the expected average frequency dierence
fbetween spin-wave fringes excited by the rst maximum of the microwave driving eld Fourier amplitude ( jBRF(k)j) in Fig. 3b of the main text. The stripline has a width w= 2:5m, such thatjBRF(k)jhas its rst node atkmin=2 2:5m1[6]. Everytime another wavelength ts within the center-to-center distance s between both striplines another fringe is observed in the signal. Therefore the condition s=n applies for every nth fringe, with the spin-wave wavelength. This means that fringes occur every
k=2 s=2 12:5m1in k-space. In the rst maximum of the excitation spectrum we would thus expectkmin
k= 5 fringes. According to the reconstructed dispersion (Fig. 3f of the main text) the frequency dierence between spin waves with wavevector kminand the minimum of the band is about 20 MHz, leading to an average frequency dierence of20 5= 4 MHz between consecutive fringes. This is on the order of the FMR linewidth of undoped YIG lms of similar thicknesses [4]. Assuming that Ga:YIG has a similar or larger linewidth, we argue that we cannot resolve fringes in the rst 23maximum of the excitation eld's Fourier amplitude because they are too narrow compared to the intrinsic spin-wave linewidth. 7 Zoomed-in spin-wave spectra displaying low-amplitude fringes Magnetic field B0 (mT)2.3 2 1.7Frequency (GHz) 110 120 -1 1d|S21|/dB0 (dB/mT) Figure S5: Detailed microwave spectrum zoomed-in on the spin-wave fringes. Low-amplitude fringes excited by the second maximum of the excitation eld's Fourier amplitude are visible at high frequencies. The actual measured data without any background-subtraction is presented ( w= 1m, s= 6m, excitation power -35 dBm). 8 Calculation of the non-linear frequency-shift coecient For Damon-Eshbach spin waves with wavevector kand frequency !k=2the non-linear four-magnon frequency-shift coecient Wkk;kk is given by [7] Wkk;kk =1 22!B+!M(Nxx;k+Nyy;k) 2!k
2
3!B+!M(2Nzz;0+Nzz;2k)
1 2 3!B+!M(Nxx;k+Nyy;k+Nzz;2k)
;(S14) 24withNij;kthe (i;j)th index of the spin-wave tensor Nk. The three-wave correction term vanishes since the spin waves propagate perpendicular to the magnetization. The precessional xyz-frame is dened such that zpoints in the plane along the magnetization, xalong the lm normal and ypoints in-plane perpendicular to zand parallel to the wavevector of the spin waves. Nkis the Fourier transform of the tensorial Green's function N(r;r0) =N(r;r0)dip+N(r;r0)ex+ N(r;r0)ani, which has components due to uniaxial anisotropy and the dipolar and exchange interac- tions Nkeikr=Z N(r;r0)eikr0d3r0=Z N(r;r0)dip+N(r;r0)ex
eikr0d3r0+Z N(r;r0)anieikr0d3r0:(S15) The contribution to Nkfrom theN(r;r0)dipandN(r;r0)excomponents in the thin-lm limit were derived earlier [7]. Following this work, N(r;r0)anidue to uniaxial anisotropy in the out-of-plane x-direction is given by N(r;r0)ani=B2? 0Ms(rr0)^x ^x: (S16) HereB2?=2K2? Msis the uniaxial out-of-plane anisotropy eld, denotes a dyadic unit vector product and(rr0) is the Dirac delta function. As a result of the dyadic product only the ( x;x) index of N(r;r0)aniis non-zero, leading to a contribution on Nxx;k Nxx;keikr=Z B2? 0Ms(rr0)eikr0d3r0=B2? 0Mseikr: (S17) 25By adding this contribution to the other components, we nd that the diagonal elements of Nkin the Damon-Eshbach conguration are given by Nxx;k=D 0Msk2+ 1fB2? 0Ms; Nyy;k=D 0Msk2+f; Nxx;k=D 0Msk2;(S18) withf= 1(1ekt)=ktandtthe thickness of the lm as before. We neglected the cubic anisotropy since it is small relative to the uniaxial anisotropy. Wkk,kk / 2π (GHz) 123 0 k (1/μm)05 1 0 Figure S6: Non-linear frequency-shift coecient Wkk;kk for Damon-Eshbach spin waves in Ga:YIG. We used the dispersion in Fig. 3f of the main text as an input, together with the extracted parameters B0= 117:5mT,= 1:3
1012J/m,2K2? Ms= 104:7mT, jj 2= 28:56MHz/mT,Ms= 1:52
104A/m andt= 45 nm. The cubic anisotropy is neglected. The positive sign of the calculated frequency-shift coecient matches the positive frequency shifts observed in the experiment. By substituting equations S18 into equation S14 we can calculate Wkk;kk for the wavevectors relevant for this work (Fig. S6). For all these wavevectors Wkk;kk is positive, explaining the positive frequency shifts of the spin waves that we observe when increasing the drive power. This is in contrast to the 26frequency shift caused by the reduction of the saturation magnetization as a result of strong driving or heating. In this simple picture a downward frequency shift is expected for in-plane magnetization (Fig. S7), highlighting the value of the Hamiltonian formalism that was used to calculate the non- linear frequency-shift coecient [7]. Magnetic field B0 (mT)500 1000 15000123Frequency (GHz) Ms 0.9 · Ms Figure S7: Expected downward frequency shift upon reduction of the saturation magnetization. Field dependence of the FMR frequency of Ga:YIG for unreduced saturation magnetization ( Ms= 1:52
104 A/m, black line) and for 10%-reduced saturation magnetization ( Ms= 1:37
104A/m, red line). The bias eld is applied in the [112]direction and the magnetic anisotropy elds are the same for both curves. The dashed line indicates the eld at which we performed our spin-wave spectroscopy measurements. Clearly a negative frequency shift is expected upon decreasing the saturation magnetization, which is in contrast to the positive frequency shifts we observe. References (1) Suhl, H. Ferromagnetic Resonance in Nickel Ferrite Between One and Two Kilomegacycles. Physical Review 1955 ,97, 555{557. 27(2) Farle, M. Ferromagnetic resonance of ultrathin metallic layers. Reports on Progress in Physics 1998 ,61, 755{826. (3) Manuilov, S. A.; Khartsev, S. I.; Grishin, A. M. Pulsed laser deposited Y 3Fe5O12lms: Nature of magnetic anisotropy I. Journal of Applied Physics 2009 ,106, 123917. (4) Dubs, C.; Surzhenko, O.; Thomas, R.; Osten, J.; Schneider, T.; Lenz, K.; Grenzer, J.; H ubner, R.; Wendler, E. Low damping and microstructural perfection of sub-40nm-thin yttrium iron garnet lms grown by liquid phase epitaxy. Physical Review Materials 2020 ,4, 024416. (5) Kalinikos, B. A.; Kostylev, M. P.; Kozhus, N.; Slavin, A. N. The dipole-exchange spin wave spectrum for anisotropic ferromagnetic lms with mixed exchange boundary conditions. Jour- nal of Physics: Condensed Matter 1990 ,2, 9861{9877. (6) Ciubotaru, F.; Devolder, T.; Manfrini, M.; Adelmann, C.; Radu, I. P. All electrical propagating spin wave spectroscopy with broadband wavevector capability. Applied Physics Letters 2016 , 109, 012403. (7) Krivosik, P.; Patton, C. E. Hamiltonian formulation of nonlinear spin-wave dynamics: Theory and applications. Physical Review B 2010 ,82, 184428. 28

2021-09-10

Yttrium iron garnet (YIG) is a magnetic insulator with record-low damping, allowing spin-wave transport over macroscopic distances. Doping YIG with gallium ions greatly reduces the demagnetizing field and introduces a perpendicular magnetic anisotropy, which leads to an isotropic spin-wave dispersion that facilitates spin-wave optics and spin-wave steering. Here, we characterize the dispersion of a gallium-doped YIG (Ga:YIG) thin film using electrical spectroscopy. We determine the magnetic anisotropy parameters from the ferromagnetic resonance frequency and use propagating spin wave spectroscopy in the Damon-Eshbach configuration to detect the small spin-wave magnetic fields of this ultrathin weak magnet over a wide range of wavevectors, enabling the extraction of the exchange constant $\alpha=1.3(2)\times10^{-12}$ J/m. The frequencies of the spin waves shift with increasing drive power, which eventually leads to the foldover of the spin-wave modes. Our results shed light on isotropic spin-wave transport in Ga:YIG and highlight the potential of electrical spectroscopy to map out the dispersion and bistability of propagating spin waves in magnets with a low saturation magnetization.

Electrical spectroscopy of the spin-wave dispersion and bistability in gallium-doped yttrium iron garnet

2109.05045v1

arXiv:1705.03220v1 [cond-mat.mes-hall] 9 May 2017Anomalous, spin, and valley Hall effects in graphene deposited on ferromagnetic substrates A. Dyrda/suppress l1and J. Barna´ s1,2 1Faculty of Physics, Adam Mickiewicz University, ul. Umulto wska 85, 61-614 Pozna´ n, Poland 2Institute of Molecular Physics, Polish Academy of Sciences , ul. M. Smoluchowskiego 17, 60-179 Pozna´ n, Poland E-mail:adyrdal@amu.edu.pl 8 October 2018 Abstract. Spin, anomalous, and valley Hall effects in graphene-based h ybrid structures arestudied theoretically withinthe Green func tion formalismand linear response theory. Two different types of hybrid systems are co nsidered in detail: (i) graphene/boron nitride/cobalt(nickel), and (ii) graphen e/YIG. The main interest is focused on the proximity-induced exchange interaction b etween graphene and magnetic substrate and on the proximity-enhanced spin-orb it coupling. The proximity effects are shown to have a significant influence on t he electronic and spin transport properties of graphene. To find the spin, anom alous and valley Hall conductivities we employ certain effective Hamiltonians wh ich have been proposed recently for the hybrid systems under considerations. Both anomalous and valley Hall conductivities have universal values when the Fermi le vel is inside the energy gap in the electronic spectrum.Anomalous, spin, and valley Hall effects in graphene deposit ed on ferromagnetic substrates 2 1. Introduction Graphene is a two-dimensional hexagonal lattice of carbon atoms. Electronic properties of pristine (or free standing) graphene have been extensively studied in recent years, mainly because of its unusual properties following from specific electronic states described by Dirac model [1, 2, 3, 4, 5]. It has been shown that theelectronicpropertiescanbestronglymodifiedwhen graphene is decorated (or functionalized) with various adatomsormoleculesattachedtoitssurfaceortoedges in graphene stripes and nanoribbons [6, 7, 8, 9, 10, 11]. Other possibilities of a significant modification of graphene electronic and magnetic properties appear in hybrid systems based on graphene deposited on various substrates (e.g. on transition metal dichalcogenides or ferromagnetic thin films) [12, 13, 14, 15, 16, 17, 18, 19, 20]. Such systems are currently of great interest both experimental and theoretical, mainly because of magnetic and spin-orbit proximity effects responsible for magnetic moment and enhanced spin-orbit interaction in the graphene layer. This, in turn, opens possibilities of spin- orbit driven phenomena in graphene-based hybrid structures at room temperatures [7, 12, 14, 19, 21]. The high-temperature experimental realizations of anomalous and spin Hall effects as well as current- induced spin polarization (or Edelstein effect) make graphene-based structures active elements of future spintronicsandspin-orbitronicsdevices–togetherwith other2Dcrystals,semiconductorheterostructures,and junctions of oxide perovskites [22]. Hexagonal two-dimensional crystals with their prominent examples such as graphene and transition metal dichalcogenides with broken inversion symmetry are currently studied very intensively, especially in the context of so-called valleytronics and also valley- based optoelectronics [23, 24, 25, 26, 27, 28, 29, 30, 31, 32]. An important property of such systems is the presence of two inequivalent ( KandK′) valleys in the corresponding electronic spectrum. Interestingly, it turned out that the valley degree of freedom can be controlled not only by circularly polarized light, but also with external magnetic and electric fields. Moreover, very promising for applications seems to be coupling of the valley and spin degrees of freedom due to spin-orbit interaction [33, 34]. Owing to this one may expect, among others, a certain enhancement of the spin and valley polarization lifetimes and also manipulation of the spin degree of freedom by valley properties. This, in turn, allows to conceive a new generation of spintronic devices which are based on chargeless and nondissipative currents. An important issue is the pure electrical genera- tion and detection of valley and spin currents. This can be realized viathe valley and spin Hall effects aswell as their inverse counterparts. In systems with a net magnetization one can also observe the anomalous Hall effect. In high quality samples (free of defects and impurities), these effects may be determined by Berry curvature of the electronic bands and may reflect topo- logical properties of the systems [35, 36, 37]. Thus, de- tailed analysis of all the Hall effects in graphene-based hybridstructuresiscrucialfortheirproperunderstand- ing. In this paper we consider two kinds of hybrid structures: (i) graphene on a few atomic monolayers of boron nitride (BN) deposited on a ferromagnetic metal like Co or Ni, and (ii) graphene deposited directly on a ferromagneticinsulating substrate (YIG). In the former case the proximity-induced exchange interaction strongly depends on the number nof atomic planes of BN, and disappears already for n= 4 such atomic planes. In the latter case, in turn, the graphene layer is deposited directly on the ferromagnetic substrate, so the exchange interaction is rather direct. Importantly, BN is a wide-gap semiconductor and therefore plays a role of energy barrier for low-energy electronic states in graphene. Spin-orbit and exchange-interaction driven phe- nomena in graphene-based hybrid structures are stud- ied within the linear response theory and Green func- tion formalism. To describe these phenomena theoret- ically we make use of the low-energy effective Hamil- tonians, that have been derived recently from first- principle calculations (see e.g. [13, 14, 18]). In par- ticular, we calculate the anomalous, spin and valley Hall conductivities. Apart from this, we also intro- duce the valley spin Hall effect. The anomalous and spin Hall effects occur due to spin-orbit coupling in the system subject to an external electric field (for review see [38, 39, 40, 41, 42]). In the case of valley and val- ley spin Hall effects, the spin-orbit interaction is not required. Electrons have anomalous velocity compo- nent (normal to external electric field) which is ori- ented in opposite direction in the two valleys (the cor- responding Berry curvatures have opposite signs). As a consequence, electrons (or holes) from the two val- leys are deflected towards opposite edges of the sam- ple. The above effects may play an important role in the graphene-based spintronics, as an effective source of spin currents and spin-orbit torques [43, 44]. These, in turn, may be responsible for spin dynamics and/or magnetic switching in the low-dimensional structures. In section 2 we present a theoretical background and describe the model and theoretical method. In sections 3 we present our results on graphene/BN/Co and graphene/BN/Ni hybrid structures. Results for graphene/YIG hybrid system are presented and discussed in section 4. Summary and final conclusions are in section 5.Anomalous, spin, and valley Hall effects in graphene deposit ed on ferromagnetic substrates 3 Figure 1. (Color online) Schematic of the system under consideration . Graphene is deposited either directly on a magnetic substr ate (YIG) or is separated from the magnetic substrate (Co, Ni) by a few atomic planes of another hexagonal crystal (BN). The un derlayer assures exchange coupling between the magnetic substrate a nd graphene and also gives rise to the spin-orbit interactio n of Rashba type. Owing to this, one may observe the Hall effects listed at the bottom of the figure. 2. Theoretical background 2.1. Model We consider graphene either deposited directly on a magnetic substrate, or separated from the magnetic substrate by a few atomic layers of another two- dimensional crystal (e.g. BN), as shown schematically in Fig.1. Influence of the substrate on magnetic and electronic properties of graphene will be taken into account in terms of certain effective Hamiltonians which have been obtained recently from results of ab- initiocalculations [13, 14, 18]. Because transport properties of graphene close to the charge neutrality pointaredeterminedmainlybyelectronsinthevicinity of Dirac points, we assume a minimal pzmodel that describes electronic and spin transport properties related to the low-energy electronic states of graphene and also takes into account the proximity-induced effects. General low-energy Hamiltonian for both ( Kand K′) Dirac points of the systems under considerations includes four terms [18] , HK(K′)=HK(K′) 0+HK(K′) ∆+HK(K′) EX+HK(K′) R.(1) The first term of the above Hamiltonian describes electronic states of pristine graphene near the K(K′) point [45], HK(K′) 0=v(±kxσx+kyσy)s0, (2) wherekxandkyare the in-plane wavevector components, while v=/planckover2pi1vFwithvFdenoting theFermi velocity. Apart from this, we use the notation according to which σ0andσare the unit matrix and the vector of Pauli matrices, σ= (σx,σy,σz), acting in the pseudospin (sublattice) space, while s0ands denote the unit matrix and vector of Pauli matrices, s= (sx,sy,sz), acting in the spin space. The second term in Eq.(1) takes into account the fact that carbon atoms from different sublattices (A and B) can feel generally different local potentials [18, 46]. Such a dependence appears for instance when graphene is deposited on a 2D material with buckled or binary (like BN) hexagonal structure. This, in turn, leads to the pseudospin symmetry breaking and gives rise to an orbital gap, ∆, in the electronic spectrum, HK(K′) ∆= ∆σzs0. (3) The third term in Hamiltonian (1) represents the proximity-induced exchange interaction between graphene and magnetic substrate, given explicitly by the formula [18] HK(K′) EX=λA EX 2(σz−σ0)sz+λB EX 2(σz+σ0)sz,(4) whereλA EXandλB EXare the exchange parameters corresponding to the sublattices A and B, respectively. Note that in the special case of λA EX=−λB EX=λEX one obtains the exchange Hamiltonian in the form: HEX=λEXσ0sz. Finally, the last term in Hamiltonian (1) describes the spin-orbit interaction of Rashba type, that appears due to the space inversion symmetry breaking in theAnomalous, spin, and valley Hall effects in graphene deposit ed on ferromagnetic substrates 4 system. This interaction takes the following general form [45]: HK(K′) R=λR(±σxsy−σysx), (5) whereλRis the Rashba parameter. Note that the so-called intrinsic spin-orbit interaction in graphene is very small and therefore it is neglected in our consideration. 2.2. Method Our key objective is to study the anomalous, spin, and valley Hall effects for graphene deposited on various substrates. Without loss of generality, we assume electric field along the axis y. The corresponding conductivities are determined by the contributions from both KandK′valleys as follows: σAHE xy=σK xy+σK′ xy (6) for the anomalous Hall effect (AHE), σV HE xy=σK xy−σK′ xy (7) for the valley Hall effect (VHE), σSHE xy=σszK xy+σszK′ xy (8) for the spin Hall effect (SHE), and σV SHE xy=σszK xy−σszK′ xy (9) for the valley spin Hall effect (VSHE). Here, σν xyand σszν xyare contributions from the valley ν(ν=K,K′) to the charge and spin conductivities, respectively. Within the zero-temperature Green functions formalism and in the linear response with respect to a dynamical electric field of frequency ω(measured in energy units), one can write the dynamical charge σν xy(ω) and spin σsz,ν xy(ω) conductivities in the form, σν xy(ω) =e2/planckover2pi1 ω/integraldisplaydε 2π/integraldisplayd2k (2π)2 ×Tr/braceleftbig ˆvν xGν k(ε)ˆvν yGν k(ε+ω)/bracerightbig , (10) σsz,ν xy(ω) =e/planckover2pi1 ω/integraldisplaydε 2π/integraldisplayd2k (2π)2 ×Tr/braceleftBig ˆjszν xGν k(ε)ˆvν yGν k(ε+ω)/bracerightBig , (11) forν=K,K′. In the above equations ˆ vν x,ydenote components of the velocity operator for the valley ν, ˆvν x,y=1 /planckover2pi1∂ˆHν ∂kx,y, whileˆjszxis the relevant component of the spin current operator. Furthermore, Gν k(ε) stands for the causal Green function corresponding to the appropriate Hamiltonian ˆHν,Gν k={[ε+µ+ iδsign(ε)]−ˆHν}−1, whereµis the chemical potential andδ→0+in the clean limit.In the following we are interested in the dc- conductivities, so we take the limit ω→0 in the above expressions. To do this let us write Tr/braceleftbig ˆvν xgν k(ε+ω)ˆvν ygν k(ε)/bracerightbig =Dν 0(ε,k,φ)+ωDν 1(ε,k,φ)+... (12) Tr/braceleftBig ˆjsz,ν xgν k(ε+ω)ˆvν ygν k(ε)/bracerightBig =Ds,ν 0(ε,k,φ)+ωDs,ν 1(ε,k,φ)+..., (13) wheregν kstands for a nominator of the Green function, φis the angle between the wavevector kand the axis y, and the terms of higher order in ωhave been omitted as their contribution vanishes in the limit of ω→0. Upon calculating the trace one finds Dν 0(ε,k,φ) = 0 andDs,ν 0(ε,k,φ) = 0. Thus, in the limit of ω→0 the expressions (10) and (11) take the form σν xy=e2/planckover2pi1 (2π)3/integraldisplay dε/integraldisplay dkkFν(ε,k), (14) σsz,ν xy=e/planckover2pi1 (2π)3/integraldisplay dε/integraldisplay dkkFs,ν(ε,k), (15) where the functions Fν(ε,k) andFs,ν(ε,k) are defined as Fν(ε,k) =Iν(ε,k)/producttext4 l=1[ε+µ−El+iδsgn(ε)]2, (16) Fs,ν(ε,k) =Is,ν(ε,k)/producttext4 l=1[ε+µ−El+iδsgn(ε)]2.(17) Here,El(l= 1−4) denote the four eigenmodes of the relevant Hamiltonian, and we introduced the following notation: Iν(ε,k) =/integraldisplay dφDν 1(ε,k,φ), (18) Is,ν(ε,k) =/integraldisplay dφDs,ν 1(ε,k,φ). (19) The integration over εin Eqs (14) and (15) can be performed in terms of the theorem of residues. As a result one finds σν xy=e2/planckover2pi1 (2π)34/summationdisplay l=1/integraldisplay dkkRν lf(El), (20) σsz,ν xy=e/planckover2pi1 (2π)34/summationdisplay l=1/integraldisplay dkkRs,ν lf(El), (21) forν=K,K′. Here, f(E) is the Fermi distribution function, while Rν landRs,ν ldenote the residua (multiplied by the factor 2 πi) of the functions Fν(ε,k) andFs,ν(ε,k), respectively, taken at ε=El−µ. Since we consider here only intrinsic (topological) contributions to the anomalous and valley Hall effects, one can express Eq.(20) alternatively in terms of theAnomalous, spin, and valley Hall effects in graphene deposit ed on ferromagnetic substrates 5 Berry curvature of electronic bands corresponding to the valley ν, σν xy=e2 /planckover2pi14/summationdisplay l=1/integraldisplaydkk (2π)2¯Ων lf(El) =e2 /planckover2pi14/summationdisplay l=1/integraldisplayd2k (2π)2Ων lf(El), (22) where Ων lis thezcomponent of the Berry curvature for thel-th subband, calculated in the vicinity of the pointν, while¯Ων listheBerrycurvatureintegratedover the angle φ,¯Ων l=/integraltextdφΩν l. Thus, the Berry curvature can be related to the residua Rν las¯Ων l= 2π/planckover2pi12Rν l.The correspondence between Kubo formulation and the approach based on topological invariants has been shownbyTholuesset al.[47, 48]and then it waswidely discussed in the literature (see review papers [36, 49]). Therefore, we only comment here that in the case of AHE and VHE, the conductivity may be nonzero even if the energy bands are described by the zero Chern number (Berry phase). This is because the local Berry curvature may be nonzero and can give rise to the anomalous or valley Hall conductivity. This is the case that we consider in this paper. Equations (20)-(22) are our general formulas which can be used to determine all the four Hall conductivities. These formulas will be applied in the following to specific hybrid systems under consideration. 3. Graphene/BN(n)/Co(Ni) Consider first graphene on a few ( n) atomic planes of hexagonal BN which is deposited on ferromagnetic Co or Ni. Since BN has a wide energy gap, it can be considered as an insulating barrier. Thus, the influence of Co (or Ni) on transport properties of graphene in the low-energy region is determined mainly by exchange interaction between graphene end Co (Ni) through the BN layer. It has been concluded from ab-initio calculations that Rashba interaction in graphene/BN/Co(Ni) hybrid system is much smaller than the exchange term and can be ruled out [18]. Therefore, we consider the limit of vanishing Rashbainteraction. Therelevantparametersextracted fromab-initio calculations for graphene/BN/Co(Ni) systems by Zollner et al[18] are given in Table 1. These parameters will be used below in our model calculations. When Rashba coupling disappears, Hamiltonian for the graphene/BN/Co(Ni) hybrid system can be reduced to the form HK(K′)=HK(K′) 0+HK(K′) ∆+HK(K′) EX. (23) The correspondingdispersionrelationsforthe Kvalley are shown in Fig.2 (top panel) for n= 1,n= 2 andTable 1. Parameters describing graphene(Gr)-based hybrid systems under considerations, taken from Ref. [18]. n ∆[meV]λA EX[meV]λB EX[meV] Gr/BN/Co 1 19.25 -3.14 8.59 2 36.44 0.097 -9.81 3 38.96 -0.005 0.018 Gr/BN/Ni 1 22.86 -1.40 7.78 2 42.04 0.068 -3.38 3 40.57 -0.005 0.017 n= 3 monolayers of BN. Splitting of the conduction and valence bands due to exchange interaction, clearly seen for n= 1 (Fig.2a), becomes reduced for n= 2 (Fig.2b) and is negligible for n= 3 (Fig.2c). This is a consequence of reduced exchange interaction when the number of atomic planes of BN increases. Note, splitting of the valence band is remarkably larger than that of the conduction band. Another interesting property of the spectrum is a relatively wide energy gap due to inversion symmetry breaking. This orbital gap is a consequence of the presence of BN layers, and its width increases with increasing number nof BN monolayers. Interestingly, the gap is much wider than thatinthefreestandinggraphene,whereitisnegligible due to a very small intrinsic spin-orbit interaction. Berry curvature integrated over the angle φis shown in Fig.2 (bottom panel) for n= 1,n= 3, and for both KandK′valleys. This figure clearly shows that the curvature of electronic bands in the Kvalley is opposite to the corresponding curvature in theK′valley. As a result one finds σK′ xy=−σK xy in the case under consideration, i.e. contributions to the anomalous Hall conductivity from individual valleys are not zero, but they have opposite signs and canceleachother. Therefore, the anomalousHall effect vanishes, σAHE xy= 0. This is rather clear as there is no spin-orbit interaction. Similarly, alsothe SHE vanishes due to the lack of spin-orbit coupling. However, the VHE effect remains nonzero, σV HE xy= 2σK xy, and from similar reasons also the VSHE is nonzero. 3.1. Valley Hall effect Due to the opposite Berry curvature of the electronic bands in the KandK′valleys, electrons from both valleys are deflected to opposite edges, giving rise to a nonzero valley Hall conductivity (see Eq. (7). Simple analytical results can be derived in a specific case of λA EX= 0. The corresponding dispersion curves around theKpoint are shown in Fig.2d. Note, the conduction band is then degenerate at k= 0. Detailed analytical calculations show that the valley Hall conductivity depends on the Fermi level µ, and bearing in mind that ∆>|λB EX|this dependence can be written as follows:Anomalous, spin, and valley Hall effects in graphene deposit ed on ferromagnetic substrates 6 -1 0 140.70 40.55 GR/BN(n=1)/Ni GR/BN(n=2)/Ni GR/BN(n=3)/Ni GR/BN(n=2)/NiE [eV] E [meV] -20 -10 0 10 20 k [106m-1] k [106m-1]-20 -10 0 10 20 K K' K K' k [107m-1]-4 -3 -2 -1 0 1 2 3 4 k [107m-1]-4 -3 -2 -1 0 1 2 3 40.8 0.4 0 -0.4 -0.8 -20 -10 0 10 20 -20 -10 0 10 20k [106m-1] 7m-1]-4 -3 -2 -1 0 1 2 3 4 k [107m-1]-4 -3 -2 -1 0 1 2 3 4Ω[104 nm2]1.5 1.0 0.5 0 -0.5 -1.0 -1.5= 0EXA 3430 3420 3410 3400(e) (f) (g) (h) k [10k [106m-1] k [106m-1] GR/BN(n=1)/Ni GR/BN(n=3)/Ni-1 -0.5 0 0.5 1 k [106m-1]
(a) (b) (c) (d) 2| | EXA 2| | EXB2 - (| |+| |) EXA EXB   Figure 2. (Color online) Energy dispersion curves around the K point f orn= 1 (a), n= 2 (b), and n= 3 (c) atomic planes of BN and for the parameters presented in Table 1. Dispersion curv es in a special case of λA EX= 0 are shown in (d). Bottom panel shows the Berry curvature ¯Ω of electronic bands for n= 1 (e,f) and n= 3 (g,h) atomic planes of BN in the vicinity of both K(e,g) and K′(f,h) Dirac points. The arrows indicate spin polarization o f the electronic bands. (i)−∆+|λB EX|< µ <∆ (Fermi level inside the gap): σV HE xy=−2e2 /planckover2pi1, (24) i.e. the valley Hall conductivity is quantized. (ii)µ >∆ (Fermi level inside the conduction bands), orµ <−(∆ +|λB EX|) (Fermi level inside the valence bands): σV HE xy=−/parenleftbigg2∆+λB EX |2µ+λB EX|+2∆−λB EX |2µ−λB EX|/parenrightbigge2 h.(25) (iii)−(∆+|λB EX|)< µ <−∆+|λB EX|: σV HE xy=−/parenleftbigg 1+2∆−|λB EX| |2µ−|λB EX||/parenrightbigge2 h. (26) In a general situation, λA EX/ne}ationslash= 0, the valley Hall conductivity was calculated numerically and is shown in Fig.3 as a function of the chemical potential µ. The valley conductivity is quantized for the Fermi level in the gap, where σV HE xy=−2e2 /planckover2pi1. The absolute value of the conductivity for µoutside the gap is reduced with increasing |µ|. The kinks appear at the points wheretheFermilevelcrossesedgesoftheconductionor valence bands, and appear in the presence of exchange splittingofthebands. Note, suchasplittingdisappears forn= 3 monolayers of BN, where the exchange interaction is vanishingly small. The kinks for positive µarelesspronouncedastheexchange-inducedsplitting of the conduction band is remarkably smaller.3.2. Valley spin Hall effect As already mentioned above, the spin Hall effect vanishes in graphene/BN/Co (Ni) systems due to vanishingly small Rashba interaction. Strictly speaking, contributions to the spin Hall conductivity from individual valleys are nonzero, however they cancel each other as the spin currents associated with theKandK′valleys are opposite. Thus, similarly to the valley Hall effect, one can define the valley spin Hall effect as the difference of spin currents from the K andK′valleys, see Eq.(9). This quantity is generally nonzero, and indicates that the net spins from the K andK′valleys are deflected to the opposite edges. As in the previoussection we analysethe full model with a finite parameter λA EX, as well as the limit λA EX= 0. For vanishing λA EXit is possible to find analytical solutions for the valley spin Hall conductivity. (i)µ >∆: σV SHE xy=e 2π/parenleftbigg2∆−λB EX |2µ−λB EX|−2∆+λB EX |2µ+λB EX|/parenrightbigg .(27) (ii)−∆+|λB EX|< µ <∆ (Fermi level is in the gap): σV SHE xy= 0, (28) i.e. the valley spin Hall conductivity vanishes. (iii)−∆−|λB EX|< µ <−∆+|λB EX|: σV SHE xy=−e 2π/parenleftbigg 1−2∆−|λB EX| |2µ−|λB EX||/parenrightbigg . (29)Anomalous, spin, and valley Hall effects in graphene deposit ed on ferromagnetic substrates 7 -2-1.6-1.2-0.8-0.4 -0.15 -0.1 -0.05 0 0.05 0.1 0.15 µ[eV] [e2/h]σxyGR/BN (n=1)/Co GR/BN (n=1)/Ni GR/BN (n=3)/Co GR/BN (n=3)/Ni Figure 3. (Color online) (a) Valley Hall conductivity as a function of the chemical potential µfor graphene/BN/Co and graphene/BN/Ni systems with n= 1 and n= 3 atomic planes of BN. (b) Schematic presentation of the VHE : electrons in the KandK′valleys are deflected in opposite orientations normal to ext ernal electric field.-                0 0        ! " # $ % & ' ( ) * 0+ , . / 1 2 3 4 5 6 7n=1 n=2 n=3x10-38 9 : ; < = > ? @VABCσ xD [EFG π] µ H I J K L M N O0 P Q R S T U W X Y Z \ ]^ _ ` a b c d e f g h i j k 0l m n o p q r s t u v µ w y z {GR/BN/Co GR/BN/Ni -1 | } ~ 0 0 ¡ ¢ £ ¤ ¥ ¦ § ¨ © ª µ « ¬ ® Figure 4. (Color online) Valley spin Hall conductivity as a function o f the chemical potential µfor graphene/BN/Co (dashed lines) and graphene/BN/Ni (solid lines) systems with n= 1 (a), n=2 (b), and n= 3 (c) atomic planes of BN. Due to small exchange coupling, the valley spin Hall conductivity for n= 3 is by three orders of magnitude smaller. The sign of conduc tivity for n= 2 is reversed due to reversed sign of the exchange parameter. (iv)µ <−∆−|λB EX|: σV SHE xy=−e 2π/parenleftbigg2∆+λB EX |2µ+λB EX|−2∆−λB EX |2µ−λB EX|/parenrightbigg .(30) Numerical results on the valley spin Hall conduc- tivity are presented in Fig.4 for the general situation, λA EX/ne}ationslash= 0, and for n= 1,n= 2andn= 3. The valley spin Hall conductivity vanishes for the Fermi level in the gap. To understand this we note first that the exchange-splittingof conduction (and also valence) bands is the same in the KandK′valleys. Since the two valence subbands in an individual valley corre- spond to oppositespin orientations,their contributions to the spin current exactly cancel each other when the Fermi level is in the energy gap. A nonzero spin cur- rent appearsthen when the Fermi level crossesthe bot- tom edge of the lower conduction subband or top edge of the higher valence subband. When |µ|groves fur- ther, the valley spin Hall conductivity decreases due to compensatingcontributionfromthe secondconduction (valence) subband. Note, the valley spin Hall conductivity for n= 2 atomic planes of BN has reversed sign in major part ofµdue to reversed sign of the exchange parameterin comparison to that for n= 1. Apart from this, the valley spin Hall conductivity for n= 3 is roughly three orders of magnitude smaller than for n= 2. This is due to a very small exchange coupling parameter for n= 3. 4. Graphene on a magnetic insulating substrate Now we consider graphene deposited directly on a magnetic insulating substrate. An important example of such a hybrid system is graphene deposited on YIG, where largeanomalousHall effect at roomtemperature has been measured recently [14, 19]. In this particular case the third term of Hamiltonian (1), corresponding to the orbital gap, is absent. However, the coexistence of proximity-induced exchange field and Rashba spin- orbit coupling is essential. Therefore, Hamiltonian (1) for the graphene/YIG system can be reduced to the following one: HK(K′)=HK(K′) 0+HK(K′) EX+HK(K′) R. (31) Moreover, one may assume λA EX=−λB EX=λEXin this particular case, so the relevant exchange HamiltonianAnomalous, spin, and valley Hall effects in graphene deposit ed on ferromagnetic substrates 8¯ °±² ³ ´µ¶ · ¸ ¹º » ¼½ ¾ ¿À -20 -10 0 10 20 -20 -10 0 10 20 -20 -10 0 10 20 -20 -10 0 10 2030 20 10 0 -10 -20 -30 Á  à 1.2 0.8Ä Å Æ 0Ç È É Ê -0.8 -1.2Ë Ì Í Î2.5 2.0 1.5 1.0 0.5 0 -0.5 -1.0 -1.5 -2.0 -2.5 Ï 3 2 1 0 -1 -2 -3Ð Ñ0.08Ò Ó Ô Õ 0Ö × Ø Ù Ú 0.080.08Û Ü Ý Þ 0ß à á â ã 0.080.08ä å æ ç 0è é ê ë ì 0.080.08í î ï ð 0ñ ò ó ô õ 0.08 0.03 0.02 0.01 0 -0.01 -0.02 -0.03ö÷ ø ù ú û ü ýþ ÿ = (
              (h)Ω [μm2] Ek    6m-1]     m-1] ! " # $ %m-1] & ' ) * +m-1] Figure 5. (Color online) (a,c,e,g) Energy dispersion curves around t heKDirac point in the graphene/YIG system for a constant Rashba parameter and exchange parameter as indicated. (b,d ,f,h) Berry curvature integrated over the angle φcorresponding to the bands shown in (a,c,e,g), respectively. reads HK(K′) EX=λEXσ0sz. (32) Figure 5 presents the energy dispersion curves for the graphene/YIG structure (top panel). The Rashba coupling was assumed there constant while the exchange parameter was changed (as indicated) from weak to strong coupling limit. Interestingly, when the exchange coupling is small, there is no energy gap in the spectrum – the gap is created when the exchange interaction is sufficiently strong. Apart from this, minima (maxima) of the conduction (valence) bands are shifted away from the Dirac points. The bottom panel in Fig. 5 shows the Berry curvature corresponding to the bands displayed in the top panel. The Berry curvature for the K′point (not shown) is the same as that for the Kpoint. Due to to this, both VHE and VSHE are absent. However, AHE and SHE conductivities do not vanish due to Rashba spin-orbit coupling,andbothcanbefoundfollowingtheapproach described in section 2. 4.1. Spin Hall effect To find the spin Hall conductivity we make use of Eq.(21). The corresponding residua can be easily evaluated and are given by the expressions RK,s 1(3)=π2λ2 Rv2(2(λ2 R+λ2 EX)+v2k2) (λ4 R+v2k2(λ2 R+λ2 EX))3/2, (33) RK,s 2,(4)=−π2λ2 Rv2(2(λ2 R+λ2 EX)+v2k2) (λ4 R+v2k2(λ2 R+λ2 EX))3/2=−RK,s 1(3).(34) Taking the above formulas into account, one can find explicit expressions for the spin Hall conductivity,which are valid in the corresponding regions of the chemical potential, as described below. These regions can be easily identified when looking at the dispersion curves in Fig. 5. (i)|µ|>/radicalbig 4λ2 R+λ2 EX: σSHE xy=∓e 8πλ2 Rv2 λ2 R+λ2 EX/parenleftbiggk2 3+ ξ3+−k2 3− ξ3−/parenrightbigg ±e 4πλ2 Rλ2 EX (λ2 R+λ2 EX)2(2λ2 R+λ2 EX)/parenleftbigg1 ξ3+−1 ξ3−/parenrightbigg ,(35) with the upper sign for µ <0 and lower for µ >0. (ii)/radicalbig 4λ2 R+λ2 EX>|µ|> λEX: σSHE xy=∓e 8πλ2 Rv2 λ2 R+λ2 EXk2 3+ ξ3+ ±e 4πλ2 Rλ2 EX2λ2 R+λ2 EX (λ2 R+λ2 EX)2/parenleftbigg1 ξ3+−1 λ2 R/parenrightbigg (36) with the upper sign for µ <0 and lower for µ >0. (iii)λEX>|µ|>/radicalbigg λ2 Rλ2 EX λ2 R+λ2 EX: σSHE xy=∓e 8πλ2 Rv2 λ2 R+λ2 EX/parenleftbiggk2 3+ ξ3+−k2 3− ξ3−/parenrightbigg ±e 4πλ2 Rλ2 EX (λ2 R+λ2 EX)2(2λ2 R+λ2 EX)/parenleftbigg1 ξ3+−1 ξ3−/parenrightbigg .(37) with the upper sign for µ <0 and lower for µ >0. (iv)−/radicalbigg λ2 Rλ2 EX λ2 R+λ2 EX< µ </radicalbigg λ2 Rλ2 EX λ2 R+λ2 EX(i.e. in the gap of electronic spectrum): σSHE xy= 0, (38) i.e. the spin Hall conductivity vanishes. In the above equations we introduced the notation:Anomalous, spin, and valley Hall effects in graphene deposit ed on ferromagnetic substrates 9 20 16 12 84 0 20 16 12 8, 0[meV]λ λ[meV]R -. λ[meV]Rλ[meV]/0 -1 -0.5 0 0.5 1 μ[meV]-1.5 -1 -0.5 0 0.5 1 1.5 161820-0.10.2-1.5-1-0.500.511.5σxy S 1 2π[e/4 ] σxySHEπ[e/4 ] σxySHEπ[e/4 ]0.2 161820-0.10.1-1.5-1-0.500.511.5 0.050.1 μ[meV]μ[me(a) (b) (c) (d) Figure 6. (Color online) Spin Hall conductivity in the graphene/YIG s ystem as a function of chemical potential and exchange parameter for fixed Rashba parameter λR= 10 meV (a) and as a function of chemical potential and Rashba parameter for fixed exchange parameter λEX= 10 meV (b). (c) and (d) represent cross-sections of the dens ity plots in (a) and (b), respectively. -2 -1.5 -1 -0.5 0 0.5 1 -40 -20 0 20 4020 16 12 8 4 0 20 16 12 8 4 0 μ[meV]0.2 161820-0.10.1-1.5-1-0.500.51 -2 0.2 161820-0.10.1-1.5-1-0.500.51 -2σxy[e2/h] σxy[e2/h] σxy[e2/h] μ[meμ[m(a) (c) (b) (d)[meV]35[meV]λ λ[meV]R 78 A9:;<> ? @ B Figure 7. (Color online) Anomalous Hall conductivity in the graphene /YIG system as function of chemical potential and exchange parameter for fixed Rashba parameter λR= 10 meV (a) and as a function of chemical potential and Rashba parameter for fixed exchange parameter λEX= 10 meV (b). (c) and (d) represent cross-sections of the dens ity plots in (a) and (b), respectively. k3±=1 v/radicalBig λ2 EX+µ2±2/radicalbig µ2(λ2 EX+λ2 R)−λ2 EXλ2 Rand ξ3±=/radicalBig λ4 R+k2 3±v2(λ2 R+λ2 EX) Variation of the spin Hall conductivity with the chemical potential µand Rashba and exchange parameters is shown in Fig. 6. For small values ofthe exchange parameter, the spin Hall conductivity depends on the chemical potential in a similar way as in graphene on nonmagnetic substrates [50]. However, when the exchange coupling increases, the spin Hall conductivity vanishes in the energy gap created by the exchange interaction around µ= 0, where σSHE xy= 0.Anomalous, spin, and valley Hall effects in graphene deposit ed on ferromagnetic substrates 10 This is clearly seen in Fig.6b and Fig.6c, where the platos correspond to the zero spin Hall conductivity in the gap. Vanishing of spin Hall conductivity in the gap is a consequence of the compensation of contributions from the two occupied valence subbands which correspond to opposite spin orientations. Width ofagivenplatodependsonthestrengthsofRashbaand exchange couplings. Outside the platos, the absolute value ofσSHE xygrows up and upon reaching a maximum decreases with a further increase in µ, tending to a universal value e/4π. 4.2. Anomalous Hall effect The anomalous Hall conductivity can be calculated in a similar way as the spin Hall conductivity. The corresponding formula for the residua, and thus also for the anomalous Hall conductivity, are rather cumbersome, so they are not presented here. Instead, we show in Fig. 7 only numerical results. First, one can note that the anomalous Hall conductivity disappears for vanishing Rashba coupling. It also vanishes when the exchange coupling is zero as the system isnonmagnetic. The mostinterestingfeatureof the AHE is its quantized value for chemical potentials in the gap formed around µ= 0 due to exchange coupling, where σAHE xy=−2e2/h. This quantized value is of intrinsic (topological) origin, and is consequence of the fact the Berry curvatures of the bands in the K andK′points are the same. 5. Summary In this paper we analyzed graphene based hybrid systems, more specifically graphene deposited on magnetic substrates. The key objective was to study the influenceofproximityeffects, especiallyofthe spin- orbit interaction of Rashba type and the proximity- induced exchange interaction. Two kinds of systems were considered: (i) graphene deposited on a few atomic monolayers of boron nitride, which in turn was deposited on a magnetic substrate (Co or Ni), and (ii) graphene deposited directly on a magnetic (insulating) substrate like YIG. To describe these systems we assumed the model Hamiltonians which were proposed recently on the basis of results obtained from ab-initio calculations. Our main interest was in the spin, anomalous and valley Hall effects. In addition, we also introduced the valey spin Hall effect. The corresponding conductivities were calculated in the linear response regime and within the Green function formalism. In the case of graphen/BN/Co(Ni) hybrid system the strength of exchange coupling is controlled by the number of atomic monolayers of BN. Moreover, the atomic structure of BN leads to a valley gap, whichin turn results in a nonzero valley Hall effect and also in a nonzero valley spin Hall effect. These effects are absent in the case when graphene is deposited directly on YIG. However, anomalous and spin Hall effects can be then observed, with universal quantized values for Fermi level in the energy gap. These universal values follow from topological properties and nonzero Berry curvature. Acknowledgments This work has been supported by the National Science Center in Poland as research project No. DEC- 2013/10/M/ST3/00488 and by the Polish Ministry of Science andHigherEducation(AD) througharesearch project ’Iuventus Plus’ in years 2015-2017(project No. 0083/IP3/2015/73). A.D. also acknowledges support from the Fundation for Polish Science (FNP). References [1] Geim A K and Novoselov K S (2007) Nature Mater. 6, 183 [2] Katsnelson M I (2007) Mater. 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2017-05-09

Spin, anomalous, and valley Hall effects in graphene-based hybrid structures are studied theoretically within the Green function formalism and linear response theory. Two different types of hybrid systems are considered in detail: (i) graphene/boron nitride/cobalt(nickel), and (ii) graphene/YIG. The main interest is focused on the proximity-induced exchange interaction between graphene and magnetic substrate and on the proximity-enhanced spin-orbit coupling. The proximity effects are shown to have a significant influence on the electronic and spin transport properties of graphene. To find the spin, anomalous and valley Hall conductivities we employ certain effective Hamiltonians which have been proposed recently for the hybrid systems under considerations. Both anomalous and valley Hall conductivities have universal values when the Fermi level is inside the energy gap in the electronic spectrum.

Anomalous, spin, and valley Hall effects in graphene deposited on ferromagnetic substrates

1705.03220v1

1 Induced ma gneto -transport properties at palladium/yttrium iron garnet interface Tao Lin, Chi Tang, and Jing Shi Department of Physics and Astronomy, University of California, Riverside, CA 92521 As a thin layer of palladium ( Pd) is directly deposited on an yttrium iron garnet or YIG (Y3Fe5O12) magnetic insulator film, Pd develops both low - and high -field magneto - transport effects that are absent in standalone Pd or thick Pd on YIG. While the low -field magnetoresistance peak of Pd tracks the coercive field of the YIG film, the much larger high-field magnetoresistance and the Hall effect do not show any obvious relationship with the bulk YIG magnetization . The distinct high -field mag neto-transport effects in Pd are shown to be caused by interfacial local moments in Pd . 2 Noble metals such as Pt and Au are preferred spin current generators or detectors [1-5] due to their strong spin -orbit interaction that results in a large spin Hall angle. The large room -temperature spin Hall effect quantified by the spin Hall angle plays an important role in spintronics. Among several interesting spin current related effect s, the spin Seebeck effect (SSE) reported in magnetic metal [6], semiconductor [7], and insulator [8, 9] has received much atte ntion. Recently, SSE in insulators in particular was challenged by a possible magnetic proximity effect (MPE) [10] existing at th e Pt/magnetic insulator interface. It was shown that the MPE, along with t he anomalous Nernst effect in the magnetized Pt interface layer can generate a significant SSE -like signal. More experiments have been carried out in different geometries and with a Au detector that is less prone to MPE [11]. At the heart of the d ebate, one key question is whether MPE exists in spin current detector material s where the inverse spin Hall effect is used . Motivat ed by these experiments , we have chosen a different material, Pd, for this study. First of all, Pd is a 4d transition metal and has large magnetic susceptibility which favors MPE. It was shown that MPE does exist at Pd/ferromagnetic metal interfaces [12]. In addition, Pd also ha s strong spin -orbit interaction and been shown to have a large spin Hall conductivity [13]. To address the issue of possible MPE in magnetic insulator - based structures, we choose Pd/ yttrium iron garnet ( YIG) as our main material system, although we have al so prepa red Pt/YIG for comparisons. In this work, we focus on Pd/YIG samples. Thin YIG films are grown on single crystal gallium gadolinium garnet (GGG) substrates with both (110) and (111) orientations using a pulsed laser deposition system. The base pres sure of the deposition chamber is 6x10-7 Torr. During growth, the chamber is back filled with ozone ( 1.5 m Torr) and the growth temperature is kept at ~ 700C. Epitaxial YIG films are obtained as indicated by the reflection high -energy elect ron diffraction (RHEED) pattern shown in Fig. 1a. For this work, YIG films with thickness ranging from 100 to 200 nm are used. Both orientations show well-defined in- plane magnetic anisotropy , indicating dominance of the shape anisotropy . Because of the similarity in both orientations , this work only include s films in (111) orientation. 3 Typical magnetic hysteresis loops are shown in Fig. 1a . The out-of-plane saturation field is ~ 2 kOe, which corresponds well to 4πM s=1780 G for YIG . After YIG films are taken out of the PLD chamber, they are immediately placed in a high-vacuum sputtering chamber where a thin Pd film is de posited. Before deposition, YIG film is lightly sputtered to provide a fresh and clean sur face. In this work, we f ocus on Pd films with a thickness range from 1.5 to 10 nm. Hall bars with the width of 200 m and length of 1000 m are patterned using ion milling. The magnetic properties of YIG films are measured with either a vib rating sample magnetometer or Quantum De sign’s magnetic property measurement system. The magneto -transport measurements are conducted in either a close -cycled refrigerator with an electromagnet (<1 T) or Quantum Design’s physical property measurement system (up to 14 T). As an in-plane magnetic field H∥ is swept along the Hall bar direction , the magnetoresistance (MR) , , of a Pd (2 nm)/YIG is shown in Fig. 2a , along with the magnetization data of YIG . Two negative peaks appear at the coercive fields of YIG. This feature resembles the ani sotropic MR effect in ferromagnets . Here the MR peak is only ~6x10-6, several orders smaller than that of the anisotropic MR in ferromagnetic conductors. MR with similar magnitude was previously reported in Pt/YIG where the YIG films are polycrystalline [10]. For comparison, a 2 nm thick Cu film deposited on YI G does not show any measurable MR signal. One possible cause of MR in Pd film is that the non -magnetic Pd film acquire s a magnetic moment whose direction is dictated by the underlying YIG film , i.e. the Pd interface layer adjacent to YIG acting as if it is magnetic . As shown in Fig. 2b, t he MR peak s are correlated with the coercive fields of YIG which do not change significantly with the temperature in this temperature range . However, the MR peak nearly doubles when the temperature is lowered to 30 K , which is consistent with reduced spin -flip scattering . We have extended MR measurements to high fields. Fig. 3a shows MR of the same Pd/YIG sample wit h the field pe rpendicular to the film , H⊥. Surprisingly, there is a much larger high-field magnetoresistance (HFMR) background that i s overwhelmingly larger than the low -field MR signal shown in Fig. 2 . At high temperatures, the positive MR is probably the usual Lorentz force induced effect. As the temperature is lowered, this 4 positive MR diminishes and turns to negative . Negative MR is usually seen in materials with random spins that can be aligned by an external field to cause suppressed scattering. At the lowest temperature, the HFMR ratio reaches ~ -10-3, nearly t wo orders greater than that of MR at low fields. The comparison between the low - and high -field MR reveals t hat in addition to the low-field phenomenon related to th e YIG magnetization reversal , there is some spin-dependent process occurring at high field s. When additional spins are aligned with high fields, the MR ratio is consequent ly enhanced. It is interesting that the temperature dependence of HFMR (inset of Fig. 3a) is markedly different from that of the low -field MR. In ferromagnetic conductors , superimposed on the ordinary Hall effect that is linear in H⊥, there is a large anomalous Hall effect (AHE) signal that is proportional to the out -of- plane magnetization component [14]. However, in the low field range (up to ~2 kOe) where the in -plane magnetization is rotated towards the perpendicular direction and therefore there should be an AHE response , we do not observe any definitive magnetization -related AHE signal. As we ramp up H⊥ further , however, an unambiguous non-linear AHE-like signal arises on the linear ordinary Hal l background (removed in Fig. 3b). At low temperature s, there is a clear saturation in Hall resistivity at the highest magnetic field. The Hall resistivity reaches ~0.17 at 5 K, equivalent to ~1x10-3 in the Hall angle. Note that the YIG magnetization saturates only with H⊥ ~ 2 kOe, but saturation of the AHE-like signal does not occur until H⊥ > 20 kOe. Therefore, similar to the HFMR effect, the high -field Hall signal also reveals a response of the magnetic moments other than those in the Pd interface layer that are possibly exchange aligned to the YIG magnetization. We fit the Brillouin function , i.e. TkJBgxxJ JxJJ JJxB BB J );21coth(21)212coth(212)( , to the AHE -like data in Fig. 3 b. Here T is the temperature, B is the Bohr magneton, and gJ is treated as a fitting parameter. The solid curves in Fig. 3b are the actual Brillouin fits. Clearly, the saturation AHE-like signal steadily increases at low temperatures. The inset shows a plot of the normalized AHE -like signal as a function of B/T, indicating that the effective magnetic moment is not a temperature -independent constant. “ gJ” dec reases from ~ 200 B at 5 room temperature to ~ 7 B at 5 K. It is known that a Fe impurity can induce a large local moment of in Pd [15, 16]; however, its temperature dependence has not yet been reported or understood. Fig. 4 a shows the Pd thickness dependence of the AHE -like signal in Pd/YIG samples . As the Pd thickness increases, the Hall magnitude sharply decreases. The inset shows the zoom -in plot of the AHE -like data for 4, 5, and 10 nm thick Pd films at room temperature . For 10 nm thick Pd, the Hall signal essentially vanishes. The rapid ly decreas ing trend of the AHE -like signal clearly demonstrates the interfacial origin of the magnetic moments that are responsible for the high -field Hall effect . Since the moments are located at the interface and it is the interface layer that produces a Hall signal , when the film thickness is much greater than the interface layer, the measured Hall voltage is quickly reduced due to the parallel resistance of the bulk Pd layer . For the same nominal 2 nm thick Pd on YIG, we have observed AHE with similar magnitude in five different samples. Fig. 4b further reveals the properties of the interface moments. First of all, Pd needs to be in direct contact with YIG. Pd on MgO doe s not produce any Hall signal; therefore, the source of the interface moments must be YIG. Second, Cu either has no interface moments or does not produce any Hall signal even if it has interface moments . We cannot distinguish these two possibilities. If the latter is true , a 6 nm thick Cu layer is sufficiently thick er than the mean -free-path so that Pd does not feel any effect from the magnetic moments at the Cu/YIG interface . Third, interface roughness seems to enhance the Hall signal. The sputter clean ed YIG surface is likely rougher than the o ne without sputter cleaning and the Hall magnitude is a factor of 5 larger in the sample with a rough interface. The above experimental facts strongly suggest that independent magnetic moments producing the high-field effects originate from the Pd/YIG int erface . On the other hand, those moments are not exc hange coupled to the YIG spins . In ferromagnetic conductors, the carriers are spin polarized and AHE arises from either extrinsic or intrinsic mechanisms due to spin -orbit interaction. But the existence of an AHE-like signal does not prove ferromagnetism. In the framew ork of AHE, the magnitude of AHE, , scales 6 with the resistivity , , either linearly or quadratically, i.e. , with n=1 or 2, depending on the microscopic mechanism [14]. In our Pd/YIG, the resistivity changes only ~ 18% but the AHE -like signal rises by a factor of 10 below 100 K. We do not expect any sharp temperature dependence of the saturation or fully aligned magnetic moments . Therefore, t he dramatic rise of mea sured AHE -like signal at low temperature s argues against the AHE mechanism for spin-polarized carriers as in regular ferromagnets. Similar high-field effects were previously found in noble met al-based dilute m agnetic alloys where the local moments can cause a left-right asymmetry to unpolarized electrons [17-19]. The Hall angle can be as large as 10-3 to 10-2. Either the spin-orbit interaction or spin -spin exchange between the local moment s and the conduction electrons can result in such a Hall angle. The former is called the skew scattering [19] and the latter the “ spin effect ” [20]. The “spin effect ” causes an enhanced ordinary Hall signal and MR, both of which vary with <S z>2, and therefo re have a zero initial slope at H =0 . This disagree s with our observation s. Our experimental data in Pd/YIG are consistent with the skew scattering picture in which unpolarized electrons are deflected by local moments via spin -orbit interaction , similar to the noble metal - based dilute magnetic alloys [21]. We should point out that Pt/YIG samples also exhibit similar characteristic high -field features as observed in Pd/YIG but with larger magnitude in the Hall signals . In summary , we have observed a low -field MR effect in Pd/YIG which tracks the bulk YIG magnetization reversal . In addition, we have also observed two different, much stronger magneto -transport effects that occur at high magnetic fields where the bulk YIG magnetization is already fully saturated. We attribute t he observed Hall effect to the scattering of conduction electrons in Pd by local magnetic moments at Pd/YIG interface. Acknowledgement: we thank F. Wang, Q. Niu, R.Q. Wu, and V. Aji for many enlight ening discussions. TL and CT were supported by DMEA/CNN; JS was supported by a NSF/EECS grant. 7 Fig. 1. (a) Normalized magnetic hysteresis loop s at 300 K of YIG film on GGG(111) substrate with an applied field in -plane (H ∥) and out -of-plane (H ⊥). Inset: RHEED patte rn of YIG film on GGG(111). (b) Schematic diagram of the patterned Hall bar. Fig. 2. (a) In -plane low -field MR of Pd(2 nm)/YIG (red sq uares), MR of Cu/YIG reference sample (black squares) , and in-plane hysteresis loop (blue squares). (b) Temperature dependence of MR ratio of Pd(2 nm)/YIG. Fig. 3. (a) Out -of-plane HFMR at different temperatures. The inset shows MR ratio at H=10 kOe. Red and blue region s represent positive and negative MR ratios respectively . (b) Field dependence of the Hall resistance R H at different temp eratures with linear background removal . Lines are the Brillouin function fits . Inset: Normalized R H as a function of B/T shows that “gJ” changes as temperature is varied . Fig. 4. (a) Pd thickness dependence of the Hall resistance R H at T=5 K. The inset shows the zoom -in data for Pd thicknesses from 4 to 10 nm. (b) RH for several reference sample s at T=5 K. All metal layers (except the Cu -layer in Pd/Cu/YIG) are 2 nm thick. The inset shows zoom -in data for reference samples. 8 -3 -2 -1 0 1 2 3-1.0-0.50.00.51.0(a) H|| M/MS Field (kOe) In-plane Out-of-plane H Figure 1 (b) 9 -60 -40 -20 020 40 60-8-6-4-20 Pd/YIG Cu/YIG M/Ms Field (Oe) (10-6) -1.0-0.50.00.51.0 M/MS(a) -100 -50 0 50 100-10-8-6-4-202 (b) (10-6) Field (Oe) 300K 200K 100K 50K 30K Figure 2 10 0 25 50 75 100-16-12-8-404 (10-4) Field (kOe) 300K 125K 100K 75K 50K 5K0 100 200 300-9-6-303 (10-4) Temperature (K)(a) -100 -50 0 50 100-200-1000100200 RH (m) Field (kOe) 300K 125K 100K 75K 50K 5K-2 0 2-1.0-0.50.00.51.0RH (norm) B/T (kOe/K)(b) Figure 3 11 -100 -50 0 50 100-200-1000100200 RH (m) Field (kOe) Pd (2nm)/YIG Pd (4nm)/YIG Pd (5nm)/YIG Pd (10nm)/YIG-100 0 100-10-50510(a) RH (m) Field (kOe) -100 -50 0 50 100-200-1000100200(b) RH (m) Field (kOe) Pd (2nm)/YIG Pd/Cu/YIG Cu/YIG Pd/MgO Pd/YIG (no cleaning)-100 0 100-20020 RH (m) Field (kOe) Figure 4 12 Refere nces: [1] E. Saito , M. Ueda, H. Miyajima , and G. Tatara , App l. Phys . Lett. 88, 182509 (2006). [2] T. Kimur a, Y. Otani, T. Sato, S. Takahashi, and S. Maekawa , Phys . Rev. Lett. 98, 156601 (2007). [3] T. Seki , Y. Hasegaw a, S. Mitani, S. Takahashi, H. Imamura, S. Maekawa, J. Nitta, and K. Takanashi, Nature Mater. 7, 125 (2008). [4] Y. Kajiwara , K. Harii, S. Takahashi, J. Ohe, K. Uchida, M. Mizuguchi, H. Umezawa, H. Kawai, K. Ando, K. Takahashi, S. Maekawa, and E. Saitoh, Nature (London) 464, 262 (2010). [5] L. Liu , R. A. Buhrman, and D. C. Ralph, ArXiv e -prints , arXiv:1 111.3702 (2011). [6] K. Uchid a, S. Takahashi, K. Harii, J. Ieda, W. Koshiba e, K. Ando, S. Maekawa, and E. Saitoh, Nature (London) 455, 778 (2008). [7] C. M. Jaworsk i, J. Yang, S. Mack, D. D. Awschalom, J. P. Heremans, and R. C. Myers, Nature Mater. 9, 898 (2010). [8] K. Uchid a, J. Xiao, H. Adachi, J. Ohe, S. Takahashi, J. Ieda, T. Ota, Y. Kajiwara, H. Umezawa, H. Kawai, G. E. W. Bauer, S. Maekawa, and E. Saitoh, Nature Mater. 9, 894 (2010). [9] K. Uchid a, H. Adachi, T. Ota, H. Nakayama, S. Maekawa, and E. Saitoh, Appl. Phys . Lett. 97, 172505 (2010). [10] S. Y. Hua ng, X. Fan, D. Qu, Y. P. Chen, W. G. Wang, J. Wu, T. Y. Chen, J. Q. Xiao, and C. L. Chien, Phys . Rev. Lett. 109, 107204 (2012). [11] T. Kikkaw a, K. Uchida, Y. Shiomi, Z. Qiu, D. Hou, D. Tian, H. Nakayama, X. –F. Jin, and E. Saitoh, Phys. Rev. Lett. 110, 067207 (2013) . [12] J. Voge l, A. Fontaine, V. Cros, F. Petroff, J. Kappler, G. Krill, A. Rogalev, and J. Goulon, Phys . Rev. B 55, 3663 (1997). [13] G. Y. Guo, J . Appl. Phys . 105, 07C701 (2009). [14] N. Nagaos a, J. Sinova, S. Onoda, A. H. MacDonald, and N. P. Ong, Rev. Mod. Phys . 82, 1539 (2010). [15] A. J. Manuel, and M. McDougald, J . Phys . C 3, 147 (1970). [16] G. Bergmann, Phys. Rev. B 23, 3805 (1981). [17] A. Hamzic, S. Senoussi, I.A. Campbell, and A. Fert, Solid State Comm. 26, 617 (1978). [18] A. Hamzic, S. Senoussi, I.A. Campbell, and A. Fert, J. Magn. Magn. Mater. 15, 921 (1980). [19] A. Fert and O. Jaoul, Phys. Rev. Lett. 28, 303 (1972). [20] M.T. Beal -Monod , and R.A. Weiner, Phys. Rev. B 3, 3056 (1971). [21] A. Fert, A. Friederich, and A. Hamzic, J . Magn . Magn . Mater . 24, 231 (1981).

2013-09-09

As a thin layer of palladium (Pd) is directly deposited on an yttrium iron garnet or YIG (Y3Fe5O12) magnetic insulator film, Pd develops both low- and high-field magneto-transport effects that are absent in standalone Pd or thick Pd on YIG. While the low-field magnetoresistance peak of Pd tracks the coercive field of the YIG film, the much larger high-field magnetoresistance and the Hall effect do not show any obvious relationship with the bulk YIG magnetization. The distinct high-field magneto-transport effects in Pd are shown to be caused by interfacial local moments in Pd.

Induced magneto-transport properties at palladium/yttrium iron garnet interface

1309.2213v1

Transition of laser -induced terahertz spin currents from torque - to conduction-electron -mediated transport Pilar Jiménez -Cavero1,2,3,4,*, Oliver Gueckstock1,2,*, Lukáš Nádvorník1,2,5 ,†, Irene Lucas3,4 Tom S. Seifert1,2, Martin Wolf2, Reza Rouzegar1,2, Piet W. Brouwer1, Sven Becker6, Gerhard Jakob6, Mathias Kläui6, Chenyang Guo7,8, Caihua Wan7, Xiufeng Han7,8, Zuanming Jin9,10, Hui Zhao11, Di Wu11, Luis Morellón3,4, Tobias Kampfrath1,2,† 1. Department of Physics, Freie Universität Berlin, Arnimallee 14, 14195 Berlin, Germany 2. Department of Physical Chemistry, Fritz Haber Institute of the Max Planck Society, Faradayweg 4-6, 14195 Berlin, Germany 3. Instituto de Nanociencia y Materiales de Aragón (IN MA), Universidad de Zaragoza -CSIC , Mariano Esquillor, Edificio I+D, 50018 Zaragoza, Spain 4. Departamento Física de la Materia Condensada, Universidad de Zaragoza, Pedro Cerbuna 12, 50009 Zaragoza, Spain 5. Faculty of Mathematics and Physics, Charles University, Ke Karlovu 3, 12116 Prague, Czech Republic 6. Institut für Physik, Joh annes Gutenberg-Universität Mainz, 55128 Mainz, Germany 7. Beijing National Laboratory for Condensed Matter Physics, Institute of Physics, University of Chinese Academy of Sciences, Chinese Academy of Sciences, Beijing 100190, China 8. Center of Materials Scienc e and Optoelectronics Engineering, University of Chinese Academy of Sciences, Beijing 100049, China 9. Shanghai Key Lab of Modern Optical System s, University of Shanghai for Science and Technology, Shanghai 200093, China 10. Department of Physics, Shanghai Univer sity, Shanghai 200444, China 11. Department of Physics and National Laboratory of Solid State Microstructures, Nanjing University, 210093, China * contributed equally to this work † E-mail: nadvornik@karlov.m ff.cuni.cz , tobias.kampfrath@fu-berlin.de Spin transport is crucial for future spintronic devices operating at bandwidths up to the terahertz (THz) range. In F|N thin -film stacks made of a ferro/ferr imagnetic layer F and a normal -metal layer N, s pin transport is mediated by (1) spin-polarized conduction electrons and/or (2) torque between electron spins. To identify a cross -over from (1) to (2), we study laser -driven spin currents in F|Pt stacks where F consists of model materials with different degrees of electrical conductivity. For the magnetic insulators YIG, GIG and 𝛾𝛾-Fe 2O3, identical dynamics is observed. It arises from the THz interfacial spin Seebeck effect (SSE), is fully determined by the relaxation of the electrons in the metal layer and provides an estimate of the spin -mixing conductance of the GIG/ Pt interfac e. Remarkably, i n the half - metallic ferrimagnet Fe 3O4 (magnetite), our measurements reveal two spin-current components with opposite direction. The slower , positive component exhibits SSE dynamics and is assigned to torque- type magnon excitation of the A - and B-spin sublattices of Fe3O4. The faster , negative component arises from the pyro-spintronic effect and can consistently be assigned to ultrafast demagnetization of e-sublattice minority -spin hopping electrons. This observation supports the magneto-elec tronic model of Fe 3O4. In general, our results provide a new route to the contact -free separation of torque- and conduction-electron- mediated spin currents . FIG. 1. (a) Schematic of photoinduced spin transport in an F|Pt stack, where Pt is platinum , and F is a magnetic layer with equilibrium magnetization 𝑴𝑴0. An ultrashort laser pulse excites the sample and induces an ultrafast spin current with density 𝑗𝑗s from F to Pt along the 𝑧𝑧 axis. In the Pt layer , 𝑗𝑗s is con verted into a transverse charge current with density 𝑗𝑗c that gives rise to the emission of a terahertz (THz) electromagnetic pulse. Schematics (1) and (2) indicate spin transfer across the F/Pt interface that is mediated by ( 1) spin-polarized conduction electrons and ( 2) spin torque, the latter coupling to magnons in F. Both (1) and ( 2) can be driven by gradients of temperature and spin accumulation. (b) Simplified schematic of the single -electron density of states vs electron energy 𝜖𝜖 of the tetrah edral A - and octahedral B-sites of the ferrimagnetic half -metal Fe 3O4. The DC conductivity predominantly arises from the B -site minority -type hopping electrons of the e-sublattice . (c) In our interpretation, optical excitation of the Fe 3O4|Pt stack triggers spin transfer through both the spin Seebeck effect ( SSE) and pyro-spintronic effect ( PSE). While the SSE current is mediated by torque between Pt and Fe 3O4 electron spins far below the Fermi level 𝜇𝜇0 [(2) in panel (a)] , the PSE transi ently increases the chemical potential of the B -site minority -spin electrons around 𝜇𝜇0 and, thus, induces a conduction- electron spin current [(1) in panel (a)] . (b) (a) PSE SSE𝒋sdue to (1) (2) Optical pumpTHz electric field𝑬 𝑧𝑧 F Pt 𝒋s 𝒋c 𝑴𝑴0 (1) (2)(c) 𝜇𝜇0𝜖𝜖 𝜖𝜖 B-sublattice: Fe2+/Fe3+ A-sublattice: Fe3+e-sub- lattice FIG. 2. THz emission from F|Pt bilayers as a function of F-layer conductivity . (a) Electrical conductivities of the studied F materials on a logarithmic scale. ( b) Electro -optic signals of THz pulses emitted from various F|Pt bilayers with F=YIG (thick and thin), GIG, 𝛾𝛾-Fe2O3, Fe 3O4 and Fe. Note the different amplitude scaling ’s. The time-axis origin is the same for all signals and was determined by the signal from Fe|Pt reference stacks (Fig. S1). The dashed vertical line marks the minimum signal for the insulating F materials YIG , GIG and 𝛾𝛾-Fe2O3, and the two black arrows label a sharp fe ature in the traces for F=Fe 3O4 and Fe. (c) Fourier amplitude spectra of the signals of panel (b) (normalized to peak height 1 ). Dashed lines show two duplicates of the spectrum of 𝛾𝛾- Fe2O3|Pt. Curves in (b) and (c) are vertically offset for clarity . FIG. 3. Ultrafast photoinduced spin transport in F|Pt stacks . (a) Curves show the spin current density 𝑗𝑗s(𝑡𝑡) in magnetic -insulator|Pt and Fe|Pt stacks , i.e., YIG(3 µm)|Pt(5 nm), GIG(58 nm)|Pt(2 nm), 𝛾𝛾- Fe2O3(10 nm)|Pt(2.5 nm) and Fe(2.5 nm)|Pt(2.5 nm), as extracted from the THz emission signals of Fig. 2b. Each signal is normalized by the pump -excitation density inside the Pt layer and by the indicated factor. (b) Spin current 𝑗𝑗s(𝑡𝑡) in Fe3O4(10 nm)|Pt(2.5 nm) along with scaled spin currents in 𝛾𝛾-Fe2O3|Pt and Fe |Pt. The violet arrows F1 and F2 mark characteristic features of the curves . Note that 𝑗𝑗s(𝑡𝑡) in Fe3O4|Pt can be well described as a linear combination of the other two spin currents (light -violet curve). (c) Same as in panel (b), but f or the THz-emission raw signals . I. INTRODUCTION Spin currents Control over spin currents is a cornerstone of spintronic technologies [1]. New functionalities in such diverse fields as energy conversion and information technologies are envisaged to benefit from the generation, processing and detection of spin currents [2-5]. An important goal is to push the bandwidth of spintronic operations to the terahertz (THz) frequency range, corresponding to ultra fast time scales [1]. A model system for the investigati on of the transport of spin angular momentum is the F|N thin-film stack of Fig. 1(a), where spin can be transmitted from a ferro- or ferrimagnetic layer F to an adjacent non-ferro/ferrimagnetic metal layer N. The spin current in F is mediated not only by (1) spin-polarized conduction electrons , which typically dominate spin transfer in metals , but also by (2 ) magnons, i.e., torque between coupled spins [6,7], which is the main transport channel in insulators . Accordingly, spin transfer across an F/N interface can be mediated by ( 1) spin-polarized conduction elec trons traversing the interface [see (1) in Fig. 1(a)] and by ( 2) spin torque between adjacent F and N regions [(2) in Fig. 1(a)]. As mechanism ( 2) results in the excitation of magnons in F [8], it can be considered as magnonic spin transfer. In general, t o drive an incoherent spin current of density 𝑗𝑗s from F to N , a difference in temperat ure or spin chemical potential (also known as spin accumulation or spin voltage) between the two layers is required [9,10] . For example, for a temperature gradient between F and N, the resulting spin current arise s from the interfacial spin-dependent Seebeck effect (SDSE) [11] for channel (1) or the interfacial spin Seebeck effect ( SSE) [12-16] for channel (2). In any case, the spin flow from F to N can be detected by conversion of the longitudinal 𝑗𝑗s into a transverse charge current with density 𝑗𝑗c [Fig. 1(a)] and measurement of the resulting voltage. For this purpose, N materials with sufficiently large inverse spin Hall effect (ISHE) , for instance Pt, are well suited. THz spin transport A powerful and ultrafast approach to deposit excess energy in F|N stacks is optical excitation by femtosecond laser pulses [Fig. 1(a)]. Measurement of the ultrafast transverse charge current 𝑗𝑗c as a function of time 𝑡𝑡 allows one to resolve elementary relaxation processes such as electron thermalization [8] and electron-spin and electron-phonon equilibration [10], but also delivers insights into spin-to -charge-current conversion [17-24] . For an insulating and pump-transparent F, temperature gradients between F and N (i.e., the SSE ) were found to be the dominant driving force of the ultrafast 𝑗𝑗s [8,25] . For metallic F, in contrast, such temperature differences (i.e., the SDSE) were concluded to make a minor contribution. Instead, spin - voltage-gradients were suggested and identified as the relevant driving force of spin transport on sub- picosecond time scales in metals [10,26-30] . More precisely, dynamic heating of F leads to a transient spin accumulation or spin voltage, which quantifies the excess of spin angul ar momentum in F. This phenomenon, which may be termed pyro-spintronic effect (PSE), induces a spin current across the F/N interface [10,28] . There are important open questions regarding the role of THz SSE and PSE. First, the SSE was so far only observed in F|Pt stacks with F made of yttrium iron garnet (YIG) . According to the microscopic model of Ref. [8], the spin-current dynamics should be fully determined by the relaxation dynamics of the Pt electrons, independent of the insulating F-layer material. This quite universal implication remains to be shown. Second, with increasing electrical conductivity of the F material, a transition from ultrafast SSE to PSE should occur , which was not yet observed. At the cross -over point , both spin transport channels (1) and ( 2) may be operative [Fig. 1(a)], and disentangling the m is crucial to maximize the overall generation efficiency of spin currents. The experimental separation of conduction-electron- and magnon-carried incoherent spin transport is challenging under quasistatic conditions [8,31,32] . However , on femtosecond time scales, SSE and PSE dominate and exhibit different temporal dynamics , as indicated by previous works [8,10] . Thus , a separation of the two effects might be possible. This work In this wor k, we study ultrafast photogenerated spin currents in F|Pt bilayers as a function of various magnetic model F-materials with different degrees of electr ic conductivity : The ferrimagnetic insulators maghemite ( 𝛾𝛾-Fe 2O3), gadolinium iron garnet (Gd 3Fe5O12, GIG) and YIG (with a thickness ranging over three orders of magnitude), the ferrimagnetic half -metal magnetite (Fe 3O4), and, for referencing purposes , the ferromagnetic metal iron ( Fe). Our study reveals that the ultrafast dynamics of the SSE is independent of the choice of the magnetic insulator (YIG, GIG, 𝛾𝛾-Fe 2O3), its thickness (3.4 nm-3 µm) and growth method. Remarkably, i n the half-metallic ferrimagnet Fe 3O4, we observe simultaneous signatures of both SSE and PSE, whose ultrafast spin currents counteract each other . The PSE current is much smaller and of opposite sign compared to Fe. We assign the PSE current in Fe 3O4 to the minority hopping electrons (Fig. 1b). II. EXPERIMENTAL DETAILS THz emission setup To launch an ultrafast spin current, the sample under study is excited with near -infrared femtosecond laser pulses (central wavel ength of 800 nm, duration of 10 fs, energy of 1 nJ, repetition rate of 80 MHz) from a Ti:sapphire laser oscillator [see Fig. 1(a)]. Part of t he energy of the incident pump pulse is instantaneously deposited in the electronic system of the Pt layer and, if metallic, of F. Any induced spin current 𝑗𝑗s(𝑡𝑡) flowing across the F /Pt interface is partially converted into a transverse charge current 𝑗𝑗c(𝑡𝑡) in Pt through the ISHE, thereby resulting in the emission of electromagnetic pulses with frequencies extending into the THz range [Fig. 1(a)] [8,17,19-24,33,34] . We detect the transient THz electric field by electro-optic sampling in a 1 mm thick ZnTe (110) crystal , resulting in the electrooptic signal 𝑆𝑆(𝑡𝑡,𝑴𝑴0) [35-37] . During the experiments, the in-plane equilibrium magnetization 𝑴𝑴0 of the sample is saturated by an external magnetic field with (magnitude ≈100 mT). We measure signals for two opposite orientations ±𝑴𝑴0. Because we are only interested in effects odd in the magnetization 𝑴𝑴0, we focus on the antisymmetric signal 𝑆𝑆(𝑡𝑡)=𝑆𝑆(𝑡𝑡,+𝑴𝑴0)−𝑆𝑆(𝑡𝑡,−𝑴𝑴0) 2. (1) All data are acquired at room temperature in air if not mentioned otherwise. Material choice For the F material in our F|Pt stacks , we choose common and spintronically relevant two-lattice ferrimagnets with increasing degree of electr ic conductivity : (i) insulating YIG (thickness 3.4 nm- 3 µm), (ii) insulating Gd 3Fe5O12 (58 nm), (iii) insulating 𝛾𝛾-Fe 2O3 (10 nm) and (iv) the half -metal Fe3O4 (10 nm) [38]. For referencing, (v) the ferromagnetic metal Fe (2.5 nm) is chosen. As N material , we choose Pt for all samples due to its large spin Hall angle [39] . The approximate F-material conductivities [40-43] are summariz ed in Fig. 2(a). The insulating F materials transfer spin angular momentum by torque [ Fig. 1(a), (1)], whereas for the metal Fe, the spin current is expected to be carried predominantly by conduction electrons [ Fig. 1(a), (2)] [10]. In this respect, Fe3O4 is special because it exhibits both localized and mobile electrons with magnetically ordered spins . More precisely, the ferrimagnet Fe 3O4 is a half -metal , since its conductivity is dominated by hopping- type transport of minority electrons. Fe3O4 possesses two sublattices A and B of , respectively, localized Fe2+/Fe3+ and Fe3+ spins, which couple antiferromagnetically [38]. In the so-called magneto-electric model, the spins of the hopping electrons are aligned predom inantly antiparallel to 𝑴𝑴0 due to exchange interaction with A and B and, thus, form the e-sublattice [44-47] . A highly simplified schematic of the electronic structure of Fe 3O4 is displayed by the spin- and site- resolved single-electro n density of states in Fig. 1(b) [48,49] . The majority (spin -up) electrons exhibit an electronic band gap with a calculated magnitude of 1.7 eV [50], while the presence of minority (spin -down) hopping electrons at the Fermi level 𝜇𝜇0 [44] make s magnetite a half -metal. The measured spin polarization at 𝜇𝜇0 amounts to -72% in Fe 3O4(001) , indicating a nonvanishing number of majority hopping electrons [Fig. 1(b)] [51]. Sample details and excitation Details on the sample fabrication can be found in the Appendix A. In brief, the YIG films are fabricated by three different techniques (pulsed-laser deposition, sputtering and liquid-phase epitaxy). The Fe|Pt reference sample is obtained by growing an Fe layer on top of half the F|Pt region for most of the samples [Fig. S1]. The THz emission signal from the resulting F|Pt|Fe regions is dominated by Pt| Fe and equals the reversed signal from an Fe|Pt layer [8] . By means of the Fe|Pt reference signals, the time axes of th e THz signals from all samples can be synchronized with an accuracy better than 10 fs. The pump electric field is approximately constant along 𝑧𝑧 [Fig. 1(a)] throughout the thin- film stack of our samples. Therefore, the locally absorbed pump-pulse energy is only proportional to Im(𝑛𝑛2), where 𝑛𝑛 is the complex -valued refractive index of the material at the pump wavelength (800 nm). While the Pt and Fe layers are strongly absorbing [Im(𝑛𝑛2)=28 and 30] [52], Fe 3O4 is weakly absorbing (2.3) [53], and YIG, GIG and 𝛾𝛾-Fe 2O3 are largely transparent to the pump beam [Im(𝑛𝑛2)≲1.5] [54,55] . III. RESULTS AND DISCUSSION Terahertz emission signals Figure 2(b) shows electro-optic signals 𝑆𝑆(𝑡𝑡) [Eq. (1)] of THz pulses emitted by the Fe |Pt, 𝛾𝛾-Fe 2O3|Pt, Fe3O4|Pt, GIG|Pt and the thinnest as well as the thickest YIG |Pt samples . THz signals from all ot her YIG sample s can be found in Fig. S2(a). Measurements of YIG(3 µm)|Pt(5 nm) [8] and Fe 3O4|Pt confirm that the THz si gnal increases linearly with the pump power [Fig. S7]. We make two observations in terms of (i) signal shape and (ii) magnitude. (i) The w aveforms from all samples with YIG , GIG and 𝛾𝛾-Fe 2O3 exhibit very similar dynamics [Fig. 2(b) and Fig. S3]. In contrast, the signal for Fe 3O4 features a steeper initial rise, a sharp notch right before the first maximum (see black arrow ) and a subsequent smaller peak . The global minimum is shifted to later times, as indicated by the dashed vertical line. This trend is even more enhanced for Fe|Pt . These observations indicate that different processes occur in the samples as the F -material conductivity increases [Fig. 2(a)] [38,40,41,56,57] . (ii) While the signals from all YIG -based samples have similar strengths [Fig. S 2(a)], the signals from the 𝛾𝛾-Fe 2O3 and Fe 3O4 samples are nearly one order of magnitude larger. The signal from Fe|Pt is even more than two orders of magnitude larger than from YIG|Pt. By Fourier transformation of the time -domain waveforms 𝑆𝑆(𝑡𝑡) [Fig. 2(b)], the normalized amplitude |𝑆𝑆(𝜔𝜔)| as a function of frequency 𝜔𝜔/2𝜋𝜋 is obtained [ Fig. 2(c)]. As expected from the time-domain data [Fig. 2(b)], the THz signal of the YIG , GIG and 𝛾𝛾-Fe 2O3 samples have approximately the same amplitude spectrum. For Fe3O4, however, a slightly blue-shifted spectrum with an increased bandwidth is found. This trend is more pronounced for the Fe |Pt spectrum . Spin current for insulat ing F materials As detailed in Appendix B, we retrieve the spin current dynamics from the measured THz signal waveforms. Figure 3(a) display s the resulting spin current density 𝑗𝑗s(𝑡𝑡) vs time 𝑡𝑡 in 𝛾𝛾- Fe2O3(10 nm)|Pt(2.5 nm), GIG(58 nm)|Pt(2 nm) and the YIG(3 µm)|Pt(5 nm) samples . We observe that (i) the 𝑗𝑗s(𝑡𝑡) in GIG|Pt, 𝛾𝛾-Fe 2O3|Pt and all YIG|Pt samples exhibit very similar temporal dynamics. (ii) The overall amplitude of the spin current in 𝛾𝛾-Fe 2O3|Pt is about one order of magnitude larger than for the YIG|Pt samples . Observations (i) and (ii) are fully consistent with the temporal shape and global amplitude of the underlying raw data [ see Fig. 2(b)]. They have three important implications. SSE dynamics .--First, it is remarkable that the optically induced spin currents in F|Pt bilayers proceed with the same dynamics , even though the magnetic layer is made of very different insulators (F=YIG , GIG and 𝛾𝛾-Fe 2O3) and covers , in the case of YIG, three different growth techniques . Note that i n these samples, the pump pulse is to the largest extent absorbed by the Pt layer. Therefore, observation (i) confirms the previous notion [8] that the ultrafast dynamics of the optically induced SSE current are solely determined by the relaxation dynamics of the electrons in the Pt layer. More precisely, the instantaneous spin current density was predicted to monitor the instantaneous state of the electronic system of N=Pt through [8] 𝑗𝑗s(𝑡𝑡)=𝒦𝒦Δ𝑇𝑇�eN(𝑡𝑡). (2) Here, 𝒦𝒦 is the interfacial spin Seebeck coefficient , and Δ𝑇𝑇�eN is the pump -induced change in a generalized temperature of the N electrons, which is also defined for nonthermal electron distributions. Importantly, Δ𝑇𝑇�eN approximately scales with the number of pump -induced electrons above the Fermi level 𝜇𝜇0. Therefore, it is relatively small directly after optical excitation, but subsequently increases by nearly two orders of magnit ude owing to carrier multiplication through electron-electron scattering [8]. The rise of 𝑗𝑗s(𝑡𝑡) on a time scale of 100 fs [Fig. 3(a)], thus, reflects the evolution of the initially nonthermal electron distribution to a Fermi -Dirac distribution. The decay is determined by energy transfer from the electrons to the phonons. Impact of YIG thickness .--Second, finding (i) also implies that the dynamics of the spin current are independent of the YIG thickness, which covers a wide range from 3.4 nm to 3 µm [Fig. S2(b)]. This result supports the notion [8] that the spin current traversing the YIG/Pt interface stems from YIG regions less than a few nanometers away from the YIG/Pt interface. It is easily understood given that magnons in YIG have a maximum group velocity of about 10 nm/ps [58] and that the major ity of the ultrafast spin-current dynamics proceed within less than 1 ps [Fig. 3(a)]. SSE amplitude. --Third, we observe that the spin current in the 𝛾𝛾-Fe 2O3|Pt sample is about 2 times higher than for the YIG|Pt or GIG|Pt sample . To understand how this observation is related to the F/Pt interface, we consider Eq. (2) and note that the SSE coefficient scales acc ording to [8] 𝒦𝒦∝𝑔𝑔r↑↓𝑀𝑀IF𝑎𝑎3. (3) Here, 𝑔𝑔r↑↓ is the real part of the spin- mixing conductance of the F/ Pt interface, 𝑀𝑀IF is the interfacial saturation magnetization, and 𝑎𝑎 is the lattice constant of F. To obtain the relative magnitude of 𝑔𝑔r↑↓, we divide the THz peak signal of each YIG, GIG and 𝛾𝛾 -Fe 2O3 sample by the deposited pump energy density , the THz impedance of the sample, and 𝑀𝑀IF𝑎𝑎3, where bulk magnetization values are assumed for 𝑀𝑀IF [59-61] (see Appendix B and Table B1). We infer that 𝑔𝑔r↑↓ is very similar in all three materials and has a relative magnitude of 1, 1 and 1.2. Thus, the spin -mixing conductance of the 𝛾𝛾-Fe 2O3/Pt and GIG/Pt interfaces approximately equals that of the YIG/Pt interface [62]. We are not aware of any previous 𝑔𝑔r↑↓ measurement of GIG/Pt . Spin current in Fe|Pt The ultrafast pump- induced spin current in the Fe|Pt reference sample is shown in Fig. 3(a) (blue curve) . It rises and decays much faster than the SSE -type spin currents in the F|Pt samples with magneti c insulator [ Fig. 3(a)]. In a previous work [10], the spin-current dynamics in F|Pt stacks with ferromagnetic metallic F was explained by the PSE: Excitation by the pump pulse leads to a sudden increase of the electron temperature of F as well as of the spin voltage Δ𝜇𝜇�s, also called spin accumulation, which quantifies the instantaneous excess of spin density in F. As the system aims to adapt the F magnetization to the excited electronic state, spin angular momentum is transferred from the electrons to the crystal lattice of F and/or to the adjacent Pt layer. Remarkably, t emperature gradients between F and Pt (i.e., the SDSE) were concluded to make a minor contribution on sub-picosecond time scales [10], resulting in the simple relationship 𝑗𝑗s(𝑡𝑡)∝Δ𝜇𝜇�s(𝑡𝑡). (4) In the case of F ermi-Dirac distributions, Δ𝜇𝜇�s equals the difference of the chemical potentials of spin- up and spin-down electrons, but the concept s of generalized spin voltage and temperature still appl y for non-thermal electron distributions [10]. The transfer of spin angular momentum out of the F electrons into the crystal lattice or the Pt layer leads to a decay of the spin voltage on time scale 𝜏𝜏es. The dynamics of 𝑗𝑗s(𝑡𝑡) is, thus, governed by 𝜏𝜏es and the relaxation of the electron excess energy of F, as parameterized by the generalized electron excess temperature Δ𝑇𝑇�eF. Quantitatively, the dynamics of Δ𝜇𝜇�s(𝑡𝑡) and, thus, 𝑗𝑗s(𝑡𝑡) can be described by [10] Δ𝜇𝜇�s(𝑡𝑡)∝Δ𝑇𝑇�eF(𝑡𝑡)−�d𝜏𝜏 𝜏𝜏es e−𝜏𝜏 𝜏𝜏es Δ𝑇𝑇�eF(𝑡𝑡−𝜏𝜏)∞ 0. (5) Following excitation by the pump [10], Δ𝑇𝑇�eF immediately jumps to a nonzero value . The spin voltage Δ𝜇𝜇�s(𝑡𝑡) and 𝑗𝑗s(𝑡𝑡) follow without delay , according to the first term of Eq. (5). Due to the subsequent transfer of spin angular m omentum out of the F electrons , the spin voltage decays with time constant 𝜏𝜏es, as forced by the second term of Eq. (5). As a consequence, the spin current in Fe|Pt rises instantaneously within the time resoluti on of our experiment (~40 fs) [10], much faster than in, for instance, YIG|Pt [Fig. 3(a)]. Its decay is predominantly determined by el ectron-spin equilibration on the time scale 𝜏𝜏es, with a minor correction due to the significantly slower electron-phonon equilibration [10]. To summarize, t he very different dynamics of SSE (magnetic -insulator|Pt) and PSE (Fe|Pt) seen in Fig. 3(a) suggest that both effects and, thus, torque- and conduction-electron- mediated spin transport can be separated. Spin current in Fe 3O4|Pt Figure 3(b) displays the spin current 𝑗𝑗s(𝑡𝑡) flowing from Fe 3O4 to the Pt layer . We observe two features with different dynamics : (F1) A fast and sharp negative dip ( see violet arrow F1), followed by (F2) a slower positive feature (arrow F2) that decays with a time constant of 0.3 ps. As Fe3O4 is a half-metal, it is interesting to compare the dynamics in Fe 3O4|Pt to those in the two F|Pt stacks with the insulator F=𝛾𝛾-Fe 2O3 and the metal F=Fe [see Fig. 3(b)]. For F=𝛾𝛾-Fe 2O3, the spin current across the F/Pt interface is mediated by spin torque, whereas for F=Fe, it is predominantly carried by spin-polarized electrons . Note that the fast feature (F1) is comparable to 𝑗𝑗s(𝑡𝑡) of Fe|Pt (blue curve), whereas the slow er feature (F2) resembles the 𝑗𝑗s(𝑡𝑡) of 𝛾𝛾-Fe 2O3|Pt (orange curve). As shown in Fig. 3(b), we are even able to reproduce the 𝑗𝑗s(𝑡𝑡) of Fe 3O4|Pt by a sum of −0.026𝑗𝑗s(𝑡𝑡) of Fe|Pt and 0.51𝑗𝑗s(𝑡𝑡) of 𝛾𝛾-Fe 2O3|Pt. We emphasize that such very good agreement is also observed for the corresponding THz electro- optic signals of Fig. 2(b), as is demonstrated in Fig. 3(c). We confirm explicitly that other signal contributions are negligible: magnetic -dipole radiation due to ultrafast demagnetization of Fe 3O4 [Fig. S4] [10,63] and signal s due to Fe contamination of Fe 3O4 by the nearby Fe reference layer, which would yield a signal similar to that from Fe|Pt [Fig. S 5(a)]. To summarize, the spin current in Fe3O4|Pt can be very well represented by a superposition of spin currents in two very different samples comprising insulating and conducting magnetic materials, respectively. This remarkable observation strongly suggests that the spin current in Fe3O4|Pt has contributions from both the PSE, i.e., through spin-polarized electrons , [see (1) in Fig. 1(a)] and the SSE, i.e., through spin torque and magnons [see (2) in Fig. 1(a)]. Physical interpretation for Fe 3O4|Pt We su ggest the following scenario to explain the coexistence of SSE and PSE in Fe 3O4. SSE. --Regarding the SSE, we note that the pump excites mainly Pt and, thus, establishes a temp erature difference between Pt electrons and Fe 3O4 magnons, leading to the SSE sp in current across the Fe 3O4/Pt interface [Fig. 1(a), (2)]. From the measured spin-current amplitudes [ Fig. 3(b)], we infer that the spin- mixing conductance of the Fe3O4/Pt interface is a factor 7.3 larger than that of YIG/Pt [see Table B1], in excellent agreement with literature [62,64,65] . The sign of the current agrees with that of YIG|Pt, suggesting the SSE in Fe 3O4 is dominated by the A and B spin-sublattices , whose total magnetization is parallel to the external magnetic field, whereas the e- sublattice is oppositely magnetized. PSE. --Regarding the PSE, we note that the pump also excites the hopping electrons of Fe 3O4, either directly by optical absorption in Fe 3O4 or by ultrafast heat transport from Pt to the int erfacial Fe 3O4 regions. Because magnetic order of the e-sublattice is understood to decrease with increasing temperature [44-47] , the spin voltage of the e-sublattice electrons changes upon arrival of the pump [Fig. 1(b),(c)] and, thus, triggers spin transfer to the crystal lattice and/or the adjacent Pt layer [Fig. 1(a), (1)] [10]. Remarkably, as the e-lattice spins are on average aligned antiparallel to the equilibrium magnetization 𝑴𝑴0 [Fig. 1(b),(c)], the PSE tends to increase the magnitude of the total magnetization in Fe 3O4, whereas in Fe, it is decreased. We, thus, interpret the observed opposite sign of the PSE currents in Fe 3O4|Pt and Fe|Pt [Fig. 2(b)] as a hallmark of the ultrafast quenching of the residual magnetization of the e-sublattice minority hopping electrons in Fe 3O4. The much smaller amplitude of the PSE current in Fe 3O4|Pt than for Fe|Pt can have several reasons . First, the transport of spin-polarized electrons requires charge conservation [66,67] and, thus, an equal back -flow of charges . However, because the Fe 3O4 spin polarization at the Fermi level is high (-72%) [51], there are less majority states permit ting the backflow of spin-unpolarized electrons from Pt to Fe 3O4 [10]. Second, the mobility of the Fe 3O4 hopping electrons is likely lower than that of the Fe conduction electrons [44,45] . Third, at room temperature, the magnetization of the e-sublattice is significantly smaller than the total Fe 3O4 magnetization [44]. The nonvanishing e-sublattice magnetization inferred here suggests that its ferro-to-paramagnetic transition covers a wide temperature range, possibly because of sample imperfections such as impurities [44] . The relaxation time of the PSE is approximately given by the electron-spin equilibration time 𝜏𝜏es. Figure 3(b) suggests that the 𝜏𝜏es values of Fe3O4 and Fe are comparable and of the order of 100 fs. This conclusion is consistent with previous measurements of ultrafast de magnetization of Fe 3O4, in which an instantaneous drop of the signal was observed directly after optical excitation [68]. It appears that the PSE dynamics does not significantly perturb the slower SSE dynamics , thereby suggest ing that the e-sublattice does not excite magnons of the A, B spins to a sizable extent on time scales below 1 ps. Indeed, in laser -induced magnetization dynamics of Fe 3O4 [68], the instantaneous signal drop was followed by a much larger component with a time constant >1 ns. To summarize, we can consistently assign the PSE current in Fe 3O4 to the demagnetization of the e-sublattice- type minority hopping electrons at the Fermi energy. Interface sensi tivity The relative values of the spin-mixing conductance 𝑔𝑔r↑↓ as inferred above need to be taken with caution because 𝑔𝑔r↑↓, 𝑀𝑀IF and, thus, the SSE are very sensitive to the F/Pt interface properties and, therefore , to the growth conditions of the F|Pt stack [17,69,70] . For instance, as observed for YIG|Pt previously [8], the spin current amplitude may vary by up to a factor of 3 from sa mple to sample. Different interface properties may also explain the amplitude variations of the THz signals between the various YIG|Pt samples studied here [ Fig. S2(b)]. For Fe 3O4|Pt, the SSE contribution is robustly observed for samples with Pt grown at room temperature. However, when the Pt deposition temperature is increased to 720 K, the SSE component disappears [Fig. S 5(b)]. We assign this effect to Pt -Fe interdiffusion at the interface, which magnetizes Pt in the vicinity of Fe, as reported previousl y [71,72] . IV. CONCLUSION We stud y ultrafast spin transport in archetypal F|Pt stacks following femtosecond optical excitation . For the ferri/ferro magnetic layer F, model materials with different degrees of electr ical conductivity are chosen. For the magnetic insulators YIG, GIG and 𝛾𝛾-Fe 2O3, our results indicate a universal behavior of the interfacial SSE on ultrafast time scales: The spin current is solely determined by the relaxation dynamics of the electrons in the metal layer, and it is localized close to the F/ Pt interface. Remarkably, i n the half -metallic ferrimagnet Fe 3O4 (magnetite), our measurements reveal two spin- current components , which exhibit opposite sign and PSE- and SSE -type dynamics. The SSE component is assigned to magnon excitation of the A, B spin sublattices [see (2) in Fig. 1(a)], whereas the PSE component can consistently be assigned to ultrafast demagnetization of e-sublattice minority -spin hopping electrons [(1) in Fig. 1(a)]. Our results show that measur ing heat-driven spin currents faster than their natural sub-picosecond formation time allows one to unambiguously separate SSE and PSE contributions by their distinct ultrafast dynamics. ACKNOWLEDGMENTS The authors acknowledge funding by the German Research Foundation through the collaborative research center s SFB TRR 227 “Ultrafast spin dynamics” (projects A05, B02 and B03), SFB TRR 173 “Spin+X” (projects A01 and B02) and project No. 358671374, the European Union H2020 program through the project CoG TERAMAG/Grant No. 681917, the Spanish Ministry of Science and Innovation through Project No. PID2020-112914RB -I00 and the Czech Science Foundation through project GA CR/ Grant No. 21-28876J . P.J.-C. acknowledges the Spanish MECD for support thr ough the FPU program (References No. FPU014/02546 and EST17/00382). APPENDIX A: SAMPLE FABRICATION All investigated F|Pt samples including film thicknesses are summarized in Fig. 2(b) and Fig. S2. YIG Films of F=YIG covering a wide range of thicknesses (3.4 nm-3 µm) are fabricated by three different methods on double-side -polished Gd3Ga3O12 (GGG) substrates . The film with the smallest thickness of 3.4 nm is epitaxially grown on GGG(111) by pulsed laser deposition (PLD) using a KrF excimer laser. The grow th temperature is 1000 K, and the oxygen pressure is 7 Pa. The growth is monitored by in situ reflection high-energy electron diffraction (RHEED). A clear RHEED pattern is observed, indicating the film is single-crystalline. The YIG films with thicknesses 5-120 nm are deposited on GGG (111) using a sputtering system (ULVAC -MPS-4000-HC7) with a base vacuum of 1× 10-6 Pa. After deposition, annealing at 1070 K in oxygen atmosphere i s carried out to further improve the crystal quality and enhance in-plane magnet ic anisotropy. Finally, a heavy metal Pt thin film is deposited on all the YIG samples . The YIG film with the largest thickness of 3 µm i s grown by liquid-phase epitaxy (LPE) on a GGG substrate (thickness of 500 µm). For details, we refer to Ref. [73] including the Supplementary Information. GIG A film of F=GIG (thickness of 58 nm) on GGG (001) is fabricated by PLD using a custom -built vacuum chamber (base pressure of 2× 10-6 Pa) and a KrF excimer laser [74]. The growth is performed at a substrate temperature of 475 °C, an O2 background pressure of 2.6 Pa and a deposition rate of 1.4 nm/min without subsequent annealing. A Pt layer (thickness of 2 nm) is deposited ex situ by sputtering deposition. The heteroepitaxial growth of GIG and the absence of impurity phases is confirmed by X-ray diffraction. 𝜸𝜸-Fe 2O3, Fe3O4 and Fe Layers of F=𝛾𝛾-Fe 2O3 and Fe3O4 (thickness of 10 nm) are epitaxially grown on MgO(001) substrates (thickness of 0.5 mm) by PLD [61]. Subsequently, all films are in-situ covered by DC sputtering with a thin film of Pt (thickness given in Fig. 2(b) and Fig. S2), followed by a thin film of Fe (thickness of 2.5 nm) on part of each of the F|Pt samples, resulting i n F|Pt and F|Pt|Fe stacks on the same substrate [see Fig. S1 ]. Deposition of adjacent F|Pt and F|Pt|Fe stacks on the same substrate significantly simplifies the THz emission experiments. The two stacks can easily be accessed by lateral shifting into the f ocus of the femtosecond pump beam, with minimal changes of optical paths [Fig. S1]. The THz signal from the F|Pt|Fe regions of each sample serves as an ideal reference that allows for accurate alignment of the setup and definition of the time-axis origin. APPENDIX B: DATA ANALYIS Extraction of spin-current dynamics In the frequency domain, the electrooptic signal 𝑆𝑆(𝜔𝜔) is related to the THz field 𝐸𝐸 (𝜔𝜔) directly behi nd the sample by multiplication with a transfer function 𝐻𝐻(𝜔𝜔) that describes the propagation of the THz wave away from the sample and the response function of the electro-optic detector, i.e., the ZnTe crystal [4,8]. Measurement of 𝐻𝐻(𝜔𝜔) using a well -understood reference emitter allows us to retrieve 𝐸𝐸(𝜔𝜔) and eventually determine the spin current 𝑗𝑗s(𝜔𝜔) through a generalized Ohm’s law [34] , 𝐸𝐸(𝜔𝜔)=𝑒𝑒 𝑍𝑍(𝜔𝜔)𝜃𝜃SH𝜆𝜆N𝑗𝑗s(𝜔𝜔). (6) Here, −𝑒𝑒 is the electron charge, 𝑍𝑍(𝜔𝜔) denotes the sample impedance, 𝜃𝜃SH∼0.1 is the spin-Hall angle of Pt, and 𝜆𝜆N=1 nm is the relaxation length of the spin current in N=Pt [75]. Relative spin mixing conductance We determine the relative spin mixing conductance using the scaling relation [8] ‖𝑆𝑆‖max∝𝑔𝑔r↑↓𝑀𝑀IF𝑎𝑎3𝐴𝐴 𝑑𝑑 𝑍𝑍. (7) Here, ‖𝑆𝑆‖max is the maximum value of the modulus |𝑆𝑆(𝑡𝑡)| of the THz emission signal , 𝑔𝑔r↑↓ is the real part of the spin- mixing conductance, 𝑀𝑀IF and 𝑎𝑎 are the saturation magnetization and lattice constant of the individual F layer, respectively. Furthermore, 𝐴𝐴 denotes the total pump absorptance of the F|Pt sample under consideration, 𝑑𝑑 is the Pt -layer thickness, and 𝑍𝑍 is the THz impedance of the stack . Note that the pump power is assumed to be absorbed in the Pt layer only . All quantities required for the estimation of 𝑔𝑔r↑↓ are taken from literature or a re measured [17]. A summary of the parameters as well as the results f or the spin mixing conductance 𝑔𝑔r↑↓ are shown in Table B1. Parameter F=Fe YIG GIG 𝜸𝜸-Fe2O3 Fe3O4 References Lattice constant 𝒂𝒂 (𝐧𝐧𝐧𝐧) 0.286 1.252 1.247 0.834 0.8396 [76] [77] [78] [79] [42] Saturation magnetization 𝑴𝑴𝐈𝐈𝐈𝐈 (𝐀𝐀 𝐧𝐧𝟐𝟐/𝐤𝐤𝐤𝐤) 222 20 5 400 400 [80] [59] [60] [61] Pt thickness 𝒅𝒅 (𝐧𝐧𝐧𝐧) 2−5 5 2 2.5 2.5 Growth F|Pt a bsorptance 𝑨𝑨 55 % 50 % 50 % 50 % 50 % Measured Conductivity 𝝈𝝈 (𝐤𝐤𝐤𝐤/𝐧𝐧) ~ 1000 ~ 0.1 ~ 0.1 ~ 0.35 ~ 20 (M) [40] [41] [42] [43] Infrared refractive index of substrate - 3.5 3.5 3.07 3.07 [81] [82] Rel. impedance 𝒁𝒁(𝐈𝐈|𝐏𝐏𝐏𝐏)/𝒁𝒁(𝐘𝐘𝐈𝐈𝐘𝐘|𝐏𝐏𝐏𝐏) - 1.0 2.5 0.6 0.1 Calculated Rel. peak signal ‖𝑺𝑺(𝐈𝐈|𝐏𝐏𝐏𝐏)‖𝐧𝐧𝐦𝐦𝐦𝐦/ ‖𝑺𝑺(𝐘𝐘𝐈𝐈𝐘𝐘|𝐏𝐏𝐏𝐏)‖𝐧𝐧𝐦𝐦𝐦𝐦 - 1.0 1.6 8.0 8.8 Measured Rel. spin mixing conductance 𝒈𝒈𝐫𝐫↑↓(𝐈𝐈/𝐏𝐏𝐏𝐏)/ 𝒈𝒈𝐫𝐫↑↓(𝐘𝐘𝐈𝐈𝐘𝐘/𝐏𝐏𝐏𝐏) - 1.00 1.04 1.18 7.30 Inferred from measurements Spin mixing conductance 𝒈𝒈𝐫𝐫↑↓ (𝟏𝟏𝟎𝟎𝟏𝟏𝟏𝟏 𝐧𝐧−𝟐𝟐): previous work - ≈1 - ~ 1 ~ 6 [62,64] [65] [83] TABLE B1. 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Schneider, T. Niizeki, R. Ramos, A. Ross, J. Cramer, E. Saitoh, and M. Kläui, Applied Physics Letters 114 (2019). Supplemental Ma terial FIG. S1. Fe|Pt reference stack. For referencing purposes, an Fe layer is grown on top of part of the F|Pt stack for F= 𝛾𝛾-Fe2O3, Fe3O4 and YIG (3µm). The regions with and without Fe can selectively be excited by the laser beam by lateral translat ion of the sample. For GIG, the stacking order of the Fe|Pt reference layers was reversed, resulting in two regions with thin films of GIG|Fe|Pt and GIG|Pt on the same substrate. The nanometer -thick YIG|Pt samples do not exhibit a reference layer. FIG. S2. Impact of film thicknesses of the YIG|Pt stacks on THz emission . (a) THz emission signals vs pump- probe delay from YIG( 𝑎𝑎)|Pt(𝑏𝑏) with varying YIG thickness ( 𝑎𝑎=3.4 nm and 𝑏𝑏=5 nm, 𝑎𝑎=5-120 nm and 𝑏𝑏=3 nm, 𝑎𝑎=3 µm and 𝑏𝑏=5 nm). The YIG thin films were grown by pulsed laser deposition , sputtering and liquid- phase epitaxy . (b) Amplitude (root mean square) of the THz signals of YIG(𝑎𝑎)|Pt(𝑏𝑏) vs YIG thickness 𝑎𝑎 normalized to the largest peak signal. FIG. S3. Back -to-back comparison of THz signals. E lectrooptic signals of THz pulses emitted from various F|Pt stacks with F=YIG ( 3 µm and 3.4 nm), GIG, 𝛾𝛾 -Fe2O3 and Fe 3O4 (also see Fig. 2b). All signals are scaled to approximately equal peak amplitude. The time axis has the same origin for all signals and was calibrated by using the signal from the Fe|Pt reference region (see Fig. S1). FIG. S4. THz e lectrooptic signals emitted from Fe 3O4|Pt (blue curve) and the Pt| Fe3O4 sample (orange curve) obtained by 180° turning of Fe3O4|Pt about an axis parallel to the magnetization 𝑴𝑴 0. The data was corrected for propagation effects of the THz pulse through the substrate and was low-pass filtered with a cut -off frequency of 6 THz. As the two signals are almost perfect ly reversed versions of each other , magnetic d ipole radiation emitted from Fe 3O4 is not a dominant contribution to the emitted THz signal. Details on the correction procedure can be found in Ref. [10]. 0.4 0.2 0 -0.2 -0.4Electrooptic signal S(t) (10-6 ) -1 0 1 Time t (ps) YIG(3µm)|Pt(5nm) YIG(3.4nm)|Pt(5 nm) GIG(58nm)|Pt(2 nm) γ-Fe2O3(10nm)|Pt(2nm) Fe 3O4(10nm)|Pt(2nm) -2 -1 0 1 2 Ti me 𝑡𝑡(ps)-6-4-20246THz signal 𝑆𝑆(𝑡𝑡)(10-7) Fe3O4 Pt Fe3O4Pt FIG. S 5. Impact of the Fe reference layer and growth conditions on the THz emission signal from Fe3O4|Pt. (a) THz signal waveforms from Fe 3O4|Pt with and without an additional Fe layer in adjacent lateral sample regions (see Fig. S1). As the two signals exhibit almost identical temporal dynamics , we exclude that a sizeable number of Fe atoms is present on top of the nominally Fe- uncovered Fe 3O4|Pt regions . (b) THz emission signals from Fe 3O4|Pt stacks for different growth temperatu res of the Pt laye r: 290 K (violet line ) and 720 K (black line). For the latter , the SSE contribution [maximum of the THz electric field at -0.4ps (violet curve ) due to slower spin current dynamics] is not observable any more, while the PSE contribution is still present . For comparison , the thick blue line shows the reversed THz signal from Fe|Pt. FIG. S6. THz spin currents measured for Fe3O4|Pt samples that were grown on different days . 0.6 0.4 0.2 0.0 -0.2 -0.4THz signal (10-6 ) -1.0 -0.5 0.0 0.5 1.0 Time t (ps)Fe|Pt × (-0.065) 290 K 720 KFe3O4|Pt0.8 0.4 0.0 -0.4THz signal (normalized) -2 -1 0 1 2 Time t (ps) w/ reference layer w/o reference layer(a) (b) -1.0-0.50.00.5Spin current j s (arb. units) 1.2 1.0 0.8 0.6 0.4 0.2 0.0 -0.2 Time t (ps) Sample 1 Sample 2 Fe3O4|Pt× (1.2) FIG. S 7. Fluence -dependence of the THz emission signal from Fe 3O4|Pt. (a) THz emission waveforms from Fe3O4|Pt for decreasing pum p power 𝑃𝑃. The pump power is controlled by a gradient -type neutral -density filter . (b) Root-mean- square amplitude of the waveforms of panel (a) as a function of pump power. The red line shows a linear fit . 1.0 0.5 0.0 -0.5THz signal (normalized) -2.0 -1.5 -1.0 -0.5 0.0 0.5 Time (ps)1.0 0.8 0.6 0.4 0.2 0.0THz signal (normalized) 100 80 60 40 20 0 Pump power P (mW)(a) (b) Fe3O4|Pt P

2021-10-11

Spin transport is crucial for future spintronic devices operating at bandwidths up to the terahertz (THz) range. In F|N thin-film stacks made of a ferro/ferrimagnetic layer F and a normal-metal layer N, spin transport is mediated by (1) spin-polarized conduction electrons and/or (2) torque between electron spins. To identify a cross-over from (1) to (2), we study laser-driven spin currents in F|Pt stacks where F consists of model materials with different degrees of electrical conductivity. For the magnetic insulators YIG, GIG and maghemite, identical dynamics is observed. It arises from the THz interfacial spin Seebeck effect (SSE), is fully determined by the relaxation of the electrons in the metal layer and provides an estimate of the spin-mixing conductance of the GIG/Pt interface. Remarkably, in the half-metallic ferrimagnet Fe3O4 (magnetite), our measurements reveal two spin-current components with opposite direction. The slower, positive component exhibits SSE dynamics and is assigned to torque-type magnon excitation of the A- and B-spin sublattices of Fe3O4. The faster, negative component arises from the pyro-spintronic effect and can consistently be assigned to ultrafast demagnetization of e-sublattice minority-spin hopping electrons. This observation supports the magneto-electronic model of Fe3O4. In general, our results provide a new route to the contact-free separation of torque- and conduction-electron-mediated spin currents.

Transition of laser-induced terahertz spin currents from torque- to conduction-electron-mediated transport

2110.05462v2

arXiv:1711.09814v1 [cond-mat.mes-hall] 27 Nov 2017Magnetic properties, spin waves and interaction between sp in excitations and 2D electrons in interface layer in Y 3Fe5O12/ AlO x/ GaAs-heterostructures L.V. Lutsev1, A.I. Stognij2, N.N. Novitskii2, V.E. Bursian1, A. Maziewski3, and R. Gieniusz3 1Ioffe Physical-Technical Institute, 194021, St. Petersburg , Russia 2Scientific and Practical Materials Research Centre, National Academy of Sciences of Belarus, 220072, Minsk, Bel arus and 3Faculty of Physics, University of Bialystok, 15-097, Bialy stok, Poland (Dated: June 25, 2018) We describe synthesis of submicron Y 3Fe5O12(YIG) films sputtered on GaAs-based substrates and present results of the investigation of ferromagnetic r esonance (FMR), spin wave propagation and interaction between spin excitations and 2D electrons i n interface layer in YIG / AlO x/ GaAs- heterostructures. It is found that the contribution of the r elaxation process to the FMR linewidth is about 2 % of the linewidth ∆ H. At the same time, for all samples FMR linewidths are high. Sputtered YIG films have magnetic inhomogeneity, which give s the main contribution to the FMR linewidth. Transistor structures with two-dimensional el ectron gas (2DEG) channels in AlO x/ GaAs interface governed by YIG-film spin excitations are des igned. An effective influence of spin excitations on the current flowing through the GaAs 2DEG chan nel is observed. Light illumination results in essential changes in the FMR spectrum. It is found that an increase of the 2DEG current leads to an inverse effect, which represents essential chang es in the FMR spectrum. I. INTRODUCTION Integration offerrites with semiconductors offers many advantages and new possibilities in microwave applica- tions such as high-speed wireless communications, active phased array antennas for radars, astronomy systems, auto radars, space electronics and satellite navigation. This integration gives significant advantages in miniatur- isation, bandwidth, speed, radioreception selectivity and the production costs of monolithic microwave integrated circuits (MMICs)1. Ferrite film growth on semiconduc- tor substrates is very important for development of new types of spintronic and spin-wave devices such as mi- crowave filters, delay lines, and spin-polarized field-effect transistors (spin-FET). At present, spin-wave devices have been realized on the basis of Y 3Fe5O12(YIG) films grown on gadolinium-gallium garnet (Gd 3Ga5O12, GGG) sub- strates2,3. Narrow-band filtration can be achieved in YIG-based one-dimensional magnonic crystals4–6. The pulsed laser deposition technique has been used to grow submicron YIG films on GGG substrates for microwave spin-wave band pass filter7. Construction of spin-wave devices on the basis of YIG films directly deposited on semiconductors is the next stage in the development of spin-wave devices. The recent progress in synthesis of nanometer YIG films of high quality on semiconductor substrates8,9and low relaxation of long-wavelength spin waves in nanometer magnetic films10,11give a possibility to construct spin-wave devices on semiconductor chips operating in the microwave frequency band. Activecontrolandmanipulationofspindegreesoffree- domin spin-FETisoneofthe mainproblemsinspintron- ics12–20. The spin transport in two-dimensional electron gas (2DEG) and large spin-orbit interaction are essen- tial for realizing spin transport devices. However, the noneffective spin injection and weak influence of the gateon electron current flowing through the transistor chan- nel are known difficulties in spin-FET design. Modulat- ing the channel conductance by using an electric field to induce spin precession is performed at low tempera- tures and has remained elusive at higher ones19,20. Fur- thermore, poor crystal quality of ferrite films sputtered on GaAs substrates has a detrimental effect on the de- vice performance1,21. At the same time, it needs to note that YIG films are regarded as perspective materials in spintronics22. In this paper we describe synthesis of YIG films de- posited on GaAs substrates with AlO xlayers by ion- beam sputtering and present results of the characteri- zation of YIG film / AlO x/ GaAs heterostructures and their interfaces (see Sec. 2). Magnetic characteristics of deposited films are deduced from the FMR X-band spec- troscopy (Sec. 3). Spin wave propagation are described in Sec. 4. In Sec. 5 we consider the influence of spin excitations in YIG films on the electron current flowing through the 2DEG channel formed at the AlO x/ GaAs interface. It is found that a high interaction between spin excitations and 2DEG channel in GaAs-based sub- strates can be achieved at the ferromagnetic resonance (FMR) frequency of YIG films. The inverse effect, the influence of the electron current on the FMR spectrum, is described in Sec. 6. This interaction is studied by a detection of S-parameters of the transistor channel at microwavefrequenciesunderthelightexposureandwith- out light illumination. The interaction is enhanced with the light exposure of the AlO x/ GaAs interface and with the microwave power increase. It is found that above the spin-wave instability threshold the increase of the 2DEG densityinduced bylightresultsinessentialchangesinthe FMR spectrum and in the S21-parameter of the channel.2 II. SAMPLE PREPARATION AND CHARACTERIZATION YIG films were deposited on GaAs substrates by the two-stage ion-beam sputtering in Ar + O 2atmo- sphere23,24. Then-GaAs substrates with thickness of 0.4 mm had the (100)-orientation. Electrical resistivity ofGaAschipswasmeasuredbythedcfour-probemethod at room temperature and was equal to 0 .9×105Ω·cm. In order to reduce elastic deformation and diffusion of Ga ions into the YIG films and to form 2DEG layer in GaAs substrates, the deposition process was produced on the amorphouslike nonstoichiometric aluminium oxide layer AlOxwith the thickness of 8–20 nm ion-beam sputtered previously on GaAs. The 2DEG layer was formed at the AlOx/ GaAs interface25,26. At the first stage a thin (30 nm) buffer YIG layer was sputtered. After anneal- ing of YIG / AlO x/ GaAs heterostructure the sputtered buffer YIG layer had a polycrystal structure. Annealing was performed in the quasi-impulse regime during 5 min at 590◦C in N 2(samples # 3, 5, 6, Table I) and air (sam- ples # 1, 2, 4) atmospheres with the pressure of 0.1 Torr. After the deposition and annealing processes at the first stage, the buffer YIG layer was polished by a low-energy (400 eV) oxygen ion beam. The polish procedure de- creased stress tension and dislocations, smoothed areas of inter-crystallite boundaries and led to reduction of the buffer YIG layerthickness to 10-16 nm. After this opera- tion the surface ofthe buffer layerwas suitable to deposit a thicker (main) YIG layer without lattice mismatch and stress tension. The main YIG layer was deposited at the second stage. After annealing during 5 min at 550◦C in N2(samples # 3, 5, 6) and air (samples # 1, 2, 4) atmo- spheres with the pressure of 0.1 Torr the sputtered main YIGlayerobtainedapolycrystalstructure. Cross-section ofthe YIG film sputteredon GaAs-basedsubstrate(sam- ple # 5) is presented in Fig. 1a. Cross-section of YIG film of deposited heterostructure was produced by ion- beam cutting on the FIB Helios NanoLab 600 Station (FEI Company, USA). YIG film surface (Fig. 1b) ex- poses the roughness caused by large-scale crystallites of the main YIG layer. The size of crystallites was in the range of 50–100 nm. The structure of YIG films was determined by the X- ray diffraction (XRD CuK α) method and by the energy- dispersive X-ray spectroscopy. The XRD spectrum con- firms the existence of the YIG phase in the sample (Fig. 2). It is found that YIG films are polycrystal and are of homogeneous phase structure. The spectroscopy meth- ods have shown that the interface layer of GaAs is en- riched with Ga due to the volatility of As ions, however, the deposited YIG films are not degraded and are not exfoliated from the GaAs substrates.TABLE I: Properties of YIG films sputtered on GaAs-based substrates. Main layer Sublayers # Thickness 4 πM−Ha4πM−Ha∆H⊥∆H|| d(nm) (Oe) (Oe) (Oe) (Oe) 1 40 1180 1464 155 106 609 -64 2 40 1209 1372 143 129 3 40 990 1278 245 207 800 609 4 250 1454 1320 73 155 1081 5 97 651 554 651 6 964 666 901 175 221 402 FIG. 1: (a) Cross-section of the YIG film sputtered on AlO x / GaAs substrate (# 5). (b) YIG film surface (# 6). III. MAGNETIC CHARACTERISTICS OF YIG FILMS Ferromagnetic resonance of sputtered YIG films was studied by the X-band electronspin resonancetechnique. Relating to samples, applied magnetic field had in-plane and perpendicular orientations. Using magnetic field3 30 40 50 60 700200400600800Intensity, I (arb.u.) 2/c113,□degree* * +++ +++ ++ FIG. 2: XRD spectrum of the sample # 4. ( ∗) marks the substrate and (+) marks the YIG film. sweeping at stabilized frequency F= 9.41 GHz, we have read the first derivative of the FMR curve with respect to the magnetic field H. FMR spectrum of the YIG film with the thickness of 40 nm (sample # 1, Table I) at per- pendicular and at in-plane magnetic fields are presented in Fig. 3. Arrows correspond to FMR peaks of YIG sublayers. In order to find magnetic characteristics of the main YIG layer and sublayers, we use the Lorentzian fitting of experimental curves. Experimental curves are fitted by the sum of first derivatives of Lorentzian curves A(H) =n/summationdisplay iC(i)∂L(i)(H) ∂H, (1) whereiis the peak number ( i= 1 is the number of the main YIG layer and i= 2,3,4,...are numbers of sublayers), C(i)is the amplitude, L(i)(H) =1 1+(H−H(i) 0)2/(∆H(i)/2)2 is the Lorentzian curve, H(i) 0is the peak position, ∆ H(i) is the FMR linewidth. From the Lorentzianfitting (1) we find peak positions H(i) 0and the FMR linewidth ∆ H(i). Differences between magnetization and uniaxial anisotropy field 4 πM−Ha(effective magnetization) of the main YIG layer and YIG sublayers are found from the FMR peak position H(i) 0||of the corresponding layer at the in-plane magnetic field27 F=γ/bracketleftBig H(i) 0||(H(i) 0||+(4πM−Ha))/bracketrightBig1/2 (2)and from the FMR peak position H(i) 0⊥at the perpendic- ular field F=γ/bracketleftBig H(i) 0⊥−(4πM−Ha)/bracketrightBig , (3) whereγ= 2.83 MHz/Oe is the gyromagnetic ratio. Tak- ing into account Eqs. (2) and (3), values of the effective magnetization 4 πM−Haof main YIG layers and YIG sublayers are found. Effective magnetizations and FMR linewidthes ∆ H⊥and ∆H||of main YIG layers are pre- sentedinTableI.Wenotethatsamples#1,2,4annealed in the air atmosphere have higher values of the effective magnetization and lower values of the FMR linewidth than samples # 3, 5, 6 annealed in the N 2atmosphere. The reason for the YIG sublayer formation is not well clear up to now and it is planned to be clarified in next study. IV. SPIN WAVES The FMR linewidth measurement is not sufficient for the determination of relaxation of spin excitations. The linewidth ∆ His formed by relaxation of spin excitations andbymagneticinhomogeneityofamagneticfilm. Inor- der to determine the relaxation parameters, one should study spin wave propagation directly. We studied the amplitude-frequency characteristicsand the relaxationof the Damon-Eshbach surface spin waves28in the in-plane oriented magnetic field. The setup is presented in Fig. 4a. The spin-wave measurement cell contains microstrip antennas. The samples are placed on the antenna struc- ture. Antennas generate and receive spin waves prop- agated in YIG films. The studied samples are irregular trapezoidal with sizes of 2 ×6mm. The distance between antennas in the cell was set to 1.2 mm. The thickness w of antennas is of 30 µm. The excited spin-wave wave- lengthkis given by the thickness wand is in the range [0,2π/w]. The antenna length is equal to 2 mm. The measurement setup contains the Rohde-Schwarz vector network analyzer ZVA-40, which generates the current flowingin the generatingantenna and detects the current induced by spin waves in the receiving one. We measure amplitude-frequency characteristics which are the trans- mission coefficient S21(the scalar gain) in the frequency range of 3.0–4.8 GHz and in the applied magnetic field H= 862 Oe with the in-plane orientation. Only for the sample # 4 we could detect the transmission coefficient S21(Fig. 4b). For other samples, the spin-wave relax- ationappearedto be muchfaster andwe could not detect the spin-wave signals on the receiving antenna. Measuring the S21-parameter, we can estimate the lower bound τ0of the spin-wave relaxation time τ(τ > τ0)9,10. For this estimation we take into account the fol- lowing approximations. (1) We suppose that |Ha| ≪4πM.4 3 4 5 6 Magnetic□field□□□H□□(kOe)-1.0-0.50.00.51.0Amplitude A (a.u.)F□=□9.41□GHz (a) 2.0 2.4 2.8 3.2 3.6 4.0 Magnetic□field□□□H□□(kOe)-1.0-0.50.00.51.0Amplitude A (a.u.)F□=□9.41□GHz (b) FIG. 3: FMR spectrum of the YIG film with the thickness of 40 nm (sample # 1, Table I) at (a) perpendicular and (b) in-plane magnetic fields. Arrows correspond to FMR peaks of YIG sublayers. (2) In order to calculate spin-wavevelocity, we substitute YIG films with inhomogeneity through thickness by ho- mogeneous films with higher velocity of propagating spin waves. (3)Theenergytransformationscurrent →spinwavesand spin waves →current in antenna structure are perfect and have not losses. TheS21-parameter with the voltage induced by spin waves on the receiving antenna and without spin waves can be written, respectively, as S21=S(0) 21+B= 10lg(U2 s+U2 0)1/2 Ug(b)(a) in outGaAs YIG metalH spin□wave Al□O23 3.2 3.6 4.0 4.4 4.8 Frequency□□□F□□(GHz)-80.02-80.00-79.98-79.96-79.94S - parameter21#□4 H□=□862□Oe B S21(0) FIG. 4: (a) Block diagram of the setup used for spin wave propagation in YIG / AlO x/ GaAs structures. (b) The S21- parameter (scalar gain) of spin waves propagated in the sam- ple # 4 in the magnetic field H= 862 Oe at the microwave powerP= 7 dBm. S(0) 21= 10lgU0 Ug, (4) whereUsis the voltage induced by spin waves on the receiving antenna in the magnetic field H= 862 Oe, U0 is the voltage on the receiving antenna without a mag- netic field and, consequently, without spin waves, Ugis the voltage on the generating antenna. We suppose that the spin-wave signals and the voltage U0are not corre- lated. The voltage Usinduced by spin waves is reduced according to Us=Ugexp/parenleftbigg−l vτ0/parenrightbigg , (5) wherevis the group velocity of spin waves, lis the dis- tance between antennas. The spin-wave velocity is given by2,3,27,28 v=π(γ4πM)2d 2F, (6)5 whereFis the frequency at the spin wave dispersion curve at which the wavevector k→0,dis the thickness of the YIG film. Solving the equations (4), (5), and (6), wefind thatthe spin-waverelaxationtime τ0= 39µs and the spin-wave damping parameter, which is given by27,29 δ0=∆ω0 ω0=1 2πFτ0, (7) is equal to 1 ·10−3. In relation (7) ∆ ω0= 1/τ0and ω0= 2πF. Taking into account this value of the spin- wave damping parameter, we can find the contribution of the relaxation process to the FMR linewidth, which is about 2 % of the linewidth ∆ H. Thus, we can suppose that the main contribution to the FMR linewidth of the sputtered YIG film is due to a magnetic inhomogeneity through the film thickness. The analogous magnetic in- homogeneityhasbeenobservedinYIGfilmssputteredon GaN substrates9. The detailed analysis of the evaluation of the spin-wave damping parameter in inhomogeneous YIG films is presented in Ref.9. V. INFLUENCE OF SPIN EXCITATIONS ON THE 2DEG CURRENT Two-dimensional electron gases are formed at oxide interfaces25,26. In order to study interaction between spin excitations in the YIG film and 2DEG in GaAs at the AlO x/ GaAs interface, we have performed the transistor structure with 2DEG channel on the samples # 3 and # 6. Electrical contacts are formed by us- ing the silver paste. We measure amplitude–frequency characteristics which are the transmission coefficient S21 and the voltage reflection coefficient S11in frequency range of 3.5–5.5 GHz and in applied magnetic fields Hup to 6 kOe with the in-plane orientation. The S- parameter matrix for the 2-port network is defined as U(out) i=SikU(in) k, whereU(out) iis the voltage wave re- flected from the i-contact and U(in) kis the incident wave at thek-contact30. The electrical resistivity of YIG films is considerably higher than the resistivity of the GaAs substrate (0 .9×105Ω·cm), consequently, the channel conductivity between contacts is due to the GaAs 2DEG interface region (Fig. 5a). In the FMR frequency band the YIG-film spin excitations give an influence on the current flowing through the GaAs channel. The mea- surement setup contains the Rohde-Schwarz vector net- work analyzer ZVA-40, which generates the current flow- ingthroughthe2DEGchannelanddetectsreflected( S11) and passed ( S21) signals. Normalized S-parametersmea- sured in sample # 3 in the magnetic field H= 1.107 kOe at the microwave power P= 10 dBm are shown in Fig. 5b. One can see that the linewidth of the transmission coefficient S21(593 MHz) is less than the linewidth of the reflection coefficient S11(1215 MHz). This differ- ence can be explained by a magnetic inhomogeneity of the YIG film over the thickness d. TheS11-parameter is3.5 4.0 4.5 5.0 5.5 Frequency□□□F□□(GHz)0.00.20.40.60.81.0Normalized S - parametersS S2111(b)(a) YIG GaAsS S11 21 1 2AlOx 2DEG FIG. 5: (a) Cross-section of YIG / AlO x/ GaAs- heterostructure with 2DEG and with contacts 1 and 2. (b) Normalized S-parameters (the transmission coefficient S21 and the voltage reflection coefficient S11) measured in the transistor structure with 2DEG channel formed on the sam- ple # 3 in the magnetic field H= 1.107 kOe and at the microwave power P= 10 dBm. formed by the inner volume, upper and YIG / AlO xin- terface regions of the YIG film near the first contact. On the contrary, the S21-parameter is formed by the 2DEG channel and the neighboring YIG / AlO xinterface re- gion. In comparison with inner volume and upper re- gions, the neighboring interface region of the YIG film givethe greatercontribution to the S21-parameter. Since magnetic parameters of the inner volume, upper and in- terface YIG regions can be different, this leads to the dif- ference between S11andS21parameters. The observed interaction between spin excitations and 2D electrons is of the electromagnetic nature. The alternating magnetic field of spin excitations induces an alternating electrical field, which influences on 2D electrons. The observed in- fluence ofspin excitationson the 2DEG currentresults in modulation of the current flowing in the 2DEG channel and, in this sense, one can say that this modulation is analogous to the action of a gate electrical potential in FET-structures.6 VI. INVERSE EFFECT. INFLUENCE OF THE CURRENT ON SPIN EXCITATIONS In order to observe the inverse effect – influence of the current on spin excitations and to enhance this in- verse effect, we have carried out the experiment under the following conditions: (1) high channel conductivity, (2) low values of the microwave frequency, at which the three-magnon decay occurs, and (3) high values of the microwave power. Increase of the 2DEG current caused by the growthofthe channel conductivity leads to the in- crease of an alternating magnetic field acted on the YIG film and at high microwave powers results in essential changes in the FMR spectrum. According to27, at the three-magnon decay of the FMR excitation at high mi- crowave powers this influence can be rather high. In the in-plane magnetic field spin excitations can decay into backward volume spin waves. In order to increase the channel conductivity in the transistor structure formed on the sample # 6, the chan- nel was exposed by a light beam ( λ= 650 nm, ε= 1.907 eV) with the photon energy εgreater than the GaAs energy band gap of 1.424 eV and less than the YIG band gap of 2.85 eV31and the AlO xband gap of 6.5 eV32. The light beam was linearly polarized with the intensity W= 81 mW/cm2. The light exposure leads to electron density increase in the GaAs 2DEG channel and it is analogous to an action of electric field in FET struc- tures. Resistance of the channel is reduced from 28.0MΩ to 16.9 MΩ. As a result of the light exposure, the local microwave intensity at neighbouring YIG / AlO xinter- face region increases. This leads to the three-magnon decay of the FMR excitation in the YIG interface layer. The magnon instability process appears at the frequency F <1.8 GHz and at the microwave power P >10 dBm. The normalized S21-parameters measured in the sample # 6 at the frequency F= 1.8 GHz and at the microwave powerP= 14 dBm under the light exposure and without light are shown in Fig. 6. Dependencies are normalized by the maximum value of the S21-parameter measured under the light exposure. The increase of electrons in the 2DEG channel induced by light leads to essential changes in the FMR spectrumand in the S21-parameter. Onecan see that an additional FMR peak bappears in applied magnetic field of 1 kOe. One can observe a decrease in the height of the peak aand a growth in the amplitude of the peak b, while decreasing frequency of the incident microwave signal and keeping the microwave power con- stant and equal to 14 dBm. Therefore, one can conclude that this leads to an increase of the thickness of the YIG layerb, where the magnon instability process occurs. VII. CONCLUSION In summary, we described synthesis of YIG films sput- tered on AlO x/ GaAs substrates, determined their mag- netic characteristics, studied properties of the spin wave0.0 0.4 0.8 1.2 Magnetic□field□□□H□□(kOe)0.00.20.40.60.81.0Normalized S - parameter2 with□light1F□=□1.8□GHz21ab YIG GaAsAlOx2DEGlight a b FIG. 6: The normalized S21-parameters measured in the sam- ple # 6 at the frequency F= 1.8 GHz and at the microwave powerP= 14 dBm (1) under the light exposure and (2) with- out light. Arrows correspond to FMR peaks of YIG layers a (without magnon instability) and b(with magnon instability). propagation and the influence of spin excitations in YIG films and 2DEG channels formed at the AlO x/ GaAs in- terface. Itisfound thatthe contributionofthe relaxation process to the ferromagnetic resonance (FMR) linewidth is about 2 % of the linewidth ∆ H. At the same time, for all samples FMR linewidths are high. It is supposed that increasing of the FMR linewidth is due to magnetic inhomogeneity of YIG films. High interaction between spin excitations and the electron current flowing through the 2DEG channel formed at the AlO x/ GaAs inter- face is achieved at the FMR frequency of YIG films. On the other hand, above the spin-wave instability thresh- old the growth of the channel conductivity induced by the light illumination results in essential changes in the FMR spectrum and in the S21-parameter of the channel. The interaction between the spin excitations in YIG film and 2DEG channel current is increased with the light ex- posure of the AlO x/ GaAs interface and with microwave power growth. The observed interaction is of great im- portance for active control and manipulation of spin de- grees of freedom in field-effect transistors at microwave frequencies. Acknowledgments This workwassupported bythe RussianScience Foun- dation(project17-12-01314)andtheRussianFoundation for Basic Research (project 15-02-06208).7 e-mail: l lutsev@mail.ru 1Z. Chen and V.G. Harris, J. Appl. Phys. 112, 081101 (2012). 2D.D. Stancil and A. Prabhakar, Spin Waves. Theory and Applications (Springer, New York, 2009). 3P. Kabos and V.S. Stalmachov Magnetostatic Waves and Their Applications (Chapman, New York, 1994). 4M. Mruczkiewicz, E.S. Pavlov, S.L. Vysotsky, M. Krawczyk, Yu.A. Filimonov, and S.A. Nikitov, Phys. Rev. B90, 174416 (2014). 5V.D. Bessonov, M. Mruczkiewicz, R. Gieniusz, U. Gu- zowska, A. Maziewski, A.I. Stognij, and M. Krawczyk, Phys. Rev. B 91, 104421 (2015). 6S.L. Vysotskii, Y.V. Khivintsev, V.K. Sakharov, G.M. Dudko, A.V. Kozhevnikov, S.A. Nikitov, N.N. Novitskii, A.I. Stognij, and Y.A. Filimonov, IEEE Magnetic Letters 8, (2017) 3706104 (2017). 7S.A. Manuilov, R. Fors, S.I. Khartsev, and A.M. Grishin, J. Appl. Phys. 105(3), 033917 (2009). 8A.I. Stognij, L.V. Lutsev, V.E. Bursian, and N.N. Novit- skii, J. Appl. Phys. 118, 023905 (2015). 9A. Stognij, L. Lutsev, N. Novitskii, A. Bespalov, O. Go- likova, V. Ketsko, R. Gieniusz, and A. Maziewski, J. Phys. D: Appl. Phys. 48, 485002 (2015). 10L.V. Lutsev, A.M. Korovin, V.E. Bursian, S.V. Gastev, V.V. Fedorov, S.M. Suturin, andN.S. Sokolov, Appl.Phys. Lett.108, 182402 (2016). 11L.V. Lutsev, Phys. Rev. B 85, 214413 (2012). 12S. Datta and B. Das, Appl. Phys. Lett. 56, 665 (1990). 13K. Hall, W.H. Lau, K. Gundogdu, M.E. Flatte, and T.F. Boggess, Appl. Phys. Lett. 83, 2937 (2003). 14J. Schliemann, J.C. Egues, and D. Loss, Phys. Rev. Lett. 90, 146801 (2003). 15J.C. Egues, G. Burkard, and D. Loss, Appl. Phys. Lett. 82, 2658 (2003). 16I.ˇZutic, J. Fabian, and S. Das Sarma, Rev. Mod. Phys. 76, 323 (2004). 17S. Sugahara, Phys. Stat. Sol. 12, 4405 (2006).18X. Lou, Ch. Adelmann, S.A. Crooker, E.S. Garlid, J. Zhang, K.S.M. Reddy,S.D. Flexner, Ch.J. Palmstrøm, and P.A. Crowell, Nature Physics 3, 197 (2007). 19H.C. Koo, J.H. Kwon, J. Eom, J. Chang, S.H. Han, and M. Johnson, Science 325, 1515 (2009). 20P. Chuang, S.-C. Ho, L.W. Smith, F. Sfigakis, M. Pepper, C.-H. Chen, J.-C. Fan, J.P. Griffiths, I. Farrer, H.E. Beere, G.A.C. Jones, D.A. Ritchie, and T.-M. Chen, Nature Nan- otechnology 10, 35 (2015). 21H. Buhay, J.D. Adam, M.R. Daniel, N.J. Doyle, M.C. Driver, G.W. Eldridge, M.H. Hanes, R.L. Messham, and M.M. Sopira, IEEE Trans. Magn. 31, 3832 (1995). 22Solid State Physics. Recent Advances in Magnetic Insula- tors – From Spintronics to Microwave Applications , Edd. M. Wu and A. Hoffmann (Academic Press, San Diego, 2013). 23G.D. Nipan, A.I.Stognij, andV.A. Ketsko, Russian Chem- ical Reviews 81, 458 (2012). 24A.I. Stognij, V.V. Tokarev, and Yu.N. Mitin, Mat. Res. Soc. Symp. Proc. 236, 331 (1992). 25H.C. Koo, H. Yi, J.B. Ko, J.D. Song, J. Chang, and S.H. Han, J. Magnetics 10, 66 (2005). 26J. Mannhart, D.H.A. Blank, H.Y. Hwang, A.J. Millis, and J.-M. Triscone, MRS Bulletin 33, 1027 (2008). 27A.G. Gurevich and G.A. Melkov, Magnetization Oscilla- tions and Waves (CRC Press, New York, 1996). 28R.W. Damon and J.R. Eshbach, J. Phys. Chem. Solids 19, 308 (1961). 29M. Sparks, Ferromagnetic-Relaxation Theory (McGraw- Hill, New York, 1964). 30J. Chomaand W.K. Chen, Feedback Networks: Theory and Circuit Applications (World Scientific, Singapore, 2007). 31P.K. Larsen and R. Metselaar, J. Solid State Chemistry 12, 253 (1975). 32S. Nigo, M. Kubota, Y. Harada, T. Hirayama, S. Kato, H. Kitazawa, andG. Kido, J. Appl.Phys. 112, 033711 (2012).

2017-11-27

We describe synthesis of submicron Y3Fe5O12 (YIG) films sputtered on GaAs-based substrates and present results of the investigation of ferromagnetic resonance (FMR), spin wave propagation and interaction between spin excitations and 2D electrons in interface layer in YIG / AlOx / GaAs-heterostructures. It is found that the contribution of the relaxation process to the FMR linewidth is about 2 % of the linewidth \Delta H. At the same time, for all samples FMR linewidths are high. Sputtered YIG films have magnetic inhomogeneity, which gives the main contribution to the FMR linewidth. Transistor structures with two-dimensional electron gas (2DEG) channels in AlOx / GaAs interface governed by YIG-film spin excitations are designed. An effective influence of spin excitations on the current flowing through the GaAs 2DEG channel is observed. Light illumination results in essential changes in the FMR spectrum. It is found that an increase of the 2DEG current leads to an inverse effect, which represents essential changes in the FMR spectrum.

Magnetic properties, spin waves and interaction between spin excitations and 2D electrons in interface layer in Y3Fe5O12 / AlOx / GaAs-heterostructures

1711.09814v1

Nanoscale X-Ray Imaging of Spin Dynamics in Yttrium Iron Garnet J. Förster,1S. Wintz,2,3J. Bailey,2,4S. Finizio,2E. Josten,5,6D. Meertens,6C. Dubs,7D. A. Bozhko,8,9H. Stoll,1,10G. Dieterle,1N. Träger,1J. Raabe,2A. N. Slavin,11M. Weigand,1,12J. Gräfe,1and G. Schütz1 1Max-Planck-Institute for Intelligent Systems, Stuttgart, Germany 2Paul Scherrer Institute, Villigen, Switzerland 3Helmholtz-Zentrum Dresden-Rossendorf, Germany 4École polytechnique fédérale de Lausanne (EPFL), Lausanne, Switzerland 5Helmhotz-Zentrum Dresden-Rossendorf, Germany 6Ernst Ruska-Centrum für Mikroskopie und Spektroskopie mit Elektronen, Forschungszentrum Jülich GmbH, Jülich, Germany 7INNOVENT e.V. Technologieentwicklung Jena, Germany 8Technische Universität Kaiserslautern, Germany 9University of Glasgow, United Kingdom 10Institut für Physik, Johannes Gutenberg-Universität Mainz, Germany 11Oakland University, Rochester, USA 12Helmholtz-Zentrum Berlin, Germany (Dated: March 5, 2019) Time-resolved scanning transmission x-ray microscopy (TR-STXM) has been used for the direct imaging of spin wave dynamics in thin film yttrium iron garnet (YIG) with spatial resolution in the sub 100 nm range. Application of this x-ray transmission technique to single crystalline garnet films was achieved by extracting a lamella (13x5x0.185 m3) of liquid phase epitaxy grown YIG thin film out of a gadolinium gallium garnet substrate. Spin waves in the sample were measured along the Damon-Eshbach and backward volume directions of propagation at gigahertz frequencies and with wavelengths in a range between 100 nm and 10 m. The results were compared to theoretical models. Here, the widely used approximate dispersion equation for dipole-exchange spin waves proved to be insufficient for describing the observed Damon-Eshbach type modes. For achieving an accurate description, we made use of the full analytical theory taking mode-hybridization effects into account. I. INTRODUCTION Spinwavesarecollectivemagneticexcitationsinferro-, ferri- and antiferromagnetic materials and an active re- search area in the field of magnetism. Recently, it was demonstrated that their quanta, magnons, show specific fundamentals of bosonic behaviour such as Bose Einstein condensation and super-fluidity1,2. And, even black hole scenarios have been predicted to occur in magnon gases3. Besides the fundamental impact of this topic, there has also been increasing interest in potential applications of spin waves as information carriers. This has led to the emergence of the field of magnonics. Compared to elec- tromagnetic waves, spin-wave wavelengths are smaller by several orders of magnitude, which fits perfectly to the lateral dimensions of 10nm1machievable by modern nanotechnology. Spin waves excellently cover the Gigahertz-regime of frequencies, which is common in today’s communications devices, allowing their creation and detection via well-developed microwave techniques. Furthermore, and in contrast to conventional electronics, spin waves can carry information without power dissipat- ing charge currents. Therefore spin waves are actively discussed as high-speed and short-wavelength informa- tion carriers for novel spintronic/magnonic devices4,5. Magnetic thin film systems exhibit three basic geome- tries for lateral spin wave propagation in their spectrum (c.f.figure 1). For in-plane magnetized films there is the backwardvolume(BV)geometrywithwavespropagatingalong the equilibrium magnetization direction, as well as the Damon-Eshbach (DE) geometry, in which the waves propagateperpendiculartoit. Forwardvolumewavesoc- cur in films magnetized out-of-plane propagating isotrop- ically in any direction in the film plane4,6–9. Finally, in addition to the fundamental modes with a quasi-uniform amplitude profile over the film thickness, all three ge- ometries possess higher order thickness modes with am- plitudeprofilesintheformofperpendicularstandingspin waves (PSSW) between the two film surfaces10. The rel- evant energy contributions that determine the dispersion relationsf(k= 2=)of the spin waves in these geome- tries are the magnetostatic and exchange interactions, which dictate the long and short wavelength regimes respectively10. The insulating ferrimagnet yttrium iron garnet (YIG) is one of the most prominent and extensively studied materials in the field of magnonics due to its exception- ally low magnetic damping and high spin wave propa- gation length, making it ideal as a model system11–13 and for possible applications4,7,14–16. Studies become sparse, however, for wavelengths and spatial features be- low 250 nm, despite the importance of this regime for potential nanoscale spintronic devices and the open ques- tions it holds. Factors such as surface effects, crystal de- fects, grain sizes orspindiffusion become moreinfluential on this scale17,18and can change spin wave behaviour compared to the well-studied microscale. For example, an increase in spin wave damping as well as the emer-arXiv:1903.00498v1 [cond-mat.mes-hall] 1 Mar 20192 gence of frequency dependent damping17,18are expected at the nanoscale. The main reason for this region be- ing less well studied lies in its experimental accessibility. Direct imaging of spin wave dynamics is conventionally performedbyopticaltechniqueslikeKerrmicroscopy19,20 and Brillouin light scattering4,5,21, which are inherently limited to a maximum spatial resolution of about 250 nm and the scattering of corresponding wavelengths22, re- spectively, rendering them unable to access nanoscale waves and devices. Another commonly used experimen- tal technique for studying spin waves is all electrical spin- wave spectroscopy using vector network analyzers23,24. While this method is not limited by the wavelength, it does not allow for a direct imaging of spin waves. It also needs comparably large samples to achieve sufficient signal to noise ratio25, limiting its access to nanoscale de- vices. Thus, from both a fundamental and applications perspective there is a clear need for the spatially resolved detection of sub-250 nm spin waves. Time-resolved scanning transmission x-ray microscopy (TR-STXM) is a technique that is able to meet these requirements26–28. Magnetic phenomena can be rou- tinelystudiedwithspatialandstroboscopictemporalres- olutions down to 20 nm and 50 ps respectively. Spin waves in metallic samples prepared as thin films on x-ray transparentsiliconnitride(SiN)membraneshavealready been imaged successfully29–33. But the lack of x-ray transparency in the bulk substrates of single-crystalline systems like YIG films on gadolinium gallium garnet (GGG) requires an appropriate thinning route for STXM investigations34,35. Therefore in the present work a thin sheet of YIG of the order of 185 nm thickness has been sliced out of a YIG thin film and its GGG substrate. The lamella was subsequently put onto an x-ray transparent SiN membrane ( cf.section II). We present TR-STXM measurements in YIG, which provide a new view on the rich and complex scenario of the spin wave characteris- tics, their interactions and coexistence in the nm range of this pivotal model system for design and understanding of future magnonic/spintronic applications. II. METHODS A YIG film of 185 nm thickness was grown by liquid phase epitaxy (LPE) on (111)-oriented GGG36. Ferro- magnetic resonance measurements showed a saturation magnetization of MS= (1432) kA=mand a Gilbert damping coefficient of = 1:3
104, which both agree well with typical values for YIG films in literature4,36–38. The film was subsequently processed using a "FEI Dual Beam System Helios NanoLab 460F1" focused ion beam (FIB). A dedicated Ga+ion milling routine39resulted in a lamella of 13x5x0.185 m3of YIG with less then 150 nm GGG attached to it. Afterwards, an "Omniprobe" micromanipulator was used to transfer the lamella to a standard SiN membrane, where it was centered on a cop- per microstrip antenna ( 2mwidth and 200 nm thick-ness) and fixated with carbon. The copper microstrip was fabricated prior to the fixation of the lamella by a combination of electron beam lithography, thermal cop- per evaporation and lift-off processing. Measurements have been carried out at the MAXY- MUS end station located at the UE46-PGM2 beam line at the BESSY II synchrotron radiation facility of Helmholtz-Zentrum Berlin. Circularly polarized x-rays were focused to 20 nm by a Fresnel zone plate. The X-ray magnetic circular dichroism (XMCD) effect40was used as magnetic contrast mechanism for imaging. For the x-ray energy the iron L 3-absorption edge was chosen, where the maximum magnetic signal fidelity was found at(7080:3) eVas a balance between XMCD strength and transmitted intensity41. The sample was mounted in normal incidence geometry, sensitive to the out-of- plane magnetization component. A quadrupole perma- nent magnet system provided an in-plane magnetic bias field in the range of 250 mT27. Spin waves were excited by the magnetic field of an RF-current flowing through the copper stripline ( cf.figure 2). Time-resolution has been achieved by using a strobo- scopic pump-and-probe technique that reaches a reso- lution of around 50 ps during the synchrotron’s regu- lar multibunch mode operation28. The raw movies from TR-STXM were normalized to enhance the dynamics. A pixel-wise fast Fourier transform (FFT) in the time- domain was subsequently used to obtain the local spin wave amplitude and phase42,43, which were then used to visualize the waves in HSV (hue, saturation, value) color space ( c.f.figure 2). A two-dimensional FFT in space was utilized to determine the corresponding wave vec- tors. See also the paper of Groß et al.33for more details on this. III. RESULTS A. Experimental results As a first step a continuous RF-current in the fre- quency range of 1:4to3:0 GHzwas used for excitation. Figure 2 shows the sample architecture and images of dynamics measured at different frequencies with an ex- ternal magnetic field of 0Hext= 25 mT applied parallel to the stripline. The picture on the left in the upper row shows the raw x-ray intensity image of the lamella. Next to it are the frames from a time-resolved normal- ized movie at 1:6 GHzexcitation frequency arranged in a time series. The frames show dynamic changes in the normal magnetization component as gray scale contrast. The vertical wavefronts of the BV type waves can clearly be seen, as well as their horizontal propagation between the time frames. From such movies, the visual represen- tation shown in the images in the lower row have been obtained, showing the color coded Fourier amplitude and phase at each pixel ( c.f.section II). As expected for spin waves in a thin film, a transition from wave front orienta-3 tion normal to the external field (BV geometry) towards orientation parallel to the external field (DE geometry) can be observed when the frequency is raised10. As is ap- parent in the first image ( 1:4 GHz) of the sequence, for thelowestfrequenciesthespinwaveswereconfinedtothe sample edges due to the locally reduced effective field be- cause of demagnetization effects as previously described inliterature44–46. Theareaofconfinementextendedfrom the edges to between 0.64 and 1.2 minto the sample. In the second image (1.6 GHz), which corresponds to the time series above, two coexisting BV waves of different wavelengths ( 1= 1:9mand2= 0:37m) are visible. Likewise, in the last two images ( 2:5 and 2:7 GHzrespec- tively), where the waves are fully in DE orientation, two DE modes of different wavelengths appear, coexisting at the same frequency (at f= 2:7 GHz:1= 2:8mand 2= 0:53m). In a second step, excitation was changed from con- tinuous sine wave to short bursts ( cf.figure 3, upper part). These bursts excited a broad spectrum of frequen- ciesand, thus, amultitudeofspinwavemodessimultane- ously (one normalized movie is attached as supplemental material). The center frequency of f= 2 GHz was cho- sen corresponding to the previously identified modes and thereby to cover a rich spin wave spectrum. The length of the burst was set to one sine period, or = 480 ps , while the downtime to the next burst was set to 31 sine periods, or approximately 0= 15 ns. Single frequency components were isolated by a Fourier transform and, as previously, visualized in figure 3. The behavior is very similar to the continuous wave experiments shown in fig- ure 2, especially for the direct comparison with the series in the middle row measured at the same field strength (0Hext= 25 mT ). A transition from the BV to the DE propagation geometry at higher frequencies can be seen at all three magnetic bias field strengths. In agree- ment with theory4,6, the spin wave spectra, and hence the transition point, shift towards higher frequencies for increasing external fields. As the antenna was oriented for DE geometry excita- tion, its field is unlikely to be the primary source of the non-DE modes in the lamella. This point is reinforced by the observation, that those waves do not originate in the antenna’s vicinity and rather from the lamella’s edges, making reflections and the aforementioned edge demag- netization fields the probable causes. Especially for the BV geometry waves the observed transition from edge confined modes to sample-wide BV modes hints at the edge fields as source. The aforementioned secondary DE mode (short wavelength) also appears to originate from the upper and lower sample edge rather than from the antenna region. B. Analytical theory To identify the specific spin waves that have been mea- sured, their dispersion f(k), wherefis the frequency andkis the magnitude of the wavevector k, was determined by a two-dimensional FFT ( cf.section II). The focus was put on the two dominant wave orientations, namely the BV geometry ( = 0) and the DE geometry ( = 90), withbeing the angle between kand the equilibrium magnetization. Pairs of fandkwere accordingly sorted by their-values and compared to analytical models of basic spin wave modes in thin YIG films. A model for spin waves in a thin ferromagnetic layer can be found in the work of Kalinikos and Slavin10, taking the following approach. An isotropic ferromagnetic film is considered, that is laterally infinite and of finite thickness dalong thezaxis (z2[d=2;d=2]). The film is magnetized in-plane by a magnetic bias field. A plane spin wave with a non- uniform vector amplitude m(z)is assumed to propagate in the film plane in the arbitrary direction: m(z;;t ) =m(z)exp[i(!tk)] (1) wheretis time and != 2f. The amplitude m(z) is then expanded into an infinite series of complete or- thogonal vector functions. For this, the eigenfunctions of the second-order exchange differential operator satis- fying the appropriate exchange boundary conditions, are chosen. For zero surface anisotropy (unpinned surface spins), which will be assumed from here on, this gives: m(z)/X nmncos n(z+d 2) (2) wheren=n d,n2N0, represents a standing wave com- ponent perpendicular to the film plane. Using equation 2, the following infinite system of algebraic equations can be obtained from the well-known Landau-Lifshitz equa- tion of motion: i! !Mmn=X n0^Wnn0mn0 (3) where!M= MS, the gyromagnetic ratio and MS is the saturation magnetization. This corresponds to equation (22) in the source paper10, where details on the square matrix ^Wcan also be found. The eigenvalues of this system give the frequency of the in-plane propagat- ing spin wave modes of the film. The mode order nhere representsastandingspinwavecomponentalongthefilm thickness given by n. Fork= 0the mode coincides with then-th order PSSW. This results in the amplitude pro- filem(z)havingnnodes along the film thickness. If only the diagonal parts ( n=n0) of ^Ware considered, which means that interactions between modes of different or- ders are neglected, an approximate dispersion equation can be explicitly formulated10:4 fn= 0 2 H+2A 0MSK2   H+2A 0MSK2+MSFnn 1=2 (4) withK2=k2+2 nand the element of the dipole-dipole matrix: Fnn= 1Pnncos2+Pnn(1Pnn)sin2MS H+2A 0MSK2 (5) whereHis the magnitude of the magnetic field, 0 the vacuum permeability, Athe exchange constant and the angle between the magnetization and k. For the fundamental zero-order mode (uniform thickness profile, no PSSW-component) P00= 11ekd kd(see appendix of the original paper10). The zero-order equation gives the dispersions of the fundamental DE and BV modes at = 90and= 0, respectively47. C. Comparing theory and experimental data In figure 4 the spin wave dispersion relation as de- duced from the experimental data of both continuous wave and burst excitations for 0Hext= 25 mT is shown. As figure 2 and 3 already suggested, it is not possible to distinguish between the dispersions measured for the two different excitation schemes, as the two data sets almost perfectly overlap. In a first step the correspond- ing theoretical dispersion curves based on the approxi- mate equation (4) were calculated ( MSgiven in section II,A= 0:36
1011J=m48) and plotted in figure 4 as dashed lines. The = 0-waves fit very well with the cal- culated fundamental BV dispersion curve (dashed black line). The influence of the exchange interaction becomes clear by means of the curve changing to a positive slope beyondk1
107rad/m (compare to exchange free curves in Ref.4,6). This also explains the two coexisting BV waves mentioned in figure 2, as they originate from the branches of the curve left and right of the apex re- spectively. Thus, it appears that equation (4) is a valid approximation for BV waves in this sample, at least in the wavelength range covered here. There seems to be no significant hybridization with higher order BV modes and or influence of the confined sample geometry besides themoriginatingfromthelateraledges. Theedgemodes, shown by the green dots, come close to the BV dispersion with a downward frequency shift of about 200 MHz. In order to explain this difference through edge demagneti- zation effects, a reduction of the effective field to about 15mTwould be necessary. According to micromagnetic simulations of the lamella’s demagnetizing field this is a reasonable value at the edges.The= 90-waves on the other hand appear to belong to two separate dispersion branches that coexist in the area between f= 2:5 to 2:9 GHz. As mentioned earlier (cf.figure 2) two modes at = 90can be seen simulta- neously in the last images ( f2:5 GHz), which already hints at this behaviour. While the analytically approxi- mated fundamental DE dispersion (dashed red line) fits the longer wavelength mode, the shorter wavelength one has to belong to a different spin wave mode featuring the same propagation direction. Obvious candidates for this are DE modes of higher orders. The dashed blue line in figure 4 represents the diagonal approximation for the first order thickness mode ( n= 1) by equation (4). It is apparent that this approximation, i.e.neglecting hy- bridization between different mode orders, is insufficient to describe the DE first order thickness mode in this par- ticular system. Thus, in a second step, numerical calculations of the zero and first order DE dispersions have been carried out using the more accurate equation system (3) and consid- ering the non-diagonal terms n6=n0of the matrix. The results are shown as solid lines (red and blue) in figure 4 and they notably diverge from the dashed analytical curves, while they agree very well with the experimental data. This strongly suggests that the two modes indeed hybridize. A closer look to the modes’ crossing point (in- setinfigure4)supportsthis, asthepresenceofhybridiza- tion effects results in a band splitting. However, compar- ing the dashed and solid curves, it can be seen that the influence of the modes’ mutual interaction reaches well beyond the crossing point. This agrees with observations made previously in 80 to 100 nm thick permalloy films29. Due to this, even the dispersion of the fundamental DE wave clearly diverges from the approximation of equation (4) below wavelengths of 600 nm. This stresses the im- portance of using the extended calculation when going below 1mwavelength. D. Micromagnetic simulation Finally,amicromagneticsimulationwascarriedoutus- ing the "MuMax 3" software developed by Vansteenkiste et al.49. For the simulation, the same material param- eters as for the dispersion calculations were used, and external field of 0H= 25mT was considered together with an RF-burst excitation similar to the experimental one with a field amplitude of 3 mT. Figure 5 shows di- rect comparisons of Fourier images from simulation and experiment at two different frequencies. Results for the dispersion along the main directions = 0and= 90 aredepictedinthelowerpartoffigure5andshowreason- able agreement with the physical measurements (dots) and theory accounting for hybridization (dashed lines). This gives reason to assume that the simulation is a good representation of the experiment and that the simulation results beyond the experimentally covered region repre- sent a viable extrapolation. The agreement of simulation5 and theory in such advanced regions further supports the theoretical approach taken. The simulation also highlights another important point. As figure 5 shows, the dispersion curves in the- ory and simulation continue beyond the experimentally observed data range. This is especially apparent for the DE-waves, which stay well below the wavenumbers mea- sured in the BV-geometry, that themselves reach a limit atk3
107rad/m (200nm). Physically, for every antenna-like spin wave source there is a sharply dimin- ishing efficiency of excitation for wavelengths below the source’s width. This limits the wavevectors that can be excited by the source at a given energy input. Since the BV-waves are likely excited by the approximately 1m widedemagnetizationfieldsonthelateraledges, thelimit forthemislowerthanforthezeroorderDE-waves, which are excited by the 2mwide copper antenna. Hence the occurence of much shorter waves in BV-geometry. IV. CONCLUSIONS Insummary,spinwavesofwavelengthsdownto200nm have been directly imaged in YIG using TR-STXM. For this, a nearly freestanding lamella was fabricated from a YIG film by focused ion beam preparation. Spin wave modesofvariousdirectionsinthesampleplanehavebeen recorded as a function of frequency and external mag- netic field. TR-STXM enabled the simultaneous deter- mination of their spatial properties, like wavefront shape, propagation direction or confinement to certain regions (e.g.the edge), and of the waves time domain features. Dynamics were excited by continuous single frequency RF-fields as well as by broad band RF-bursts. The ob- served BV waves agree very well with a simple diagonal approximation of the analytical expression for the dis- persion relation10. This approach still held reasonably for the zero order DE mode up to k6
106rad/m. However, a second DE dispersion branch was observed, leading to the coexistence of two DE modes with strongly different wavelengths in the frequency range between 2.5 and 2.9 GHz. The diagonal approximation does not correctly describe the second mode, neither as DE zero order nor as its first higher order thickness mode. A more rigorous numerical calculation based on the full set of equations10was necessary and provided an excellent match with the experimental findings. It can be con- cluded that hybridization between different mode orders plays a major role in this system for the formation of spin waves propagating in the DE geometry. Micromag- netic simulations have been done and fit well with the experimental data and the calculated dispersions, indi- cating their potential to predict the observed magnonic scenario in the system studied. While all analytic calcu- lations assumed a laterally infinite film, it appears that the measured wavelengths were sufficiently small com- pared to the dimensions of the sample to still warrant this assumption.The presented work demonstrates that TR-STXM is a powerful and versatile tool for high resolution imaging of magnetization dynamics in real space and time domain. ItclearlydemonstratesitsapplicabilitytoYIGthinfilms, making these accessible to space and time resolved spin wave studies beyond optical resolution limits. This opens up a pathway to directly image nanoscaled spin dynam- ics in YIG and other single crystalline materials and will have an important impact for fundamental magnonic re- search and applications in nano devices. ACKNOWLEDGMENTS We thank HZB for the allocation of synchrotron radi- ation beamtime. Michael Bechtel is gratefully acknowl- edged for support during beamtimes. J.B. is supported fromtheEuropeanUnion’sHorizon2020researchandin- novation programme under the Marie Skłodowska-Curie grant agreement No.66566. C.D. acknowledges the fi- nancial support by the Deutsche Forschungsgemeinschaft (DU 1427/2-1). A.N.S. was supported by the Grant Nos. EFMA-1641989 and ECCS-1708982 from the Na- tional Science Foundation (NSF) of the USA, and by the Defense Advanced Research Projects Agency (DARPA) M3IC Grant under Contract No. W911-17-C-0031.6 FIG. 1. Overview of the basic spin wave mode geometries in a magnetic thin film of thickness d. Green arrows symbolize the magnetization vector M, while the black ones show the wave vector k.7 FIG. 2. Upper part: Schematics of the sample used for the experiments. Gray: Silicon nitride membrane (Silson Ltd). Red cuboid: YIG Lamella, dimensions: 1350:185m3. The magnetic bias field Hextwas oriented in the sample plane parallel to the copper stripline. Lower part: TR-STXM measurements of the sample at 0Hext= 25 mT and frequencies from 1:4 to2:7 GHz. Picture (a) in the upper row shows a raw x-ray image of the lamella (dark gray rectangle) and the stripline. The image series next to it displays the frames of a time-resolved movie at 1:6 GHz, showing the normalized (see section II) out-of-plane magnetization in arbitrary units. The color images in the lower row have been obtained from such movies by gaining the local Fourier amplitude and phase ( c.f.section II) of each pixel’s time-evolution and visualizing it in the HSV color space (color code on the upper left). Wave fronts visibly change from backward volume orientation, through diagonal intermediate states, to Damon-Eshbach direction as the frequency increases. Wave vector directions are indicated in the images with the corresponding wave lengths stated above. .8 FIG. 3. Top part: RF-burst signal used for excitation of the sample (Duration: 480 ps, repetition time: 15.4 ns, voltage amplitude: 2V). Main part: Results of burst measurements analogue to figure 2 at three different magnetic bias fields Hext. Spin waves shift from the backward volume to the Damon-Eshbach propagation geometry as the frequency is raised, the transition point and general spectrum shifting to higher frequencies at greater field strength.9 FIG. 4. Plot of experimental dispersion data at 0Hext= 25 mT for both continuous wave experiments ( c.f.figure 2) and the RF-burst experiment ( c.f.figure 3). Dots represent measured data sorted by propagation direction of the waves. Black and green dots show backward volume (BV) propagation ( = 0) with the green dots marking those confined to the sample edges. The red-blue dots represent Damon-Eshbach(DE) propagation ( = 90). Dashed lines show theoretical dispersion calculated using the approximate equation 4 (no hybridization). The red and blue lines show the zero and first order DE dispersions, while the black line stands for the BV mode. The solid lines represent DE dispersion based on numerical calculations according to equation 3, taking into account the hybridization of modes. The inset shows a magnification of the avoided crossing region of the two branches.10 FIG. 5. Upper part: Direct comparison of Fourier images of the experiment and the micromagnetic simulation at 0H= 25 mT. Lower part: Heatmap of the spatial Fourier transform of the simulation for the two main directions = 0and = 90with the corresponding experimental data (dots) and theoretical dispersion relations (dashed lines) for the Damon- Eshbach modes of order zero and one based on equation 3, as well as for the backward volume mode based on equation 4.11 1S. O. Demokritov, V. E. Demidov, O. Dzyapko, G. A. Melkov, A. A. Serga, B. Hillebrands, and A. N. Slavin, Nature 443, 430 (2006). 2D. A. Bozhko, A. A. Serga, P. Clausen, V. I. 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2019-03-01

Time-resolved scanning transmission x-ray microscopy (TR-STXM) has been used for the direct imaging of spin wave dynamics in thin film yttrium iron garnet (YIG) with spatial resolution in the sub 100 nm range. Application of this x-ray transmission technique to single crystalline garnet films was achieved by extracting a lamella (13x5x0.185 $\mathrm{\mu m^3}$) of liquid phase epitaxy grown YIG thin film out of a gadolinium gallium garnet substrate. Spin waves in the sample were measured along the Damon-Eshbach and backward volume directions of propagation at gigahertz frequencies and with wavelengths in a range between 100~nm and 10~$\mathrm{\mu}$m. The results were compared to theoretical models. Here, the widely used approximate dispersion equation for dipole-exchange spin waves proved to be insufficient for describing the observed Damon-Eshbach type modes. For achieving an accurate description, we made use of the full analytical theory taking mode-hybridization effects into account.

Nanoscale X-Ray Imaging of Spin Dynamics in Yttrium Iron Garnet

1903.00498v1

Electrical detection of unconventional transverse spin-currents in obliquely magnetized thin lms Pieter M. Gunnink,1,Rembert A. Duine,1, 2and Andreas R uckriegel1 1Institute for Theoretical Physics and Center for Extreme Matter and Emergent Phenomena, Utrecht University, Leuvenlaan 4, 3584 CE Utrecht, The Netherlands 2Department of Applied Physics, Eindhoven University of Technology, P.O. Box 513, 5600 MB Eindhoven, The Netherlands (Dated: July 7, 2020) In a typical experiment in magnonics, thin lms are magnetized in-plane and spin waves only carry angular momentum along their spatial propagation direction. Motivated by the experiments of Bozhko et al. [Phys. Rev. Research 2, 023324 (2020)], we show theoretically that for obliquely magnetized thin lms, exchange-dipolar spin waves are accompanied by a transverse spin-current. We propose an experiment to electrically detect this transverse spin-current with Pt strips on top of a YIG lm, by comparing the induced spin-current for spin waves with opposite momenta. We predict the relative dierence to be of the order 104, for magnetic elds tilted at least 30out of plane. This transverse spin-current is the result of the long range dipole-dipole interaction and the inversion symmetry breaking of the interface. Introduction. Magnons, or spin waves, are able to transport angular momentum over long distances along their propagation direction [1, 2]. This has opened the way to novel signal processing devices which could re- place conventional electronic devices [3{5]. In recent years, multiple applications have been explored, such as wave-based computing [6, 7], three-terminal transistors [8], logic gates [9, 10] and novel non-linear eects [11, 12]. The manipulation of spin waves is still an ongoing area of research and a full toolbox for controlling spin waves is yet to be developed [13]. In this work we consider an alternative approach to control the spin current in a magnetized thin lm: by tilting the magnetic eld out of plane. This breaks the inversion symmetry and allows a spin current to ow transverse to the propagation direc- tion of the spin waves, transporting angular momentum along the lm normal. This mechanism for generating a transverse spin- current was rst proposed by Bozhko et al. [14], who used a micromagnetic approach to calculate the exchange spin-current in a thin lm of Y 3Fe2(FeO 4)3(YIG), with- out considering spin absorption at the boundaries. They argued that this spin current is non-zero if the magnetic eld is tilted out of plane. However, this transverse spin- current can only be detected with an attached spin sink, such as a heavy metal strip. The interaction with the spin sink in uences the physics of the problem signi- cantly. Moreover, only the transfer of angular momen- tum by the exchange interaction was considered. The dipole-dipole interaction is also capable of transporting angular momentum and therefore needs to be taken into account for a complete description of this system. In this work we propose an experiment where the trans- verse spin-current in an obliquely magnetized thin lm is detected electrically. We consider, within linear spin- wave theory, a thin ferromagnetic lm with two leads at- tached, which pick up the transverse spin-current inducedby left- and right-moving spin waves via the inverse spin- Hall eect (ISHE) [15]. A transverse spin-current would transport more angular momentum into the right spin sink than into the left spin sink, or vice versa. This is equivalent to the experimentally harder to realize system with leads attached to the top and bottom. We propose to compare the spin current picked up by the left and right lead, in order to exclude any usual spin pumping eects, which are also present for an in-plane magnetic eld [16]. In order to further understand the origin of the transverse spin-current we show in the supplemen- tal material [17] with a magnetostatic calculation, that the symmetry breaking at the interface is carried by the dipole-dipole interaction. Method. The setup we consider is a thin lm of fer- romagnetic YIG, where coherent spin-waves are excited using a coplanar waveguide [18], as depicted in Fig. 1. The wavevector ( k) of the excited magnons is controlled by the grating of the antenna and the frequency ( !) of the excited magnons by the frequency of the driving eld. To the right and left of this antenna two platinum (Pt) leads are placed which function as spin sinks via the in- verse spin-Hall eect and pick up the transverse spin- current induced by the spin waves with opposite mo- menta. The distance between the Pt leads and the copla- nar waveguide is assumed to be such that the signal is strong enough to measure small variations. Structures with a separation distance of 3 mm are possible [19], but the magnon diusion length of = 9:4m in YIG [1] indicates that shorter distances would be preferable. The spin dynamics are governed by the semi-classical Landau-Lifshitz-Gilbert (LLG) equation: @tSi=Si @H @Si+hi(t)i S@tSi ; (1) where we describe YIG as a Heisenberg ferromagnet with eective spin S, on a cubic lattice. Including both the ex-arXiv:2003.12520v4 [cond-mat.mes-hall] 6 Jul 20202 FIG. 1. The setup considered, with a coplanar waveguide in the middle, exciting spin waves in two opposite directions in a thin ferromagnetic lm with thickness d. Two heavy-metal leads pick up the spin current induced by these left- and right- moving spin waves. The magnetic eld is tilted out of plane at an angleHwith the plane and the magnetization has angle Mwith the plane. change and dipole-dipole interactions our eective Hamil- tonian [20] is H=1 2X ijJijSi
SjHe
X iSi 1 2X ij;i6=j2 jRijj3h 3 Si
^Rij Sj
^Rij Si
Sji ; (2) where the sums are over the lattice sites Ri, withRij= RiRjand ^Rij=Rij=jRijj. We only consider nearest neighbour exchange interactions, so Jij=Jfor near- est neighbours and 0 otherwise. Here = 2Bis the magnetic moment of the spins, with B=e~=(2mec) the Bohr magneton. Heis the external magnetic eld, which we take strong enough to fully saturate the ferromagnet. To the top of the thin lm we attach a spin sink to detect the spin waves, which introduces an interfacial Gilbert damping L i, which is only non-zero for sites at the top interface of the ferromagnet [16]. The total Gilbert damping is then i=B+L i, whereBis the bulk Gilbert damping. Furthermore, hi(t) is the circu- larly polarized driving eld, which we take to be uniform throughout the lm. Within linear spin-wave theory, the LLG has been shown to be fully equivalent to the non- equilibrium Greens function formalism [21]. We consider a thin lm, innitely long in the y;zdi- rections and with a thickness d=Nain thexdirection, whereais the lattice constant and Nis the number of layers. The magnetic eld is tilted at an angle Hwith respect to the lm, as shown in Fig. 1. The magnetization is tilted by an angle M, as determined by minimizing the energy given by Eq. (2) for a classical, uniform spin conguration: @ @M MsHecos (MH)2M2 scos2H = 0; (3)whereMs=S=a3is the saturation magnetization and He=jHej. We have two reference frames, one aligned with the thin lm as described above and one where the zaxis is aligned with the magnetization M. We work in the reference frame of the lattice and rotate the spin opera- tors, such that Si!R1 y(M)Si, whereRy(M) is a rotation around the y-axis by angle Mand Siare the rotated spin operators, with the Sz icomponent pointing along the magnetization M. We linearize in the deviations from the ground state, bi=1 2p 2SSx i+iSy i
and assume translational invari- ance in the yz-plane. The equation of motion for bibe- comes in frequency space: G1 k(!) k(!) =hk(!); (4) wherek= (ky;kz) and we have introduced the driving eld hk(!) = (hk(!);:::;h k(!)|{z} Nelements;h k(!);:::;h k(!)|{z} Nelements)T;(5) wherehk(!) =hx+ihyis the Fourier transform of the rotated driving eld. Furthermore, the magnon state vec- tor is k(!) = bk(!;x 1);:::;b k(!;xN); b k(!;x 1);:::;b k(!;xN)
T(6) and the inverse Green's function is G1 k(!) =3(1 +i3)!3Hk; (7) where we have introduced 3= diag (1;:::;1;1;:::;1), = diag (1;:::;N;1;:::;N) and Hk=AkBk By kAk ; (8) which is the Hamiltonian matrix within linear spin-wave theory, with the amplitude factors [ Ak]ij=Ak(xixj) and [Bk]ij=Bk(xixj). The dispersion is obtained by diagonalizing the inverse Green's function (4) in the ab- sence of damping and spin pumping. The full expressions for the amplitude factors Ak;Bkand the dispersions for dierent tilting angles of the magnetic eld are given in the supplemental material [17]. From the equation of motion, Eq. (4), the total spin- current injected into the lead is obtained from the conti- nuity equation for the spin: @tSz i+X jIex i!j+X jIdipdip i!j =I i+Ih i: (9) The explicit form of the terms is given in the supple- mental material [17]. We nd a source and sink term,3 TABLE I. Parameters for YIG used in the numerical calcula- tions in this work. Note that Sfollows from S=Msa3=. Quantity Value N 400 a 12:376A [22] S 14.2 4Ms 1750 G [23] J 1:60 K [24] B7104[25] L7103[25] He 2500 Oe hx;hy 0:01He providing angular momentum via the driving eld ( Ih i) and dissipating angular momentum to the lattice and the lead via the Gilbert damping ( I i). There are two ways angular momentum can be transferred through the lm. Firstly, there is a spin current transferring angular mo- mentum between adjacent sites ( Iex i!j), which is driven by the exchange interaction. The dipole-dipole interac- tion also transports angular momentum ( Idipdip i!j), but because the dipole-dipole interaction is non-local, angu- lar momentum is transferred from and to all other sites. It is therefore not possible to write this as a local diver- gence and thus as a current. Also note that the dipole- dipole interaction couples the magnons to the lattice, which means that a non-zero dipole-dipole contribution is accompanied by a transfer of angular momentum from and to the lattice. The measurable quantity is the angular momentum ab- sorbed by the spin sink in the attached lead, which is proportional to the voltage generated by the ISHE, and is given by I L(k;!) = 2LIm [b k(x1)@tbk(x1)]: (10) We are interested in the relative dierence between the spin currents induced by the left- and right-moving spin waves in order to show a transverse spin transport, which we dene as
(jkj;!) =I L(k;!)I L(k;!) max [jI L(k;!)j;jI L(k;!)j]: (11) In the next section we consider this quantity in detail. Results. The parameters used throughout this work are summarized in Table. I. In Fig. 2 we show the dif- ference between the spin current induced by left- and right-moving spin waves for dierent tilting angles of the magnetic eld. For a magnetic eld either completely in- or out of plane there is no dierence between the left and right lead (not shown). As we tilt the mag- netic eld out of plane a small dierence becomes vis- ible, which peaks at
= 1 :25104forH= 60 and 2:5< k < 12:5m1. As the tilting angle is fur- ther increased the distribution of
shifts slightly, withthe most notable change the movement of the maximum, which moves towards smaller wavevectors. We found that the relative dierence
increases linearly with the bulk Gilbert damping constant. In order to measure this ef- fect it might therefore be benecial to use a YIG thin lm with deliberately introduced impurities such as rare- earth ions, to increase the damping [26], or even use a dierent ferromagnetic material with a higher Gilbert damping. Numerically, we found that the relative dierence
is non-zero even when the exchange coupling is articially turned o, which indicates that only the dipole-dipole interaction is responsible for this eect. In the supple- mental material [17] we show a full magnetostatic deriva- tion of the eigenmodes for an obliquely magnetized thin lm with only dipole-dipole interactions. Even though the energies are inversion-symmetric, we nd that the eigenmodes explicitly depend on kzsin (2M); (12) which introduces an asymmetry between left- and right- moving spin waves if the magnetic eld is tilted out of plane. A complete description of this problem also re- quires the inclusion of the exchange coupling, as was done in our numerical calculations. However, ignoring the exchange coupling allows us to demonstrate that the origin of the asymmetry between left- and right-moving spin waves lies in the the long range dipole-dipole inter- action carrying the inversion symmetry breaking of the interface. Bozhko et al. [14] suggested a partial-wave picture to explain the transverse spin-current. They reason that the prole along the lm normal is made up by two par- tial waves, which have opposite momenta kxand equal frequency!if the lm is magnetized in-plane, thus can- celling any transfer of angular momentum or energy. As the magnetic eld is tilted out of plane the two partial waves would, in this picture, no longer have opposite mo- menta, but still have the same frequencies. This would then allow for angular momentum transfer, but not en- ergy transfer. With the magnetostatic calculation we are able to show that this picture is incomplete: the am- plitudes of the two partial waves are asymmetric, not their momenta. This therefore allows both energy and angular momentum transfer, which we have conrmed numerically by evaluating h@tEi. We found numerically that the region in k-space where the relative dierence
is signicant has a lower bound related to the thickness of the thin lm. Decreasing the thickness shifts the distribution as seen in Fig. 2 towards larger wavevectors. This can be traced to the fact that the long-wavelength magnetostatic magnon modes are standing waves [17], with wavevectors kx, wherekxis proportional to kz. The standing waves need to have a wavevector big enough to t at least one wavelength into the system, thus requiring that kz&kL, where4 5 10 15 k(µm−1)24681012ω(GHz)(a) 5 10 15 k(µm−1)(b) 5 10 15 k(µm−1)(c) 0.000.250.500.751.001.25∆×10−4 FIG. 2. The relative dierence
between the spin current induced by left- and right-moving spin waves, as dened in Eq. (11), as a function of kand!, for three dierent tilting angles of the magnetic eld. The spin waves travel parallel to the in-plane projection of the magnetic eld, such that k=k^z. (a)H= 30;M= 18, (b)H= 60;M= 40and (c) H= 80;M= 64. The peak dierence is
( k= 7:5µm1;!= 4 GHz) = 1 :25104, when the eld is tilted at an angle H= 60. For a magnetic eld completely in- or out of plane (not shown) there is no discernible dierence. kL= 2=d. The reason for this coupling of the in-plane and out of plane directions is the long-range nature of the dipole-dipole interaction, ensuring that within our system the divergence of the magnetic eld is zero, i.e., r
B= 0. The maximum value of
does not change depending on the thickness of the lm, only the location of the maximum. We have conrmed this numerically for the range 60d480 nm. For even thinner lms the maximum value of
becomes lower. Excitation of magnons is only possible for values of ! determined by the spin-wave dispersion, with a minimum given by the lowest mode. We therefore show in Fig. 3 for xedk= 7:5µm1the evolution of the relative spin- current dierence
as the magnetic eld is tilted out of plane, with a driving at frequency !corresponding to the lowest mode in the spin-wave dispersion. Also shown is the frequency of the lowest mode as a function of mag- netic eld tilt angle. It is clear that up to some critical value of the magnetic eld angle
increases linearly, af- ter which it falls o rapidly. It is also clear that the low- est mode is capable of transferring angular momentum along the lm normal. This is contrary to the statements made by Bozhko et al. [14], who predicted that the lowest mode, which has an uniform prole, would not induce a transverse spin-current. This is most likely due to the fact that in their work only the exchange current is con- sidered, whereas we have taken all current contributions into account. Another possible explanation is their ex- pansion in eigenfunctions of the second-order exchange operator, which might have failed to properly take the dipole-dipole interaction into account. The dierent contributions to the transverse angular momentum transport, as dened in Eq. (9), are shown in Fig. 4 for left- and right-moving spin waves. We have set the bulk and interface damping to zero in order to 45678 ω(GHz) 0 20 40 60 80 φH(◦)0.000.250.500.751.001.25∆×10−4FIG. 3. Relative dierence between the spin current induced by left- and right-moving spin waves,
, as dened in Eq. (11), as a function of magnetic eld tilt angle H, for!correspond- ing to the lowest mode in the spin-wave dispersion and xed k= 7:5µm1(solid line). Also shown is the frequency of the lowest mode as a function of the tilt angle (dashed line). clearly show the exchange, dipole-dipole and driving con- tributions to the transfer of spins along the lm normal. Firstly, we can see that there is a transport of angular momentum, even in the case of no spin absorption at the boundary, which agrees with the results by Bozhko et al. [14]. All contributions are zero in the case of an in-plane magnetic eld (not shown)|if no spin sinks are attached. We can see that every contribution switches sign between left- and right-moving spin waves, as would be expected from symmetry. From this gure it is clear that the exchange spin current is not the only way the system transfers angular momentum. In fact, the contri- butions from the dipole-dipole interaction are larger than those of the exchange current. This shows that it is nec- essary to consider both interactions in order to gain a full5 0 50 100 xi0Ii(a.u.)−k 0 50 100 xiExchangeDriving Dipole-dipole+k FIG. 4. The dierent contributions to the transfer of angular momentum along the lm normal, where Ii=P jIi!jfor the exchange and dipole-dipole interaction. The damping plays a negligible role in the transport of angular momentum, so it is turned o to illustrate the eects of the other contributions. The thickness of the thin lm is reduced to N= 100 in order to better illustrate the variation through the lm. The mag- netic eld is tilted out of plane with angle M= 60and the wavevector and driving frequency are xed at k= 30 µm1, != 4 GHz. understanding of the transport of angular momentum in the transverse direction. Also note that since the dipole- dipole contribution is non-zero there is a nite torque on the system, which could be measured in a cantilever experiment [27]. Conclusion and Discussion. In this work we have shown, using microscopic linear spin-wave theory, that there is a ow of angular momentum, or spin current, along the lm normal in obliquely magnetized thin lms. This can be measured using an antenna-detector setup, where the spin current induced by the left- and right- moving spin waves will be dierent, proving the existence of a transverse spin-current. This eect can be used as a way to manipulate the spin current owing along the lm normal, for example by controlling the magnetic eld angle. We have also demonstrated that this spin current is the result of the dipole-dipole interactions in the lm, which carry the inversion breaking at the interface. We have not considered explicitly the interactions of the spin waves with the lattice. The dipole-dipole in- teractions couple the magnons to the lattice and there- fore angular momentum can be transferred from and to the phonons, which can also transport angular momen- tum [28{30]. A more complete description of the system should therefore include these phonon-magnon interac- tions, but this is beyond the scope of this article. R.D. is member of the D-ITP consortium, a program of the Dutch Organization for Scientic Research (NWO) that is funded by the Dutch Ministry of Education, Cul- ture and Science (OCW). This project has received fund- ing from the European Research Council (ERC) under the European Unions Horizon 2020 research and inno-vation programme (grant agreement No. 725509). This work is part of the research programme of the Founda- tion for Fundamental Research on Matter (FOM), which is part of the Netherlands Organization for Scientic Re- search (NWO). It is a pleasure to thank Alexander Serga and Huaiyang Yang for discussions. p.m.gunnink@uu.nl [1] L. J. Cornelissen, J. Liu, R. A. Duine, J. B. Youssef, and B. J. van Wees, Nature Physics 11, 1022 (2015). [2] B. L. Giles, Z. Yang, J. S. Jamison, and R. C. Myers, Physical Review B 92, 224415 (2015). [3] R. L. Stamps, S. Breitkreutz, J. L. Akerman, A. V. Chu- mak, Y. Otani, G. E. W. Bauer, J.-U. Thiele, M. Bowen, S. A. Majetich, M. Kl aui, I. L. Prejbeanu, B. Dieny, N. M. Dempsey, and B. Hillebrands, Journal of Physics D: Ap- plied Physics 47, 333001 (2014). [4] G. Csaba, A. Papp, and W. Porod, Physics Letters A 381, 1471 (2017). [5] S. Klingler, P. Pirro, T. Br acher, B. Leven, B. Hille- brands, and A. V. Chumak, Applied Physics Letters 106, 212406 (2015). [6] A. Khitun, M. Bao, and K. L. Wang, Journal of Physics D: Applied Physics 43, 264005 (2010). [7] A. Khitun and K. L. Wang, Journal of Applied Physics 110, 034306 (2011). [8] A. V. Chumak, A. A. Serga, and B. Hillebrands, Nature Communications 5, 1 (2014). [9] T. Schneider, A. A. Serga, B. Leven, B. Hillebrands, R. L. 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[17] See Supplemental Material for the full form of the am- plitude factors in Eq. (8), the dispersions, the details of the magnetostatic calculation and the explicit form of the terms in the continuity equation for the spin in Eq. (9). [18] L. Fallarino, M. Madami, G. Duerr, D. Grundler, G. Gubbiotti, S. Tacchi, and G. Carlotti, IEEE Trans- actions on Magnetics 49, 1033 (2013). [19] A. V. Chumak, A. A. Serga, M. B. Jung eisch, R. Neb, D. A. Bozhko, V. S. Tiberkevich, and B. Hillebrands,6 Applied Physics Letters 100, 082405 (2012). [20] V. Cherepanov, I. Kolokolov, and V. L'vov, Physics Re- ports229, 81 (1993). [21] J. Zheng, S. Bender, J. Armaitis, R. E. Troncoso, and R. A. Duine, Physical Review B 96, 174422 (2017). [22] S. Geller and M. A. Gilleo, Journal of Physics and Chem- istry of Solids 3, 30 (1957). [23] B. R. Tittmann, Solid State Communications 13, 463 (1973). [24] A. Kreisel, F. Sauli, L. Bartosch, and P. Kopietz, The European Physical Journal B 71, 59 (2009). [25] M. Haertinger, C. H. Back, J. Lotze, M. Weiler, S. Gepr ags, H. Huebl, S. T. B. Goennenwein, andG. Woltersdorf, Physical Review B 92, 054437 (2015). [26] V. Sharma and B. K. Kuanr, Journal of Alloys and Com- pounds 748, 591 (2018). [27] K. Harii, Y.-J. Seo, Y. Tsutsumi, H. Chudo, K. Oy- anagi, M. Matsuo, Y. Shiomi, T. Ono, S. Maekawa, and E. Saitoh, Nature Communications 10, 1 (2019). [28] S. Vonsovskii and M. Svirskii, Soviet Physics, Solid State 3, 1568 (1962). [29] A. T. Levine, Il Nuovo Cimento 26, 190 (1962). [30] L. Zhang and Q. Niu, Physical Review Letters 112, 085503 (2014). [31] J. H. P. Colpa, Physica A: Statistical Mechanics and its Applications 93, 327 (1978).1 Supplemental Material: Electrical detection of unconventional transverse spin currents in obliquely magnetized thin lms MAGNETOSTATIC CALCULCATIONS Our goal is to derive the eigenfunctions for the thin lm geometry as depicted in Fig. 1 in the main text. We know from the numerics that the dipole-dipole interaction alone is sucient to give a transverse spin-current, so we ignore the exchange interaction in this derivation. This considerably simplies the work needed and allows us to nd a completely analytical expression for the eigenfunctions. We start from the Landau-Lifshitz-Gilbert equation (LLG) @tS(x;r;t) =S(x;r;t)h He S@tS(x;r;t)i ; (S1) wherer= (y;z). The classical ground state is ^n=hSi S= sinM^x+ cosM^z; (S2) with the angle Mis determined by Eq. (3) in the main text. We write the solution to the LLG as uctuations on this ground state with =1 Sp 2 ^a+i^b
S(r;t); (S3) where ^a;^bare orthogonal unit vectors chosen such that by ^a^b=^n. The eective magnetic eld is given by He=H+HD;HD=H(0) D+r; (S4) whereHDis the dipolar eld with a static component H(0) Dand a dynamic component, r.His the external eld. We transform to Fourier space with the relations (x;r;!) =Zd2k (2)2eik
r (x;k;!);  (x;r;!) =Zd2k (2)2eik
r(x;k;!): (S5) We only consider the situation where ky= 0, sok=k^z. Outside the lm the dynamics of the dipolar eld are governed by k2+@2 x
(x;k;!) = 0; xd 2(S6) which has solutions (x;k;!) =( d 2;k;!
ejkj(xd=2); xd 2;  d 2;k;!
ejkj(x+d=2); xd 2:(S7) The boundary conditions for at the top and bottom of the thin lm are @x(x;k;!) x=d 20++ 4MS1p 2" ^x
 ^ai^b  d 2;k;! +^x
 ^a+i^b  d 2;k;!# =jkj d 2;k;! (S8) and the bulk equation of motion for is k2+@2 x
(x;k;!) + 4MS1p 2h ^ai^b
(ik+^x@x) (x;k;!) + ^a+i^b
(ik+^x@x) (x;k;!)i = 0;jxjd 2:(S9)2 For the magnon eld we have the bulk equation of motion h (1 +i)!H
^n4MS(^x
^n)2i (x;k;!) +hD(x;k;!) = 0: (S10) This gives the solution (x;k;!) =G(!)hD(x;k;!); (S11) where G(!) =h (1 +i)!+H
^n+ 4MS(^x
^n)2i1 : (S12) For brevity we dene
G(!)G(!) +G(!): (S13) From the bulk equation of motion the solution for the potential is (x;k;!) =+eqx+eqx; (S14) where q=jkzjs a(k;!) b(k;!); (S15) with a(k;!) = 1 + 2Ms
G(!) sin2M; (S16) b(k;!) = 1 + 2MS
G(!) cos2M: (S17) From the boundary conditions in Eq. (S8) we then have the matrix equation (F+(k;!) +jkj)eqd 2(F(k;!) +jkj)eqd 2 (F+(k;!)jkj)eqd 2(F(k;!)jkj)eqd 2!+  = 0 (S18) where F(k;!) =iMS
G(!)kzsin (2M)q 2MS
G(!) cos2M+ 1
: (S19) The solutions for the potential are then +=(F(k;!) +jkj) (F+(k;!) +jkj)eqd(S20) which gives for the magnon eld (x;k;!) =G(!)" qcosM(F(k;!) +jkj) (F+(k;!) +jkj)eqxqd+eqx +ikzsinM (F(k;!) +jkj) (F+(k;!) +jkj)eqxqd+eqx# :(S21) BecauseF(k;!) depends linearly on kzsin (2M), the eigenfunctions for magnons travelling in kzdirections dier whenever sin (2 M)6= 0. This behaviour is in agreement with our numerics, which show that the dierence between the transverse spin-current induced by left- and right-moving spin waves vanishes if the magnetization is either completely in- or out of plane. Ultimately the source of the linear term is therefore the boundary conditions in Eq. (S8). Because the dipole-dipole interaction is a long-range interaction the boundary conditions interact with all the spin-waves in the thin lm, carrying the inversion breaking at the interface. This thus allows a transverse spin current to ow.3 0 10 20 k(µm−1)24681012E(GHz)(a) 0 10 20 k(µm−1)24681012(b) 0 10 20 k(µm−1)24681012(c) 0 10 20 k(µm−1)24681012(d) FIG. S1. Spin wave dispersion of a YIG lm with thickness d= 400a0:48m for increasingly tilted magnetic eld. The spin waves travel parallel to the in-plane projection of the magnetic eld, such that k=k^z. (a)H=M= 0, (b) H= 30;M= 18, (c)H= 60;M= 40and (d)H= 80;M= 64. DISPERSION We diagonalize the Hamiltonian in Eq. (8) in the main text, in the absence of damping and spin pumping, from which we obtain the spin-wave energies [31]. The spin-wave spectra are shown in Fig. S1 for multiple tilt angles of the magnetic eld, for spin waves propagating parallel to the in-plane projection of the magnetic eld, along the kz direction. The parameters used for these spectra are summarized in Table. I in the main text. We show the regime of wavevectors where both dipole-dipole interactions and the exchange interaction are of roughly equal magnitude. The exchange interaction dominates for large wavevectors and gives a quadratic wavevector dependence, curving the bands upwards. For small wavevector the dipole-dipole interaction is the dominant term in the Hamiltonian, which suppresses the quadratic behavior. Comparing these dispersion with both the numerical and experimental results [14] the general shape of the dispersions matches well, and the same shift down in energy is observed as the magnetic eld is tilted. COMPLETE AMPLITUDE FACTORS The amplitude factors in Eq. (8) in the main text are Ak(xij) =X rijeik
rA(xixj;r); =ij" cos (HM)h+SX n sin2MDxx 0(xin) + cos2MDzz 0(xin) + sinMcosMDxz 0(xin)
# S 2 cos2MDxx k(xij) +Dyy k(xij) + sin2MDzz k(xij)2 sinMcosMDxz k(xij) +SJk(xij);(S22) Bk(xij) =X rijeik
rB(xixj;r); =S 2h cos2MDxx k(xij)Dyy k(xij) + sin2MDzz k(xij)cosMsinMDxz k(xij) +isinMDyz k(xij)icosMDxy k(xij)i ; (S23) where Jk(xij) =J[ij(6j1jN2 cos(kya)2 cos(kza))ij+1ij1] (S24) andrij= (yij;zij).4 The dipole-dipole interaction is written as a tensor D k(xij) =X rijeik
rijD ij; (S25) where D ij=2(1ij)@2 @R ij@R ij1 jRijj: (S26) For small wavevectors the sums in Eq. (S25) are slowly converging, so we use the Ewald summation method as outlined by Kreisel et al. [24]. With this method the sums are split in two parts: one sum over real space and a one sum over reciprocal space. These sums are much faster to converge. We rst write the sums as a derivative of Ik(xij) =2X yij;zijei(kyyij+kzzij) (x2 ij+y2 ij+z2 ij)5=2; (S27) such that we have Dxx k=@2 @k2z+@2 @k2y+ 2x2 ij Ik(xij); (S28) Dyy k=@2 @k2z2@2 @k2yx2 ij Ik(xij); (S29) Dzz k=@2 @k2y2@2 @k2zx2 ij Ik(xij); (S30) Dxy k= 3ixij@ @kyIk(xij); (S31) Dxz k= 3ixij@ @kzIk(xij); (S32) Dyz k= 3@ @kz@kyIk(xij): (S33) Note the symmetries Dyy k=Dzz k(ky!kz;kz!ky) andDxz k=Dxy k(ky!kz;kz!ky), so we need not derive the full form of all dipolar sums. Then, after applying the Ewald summation, we have Dxx k(xij) =2 a2X g8p" 3pep2q2jk+gjf(p;q) 42 3r "5 X r jrijj23x2 ij
cos (kyyij) cos (kzzij)'3=2(jrijj2"); (S34) Dyy k(xij) =2 a2X g4p" 3pep2q2(ky+gy)2 jk+gjf(p;q) 42 3r "5 X r jrijj23y2 ij
cos (kyyij) cos (kzzij)'3=2(jrijj2"); (S35) Dxy k(xij) =i2 a2sig(xij)X g(ky+gy)f(p;q) +i4"5=22 pxijX rsin(kyyij) cos(kzzij)'3=2(jrijj2"); (S36) Dyz k(xij) =2 a2X g(ky+gy)(kz+gz) jk+gjf(p;q) + 4"5=22 pX ryijzijsin(kyyij)sin(kzzij)'3=2(jrijj2"); (S37)5 where '3=2(x) =ex3 + 2x 2x2+3pErfc (px) 4x5=2(S38) andq=xijp",p=jk+gj=(2p") andf(p;q) =e2pqErfc(pq) +e2pqErfc(p+q). The sums are either over the real space lattice or the reciprocal lattice, where the reciprocal lattice vectors are gy= 2m,gz= 2n,fm;ng2Z. "determines the ratio between the reciprocal and real sums. We choose "=a2, such that 2 pq1 and exp[2pq] converges quickly. CURRENT CONTRIBUTIONS In the continuity equation for the angular momentum in the main text, Eq. (9), the explicit form of the terms is I i(k;!) = 2iIm [b k(xi)@tbk(xi)] (S39) Ih i(k;!) =p 2SIm [hib k(xi)]; (S40) Iex i!j(k;!) =i(1ij)SJk(xij)b k(xi)bk(xj): (S41) Idipdip i!j(k;!) =i" (1ij)Adip k(xij)b k(xi)bk(xj) Bk(xij) 2bk(xi)bk(xj) +B k(xij) 2b k(xi)b k(xj)# ; (S42) whereAdip k(xij) =Ah=J=0 k (xij), i.e., only the contributions from the dipole-dipole interaction. Note that Bk(xij) already includes only dipole-dipole interactions.

2020-03-27

In a typical experiment in magnonics, thin films are magnetized in-plane and spin waves only carry angular momentum along their spatial propagation direction. Motivated by the experiments of Bozhko et al. [Phys. Rev. Research 2, 023324 (2020)], we show theoretically that for obliquely magnetized thin films, exchange-dipolar spin waves are accompanied by a transverse spin-current. We propose an experiment to electrically detect this transverse spin-current with Pt strips on top of a YIG film, by comparing the induced spin-current for spin waves with opposite momenta. We predict the relative difference to be of the order $10^{-4}$, for magnetic fields tilted at least $30^{\circ}$ out of plane. This transverse spin-current is the result of the long range dipole-dipole interaction and the inversion symmetry breaking of the interface.

Electrical detection of unconventional transverse spin-currents in obliquely magnetized thin films

2003.12520v4

arXiv:1711.07517v1 [cond-mat.mtrl-sci] 20 Nov 2017Temperature dependent relaxation of dipole-exchange magn ons in yttrium iron garnet films Laura Mihalceanu,1,∗Vitaliy I. Vasyuchka,1Dmytro A. Bozhko,1Thomas Langner,1 Alexey Yu. Nechiporuk,2Vladyslav F. Romanyuk,2Burkard Hillebrands,1and Alexander A. Serga1 1Fachbereich Physik and Landesforschungszentrum OPTIMAS, Technische Universit¨ at Kaiserslautern, 67663 Kaisersla utern, Germany 2Faculty of Radiophysics, Electronics and Computer Systems , Taras Shevchenko National University of Kyiv, 01601 Kyiv, U kraine (Dated: August 31, 2021) Low energy consumption enabled by charge-free information transport, which is free from ohmic heating, and the ability to process phase-encoded data by na nometer-sized interference devices at GHz and THz frequencies are just a few benefits of spin-wave-b ased technologies. Moreover, when approaching cryogenic temperatures, quantum phenomena in spin-wave systems pave the path towards quantum information processing. In view of these ap plications, the lifetime of magnons— spin-wave quanta—is of high relevance for the fields of magno nics, magnon spintronics and quantum computing. Here, the relaxation behavior of parametricall y excited magnons having wavenumbers from zero up to 6 ·105radcm−1was experimentally investigated in the temperature range f rom 20K to 340K in single crystal yttrium iron garnet (YIG) films epit axially grown on gallium gadolinium garnet (GGG) substrates as well as in a bulk YIG crystal—the m agnonic materials featuring the lowest magnetic damping known so far. As opposed to the bulk Y IG crystal in YIG films we have found a significant increase in the magnon relaxation rate be low 150K—up to 10.5 times the refer- ence value at 340K—in the entire range of probed wavenumbers . This increase is associated with rare-earth impurities contaminating the YIG samples with a slight contribution caused by coupling of spin waves to the spin system of the paramagnetic GGG subst rate at the lowest temperatures. The fields of spintronics and magnonics promote the realization of faster data processing technologies with lower energy dissipation by complementing or even replacing electron charge-based technologies with spin degree of freedom based devices [ 1–3]. Simultaneously, novel fascinating magnetic phenomena—such as, e.g., room-temperature Bose-Einstein magnon condensates [4–6], magnon vortices [ 7] and supercurrents [ 8–11]— open a whole new range of research areas [ 3,12] both for basic and applied spin physics. For these purposes many novel materials have been designed and investigated [13–15] whereupon one of the most outstanding ones so far is the insulating ferrimagnet yttrium iron garnet (Y3Fe5O12, YIG). Since its discovery in 1956, YIG has served as a prime example material for its microwave, optical, acoustic, and magneto-optical properties [ 16] in a wide range of experiments and applications. Nowadays, single crystal YIG films epitaxially grown on gadolinium gallium garnet (Gd 3Ga5O12, GGG) substrates [ 17–19] dominate in theoretical and experimental studies [ 5–8,20–24]. Their pertinence ranges from building of devices like microwave YIG oscillators, filters, delay lines, phase shifters, etc. [ 25] up to the latest high-profile research as in magnonics [ 26,27], spintronics [ 1,28] and quantum computing [ 29,30]. Consequently it has become clear that a deep understanding of the magnetic damping properties, determining the magnon lifetimes, is of crucial importance throughout these fields. Given its high Curie temperature at 560K, YIG is applicable at ambient temperatures, where its exceptional low Gilbertdamping parameter of down to 10−5[31] enables a long spin precession lifetime. Furthermore, in quantum computing YIG is used at very low temperatures for the coupling of single magnons to superconductive qubits in microwavecavities for the storage of information [ 32–36]. All of this in connection with arising demands on minia- turization of magnonic devices motivates our studies of the damping behavior in YIG films towards cryogenic temperatures in a wide range of spin-wave wavelengths. Up to now the temperature dependence of magnetic damping in YIG has been examined only for long- wavelength dipolar magnons with wavenumbers q→0 [37,38]. There exist numerous ways to measure the relaxation behavior of a precessing magnetic moment in different ranges of magnon wavenumbers. Among the most established are the technique of ferromagnetic resonance (FMR) [ 39], the measurement of thresholds of parametric excitation of magnetic oscillations and waves [40], the determination of spin-wave relaxation time from direct observation of magnetization decay by means of time-resolved Brillouin light scattering spectroscopy [5,41,42] and the magneto optical Kerr effect [ 43], the magnetic-resonance force microscopy [ 44], and echo- methods [ 45–47]. However, only parametric excitation techniques allow for the effective excitation and probing of dipolar-exchange magnons with wavenumbers up to q≤106radcm−1[48,49]. For example, the parametric pumping process of the first order, as described by Suhl [50] and Schl¨ omann et al.[51], resembles a form of spin-wave excitation when either a magnon of an externally driven magnetization precession or a photon2 Magnetic field H (Oe)(arb. units) 0 1 2 3-3 -2 -1h~ǁ ωp Pq⊥H q
H Frequency (2πGHz) ωp 2 ωFMR Hcy zxh~⊥H14 YIG8 qωp q⊥H(a) (c)(b) FIG. 1. (a) Sketch of the experimental setup. The spin sys- tem of a YIG sample, which is placed on top of a microstrip resonator, is driven by a microwave Oersted field with com- ponents oriented perpendicular ( h⊥ ∼) and parallel ( h/bardbl ∼) to the bias magnetic field H. (b) Schematic illustration of the para- metric pumping process in an in-plane magnetized YIG film. The transversal (red curve) and longitudinal (blue curve) lowest magnon branches are calculated for H= 1600Oe. The purple area contains the magnon branches with wavevectors lying in the film plane in the angle range between 0 and 90◦ relative to the field H. Two arrows show the splitting of a microwave photon in two magnons at half of the pumping fre- quencyωp/2. For the given bias magnetic field the magnons are excited on the transversal dispersion branch. (c) Depen - dence of the threshold power Pthrof parametric instability on the bias magnetic field Hmeasured in a 53 µm-thick YIG film at 60K. The minima of the threshold curve at H=Hc corresponds to the excitation of magnons with wavenumbers q→0 near the frequency of the ferromagnetic resonance. AtH < H cdipolar-exchange magnons corresponding to the transversal dispersion branch are directly excited by the parallel component h/bardbl ∼of the pumping Oersted field. For H > H cthe magnons from the purple spectral area (panel (b)) are excited bythe precessing magnetization drivenby t he perpendicular component h⊥ ∼of the pumping Oersted field. of a pumping microwave magnetic field with wavenum- bersqp≈0 splits into two magnons with opposite wavevectors qand−qat half of the pumping frequency ωp/2. Thus, a rather spatially uniform microwave magnetic field can generate short-wavelength magnons, whose wavenumbers are determined by the applied bias magnetic field Hand are bounded above only by the chosen pumping frequency ωp. In our experiments, we investigated parametrically ex- cited magnons in in-plane magnetized YIG films of thick- nesses of 5.6 µm, 6.7µm and 53 µm, which were epitax- ially grown in the (111) crystallographic plane on GGG substrates of 500 µm thickness. In addition, the GGG substrate was mechanically polished away from the orig- inally 53 µm-thick sample down to a 30 µm-thick YIG film. This sample was used to reveal a possible contribu-tionoftheinteractionbetweentheferrimagneticYIGand paramagnetic GGG spin systems to the magnon damp- ing. The YIG samples with lateral sizes of 1 ×5mm2 prepared by chemical etching on the 5 ×6mm2large GGG substrates were magnetized along their long axis to avoid undesirable influence of static demagnetizing on the value of the internal magnetic field. The experimental realization is provided by the mi- crowave setup shown in Fig. 1(a). The setup is attached on a highly heat-conducting AlN substrate at the bot- tom and is allocated inside a closed cycle refrigerator system. A microwave pumping pulse of 10 µs duration at a frequency ωpof 2π·14GHz with a 10ms repetition rate and a maximal pumping power Ppof 12W feeds a 50µm-wide microstrip resonator capacitively coupled to a microwave transmission line. The microwave Oersted fieldhpinduced by the resonator drives the magnetiza- tionofaYIG-filmsampleplaced ontopofthe microstrip. Subsequently the signal reflected by the resonator is for- warded to an oscilloscope. When the threshold field condition hp=hthris ful- filled, the action of the microwave Oersted field com- pensates the spin-wave damping and gives rise to the parametric instability process, where a selected magnon mode, which has the lowest damping and the strongest coupling to the pumping, grows exponentially in time. The arising mode increasingly absorbs the microwaveen- ergy accumulated in the pump resonator. This process detunes the resonator and, thus, changes the level of the reflected signal passed to the oscilloscope. As a result, a kink appearing at the end of the reflected pump pulse indicates the threshold microwave power Pp=Pthrre- quiredfortheparametricexcitationprocess[ 52].Pthrcan bedeterminedformagnonmodesoverthewide q-spectral range by changing the magnetic field H, which leads to a vertical shift of the dispersion curve (Fig. 1(b)) along frequency axis and results in a characteristic threshold curve shown in Fig. 1(c). In order to understand the shape of this curve one needs to consider that the overall threshold power Pthr is determined by instabilities of magnons excited by the components of the microwaveOersted field oriented both perpendicular h⊥ ∼(blue arrow in Fig. 1(a)) and parallel h/bardbl ∼(red arrow in Fig. 1(a)) to the bias magnetic field H[52]. At the critical field H=Hcspin waves with q→0 are excited near the frequency of the ferromag- netic resonance: ωp/2≈ωFMR. In Fig. 1(c) this situ- ation corresponds to the minima of the threshold curve Pthr(H). The threshold power at H≤Hcis dominated by direct parametric interaction of the parallel field com- ponenth/bardbl ∼with the lowest thickness mode corresponding to the transversal magnon dispersion branch (red curve in Fig.1(b)) [53]. As this mode is characterized by the largest precession ellipticity, the longitudinal component mzoftheprecessingmagneticmomentstronglyoscillates alongthedirectionofthemagneticfield Hwithfrequency3 0.10.1Threshold power Pthr (W) Wavenumber q Threshold power Pthr (W) (105 rad cm-1) FIG. 2. Threshold curves Pthr(H) at different temperatures in the range 340 −180K (a) and 180 −20K (c). (b) Wavenumber in the wide temperature range. All present data is recorded a nd calculated for a 53 µm-thick YIG film grown on top of a GGG substrate. ωpand thus effectively couples with the parallel compo- nenth/bardbl ∼of the pumping field. With decreasing external magnetic field, the threshold power slowly increases due to an increase in wavenumbers of the excited magnons and a related decrease in the precession ellipticity [ 53]. The strong increase in Pthrat the magnetic field Hbe- low 100Oe is caused by transition of the homogeneously magnetized YIG film to a multi-domain state. AboveHcnomagnonswithwavevectors q⊥Hexistat ωp/2 and the parametricpumping excites magnonsprop- agating at angles θq<90◦relative to the field H. These magnons escape the narrow pumping area above the mi- crostripresonator(Fig. 1(a))andtherelatedenergyleak- age results in the sharp jump up in the threshold power justabove Hc. Thisconfinementeffecttogetherwithgen- eral reduction in the precession ellipticity caused by the decrease of θqleads to a further transition from the par- allel to the perpendicular pumping regimes for H > H c [52]. Finally, the threshold power Pthr→ ∞when the bottom of the magnon spectrum is shifted above ωp/2. For determining the magnon relaxation behavior the pumping regime H≤Hcis of main interest in this re- port as the wavenumbers of the parametrically excited magnons can be unambiguously calculated in this case.Henceforth we approximate hp≃h/bardbl ∼. Figure2presents the dependencies Pthr(H) recorded for a number of temperatures in the range from 340K to 180K (Fig. 2(a)) and from 180K to 20K (Fig. 2(c)) for the 53 µm-thick film. The dotted arrows indicate the shift of both the critical threshold power Pthr(Hc) and Hcwith temperature. One can see that Fig. 2(a) shows adecrease in the threshold power with decreasing tem- perature from 340K to 180K. On the contrary, Fig. 2(c) reveals a strong increase in the threshold power with fur- ther temperature decrease from 180K to 20K. At the same time, the experimentally determined critical field Hcmonotonically decreases towards lower temperatures along the whole temperature range. This decrease of Hcrelates to an upward frequency shift of the magnon spectrum caused by a temperature dependentincreaseinthesaturationmagnetization4 πMs as well as by changes of the cubic Hc aand uniaxial Hu a anisotropy fields of the YIG film [ 54,55]. The field de- pendenceofthewavevectorspectralrangefortheperpen- dicular spin-wave branch can be calculated using Eq.7.9 from Ref. [ 55]: ω=γ/radicalbig (H+Dq2)(H+Dq2+4πMs−Hca−Hua),(1)4 whereω=ωp/2,γ= 1.76·107Oe−1s−1the gyro- magnetic ratio, and the nonuniform exchange constant D= 5.2·10−9Oecm2are considered to be not varying with temperature [ 56]. The difference 4 πMs−Hc a−Hu a is defined from the measured values of Hc(T,q= 0). An expected demagnetizing effect caused by a stray mag- netic field induced at low temperatures in YIG films by the paramagneticGGG substrate can be neglected in our case of laterally extended samples [ 57]. The calculated dependencies of the magnon wavenum- berq=q(H) for different temperatures are shown in Fig.2(b). The vertical dashed lines in Fig. 2correlate the threshold curves with the corresponding wavenum- ber atHc. As is shown, in our experiment spin waves are probed by parametric pumping in the wavenumber range from zero to 6 ·105radcm−1. The variation of the saturation magnetization directly affectsthecouplingbetweenthemicrowavepumpingfield /vectorh/bardbl ∼and the longitudinal component mzof the precess- ing magnetic moment M. As a result, the threshold field hthris influenced by two temperature dependent phys- ical quantities: the spin-wave relaxation rate and the parametric coupling strength. These influences can be estimated using the relation for the threshold field [ 55]: hthr= min/braceleftbiggωp∆Hq ωMsin2θq/bracerightbigg , (2) whereωM=γ4πMs, andθqis the angle between the magnon wavevector qand the magnetization direction. For the parametric excitation near and above the FMR frequency( H≤Hc), we canapproximate θq≈90◦. ∆Hq is the width of a linear resonance curve of the parametri- cally excited magnon mode with the wavenumber q. It is defined as ∆ Hq= 1/(γTq) = Γq/γ, whereTqand Γqare the magnon lifetime and the spin-wave relaxation rate. It is known that the saturation magnetization 4 πMs for bulk YIG crystals demonstrates a rather non-linear change with temperature [ 58], which can be calculated using the two-sublattice model described in Ref.[ 59] as it is shown by the red solid line in Fig. 3(a). By-turn, the temperature behaviorof the cubic anisotropyfield can be approximated [ 57] as Hc a=Hc a(0)+αT3 2, (3) withHc a(0) =−147Oe and α= 2.175·10−2OeK−1.5. The slopes of the 4 πMs(T)−Hc a(T)−Hu a(T) curves determined for all films at room temperatures are in good agreement with previously reported results [ 49,60]. However, due to the unknown contribution of the uni- axial anisotropy [ 61], which is caused by a tempera- ture dependent mismatch between YIG and GGG crys- tal lattices, the calculated temperature dependencies for both 4πMsand for 4 πMs−Hc a(dashed blue line in Fig.3(a)) significantlydivergefromthe experimentaldif- ference 4 πMs−Hc a−Hu a(see, e.g., the data for the53µm-thick YIG film shown by red circles in Fig. 3(a)). At the same time, the substrate-free YIG sample of 30µm thickness prepared from the 53 µm-thick YIG film demonstrates good agreement between experimentally measured (empty blue circles) and theoretically calcu- lated valuesof4 πMs−Hc a. In the caseofthe thinner YIG films the experimental data for 4 πMs−Hc a−Hu afollow the general trend of the calculated saturation magnetiza- tion 4πMs(see Fig. 3(a)). This agreement evidences the applicability of the chosen model for our YIG films and allows us to use the theoretical magnetization values for the calculation of the temperature dependent parametric coupling strength. By assuming initially the value of ∆ Hqin Eq.2to be constant over the entire temperature range and tak- ing into account the theoretical values of 4 πMs(T) we have calculated the normalized (with respect to 340K) temperature dependence of the threshold field, which is solely determined by the change in the parametric cou- pling strength. This dependence is shown by the circles in Fig.3(b). Theexperimental threshold field hexp thr, which contains information about the relaxation of parametrically ex- cited magnons, can be found from the measured thresh- old powers using the relation hexp thr=C√Pthr. The value ofCdepends on the pumping frequency ωp, the geom- etry and the quality factor of the pumping resonator. As the resonance frequency and the quality factor of our microstrip resonator do almost not change with temper- ature we assume Cto be constant. Theexperimentalvaluesofthedimensionlessthreshold fieldnormalizedtothe referencevalueatthe temperature of 340K are plotted in Fig. 3(b) (squares) for magnons excited near the FMR frequency ( H=Hc). Its behav- ior is visibly non-monotonic: down to 180K the thresh- old field hexp thrslightly decreases, while below 180K it in- creases up to 6.5 times compared to the reference value. The comparisonof the calculated ( hthr(T), circles)and experimental ( hexp thr(T), squares) threshold dependencies clearly evidences that at high temperatures ( T≥180K) the experimental dependencies are mostly determined by the variation in the parametric coupling strength. On the contrary, the strong increase of hexp thrin the low- temperature range ( T <180K) is caused by the spin- wave relaxation. Figure3(c) shows the normalized relaxation rate Γ q calculated at Hcwith help of Eq. 2usinghexp thr(T) and theoretically calculated 4 πMs(T). It becomes evident that for the temperature decrease from 180K to 20K the relaxation rate Γ qincreases up to about 10.5 times for the 53 µm-thick YIG film while the thinner films ex- hibit the same trend. The same relaxation behavior, as it is clear from the nearly wavenumber-independent ver- tical shift of all threshold curves (see, e.g., Fig. 2), is ob- served in all range of probed magnon wavenumbers up to 6·105radcm−1. The strong increase of the relaxation5 M (T)   M (T   M (T    )u 642 0     1   YIG(5.6)/GGG(500) YIG(6.7)/GGG(500)YIG(53)/GGG(500) YIG(30)M M (T) 10(a) (b) (d) (c) 10 YIG(53)/GGG(500)Relaxation rate Relaxation rate YIG(bulk) ultrapure FIG. 3. (a) Saturation magnetization plotted as a function o f temperature compared to theoretical calculations. (b) Te m- perature dependence of the threshold pumping field for magno ns parametrically excited near the FMR frequency at H=Hc. Squares – the hexp thrvalues are determined using the measured threshold powers Pthr. Circles – the hthrvalues are calculated using Eq. 2for the experimentally determined values of 4 πMS(T) on the assumption that ∆ Hq=const. (a) and (b) are determined for the 53 µm-thick sample. (c) Normalized relaxation rate obtained fo r YIG films of the thicknesses of 5.6 µm, 6.7µm and 53 µm epitaxially grown on a GGG substrate of 500 µm thickness. (d) Comparison of the normalized relaxation rates of 53 µm-thick YIG on GGG, 30 µm-thick substrate-free YIG and an ultrapure bulk YIG sample measured at H=Hc. The number in brackets corresponds to the material layer thi ckness in micrometers. rate is considered to be atypical for pureYIG, for which a monotonic decrease of Γ qis expected with decreasing temperature [ 62]. The revealed relaxation behavior at low temperatures can be related either to the contribution of fast-relaxing rare-earth ions contaminating the chemical composition of YIG [63,64] or to the magnetic losses caused by the dipolar coupling of magnons with the spin-system of the paramagnetic GGG substrate [ 65,66]. In order to clar- ify the origin of the increased relaxation we replicate our measurements on the 53 µm-thick YIG sample after pol- ishing the GGG side down to a 30 µm substrate-free YIG film. The comparison of the relaxation rate is shown in Fig.3(d). Both YIG film samples demonstrate a strong increase in the magnon relaxation rate for decreasing temperatures, starting from approximately 150K. This fortifies the assumption of the prevailing contribution of fast-relaxing rare-earth ion impurities in epitaxial YIG films at lowtemperatures. Belowapproximately80Kthe relaxation rate of the 53 µm-thick YIG sample increases fasterincomparisonwiththepolishedsubstrate-freeYIG film. This difference can be attributed to coupling of YIG’s ferrimagnetic spin system with the electron spins of Gd3+ions of paramagneticGGG. The coupling is sup- posed to be proportional to 1 /T[65] and leads to addi-tional low-temperature energy losses for all YIG films placed on GGG substrates. For comparison, we measured the temperature- dependent magnon damping in an impurity- and GGG- free bulk YIG sample. Similar to the experiments with YIGfilms, thesemeasurements,whichwereperformedby meansoftheparallelparametricpumpingtechniqueinan ultrapure YIG crystal of the size of (1 ×1×3)mm3, show the same damping behavior in a wide range of magnon wavenumbers. However, in contrast to the experiment with YIG films the relaxation rate Γ qmonotonically de- creases with decreasing temperature. It is clearly shown by the line denoted by semi-filled circles in Fig. 3(d). It further supports our assumption about the significant in- fluence of chemical contaminations on magnon damping in epitaxial YIG films at cryogenic temperatures. In conclusion, in the temperature range from 20K to 340K we have investigated the relaxation of parametri- cally excited dipole-exchange magnons in YIG films of 5.6µm, 6.7µm, and 53 µm thickness grown on a GGG substrate by liquid phase epitaxy. We have found that at cryogenic temperatures the magnon lifetime strongly decreases for all film thicknesses. By comparing the substrate-free YIG with the YIG/GGG samples the ob- served relaxation behavior could be related to the mag-6 netic dampingcausedby the couplingofmagnonsto fast- relaxing rare-earth ions inside the YIG film ( T<∼150K) and to the paramagnetic spin system of GGG substrates (T<∼80K). Comparison of these results with the data obtained from the ultrapure bulk YIG crystal shows that in order to sustain a long magnon lifetime low- temperature magnetic experiments in YIG must be per- formed in chemically-pure and substrate-free samples. Financial support from the Deutsche Forschungsge- meinschaft (project INST 161/544-3 within SFB/TR49, projects VA 735/1-2 and SE 1771/4-2 within SPP 1538 “Spin Caloric Transport”, and project INST 248/178-1) is gratefully acknowledged. ∗mihalcea@rhrk.uni-kl.de [1] K. Sato and E. Saitoh, Spintronics for Next Generation Innovative Devices (Wiley, Chichester, 2015). [2] V.V. Kruglyak, S.O. Demokritov, and D. Grundler, Magnonics ,J. Phys. D: Appl. Phys. 43, 264001 (2010). [3] A.V. Chumak, V.I. Vasyuchka, A.A. Serga, and B. Hille- brands,Magnon spintronics ,Nat. Phys., 11, 453 (2015). [4] S.O. Demokritov, V.E. Demidov, O. Dzyapko, G.A. Melkov, A.A. Serga, B. Hillebrands, and A.N. 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2017-11-20

Low energy consumption enabled by charge-free information transport, which is free from ohmic heating, and the ability to process phase-encoded data by nanometer-sized interference devices at GHz and THz frequencies are just a few benefits of spin-wave-based technologies. Moreover, when approaching cryogenic temperatures, quantum phenomena in spin-wave systems pave the path towards quantum information processing. In view of these applications, the lifetime of magnons$-$spin-wave quanta$-$is of high relevance for the fields of magnonics, magnon spintronics and quantum computing. Here, the relaxation behavior of parametrically excited magnons having wavenumbers from zero up to $6\cdot 10^5 \mathrm{rad~cm}^{-1}$ was experimentally investigated in the temperature range from 20 K to 340 K in single crystal yttrium iron garnet (YIG) films epitaxially grown on gallium gadolinium garnet (GGG) substrates as well as in a bulk YIG crystal$-$the magnonic materials featuring the lowest magnetic damping known so far. As opposed to the bulk YIG crystal in YIG films we have found a significant increase in the magnon relaxation rate below 150 K$-$up to 10.5 times the reference value at 340 K$-$in the entire range of probed wavenumbers. This increase is associated with rare-earth impurities contaminating the YIG samples with a slight contribution caused by coupling of spin waves to the spin system of the paramagnetic GGG substrate at the lowest temperatures.

Temperature dependent relaxation of dipole-exchange magnons in yttrium iron garnet films

1711.07517v1

arXiv:2305.09270v1 [physics.optics] 16 May 2023Tunable all-optical logic gates based on nonreciprocal top ologically protected edge modes Jie Xu1,2,7, Panpan He3,4, Delong Feng1,2, Yamei Luo1,2, Siqiang Fan5, Kangle Yong1,2,8, Kosmas L. Tsakmakidis6,9 1School of Medical Information and Engineering, Southwest Medical University, Luzhou 646000, China 2Medical Engineering & Medical Informatics Integration and Transformational Medicine of Luzhou Key Laboratory, Luzhou 646000, China 3Luzhou Key Laboratory of Intelligent Control and Applicati on of Electronic Devices, Luzhou Vocational & Technical College, Luzhou 646000, Chin a 4School of Electrical and Electronic Engineering, Luzhou Vocational & Technical College, Luzhou 646000, Chin a 5Chongqing Key Laboratory of Photo-Electric Functional Mat erials, Chongqing 401331, China 6Section of Condensed Matter Physics Department of Physics N ational and Kapodistrian University of Athens Panepistimioupolis, At hens GR-157 84, Greece 7xujie011451@163.com 8Kangle@swmu.edu.cn and 9ktsakmakidis@phys.uoa.gr Abstract All-optical logic gates have beenstudiedintensively fort heirpotential toenablebroadband,low-loss, and high-speed communication. However, poor tunability ha s remained a key challenge in this field. In this paper, we propose a Y-shaped structure composed of Yt trium Iron Garnet (YIG) layers that can serve as tunable all-optical logic gates, including, bu t not limited to, OR, AND, and NOT gates, by applyingexternal magnetic fields to magnetize the YIG lay ers. Our findingsdemonstrate that these logic gates are based on topologically protected one-way ed ge modes, ensuring exceptional robustness against imperfections and nonlocal effects while maintainin g extremely high precision. Furthermore, the operating band of the logic gates is shown to be tunable. I n addition, we introduce a straightfor- ward and practical method for controlling and switching the logic gates between ”work”, ”skip”, and ”stop” modes. These findings have important implications fo r the design of high-performance and precise all-optical integrated circuits. 1I. INTRODUCTION Since the invention of the transistor in 1947, human society has exp erienced an unprece- dented boom in electronic communications based on electrical signals to meet the needs of everyday life and scientific research [1, 2]. However, with the deve lopment of integrated cir- cuits, transistors are becoming increasingly miniaturized, resulting in increased energy waste. Additionally, electronic communication still suffers from defects suc h as high error rates and cross-talk [3]. On the other hand, optical communication has advan tages such as high-speed signal processing, error-free transmission [4], parallel computat ion [5], and low loss [6], making it a potential candidate for the next-generation communication te chnology. In recent decades, the concept of integrated optical circuits has been introduced, g reatly developed, and studied. All-optical logic gates (LGs) are an important component of integra ted optical circuits and have received considerable attention in recent years, with inte resting results in this field. Researchers have constructed various types of all-optical LGs, such as photonic crystal and Mach-Zehnder interferometer structures, using nonlinear proc esses [7–9] and/or interferome- try [10–13], and have implemented all basic logic operations. However , most LGs suffer from low contrast ratios (CRs), typically less than 30 dB. This is understa ndable because reflec- tions are unavoidable in conventional optical LGs, and imperfection s in their manufacturing affect the accuracy of the gates to some extent, particularly in no nlinearity-based LGs [14–16]. In many studies on sub-wavelength all-optical LGs, researchers o ften neglect the impact of nonlocal effects on logical operations. While this is generally true in ne ar-wavelength cases, non-local effects should be considered when the device’s scale is sub wavelength or even deep- subwavelength. In fact, the impact of nonlocal effects on nonrec iprocal/one-way surface mag- netoplasmons (SMPs) has been widely discussed in the past several years [17–19]. SMPs are edge modes sustained in magneto-optical (MO) heterostructure s, and many interesting and meaningful results, such as slow light [20–22], overcoming the time- bandwidth limit [23], and rainbow trapping [24, 25], have been discovered. Recently, we prop osed a method to imple- ment (sub-wavelength) all-optical logic operations using one-way S MP modes [26]. This type of one-way electromagnetic (EM) mode has been proven to be topolog ically protected [27, 28] in the microwave regime by several research groups, and no significa nt impact of non-local effects has been observed. Therefore, in this paper, we focus on such no nlocality-immune SMPs to study tunable LGs. Additionally, since guided wave modes have only on e transmission direc- tion, the problem of preparation process defects is well overcome , and unidirectional modes are immune to backscattering. More importantly, all-optical LGs ba sed on unidirectional EM modes theoretically have an infinite contrast ratio, which means unp aralleled accuracy. Note that in Ref.[26], the designed all-optical LGs relied on Yttrium Iro n Garnet (YIG) with remanence. Consequently, although unidirectional SMPs-bas ed all-optical LGs were im- plemented using MO heterostructures, their lack of tunability hinde red their application in future integrated optical circuits. In this paper, we propose a Y- shaped structure composed of three YIG layers under different bias magnetic fields and theoretica lly analyze the dispersion relation in the three arms, which are all YIG-YIG heterostructure s. We observe interesting 2phenomena, such as reverse propagation direction, and close and /or reopen one-way regions. More importantly, we discover highly tunable characteristics of the Y-shaped structure and the LGs, which are confirmed by full-wave simulation. Our proposed ( subwavelength) tunable LGs have the potential to be applied in the design of high-performan ce and programmable integrated optical circuits. II. PHYSICAL MODEL AND TOPOLOGICALLY PROTECTED SMPS The Y-shaped configuration is a commonly used physical model in the field of all-optical LGs, which has been extensively studied in recent decades [11, 29–3 2]. In Fig. 1(a), we propose a Y-shaped YIG-based model that enables tunable all-optical logic o perations. The model comprises three straight arms, each containing two layers of YIG. Unlike our previous work [26], where YIG with remanence was used, all the YIG layers in this stu dy are subjected to an external magnetic field (H 0) to further enhance the tunability of the LGs. It should be noted that metals can always be considered as perfect electric con ductor (PEC) walls in the microwave regime [33]. For simplicity, as shown in Fig. 1(b), the arm with YIG layers having the same magnetization is referred to as ’EYYE-s’, where ”E” repre sents the PEC boundary, ”Y” represents YIG, and ”s” symbolizes the same magnetization dir ection. Similarly, the structure with YIG layers having opposite magnetization directions is labeled ’EYYE-r’. To achieve basic logic operations based on one-way modes, the key is to establish two separate one-way channels that allow efficient transfer of the EM wave/signa l. The question then arises as to how to design suitable arms and how to efficiently tune the struc ture according to our needs. input-1 input-2outputd One-way channel-1 One-way channel-2(a)(b) (c)EYYE-rEYYE-s metal YIGxy d H0 A BCAB C A C B C FIG. 1. (a) The schematic of the Y-shaped structure of all-op tical logic operations. (b) Two types of arms are shown, i.e. the ’EYYE-s’ and the ’EYYE-r’. (c) Pre-d esigned two one-way channels in our proposed structure. Note that, in this paper, we use ωa 0,ωb 0, andωc 0to clarify the procession angular frequencies ( ω0) for green-colored YIG, yellow-colored YIG and blue-color ed YIG layers, respectively. To achieve this, one must first study the dispersion relation of the S MPs in those arms. The ’EYYE-r’ contains two layers of YIG with two different relative perme ability and for the lower 3(¯µa) and upper (¯ µb) YIG, we have ¯µa= µa 1−iµa 20 iµa 2µa 10 0 0 1 ,¯µb= µb 1iµb 20 −iµb 2µb 10 0 0 1 (1) whereµ1= 1+ωm(ω0−ivω) (ω0−ivω)2−ω2andµ2=ωmω (ω0−ivω)2−ω2.ω,ωm,νandω0=µ0γH0refer respectively to the angular frequency, the characteristic circular frequency , the damping factor, and the procession angular frequency[33]. Please note that the superscr ipts ’a’ and ’b’ represent the lower and upper layers, respectively. In this paper, we assume tha t the magnetic-field direction in the lower layer is permanently oriented in the -z direction. By applyin g Maxwell’s equations and three boundary conditions in the ’EYYE-r’ arm, one can easily ca lculate the dispersion relation of the SMPs sustained on the YIG-YIG interface. The dispe rsion relation takes the following form µa v/bracketleftbiggµb 2 µb 1k+αb tanh(αbd)/bracketrightbigg +µb v/bracketleftbiggµa 2 µa 1k+αa tanh(αad)/bracketrightbigg = 0 (2) whereαa=/radicalbig k2−εmµa vk2 0,αb=/radicalbig k2−εmµb vk2 0, andµv=µ1−µ2 1/µ2. Equation (2) re- veals that the SMPs in the ’EYYE-r’ arm exhibit different propagation properties for opposite wavenumbers, i.e., k1=−k2, which is a well-known nonreciprocity effect. More importantly, adjusting the external magnetic field can create a special one-wa y region where the waves prop- agate in only one specific direction. The asymptotic frequencies (AF s) of the SMPs in the ’EYYE-r’ arm can be derived and calculated from Eq. (2). We found f our AFs, which can be described by the following equations: ω(+) sp= ω(+1) sp=ωa 0+ωm ω(+2) sp=ωb 0+ωm(3) ω(−) sp= ω(−1) sp=(ωa 0+ωb 0+ωm)+√ (ωa 0+ωb 0+ωm)2−2(2ωa 0ωb 0+ωa 0ωm+ωb 0ωm) 2 ω(−2) sp=(ωa 0+ωb 0+ωm)−√ (ωa 0+ωb 0+ωm)2−2(2ωa 0ωb 0+ωa 0ωm+ωb 0ωm) 2(4) ω+ spandω− spindicate the AF as k→+∞andk→ −∞, respectively. In fact, the value of ω+ sp corresponds to the zero point of µa vorµb v. Similarly, the dispersion relation of the SMPs in the ’EYYE-s’ arm can be directly obtained from Eq. (2) by replacing µb 2,µb 1,µb v, andαbwith−µc 2, µc 1,µc v, andαc, respectively. In this case, the permeability (¯ µc) of the upper YIG has the same form as ¯µa, and the corresponding dispersion equation can be written as follow s: µa v/bracketleftbigg−µc 2 µc 1k+αc tanh(αcd)/bracketrightbigg +µc v/bracketleftbiggµa 2 µa 1k+αa tanh(αad)/bracketrightbigg = 0 (5) 4k/km50 /g90//g90m 0 -50012 /g90/g3 s/g90/g3 sp (+) /g90/g3 sp (-) k/km50 /g90//g90m 0 -50012(a) k/km50 0 -50(b) (c) [0.4,0.3,r] [0.6,0.3,r][0.6,0.4,s] 10 010 210 410 6 0 10 20 30 |E| (V/m) x (mm)(d)/g90//g90m 012 air hole[0.6,0.4,s][0.4,0.3,r] [0.6,0.3,r] A BC FIG. 2. (a-c) The dispersion diagrams of three arms are shown , in which the lower YIG has ω0values of 0.6ωm, 0.4ωm, and 0.6ωm, respectively, while the upper YIG has ω0values of 0 .3ωm, 0.3ωm, and 0.4ωm”, respectively. Note that ’r’ and ’s’ indicate the ’EYYE-r’ arm and ’EYYE-s’ arm, respectively, and the magnetization orientation of the lower YIG is perman ently -z. The cyan lines represent the edge of the bulk zones, while the black arrows indicate th e location of ω=ωs. Stars show the corresponding asymptotic frequencies in each case. (d) The simulated electric field distributions of the three cases are shown for f= 0.8fm. The other parameters are (a-c) d= 0.02λm,ν= 0 and (d) ν= 0.001ωm. There are also four potential AFs in the ’EYYE-s’ arm, which have th e following form: ω(+) sp= ω(+1) sp=ωa 0+ωm ω(+2) sp=(ωc 0−ωa 0)+√ (ωc 0−ωa 0)2+2(2ωa 0ωc 0+ωa 0ωm+ωc 0ωm) 2(6) ω(−) sp= ω(−1) sp=ωc 0+ωm ω(−2) sp=(ωa 0−ωc 0)+√ (ωc 0−ωa 0)2+2(2ωa 0ωc 0+ωa 0ωm+ωc 0ωm) 2(7) Based on Eqs. (4) and (5), we plot the dispersion curves for the SM Ps in both the ’EYYE-r’ and’EYYE-s’ arms as d= 0.02λm(λm= 2πc/ωm=c/fm)andν= 0 (lossless condition). Three different values of ω0(H0) are applied in the three arms, and for convenience, we introduce a simple notation- ’[ α,β,θ]’ - inwhich αandβrepresent the absolute values of the normalized ω0 ( ¯ω0=ω0/ωm) for the lower and upper YIG, while θcould be either ’r’ referring to the ’EYYE-r’ arm or ’s’ referring to the ’EYYE-s’ arm. For example, [0.6, 0.3, r] in F ig. 2(a) implies that the dispersion curve is calculated in the ’EYYE-r’ arm, where ωa 0= 0.6ωmandωb 0= 0.3ωm. In 5Fig. 2(a), the green and blue stars represent ω(+) spandω(−) sp, respectively. The red and black lines indicate the dispersion curves of SMPs on the YIG-YIG interfac e, and due to the deep- subwavelength thickness of the YIG layers, the bulk zones are sign ificantly compressed [24, 34]. Therefore, it is believed that almost all the SMPs on the red and black lines are one-way EM modes except for the SMPs located near the resonant frequencie s of YIG ( ωs=/radicalbig ω2 0+ω0ωm), which are marked by black arrows. As depicted in the inset of Fig. 2(a ), the case of [0.6, 0.3, r] can be treated as one of the input arms (arm ’A’) of the Y-shaped heterostructure. We also calculate the dispersion curves for the other arms in Figs. 2(b) and 2(c). As a result, similar to the first case, there are two one-way regions in both cases. Ho wever, in the case of [0.6, 0.4, s], the EM waves within the lower one-way region have negative group velocities ( vg<0). Based on Eqs. (3), (4), (6), and (7) as mentioned earlier, the one -way regions are defined by the AFs (green and blue stars in Fig. 2). For the three cases disc ussed above, the regions are: (a) [0.428 fm, 1.3fm] and [1.472 fm, 1.6fm], (b) [0.3475 fm, 1.3fm] and [1.3525 fm, 1.4fm], and (c) [0.766 fm, 0.966fm] and [1.4 fm, 1.6fm]. Therefore, to design two one-way channels, the frequencies used must be located within the [0.766 fm, 0.966fm] region (the red line region in Fig. 2(c)). In addition, the loss effect and the robustness of the one-way propagation of SMPs are examined using full-wave simulations, as illustrated in Fig. 2(d ). In this case, we consider ν= 0.001ωmandf= 0.8fm, and air holes with a radius of r= 0.5 mm (∼0.008λm) are placed on the YIG-YIG interface. The simulation results show go od agreement with the theoretical analysis, and the imperfections have a negligible impact o n the one-way SMPs. It is also worth noting that recent studies have questioned the robus tness of one-way SMPs, with the nonlocal effect being a major focus of these works[17, 18]. He re, we emphasize that the one-way SMPs studied in this paper are theoretically topologically pro tected, which has been theoretically demonstrated[35, 36] and experimentally proved[27, 37] by many groups. This nonlocality-immune property is particularly evident in cases where th e waveguide is relatively thick or the wavenumber (k) is relatively small[19]. Our proposed logic g ates in this paper are believed to be largely unaffected by nonlocal effects, given the tuna bility of the SMPs, which will be discussed in the next subsection. III. TUNABLE ALL-OPTICAL LOGIC GATES Inourtheoreticalanalysis, wehaveshownthataY-shapedstruc tureconsistingofmagnetized YIG layers can support two independent one-way channels, making it suitable as a logical gate[26]. More importantly, benefiting from the tunability of the top ologically protected one- way SMPs, the proposed LGs should be easily tunable by changing the bias magnetic fields. In the following sections, we demonstrate the tunability of our prop osed logical gates in detail. Firstly, we study the impact of H 0on AFs, which always define the one-way regions. As displayed in Fig. 3(a), four AFs in the ’EYYE-r’ arm ’A’ (’C’), are plotte d as a function of ωa 0(ωc 0) andωb 0. Figure 3(b) depicts the similar relationship between AFs and ωaandωc. To differentiate between the four distinct AFs in Eqs. (3, 4) (’EYYE-r’) and (6, 7) (’EYYE-s’), 6/g90//g90m 01sp 2 01 10.2 0.4 0.6 0.8 0.5 /g90/g3/g3/g3/g3//g90m 0a/c /g90/g3/g3/g3/g3//g90m 0b 01 10.2 0.4 0.6 0.8 0.5 /g90/g3/g3/g3/g3//g90m 0a /g90/g3/g3/g3/g3//g90m 0c/g90sp (-1)/g90sp (-2)/g90sp (+1)/g90sp (+2)(a) (b) (c)/g90//g90m sp 012 /g90/g3/g3/g3/g3//g90m 0b[0.6,x,r] 0 1 0.2 0.4 0.6 0.8 /g90//g90m sp 012 /g90/g3/g3/g3/g3//g90m 0c0 1 0.2 0.4 0.6 0.8[0.6,x,s] /g90//g90m sp 01 /g90/g3/g3/g3/g3//g90m 0b0 0.1 0.02 0.04 0.06 0.08[0.1,x,r] 2(d) (e)EYYE-r EYYE-s V > 0g V < 0g/g90/g3/g3/g3/g3 //g90m 0c /g90/g3/g3/g3/g3//g90m 0b0 0.6(f) /g90/g3/g3/g3/g3 = 0.60a/g90m 0.10.6 0.10.6 0.1 -0.02 -0.22 -0.15 -0.45 0 0.60.10 -0.2 0 -0.25 (f-1) (f-2)(f-3) (f-4) FIG. 3. (a,b) The asymptotic frequencies (AFs) are plotted a s a function of ωa 0(orωc 0) andωb 0for (a) the ’EYYE-r’ arm and (b) the ’EYYE-s’ arm. (c-e) AFs are plott ed as a function of ωb 0when (c,d) ωa 0= 0.6ωmand (e)ωc 0= 0.1ωm. (f) Four constructed equations ( y1,y2,y3, andy4) as shown in Eq. (8) are plotted as functions of ωb 0andωc 0whenωa 0= 0.6ωm. we use the names ω(+1) sp,ω(+2) sp,ω(−1) sp, andω(−2) sp. Notably, as ω0(H0) changes, the values of AFs and their numerical relationships may also change. This can lead to a r eversal of the group velocity and the transmission direction of EM signals in LGs. Therefor e, any changes in the AFs can affect the functionality of the LGs. To illustrate the changes in AFs and one-way regions, we set the lowe r YIGω0to 0.6ωm and assume 0 < ω0< ωmfor the upper YIG in both ’EYYE-r’ (Fig. 3(c)) and ’EYYE-s’ (Fig. 3(d)) arms. As ω0(ωb 0) of the upper YIG varies from 0 to 0 .6ωm, the lower one-way region gradually widens, while the upper one-way region becomes smaller and eventually closes at ωb 0= 0.6ωm. The black dashed line represents the [0.6, 0.3, r] case discussed ea rlier in Fig. 2(a), in which two clear one-way regions are present (excluding the local area near ω=ωs). Asωb 0increases further, for the ’EYYE-r’ arm, the first one-way regio n is compressed slightly, while a new one-way region bounded by ω(−1) spandω(+2) spemerges with a forward propagation direction ( vg>0).The inset of Fig. 3(c) displays a zoomed-in dispersion curve for th e case of ωb 0= 0.8ωm(blue line). In contrast, the ’EYYE-s’ arm behaves differently. As s hown in Fig. 3(d), when ωc 0(in the upper YIG) is increased, the propagation direction of SMPs in the lower one-way region changes from backward ( vg<0) to forward ( vg>0), and the one-way region closes and reopens. Similar phenomena of reversing propagation dir ection and close-reopen one-way regions are observed in the higher regime as well. The black a nd blue dashed lines in Fig. 3(d) indicate cases where ωc 0= 0.4ωmandωc 0= 0.8ωm, respectively, with ωa 0= 0.6ωm in both cases. The insets in Fig. 3(d) show the reversed one-way re gions and the dispersion curves of SMPs. 7The question of whether the group velocity will reverse in one-way s ystems can be answered by determining if the system’s symmetry or chirality is broken. As dem onstrated in Figure 3(c), within the lower one-way region of the ’EYYE-r’ arm, the SMPs can propagate only in the forward direction, regardless of which layer has a higher H 0(ω0). We consider two conditions, [0.6, 0.4, r] and [0.4, 0.6, r], where the propagation direct ions are the same. This is becausethesecondcasecanbetreatedastheentiresystemoft hefirstcaserevolving180degrees around the propagation direction, and thus the system’s symmetr y/chirality is conserved. In contrast, [0.6, 0.4, s] and [0.4, 0.6, s] have opposite propagation dir ections because they cannot be obtained by simply rotating each other, and thus the system’s sy mmetry/chirality is broken when changing ω0accordingly. Toachieve arelatively broadone-wayband, itisnecessary that ωc 0inarm’B’issmall enough, as shown in Figure 3(d). Thus, we select ωc 0= 0.1ωm(marked by the red dashed line in Figure 3(d)), and Figure 3(e) depicts the corresponding AFs and one-wa y regions as functions of ωb 0. For this case, ω(−2) sp≃0.048ωmandω(+2) sp=ωm+ωb. Withωa 0= 0.6ωmandωc 0= 0.1ωmfixed, the only remaining unknown parameter in the Y-shaped structure is ωb 0. Ideally, we aim for the whole one-way region with vg<0 in arm ’B’ to be the working band of the LGs. Based on our calculations, we can achieve this goal if 0 < ωb 0<0.31ωm. However, in most cases, to ensure that the entire one-way region of arm ’B’ is the working band of LGs, we need to ensure that ω(−2) sp(the blue line in Figure 3(d)) and ω(+2) sp(the green line in Figure 3(d)) are both inside the one-way regions with vg>0 in arms ’A’ and ’C’. To accomplish this, we construct the following equations: y1= (ω(−2) spB−ω(−2) spA)(ω(−2) spB−ω(+2) spA) y2= (ω(−2) spB−ω(−2) spC)(ω(−2) spB−ω(+2) spC) y3= (ω(+2) spB−ω(−2) spA)(ω(−2) spB−ω(+2) spA) y4= (ω(+2) spB−ω(−2) spC)(ω(−2) spB−ω(+2) spC)(8) where ’A/B/C’ represent arm ’A’/’B’/’C’. In this context, it is worth no ting that arms ’A’ and ’C’ belong to the ’EYYE-r’ type, while arm ’B’ belongs to the ’EYYE-s’ t ype. The AFs are represented by ω(+) spandω(−) sp, which are given by Eqs. (3), (4), (6), and (7). Equation (8) determines whether ω(−2) spin arm ’B’ lies within the one-way region of arm ’A’, which occurs fory1<0. Ify1<0,y2<0,y3<0, andy4<0 at the same time, it means that the entire one-way region with vg<0 in arm ’B’ lies within the one-way regions of both arms ’A’ and ’C’. Figure 3(f) represents the functions of y1((f-1)),y2((f-2)),y3((f-3)), and y4((f-4)) based on ωb 0andωc 0whenωa 0= 0.6ωm. We observe that y1,y2, andy4are always negative, while y3can be positive for relatively large ωb 0and small ωc 0. Therefore, we set ωa 0= 0.6ωmandωc 0= 0.1ωm, and keep ωb 0relatively small, such as ωb 0= 0.1ωm(marked by red balls in Fig. 3(f)), to ensure that the entire one-way region in arm ’B’ corresponds to the workin g band of LGs. Figures 4(a)-4(c) present dispersion curves for the scenario wh ereωa 0= 0.6ωmandωb 0= ωc 0= 0.1ωm. Similar to Fig. 2, there is a one-way region with vg>0 in arm ’A’ (depicted as a red-line region in Fig. 4(a)) and arm ’C’ (depicted as a red-line reg ion in Fig. 4(b)). 8Additionally, there is a one-way region with vg<0 in arm ’B’ (depicted as a red-line region in Fig. 4(c)). Moreover, the backward one-way region is much large r than that illustrated in Fig. 2(c). Consequently, the working band of the LGs in this situatio n should be significantly broader. Figs. 4(d) and 4(f) show the coupling effect between arm s whenf= 0.8fm, which falls within the one-way regions of interest. Consequently, two one-wa y channels (’A-C’ and ’B-C’) are established, while the EM signal cannot propagate from arm ’A’ t o arm ’B’. It should be noted that the first part ([0.1, 0.6, s]) of the ’B-C’ channel differs f rom that of the ’A-B’ channel ([0.6, 0.1, s]) due tothe geometrical relationship between the arms. As per symmetry, the SMPs in the [0.1, 0.6, s] and [0.6, 0.1, s] structures must have opposite pro pagation directions. In the simulations, the EM signal can transfer efficiently in the one-way cha nnels, while the forward transferring signal halts at the interface of arm ’A’ and arm ’B’. k/km50 /g90//g90m 0 -50012 (a) (b) (c)[0.6,0.1,r] [0.1,0.1,r] [0.6,0.1,s] k/km50 /g90//g90m 0 -50012k/km50 /g90//g90m 0 -50012 (d) [0.6,0.1,r]-[0.6,0.1,s][0.6,0.1,r]-[0.1,0.1,r] 10 010 210 410 6 0 10 20 30 |E| (V/m) x (mm)[0.1,0.6,s]-[0.1,0.1,r] [0.6,0.1,s] [0.1,0.1,r] [0.1,0.1,r](e) [0.6,0.1,r] [0.6,0.1,r] [0.1,0.6,s]BC A A B C BA C FIG. 4. (a,b) Dispersion diagrams of three arms with optimiz ed parameters, ωa 0= 0.6ωm,ωb 0= 0.1ωm andωc 0= 0.1ωm. (d,e) The simulated electric field distribution obtained f rom coupling simulations containing two arms, with each arm being either the ’EYYE-r’ type or ’EYYE-s’ type. 9(a) (c) input-1 input-2output(b) input-1 input-2outputOR operation AND operation NOT operation ‘1’: robust one-way signal ‘0’: no signal ‘0’: robust one-way signal ‘1’: no signal Positive logic! Negative logic! input: positive logic output: negative logica output: positive logic input: negative logicbOR operation input-1 input-2 output ‘1’ ‘0’ ‘1’ ‘1’ ‘1’ ‘0’ AND operation input-1 input-2 output ‘0’ ‘1’ ‘0’ ‘0’ ‘0’ ‘1’ NOT operation-a input-1 input-2 output ‘1’ ‘0’ ‘1’ ‘0’air hole FIG. 5. (a) Theory of all-optical logic operation using the p ositive and/or negative logic. (b) Numerical simulations in the Y-shaped module as f= 0.8fm, and air holes with r= 0.5 mm were set on the YIG-YIG interfaces to verify the robustness of logic o perations. (c) The truth tables of the OR, AND, and NOT operations. IV. REALIZATION OF BASIC LOGIC GATES The Y-shaped structure that is designed canfunction within the on e-way region andperform as basic logic gates, including, but not limited to, OR, AND, and NOT gat es, as presented in Fig. 5. During logical operations, arms ’A’ and ’B’ are considered as t wo input ports, with arm ’C’ regarded as the output port. Primarily, the structure ope rates as a natural OR gate, where any input one-way EM signal can and must propagate to the o utput port. If we consider the presence of the EM signal as logic ’1’ and the absence of the EM s ignal as logic ’0,’ i.e., positive logic, then the Y-shaped structure functions as a broad O R gate that can be adjusted by external magnetic fields. Negative logic (where the presence of the EM signal is recognized as logic ’0’) is used for the AND gate and NOT gate. In the AND operatio n, any input EM signal is treated as logic ’0,’ resulting in the output EM signal also being logic ’0.’ However, the NOT operation employs negative logic in either the input or output port, with positive logic used in the remaining port. Figure 5(b) depicts simulations for th e Y-shaped structure withf= 0.8fmwhen the EM signal is excited in only one of the input ports. Air holes wit h r= 0.5 mm were set as imperfections on the YIG-YIG interface to show th e robust function of our proposed LGs. As a result, the LGs work fine and perform ex tremely high CR which is larger than 200 dB (infinity in theory). Figure 5(c) shows the corr esponding truth tables of the OR, AND, and NOT operations. The Y-shaped structure is designed to function within the one-way region and can serve as basic logic gates, including but not limited to OR, AND, and NOT gates, a s illustrated in Fig. 5. During logical operations, arms ’A’ and ’B’ are the input ports, wh ile arm ’C’ is the output port. The structure operates as a natural OR gate, where any in put one-way EM signal must 10+0.6 /g90/g3 m -0.1 /g90/g3 m+0.1 /g90/g3 m(a) (b) logic gate (“work”) logic gate (“stop”) logic gate (“stop”) logic gate (“skip”) +0.6 /g90/g3 m +0.6 /g90/g3 m+0.6 /g90/g3 m+0.1 /g90/g3 m +0.1 /g90/g3 m+0.1 /g90/g3 m-0.1 /g90/g3 m -0.1 /g90/g3 m-0.1 /g90/g3 m /g90/g3/g3/g3/g30c/g90/g3/g3/g3/g30a /g90/g3/g3/g3/g30b(c) logic gate (“work”) logic gate (“stop”) logic gate (“skip”) logic gate (“stop”) ‘0’ ‘1’‘1’ ‘0’‘0’ ‘0’ ‘1’ ‘1’ FIG. 6. (a) Switch theory for tunable LGs. The red arrow refer s to the electromagnetic wave path that allows passage. (b,c) Simulation for tunable LGs by swi tching the external magnetic fields for input signals (b) [’1’, ’0’] and (c) [’0’, ’1’]. Input ’1’ cou ld be alternatively programmed to ’0’ (”stop” mode) or ’1’ (”skip” or ”work” mode.) propagate to the output port. If we assume that the presence o f the EM signal is logic ’1’ and its absence is logic ’0’ (positive logic), then the Y-shaped struct ure functions as a versatile OR gate that can be externally adjusted by magnetic fields. Howeve r, the AND and NOT gates use negative logic, where the presence of the EM signal is rec ognized as logic ’0.’ In the AND operation, any input EM signal is treated as logic ’0,’ resulting in th e output EM signal also being logic ’0.’ On the other hand, the NOT operation employs nega tive logic in either the input or output port, with positive logic used in the remaining port . Figure 5(b) shows simulations of the Y-shaped structure with f= 0.8fmwhen the EM signal is excited in only one of the input ports. Imperfections on the YIG-YIG interface w ere introduced as air holes withr= 0.5 mm to demonstrate the robustness of our proposed LGs. As a re sult, the LGs exhibits high CR of over 200 dB (infinity in theory). Figure 5(c) prese nts the corresponding truth tables for the OR, AND, and NOT operations. As mentioned earlier in Fig. 3, we can control the LGs based on the Y- shaped structure with external magnetic fields (H 0). By changing the direction(s) of H 0, we can reverse the propagation direction of the one-way SMPs, as illustrated by the low er one-way regions in Fig. 3(c,d), such as [0.6, 0.4, r] transitioning to [0.6, 0.4, s]. Another nota ble case is that changing the direction(s) of H 0cancause the previous one-way regionto close, such as[0.6, 0.6, r] shifting towards [0.6, 0.6, s]. In this scenario, there areno one-way regions , andthe entire band becomes a band gap, preventing the propagation of EM signals. Additionally, a ltering the value of H 0, either by increasing or decreasing it, can significantly affect the logic operations. Therefore, the operating band of our proposed LGs can be easily tunned by changin g the external magnetic fields. Besides, we suggest an innovative approach to tune LGs, as demon strated in Fig. 6. This method achieves three modes of LGs by switching H 0orω0, namely the ”work,” ”stop,” and 11”skip” modes. As initial magnetic-field parameters, we set ωa 0= 0.6ωm,ωb 0=−0.1ωm, and ωc 0= 0.1ωm. It is noteworthy that the ’-’ sign indicates the external magnetic field is in the +z direction. Upon exchanging ωb 0(Hb 0) andωc 0(Hc 0), it is easy to calculate that SMPs with f= 0.8fmhave opposite propagation directions in the original arms. Similarly, e xchanging ωc 0 (Hc 0) andωa 0(Ha 0) reverses the propagation direction of SMPs in arm ’B,’ whereas the direction remains unchanged in other arms. Moreover, exchanging ωa 0(Ha 0) andωb 0(Hb 0) reverses the propagation direction of SMPs in arm ’A,’ whereas the direction remain s unaltered in other arms. The simulation of the three modes of LG is illustrated in Figs. 6(b ) and 6(c). The orig- inal mode is designated the ”work” mode since it can work as LGs. Two methods can achieve the ”stop” mode with EM signals being halted, and one method can acc omplish the ”skip” mode with EM signals skipping the present calculation. Therefore, ou r proposed Y-shaped LGs offers rich manipulation possibilities and is promising for programma ble optical commu- nication/devices. In contrast, traditional all-optical LGs typically operate at fixed frequencies and can be challenging to tune. V. CONCLUSION In summary, we have devised a Y-shaped structure made of YIG lay ers with distinct mag- netizations. The arms of the structure were categorized into two types: the ’EYYE-r’ type with opposing magnetization directions and the ’EYYE-s’ type with ide ntical magnetization directions. Our theoretical analysis of the ’EYYE-r’ and ’EYYE-s’ a rms led to the construction of two one-way channels capable of supporting topologically protec ted one-way SMPs. Further- more, the implementation of basic logic gates, such as OR, AND, and N OT gates, is achieved through these broadband and topological one-way SMPs, resultin g in highly robust (resistant to backscattering and imperfections) and precise (theoretically in finite contrast ratio) LGs. In addition, we explored the tunability of these LGs. By adjusting exte rnal magnetic fields, the one-way region can be easily modulated, either broadened or narro wed, the propagation direc- tions of the SMPs within the region can be completely reversed, or th e region can be closed. Given the intriguing tunability of the operating band of the Y-shaped LGs. In addition, we proposed a potential application for the structure/LGs: by switc hing external magnetic fields, three switchable modes (”work”, ”skip”, and ”stop”) can be achie ved. Our proposed LGs, based on magnetized YIG, may pave the way for tunable all-optical lo gic operations and hold promise for high-efficiency programmable optical communication circ uits. ACKNOWLEDGEMENT This work was supported by the National Natural Science Foundat ion of Sichuan Province (No. 2023NSFSC1309), andtheopenfundofLuzhouKey Laborat oryof Intelligent Control and Application of Electronic Devices (No. ZK202210), Sichuan Science a nd Technology Program (No. 2022YFS0616), the Science and Technology Strategic Coope ration Programs of Luzhou 12Municipal People’s Government and Southwest Medical University (N o. 2019LZXNYDJ18). J.X., K.Y., and Y.L. thanks for the support of the Innovation Labora tory of Advanced Medical Material&PhysicalDiagnosisandTreatmentTechnology. K.L.T.was supportedbytheGeneral Secretariat for Research and Technology (GSRT) and the Hellenic F oundation for Research and Innovation (HFRI) under Grant No. 4509. 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2023-05-16

All-optical logic gates have been studied intensively for their potential to enable broadband, low-loss, and high-speed communication. However, poor tunability has remained a key challenge in this field. In this paper, we propose a Y-shaped structure composed of Yttrium Iron Garnet (YIG) layers that can serve as tunable all-optical logic gates, including, but not limited to, OR, AND, and NOT gates, by applying external magnetic fields to magnetize the YIG layers. Our findings demonstrate that these logic gates are based on topologically protected one-way edge modes, ensuring exceptional robustness against imperfections and nonlocal effects while maintaining extremely high precision. Furthermore, the operating band of the logic gates is shown to be tunable. In addition, we introduce a straightforward and practical method for controlling and switching the logic gates between "work", "skip", and "stop" modes. These findings have important implications for the design of high-performance and precise all-optical integrated circuits.

Tunable all-optical logic gates based on nonreciprocal topologically protected edge modes

2305.09270v1

Current-induced switching of YIG/Pt bilayers with in-plane magnetization due to Oersted elds Johannes Mendil,1,)Morgan Trassin,1Quingquing Bu,1Manfred Fiebig,1and Pietro Gambardella1 Department of Materials, ETH Zurich, 8093 Zurich, Switzerland (Dated: 25 April 2019) We report on the switching of the in-plane magnetization of thin yttrium iron garnet (YIG)/Pt bilayers induced by an electrical current. The switching is either eld-induced and assisted by a dc current, or current- induced and assisted by a static magnetic eld. The reversal of the magnetization occurs at a current density as low as 105A/cm2and magnetic elds of 40T, two orders of magnitude smaller than in ferromagnetic metals, consistently with the weak uniaxial anisotropy of the YIG layers. We use the transverse component of the spin Hall magnetoresistance to sense the magnetic orientation of YIG while sweeping the current. Our measurements and simulations reveal that the current-induced eective eld responsible for switching is due to the Oersted eld generated by the current owing in the Pt layer rather than by spin-orbit torques, and that the switching eciency is in uenced by pinning of the magnetic domains. The possibility of manipulating the magnetization of planar structures using electrical currents opens excit- ing perspectives in spintronics. Electrical currents can aect the magnetization of thin lms through the Oer- sted magnetic eld,1{5spin transfer torques,6and spin- orbit torques.7Previous work has focused on magneti- zation switching and domain wall dynamics induced by spin-orbit torques in metallic ferromagnets adjacent to a heavy metal layer.8{15Recently, investigations extended towards insulating ferrimagnetic garnets, which, owing to the low magnetic damping, are particularly appealing for generating and transmitting spin waves16{19as well as for magnetization switching.20{22The most prominent expo- nent of this material class is yttrium iron garnet (YIG). Extensive work on the interplay of current-induced ef- fects and magnetization dynamics in YIG/Pt bilayers demonstrated ecient spin-wave excitations,23{27spin- wave amplication,28,29and the control of magnetiza- tion damping.30So far, however, no attempt at current- induced magnetization switching of YIG has been re- ported. Two plausible reasons for the scarcity of results in this area are the extreme sensitivity of YIG to mag- netic elds, which makes it dicult to control the in- termediate magnetization states, as well as to the need to utilize YIG lms with uniaxial in-plane anisotropy, which is required to achieve binary switching. Indeed, the electrical switching of garnet insulators has been re- ported only for thin lms with relatively large perpen- dicular anisotropy, such as thulium iron garnet layers in combination with either Pt or W.20{22 In this paper, we investigate the reciprocal eects of current and magnetic eld on the switching of YIG/Pt bilayers with in-plane magnetic anisotropy. We demon- strate eld-induced switching assisted by a dc cur- rent as well as current-induced switching assisted by a static magnetic eld at extremely low current density (105A/cm2) and bias elds (40 60T). We fur- ther show that the magnetization reversal can be sensed )Electronic mail: johannes.mendil@mat.ethz.chelectrically by measuring the transverse component of the spin Hall magnetoresistance (SMR)31{33and adding an ac modulation to the dc current inducing the switching. Current and thickness dependent measurements reveal that the eective switching eld is consistent with the Oersted eld generated by the current owing in the Pt layer. No signicant eect of spin-orbit torques was de- tected in the current range from 1 to 8 105A/cm2in- vestigated in this work. Our results are relevant for the operation of YIG-based spintronic devices at very low current density in the thin lm regime. YIG layers with thickness between 6 and 7 nm were grown epitaxially by pulsed laser deposition on (111)- oriented gadolinium gallium garnet substrates, followed by in-situ magnetron sputtering of a 3 nm thick poly- crystalline Pt lm with a sheet resistance of 160 . For electrical measurements, the samples were patterned into Hall bars using optical lithography followed by Ar-ion milling [Fig. 1 (a)]. The current line is 50 m wide and is oriented along the [1 10] crystal direction of the substrate. The separation between two consecutive Hall arms is 500m. The YIG layers have in-plane magnetization with saturation value Ms= (1:00:2)105A/m, which is smaller by about 30% compared to the Msof bulk YIG. This reduced Ms, typical for very thin YIG, is assigned to the diusion of Gd atoms from the substrate into YIG.34 In addition to the shape anisotropy, the layers have a rather strong easy plane anisotropy, corresponding to an eective isotropic anisotropy eld of about 75 mT, and a weaker in-plane uniaxial anisotropy, corresponding to an in-plane anisotropy eld BK4050T, which is not correlated to a specic crystal direction. The ori- gin of the uniaxial in-plane anisotropy in the epitaxial YIG(111) layers is attributed to local strain variations in- troduced during the microfabrication process. A detailed structural and magnetic characterization of our samples is reported in Ref. 34. To sense the magnetic orientation and current-induced eective elds, we performed harmonic Hall voltage measurements,7,35whereby an ac current with a fre- quency of 10 Hz and current density j= 105A/cm2isarXiv:1904.10517v1 [cond-mat.str-el] 23 Apr 20192 sent through the Hall bar while the transverse resistance is acquired and decomposed into its harmonic compo- nents. To derive the orientation of the in-plane magneti- zation, it is sucient to consider the rst harmonic Hall resistanceRxyas a function of the direction of the ex- ternal magnetic eld Bext. The azimuthal angles of Bext and magnetization are 'Band', respectively, dened with respect to the current direction. The correspond- ing polar angles are Band[see Fig. 1 (a)]. Bextis measured by a calibrated Hall sensor placed next to the sample, without correction for the earth's magnetic eld. All experiments are performed at room temperature. Figure 1 (b) shows Rxyof YIG(6 nm)/Pt(3 nm) mea- sured as a function of 'BforBext= 7 mT (green curve) and 60T (black curve). As =B==2, the Hall resistance is determined by the planar Hall-like contribu- tion from the SMR31,32 Rxy=R?sin(2'); (1) whereR?denotes the transverse SMR coecient. If the magnetization is saturated parallel to the eld, we have that'='BandRxy=R?sin(2'B), in agreement with the measurement performed at Bext= 7 mT. Con- versely, for Bext= 60T, that is, comparable or smaller thanBK, we observe signicant deviations from the sat- urated behavior. These deviations consist in a reduction of the signal amplitude, due to '6='B, and two abrupt jumps separated by 180. We attribute these jumps to the sudden switch of the magnetization from the positive to the negative direction (relative to the easy axis) as Bextcrosses the hard axis, consistently with the uniaxial in-plane anisotropy of our lms. In order to support this hypothesis and quantify BK, we performed macrospin simulations based on the mag- netic energy functional E=M
Bext+MsBKsin2(''EA)M
BI;(2) where the rst two terms on the right hand side cor- respond to the Zeeman energy and uniaxial in-plane anisotropy energy, respectively, and the last term repre- sents the interaction between the magnetization Mand the current-induced magnetic eld BI, which we will dis- cuss later on. Minimization of Efor a given set of 'Bat constantBextandBI= 0 yieldsBKand a set of values ', which we use to simulate Rxyusing Eq. (1). The best t between simulations and data is achieved for BK= 40T and an easy axis 'EA= 63. TheRxycurves calculated using these parameters are shown in Fig. 1 (c) for dier- ent values of Bext. The simulations reproduce fairly well the main features of the Hall resistance measurements, namely the lineshape, the amplitude and position of the jumps, and their separation by 180. We thus conclude that the macrospin model is appropriate to describe the behavior of the magnetization, at least in the Hall cross region probed by Rxy. SinceBextandBKare in the range of tens of T, we expect that any additional current-induced eld BIshould have a pronounced impact on the orientation of the magnetization, even for very small current densities. To prove this point, we added a dc oset to the ac cur- rent and measured Rxyat low eld as a function of 'B. For a dc oset of 8 105A/cm2, we observe that the angle'Bat which the magnetization switches shifts by an amount
'. The sign of
'depends on the polarity of the dc current, as shown by the red and blue curves in Fig. 1 (b). Such a shift is attributed to the action of a dc eldBI, which assists Bextsuch as to favor or hinder the switching of the magnetization in proximity of the hard axis [Fig. 1 (d)]. Accordingly, in the rst hemicycle (0'B<180), a negative (positive) current shifts the magnetization reversal towards smaller (larger) 'B, whereas, in the second hemicycle (180'B<360), the opposite eect occurs. (d) jB BIDj BextMx yz(a) 0 90 180 270 360 j (deg)B-20R (mW) xy20(b) 0 20 10 150 170330 350 j (deg)B(e)5 2 j(10 A/cm ) dc -8-7-6-5 +5+6+7+8 0 I- I+I-0 90 180 270 360 j (deg)B-20R (mW) xy20(c) 0 20 10 150 170330 350 j (deg)B(f) B(mT)I -20-15-10-5 5101520 0 B<0I B>0IB<0IHAR (mW) xy FIG. 1. (a) Schematics of the YIG/Pt Hall bar with the coordinate system. (b) Rxyof YIG(6 nm)/Pt(3 nm) measured as a function of 'BatBext= 7 mT (green line) and 60 T (black line). The red an blue lines are measured at Bext= 60T in the presence of a dc oset of 8 and -8 105A/cm2, respectively. (c) Macrospin simulations of the data shown in (b). (d) Diagram showing the combined eect of BIand Bexton magnetization switching in proximity of the hard axis (HA). (e) Detail of the shift of Rxyas a function of dc oset and (f) macrospin simulations. In order to quantify BI, we performed a series of mea- surements for positive and negative dc osets, shown in3 Fig. 1 (e). We then tted the Rxycurves using the en- ergy functional from Eq. (2) while keeping BKand'EA equal to the values determined in the absence of a dc cur- rent andBIas the only free parameter. The simulations, shown in Fig. 1 (f), reproduce well the current-dependent switching observed in Fig. 1 (e). Overall, the model sup- ports the presence of a eld BIkyfor a dc current jdckx, which has the same symmetry as the Oersted eld expected from the current owing in the Pt layer. The current dependence of BI, reported in Fig. 2, further shows that BIscales linearly as a function of jdcand that its amplitude is comparable with the Oersted eld calcu- lated from Amp ere's law as BOe=0jdctPt=20:19 mT forjdc= 107A/cm2(thin black line), where tPtis the thickness of Pt and 0denotes the vacuum permeability. 0 2 4 6 8 10 |j| dc5 2 (10 A/cm )B (mT)I10 0 -10 4 6 8 29 9000.10.20.3 t (nm)YIG7 -2 B /j (mT/10 Acm ) FL+Oe DC induced shift BOe(b) j >0dc BOe fitsj <0dc harmonic Hall FIG. 2. Current dependence of BIin YIG(6 nm)/Pt(3 nm) for positive and negative dc osets. The red and blue lines are linear ts to the data. The thin black lines show the Oersted eld calculated from Amp ere's law. The presence of uniaxial in-plane anisotropy and the niteBIallow us to switch the YIG magnetization by ramping the dc current in Pt. To enable the current- induced switching, we select a conguration in which the magnetization is bistable, namely the hysteretic region ofRxyshown by the red curve in Fig. 3 (a). We thus xBext= 34T at'B= 160when sweeping from 360to 160which corresponds to the point indicated by the dashed line in Fig. 3 (a). In this conguration, the magnetization is tilted towards the hard axis. We then ramp the dc current towards positive values and si- multaneously record Rxy[red curve in Fig. 3 (b)]. From our former analysis, we expect that BIinduces a tilt
'that will eventually lead to switching. Indeed, when reachingjdc= 5105A/cm2, we observe a step-like decrease of Rxyindicating the reversal of the magneti- zation, followed by a parabolic-like increase of Rxyat higher current, which we assign to a tilt of the magne- tization in areas close to the Hall cross that have not switched. When sweeping the current back to zero, Rxy remains in the low resistance level (black curve). More- over, the resistance switches back to the initial value at jdc=2105A/cm2. This behavior is similar to that reported for the current-induced switching of strained GaMnAs layers, with the dierence that BIin GaMnAsoriginates from spin-orbit coupling rather than by the Oersted eld.36 Figures 3 (c) and (d) further show that the switching is reproducible for a sequence of positive and negative current pulses. In particular, the high and low levels ofRxyreproduce the full excursion of the Rxysignal atBext= 34T [red curve in Fig. 3 (a)] and persist at zero dc current conrming the remanent character of the switching. Moreover, applying two consecutive pulses with the same current polarity does not lead to an additional increase or decrease of Rxy, suggesting that the switching occurs between well-dened magnetization states, suggesting that the reversal process involves a ma- joritary and reproducible portion of the magnetic layer in the proximity of the Hall cross. Additional eects due to Joule heating are neglected, since the temperature in- crease derived from measurements of the resistivity dur- ing current injection is lower than 1 K. 0 90 180 270 360-4-202 -10 -5 0 5 101234 -808R (mW) xy j (deg) B 4.8 mT 34 mT fwd bkw(b) c)( 0 8 16 24 32 time (sec)24(d)(a) 0 90 180 270 360-4-2024 -10 -5 0 5 101234 24 0 8 16 24 32-808Rxy (mW) jB (deg) 4.8 mT 340 mT jdc (105 A/cm2) up down upRxy (mW) jdc (105 A/cm2) time (sec) R (mW) xy5 2 j (10 A/cm ) xc5 2 j (10 A/cm ) dc FIG. 3. (a) Rxyof YIG(7 nm)/Pt(3 nm) as a function of 'B atBext= 4:8 mT (black line) Bext= 34T (red line). The dashed line indicates the value of 'Bused for current-induced switching. (b) Rxyduring a forward (red curve) and backward dc current sweep (black curve). (c,d) Current sequence and Rxymeasured at 'B= 160andBext= 34T. Before concluding, we discuss the origin of the current- induced eld BI. As seen in Fig. 2, BIis only slightly smaller than BOe, suggesting that BIis dominated by the Oersted eld, with possibly a small opposing spin- orbit eective eld at the interface with Pt.7This con- clusion is consistent with earlier work on the current- induced ferromagnetic resonance of YIG/Pt bilayers.37 As the Oersted eld acts on the entire magnetic vol- ume of YIG, we also expect that BIdoes not depend on the YIG thickness ( tYIG). Measurements of the an- gular shifts
'as a function of tYIG, however, give val-4 ues ofBIthat vary signicantly between tYIG=3.5 nm and 7 nm, and nally saturate to about 0.05 mT/(107 Acm2) fortYIG9 nm (dotted circles in Fig. 4), which is much smaller than BOe0:19 mT/(107Acm2) (gray line in Fig. 4). Whereas the increase of BIbe- tweentYIG=3.5 nm and 4.5 nm can be attributed to a reduction of the interfacial spin-orbit eective eld, which has opposite direction relative to BOeand scales as 1/tYIGMs, the monotonic decrease of BIobserved at tYIG>4:5 nm has apparently no explanation. Further- more, harmonic Hall voltage measurements7,35ofBIper- formed on thick YIG samples yield values of BIthat are consistent with BOe(dotted triangles in Fig. 4), in clear contrast with BIdetermined from the angular shifts of the hysteretic Rxycurves. This apparent discrepancy can be reconciled by taking into account the pinning of do- main walls, which in uences the magnetization reversal and hence the values of BIdetermined using the angular shift method. Indeed, x-ray photoelectron emission mi- croscopy shows that the domain morphology of YIG un- dergoes a transition around tYIG= 9 nm, changing from an irregular elongated pattern to 100 m-wide pinned zigzag domains.34We therefore conclude that BIorigi- nates mostly from the Oersted eld, and that its eect on the magnetization is highly sensitive to the local pin- ning eld, which depends strongly on tYIG. Finally, we note that the harmonic Hall voltage measurements were not feasible in the thinner samples due to additional ef- fects overlapping with the Oersted eld and spin-orbit torques, which are likely due to the small coercivity of the layers and prevent a reliable analysis of the data. 0 2 4 6 8 10 |j| DC5 2 (10 A/cm )B (mT) FL+Oe10 0 -10 4 6 8 29 9000.10.20.3 t (nm)YIGB (mT)I DC induced shift BOe(b)j>0DC j<0DCj >0DC BOe fitsj <0DC harmonic Hall(a) FIG. 4. Thickness dependence of the current-induced eective eldBImeasured by the angular shift method (dotted circles) and harmonic Hall voltage measurements (dotted triangles). The data are shown for a current density jdc= 107A/cm2. In summary, we have shown that the current-induced eective eld BIis sucient to reversibly manipulate the direction of the magnetization in YIG/Pt bilayers with in-plane anisotropy in the presence of a weak static ex- ternal eld. In YIG lms thicker than 4 nm, BIis con- sistent in sign and magnitude with the Oersted eld gen- erated by the current owing in the Pt layer. Current- induced switching is achieved at an extremely small cur- rent density (2105A/cm2), which is two orders of mag-nitude smaller compared to the dc current switching of metallic ferromagnets such as Pt/Co38, ferrimagnets such as Pt/GdCo39, and even thulium iron garnet/Pt.20We attribute this dierence to the extremely small uniax- ial anisotropy and depinning eld of YIG compared to ferro- and ferrimagnets with perpendicular anisotropy. The switching eciency decreases in lms thicker than 7 nm, which we attribute to a change of the domain mor- phology and increased pinning of the magnetic domain walls.34Strain engineering of YIG thin lms may be used to further tailor the magnetic anisotropy40and hence the switching behavior of YIG in response to current-induced elds of either Oersted or spin-orbit origin. Our results should also be taken as a cautionary warning about the possible undesired switching of YIG at current densities commonly used to excite and sense the magnetization of Pt/YIG bilayers. ACKNOWLEDGMENTS We acknowledge nancial support by the Swiss Na- tional Science Foundation under grant no. 200020- 172775. We thank Can Onur Avci for valuable discus- sions. 1D. Morecroft, I. A. Colin, F. J. Casta~ no, J. A. C. Bland, and C. A. Ross, Phys. Rev. B 76, 054449 (2007). 2L. Yuan, D. S. Wisbey, S. T. Halloran, D. P. Pappas, F. C. S. da Silva, and H. Z. Fardi, Journal of Applied Physics 106, 113919 (2009). 3V. Uhl  r, S. Pizzini, N. Rougemaille, V. Cros, E. Jim enez, L. Ranno, O. Fruchart, M. Urb anek, G. Gaudin, J. Camarero, C. Tieg, F. Sirotti, E. Wagner, and J. Vogel, Phys. Rev. B 83, 020406 (2011). 4C. Nam and B.-K. Cho, Applied Physics Express 4, 113004 (2011). 5A. Fuhrer, S. Alvarado, G. Salis, and R. Allenspach, Applied Physics Letters 98, 202104 (2011). 6A. Brataas, A. D. 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2019-04-23

We report on the switching of the in-plane magnetization of thin yttrium iron garnet (YIG)/Pt bilayers induced by an electrical current. The switching is either field-induced and assisted by a dc current, or current-induced and assisted by a static magnetic field. The reversal of the magnetization occurs at a current density as low as $10^5$~A/cm$^{2}$ and magnetic fields of $\sim 40$~$\mu$T, two orders of magnitude smaller than in ferromagnetic metals, consistently with the weak uniaxial anisotropy of the YIG layers. We use the transverse component of the spin Hall magnetoresistance to sense the magnetic orientation of YIG while sweeping the current. Our measurements and simulations reveal that the current-induced effective field responsible for switching is due to the Oersted field generated by the current flowing in the Pt layer rather than by spin-orbit torques, and that the switching efficiency is influenced by pinning of the magnetic domains.

Current-induced switching of YIG/Pt bilayers with in-plane magnetization due to Oersted fields

1904.10517v1

arXiv:1912.13111v1 [quant-ph] 30 Dec 2019Hybrid nanophotonic-nanomagnonic SiC-YiG quantum sensor: II/ optical fiber based ODMR and OP-PELDOR experiments on bulk HPSI 4H-SiC. J´ erˆ ome Tribollet∗ Institut de Chimie de Strasbourg, Strasbourg University, U MR 7177 (CNRS-UDS), 4 rue Blaise Pascal, CS 90032, F-67081 Strasbourg Cedex, Fra nce E-mail: tribollet@unistra.fr Abstract Here I present my first fiber based coupled optical and EPR expe riments associated to the development of a new SiC-YiG quantum sensor that I rece ntly theoretically de- scribed (arXiv:1912.11634). This quantum sensor was desig ned to allow sub-nanoscale single external spin sensitivity optically detected pulse d electron electron double reso- nance spectroscopy, using an X band pulsed EPR spectrometer , an optical fiber, and a photoluminescence setup. First key experiments before the demonstration of ODPEL- DOR spectroscopy are presented here. They were performed on a bulk 4H-SiC sample containing an ensemble of residual V2 color centers (spin S= 3/2). Here I demon- strate i/ optical pumping assisted pulsed EPR experiments, ii/ fiber based ODMR and optically detected RABI oscillations, and iii/ optical pum ping assisted PELDOR ex- periments, and iv/ some spin wave resonance experiments. Th ose experiments confirm the feasability of the new quantum sensing approach propose d. 1Introduction I recently presented the theory1of a new SiC-YiG quantum sensor and the associated state of art optically detected pulsed double electron electron spin reson ance spectroscopy (OD- PELDOR), allowing sub-nanoscale single external spin sensing. This n ew methodology re- quires only the use of a standard X band pulsed EPR spectrometer,2as well as an optical fiber and a new SiC-YiG quantum sensor. The fiber and the quantum s ensor can be both introduced in a standard EPR tube. Here I present my first combined pulsed EPR and optical experiment s, all performed on a commercially available bulk 4H-SiC HPSI sample, naturally containing a dilu ted ensemble of V2 color centers spin probes.3–7The aim of those first experiments is to demonstrate the relevance and feasability of interfacing a standard optical set up for photoluminescence excitation and collection with a commercial pulsed EPR/ pulsed ELDOR E LEXYS E 580 spectrometer operating at X band from Bruker, by means of a sing le optical fiber (or a fiber bundle). This setup allows to perform ODMR and optical pumping assisted PELDOR (puulsed electron electron double resonance) experiments,2which are key intermediate ex- periments to perform, before the demonstration of pulsed ODPEL DOR experiments with a SiC-YiG quantum sensor. The whole experimental setup I used cor responds to the one described on fig.2 of my previous theoretical work,1the coupler between the SiC sample and the optical fiber being here a GRIN microlens. The optical fiber, the GRIN microlens and the 4H-SiC sample are all introduced in an EPR tube, which itself is in serted inside the pulsed EPR resonator, a flexline resonator from Bruker (MD5 o r MS3 depending on experiments). The pulsed EPR resonator itself is introduced inside a n Oxford CF935 con- tinuousflowcryostat forpulsedEPR spectroscopy atvariabletem perature(4K-300K).When necessary, a 785 nm laser was used for optical pumping of V2 spins a nd for optical excita- tion of the V2 color center photoluminescence, centered around 9 15 nm at low temperature. This photoluminescence was detected, after optical filtering, by a silicon photodiode, in all presented ODMR experiments. Excitation and collection of the phot oluminescence of the 2SiC sample was performed using the same optical fiber by means of a d ichroic mirror. A lock in amplifier or a transient recorder were used for data acquisitio n, which were visualized ontheXEPR softwareofBruker providedwiththeELEXYSE580puls edEPRspectrometer. Optical pumping assisted EPR and ODMR characteri- zation of V2 spins in bulk 4H-SiC First, I demonstrate on fig.1 that the Electron Paramagnetic Reso nance (EPR) rotational pattern of the V2 color centers spins in bulk 4H-SiC can be recorded , under optical pumping conditions with this experimental setup, allowing to check the zero fi eld splitting andg factor of those paramagnetic centers,3–7and finally to identify them. 3310 3320 3330 3340 3350 3360 3370 B0(G)angle (°)55° 35° 20° 0°90° 80°85° Figure 1: CW EPR rotational pattern of the V2 spins in bulk HPSI 4H-S iC recorded at room temperature and at X band (f=9.369 GHz) under continuous o ptical pumping, with a laser at 785 nm providing a power of P=39 mW at the outpout of optic al fiber. The nul angle correspond to the external magnetic field aligned along the c a xis of 4H-SiC. This rotational pattern was obtained here at X band and room temp erature, using cw 3EPR under continuous optical pumping with a 785 nm laser. The optica l pumping effect is clearly seen on the shape of the EPR spectrum of fig.1: the left side positive signal correspondtoanEPRtransitionwithinducedabsorption,whilethen egativeoneonrightside correspondstostimulatedemissionassociatedtopopulationinvers iononthisEPRtransition. The rotational pattern of fig.1 can be well reproduced (except th e optical pumping effects) by a numerical simulation with Easyspin,8as shown on fig.2, considering a spin S=3/2 with a zero field splitting D = 35 MHz and an isotropic g factor g = 2.0028, con firming previously obtained magnetic parameters of the V2 spin hamiltonian in 4H-SiC.3–7 Figure 2: Numerical simulation with Easyspin of the V2 rotational pat tern in 4H-SiC, considering a spin S=3/2 with a zero field splitting D= 35 MHz and an isotr opic g factor g=2.0028; f=9.369 GHz; the linewidth is 3G here. The nul angle corres pond to the external magnetic field aligned along the c axis of 4H-SiC. Derivative of absorpt ion curves are shown here. The spin state populations assumed here are those of the th ermal equilibrium. Secondly, I demonstrate on fig.3, that several EPR experiments o n V2 color centers spins in bulk 4H-SiC and under continuous optical pumping are possible with t his fiber based ODMR setup, that is, from top to bottom, room temperature cw EP R, room temperature 4pulsed EPR, room temperature ODMR, and 90K ODMR. The two first e xperiments benefits from the fact that a large ensemble of V2 spins is present in this 4H SiC HPSI sample allowing adirect detection oftheEPR signal here, without anyuseof thephotoluminescence, the optical fiber being however used for optical pumping of the V2 s pins. However, the last two experiments presented on fig.3 are true ODMR experiments , meaning that the photoluminescence of the V2 spin probe is the recorded signal along the vertical axis of those two ODMR experiments. 3290 3300 3310 3320 3330 3340 3350 3360 3370-0.500.5EPR (a.u.) 3270 3280 3290 3300 3310 3320 3330 3340 3350 3360-50510EPR (a.u.)106 3400 3450 3500 35500510ODMR (a.u.)105 3400 3450 3500 3550 B0(G)01020ODMR (a.u.)106 Figure 3: From top to bottom and under continuous optical pumping at 785 nm: a/room temperaturecwEPRspectrum(f=9.320GHz, MS3, 36mWat785nm) , b/roomtemperature field sweep pulsed EPR spectrum (f=9.308 GHz, MS3, 36 mW at 785 nm, recorded at 2τ≈2.4µs), c/ room temperature ODMR spectrum (f=9.743 GHz, MD5, 36 mW a t 785 nm), and d/ 90K ODMR spectrum (f=9.746 GHz, MD5, 30 mW at 785 nm) . The static magnetic field B0 is applied along the c axis of 4H-SiC. Note that the two ODMR spectrum of the V2 spins in 4H SiC presented here have been obtained under continuous optical pumping. To sensitively detect t he ODMR spectrum, a train of periodic microwave pulses is send on the sample, such that in stead of having a constant rate of photoluminescence under continuous optical e xcitation, one obtains a periodic modulation of the photoluminescence signal following the per iod of the microwave 5pulses, but onlywhen aparamagnetic resonance oftheV2spins is ex cited by themicrowaves. 0 50 100 150 200 250 300 350 time (ns)012EPR (a.u.)106 0 50 100 150 200 250 300 3500123EPR (a.u.)106 0 50 100 150 200 250 300 3500123EPR (a.u.)106 Figure 4: ODMR detected coherent RABI oscillations of V2 spins in 4H S iC recorded at different microwave power attenuation (from top to bottom: 10 dB , 5 dB, 0 dB attenuation, MD5, B0= 3457 G, f= 9.746 GHz), for various duration of the nutatio n microwave pulse applied (time). Periodically cycling this experiment, again allows to obta in a modulated amount of photoluminescence easily detected by a lock in amplifier. Th ose curves demon- strate the quantum coherent control of V2 spins by microwave pu lses (nutation or single qubit quantum gate performed over a spin ensemble) and also their o ptical detection by photoluminescence, in a single ODMR nutation experiment. Such paramagnetic resonance induces a change of spin state popu lations in the ground state, further converted in a change of the amount of photolumin escence collected under optical excitation. Such periodic photoluminescence signal is then c ollected by a photodiode and then send into a lock in amplifier allowing an efficient extraction of th e amount of pho- toluminescence modulated at a frequency inversely proportionnal to the period between two successive microwave Pi pulses applied on V2 spins. Note that EPR nu tation experiments, also called RABI oscillations measurements, were generally performe d on V2 spins ensemble under optical pumping before such experiments, either using direc t detection (not shown) or photoluminescence detection of the RABI oscillation as shown on fig.4 , in order to check the appropriate microwave Pi pulse parameters (choosing microwave p ower and pulse duration). 6Once the train of microwave Pi pulses resonant with the V2 spins has been adjusted, the ODMR measurement is easy to perform with the lock in amplifier. Spin decoherence and spin relaxation time of bulk V2 spins Thirdly, I demonstrate on fig.5 how to measure at room temperatur e the spin coherence time T2, the longitudinal relaxation time T1 and the optical pumping time To p of the bulk V2 spins with appropriate experiments combining an optical pumping puls e and appropriately synchronized and time delayed microwave pulses. At room temperat ure and with those optical pumping conditions, I find Top= 139µs,T1= 354µs, andT2= 48µs. Those values of T1andT2are in good agreement with values reported for bulk V2 spins in 4HSiC at room temperature.3–7One note that it was also previously reported that the optical pumping time Topdepends on the optical power send on the V2 spins9(and thus on the laser and coupler used) and on the temperature which contr olsT1. One can also note here that for using optimally the SiC-YIG quantum sensor which I described in my previous theoretical work,1a spin coherence time for an isolated sub-surface V2 spin probe ofT2= 12.5µswas assumed. Thus, ODPELDOR spectroscopy should be feasible if t he SiC surface defects can be made sufficiently silent, either by surface p assivation or by cryogenic cooling of the whole quantum sensor, in order that the T2 of sub sur face V2 has a value on the order of the one of bulk V2 spins found here. 70.5 1 1.5 2 2.5 3 3.5 4 4.5 106-5051015EPR sig. (a.u.)106 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 105-50510EPR sig. (a.u.)106Top(RT) = 139 µs T1(RT) / 2 = 177 µs time (ns)time (ns) T2e(RT) = 48 µs Figure 5: Top curve: measurement of TopandT1of bulk V2 spins in 4H-SiC at room temperature. A long optical pumping pulse of 1 ms at 785 nm and 36 mw optical power at the fiber output is used and has a fixed temporal position in the se quence. A standard direct detection spin echo sequence (π 2τ πτ echo ) is synchronized with the optical pumping pulse and globally translated in time through the optical pumping pulse , allowing to follow the time evolution of the spin state populations associated to a given EPR transition of the V2 spins (B0= 3297 G , f=9.308 GHz, MS3), before , during and after the optical pumping pulse. The data are in blue, the two monoexponential fit are in red, p roviding: Top= 139µs andT1= 354µs. Bottom curve: Standard Hahn spin echo decay curve (π 2τ π τ echo ) recorded on one transition of the V2 (B0= 3297 G , f=9.308 GHz, MS3 ), where the spin echo is recorded at various delays 2 τ, the first microwave pulse starting 20 µsafter a long optical pumping pulse of 900 µsat 785 nm and with 36 mw of optical power at the fiber output. The data are in blue, the monoexponential fit is in red, prov iding:T2= 48µs. Inset: pulses sequences applied: pink: optical pulse; red: microwa ve pulse; dark:spin echo for direct detection. The third arrow compared to second one sho ws which delay parameter vary. Optical pumping assisted PELDOR spectroscopy and quantum sensing of carbon related defects by bulk V2 spins Finally, thelastkey experiment combining opticalandEPRtoolswhich is demonstratedhere andwhichallowstogoastepfurthertowardsfiberbasedODPELDOR quantumsensing with a SiC-YIG quantum sensor, is optical pumping assisted PELDOR expe riment, as shown on fig.6. It is a pump-probe like two microwave frequencies experiment c ombined with optical 8pumping, as explained in my previous theoretical work.1It is conveniently implemented here by interfacing, through an optical fiber, the capabilities of the com mercial pulsed ELDOR spectrometer (ELEXYS E580) and the ones of the outside optical setup. The two PELDOR experiments presented here are four-pulse DEER (double electro n electron resonance) exper- iments2combined, either with a continuous optical pumping and direct EPR de tection of the stimulated echo (top spectrum), or with a transient optical pu mping pulse and a direct detection of the refocused echo (bottomspectrum). When the p ump frequency fp is resonant with any spin specy present in the sample and physically close to the V2 spins probe, then a driven decoherence effect occurs producing a reduced spin echo of the V2 spins probes, and thus a dip in the PELDOR spetrum. As the stimulated echo has a lar ger amplitude than the refocused echo, the PELDOR spectrum of fig.6 a/ (top) h as a better signal to noise ratio than the one of fig.6 b/ using the refocused echo, which is gene rally used in structural biology.2The comparison also shows that continuous optical pumping seems n ot to induce a decrease of the signal to noise ratio of such PELDOR experiment. The two obtained PELDOR spectrum correspond to the one expect ed. The preparatory experiment was the field sweep spectrum of fig.3 b/, thus the PELDO R spectrum (versus fp) should reflect somehow this field sweep spectrum (versus B0), but with of course an inverted order of the resonance lines. Of course, when fp=fs=9.3 08 GHz, the V2 spins feel an accelerated driven decoherence due to themselves, such that the probing EPR line chosen is always seen in such a PEDLOR spectrum versus fp. However, one c an also distinguished on both PELDOR spectrum another dip occuring at fp= 9.243 GHz. Th is resonance line is 65 MHz below fs, corresponding to a resonnace line 23.2 G above the lo w field EPR line of the V2 shown on the field sweep spectrum of fig.3 b/. This line was thus also present on fig.3 b/ and correspond to other kinds of intrinsic defects in 4H-SiC havin g a g factor also very close to the one of the V2, that is close to g=2.0028. This line is often a ttributed to carbon related defects in bulk 4H-SiC. That means that the dip seen at fp= 9 .243 GHz corresponds to the additionnal decoherence effect felt by the V2 spins probes d ue to the microwave 99.15 9.2 9.25 9.3 9.35 9.4246810EPR sig. (a.u.)106 9.15 9.2 9.25 9.3 9.35 9.4-2024EPR sig. (a.u.)106 fpump (GHz)fpump (GHz) Figure 6: PELDOR experiments of the type four-pulse DEER experim ents combined, either with a continuous optical pumping and direct EPR detection of the st imulated echo (a/ top spectrum), or with a transient optical pumping pulse and a direct de tection of the refocused echo (b/ bottom spectrum). B0 is parallel to the c axis of 4H-SiC. In both PELDOR experiments, B0=3297 G, T= 300K, and the probe microwave frequ ency is fs= 9.308 GHz, such that the low field EPR line of the V2 spins seen on fig.3 b/ is here res onantly excited at fs. The microwave pump frequency fp is varied in the range [9.15; 9.4] GHz, thus over 250 MHz, in 250 steps of 1 MHz. Inset: pulses sequences applied: pink: o ptical pumping (first arrow); red: microwave pulse at fs (second arrow) and fp (third a rrow); dark: the spin echo used for direct detection, ie the one integrated. driven manipulation of the carbon related defects located nearby t hem, producing through dipolar couplings, a fluctuating local magnetic field on the sites of the V2 spins probes. This is exactly the principle at the heart of quantum sensing as explain ed in my previous theoretical work on the SiC YIG quantum sensor.1The main difference is that here the target spin bath is 3D, whereas in ODPELDOR spectroscopy applied t o structural biology, the target spin bath is 2D. Here also, direct detection is used instea d of the highly sensitive photoluminescence detection. As a last remark, one notes that the third EPR line seen on the field sw eep spectrum of fig.3 b/ is not seen here in the PELDOR spectrum. The reason is assum ed to be the limited 10bandwidth available in the presented PELDOR experiments performe d with a standard MS3 flexline resonator from Bruker. The MS3 cavity is known to have a bandwidth at half maximum of its microwave reflexion curve nearly equal to 100 MHz . As the central frequency of the cavity is here set equal to the resonant freque ncy of the V2 spins probes at fs=9.308 GHz under B0=3297 G, then any other resonant EPR line loc ated much beyond +/- 50 MHz from fs=9.308 GHz can not be observed in the PELDOR spe ctrum by lack of microwave power at the associated pump frequency entering the c avity at those frequencies (the microwave power is reflected). The high field EPR line of the V2 sp ins occurs at B0= 3345Gonfig.3b/ andshouldthusappearsat9243-65=9178MHz, wh ichisthusnot possible here. Those two room temperature optical pumping assisted PELD OR experiments thus: i/ clearly show that V2 spins probes, which are photoluminescent and which can thus be optically detected in an ultra sensitive manner by ODMR, can sense by microwave driven decoherence effectssomeparamagneticcenters locatednearby themwhicharethemselves not photoluminescent, thus providing a way to considerably increase th e sensitivity of standard pulsed EPR spectrometers if the proposed SiC-YIG quantum senso r can be fabricated; ii/ they also demonstrate that using one single V2 EPR line and targeting a spin label EPR line located nearby the V2 EPR line, it should be clearly possible to perfo rm ODPELDOR spectroscopy applied to structural biology with the SiC-YIG quant um sensor I previously proposed.1 Spin wave resonance experiments on model ferromag- netic nanostripes of Permalloy Finally, I present in this section some test spin wave resonance expe riments, performed not on YIG nanostripes at X band, but on the more easily accessible Perm alloy nanostripes at Q band (34 GHz). Those experiments were numerically simulated follow ing the theoretical approach I previously presented in the context of quantum compu ting with an array of spin 11qubits in SiC located nearby a permalloy ferromagnetic nanostripe.10 1 1.1 1.2 1.3 1.4 1.5 B0 (G) 104-1-0.500.51EPR (a.u.) Figure 7: Spin Wave Resonance (SWR) spectrum of an ensemble of Pe rmalloy ferromag- netic nanostripes (thickness T=100 nm, width w= 300 nm, and length L= 100µm): in red: derivative spectrum, as measured at Q band (f= 34 GHz) with a magn etic field applied in the plane of the nanostripes, along the width w; in blue: absorption s pectrum, as numeri- cally simulated without any free parameter (and without considering the different oscillator strengths of the various SWR). For the numerical simulation, I used the saturation magnetization ( 11700 G) and g fac- tor (2.00) known for Permalloy and the dimension of the Py nanostrip es (thickness T=100 nm, width w= 300 nm, and length L= 100µm). As it can be seen on fig.7, the experi- mentally observed spin wave resonance spectrum and the theoret ical one match quite well, the six main spin wave resonance being obtained and having resonant magnetic fields close to the experimental ones, with an error on the order of one or few spin wave resonance linewidth. This is quite satisfactory considering the fact that there are no free parameter in the theoretical simulation presented here. 12Conclusion In this article, I have presented my first experiments towards the development of a new SiC- YiG hybrid quantum sensor compatible with a standard X band pulsed E PR spectrometer widely used worldwide. The measured T2 spin coherence time of 48 µsat room temperature found here for bulk V2 spins in 4H-SiC confirm the high potential of th ose solid state spin qubits for quantum sensing application. My successful optically det ected magnetic reso- nance experiments, as well as my optical pumping assisted PELDOR e xperiments confirm the relevance of this new experimental approach for state of art quantum sensing at the single spin sensitivity and with sub nanoscale resolution, interfacing a standard photolumi- nescence setup with a standrad pulsed ELDOR spectrometer by me ans of an optical fiber. This experimental approach and the related SiC-YIG quantum sens or should thus be of great interest for all the biophysicists, chemists and physicists wh ich are already worldwide pulsed EPR user. The next challenges towards the practical demon stration of such a new experimental approach for state of art quantum sensing are i/ th e fabrication by ion im- plantation of sub-surface quantum coherent isolated V2 spin prob es, ii/ the fabrication of 4H-SiC nanophotonic structures for the efficient excitation and co llection of V2 spin probes photoluminescence, and iii/ the fabrication of appropriate YIG nano magnonic structures for the investigation of the depth profile of sub surface V2 color cente r spins created by ion implantation. Acknowledgments The author thanks the University of Strasbourg and CNRS for the reccurent research fund- ings. The author also thanks the STNano central of technology in S trasbourg for fabricating and providing the model permalloy nanostripes studied here. 13References (1) J. Tribollet, Hybrid nanophotonic-nanomagnonic SiC-YiG quantu m sensor: I/ theo- retical design and properties, Arxiv:1912.11634; submitted on 25 1 2 2019, 39 pages (2019). (2) A. Schweiger et al., Principles of pulse electron paramagnetic res onance, Oxford Uni- versity Press, Oxford UK; New York (2001). (3) H. Kraus et al., Scientific Reports 4, 5303 (2014). (4) M. Widmann et al., Nature Materials 14, 164 (2015). (5) P.G. Baranov et al., Materials Science Forum 740, 425 (2013). (6) Franzsiska Fuchs, PhD thesis Wurzburg University (2017). (7) S.A. Tarasenko et al., Phys. Status Solidi B 255, 1700258 (2018). (8) S. Stoll et al., Journal of Magnetic Resonance 178, 42 (2006). (9) M. Fischer et al., Phys. Rev. Applied 9, 54006 (2018). (10) J. Tribollet et al., Eur. Phys. J. B. 87, 183 (2014). Competing financial interests The author declare that he has no competing financial interests. 14

2019-12-30

Here I present my first fiber based coupled optical and EPR experiments associated to the development of a new SiC-YiG quantum sensor that I recently theoretically described (arXiv:1912.11634). This quantum sensor was designed to allow sub-nanoscale single external spin sensitivity optically detected pulsed electron electron double resonance spectroscopy, using an X band pulsed EPR spectrometer, an optical fiber, and a photoluminescence setup. First key experiments before the demonstration of ODPELDOR spectroscopy are presented here. They were performed on a bulk 4H-SiC sample containing an ensemble of residual V2 color centers (spin S=3/2). Here I demonstrate i/ optical pumping assisted pulsed EPR experiments, ii/ fiber based ODMR and optically detected RABI oscillations, and iii/ optical pumping assisted PELDOR experiments, and iv/ some spin wave resonance experiments. Those experiments confirm the feasability of the new quantum sensing approach proposed.

Hybrid nanophotonic-nanomagnonic SiC-YiG quantum sensor: II/ optical fiber based ODMR and OP-PELDOR experiments on bulk HPSI 4H-SiC

1912.13111v1

Direct observation of magnon-phonon coupling in yttrium iron garnet Haoran Man,1,Zhong Shi,2, 3,Guangyong Xu,4Yadong Xu,2Xi Chen,5 Sean Sullivan,5Jianshi Zhou,5Ke Xia,6Jing Shi,2,yand Pengcheng Dai1, 6,z 1Department of Physics and Astronomy, Rice University, Houston, Texas 77005, USA 2Department of Physics and Astronomy, University of California, Riverside, California 92521, USA 3School of Physics Science and Engineering, Tongji University, Shanghai 200092, China 4NIST Center for Neutron Research, National Institute of Standards and Technology, Gaithersburg, Maryland 20899, USA 5Materials Science and Engineering Program, Texas Materials Institute, The University of Texas at Austin, Austin, Texas 78712, USA 6Department of Physics, Beijing Normal University, Beijing 100875, China (Dated: September 13, 2017) The magnetic insulator yttrium iron garnet (YIG) with a ferrimagnetic transition temperature of 560 K has been widely used in microwave and spintronic devices. Anomalous features in the spin Seeback eect (SSE) voltages have been observed in Pt/YIG and attributed to the magnon-phonon coupling. Here we use inelastic neutron scattering to map out low-energy spin waves and acoustic phonons of YIG at 100 K as a function of increasing magnetic eld. By comparing the zero and 9.1 T data, we nd that instead of splitting and opening up gaps at the spin wave and acoustic phonon dispersion intersecting points, magnon-phonon coupling in YIG enhances the hybridized scattering intensity. These results are dierent from expectations of conventional spin-lattice coupling, calling for new paradigms to understand the scattering process of magnon-phonon interactions and the resulting magnon-polarons. Spin waves (magnons) and phonons are propagating disturbance of the ordered magnetic moment and lat- tice vibrations, respectively. They constitute two fun- damental quasiparticles in a solid and can couple to- gether to form a hybrid quasiparticle [1, 2]. Since our current understandings of these quasiparticles are based on linearized models that ignore all the high-order terms than quadratic terms and neglect interactions among the quasiparticle themselves [3], magnons and phonons are believed to be stable and unlikely to interact and break- down for most purposes [4]. Therefore, discovering and understanding how the otherwise stable magnons and phonons can couple and interact with each other to in- uence the electronic properties of solids are one of the central themes in modern condensed matter physics. In general, spin-lattice (magnon-phonon) coupling can modify magnon in two dierent ways. First, the static lattice distortion induced by the magnetic order may af- fect the anisotropy of magnon exchange couplings, as seen in the spin waves of iron pnictides with large in- plane magnetic exchange anisotropy [5]. Second, the dynamic lattice vibrations interact with time-dependent spin waves may give rise to signicant magnon-phonon coupling [6, 7]. One possible consequence of such cou- pling is to create energy gaps in the magnon dispersion at the nominal intersections of the magnon and phonon modes [8, 9], as seen in antiferromagnet (Y,Lu)MnO 3 [10]. Alternatively, magnon-phonon coupling may give rise to spin-wave broadening at the magnon-phonon crossing points [11]. In both cases, we expect the in- tegrated intensity of hybridized excitations at the inter- secting points to be the sum of separate magnon and phonon scattering intensity without spin-lattice coupling[8]. Finally, if magnon and phonon lifetime-broadening is smaller than their interaction strength, the resulting mixed quasiparticles can form magnon polarons [6, 7]. Here we use inelastic neutron scattering to study low- energy ferromagnetic magnons and acoustic phonons in the ferrimagnetic insulator yttrium iron garnet (YIG) with chemical formula Y 3Fe5O12[Figs. 1(a)-1(d)] [12{ 14]. At zero eld and 100 K, we conrm the quadratic wave vector dependence of the magnon energy, E=Dq2, whereDis the eective spin wave stiness constant and qis momentum transfer (in A1or 1010m1) away from a Bragg peak [Fig. 1(e)] [13{19]. We also conrm the linear dispersion of the TA phonon mode [Fig. 1(e)]. Upon application of a magnetic eld H0, a spin gap of the magnitude gH0(g2 is the Land e electron spin g- factor) opens and lifts up spin waves spectra away from the eld-independent phonon dispersion [Figs. 1(f) and 1(g)] [13, 14]. By comparing the zero and 9.1 T eld wave vector dependence of the spin wave spectra, we nd that instead of splitting and opening up gaps at the spin wave and acoustic phonon dispersion intersecting points, hy- bridized magnon polaron scattering at the intersecting points has larger intensity at zero eld and magnons re- main unchanged at other wave vectors as shown schemat- ically in the bottom panels of Figs. 1(f) and 1(g). This is dierent from the expectations of conventional magnon- phonon interaction, where hybridized polaronic excita- tions at the crossing points should have the sum of sep- arate magnon and phonon scattering intensity, and be- come broader in energy due to the repulsive magnon- phonon dispersion curves [8{11]. Our results thus reveal a new magnon-phonon coupling mechanism, calling for a new paradigm to understand the scattering process ofarXiv:1709.03940v1 [cond-mat.str-el] 12 Sep 20172 aaaa/2 H0 Q = (2,2,0) + qQ = (4,0,0) + q magnon scan phonon scan KHL Fe ‘d’Fe ‘a’ Y ‘c’ 0 2 4 6 8 10 1202468 Energy (meV) q (108 m-1)magnon 9.1 Tmagnon 0 Tmagnon Energy scan Q = (2,2,0) + q phonon Energy scan Q = (4,0,0) + qLA phonon TA phonon magnon phonon Energy Energy9.1 T0 T 9.1 T0 TEnergy q q χ’’ Energy EnergyEnergy q q χ’’ 9.1 T 0 T 9.1 T 0 T TALA TA TA TALALA magnon phonon 9.1 T0 T 9.1 T0 Te f ga b d c H 32333435 28293031 V (µV)LAV (µV)TA µ0H (T) µ0H (T)µ0H (T) µ0H (T) 1.5 2.0 2.5 3.0 3.5 8.0 8.5 9.0 9.5 10.02.0 2.5 3.0 8.5 9.0 9.5 10.00.2 0.10.00.1 0.0V (µV) FIG. 1: (a) The full unit cell of YIG comprises eight cubes that are related by glide planes to the basic cube shown in the gure. (b) The corresponding reciprocal space with the [H;K; 0] scattering and vertical magnetic eld H0. The red and green solid circles mark the positions of reciprocal space where we probe spin waves and acoustic phonons, respectively. (c) A picture of the Pt/YIG device used for SSE measure- ments. (d) SSE voltage in the eld ranges where anomalous features appear at 100 K. (e) Magnon and phonon disper- sions of YIG at 100 K and dierent magnetic elds. The black squares and solid red circles are data from 0 T and 9.1 T measurements, respectively. The q4:5108m1point corresponds to
Q= 0:062 in Fig. 2(d). The black and red solid lines are quadratic ferromagnetic spin wave t to the data. The blue and red boxes indicate magnon-phonon crossing points. The green and blue solid lines are TA (with phonon velocity C?3:9103m/s)and LA ( Cjj7:2103 m/s) phonons, respectively [31]. (f,g) The expanded view of the blue and red boxes in (e), respectively. The bottom panels in (f,g) summarize the results obtained in our measurements on the magnetic eld eect on spin waves, hybridized excita- tions, and TA phonons. magnon-phonon interactions and the resulting magnon polarons [31]. We chose to study magnon-phonon coupling in YIG because it is arguably the most important material used in microwave and recent spintronic devices [20]. In ad- dition to having a ferrimagnetic ordering temperature of560 K suitable for room temperature applications, YIG can be grown with exceptional quality, and has the lowest Gilbert damping of any known materials and a narrow magnetic resonance linewidth allowing transmis- sion of spin waves over macroscopic distances [21{23].The spin Seebeck eect (SSE), which allows spin cur- rents produced by thermal gradients in magnetic mate- rials to be transmitted and converted to charge voltages in a heavy metal such as Pt, is one of the most techno- logically relevant thermoelectric phenomena to be used in `spin caloritronic' devices [24{29]. In the case of a Pt lm on the surface of a polished single-crystalline YIG slab (Pt/YIG) [Fig. 1(c)] [30], anomalous features in magnetic eld dependence of the SSE voltages at low temperatures are attributed to the magnon-phonon in- teraction at the \touching" points between the magnon and transverse acoustic (TA) and longitudinal acoustic (LA) phonon as magnon dispersion curve is lifted by the applied eld while phonon is not aected by the eld [Fig. 1(d)] [31]. While we nd no anomaly at the magnon and TA/LA acoustic phonon touching points, our data reveal clear evidence for magnon-phonon interaction at zero eld, consistent with the formation of magnon po- larons. Our neutron scattering experiment was carried out at NIST center for neutron research, Gaithersburg, Mary- land [32]. The full body-centered-cubic unit cell of YIG with space group Ia3dcomprises eight cubes that are re- lated by glide planes to the basic cube as shown in Fig. 1(a), where the metallic atomic sites are labelled as `a', `d', and `c' [13]. Using the cubic lattice parameter of a=b=c= 12:376A, we dene momentum transfer Qin three-dimensional (3D) reciprocal space in A1as Q=Ha+Kb+Lc, whereH,K, andLare Miller indices and a=^a2=a,b=^b2=a,c=^c2=a[Figs. 1(a) and 1(b)]. Consistent with Ref. [31], the magnetic eld dependence of SSE voltage on our Pt lm on YIG contains two anomalous features at 2.5 T and 9.1 T [Figs. 1(c)-1(e)] [32{36]. The sample for neutron scattering experiments was ori- ented withaandb(a)-axis of the crystal in the horizontal [H;K; 0] scattering plane [Fig. 1(b)] and mounted inside a 10 T vertical eld magnet. In this geometry, we mea- sured magnon dispersion around (2 ;2;0) and phonon dis- persion around (4 ;0;0). The momentum transfers Qat these wave vectors are Qmagnon = (2 +
Q;2 +
Q;0) andQphonon = (4;
Q;0) for TA phonon [Fig. 1(b)]. For convenience, we calculate relative momentum trans- fer asq= 2p 2
Q=a for magnon and q= 2
Q=a for phonon. We chose (2 ;2;0) for magnetic and (4 ;0;0) for phonon measurements because of their huge dierences in nuclear structure factors [4.75 at (2 ;2;0) versus 50.5 at (4;0;0)], which is directly related to the acoustic phonon intensity. Although we expect to nd mostly magnetic scattering at (2 ;2;0) and phonon scattering at (4 ;0;0), the nite Fe3+magnetic form factor of jF(Q)jmeans that there are still magnetic contributions to the phonon scattering at (4 ;0;0) (jF(2;2;0)j2=jF(4;0;0)j21:86). Magnetic neutron scattering directly measures the magnetic scattering function S(Q;E), which is propor- tional to the imaginary part of the dynamic susceptibil-3 9.1 T 0 T9.1T 100K 0T 100K9.1 T 0 T 24 0Energy (meV) 0 5 10 0 5 10 9.1 T 0 T Q=(2+/uni0394Q, 2+/uni0394Q, 0)/uni0394Q = 0.062/uni0394Q = 0.112 /uni0394Q = 0.092/uni0394Q = 0.152 /uni0394Q = 0.062/uni0394Q = 0.112 /uni0394Q = 0.092/uni0394Q = 0.132/uni0394Q = 0.152 /uni0394Q = 0.032 /uni0394Q = 0.0120123 5K 100K χ’‘(q, E) (a.u.)0T 5K0 T 0 2 4 6 8 100.00.20.40.60.81.01.21.41.61.82.0 0 2 4 6 8 100.00.20.40.60.81.01.21.41.61.82.0 Energy (meV) Energy (meV)Instrument Resolutiona b c d eq (108 m-1) q (108 m-1)0 2 4 6 8 Energy (meV)/uni0394Q = 0.092 χ’’ (a.u.) χ’’ (a.u.) χ’‘(q, E) theoretical (a.u.) /uni0394Q = 0.1320 0.05 0.10 0.15 0 0.05 0.10 0.15/uni0394Q (r.l.u) /uni0394Q (r.l.u) q = (/uni0394Q, /uni0394Q, 0)χ’‘(q, E-gH0) FIG. 2: (a) Schematic illustration of the expected magnon dispersions at 0 T and 9.1 T for a simple ferromagnet. (b) The expected temperature, magnetic eld dependence of low- energy00(Q;E) for simple ferromagnet obtained from SpinW software package [39]. Here the magnetic eld induced spin gapgH0has been subtracted in the 9.1 T 00(q;EgH0) (red). The upper and bottom units are
Qandq, respectively. (c) Our estimated 00(Q;E) with Q= (2:092;2:092;0) at 5 K and 100 K after correcting measured S(Q;E) for the background and Bose-population factor. (d,e) The estimated 00(Q;E) at 0 T and 9.1 T, respectively, after correcting for background and Bose population factor. Scans at dierent wave vectors are lifted up by 0.3 sequentially. The black and red arrows marks the peak positions at 0 T and 9.1 T, respectively. ity00(Q;E) throughS(Q;E)/jF(Q)j200(Q;E)=[1 exp(E kBT)], whereEis the magnon energy, kBis the Boltzmann constant [2]. Although YIG is a ferrimag- net, its low-energy spin waves can be well described as a simple ferromagnet [17]. In the hydrodynamic limit of long wavelength (small- q) and small energies, we expect E=
0+gH0+Dq2for spin wave dispersion, where
0is the possible intrinsic spin anisotropy gap, gH0is the size of the magnetic eld induced spin gap, and Dis in units of meV A2[Fig. 2(a)] [13{16]. In addition, for a pure magnetic ordered system without spin-lattice interaction, we expect that 00(Q;E) to be independent of temper- ature at temperatures well below the magnetic ordering temperature and applied magnetic eld after correcting for the eld-induced spin gap gH0[Fig. 2(b)] [37{39]. To determine if temperature and magnetic eld de- pendence of spin waves in YIG follow these expecta- tions, we measured wave vector dependence of magnon -2 0 2 4 60.00.51.01.52.02.53.03.5 0 2 4 6 8 100.00.51.01.52.02.53.03.5 0 2 4 6 8 100.00.51.01.52.02.53.03.5 2 4 6 8 100.00.51.01.52.02.5/uni0394Q = 0.062 /uni0394Q = 0.112 /uni0394Q = 0.132 /uni0394Q = 0.1529.1 T0 TIntensity (a.u.)Intensity (a.u.) Intensity (a.u.) Intensity (a.u.)Q=(2+/uni0394Q, 2+/uni0394Q, 0) 1.05 meVa b c dEnergy (meV) Energy (meV) Energy (meV) Energy (meV)FIG. 3: (a,b,c,d) Comparison of the estimated 00(Q;E) as a function of increasing wave vector at 0 T (black) and 9.1 T (red). The 9.1T data is shifted by 1.05 meV to accommo- date the eld induced energy shift. Light red dots represents the original data position of the 9.1 T data. The horizontal bars are estimated instrumental energy resolution based on magnon dispersion at 100 K. energy of YIG at dierent temperatures and magnetic elds. Figure 2(c) shows our estimated constant- Qscans [Q= (2:092;2:092;0) or
Q= 0:092 rlu] of00(Q;E) at 5 K (lled black squares) and 100 K (lled orange circles). Consistent with the expectation, we see that 00(Q;E) at these two temperatures are identical within the er- rors of the measurement. Figure 2(d) shows constant- Qscans of spin waves of YIG at 100 K and 0 T. At Q= (2:062;2:062;0) or
Q= 0:062,00(Q;E) has a clear peak in energy that is slightly larger than the in- strumentation resolution (horizontal bar). With increas- ing
Q, the peak in 00(Q;E) moves progressively to higher energies. We have attempted but failed to t the spin wave spectra with a simple harmonic oscillator gen- erally used for a ferromagnet [38]. This may be con- sistent with recent inelastic neutron scattering study of YIG that reveals the need to use long range magnetic exchange couplings to t the overall spin wave spectra [19]. By tting the spin wave spectra at zero eld with an exponentially modied Gaussian peak function [32], we obtain the magnon dispersion curve as shown in Fig. 1(e). Fitting the dispersion curve with E=
0+Dq2 yields
00 andD= 58060 meV A2, consistent with earlier work giving D533 meV A2[14].4 0.00 0.05 0.10 0.15012345 FWHM 0 T FWHM 9.1T ∆Q (r.l.u)0 2 4 6 8 100.00.51.01.52.0Energy (meV) Energy (meV)9.1 T0 T Q=(4, 0.2, 0) 9.1 T0 T 2.5 TQ=(4, 0.1, 0) 9.1 T0 T 2.5 T5 T Magnetic Field (T)Q=(2+/uni0394Q, 2+/uni0394Q, 0)Q=(4, 0.1, 0)a b c d Integrated Area CNTS/105 monitorCNTS/105 monitor Energy(meV)0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.00.00.51.01.52.02.53.0 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 5.00.00.20.40.60.81.01.2 FIG. 4: (a) Energy scan of S(Q;E) atQ= (4;0:2;0), a position far away from magnon-phonon crossing points, and 100 K to probe TA phonon at 0 T and 9.1 T (b) Energy scan ofS(Q;E) to probe magnon-phonon hybridized excitations at Q= (4;0:1;0) near magnon-phonon crossing point at 0 T, 2.5 T, and 9.1 T. (c) Magnetic eld dependence of the integrated intensity of magnon-phonon hybridized excitations at 100 K andQ= (4;0:1;0). (d) FWHM of the magnon at 0 T and 9.1 T as a function of
Q. Upon application of a 9.1 T eld at 100 K, we expect the magnon dispersion curve to be lifted by gH01 meV. This would be consistent with the observation of a sharp gap below 1.05 meV in constant- Qscan at Q= (2:012;2:012;0) (
Q= 0:012) [Fig. 2(e)]. Constant- Qscan at Q= (2:032;2:032;0) shows similar behavior. Figure 2(e) also shows constant- Qscans at identical wave vectors as those in Fig. 2(d) at 0 T. Using data in Fig. 2(e), we plot the magnon dispersion at 9.1 T eld in Fig. 1(e). Consistent with the expectation, we see a clear gH0 upward shift in magnon energy but the spin wave stiness Dremains unchanged. To quantitatively determine the magnetic eld eect on00(Q;E) of YIG, we compare 00(Q;E) at 0 T with those at 9.1 T. Figure 3(a)-3(d) summarizes the energy dependence of 00(Q;E) after down shifting the 9.1 T data bygH0= 1:05 meV. At
Q= 0:062, the scan along the red arrow direction near the magnon-phonon cross- ing point as shown in Fig. 1(f), we see that 00(Q;E) at 9.1 T eld is lower in intensity compared with those at 0 T. On moving to
Q= 0:10 with no magnon-phonon crossing,00(Q;E) at 0 T and 9.1 T are virtually iden-tical as expected. At the second magnon-phonon cross- ing point with
Q0:13 [see red arrow in Fig. 1(g)], the dierences between 00(Q;E) at 0 T and 9.1 T are even more obvious, with intensity at 0 T considerably larger than that at 9.1 T [Fig. 3(c)]. Finally, on moving to
Q= 0:152 well above the magnon-phonon crossing point wave vectors [Fig. 1(e)], we again see no obvious dierence in 00(Q;E) between 0 T and 9.1 T. Figure 3 shows that magnetic eld dependence of 00(Q;E) is highly wave vector selective, revealing clear magnetic eld induced intensity reduction in 00(Q;E) at wave vectors associated with magnon-phonon cross- ing points while having no eect at other wave vectors. To conrm the presence of TA phonon and determine its magnetic eld eect, we carried out TA phonon mea- surements near (4 ;0;0), which has a rather large nu- clear structure factor compared with (2 ;2;0). Figure 4(a) shows energy scans of at Q= (4;0:2;0) and 100 K, which is along the green arrow direction in Fig. 1(g) and far away from the magnon dispersion. The spectra reveal a clear magnetic eld independent peak at E3 meV, conrming the TA phonon nature of the scattering. Fig- ure 4(b) shows similar energy scan at Q= (4;0:1;0) and 100 K, which is along the green arrow direction and near the magnon-phonon crossing point in Fig. 1(f). At 0 T, we see a peak around E1:7 meV consistent with dispersions of magnon and TA phonon. With increas- ing eld to 2.5 T and 9.1 T, the intensity of the peak decreases, but its position in energy remains unchanged [Fig. 4(b)]. Figure 4(c) shows magnetic eld dependence of the integrated intensity, conrming the results in Fig. 4(b). Since the energy of the magnon should increase with increasing magnetic eld, the eld independent na- ture of the peak position in Fig. 4(b) suggests that the mode cannot be a simple addition of magnon and phonon, but most likely arises from hybridized magnon polarons [6, 7]. Figure 4(d) shows the full width at half maximum (FWHM) of the magnon width at 0 T and 9.1 T. Within the errors of our measurements, we see no energy width change in the measured wave vector region. Our results provided compelling evidence for the pres- ence of magnon-phonon coupling in YIG at the magnon- phonon crossing points at zero eld. This is clearly dif- ferent from the SSE measurements, where anomalies are only seen at the critical elds that obey \touch" condition at which the mangnon energy and group velocity agree with that of the TA/LA phonons. When the applied eld is less than the critical eld, the magnon disper- sion has two intersections with TA/LA phonon modes. When the applied eld is larger than the critical eld, the magnon dispersoin is separated from the TA/LA phonon modes. In the theory of hybrid magnon-phonon excita- tions [6, 7], the SSE anomalies occur at magnetic elds and wave vectors at which the phonon dispersion curves are tangents to the magnon dispersion, where the ef- fects of the magnon-phonon coupling are maximized [40].5 While our ndings of a novel magnon-phonon coupling at zero eld are consistent with the formation of magnon- polarons in YIG [6, 7], they are not direct proof that magnon-polaron formation alone causes anomalous fea- tures in the magnetic eld and temperature dependence of the SSE. Other eects, such as spin diusion length, acoustic quality of the YIG lm, and magnon spin con- ductivity also play an important role in determining the SSE anomaly [41]. Regardless of the microscopic origin of the SSE anomaly, our discovery suggests the need to understand why magnon-phononok interaction and the resulting magnon polarons enhance the hybridized exci- tations at the magnon-phonon intersection points. The neutron scattering work at Rice is supported by the U.S. DOE, BES de-sc0012311 (P.D.). The materials work at Rice is supported by the Robert A. Welch Foun- dation Grant No. C-1839 (P.D.). The work at UCR (J.S. and Z.S.) is supported as part of the SHINES, an En- ergy Frontier Research Center funded by the U.S. DOE, BES under Award No. SC0012670. YIG crystal growth at UT-Austin is supported by the Army Research Oce MURI award W911NF-14-1-0016. These authors made equal contributions to this paper yElectronic address: jings@ucr.edu zElectronic address: pdai@rice.edu [1] W. Heisenberg, Z. Phys. 49, 619 (1928). [2] S. W. Lovesey, Theory of Thermal Neutron Scattering from Condensed Matter (Clarendon, Oxford, 1984), Vol. 2, Chap. 9. [3] L. D. Landau, Sov. Phys. JETP 3, 920 (1957). [4] M. E. Zhitomirsky and A. L. Chernyshev, Rev. Mod. Phys. 85, 219 (2013). [5] Jun Zhao, D. T. Adroja, Dao-Xin Yao, R. Bewley, Shil- iang Li, X. F. Wang, G. Wu, X. H. 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[16] C. M. Srivastava and R. Aiyar, J. Phys. C: Solid StatePhys. 20, 1119 (1987). [17] J. Barker and G. E. W. Bauer, Phys. Rev. Lett. 117, 217201 (2016). [18] L. S. Xie, G. X. Jin, L. X. He, G. E. W. Bauer, J. Barker, and K. Xia, Phys. Rev. B 95, 014423 (2017). [19] A. J. Princep, R. A. Ewings, S. Ward, S. T oth, C. Dubs, D. Prabhakaran, A. T. Boothroyd, arXiv:1705.06594. [20] A.V. Chumak, V. I. Vasyuchka, A. A. Serga, and B. Hille- brands. Nature Physics 11, 453461 (2015); [21] A. A. Serga, A.V. Chumak, and B. Hillebrands. J. Phys. D: Appl. Phys. 43 264002 (2010). [22] A.V. Chumak, , A. A. Serga, and B.Hillebrands. Nat. Commun. 5, 4700 (2014). [23] Y. Kajiwara, K. Harii, S. Takahashi, J. Ohe, K. Uchida, M. Mizuguchi, H. Umezawa, H. Kawai, K. Ando, K. Takanashi, S. Maekawa, and E. Saitoh, Nature (London) 464, 262 (2010). [24] Gerrit E. W. Bauer, Eiji Saitoh, and Bart J. van Wees, Nat. Mater. 11, 391 (2012). [25] K. Uchida, S. Takahashi, K. Harii, J. Ieda, W. Koshibae, K. Ando, S. Maekawa, and E. Saitoh, Nature 455, 778 (2008). [26] K. Uchida, J. Xiao, H. Adachi, J. Ohe, S. Takahashi, J. Ieda, T. Ota, Y. Kajiwara, H. Umezawa, H. Kawai, G. E. W. Bauer, S. Maekawa, E. Saitoh, Nat. Mater. 9, 894 (2010). [27] Ken-ichi Uchida, Hiroto Adachi, Takeru Ota, Hiroyasu Nakayama, Sadamichi Maekawa, and Eiji Saitoh, Appl. Phys. Lett. 97, 172505 (2010). [28] C. M. Jaworski, J. Yang, S. Mack, D. D. Awschalom, R. C. Myers, and J. P. Heremans Phys. Rev. Lett. 106, 186601 (2011). [29] A. Homann and S. D. Bader. Opportunities at the Fron- tiers of Spintronics. Phys. Rev, Appl. 5, 047001 (2015). [30] Z. Qiu, K. Ando, K. Uchida, Y. Kajiwara, R. Takahashi, H. Nakayama, T. An, Y. Fujikawa, and E. Saitoh, Appl. Phys. Lett. 103, 092404 (2013). [31] T. Kikkawa, K. Shen, B. Flebus, R. A. Duine, Ken-ichi Uchida, Z. Qiu, G. E.W. Bauer, and E. Saitoh, Phys. Rev. Lett. 117, 207203 (2016). [32] For additional data and analysis, see supplementary in- formation. [33] Z. Jiang, C.-Z. Chang, M. R. 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2017-09-12

The magnetic insulator yttrium iron garnet (YIG) with a ferrimagnetic transition temperature of $\sim$560 K has been widely used in microwave and spintronic devices. Anomalous features in the spin Seeback effect (SSE) voltages have been observed in Pt/YIG and attributed to the magnon-phonon coupling. Here we use inelastic neutron scattering to map out low-energy spin waves and acoustic phonons of YIG at 100 K as a function of increasing magnetic field. By comparing the zero and 9.1 T data, we find that instead of splitting and opening up gaps at the spin wave and acoustic phonon dispersion intersecting points, magnon-phonon coupling in YIG enhances the hybridized scattering intensity. These results are different from expectations of conventional spin-lattice coupling, calling for new paradigms to understand the scattering process of magnon-phonon interactions and the resulting magnon-polarons.

Direct observation of magnon-phonon coupling in yttrium iron garnet

1709.03940v1

Octave-Tunable Magnetostatic Wave YIG Resonators on a Chip Sen Dai, Sunil A. Bhave ,Senior Member, IEEE , and Renyuan Wang ,Member, IEEE Abstract —We have designed, fabricated, and charac- terized magnetostatic wave (MSW) resonators on a chip. The resonators are fabricated by patterning single-crystalyttrium iron garnet (YIG) film on a gadolinium galliumgarnet (GGG) substrate and excited by loop-inductor transducers. We achieved this technology breakthroughby developing a YIG film etching process and fabricatingthick aluminum coplanar waveguide (CPW) inductor looparound each resonator to individually address and excite MSWs. At 4.77 GHz, the 0.68-mm 2resonator achieves a quality factor ( Q)>5000 with a bias field of 987 Oe. We also demonstrate YIG resonator tuning by more than one octave from 3.63 to 7.63 GHz by applying an in-plane externalmagnetic field. The measured quality factor of the resonatoris consistently over 3000 above 4 GHz. The micromachiningtechnology enables the fabrication of multiple single-and two-port YIG resonators on the same chip with allresonators demonstrating octave tunability and high Q. Index Terms —Magnetostatic wave (MSW), micromachin- ing, resonator, spin wave, yttrium iron garnet (YIG). I. INTRODUCTION THE advent of 5G and the desire for large bandwidth has brought the 3–30-GHz band into prominence [1]. RF MEMS piezoelectric film bulk acoustic resonators (FBARs) [2], the gold standard of 4G filter technology, do not scale favorably with 5G RF communication [3] because of reduced thickness, high metal resistance, and challenging lithography. On the other hand, electromagnetic (EM) wave-based resonators, such as microstrip lines, 3-Dmicromachined coaxial lines, and evanescent cavities [4], [5], are too large for chip-scale integration. Magnetostatic wave (MSW) resonators and filters are a promising technology to fill this gap [6]. MSWs exist in Manuscript received January 27, 2020; accepted June 1, 2020. Date of publication June 4, 2020; date of current version October 26, 2020. This work was supported by Defense Advanced ResearchProjects Agency (DARP A) under Contract HR0011-19-C-0017. Purdueco-authors also acknowledge s upport under Semiconductor Research Corporation (SRC) Contract 2018-LM-2830. The views, opinions and/orfindings expressed are those of the author and should not be interpretedas representing the official views or policies of the Department of Defense or the U.S. Government. This manuscript was approved for public release; distribution statement A; distribution unlimited. This manuscriptis not export controlled per ES-FL-011720-0016. (Corresponding author: Renyuan Wang.) Sen Dai is with the Department of Physics and Astronomy, Purdue University, West Lafayette, IN 47907 USA. Sunil A. Bhave is with the School of Electrical and Computer Engineer- ing, Purdue University, West Lafayette, IN 47907 USA. Renyuan Wang is with FAST Labs, BAE Systems, Inc., Nashua, NH 03060 USA (e-mail: renyuan.wang .@ .baesystems.com).ferromagnetic/ferrimagnetic materials. It is a lattice wave where the lattice consists of electron spin precessions. These waves possess two salient features making them attractive forrealizing chip-scale resonators and filters in the super-high- frequency (SHF) band. First, the group velocity is on the order of 1000 km/s and is a strong function of magneticbias applied to the material [7]. Therefore, the device size does not scale to extremely small dimensions with increased operating frequency. Second, the material loss limited Qof MSW resonator is theoretically frequency independent. Single- crystal yttrium iron garnet (YIG) exhibits the lowest dampingfor MSW, with a material loss limited Q>10 000 for frequencies in the UHF to Kabands [8], [9] has demonstrated a YIG MSW resonator can reach Q>3000 in the X-band. State-of-the-art MSW resonators are constructed from bond- ing a YIG-on-gadolinium gallium garnet (GGG) substrate onto another low dielectric loss substrate with λ/4 orλ/2 planar transmission lines to excite the MSW resonant mode [6], [9].Such an approach leads to a centim eter-scale resonator forfeit- ing the YIG’s advantage over a conventional EM resonator. This poses challenges in monolithic integration and miniatur- ization of multiple MSW devices and limits its applicationin higher order MSW filters and multiplexers. In this article, we designed a novel MSW resonator that consists of a YIG thin-film mesa as well as a new loop-inductor transducer struc- ture to efficiently excite the MSW [see Fig. 1(a) ], leading to significantly reduced resonator size. We also developed a new microfabrication process to fabricate multiple MSW resonators on the same chip, as shown in Fig. 1(b) . The combination of novel microfabrication process with the significantly reduced resonator size provides the freedom to design different MSW resonators with different wavelengths on a single chip so that they have different resonant frequencies even with the same magnetic bias. This provides the potential of monolithic high-order MSW filters and multiplexers with a small externalmagnet. II. R ESONATOR DESIGN AND MODELING In this work, 3- μm liquid-phase epitaxy (LPE) YIG film grown on 500- μm (111)-oriented GGG substrate was used. The saturation magnetization ( Ms)of YIG is ∼1750 G, and the gyromagnetic ratio γμ 0is 2.8 MHz/G. A thin-film ferromagnetic/ferrimagnetic structure can sup- port three different types of MSW—magnetostatic forward volume wave (MFVW), magnetostatic backward volume wave (MBVW), and magnetostatic surface wave (MSSW) [7]. Withthe in-plane external magnetic bias field applied in parallelFig. 1. (a) Top view of the chip with multiple MSW YIG resonators that are designed to operate from 4.54 to 4.58 GHz with the same 900-Oe dc magnetic bias [Devices ①–⑥].(b)Schematic of MSW res- onator ①as marked in red. with the wavenumber k, MBVW will be excited in the YIG film. Suppose that the YIG film has an infinite lateral size, thus ignoring the in-plane boundary, the wave amplitude distributes sinusoidally through the volume of the film, and the dispersionrelation of the lowest order mode ( n=1) is ω 2=ω0/bracketleftbigg ω0+ωM/parenleftbigg1−e−kd kd/parenrightbigg/bracketrightbigg (1) where ω0=μ0γHdc eff,ωM=μ0γMS,ωis the frequency of the MBVW, kis the wave vector of the MBVW and has an in-plane direction, dis the film thickness, γis the gyromagnetic ratio that is fundamentally related to electroncharge to mass ratio, μ 0is the vacuum permeability, Hdc effis the magnitude of the effective torque-exerting dc bias field internalto the material, and M Sis the magnitude of the saturation magnetization [10]. For the lowest order mode considered inthis manuscript, kd1. Therefore, for a fixed wavelength, the frequency tuning sensitivity is dω dHdc eff=dω dω0dω0 dHdc eff≈1 22ω0+ωM /parenleftbigω2 0+ω0ωM/parenrightbig1 2μ0γ. (2) The saturation magnetization of YIG film is ∼1750 G [11], and the gyromagnetic ratio γμ 0is 2.8 MHz/G.In contrast with conventional method of exciting MBVW through centimeter long transmission lines, a simple planar loop-inductor design is proposed. The inductor loop generates an RF magnetic field perpendicular to the YIG film. As theexcitation efficiency of MSW depends on the overlap integral between the RF H-field and the MSW mode profile, the induc- tor loop provides strong coupling as well as realizes a much smaller resonator size. A simple analogy between bulk acoustic wave and MBVW could be used to further illustrate the working principle of MBVW resonator and the function of inductor loop as follows. 1)An external magnetic bias Habove the saturation field is needed to align the magnetic spin momentum ( M) with the external magnetic bias, which corresponds to the poling process of piezomate rial using an electric field Eto reach saturation polarization ( P)regime. 2)An open circuit, as a capacitor, is used to excite theacoustic wave and the driving force is the charge- induced electric field Eperturbation on the polariza- tion P. In comparison, a short circuit, as an inductor loop we proposed here, is used to excite the spin waveand the driving force is the RF current-induced magnetic field Hperturbation on the magnetic momentum M. 3)In piezomaterial, t his electric field ( E)-induced pertur- bation on Pinduces a coherent movement of atoms of the lattice out of their equilibrium positions, which is called an acoustic wave. In ferromagnetic/ferrimagnetic material, the magnetic field ( h)-induced perturbation on Minduces a coherent spin precession, which is called MSW. Despite the similarities between acoustic wave and MSW, MSW resonator possesses a few advantages over acoustic wave resonators in the SHF band. First, typical magnetic wave velocity is around 1 ×10 5to 1×106m/s, leading to a device size of a few millimeters to submillimeters and is easily tunedby an external magnetic bias field to fit different sizes and dif-ferent frequencies. However, a typical acoustic wave velocity is around 5000–10 000 m/s, requiring a submicrometer-level fabrication and is hard to tune because it requires changingYoung’s modulus of the piezomaterial. Second, in many types of acoustic resonators, the material loss limited resonator f×Qproduct is generally a constant, resulting in intrinsic quality factor degradation at higher frequency [12]. For ferri- magnetic/ferromagnetic material, the material loss limited Q of MSW resonator is theoreti cally frequency independent, as no thermoelastic damping exists in coherent spin pre-cession. Finally, while the acoustic wave exists in all solidmaterials, MSW only exists in ferromagnetic/ferrimagnetic materials. Therefore, the material and design of the transducer for acoustic wave devices need to be carefully optimizedto achieve a balance between material acoustic loss, energy confinement, and electrical resi stance. On the other hand, for MSW, many low resistivity metal materials do not supportMSW transportation. This significantly relaxes the constraint for optimizing toward low parasitic electrical resistance fromthe transducers. The designed MSW resonator devices consist of two YIG film mesas wrapped around by electrode loop inductorsFig. 2. Device performance simulation using HFSS with the magnetic bias of 950 Oe. The simulated structure mimics the exact physical construct of actual MBVW resonator where two YIG islands sit on a GGG substrate with loop inductors wrapped around them. Inset: schematic of the simulated structure. (seeFig. 2 ). As will be described in Section III, the YIG mesas are formed by a unique ion mill etching technique. Because the GGG substrate does not support spin-wave transportation, this provides 3-D energy confinement of the MSW energy (Fig. 3 ) similar to that of AlN FBARs. A single YIG film island and its associated loop inductor can be considered as a single MSW resonator, and the device in Fig. 2 can be considered as two MSW resonators connected in parallel. Sucha balanced structure is to facilitate the characterization of thesedevice, to avoid the parasitics from converting the balanced RF signal from the coplanar waveguide (CPW) to a single-ended resonator structure. The width of the YIG resonator is 200 μm, and the length of the YIG resonator is 1700 μm. The width of the CPW trace is 100 μm. As submicrometer-thick epitaxial YIG films tend to exhibit higher intrinsic damping for MSW due to the crystal defects [13], we opt for a 3-μ m-thick YIG film to optimize toward high Qoperation. In addition, the transducer is intentionally separated from the YIG mesaby 5μm to prevent potential MSW damping caused by spin- wave pumping from YIG to metallic electrode material [14]. III. F ABRICATION The key process steps for the MSW resonator fabrication are shown in Fig. 4 . A layer of photoresist was patterned on the YIG film as mask for ion mill etching. As the etching selectivity between photoresist and YIG was tested to be close to 1:1, the photoresist thickness was chosen to be 5μm in order to ensure sufficient masking as well as prevent the transfer of PR surface morphology to the YIG etching sidewall. With an op timized recipe, we were able to achieve an etching rate of 32.6 nm/min and at the sametime retain a vertical sidewall angle with intermissive cooling cycles to avoid PR burning. After ion milling, the chip wastransferred to acetone and sonicated for 30 min to remove the hardened PR mask. The hardening of PR was due to Ar plasma exposure and high-temperature during etching. Another 30 min Fig. 3. Mode profile (time-varying component of the magnetization vector) of the MSW in the proposed structure. Fig. 4. (a) 3-µm single-crystal (111) YIG is grown on the GGG substrate. (b)Photoresist with 5- µm thickness has been patterned as a mask for YIG ion milling. (c)YIG is etched by ion milling at an etching rate of 32.6 nm/min with cooling cycles. (d)PR mask and resputtered YIG is removed for clean YIG patterning. (e)Thick photoresist mask has been patterned through lithography. (f)2µm of Al is e-beam deposited followed by a liftoff process. of phosphoric acid soak at 80◦C was implemented to remove most of the resputtered YIG around the sidewall. The SEMof the etched YIG sidewall is shown in Fig. 5 .A2 -μm-thick Al electrode was defined by the liftoff process to minimizeelectrical resistive loss. To ensure the proper lift-off of the2-μm-thick metal, we developed an ultrathick PR recipe. The Al electrode is deposited using e-beam evaporation with10-min cooling cycles between each of the 400-nm depositionintervals to avoid PR overheating and repeated metal soak process to facilitate source metal reflow. Fig. 5 shows the SEM of the fabricated MW resonator device as well as the top viewof one resonator device. IV. M EASUREMENT A three-axis projection magnet (GMW Magnet System 5201 Model) is used to generate the magnetic bias field and thedevice is placed 2 mm above the center of the magnet projector to ensure that only in-plane magnetic field is applied to theFig. 5. SEM images of YIG ion milling etching sidewall view (top) and top view of the MSW resonator (bottom). Fig. 6. Photograph of the MSW resonator testing setup (left) and zoomed-in view of RF probe station (right). device. The magnetic field is calibrated using a three-axis Hall sensor. The measured magnetic field uniformity is within ±1% over an area of 20 mm2, where the device is placed to ensure that it is uniformly biased. The scattering parameters of the devices were measured using a network analyzer (Agilent PNA-L N5230A) with 20 000 sampling points in a 2-GHz scan with a resolution bandwidth of 100 kHz (see Fig. 6 ). The impedance (as measured, no deembedding was performed) of a one-port resonator from 4.0 to 5.5 GHzwith a magnetic bias of 987 Oe is measured and shown in Fig. 7 . The measured response matches with our simulations. The main resonant frequency is at 4.770 GHz and the Fig. 7. Magnitude (top) and phase (bottom) of the impedance of the one-port MSW resonator. 3-dB bandwidth is 0.907 MHz, resulting in a quality factor Q=fresonance /f3d Bof 5259. To the best of our knowledge, this is the highest Qdemonstrated by MSW devices to date. The impedance at resonance is 237 , which translates to an impedance of 474 for single YIG island resonator. The figure of merit (FOM) f×Qis 2.51×1013surpasses that of acoustic wave resonator counterparts [15]. From the measured 4.770-GHz resonance under 987-Oe magnetic bias, we could back-calculate that at 950 Oe, the resonance should be at 4.653 GHz using (1). The mismatch inresonant frequency is due to the fact that the effective magnetic bias from shape anisotropy and magnetocrystalline anisotropy was not accounted for in the simulation. The resonator can bemodeled as a parallel RLC circuit in series with a resistance and an inductance, as shown in Fig. 8 . The series inductance and resistance are from the electrical loop inductance and the electrical resistance of the transducer, respectively. The parallel RLC is the equivalent circuit for the resonance in the MSW domain. The extracted coupling factor is 0.206% (k 2 t=(π2/4)((fp−fs)/fp))with a resistance of 4.66  from the inductor loop. This higher-than-expected electricalFig. 8. Curve fitting of the resonant peak. Inset: equivalent circuit model. Fig. 9. Measured |Z11|of the MSW resonator under different magnetic bias fields from 987 to 1860 Oe. resistivity causes additional loading of the quality factor and explains the deviation of measured impedance (237 )from the simulation result (810 ). |Z11|of the one-port MSW resonator under different magnetic biases is measured and plotted in Fig. 9 . The dc magnetic bias was stepped from 705 to 1860 Oe, the frequencyof the main resonance increased accordingly by more than oneoctave from 3.630 to 7.626 GHz. Qcalculated from the 3-dB bandwidth as a function of magnetic bias is plotted in Fig. 10 , as well as the frequency of the main resonance a functionof magnetic bias. As shown in Fig. 10 , the tuning efficiency changes from 3.367 to 2.96 MHz/Oe as the magnetic biasincreases from 705 to 1860 Oe, which is consistent with (2). The parasitic resonance from the pad’s parasitic capacitance and the inductance of the loop ge nerates a strong resonating current in the inductor, thus boosting the coupling of MSW. As the electrical impedance of MSW is determined by coupling and Q, this leads to large impedance variation, which Fig. 10. MSW resonator resonance frequenc y and extracted quality factor. is also reflected in Fig. 9 . Interestingly, Qvaries with applied magnetic bias. At low bias, the device structure supportsspin-wave modes that are half of the frequency of the main MBVW mode, and therefore, parametric spin-wave pumping by the MBVW is allowed, which degrades Q[16]. As the bias increases, these half-frequency spin-wave modes are no longerpermitted, and Qincreases. On the other hand, due to the unexpected high resistivity from the electrodes and the contact resistance, at even higher bias (therefore higher resonant frequency), the electrical resist ance becomes significant and starts to limit Q.This can be prevented by switching to a different electrode material an d better process control. Except in the spin-wave pumping regime, the extracted Qvaluers are consistently above 3000, and the highest Qvalue achieved is 5259. As predicted in simulations, a salient feature of MBVW resonators is that the spurious modes appear at the lowerfrequency side of the main re sonance due to the abnormal dispersion of MBVW, which is quite similar to bulk acoustic wave devices with a type II dispersion [17]. The measured temperature coefficient of frequency (TCF) of the MSW resonator under different biases is shown in Fig. 11 . Interestingly, the TCF varies from −996 to −1440 ppm/K as the dc magnetic bias decreases. This is because among theeffects that contribute to TCF (such as temperature depen- dence of magnetocrystalline anisotropy, thermal stress-inducedmagnetocrystalline anisotropy, and magnetization), the domi- nating effect is the temperature dependence of the saturation magnetization. As shown previ ously, the resonant frequency can be approximated by ω=(ω 0(ω0+ωM))1/2forkd1. Therefore, the TCF can be approximated by dω dT=1 2 1+ωM ω0·dωM dT. (3) As shown in (3), the TCF increases as the dc bias decreases.Although the TCF of MSW devices is higher than that of typical BAW and SAW devices, which typically ranges from a few tens of ppm/K to ∼100 ppm/K, the temperature stabilityFig. 11. Measured relationship between resonant frequency and temperature under different biases within the operating frequency range (with linear fitting). TABLE I COMPARISON OF THISWORK WITHSTATE -OF-THE-ART MSW R ESONATORS can be improved by leveraging the tunability of MSW devices with the tradeoff with system complexity. Table I summarizes the performance of the device presented in this article, and itscomparison with state of the art. With our novel design and fabrication method, much higher Qis achieved with much smaller sizes, and similar tuning efficiency and TCF. V. C ONCLUSION A novel MSW resonator structure consists of patterned YIG film and inductor loop transducer has been designed and fabricated with a novel microfa brication process on a chip. The lateral dimension of a single-patterned YIG structure is 1700μm×200μm. The designed MSW resonator is tuned from 4.787 to 7.626 GHz, with measured quality factor atresonance frequency higher t han 3000 across the whole tuning range. The f×Qproduct of these devices is significantly higher than their acoustic counterparts. The small device sizemade possible by our novel fabrication process and transducer design enables single-chip integration of multifrequency devices, which facilitates the realization of chip-scalehigh-order MSW filters, multiplexers, circulators [22],microwave-to-optical converters [23], and quantum coherent spin-magnon transducer [24]. The YIG micromachining process and resonator design are not limited to the GGGsubstrate. They can be directly ported to layer-transferred YIG thin-film technologies [25]. Lev eraging the recent advances in integrated magnetic materials, a small permanent magnet (such as a screen-printed magnet [26]) is sufficient for providing homogeneous magnetic bias for these devices to operate in SHF bands because of the small size of these resonators, thus enabling a chip-scale SHF multiplexing solution. A CKNOWLEDGMENT The authors would like to thank the staff at Purdue’s Birck Nanotechnology Center for their technical support. They would also like to thank Tingting Shen for discussionsabout fabrication and Yiyang Feng for discussions about SEM imaging. R EFERENCES [1]World Radiocommunication Conference 2019 . Accessed: Nov. 25, 2019. [Online]. Available: https://www.itu .int/en/ITU-R/conferences/wrc/2019/ Pages/default.aspx [2]R. Ruby, “The ‘how & why’a deceptively simple acoustic resonator became the basis of a multi-billion dollar industry,” in Proc. IEEE 30th Int. Conf. Micro Electro Mech. Syst. (MEMS) , Jan. 2017, pp. 308–313. [3]S. Mahon, “The 5G effect on RF filter technologies,” IEEE Trans. Semicond. Manuf. , vol. 30, no. 4, pp. 494–499, Nov. 2017. [4]C. Hermanson, R. Reid, and W. Stacy, “Ultra-compact four-channel 5–18 GHz switched filter bank utilizing polystrata microfabrication and 3D packaging,” in Proc. Int. Symp. Microelectron. , 2017, no. 1, pp. 000040–000045. [5]M. S. Arif, W. Irshad, X. Liu, W. J. Chappell, and D. Peroulis, “A high- Q magnetostatically-tunable all-silicon evanescent cavity resonator,” inIEEE MTT-S Int. Microw. Symp. Dig. , Jun. 2011, pp. 1–4. [6]G.-M. Yang, J. Wu, J. Lou, M. Liu, and N. X. Sun, “Low-loss magnetically tunable bandpass filters with YIG films,” IEEE Trans. Magn. , vol. 49, no. 9, pp. 5063–5068, Sep. 2013. [7]W. S. Ishak and K.-W. Chang, “Tunabl e microwave resonators using magnetostatic wave in YIG films,” IEEE Trans. Microw. Theory Techn. , vol. MTT-34, no. 12, pp. 1383–1393, Dec. 1986. [8]W. J. Keane, “Narrow-band YIG filters aid wide-open receivers,” MicroWaves , vol. 17, pp. 50–54, Sep. 1978. [9]R. Marcelli, P. De Gasperis, and L. Marescialli, “A tunable, high Q magnetostatic volume wave oscillator based on straight edge YIG res- onators,” IEEE Trans. Magn. , vol. 27, no. 6, pp. 5477–5479, Nov. 1991. [10] D. D. Stancil, Theory of Magnetostatic Waves . New York, NY , USA: Springer, 2012. [11] J. L. V ossen, Thin Films for Advanced Electronic Devices: Advances in Research and Development . New York, NY , USA: Academic, 2016. [12] R. Tabrizian, M. Rais-Zadeh, and F. Ayazi, “Effect of phonon inter-actions on limiting the f.Q product of micromechanical resonators,” in Proc. TRANSDUCERS-Int. Solid-State Sensors, Actuat. Microsyst. Conf. , Jun. 2009, pp. 2131–2134. [13] G. Gurjar, V . Sharma, S. Patnaik, and B. K. Kuanr, “Structural and magnetic properties of high quality single crystalline YIG thin film:A comparison with the bulk YIG,” in Proc. DAE Solid State Phys. Symp. Melville, NY , USA: AIP Publishing LLC, 2019, vol. 2115, no. 1, Art. no. 030323. [14] S. A. Manuilov et al. , “Spin pumping from spinwaves in thin film YIG,” Appl. Phys. Lett. , vol. 107, no. 4, Jul. 2015, Art. no. 042405. [15] M. Ghatge, G. Walters, T. Nishida, a nd R. Tabrizian, “High-Q UHF and SHF bulk acoustic wave resonators with ten-nanometer Hf 0.5Zr0.5O2 ferroelectric transducer,” in Proc. 20th Int. Conf. Solid-State Sensors, Actuat. Microsyst. Eurosensors XXXIII (TRANSDUCERS EUROSEN-SORS XXXIII), Jun. 2019, pp. 446–449. [16] D. M. Pozar, Microwave Engineering . Hoboken, NJ, USA: Wiley, 2009. [17] H. Suhl, “The nonlinear behavior of ferrites at high microwave signal levels,” Proc. IRE, vol. 44, no. 10, pp. 1270–1284, Oct. 1956.[18] J. P. Castéra and P. Hartemann, “Magnetostatic wave resonators and oscillators,” Circuits, Syst., Signal Process. , vol. 4, nos. 1–2, pp. 181–200, Mar. 1985. [19] K.-W. Chang and W. S. Ishak, “Magnetostatic forward volume wave straight edge resonators,” in IEEE MTT-S Int. Microw. Symp. Dig. , Jun. 1986, pp. 473–475. [20] J. M. Owens, C. V . Smith, E. P. Snapka, and J. H. Collins, “Two-port magnetostatic wave resonators utilizing periodic reflective arrays,” in IEEE MTT-S Int. Microw. Symp. Dig. , Jun. 1978, pp. 440–442. [21] W. R. Brinlee, J. M. Owens, C. V . Smith, and R. L. Carter, “‘Two-port’ magnetostatic wave resonators utilizing periodic metal reflective arrays,” J. Appl. Phys. , vol. 52, no. 3, pp. 2276–2278, 1981. [22] N. Zhu, X. Han, C.-L. Zou, M. Xu, and H. X. Tang, “Magnon-photonstrong coupling for tunable m icrowave circulators,” P h y s .R e v .A ,G e n . Phys., vol. 101, no. 4, Apr. 2020, Art. no. 043842. [23] N. Zhu et al. , “Waveguide cavity optomagnonics for broadband multimode microwave-to-optics conversion,” 2020, arXiv:2005.06429 . [Online]. Available: http://arxiv.org/abs/2005.06429 [24] D. R. Candido, G. D. Fuchs, E. Johnst on-Halperin, and M. E. Flatté, “Predicted strong coupling of solid-state spins via a single magnonmode,” 2020, arXiv:2003.04341 . [Online]. Available: http://arxiv.org/ abs/2003.04341 [25] H. S. Kum et al. , “Heterogeneous integration of single-crystalline complex-oxide membranes,” Nature , vol. 578, no. 7793, pp. 75–81, Feb. 2020. [26] T. Speliotis et al. , “Micro-motor with screen-printed rotor magnets,” J. Magn. Magn. Mater. , vol. 316, no. 2, pp. e120–e123, Sep. 2007. Sen Dai received the B.Sc. degree in physics from the University of Science and Technologyof China, Hefei, Anhui, China, in 2014. He is currently pursuing the Ph.D. degree with the Department of Physics and Astronomy, PurdueUniversity, West Lafayette, IN, USA. He joined Prof. Sunil Bhave’s OxideMEMS Lab in January 2017. His research is focused onRF MEMS resonators and micromachining ferritecomponents. Sunil A. Bhave (Senior Member, IEEE) received the B.S. and Ph.D. degrees in electricalengineering and computer sciences from the University of California at Berkeley, Berkeley, CA, USA, in 1998 and 2004, respectively. He was a Professor with Cornell University, Ithaca, NY , USA, for ten years and worked at Analog Devices, Woburn, MA, USA, for five years. In April 2015, he joined the Departmentof Electrical and Computer Engineering, Purdue University, West Lafayette, IN, USA, where he is currently the Associate Director of operations at the Birck Nanotechnology Center. He is a Co-Founder of Silicon Clocks, Fremont,CA, USA, that was acquired by Silicon Labs, Austin, TX, USA, in April 2010. His research interests focus on the interdomain coupling in optomechanical, spin-acoustic, and color center-MEMS devices. Dr. Bhave received the NSF CAREER Award in 2007, the DARP A Y oung Faculty Award in 2008, the IEEE Ultrasonics Society’s Y oungInvestigator Award in 2014, and the Google Faculty Research Awardin 2020. His students have received best paper awards at the IEEEPhotonics 2012, the IEEE Ultrasonics Symposium 2009, and IEDM 2007. Renyuan Wang (Member, IEEE) received the B.S. degree from the Harbin Institute of Technol- ogy, Harbin, China, in 2007, the M.S. degree from the University of Massachusetts at Dartmouth,North Dartmouth, MA, USA, in 2010, and the Ph.D. degree from Cornell University, Ithaca, NY , USA, in 2014. From 2015 to 2017, he was a Research and Development Engineer with the BAW RnD Group, Qorvo, Apopka, FL, USA. He then joined the FAST Labs, BAE Systems, Inc., Nashua, NH 03060 USA, where he is currently a Senior Scientist. He worked in developing high-dynamic-range coherent RF photonic radar front ends, lithium niobate thin-film devices for applications in RF MEMS,optomechanics, nonlinear optics, and inertial measurement units, as wellas aluminum nitride bulk acoustic w ave resonators and filters for personal mobile wireless devices. His current research interests focus on MEMSdevices exploiting ferroelectric, ferro/ferrimagnetic materials, and theirintercouplings.

2020-10-24

We have designed, fabricated, and characterized magnetostatic wave (MSW) resonators on a chip. The resonators are fabricated by patterning single-crystal yttrium iron garnet (YIG) film on a gadolinium gallium garnet (GGG) substrate and excited by loop-inductor transducers. We achieved this technology breakthrough by developing a YIG film etching process and fabricating thick aluminum coplanar waveguide (CPW) inductor loop around each resonator to individually address and excite MSWs. At 4.77 GHz, the 0.68 square mm resonator achieves a quality factor Q > 5000 with a bias field of 987 Oe. We also demonstrate YIG resonator tuning by more than one octave from 3.63 to 7.63 GHz by applying an in-plane external magnetic field. The measured quality factor of the resonator is consistently over 3000 above 4 GHz. The micromachining technology enables the fabrication of multiple single- and two-port YIG resonators on the same chip with all resonators demonstrating octave tunability and high Q .

Octave-Tunable Magnetostatic Wave YIG Resonators on a Chip

2010.12732v1

Magnon-assisted magnetization reversal of Ni81Fe19nanostripes on Y 3Fe5O12with different interfaces Andrea Mucchietto,†Korbinian Baumgaertl,‡and Dirk Grundler∗,‡,¶ †´Ecole Polytechnique F´ ed´ erale de Lausanne (EPFL), 1015 Lausanne, Switzerland, Laboratory of Nanoscale Magnetic Materials and Magnonics, Institute of Materials (IMX ) ‡´Ecole Polytechnique F´ ed´ erale de Lausanne (EPFL), 1015 Lausanne, Switzerland, Laboratory of Nanoscale Magnetic Materials and Magnonics, Institute of Materials (IMX) ¶’Ecole Polytechnique F´ ed´ erale de Lausanne (EPFL), 1015 Lausanne, Switzerland, Institute of Electrical and Micro Engineering (IEM) E-mail: dirk.grundler@epfl.ch Abstract Magnetic bit writing by short-wave magnons without conversion to the electri- cal domain is expected to be a game-changer for in-memory computing architectures. Recently, the reversal of nanomagnets by propagating magnons was demonstrated. However, experiments have not yet explored different wavelengths and the nonlinear excitation regime of magnons required for computational tasks. We report on the magnetization reversal of individual 20-nm-thick Ni 81Fe19(Py) nanostripes integrated onto 113-nm-thick yttrium iron garnet (YIG). We suppress direct interlayer exchange coupling by an intermediate layer such as Cu and SiO 2. Exciting magnons in YIG with wavelengths λdown to 148 nm we observe the reversal of the integrated ferromagnets in a small opposing field of 14 mT. Magnons with a small wavelength of λ= 195 nm, 1arXiv:2312.15107v1 [cond-mat.mes-hall] 22 Dec 2023i.e., twice the width of the Py nanostripes, induced the reversal at an unprecedentedly small spin-precessional power of about 1 nW after propagating over 15 µm in YIG. Considerations based on dynamic dipolar coupling explain the observed wavelength dependence of magnon-induced reversal efficiency. For an increased power the stripes reversed in an opposing field of only about 1 mT. Our findings are important for the practical implementation of nonvolatile storage of broadband magnon signals in YIG by means of bistable nanomagnets without the need of an appreciable global magnetic field. Keywords spin waves, magnetization reversal, YIG, broadband spectroscopy, magnetic interfaces 2Collective spin excitations in a magnetically ordered material are called spin waves (SWs) or, in quantum-mechanical terms, magnons. By means of SWs, angular momentum is trans- ferred without electrical charge motion, hence no Joule heating is generated. Therefore, SWs represent a new paradigm for signal processing at low power consumption and for a non- charge-based beyond-CMOS technology.1–3In magnonic applications, microwave signals are applied to integrated coplanar waveguides (CPWs) and excite coherent SWs in the adjacent magnetic layer. Grating couplers consisting of ferromagnetic nanoelements [Fig. 1(a)] have proven to enhance the microwave-to-magnon coupling at GHz frequencies if integrated to CPWs.4–7They emit and detect magnons with wavelengths λdown to below 50 nm in ferri- magnetic yttrium iron garnet (YIG).8,9Wang et al. explored the spin wave emission from a ferromagnetic stripe into YIG.10They explained the strong spin-wave signal in the underlying YIG by prominent dipole-dipole interaction without assuming spin currents. Recently, it has been reported that dipolar SWs reversed 100-nm-wide ferromagnetic nanostripes deposited directly on YIG after propagating over 25 micrometers.11The magnon-induced switching of Ni81Fe19(Py) nanostripes on YIG occurred in the linear excitation regime at low microwave power. However, the wavelength λused for switching remote nanostripes was a few microm- eters long. Such value of λis not adequate for nanomagnonic in-memory computing in either the linear or nonlinear excitation regime.12–14 In this work we report remote switching of 100-nm-wide Py nanostripes by magnons with λdown to 148 nm in YIG. We explore different interfaces and, both, the linear and nonlinear excitation regime. The Py nanostripes were integrated on an intermediate layer of either Cu or SiO 2on YIG. Thereby we suppressed the direct exchange coupling15between Py and YIG. Using the identical nanostripe design, we compare our results to Ref.11in which an intermediate layer between Py and YIG was avoided. Using broadband spectroscopy [Fig. 1(d) to (f)] and spatially resolved Brillouin light scattering (BLS) we acquire magnon spectra before and after exciting propagating magnons of different λin YIG. We observe irreversible changes in BLS spectra which indicate reversed states of Py magnetization vectors MPy 3xzCPW1SW irf YIGCPW2 ΔS ΔS -40 1f [GHz]Py stripes hrf(a) (b) (c)yx Bare YIG →MYIG→MPy Hµ→ →MYIG→MPyPy stripes21 1112 0 -140 Field [mT]-40 5 -5Field [mT]40 k1k1k1 +G 0Hµ→ 0µ H0c212 0 12 0Hµ→ 0 µ H0c1(d) (e) (f)k1+PSSW1 µ H0c2 µ H0c1 µ H0c2Figure 1: (a) Schematic device with the two CPWs, Py stripes (gratings) and microwave tips connected to a VNA. A current irfat frequency firris injected into CPW1. The generated fieldhrfexcites magnons. Sketches of the (b) anti-parallel (AP) and (c) parallel (P) magnetic configuration of Py nanostripes and YIG. Color-coded spectra ∆ S11(left) and ∆ S21(right) taken as a function of field from -90 mT to +90 mT on sample A for powers Pirrof (d) −30 dBm, (e) −15 dBm, and (f) 0 dBm. The horizontal arrow in (d) indicates the magnetic field sweep direction. We display ∆ S, i.e., the difference of scattering parameters Sthat are taken at subsequent field values. Intense and dark colors indicate magnon resonances. Labels and symbols highlight specific resonances and critical fields. The black (red) vertical lines indicate 24 mT (40 mT). In (f, right) the AP branch is not resolved indicating HC1is (close to) zero. 4which we attribute to magnon-induced switching in a small opposing field. We analyze the power absorbed by the precessing spins in YIG and find that propagating magnons whose wavelength is twice the nanostripe width show the minimum power level of about 1 nW representing the highest reversal efficiency. Our findings go beyond earlier reports in that we (i) demonstrate experimentally that dynamic dipolar coupling between Py and YIG is suffi- cient for magnon-induced reversal, (ii) explain the wavelength dependent reversal efficiency, and (iii) report switching by propagating magnons with a wavelength of only 148 nm. (iv) Our BLS data reveal the magnon-induced reversal in the non-linear excitation regime which we attribute to parametrically pumped magnons. Our findings pave the way for future in- memory computation in linear and non-linear nanomagnonics with materials combinations which do not require direct exchange coupling between the magnetic elements. Results We fabricated one-dimensional (1D) periodic arrays of Py nanostripes (gratings) on 113-nm- thick YIG which was commercially available from the same supplier as in Refs.7,16,17The stripes consisted of 20-nm-thick Py and were 100 nm wide. They are arranged with a period ofp= 200 nm. The stripe lengths were consistent with Ref.11and alternated between 25 and 27 µm. The total width of a grating amounted to wGC= 10 µm. They were fabricated on YIG with a 5-nm-thick intermediate layer of either Cu (sample A) or SiO 2(sample B). We introduced the intermediate layers to intentionally modify the coupling between Py and YIG compared to Ref.11where Py had been deposited directly on YIG. The intermediate layers suppress the exchange coupling. Moreover, in sample B the SiO 2spacer avoids the spin pumping mechanism thus allowing for only dipolar coupling between Py and YIG. In sample A, the Cu spacer thickness is smaller than the spin diffusion length18and a spin pumping related torque could occur in addition to dipolar coupling.19In the following we denote the 5sample without an intermediate layer used in Ref.11as sample C. In the coplanar waveguides (CPWs) [Fig. 1(a)], the Au lines (gaps) were 2.1 µm (1.4 µm) wide. The distance between signal lines of two parallel CPWs was 15 µm. A finite element analysis using COMSOL Multiphysics provided the inhomogeneous radiofrequency (rf) field hrfof the CPWs (Fig. S1). Without the lattice of nanostripes, they excited and detected most efficiently spin waves with a wave vector kofk1= 0.87 rad /µm. An in-plane magnetic field Hwas applied to realize specific magnetic histories and controlled different relative orientations of magnetiza- tion vectors in Py ( MPy) and YIG ( MYIG) [Fig. 1(b) and (c)]. We performed broadband measurements (Methods) of scattering parameters ∆ S11(reflection) and ∆ S21(transmis- sion) with port 1 and port 2 of a vector network analyzer (VNA) connected to CPW1 and CPW2, respectively. We observed several resonant branches above the k1excitation in Fig. 1(d) to (f). Considering Refs.,5,7,8the additional high-frequency branches reflected grating coupler modes such as k1+G, with G= 2π/p, different orders of perpendicular standing spin waves (PSSWs), and the magnetic resonance in the Py nanostripes. The latter one was the prominent high-frequency branch in ∆ S11which started at about 10.5 GHz at -90 mT in Fig. 1(d). In Figs. S2 and S3 we report further spectra from which we extracted the quasi-static characteristics of samples A and B. We applied BLS microscopy ( µBLS) in that we focussed laser light for inelastic light scattering on Py nanostripes in the different gaps of a CPW. The laser spot diameter was about 400 nm. We note that the resonance frequency of Py nanostripes was high (low) if MPywas parallel (anti-parallel) to the applied field H (see below).20The BLS microscopy was used to gain spatially resolved information about Py nanostripe reversal and explore the non-linear regime which was not achieved in Ref.11 For obtaining the spectra ∆ S11and ∆ S21of sample A in Fig. 1(d) to (f) we applied H along the y-direction of sample C. We measured the scattering parameters in the following order: S11,S21,S22andS12. In the following we focus on spin waves which were excited at CPW1 and propagated to CPW2, i.e., we report spectra S11andS21, respectively. The spectra S22andS12showed consistent features when considering nonreciprocity and apply- 6ing an inverted magnetic history. We varied µ0Hfrom -90 mT to +90 mT [indicated by the black horizontal arrow in Fig. 1(d)] in steps of 2 mT. The nonreciprocal spin wave characteristics led to the large signal-to-noise ratios at positive H. The same measurement protocol was repeated for different VNA powers Pirr=−30,−15 and 0 dBm in Fig. 1 (from top to bottom). At small power, we interpreted the branches of Fig. 1(d) such that at small positive µ0Hbelow µ0HC1= 26 mT [Fig. 2(a)] the magnetization vectors of YIG and Py nanostripes were anti-parallel (AP) [Fig. 1(b)], in agreement with Co nanostripes on YIG reported in Ref.8In this field regime, the branch with negative slope d f/dH in ∆S11[marked by a grey circle in Fig. 1(d)] was attributed to the ferromagnetic resonance inside the Py nanostripes. Their magnetization vectors MPypointed still in −y-direction and against the applied positive field H. They were anti-parallel also with MYIGas YIG had a coercive field≤2 mT. At small applied power Pirr=−30 dBm, several of the grating coupler modes gained abruptly a pronounced signal strength at 40 mT (indicated by the red dashed line). We attributed this observation to the critical field µ0HC2[Fig. 2(a)] at which the reversal of the Py nanostripes underneath CPW2 (i.e., the detector CPW) occurred. For µ0H > 40 mT, all the detected branches in Fig. 1(d) were similar to the ones at the correspondingly large negative fields. These branches indicated that the magnetization vectors of Py nanostripe lattices underneath both CPWs were now parallel (P) with HandMYIG. Correspondingly, the transmission data showed the richest spectra of grating coupler modes [Fig. 1(d) on the right]. For the spectra ∆ S11shown in Fig. 1(e), we used a larger power Pirrof -15 dBm. In the transmission spectra ∆ S21[right panel in Fig. 1(e)], the AP branch ended at a smaller field value µ0HC1= 14 mT and the region P started near µ0HC2= 22 mT instead of 40 mT. The Py nanostripes underneath CPW1 and CPW2, respectively, experienced a smaller field region of anti-parallel alignment with MYIGcompared to Fig. 1(d). This observation indicated that the larger VNA power Pirrused for broadband spectroscopy led to the reversal of Py nanostripes underneath both CPW1 and CPW2. 7The onset of the P region occurred at an even smaller Hin Fig. 1(f) when Pirrof 0 dBm was used. The branches attributed to grating coupler modes in the P region showed a weak signal strength already at µ0H= 2 mT which increased with increasing H. This means that close to zero field the magnetization vector MPyunderneath CPW1 pointed into the +y-direction (P configuration), i.e., µ0HC1≤2 mT. The reversal field of the Py nanostripes under CPW1 of sample A was hence reduced by about 26 mT when applying Pirr= 0 dBm (1 mW) compared to Pirr=−30 dBm (1 µW). Such a large reduction of µ0HC1was not reported in Ref.11(sample C) which had Py directly deposited on YIG. To characterize the power-dependent switching field distribution for samples A and B 50 0 μ 0H C [mT](a) (b) 25 Pirr [dBm]-30 -15 0 Pirr [dBm]-30 -15 0μ0HC2 μ0HC1μ0HC2 μ0HC1 Figure 2: For samples (a) A and (b) B the critical fields µ0HC1andµ0HC2realizing 50% of the maximum signal strengths of the two relevant magnon branches are shown as a function ofPirr. The error bar refers to fields needed to achieve 30% and 70% of the maximum signal strengths of branches AP and P. (Fig. 2) we adopted the methodology developed in Ref.11We evaluated VNA spectra taken at many different powers Pirrand analyzed the field-dependent signal strengths of the first GC branch in the AP and P state. In such experiments, we applied irfcovering a broad frequency regime from about 10 MHz to 20 GHz. Assuming that the magnon mode for most efficient switching resided in this frequency regime, we obtained the minimum critical field values for reversal at a given Pirr. At each value of Pirr, we extracted the critical field values µ0HC1andµ0HC2that corresponded to 50% of the maximum signal strengths of the AP and P branches, respectively (symbols in Fig. 2). The difference ( HC2−HC1) reflected 8the distribution of switching fields of nominally identical Py nanostripes underneath the two CPWs. In sample A (Fig. 2a), the switching fields were distributed over a larger field range than in sample B (Fig. 2b) for Pirr<−20 dBm. We first consider the critical fields µ0HC1extracted from the AP branches of both sam- ples A and B. At low power, Pirr≤-25 dBm, µ0HC1is comparable within error bars in both samples. µ0HC1decreases to 0 ( ±1) mT in sample A (B) when Pirr≥-5 dBm (-10 dBm). We now focus on the critical fields of the P branch, i.e. µ0HC2. For Pirr<−20 dBm the critical fieldµ0HC2is larger for sample A than for sample B (cf. Fig. 2a and 2b). µ0HC2decreases asPirrincreases up to -5 dBm. µ0HC2is reduced to 2 mT in sample B when Pirr≥-10 dBm and it maintains this small value at larger Pirr. This is the smallest value so far detected for reversal of Py nanostripes on YIG induced by propagating magnons. This finding is one of the key achievements of this work. Near-zero critical fields were not be observed in Ref.11 For comparison, in sample A, the critical field µ0HC2is about 15 mT at −5 dBm. At the largest Pirr, it has increased again (Fig. 2a) and reaches ≈22 mT. Considering Ref.11we attribute this increase in critical fields of Py nanostripes underneath CPW2 at high Pirrto the nonlinear regime of magnon excitation underneath CPW1 with enhanced magnon scat- tering. Because of the scattering processes the magnon amplitudes after 15 µm are below the threshold for complete reversal of the nanostripe array under CPW2. The incomplete reversal at high power is observed for both samples A and C where spin pumping is allowed. In sample B (Fig. 2b) we do not observe an increase in critical fields at large powers. Instead, we find the largest reduction in critical fields HC1andHC2in sample B. Here, a 5-nm-thick SiO 2spacer rules out that spin pumping is relevant for the efficient magnon-induced reversal. We note that the insertion of both the SiO 2and Cu spacer excludes the transfer of exchange magnons which was assumed in Refs.21–24Our experiments highlight the importance of dy- namic dipolar coupling between Py and YIG when developing a microscopic understanding of the magnon-induced reversal mechanism. To quantify the power level at which a specific spin wave mode in YIG reversed nanos- 9P P37 37 fsens [GHz] fsens [GHz]-20 5Pirr [dBm] -20 5Pirr [dBm]2 8firr [GHz]-226-226 Pirr [dBm] Pirr [dBm] 0MAX MAX 0AP branch P branch (a) (d)(b) (c) (e) (f) 2 8firr [GHz]Figure 3: (a) Mag( S21) recorded on sample C (Cu spacer) between fsens= 3 and 7 GHz at +14 mT with a power of -25 dBm after applying a microwave signal with firr= 1.75 GHz to CPW1 for increasing power Pirr. Switching yield maps at +14 mT for sample C displaying color-coded (b) Mag( S11) and (c) Mag( S21) integrated as a function of fsensfor the AP and P branch respectively. The frequency integration range for the P branch is highlighted by the red dashed lines in (a). To extract the switching yield map for the AP branch, the first GC mode branch in the Mag( S11) spectrum is used. Panels (d) to (f) show the corresponding dataset for sample B (SiO 2spacer). Arrows indicate local minima in the power threshold inducing stripe reversal by specific magnons discussed in the text. tripes we followed the concept of switching yield maps (Methods) introduced in Ref.11 (Fig. 3). The samples were first saturated at -90 mT applied along the y-axis. Then, the field was gradually increased to +14 mT and kept constant. We provided powers Pirr ranging from -25 to +6 dBm with +1 dBm steps within a 0.25-GHz-wide frequency window starting at a specific frequency firr. After each power step and corresponding irradiation for 1 msec, the VNA power level was reduced to −25 dBm and the transmitted signal ( S21) was recorded as a function of frequency fsensranging from 3 to 7 GHz. Figure 3 displays such datasets in panels (a) and (d) as well as gray-scaled switching yield maps performed at +14 mT for sample A (top row) and sample B (bottom row). The maps labelled by AP (P) branch in Fig. 3b and e (Fig. 3c and f) reflect the reversal of Py nanostripes under- 10neath CPW1 (CPW2) of sample A and B, respectively. From these maps, we extracted the critical power levels for magnon-assisted switching at CPW1 and CPW2 which we denote byPC1andPC2, respectively. In Fig. 4, we particularly display the critical power values extracted near the local minima indicated by arrows in Fig. 3 reflecting modes k1,k1+G andk1+G+kPSSW1 (from left to right). When exciting the k1mode near 2 GHz in sample 103PC,prec [nW]k1+G+kPSSW1 10-2k1+G k1100 10-2PC [mW] firr [GHz]2 5 firr [GHz]2 5(a) (b) (c) (d) Figure 4: For samples A (red), B (blue), and C (black, magenta) the critical powers (a) PC1and (b) PC2, (c) PC1,precand (d) PC2,precare depicted for irradiation frequencies firrin half-logarithmic graphs. In (a) we label magnon modes by k1,k1+Gandk1+G+kPSSW1 . In (d) the values of sample C (magenta symbols) are scaled by a factor ρto correct for the different propagation path length and decay of magnon amplitudes between CPWs (Methods, paragraph C). Connecting lines are guide to the eyes. A and B at +14 mT, we require PC1between 30 and 40 µW for the reversal of 50% of the nanostripes below CPW1 [red symbols in Fig. 4(a)]. This power value is only about a factor of three larger than the one of sample C published in Ref.11[black symbol near 1.5 GHz in Fig. 4(a)]. For all samples, PC1andPC2increase with increasing mode frequency. For the reversal underneath CPW1 by means of the GC mode k1+G(excited between 2.75 and 3.5 GHz) VNA powers PC1of 400 to 800 µW are required. The reversal of Py nanostripes 11underneath CPW2 is achieved at a further increased power level PC2of up to 2.5 mW. For sample C, PC2was larger than 3 mW and not determined. To compare different samples, it is instructive to consider the power values PC1,prec[Fig. 4(c)] and PC2,prec[Fig. 4(d)] which quantify the power absorbed by the spin-precessional (prec) motion in YIG at the emitter CPW (Methods). These values consider that only part of the rf power applied by the VNA is absorbed by the spin system and converted into magnons. These values, taken at the same field, allow us to compare the different samples independent of the individual efficiency of microwave-to-magnon transduction. In case of the long-wavelength modes k1existing near 2 GHz in samples A and B, we observe reversal at power levels PC1,precbetween 4 to 10 nW [Fig. 4(c)]. PC2,precfor modes k1is only slightly larger attributed to a weak decay of the magnon mode between emitter and detector CPW. At larger excitation frequencies firrbetween 3 and 3.5 GHz corresponding to the first GC mode resonance k1+G, power values PC1,precare smaller by up to two orders of magnitude compared to modes k1in Fig. 4(c). Here, sample B realizes the smallest values PC1,precdown to about 0.5 nW. Note that despite larger coercivities in sample A the magnon-induced re- versal underneath CPW1 via mode k1+Gis realized at a smaller power than in sample C with the direct interface between Py and YIG. A similar small value of about 1 nW is found in Fig. 4(d) for reversal underneath CPW2, suggesting a weak decay of magnon amplitudes after a path of 15 µm. The key finding of Fig. 4(d) is that the mode k1+ 1Gin sample B is most efficient in terms of PC2,precand nanostripe reversal underneath CPW2. Considering its intermediate layer to be an insulator (SiO 2) the dipolar coupling between magnons in YIG and Py provides the torque for the reversal. We note that in sample B we observe nanostripe reversal by a further mode with wave- length λ= 148 nm corresponding to a wavevector k=p (k1+G)2+k2 PSSW1 (kPSSW1 is the first quantized magnon mode across the YIG thickness). When exciting this mode in the frequency range 4.75 to 5.25 GHz, we extract PC2,prec= 1.3 to 5.4 nW. These power values are increased compared to PC2,precfound near 3 GHz. For sample B, we observe the smallest 12spin-precessional power PC1,precfor reversal in Fig. 4(c) at 5.25 GHz. We explain the small value by the combined effect of magnon-induced reversal and microwave-assisted switching near the eigenresonance of the Py nanostripes underneath CPW1 before their reversal at +14 mT. For the nanostripes underneath CPW2 ( PC2,prec) the microwave-assisted switching does not play a role due their large separation from the CPW1 attached to the rf source. In the following we apply micro-focus BLS to sample A [Fig. 5(a)] and gain spatially y xH 4 1 0 30 k [rad/µm]f [GHz] 2 4 6 8 101.0E-42.0E-43.0E-4 Freq. (GHz) f [GHz]2 6 10Y’ XYirf CPW1magnon CPW2Py/Cu stripes5.9µm Pos 2.1 Laser(a) (b) (c)BLS pos. 2.12.2 24 mT, 4.3 GHz, Pos. 2.1 2 4 6 8 101.0E-42.0E-43.0E-4 Freq. (GHz)0300 BLS counts [arb. units] f [GHz]2 6 10XY(d) 24 mT, 1.25 GHz, Pos. 2.1 Figure 5: (a) Sketched cross-section of the device (top). The CPW lines (yellow) are on top of the stripes (dark grey) which have been fabricated on YIG (green). In BLS we detected thermally excited magnons in Pos. 2.1 (microscopy image) and 2.2. (b) Magnon dispersions in the thin YIG at 24 (solid lines) and 2 (dotted lines) mT calculated via the Kalinikos-Slavin formalism for two limiting configurations, i.e. Damon-Eshbach (blue lines) and backward volume (red lines) configuration. The horizontal dashed green line indicates 1.25 GHz which is below (inside) the magnon band at 24 mT (2 mT). Magnon spectra in Pos 2.1 at 24 mT before (black) and after (red) applying irfto CPW1 for Pirr= 16 dBm with (c) 4.3 GHz and (d) 1.25 GHz. Labels X, Y and Y’ indicate characteristic resonant modes. resolved information about the magnon modes that modify the magnetization vectors MPy of Py stripes. We do not evaluate absolute power values here as the BLS setup has not allowed for calibration, and CPWs were wire-bonded. We discuss BLS spectra reflecting the incoherent magnons excited thermally at room temperature. We compare spectra taken 132 4 6 8 101.0E-42.0E-43.0E-4 Freq. (GHz)2 4 6 8 101.0E-42.0E-43.0E-4 Freq. (GHz) 2 4 6 8 101.0E-42.0E-43.0E-4 Freq. (GHz)2 4 6 8 101.0E-42.0E-43.0E-4 Freq. (GHz)2 4 6 8 101.0E-42.0E-43.0E-4 Freq. (GHz) Pos 2.2 LaserCPW2 2 10 f [GHz] 2 10 f [GHz] 2 10 f [GHz] 2 10 f [GHz] 2 10 f [GHz]BLS counts [arb. units] 10030024 mT, 4.3 GHz, λ = 195 nm, Pos. 2.224 mT, 5 GHz,parametric p., Pos. 2.22 mT, 1.25 GHz,λ = 7.22 µm, Pos. 2.2(b) (d) (f)(a)24 mT, 5 GHz, parametric p., Pos. 2.22 mT, 1.25 GHz,λ = 7.22 µm, Pos. 2.2(c) (e)Figure 6: (a) Optical image when positioning the laser at Pos. 2.2. (b) Magnon spectra taken in Pos. 2.2 at 24 mT before (black) and after (red) applying irfto CPW1 with firr= 4.3 GHz. The reversal of Py nanostripes by magnons k1+Gis evidenced. Magnon spectra in (c) Pos. 2.1 and (d) Pos. 2.2 before (black) and after (red) applying irfwith firr= 5 GHz at CPW1. The black (red) arrows highlight characteristic modes (changes). Thermal magnon spectra in (e) Pos. 2.1 and (f) Pos. 2.2 before (black) and after (red) emitting magnons with k1 by applying irfwith firr= 1.25 GHz at CPW1. In all these experiments the irradiation power was Pirr= 16 dBm. The legends list relevant parameters and highlight the parametric pumping (p.) experiments. before (black curves) and after (red curves) applying microwaves to CPW1. We explore dif- ferent fields Hmodifying the spin-wave dispersion relation in YIG [Fig. 5(b)] and different firr. The laser wavelength (power) was 473 nm (0.8 mW). Given the laser spot diameter of about 400 nm, we collected the Stokes’s signal of magnons from up to two Py nanostripes and the underlying YIG. Each spectrum in Fig. 5(c) and (d) had an acquisition time of approximately 2 hours. The spectrum shown as the black curve in Fig. 5(c) displays magnon resonances existing in the gap of CPW2 in Pos. 2.1 for µ0H= 24 mT after saturation along −y-direction using µ0H=−84 mT and before applying irfto CPW1. The frequencies of resonances marked X and Y indicate that the Py nanostripes are anti-parallel to H[Fig. 1(b)]. They are consis- 14tent with the frequencies marked by brown and grey circles, respectively, in Fig. 1(d). After applying irfatfirr= 4.3 GHz to CPW1 with a nominal irradiation power Pirrof up to 39.8 mW (16 dBm) the red spectrum was obtained at the same position. Due to wire-bonded connections we expected the power in CPW1 to be a few dB lower than the nominal value. The red spectrum is markedly modified compared to the black curve: the resonance peaks X and Y reduced to the noise level, and a higher frequency resonance Y’ was resolved. The new peak in the red spectrum was consistent with the branch existing in the P configuration of the sample above the grey circle in Fig. 1(f). The microwave current applied to CPW1 with firr= 4.3 GHz hence led to the reversal of Py nanostripes at the remotely located CPW2. To investigate if heating of CPW1 by irfinitiated the reversal we followed the same mea- surement protocol as applied in Fig. 5(c) but changed the rf signal frequency to 1.25 GHz. The signal irfwas applied for two hours, before taking the red spectrum in Fig. 5(d). The red spectrum is found to contain the identical resonances as the black spectrum, i.e., MPy was not changed by applying an rf signal at 1.25 GHz. We explain the different spectra (red) in panels (c) and (d) of Fig. 5 by the dispersion relation of YIG at 24 mT displayed in Fig. 5(b). At firr= 1.25 GHz (green dashed line), magnons are not emitted into YIG as all allowed magnon bands reside at higher frequencies. This is different for firr= 4.3 GHz. Here, a Damon-Eshbach (DE) mode is allowed and excited at CPW1. It propagates to CPW2 as evidenced by the transmission spectra shown in Fig. 1. The allowed mode is the grating coupler mode k1+Gwith λ= 195 nm. The characteristic resonance Y’ in the red spectrum of Fig 5(c) evidences the reversal of Py nanostripes in Pos. 2.1 by the propagating magnon mode. In Pos. 2.2 located a few micrometers further away from CPW1 (Fig. 6a), we detected a modified spectrum (red) at 24 mT as well (Fig. 6b). Hence, the reversal of Py nanostripes was induced in both gaps by the short-wave magnon mode k1+Gexcited at CPW1 at firr= 4.3 GHz. When applying a microwave signal with firr= 5 GHz to CPW1 (after again initializing sample A at -84 mT), we observed modified spectra (red) taken at Pos. 2.1 [Fig. 6(c)] and 15Pos. 2.2 [Fig. 6(d)]. Note that the directly excited grating coupler mode did not exist at 5 GHz. Still, the finding is different from the experiment conducted with firr= 1.25 GHz in Fig. 5(d). We attribute the observed reversal of Py nanostripes to magnons which were excited by parametric pumping at CPW1 (Supplementary information). Their frequency reads fm=firr/2 = 2 .5 GHz, which was above the k1resonance ( fk1= 2.3 GHz) at 24 mT and inside the allowed magnon band for propagation. Such magnons hence reached CPW2 and could explain the observed reversal. We note that the phase-sensitive voltage detection of the VNA experiment does not allow us to evidence the magnons created by parametric pumping because of their shifted frequency. We resolve them by BLS as presented in detail in Fig. S6. We performed experiments also at 2 mT after initializing the sample at -84 mT. In Pos. 2.1 [Fig. 6(e)] we observed that the magnon spectrum (red) was modified after applying irf with firr= 1.25 GHz. At 2 mT, spin waves at this small frequency were allowed [dotted lines in Fig. 5(b)] and possessed a wave vector k1with λ= 7222 nm. Excited at CPW1, the magnon mode changed the Py nanostripes underneath CPW2 at Pos. 2.1, but not at Pos. 2.2 [Fig. 6(f)]. We assume that at the small field the excitation of the k1mode was in the nonlinear regime as well, but additional parametric pumping did not take place at the small frequency. Instead, the amplitude of magnons decayed due to enhanced scattering and was below the threshold for reversal in Pos. 2.2. The excitation of propagating magnons with a too high microwave power hence led to an incomplete reversal of nanostripes below CPW2. This finding is consistent with the non-monotonous variation of critical fields with applied microwave power reported in Ref.11and as shown in Fig. 2. BLS studies on magnon-induced reversal in sample B are shown in Fig. S7 and support the findings reported for sample A. We now discuss the roles of the intermediate (spacer) layers. The critical fields displayed in Fig. 2 indicate that the insertion of the Cu spacer between the Py and YIG increased the switching field distribution and the coercivity of individual Py nanostripes compared to the SiO 2spacer. Our spatially resolved BLS data demonstrate that the precessing magnetiza- 16tion of allowed spin waves in YIG creates a torque leading to an irreversible change of the nanostripe magnetization. The insertion of the Cu spacer excludes the transfer of exchange magnons as the main mechanism in contrast to assumptions made in Refs.21–24Still, the Cu spacer thickness is smaller than the spin diffusion length.18In this case, forced spin preces- sion in YIG and concomitant spin pumping into Py might introduce an additional torque.19 We noticed however significantly larger critical fields and a reduced switching efficiency at high power for sample A with Cu spacer compared to sample B with SiO 2spacer. The data do not support the spin pumping effect to be relevant for reversal. Strikingly, we find the largest reduction in the critical fields HC1andHC2in sample B with the 5-nm-thick insulating spacer which avoids the spin-pumping torque. Thereby, we assume that dipolar coupling alone allowed for nanostripe reversal at a small power level. In the following we discuss the possible origin for the observed variation of spin-precessional power for magnon-induced reversal. We focus on Fig. 4(d), where we exclude direct microwave-assisted switching of nanostripes by the applied rf signal. In sample B, we observe the smallest spin-precessional power when the propagating magnons exhibit a wavelength of 195 nm underneath the gratings.7This value is (very close to) twice the width of a Py nanostripe. We argue that such a relation between the magnon wavelength and nanostripe width ensures the highest possible repetition rate by which a maximum in the dipolar stray field of a DE mode exerts a torque on the Py magnetization vector MPy. For a shorter magnon wavelength a partial cancellation of the dynamic dipolar field occurs underneath a nanostripe, and the dynamic dipolar coupling is reduced. For a long wavelength the repe- tition rate is small by which the maxima of the dynamic stray field pass by the nanostripe and produce the relevant torque. These considerations motivate the observed minimum in Fig. 4(d) as a function of firr, i.e., magnon wavelength. The slightly increased power levels needed for reversal in sample A incorporating the Cu spacer might indicate that additional spin pumping or an eddy current effect reduced the total torque. Further studies on stripes with different spacer layers and of e.g. different lengths and widths are needed to engineer 17their own eigenresonance frequency and explore in detail the hypothesis of a wavelength- dependent reversal mechanism drawn from the presented experiments. Conclusions We reported magnon-induced reversal in Py/YIG hybrid structures with different interme- diate layers. We quantified and compared the power values for magnon-induced switching. Reversal of 100-nm-wide Py stripes was achieved by means of propagating magnons with wavelengths ranging from 148 nm to 7222 nm. Their excitation was realized both in the lin- ear and non-linear regime. The non-linear parametric pumping was evidenced by local BLS microscopy. In an opposing field of 14 mT a spin-precessional power of the order of 1 nW was enough to reverse the up to 27 µm long Py nanostripes after magnon propagation over 15µm. The absence of interlayer exchange coupling due to a spacer layer between Py and YIG led to nanostripes with partly enhanced coercive fields compared to Py stripes directly integrated on YIG. The enhanced coercive fields are advantageous in terms of a nonvolatile memory of magnon signals. Importantly, with increasing power, we achieved a reduction of switching fields of nanostripes to (nearly) zero mT. Our results promise that nonvolatile magnon-signal storage in magnetic bits is feasible for wave-logic circuits and neural networks performing computational tasks at different magnon frequencies. Considerations based on dynamic dipolar coupling suggest that the power for magnon-induced storage might be min- imized when the width of the magnetic bit equals half the wavelength of the magnon. Methods A. Sample fabrication . Devices are fabricated on 113-nm-thick YIG originating from the same wafer. The YIG had been deposited by liquid phase epitaxy on a 3-inch wafer and purchased by the company Matesy GmbH in Jena, Germany. The spacer is fabricated by DC sputtering of 5-nm-thick Cu on YIG. Then 20-nm-thick Py (Ni 81Fe19) is deposited via 18electron beam evaporation on the YIG. The gratings were written with electron beam lithog- raphy (EBL) using hydrogen silsesquioxane (HSQ) as negative resist and then transferred into the Py/Cu by ion beam etching. We etch both layers of Py and Cu. CPWs are fabri- cated via lift-off processing after EBL and Ti/Au (5 nm / 120 nm) evaporation. For sample B, the SiO 2layer is deposited by e-beam evaporation and the following steps to fabricate stripes and CPW are unchanged. B. Broadband VNA spectra . The broadband spectroscopy data ∆ Sαβ(α, β = 1, 2) are obtained by nearest-neighbor subtraction of raw linear magnitude signals, i.e. ∆ Sαβ(f, H i) = Sαβ(f, H i+1)−Sαβ(f, H i). The magnetic field step is 2 mT. The linear magnitude signal Mag( Sαβ) is obtained from the quadrature sum of real (Re( S)) and imaginary (Im( S)) parts of median-subtracted signals. Re( S) (Im( S)) is obtained by the raw real (imaginary) part after removing at each measured frequency its median value across all applied magnetic fields. C. Switching yield maps . To build switching yield maps (Fig. 3) we have followed the methodology described in Ref.11To get µ0HC= 14 mT by the emission of the magnon mode k1(in Ref.11this field HCwas labelled HC2.) We required ( −12±1) dBm (63.1 µW) at CPW1. We compare this value to 58.4 µW needed in Ref.11. The separation between CPW1 and CPW2 was larger by 20 µm in Ref.11compared to the present samples. However the previously reported critical field was only +28 mT compared to +40 mT in Fig. 1(d). This larger coercive field of nanostripes with the Cu underlayer used here might explain that a similar power level for reversal was needed though the propagation length of magnons was shorter. To characterize the critical power levels featuring the switching at CPW1 (CPW2) we focus on the frequency branch 4.5 ÷4.7 (3.9 ÷4.1) GHz of the S11(S21) spectra (cf. Fig. 3). To evaluate the critical precessional power PC,precwe first extract the minimum critical power PCfor the same frequency. The overall irradiation frequency firrrange is divided into sub- intervals of 250 MHz width. We record PCand the frequency sub-interval δfPCthat achieves 19PC. These measurements are conducted with 1 kHz bandwidth and 250 MHz frequency reso- lution. Examples of such measurements are reported in Fig. 3. The magnetic field is 14 mT. To obtain the relevant Mag(S 11) signal we acquire field-dependent reflection spectra with 0.1 kHz bandwidth and 3.3 MHz frequency resolution. The VNA power for these measurements is labelled Pb.Pbequals -25 (-10) dBm for datasets that are analysed to evaluate PC1,prec (PC2,prec). The magnetic field is swept from -90 mT to positive fields larger than µ0HC2. With these datasets we define for both real and imaginary parts a median value across all applied magnetic fields at each frequency point. Then the linear magnitude signal Mag( S11) is constructed as described in paragraph B of the Methods section. We focus on the fre- quency range defined by the previously found δfPCand consider the reflection spectrum in the same range. Inside this frequency range we identify the frequency value f∗that achieves the local maximum of Mag( S11): (Mag( S11))∗. This represents the maximum absorbed en- ergy by the spin system. The critical spin precessional power is then evaluated by PC,prec= PC·[(Mag( S11))∗]2. D. BLS measurement protocol . To acquire the BLS spectra we initialize the system by applying -82 mT with a permanent magnet we then gradually increased the field to reach the targeted positive value. In so doing the system reaches the AP state. Thermal magnon spectra are acquired before injection of any rf signal at CPW1. We apply rf signal at CPW1 at a fixed frequency for increasing nominal powers. At each power step, we record the BLS signal while having the rf on. The rf irradiation at each power level is approximately 2 hours long. At the end of the experiment, after switching off the rf generator, the thermal magnon spectra is measured again and compared to the one acquired in the ’as-prepared’ AP state. To minimize spatial drift and maintain the same position of the laser spot we used a feedback system with image recognition acting every 5 minutes. For the BLS experiments the sample is wire-bonded to a PCB. The power levels that we discussed for BLS measurements are meant as nominal values. 20Supporting Information Available Numerical evaluation of the excitation spectrum of the CPW inhomogeneous dynamic field conducted by combining COMSOL simulation and FFT analysis. Broadband reflection spin wave spectra at PVNA= -25 dBm. Protocol for evaluation of nearly zero critical fields for nanomagnet reversal. Experimental datasets acquired with Brillouin light scattering microscopy ( µBLS) at the emitter CPW (CPW1) investigating magnon-induced magnetization reversal of the nano- magnets beneath CPW1. µBLS datasets of the Py nanostripes during continous-wave excitation, at different power levels, of multiple magnon modes in the underlying YIG. µBLS experiments reporting magnon-induced magnetization reversal for another device with hybrid interface Py/SiO 2/YIG. Inductive broadband spectroscopy measurements at -25 dBm acquired for samples A, C and B for comparison of reflection and transmission spectra. Acknowledgments The authors have used the colour maps for visualization of the VNA data provided by Fabio Crameri. The Scientific colour map bam25is used in this study to prevent visual distortion of the data and exclusion of readers with colour-vision deficiencies.26The authors acknowledge experimental support by Ping Che and discussions with Shreyas Joglekar and Mohammad Hamdi. Funding The research was supported by the SNSF via grant number 197360. 21Author contributions D.G., K.B. and A.M. planned the experiments and designed the samples. A.M. prepared the samples and performed the experiments together with K.B. A.M. and D.G. analyzed and interpreted the data. A.M. and D.G. wrote the manuscript. All authors commented on the manuscript. Competing interests The authors declare that they have no competing interests. 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2023-12-22

Magnetic bit writing by short-wave magnons without conversion to the electrical domain is expected to be a game-changer for in-memory computing architectures. Recently, the reversal of nanomagnets by propagating magnons was demonstrated. However, experiments have not yet explored different wavelengths and the nonlinear excitation regime of magnons required for computational tasks. We report on the magnetization reversal of individual 20-nm-thick Ni81Fe19 (Py) nanostripes integrated onto 113-nm-thick yttrium iron garnet (YIG). We suppress direct interlayer exchange coupling by an intermediate layer such as Cu and SiO2. Exciting magnons in YIG with wavelengths {\lambda} down to 148 nm we observe the reversal of the integrated ferromagnets in a small opposing field of 14 mT. Magnons with a small wavelength of {\lambda} = 195 nm, i.e., twice the width of the Py nanostripes, induced the reversal at an unprecedentedly small spin precessional power of about 1 nW after propagating over 15 {\mu}m in YIG. Considerations based on dynamic dipolar coupling explain the observed wavelength dependence of magnon-induced reversal efficiency. For an increased power the stripes reversed in an opposing field of only about 1 mT. Our findings are important for the practical implementation of nonvolatile storage of broadband magnon signals in YIG by means of bistable nanomagnets without the need of an appreciable global magnetic field.

Magnon-assisted magnetization reversal of Ni81Fe19 nanostripes on Y3Fe5O12 with different interfaces

2312.15107v1

Springer Nature 2021 L ATEX template Edge spin wave transmission through a vertex domain wall in triangular dots Diego Caso1and Farkhad Aliev1,2 1Departamento de F sica de la Materia Condensada C03, Universidad Aut onoma de Madrid, Cantoblanco, Madrid, 28049, Spain. 2Instituto Nicol as Cabrera (INC) and Condensed Matter Physics Institute (IFIMAC), Universidad Aut onoma de Madrid, Cantoblanco, Madrid, 28049, Spain. Abstract Spin waves (SWs), being usually re ected by domain walls (DWs), could also be channeled along them. Edge DWs yield the interesting, and potentially applicable to real devices property of broadband spin wave connement to the edges of the structure. Here, we investigate through numerical simulations the propagation of quasi one-dimensional spin waves in triangle-shaped amorphous YIG ( Y3Fe5O12) micron sized ferromagnets as a function of the angle aperture. The edge spin waves (ESWs) have been propagated over the corner in triangles of 2 microns side with a xed thickness of 85 nm. Parameters such as supe- rior vertex angle (in the range of 40-75) and applied magnetic eld have been optimized in order to obtain a higher transmission coe- cient of the ESWs over the triangle vertex. We observed that for a certain aperture angle for which dominated ESW frequency coincides with one of the localized DW modes, the transmission is maximized near one and the phase shift drops to =2indicating resonant trans- mission of ESWs through the upper corner. We compare the obtained results with existing theoretical models. These results could contribute to the development of novel basic elements for spin wave computing. 1arXiv:2201.04054v2 [cond-mat.other] 18 Apr 2022Springer Nature 2021 L ATEX template 2 Edge spin wave transmission through a vertex domain wall in triangular dots 1 Introduction Despite the growing success of magnonics in recent years, particularly due to its potential application for spin wave computing and signal processing [1], the currently available spin-wave devices based on ferromagnetic strip lines are restricted in the frequency span and have none or limited capability in the redirection of SWs [1, 2]. Their typical operating frequencies are below few GHz, and the propaga- tion is conned to linear coplanar waveguides. Magnon excitation is usually achieved through the coupling between the magnetization and the magnetic eld due to microwave (mw) currents. The eld is however dicult to conne to sub-micrometre volumes. This impedes the use of the traditional inductive cou- pling methods. Lara et al. [3] proposed a technologically new approach which could lead to a radical enhancement of the coupling in small magnonic struc- tures, ultimately promising a full integration of the SW devices into CMOS technology. This radically new pathway is based on the excitation and propagation of a new class of localized quasi-one-dimensional spin waves, the so called Win- ter magnons (WMs) [4]. These spin waves are analogous to the displacement waves of strings and could be excited in a wide class of patterned magnetic nanostructures possessing domain walls. Localized WMs have been identi- ed experimentally in dierent ferromagnetic structures with DWs. Winter magnons have been excited in circular magnetic dots in double vortex states, being localized along DWs connecting vortex cores with half-antivortices [5]. The experiments have been supported by numerical modelling and analyt- ical theory. Garcia-Sanchez et al. [6] evidenced theoretically and through micromagnetic simulations non-reciprocal channeling of WMs in ultrathin fer- romagnetic lms along N eel-type DWs arising from a Dzyaloshinskii-Moriya (DMI) interaction. Other possible application of WMs include spin wave diode [7] or SW propagation along DWs using recongurable spin-wave nanochannels or spin textures [8{10]. Park et al. [11] and Osuna Ruiz et al. [12] investigated the propagation of WMs in patterned structures involving both DWs and sin- gle vortex states. WMs excited in exchange-coupled ferromagnetic bilayers have been suggested for their implementation in emerging spin wave logic and computational circuits [13]. An entirely dierent proposal of SW transmission and processing uses Winter magnons conned to edge DWs in ferromagnetic geometries such as triangles or rectangles in congurations created by an in-plane (IP) bias eld [3, 14]. The resulting state indeed supports the excitation and detection of SWs locally in magnonic logic gates, leading to a natural decrease in device dimensions. Control over edge-localized SWs was accomplished by Zhang et al. [15] by placing magnetic structures adjacently to a propagating microstrip. Additionally, micromagnetic simulations done by Gruszecki et al. [16] demon- strated that the edge of a structure with locally conned SWs could be used to excite plane waves with twice its frequency and less than half its wavelength.Springer Nature 2021 L ATEX template Edge spin wave transmission through a vertex domain wall in triangular dots 3 Our work investigates numerically the excitation, propagation and control of edge spin waves in micron sized YIG triangles with dierent apertures. We have observed a resonant enhancement of the ESW transmission for certain aperture angles accompanied by a =2 phase shift of the propagated spin wave. We link this eect to the interaction of the surface spin waves with the vertex domain wall. 2 Results and Discussion 2.1 Optimization of the Exchange Energy Channels with Applied Bias Field To get the ground state magnetization distribution a static bias eld is applied parallel to the base of the triangle. Once the system is relaxed, the exchange energy distribution will be similar to Fig. 1a. Under a DC eld parallel to the base of the triangle, the edge magnetic moments rotate and therefore the internal magnetic eld distribution is minimized near the lateral dot edges. Edge spin waves conned to this potential well can propagate along the nanostructure edges [3]. Depending on the strength of the magnetic eld, the exchange energy will be accumulated in a larger or lesser degree in the lateral edges of the triangle, shaping two excess exchange energy edge channels [3]. The exchange energy channels boundaries are determined here by performing exchange energy cross sections at the middle of the in-plane size of the dot (see inset of Fig. 1a), and established where the exchange energy density in the channels is reduced by 90% from its maxima, determining their width, as presented in Fig. 1b). The maximum value of exchange energy density in the channels determines the exchange energy in the channel (C exch:), illustrated in Fig. 1c, and the delo- calized energy (D exch:) shown on Fig. 1d is the total summatory of exchange energy outside the channel boundaries. Both quantities are normalized by the maximum value achieved over the applied eld. The accumulation of exchange energy on the lateral edges of the triangle with magnetic eld is a phenomena re ected on Figs. 1b - 1d: at small elds the exchange energy density progres- sively transfers from being delocalized (exchange energy outside the channel) to being part of the exchange energy channels. Logically, when stronger eld is applied the edge exchange energy channels are being narrowed and there is a better localization for the possible propagation of ESWs (see Fig. 1b). For elds greater than 800 Oe however, the exchange energy localized in the edges drops, Fig. 1c. This results in a weakening of the channels, which is not desirable for SW propagation. Hence, we chose to use 1 kOe applied eld for our micromagnetic simu- lations. The application of this IP eld results in reasonably high exchange energy in the channels for the SW propagation, well localized at the edges of the triangle, as well as reasonably low delocalized exchange energy in the bulk of the system Fig. 1d, which implies that the spin waves will less likelySpringer Nature 2021 L ATEX template 4 Edge spin wave transmission through a vertex domain wall in triangular dots Fig. 1 (a) Shows the exchange energy density (E exch: dens.) distribution of the YIG triangle with an applied 1 kOe IP magnetic eld parallel to the base of the geometry. Inset shows a cross section of the exchange energy density at the middle of the dot, indicating the way to evaluate the exchange energy channel width. Part (b) represents the exchange energy channel width vs. the IP applied eld. (c) Indicates how the normalized to maximum value exchange energy in the channel (C exch: ) varies with the intensity of the applied eld. Part (d) shows the normalized to zero eld delocalized exchange energy (D exch: ) against the applied eld. Dashed lines in (b), (c) and (d) indicate the optimal applied eld (1 kOe) used in the dynamic simulations. Parts (e) and (f) correspond to the zoomed area surrounded by the red dashed rectangle in (a). (e) Illustrates the variation of the top corner static M y component magnetization prole in the bulk normalized by the saturation magnetization (Ms). Part (f) reveals an out-of-plane (OOP) prole, limited by the base of the dashed red rectangle in (a), of the M zmagnetization component normalized by M s. travel through the non-edge region of the triangle and cause interferences on the opposite edge from the source. Relaxing the magnetic system with no applied eld yields in a conguration with an excess of exchange energy as a perpendicular barrier from the middle of the triangles base up to the top corner, which could potentially also be used to propagate SWs. When the 1kOe IP eld is applied and the system is relaxed, the remnants of this barrier survive in the top corner. This is related to a domain wall that is originated in the top corner in this particular conguration (see Figs. 1e, f). After the bias eld is applied and the system is relaxed, the static magneti- zation distribution is mostly IP saturated in the direction of the eld. However, close to the edges of the triangle the magnetization becomes increasingly par- allel to them due to the minimization of stray elds. In the upper corner, this translates as a complete change of the IP magnetization component perpen- dicular to the base (Y axis) from one side to another of the triangle, leading to a soft 30N eel-type DW in the top vertex (see Figs. 1e, f). However, this statement is accurate only for the bulk of the structure, since the magnetiza- tion has an increasing out-of-plane (OOP) component in the surfaces of the triangle (see Fig. 1f), leading to a state were the DW minimum-energy cong- uration is intermediate between Bloch and N eel. The resulting state is close to having a weak DMI-like eect [17]. This topological anomaly has in itself itsSpringer Nature 2021 L ATEX template Edge spin wave transmission through a vertex domain wall in triangular dots 5 own magnetic texture and it is of interest to understand spin wave propagation through such structures. The DW spin conguration is head-to-tail, however, extreme high-re ection and low-transmission eects that would occur in 90 N eel-type head-to-tail DWs are negligible due to the softness of the DW, lead- ing to an almost transparent structure in terms of transmission [18]. The width of the DW is measured at 180 nm. 2.2 Propagation of the Edge Spin Waves The generated DW has its own associated eigenmodes that can be dynamically stimulated, this is key to understand SW transmission and further eects that will be discussed later on. After a 20 ns sinc-shaped pulse (see Methods), we found that the response of the whole system to the pulse resulted in clear distinguishable eigenmodes (bulk modes from now on). For all of the analyzed angles, from 40 to 75 degrees, the observed eigenmodes were restricted in the 3 GHz to 5 GHz range (see Fig. 2a for the modes of a 49triangle). The mode of the highest amplitude, however, is closer to 4.4 GHz, slightly oscillating for the dierent triangle apertures (Fig. 2b). The analysis of the eigenfrequencies is also done for local known specic magnetic structures such as the edges or the upper vertex DW. This allows to dierentiate the propagation of the spin waves through three dierent mag- netic structures with their own modes: bulk, edges, and the DW. Local analysis of the eigenmodes displays that generally f(bulk) >f(edges)>f(DW) (f being the frequency of the modes) (see Fig 2c). However, these modes are not over- whelmingly separated (all of them are between 1 and 5 GHz), and one can exploit this fact to excite the system at matching frequencies between two -or more- of these structures, resulting in a resonance-like system in which energy is being pumped from the magnetic structures to the spin wave to boost and perpetuate the propagation. Local excitation of the spin waves is done at the left corner of the triangle (see Methods), where the magnetization is con icted between being parallel to the left edge or to the base, which is the bulk magnetization direction. Our micromagnetic simulations indicate that it is possible to excite edge spin waves propagating either from the left or the right vertex with a dierence in transmission below 5% and a dierence in the excited wavelength below 2%. Interestingly, the amplitude of the propagated SW was found to be about twice larger when ESWs are excited from the left vertex (conguration discussed in this manuscript). The ESW intensity dierence could be due to the dierence in the angle between the mw excitation (directed perpendicularly to the tri- angle side) and the direction of the static local magnetization in left and right corners. The used frequencies for the mw excitations are the most intense detected bulk modes (red line in Fig. 2b). However, these excitations are directed eec- tively perpendicular to the left edge of the triangle. Thanks to this and to theSpringer Nature 2021 L ATEX template 6 Edge spin wave transmission through a vertex domain wall in triangular dots Fig. 2 (a) Average modes of the whole YIG triangle's OOP magnetization component visualized in the particular case of a 49aperture. The most intense mode (in this case 4.39 GHz) is to be excited in the left corner of the triangle. Inset shows a sketch of magnetic eld distribution in the system: a 1 kOe DC eld parallel to the base of the triangle and a smaller AC pulse perpendicular to it. (b) Frequency of the mode with the highest frequency in the DW and the most intense bulk mode (corresponding to the posterior SWs excitation frequency) against the angle aperture. For a wide range of angles in the high transmission region or fairly close to it, some DW modes frequencies agree with bulk modes, i.e, when exciting with this frequency they couple and result on a resonant system, which amplies the transmission through the DW. Inset of (b) is an enhancement in the high transmission regime, showing the overlapping of modes. (c) Two most prominent modes in the bulk, edges and DW against the corner aperture from 40 to 60 degrees. In light green is highlighted the high transmission regime. Inset for each graph in (c) corresponds to a typical prole of the most intense mode for the bulk, edges and DW. exchange energy channels, we can "trick" the spin wave to being almost com- pletely localized in the edges of the system (see inset of Fig. 3a), even though the excited modes are present in the whole system. Spin waves propagated from the left to the right corner of the triangle (see Supplementary videos) show a range of aperture angles in which there is a peak in transmission (see Methods section for a description of the transmission analysis), even slightly surpassing the value of 1 for 49, which means the edge localized spin wave is a bit more intense in the right side of the triangle than on the left one, where the source is placed (see Fig. 3a). We tentatively explain the transmission coecient exceeding one obtained for the aperture angle of 49 degrees, as a result of ESWs and DW modes excited through their resonant interaction with bulk modes. This is backed up by the fact that at the high transmission regime angles some of the highest frequency DW modes coincide with the most intense main modes of the whole system (Fig. 2b), probably providing an enhanced excitation of the upper vertex DW by the delocalizedSpringer Nature 2021 L ATEX template Edge spin wave transmission through a vertex domain wall in triangular dots 7 Fig. 3 (a) Transmission coecient and wavelength of the ESW for an upper vertex angle aperture between 40 and 75 degrees analyzed in the bulk of the triangle. The distinct trans- mission peak at 49-50indicates that the spin wave propagates almost perfectly through the edges of the triangle. Inset shows the ESW propagating through the edges a for 49 degree angle dot. The excited frequencies correspond in each case to the most intense bulk modes. (b) Phase shift of the ESW induced at the upper vertex DW. At the 49-50aperture angle, which is the high transmission range, the ESW experiences a /2 phase shift, indicating the possible existence of resonance. (c) Normalized DW asymmetry between the two lateral sides of the triangle in the propagation of the ESW indicates a more asymmetrical propa- gation for the high transmission angles. Insets reveal an enhancement of the upper vertex DW, showing a snapshot of the SWs once the propagation is steady for three dierent angle apertures. Enhanced area is marked in red in the top inset, which constitutes approximately only 17.5% of the in-plane length of the whole dot. SWs and therefore eectively boosting the ESWs amplitude at the right side of the triangle. To verify the possible resonant transmission for specic frequencies we per- formed a simulation of a 49 degree triangle excited at an arbitrary frequency of 4.35 GHz (varying less than a 2% the frequency of the resonant mode), which does not correspond to any mode in the system. Although the ESW signal is propagating (probably because the excitation is close in frequency to other modes), the resulting transmission dropped from about 1, obtained for the most intense mode of 4.39 GHz, to 0.36. The strong dependence in ESW transmission on the aperture angle remarks the importance of the upper vertex DW topology in the transmission process. As we mentioned, an increment in transmission of the SWs in a narrow range of top corner aperture angle could be due to the concordance between the most intense bulk SW modes and the highest frequency DW modes (see Fig. 2b). This correlation is dicult to disregard, and backs up the expected result in a resonant system with two sources. A deep analysis of the process in the upper vertex DW when the SW is propagated helps to understand these eects: for the high transmission angle range, the SWs experiments a phase shift (see Methods) close to /2 (marked in red in Fig. 3b), which is in agreement with some mechanical systems in resonance and previous recorded phase shifts in N eel-type DWs [19].Springer Nature 2021 L ATEX template 8 Edge spin wave transmission through a vertex domain wall in triangular dots The calculated DW asymmetry (see Methods section) along an axis parallel to the Y axis that divides the upper vertex in two (see inset of Fig. 3c) is also maximum for the angles of high transmission coecient, which is highly correlated to the /2 phase shift (Fig. 3c). At 49(see inset of Fig. 3c), the incoming and departing signals at the corner do not interfere with each another (which happens due to the /2 phase shift), showing the high eciency of the local DW SW propagation for this angles. However, for an angle outside of the high transmission range this statement is not true since the incoming and departing SWs collide at the corner, resulting in a transmission not as ecient. 2.2.1 Dispersion Relations Analyzed SW dispersion (see Methods) reveals two dierent dynamic regimes: low-frequency modes, which are constant in frequency along all wavenumbers, and a parabolic mode at higher frequencies (see Fig. 4a). These regimes are both present when the dispersion is represented from the data obtained in the edge region. Dierent performed tests reveal that the intensity of the dynamic regimes is dependent on the path chosen for the 2D Fourier Transform analysis, which generates the dispersion relations: a path that crosses through the left edge of the structure is associated with the two dynamic regimes being present (Fig. 4a). However, they become less intense if the path is separated from the edges (see Fig. 4b). If the path crosses the triangle from the base all the way to the upper vertex the dispersion relation only shows the constant frequency modes through all wavenumbers (see Fig. 4c). These frequency constant modes (coinciding with some of the modes presented in Fig. 2a) most probably origi- nate from the localized DW modes along the analyzed vertical line (remarked by a dashed line in Fig. 1f) in contrast with delocalized parabolic-type modes linked to edge spin waves. Here, the positive k vector branch of the parabolic mode corresponds to the ESWs emitted by the left vertex while the negative k branch describes ESWs emitted by the top corner in direction to the left vertex. In this conguration, these two branches have an equivalent intensity, meaning that the propagation of ESWs is in both directions is comparable. Their approximately parabolic dispersion relation also corroborates the predominantly 1D character of SWs along the edge with an IP eld paral- lel to the base. Indeed, the exact solution for SDWs in a 1D ferromagnetic chain predicts a parabolic k dispersion [20], valid except in the region of k →0, where the dipolar contribution dominates. The parabolic nature of the dispersion relation of the ESWs also matches the expected for the analytical model proposed by Lara et al. [3] (see Methods) if the exchange length in the edges is more (an order of magnitude) than the expected value at the bulk (see inset of Fig. 4a). This is reasonable since the exchange energy is predomi- nantly accumulated in the edges, which would generate a stronger coupling of the magnetic moments in this region, and thus, a longer coupling distance.Springer Nature 2021 L ATEX template Edge spin wave transmission through a vertex domain wall in triangular dots 9 Fig. 4 Dispersion relation in a 49 degree angle aperture triangle for (a) the left edge of the triangle from the left vertex all the way to the top. Both dynamical regimes are represented in the chosen path: the low-frequency modes constant in frequency and the parabolic ones, at higher frequencies. (b) The left side of the dot with a separation of 80nmfrom the edge. Although noticeable, both mode regimes are less intense than in the case of the left edge analysis. The black dashed line indicates the less intense parabolic dispersion. (c) A straight line perpendicular to the base of the dot, separating it in two halves. In this particular case only the low-frequency modes are present. These results reveal a direct correlation between the low frequency modes and the top corner of the triangle, associated with the DW, and the parabolic mode with the edge of the system 2.3 Edge Spin Waves Phase Shift at the Vertex Domain Wall Several studies have previously considered theoretically spin wave propagation through a domain wall. Hertel et al. [22] predicted that there is a proportional- ity between the SW phase shifted produced by DW and the angle by which the magnetization rotates inside this domain wall (
=
f/2), meaning that the phase of spin waves with k-vector perpendicular to a domain wall changes by a factor of /2 when the wave propagates in a ferromagnetic layer through a transverse wall. The changes of phase shift of the spin wave with upper vertex aperture angle observed here suggests a possible topological variability of the DW magnetic texture with angle aperture once the excitation reaches a steady state, as conrmed by our results. The change in propagation direction near the vertex could also be a potential cause of the background variation in phaseSpringer Nature 2021 L ATEX template 10 Edge spin wave transmission through a vertex domain wall in triangular dots and amplitude of the transmitted edge waves outside the region where reso- nant enhancement of the transmission and abrupt change in the phase shift is observed. However, the model [22] also suggests that at the high transmission range, in which the phase shift is almost exactly /2, the DW becomes com- pletely transverse. This scenario must be discarded due to the analysis done at the DW, which does not show a situation with a particularly hard DW, even when the spin wave is stabilized. On the other hand, Bayer et al. [23] predicted analytically that the spin- wave transport through an innitely extended one-dimensional Bloch-type domain wall induces a nite phase shift without re ection. We deduce then that the slight OOP component in the DW plays a role in the transmission of the wave. Indeed, when the excitation stabilizes, the spin wave propagating on each side of the dot have opposing OOP magnetization component, meaning that is key for the energy minimization in the system. Further studies involv- ing more precise control over the magnetic texture of the vertex domain wall, for example using an underlying material with spin orbit coupling (such as a Ptunderlayer) are needed to explore the direct link between the vertex DW internal magnetic texture and the spin wave transmission through it. 3 Conclusions Detailed investigation on the in uence of the upper vertex aperture on trans- mission and dephasing of edge spin waves in amorphous YIG ferromagnetic triangles shows the possibility of ne tuning of the edge spin wave transmis- sion between two remote corners. The results suggest that an aperture angle of 49-50triggers a high transmission response of the spin wave propagation sys- tem. Local analysis of the eigenmodes reveals the following relation: f(bulk) > f(edges)>f(DW) for the vast majority of those SW modes branches. In some instances, however, the local modes seemingly overlap each other, resulting in an energy pumping into the upper vertex DW structure. The maximum DW asymmetry is found at the high transmission range, as well as a phase shift of/2, characteristic of resonant systems and N eel-type DWs. The analysis of the SW dispersion relations reveal that the DW-related modes are constant in frequency, whereas the ESWs modes have a parabolic behavior. The best t to the analytical model [3] suggests a possible increase of the exchange length along the edge DW. We conclude that the high transmission mechanism is greatly due to the specic DW topology on each angle aperture, which could indeed raise resonance-like interactions between local and bulk SW modes in the triangular ferromagnetic structure. We believe that this work could con- tribute to better understand the SW propagation through topological objects and/or magnetic textures. Methods The micromagnetic simulations were carried out using the MuMax3 code [21]. The typical YIG parameters were used: saturation mag- netization M s= 130 kA/m, exchange stiness constant A ex= 3.51012 J/m and damping constant = 2.8103. The base of the used trianglesSpringer Nature 2021 L ATEX template Edge spin wave transmission through a vertex domain wall in triangular dots 11 was 2m and its thickness 85 nm. Discretization was set at 7.8 6.74.2 nm per cell (256 25620 cells) for a standard 60angle triangle, enough to precisely characterize the upper vertex DW. No anisotropies were added to our structure. To nd the eigenfrequencies of a magnetic system, a 2 Oe, 20 ns, sinc-shaped IP magnetic eld pulse was applied uniformly perpendicularly to both the base of the triangle and the direction of the applied static eld. After relaxing the system, the spin wave eigenmode spectrum is obtained using the Fourier Transform of the OOP component of the magnetization. Know- ing the eigenfrequencies, which appear as peaks in the absorption spectrum, a local excitation at a single eigenfrequency can be applied locally to observe the response of the magnetic system (i.e., to observe the propagation of spin waves away from the source), for as long as it may be necessary for the spin waves to reach a target area. The excitation source was localized in the left vertex of the triangle, in a small volume of just three cells over all the thickness of the triangle (in-plane area is 256 256 cells2). MW excitation is also directed perpendicularly to the left edge of the structure to ease the propagation and help localize the SW into the edges. To characterize the DW asymmetry we rst re ect the propagation of one side into the other of the DW, thus creating a map in which completely sym- metric SWs maxima or minima cancel each other. Then, we estimate the DW asymmetry as the resulting magnetization in the re ected map as follows: DW asymmetry =PjMzj, the summatory being over all the cells in the map. Dispersion relations are calculated from the simulations using a 2D Fourier Transform of the OOP magnetization component along a desired path after the sinc-shaped pulse is applied. The proceeding involves using the output magne- tization les from the pulse simulation to record a 1D path of magnetization for each assigned time (keeping only the magnetization from the path that has been chosen). An n m matrix should emerge from this, where n is the number of cells in the path and m the number of time points in the pulse simulation. Then, the 2D Fourier transform is used to switch it into the reciprocal space, which is what is presented in Fig. 4. Since the two signals from left and right edge-localized propagated spin waves share one fundamental frequency, its phase dierence can be analyzed the same way as two AC signals: we rst remove the DC part, i.e, the oset. Then we perform a Fourier Transform on both signals. Since the returned values of the Fourier Transform are in terms of magnitude and phase, the phase angle of each signal can be extracted numerically. The phase shift is calculated in the manuscript as the phase encountered in the left edge subtracted from the phase at the right edge of the dot once the signal reaches equilibrium. Transmission has been computed by analyzing the signals of the ESWs (once the steady state in the propagation has been reached) by perform- ing a Fourier Transform of the edge spin wave signals on both sides of the dot (approximately 15 nanometers away from the edge) and determining the amplitude of the Fourier Transform peaks quotient. This method allows theSpringer Nature 2021 L ATEX template 12 Edge spin wave transmission through a vertex domain wall in triangular dots almost total elimination of interferences in the process of analysis that would be characterized by undesirable frequencies in the FT. The analytical t of the ESWs dispersion relation in Fig. 4a was supported by the model proposed by Lara et al. [3] for edge localized spin waves in magnetic dots, valid only when the dot's aspect ratio is (thickness/in-plane size)<<1: !2() =!2 M[1 +l2 e2+ (12)h][l2 e2+ (12)h] (1) Whereis the wavevector along the edge of the dot, h=H=4Mscorre- sponds to the reduced bias magnetic eld parallel to the base of the triangle, le=p A=2M2sis the exchange length, and , which is a parameter of the model that quanties the ratio of spatial decays between dynamic and static magnetizations, has to be less than one to assure that the frequency of the SWs increases with decreasing h. Acknowledgments Authors acknowledge Ahmad Awad, C esar Gonz alez- Ruano and Antonio Lara for discussions. The work in Madrid was supported by Spanish Ministerio de Ciencia (RTI2018-095303-B-C55) and Consejer a de Educaci on e Investigaci on de la Comunidad de Madrid (NANOMAGCOST- CM Ref. P2018/NMT-4321) Grants. FGA acknowledges nancial support from the Spanish Ministry of Science and Innovation, through the "Mar a de Maeztu" Program for Units of Excellence in R&D(CEX2018-000805-M) and "Acci on nanciada por la Comunidad de Madrid en el marco del convenio plurianual con la Universidad Aut onoma de Madrid en L nea 3: Excelencia para el Profesorado Universitario". D.C. has been supported by Comu- nidad de Madrid by contract through Consejer a de Ciencia, Universidades e Investigaci on y Fondo Social Europeo (PEJ-2018-AI/IND-10364) References [1] A. V. Chumak, et al (2022) Roadmap on spin- wave computing. 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Journal of Applied Physics, 114:213905 https://doi.org/10.1063/1.4839315Springer Nature 2021 L ATEX template 14 Edge spin wave transmission through a vertex domain wall in triangular dots [15] Z. Zhang, M. Vogel, M. B. Jung eisch, A. Homann, Y. Nie, V. Novosad (2019) Tuning edge-localized spin waves in magnetic microstripes by proximate magnetic structures. Phys Rev B 100:174434 https://doi.org/10.1103/PhysRevB.100.174434 [16] P. Gruszecki, I. L. Lyubchanskii, K. Y. Guslienko, M. Krawczyk (2021) Local non-linear excitation of sub-100 nm bulk-type spin waves by edge-localized spin waves in magnetic lms. Appl Phys Lett 118:062408 https://doi.org/10.1063/5.0041030 [17] F.J. Buijnsters, Y. Ferreiros, A. Fasolino, M.I. Katsnelson (2016) Chirality-dependent transmission of spin waves through domain walls. Phys Rev Lett 116:147204 https://doi.org/10.1103/PhysRevLett.116.147204 [18] S. J. H am al ainen, M. Madami, H. Qin, G. Gubbiotti, S.V. Dijken (2018) Control of spin-wave transmission by a programmable domain wall. Nat Commun 9:1 https://doi.org/10.1038/s41467-018-07372-x [19] O. Wojewoda et al (2020) Propagation of spin waves through a N eel domain wall. Appl Phys Lett 117:022405 https://doi.org/10.1063/5.0013692 [20] G. Gruner (1994) The dynamics of spin-density waves. Rev Mod Phys 66:1 https://doi.org/10.1103/RevModPhys.66.1 [21] A. Vansteenkiste, J. Leliaert, M. Dvornik, M. Helsen, F. Garcia-Sanchez, B. V. Waeyenberge (2014) The design and verication of MuMax3. AIP Advances 4:107133 https://doi.org/10.1063/1.4899186 [22] R. Hertel, W. Wulfhekel, J. Kirschner (2004) Domain-wall induced phase shifts in spin waves. Physical Review Letters 93:257202 https://doi.org/10.1103/PhysRevLett.93.257202 [23] C. Bayer, H. Schultheiss, B. Hillebrands, R. Stamps (2005) Phase shift of spin waves traveling through a 180bloch domain wall. IEEE Transactions on Magnetics 41:10 https://doi.org/10.1109/TMAG.2005.855233 [24] A. V. Chumak, V. I. Vasyuchka, A. A. Serga, B. Hille- brands (2015) Magnon Spintronics. Nature Phys 11:453{461 https://doi.org/10.1038/nphys3347 [25] C. Liu et al (2018) Long-distance propagation of short-wavelength spin waves. Nat Commun 9:738 https://doi.org/10.1038/s41467-018-03199-8 [26] S. Macke, D. Goll (2010) Transmission and re ection of spin waves in the presence of N eel walls. J Phys: Conf Ser 200:042015Springer Nature 2021 L ATEX template Edge spin wave transmission through a vertex domain wall in triangular dots 15 [27] M. P. Kostylev, A. A. Serga, T. Schneider, T. Neumann, B. Leven, B. Hillebrands, R. L. Stamps (2007) Resonant and nonresonant scat- tering of dipole-dominated spin waves from a region of inhomoge- neous magnetic eld in a ferromagnetic lm. Phys Rev B 76:184419 https://doi.org/10.1103/PhysRevB.76.184419 [28] P. Borys, O. Kolokoltsev, N. Qureshi, M. L. Plumer, T. L. Monchesky (2021) Unidirectional spin wave propagation due to a saturation magnetization gradient. Phys Rev B 103:144411 https://doi.org/10.1103/PhysRevB.103.144411

2022-01-11

Spin waves (SWs), being usually reflected by domain walls, could also be channeled along them. Edge domain walls yield the interesting, and potentially applicable to real devices property of broadband spin waves confinement to the edges of the structure. Here we investigate through numerical simulations the propagation of quasi one-dimensional spin waves in triangle-shaped amorphous YIG ($Y_3Fe_5O_{12}$) micron sized ferromagnets as a function of the angle aperture. The edge spin waves (ESWs) have been propagated over the corner in triangles of 2 microns side with a fixed thickness of 85 nm. Parameters such as superior vertex angle (in the range of 40$^\circ$-75$^\circ$) and applied magnetic field have been optimized in order to obtain a higher transmission coefficient of the ESWs over the triangle vertex. We observed that for a certain aperture angle for which dominated ESW frequency coincides with one of the localised DW modes, the transmission is maximized near one and the phase shift drops to $\pi/2$ indicating resonant transmission of ESWs through the upper corner. We compare the obtained results with existing theoretical models. These results could contribute to the development of novel basic elements for spin wave computing.

Edge spin wave transmission through a vertex domain wall in triangular dots

2201.04054v2

Nonlocal detection of interlayer three-magnon coupling Lutong Sheng,1,Mehrdad Elyasi,2,Jilei Chen,3, 4,Wenqing He,5,Yizhan Wang,5Hanchen Wang,1, 4 Hongmei Feng,6Yu Zhang,5Israa Medlej,3, 4Song Liu,3, 4Wanjun Jiang,6Xiufeng Han,5 Dapeng Yu,3, 4Jean-Philippe Ansermet,7, 3Gerrit E. W. Bauer,2, 8, 9, 10and Haiming Yu1, 4,y 1Fert Beijing Institute, MIIT Key Laboratory of Spintronics, School of Integrated Circuit Science and Engineering, Beihang University, Beijing 100191, China 2WPI Advanced Institute for Materials Research, Tohoku University, Sendai 980-8577, Japan 3Shenzhen Institute for Quantum Science and Engineering, Southern University of Science and Technology, Shenzhen 518055, China 4International Quantum Academy, Shenzhen 518048, China 5Beijing National Laboratory for Condensed Matter Physics, Institute of Physics, University of Chinese Academy of Sciences, Chinese Academy of Sciences, Beijing 100190, China 6State Key Laboratory of Low-Dimensional Quantum Physics and Department of Physics, Tsinghua University, Beijing 100084, China 7Institute of Physics, Ecole Polytechnique F ed erale de Lausanne (EPFL), 1015, Lausanne, Switzerland 8Institute for Materials Research, Tohoku University, Sendai 980-8577, Japan 9Center for Spintronics Research Network, Tohoku University, Sendai 980-8577, Japan 10Zernike Institute for Advanced Materials, University of Groningen, 9747 AG Groningen, Netherlands (Dated: September 7, 2022) A leading nonlinear eect in magnonics is the interaction that splits a high-frequency magnon into two low-frequency ones with conserved linear momentum. Here, we report experimental observation of nonlocal three-magnon scattering between spatially separated magnetic systems, viz. a CoFeB nanowire and an yttrium iron garnet (YIG) thin lm. Above a certain threshold power of an applied microwave eld, a CoFeB Kittel magnon splits into a pair of counter-propagating YIG magnons that induce voltage signals in Pt electrodes on each side, in excellent agreement with model calculations based on the interlayer dipolar interaction. The excited YIG magnon pairs reside mainly in the rst excited ( n= 1) perpdendicular standing spin-wave mode. With increasing power, the n= 1 magnons successively scatter into nodeless ( n= 0) magnons through a four-magnon process. Our results help to assess non-local scattering processes in magnonic circuits that may enable quantum entanglement between distant magnons for quantum information applications. Nonlinear eects are ubiquitous in a variety of phys- ical systems, such as lasers [1], electron beams [2], cold atoms [3], and water waves [4]. Magnons are the quanta of spin waves, the collective excitations of the magnetic order. In magnon spintronics or magnonics [5{9] they are employed as information carriers for low-power process- ing and transmission [10{12]. Non-linearities in the mag- netization dynamics are known for many decades [13{ 22]. New nonlinear phenomena have been discovered in magnetic textures [23{27], nanoscale magnets [28{ 30] and hybrid systems [31, 32]. Non-linearity provides rich physics [33] and is relevant for technological appli- cations [34], such as mechanical force sensors [35]. Opti- cal non-linearities enable the coupling of LC resonators with a superconducting qubit [36]. A nonlinear mag- netostrictive interaction may generate magnon-photon- phonon entanglement in cavity magnetomechanics [37]. Non-linearities generate magnon interaction that leads to \squeezing" of magnon amplitudes [38] and continuous variable quantum entanglement [39]. The latter is dis- tillable [40] only when non-local and quantum-entangled over a distance. Separation of entangled magnons ap- pears to be a formidable task, but could be useful in information technologies such as quantum key distribu-tion [41] and quantum teleportation [42]. The leading nonlinear eect in magnetic systems is the three-magnon interaction [17{25], in which a magnon with energy ~!and zero momentum (Kittel mode) de- cays into two lower energy (frequency) magnons ~!=2 with opposite wave vectors ( ~kand~k). It has been observed in yttrium iron garnet (YIG) lms [17{20], spin- valve nanocontacts [21] and magnetic vortices [23, 24]. In all studies, the three magnons are part of the same magnet. Indeed, the magnon interactions are ususally assumed to be very short-ranged. However, this is not self-evident, since the long-range dipolar interaction con- tributes as well. A nonlocal interaction between dier- ent material systems, e.g. a magnon of system A that decays into two magnons in system B, would oer ad- ditional functionality for hybrid [43] and 3D magnon- ics [44]. Here, we demonstrate interlayer three-magnon interactions in a CoFeB jYIG hybrid nanostructure, in which a magnon in a CoFeB nanowire splits into two counter-propagating magnons in a YIG thin lm. We excite the magnetic system that consists of a CoFeB nanowire (200 nm wide, 30 nm thick and 100 m long) on top of a YIG lm with a thickness d= 80 nm [see Fig. 1(a)] by the microwaves emitted from a goldarXiv:2209.01875v1 [cond-mat.mes-hall] 5 Sep 20222 (a) CoFeB VLMicrowave VRVLVNA H(b) HθPt Pt+ 2 µmYIG+d 9101112(c) 91011f (GHz) -40 -20 020 40 field (mT)10 mW 9101112 91011f (GHz) -40 -20 020 40 field (mT)32 mW 5 0 -5(µV)(d) Low excitation High excitation 12 12 VLVL+k -kVR yx zjsĉ â₁s â₁ FIG. 1. (a) Schematic (side view) of spin pumping by a CoFeB nanowire detected non-locally by two Pt electrodes on the left (VL) and right ( VR) sides. Magnetic eld His applied in theyzplane with an angle = 45with respect to the z axis.sdenotes the distance between the CoFeB wire and a Pt bar. (b) Optical microscopic image of the three-terminal device. The center line is a CoFeB nanowire covered by a gold microwave stripline antenna. The Pt electrodes on both sides detect nonlocal spin pumping from the CoFeB wire. (c) VL measured with eld swept from negative to positive values. At an input microwave power of 10 mW, we are still in the linear regime. The green arrow marks the spin pumping when the magnetizations are anti-parallel. (d) VLmeasured at a high input power of 32 mW. The eld sweep is the same as in (c). Light blue arrows indicate a mode associated with the interlayer three-magnon coupling. We attribute another mode (orange arrows) to a secondary nonlinear process. stripline antenna [45] (not shown) above. The width of CoFeB wire is characterized by the scanning electron microscope (SEM) shown in the Supplementary Material (SM) Sec. I [46]. The YIG thin lms are deposited on gadolinium gallium garnet substrates by radio-frequency magnetron sputtering. We detect propagating magnons by their spin pumping [47{51] into Pt contacts placed on each side of CoFeB at a distance s= 2:5m, in which the inverse spin Hall eect (ISHE) generates a transverse voltage. Figure 1(b) is a microscopic image of the device. A magnetic eld applied at an angle = 45[49] with respect to the nanowire direction allows both ecient excitation of the nanowire (maximal for 0) and detection by ISHE (best for 90). Figure 1(c) shows the ISHE voltage at the left Pt electrode ( VL) as a function of excitation frequency and applied eld for a small microwave power of 10 mW, which is safely in the linear regime. The red (blue) color represents negative (positive) voltage response. The relativelystrong negative ISHE voltage (  6V, marked by the green arrow) corresponds to the antiparallel magnetization of the CoFeB and YIG layers with large interlayer dipolar coupling [52]. Figure 1(d) shows VL under high excitation (microwave power 32 mW). We observe additional modes as indicated by the blue and orange arrows in Fig. 1(d). We argue below that an interlayer three-magnon process causes the former ones and attribute them to parametric pumping of the rst excited perpendicular standing spin waves (PSSWs) in YIG by the stray eld of the CoFeB Kittel mode [blue arrows in Fig. 2(a)]. The latter one (orange arrows) should originate from an intralayer four-magnon scattering (orange dashed arrows in Fig. 2(a)) following the interlayer three-magnon process (blue arrows). The comparison of VLin Figs. 1(c) and (d) with VRin the SM Sec. II [46] conrms the strong chirality of the linear- response modes (areas marked by green arrows) [53, 54], while the nonlinear signals (blue and orange arrows) are nearly equally strong on both sides. In the vicinity of the CoFeB resonance ( 10 GHz, see microwave re ection spectraS11in the SM Sec. III [46]), the CoFeB wire switches more easily under high excitation. Figure 2(a) shows the dispersion of the nodeless ( n= 0) and single-node ( n= 1) perpendicular standing spin waves (PSSWs). The frequency of a spin wave with mo- mentumkin modenreads [55, 56] fn(k) = 0Ms 2exs k2+n d2 k2+n d2 +1 ex : (1) In calculations, we use the YIG exchange constant ex = 31016m2, saturation magnetization Ms= 140 kA/m [57] and lm thickness d= 80 nm. According to our modelling explained below, the CoFeB Kittel mode at10 GHz couples primarily with the high-ksingle-node mode as indicated by the green arrow in Fig. 2(a). This linear interlayer magnon coupling is strongly enhanced in the antiparallel conguration, here in the eld interval 0-20 mT [52, 54]. The linear process of spin pumping by this \two-magnon" scattering is strongly unidirectional due to the interlayer dipolar interaction [53, 54], with voltage signals in the left Pt electrodeVL[green arrow in Fig. 1(d)] but not in the right oneVR[Fig. 2(b)]. In this process, the nanometric width of CoFeB wire generates a broad kdistribution and thus enables ecient scattering between a k= 0 CoFeB magnon and a high- k n= 1 YIG magnon given by the dispersion in Fig. 2(a). We derive the interlayer magnon-magnon coupling strength by the magnetodipo- lar interaction for the n= 0 andn= 1 modes with +kandkwave vectors in the SM Sec. IV-C [46]. The chiral spin pumping signal scales linearly with the3 2468101214 field (mT)0 -50f (GHz) 2468101214(a) VR(b) -100 50 100k (rad/µm) 60 40 20 0 -20 -40 -6010 5∆kXf (GHz) YIG n = 0YIG n = 1CoFeB 02 -4-2VISHE (µV) 8 910 11 12 f (GHz)4 +14 mT -14 mT1f ½f2f3f n = 12(n = 1)(c) FIG. 2. (a) Spin-wave dispersion of the nodeless ( n= 0 black curve) and single-node ( n= 1 red curve) modes for a YIG lm with thickness d= 80 nm and applied eld of 10 mT at an angle= 45. The spin pumping induced by the inter- layer two-magnon scattering process (green arrow) is unidi- rectional, while spin pumping by the interlayer three-magnon scattering (blue arrows) is not. Dashed orange arrows: In- terband secondary four-magnon scattering from the n= 1 to n= 0 mode. (b) Nonlocal ISHE voltage signals measured at the right Pt electrode VRat an input power of 32 mW. The black arrows denote excitations of n= 0 modes at frequen- cies1 2f,f, 2fand 3fmodes. The red arrows indicate direct excitation of the n= 1 modes, while the blue arrows indicate their parametric pumping. The eld is swept from negative to positive values. (c) The blue (red) lineplot presents the frequency-dependent ISHE voltage extracted from (b) at an applied eld of 14 mT (-14 mT). The blue open squares (red open circles) are extracted under the same conditions for a bare YIG lm without CoFeB wire. microwave power and can be detected down to 1 mW (see SM Sec. V [46]). At powers above 10 mW, nonlinear eects emerge. Figure 2(b) shows the spin pumping signals measured at the right Pt electrode at 32 mW. We observe multiple new features associated with the n= 0 mode including parametric pumping of the fmode at 2fmicrowave excitation [48] and a triple-frequency (3f) [21, 27] mode, but also second harmonic generation at microwave frequencies1 2f. Then= 1 PSSW mode is observed in Fig. 2(b) indicated by the red arrows. By varying the lm thickness from 80 nm to 40 nm, then= 1 mode shifts to 12 GHz as shown in the SM Sec. VI [46]. Evidence for parametric pumping of the highern= 1 mode appears at high frequencies around 10 GHz as marked by blue arrows. Figure 2(c) shows two lineplots at +14 mT (blue squares) and -14 mT (reddots) with positive and negative ISHE voltage signals around 10 GHz. Open blue squares and open red circles show data obtained from a bare YIG sample without CoFeB wire on top (see SM Sec. VII [46]). If we replace the 200 nm-wide CoFeB wire by a 800 nm-wide one, the CoFeB Kittel mode frequency drops signicantly and no longer matches twice the frequency of the n= 1 mode. As a result, no signal is observed around 10 GHz (see SM Sec. VII [46]). The signals in Fig. 2(b) marked by blue arrows are therefore caused by parametric pumping ofn= 1 YIG magnons by the stray eld from the CoFeB dynamics but not the stripline. Our experiments uniquely combine the advantages of microwave and electrical magnon transport studies. The observable is S21(!1;!2), the bichromatic scatter- ing matrix of a magnon injected at frequency !1at con- tact/stripline 1 to a magnon with frequency !2at con- tact/stripline 2. Propagating magnon spectroscopy [7] studies the coherent magnons at frequency !;in terms ofS21(!;!). The electrical injection and detection of magnons by heavy metal contacts [8] is the method of choice to study diuse magnon transport. However, senses onlyR jS21(!1;!2)jd!1d!2;so all spectral infor- mation is lost. Here we measure the coherent response to inductive magnon injection at frequency !and elec- tric detection at a distant contact, i.e.R jS21(!;! 2)jd!2. In the linear regime this does not provide new informa- tion. However, the emergence of a magnon frequency comb leads to an increased signal at a magnon resonance !=!k, while new signals due to parametric pumping emerge when != 2!k. When the magnon decay length is larger than the contact distance, we can interpret the experiments simply in terms of the magnon spectrum generated by the microwaves under the stripline since the electrical detection is not sensitive to the propaga- tion phase. Here we model the observed non-linearities by the leading terms in the Holstein-Primako expansion with Hamiltonian ^H=^H(0) C+^H(0) Y+^H3M CY; (2) Here ^H(0) C="Ccycand ^H(0) Y=P kn"knay knaknare the excitations of the Kittel mode in the magnetic wire and spin waves in mode knof the lm. In principle, all states may be excited by the microwaves emitted by the stripline with mode-dependent eciencies. The leading non-linear term is the 3-magnon interaction ^H3M CY. In the following, we model the nonlinear excitations ob- served around the CoFeB nanowire resonance frequency (10 GHz) by the magneto-dipolar eld of the nanowire in the YIG thin lm with an interlayer 3-magnon interac- tion ^Hh3Mi CY =D(n) ~k+~k^cy^an;~k+^an;~k+ H:c:, where ^c(^an;~k) is the annihilation operator of the CoFeB nanowire Kit- tel mode magnon (YIG magnon of n= 0;1 with wave4 vector~k),D(n) ~k+~kis a coecient, and ~k+~k(see SM Sec. IV-A [46]). The eciency of the parametric ex- citation scales with the ellipticity of the excited magnon pairs, which decreases with k, sojD(1) ~k+~kj>jD(0) ~k+~kj. Therefore, the CoFeB Kittel mode excites n= 1 YIG magnon pairs at a lower threshold than that of n= 0 pairs. The parallel magnetic pumping by the Zeeman interaction 0 ~ mYIG k
~hdipdoes not depend on the po- larization of neither the magnon nor the dipolar eld, in contrast to the chiral spin pumping [53, 54]. The 3- magnon interaction is therefore not chiral and the signals in both Pt contacts are nearly the same [see Figs. 2(b) and (c)]. power (mW)VISHE (µV) 0 10 20 30 4001234Linear Nonlinear FIG. 3. ISHE peak voltages of the right Pt electrode as a function of the input microwave power measured at 5 GHz for the low- k n= 1 PSSW mode (black open squares) and at 10 GHz for the nonlinear modes (light blue open circles and orange open triangles). The red line is a linear t up to 20 mW. The dark blue double-sided arrow marks the de- viation from linearity. The light green area represents the nonlinear regime of interlayer three-magnon processes (light blue open circles) above 20 mW (blue dashed line). An ad- ditional feature (orange arrows in Fig. 1(c) and orange open triangles) observed above 32 mW (orange dashed line) is at- tributed to a secondary four-magnon process as indicated by orange dashed arrows in Fig. 2(a). In Fig. 3 we address the power dependence of the signals at 5 GHz and 10 GHz. We focus on the VISHE on the right Pt electrode for input powers from 1 mW to 47 mW (see SM Sec. VIII [46] for the raw data). The signal attributed to the interlayer three-magnon inter- action at microwave powers above 20 mW (light green area in Fig. 3) are nearly the same in both contacts. The signal associated to the excitation of the low- k n= 1 mode deviates from a linear power dependence (black open squares) at 20 mW, which we interpret an evidence for the Suhl instability [13]. The spin pumpingsignal drops above the threshold, because the con uence scattering opposes the Kittel magnon decay in the wire. When the power reaches 32 mW, an additional mode (orange arrows in Fig. 1(d)) emerges. We attribute this additional mode to a four-magnon process [30, 58] during which twon= 1 YIG magnons scatter to two n= 0 YIG magnons, i.e. ay 1ay 1a0a0+ H:c:(orange dashed arrows in Fig. 2(a)). The drop in the n= 1 mode intensity (black open squares) accompanies a new signal (orange open triangles), similar to a four-magnon scattering signal reported for a YIG nanoconduit [30]. We explain the increased slope of the orange mode as a function of eld as follows, see also SM Sec. IV-B [46]. The four-magnon interaction scales like / j~k0j2, where~k0are the momenta of the n= 0 magnons degenerate with the n= 1 magnons that are eciently excited by the CoFeB Kittel mode when their ~kis small. The amplitude of then= 1 magnons therefore decreases with excitation frequency higher than 2 min!1;~k, butj~k0jof the n= 0 magnons increases. The secondary maximum of the spin pumping signals seen in experiments and calcu- lations reveals that the four-magnon scattering can win this competition in a narrow frequency interval. We do not observe indications for an intralayer three-magnon process in which one n= 1 YIG magnon splits into two n= 0 YIG magnons, because the overlap integrals are suppressed due to the dierent parity of the standing wave amplitudes (see SM Sec. IV-B [46]). Finally, we demonstrate in Fig. S6 of the SM [46] excellent agreement of the calculated resonance energies with the observed spectra at both low and high excitation powers. In conclusion, we detect nonlinear interlayer magnon interactions in a hybrid magnetic nanostructure (YIGjCoFeB) by nonlocal spin pumping. The leading nonlinearity is a three-magnon process in which one CoFeB Kittel magnon splits into a pair of single-node (n= 1) YIG magnons with opposite wave vectors (+ k andk). By comparing the ISHE voltage signals of left and right Pt electrodes, we nd nearly symmetric magnon emission in both directions in contrast with the almost perfect chirality of linear excitations in agreement with model calculations based on purely magnetodipo- lar couplings. The theoretical analysis also indicates that the nonlinear interlayer coupling with single-node (n= 1) YIG magnons dominates over that with nodeless (n= 0) ones. We attribute an additional signal at even higher power to a cascade of interlayer three-magnon and intralayer four-magnon processes. Understanding the dynamics in hybrid magnetic systems may help engineer dissipation and cross talk in nanomagnonic devices, which is a necessary step in the prospect of quantum magnonics for entanglement distillation through nonlinear coupling of local nanowire magnons and paires of long-distance propagating magnons.5 ACKNOWLEDGMENTS We thank D. Wei for helpful discussions. The authors acknowledge support from the NSF China under Grants 12074026, 12104208 and U1801661, the National Key Research and Development Program of China Grants 2016YFA0300802 and 2017YFA0206200, and JSPS Kak- enhi Grants # JP19H00645 and 21K13847. These authors contributed equally to this work. yhaiming.yu@buaa.edu.cn [1] Th. Udem, R. Holzwarth, and T. W. H ansch, Optical frequency metrology. Nature 416, 233 (2002). [2] Y. C. Mo, R. A. Kishek, D. Feldman, I. Haber, B. Beau- doin, P. G. O'Shea, and J. C. T. Thangaraj, Experimen- tal observations of soliton wave trains in electron beams. Phys. Rev. Lett. 110, 084802 (2013). [3] Z. Dutton, M. Budde, C. Slowe, and L. V. Hau, Obser- vation of quantum shock waves created with ultra- com- pressed slow light pulses in a Bose-Einstein condensate. Science 293, 663 (2001). [4] G. G. Rozenman, W. P. Schleich, L. Shemer, and A. Arie, Periodic wave trains in nonlinear media: talbot revivals, akhmediev breathers, and asymmetry breaking. Phys. 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[46] See Supplemental Material for the SEM image of the CoFeB wire, the spin pumping signals measured at the right Pt electrode, S11re ection spectra measured by VNA of bare YIG and YIG/CoFeB samples, theoreti- cal derivation and discussion, spin pumping signals un- der dierent microwave power at the right Pt electrode, single-node mode in 80 nm and 40 nm YIG thin lms,spin pumping measurements on control samples, power- dependent measurements. [47] Y. Tserkovnyak, A. Brataas, G. E. W. Bauer, and B. I. Halperin, Nonlocal magnetization dynamics in ferromag- netic heterostructures. Rev. Mod. Phys. 77, 1375 (2005). [48] C. W. Sandweg, Y. Kajiwara, A. V. Chumak, A. A. Serga, V. I. Vasyuchka, M. B. Jung eisch, E. Saitoh, and B. Hillebrands, Spin pumping by parametrically excited exchange magnons. Phys. Rev. Lett. 106, 216601 (2011). [49] O. d'Allivy Kelly et al. , Inverse spin Hall eect in nanometer-thick yttrium iron garnet/Pt system. Appl. Phys. Lett. 103, 082408 (2013). [50] M. B. 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2022-09-05

A leading nonlinear effect in magnonics is the interaction that splits a high-frequency magnon into two low-frequency ones with conserved linear momentum. Here, we report experimental observation of nonlocal three-magnon scattering between spatially separated magnetic systems, viz. a CoFeB nanowire and an yttrium iron garnet (YIG) thin film. Above a certain threshold power of an applied microwave field, a CoFeB Kittel magnon splits into a pair of counter-propagating YIG magnons that induce voltage signals in Pt electrodes on each side, in excellent agreement with model calculations based on the interlayer dipolar interaction. The excited YIG magnon pairs reside mainly in the first excited (n=1) perpdendicular standing spin-wave mode. With increasing power, the n=1 magnons successively scatter into nodeless (n=0) magnons through a four-magnon process. Our results help to assess non-local scattering processes in magnonic circuits that may enable quantum entanglement between distant magnons for quantum information applications.

Nonlocal detection of interlayer three-magnon coupling

2209.01875v1

arXiv:1302.6697v1 [cond-mat.mes-hall] 27 Feb 2013Optimization of the yttrium iron garnet/platinum interfac e for spin pumping-based applications M. B. Jungfleisch,a)V. Lauer, R. Neb, A. V. Chumak, and B. Hillebrands Fachbereich Physik and Landesforschungszentrum OPTIMAS, Technische Universit¨ at Kaiserslautern, 67663 Kaiserslautern, Germany (Dated: 28 February 2013) The dependence of the spin pumping efficiency and the spin mixing cond uctance on the surface processing of yttrium iron garnet (YIG) before the platinum (Pt) deposition ha s been investigated quantitatively. The ferromagnetic resonance driven spin pumping injects a spin polarize d current into the Pt layer, which is transformed into an electromotive force by the inverse spin Hall eff ect. Our experiments show that the spin pumping effect indeed strongly depends on the YIG/Pt interface co ndition. We measure an enhancement of the inverse spin Hall voltage and the spin mixing conductance of more than two orders of magnitude with improved sample preparation. In the last decades, there was rapidly increasing inter- est in the field ofspintronics. The promisingaim is to ex- ploit the intrinsic spin of electrons to build efficient mag- netic storage devices and computing units.1An emerg- ing sub-field of spintronics is magnon spintronics, where magnons, the quanta of spin waves(collective excitations ofcoupledspinsinamagneticallyorderedsolid), areused to carry and process information.2Magnons pose a num- ber of advantages to conventional spintronics. One of them isthe realizationofinsulator-baseddeviceswith de- creased energy consumption: Since spin-wave based spin currentsin insulatorsarenot accompaniedbychargecur- rents, parasitic heating due to the movement of electrons can be excluded.3 In order to combine magnon spintronics and charge- based electronics, it is necessary to create effective con- verters, which transform spin-wave spin currents into conventional charge currents. The combination of the spinpumping4,5andtheinversespinHalleffect(ISHE)6,7 turned out to be an excellent candidate for this purpose. Spin pumping refers to the generation of spin polarized electron currents in metals by the magnetization pre- cession in an adjacent ferromagnetic layer, whereas the ISHE transforms this spin current into a conventional charge current. In the last years,hetero-structuresconsistentofamag- netic insulator yttrium iron garnet (YIG) film and an adjacent platinum (Pt) layer2attracted considerable at- tention. Since YIG is an insulator with a band gap of 2.85 eV8no direct transition of spin polarized electron currentsfrom the YIG intothe Ptlayerispossible. Thus, spin pumping is the only method applicable in these structures to inject spin currents into the Pt layer. It has been shown, that standing9as well as propagating10 magnonsin awide rangeofwavelengthsfromcentimeters to hundred nanometers11,12can be efficiently converted into charge currents using a combination of spin pump- ing and ISHE. Furthermore, the Pt thickness dependence a)Electronic mail: jungfleisch@physik.uni-kl.deon the ISHE voltage from spin pumping13and non-linear spin pumping14have been investigated. Since spin pumping is an interface effect, it is of cru- cial importance to investigate how to control and ma- nipulate the YIG/Pt interface condition in order to ob- tain an optimal magnon to spin current conversion effi- ciency. Recently, the influence of Ar+ion beam etching on the spin pumping efficiency in YIG/Au structures was investigated.15It was shown, that the spin mixing con- ductance determined by the Gilbert damping constant can be increased by a factor of 5 using Ar+etching. Nev- ertheless, there are no systematic, quantitative studies of the influence of the YIG/Pt interface treatment on the ISHE voltage from spin pumping up to now. In this Letter, we present our results on the influence of processing of the YIG film surface before the Pt de- position on the spin pumping efficiency. We measure the ferromagnetic resonance (FMR) spectra using con- ventional microwave techniques, as well as the inverse spin Hall voltage, which allows us to calculate the spin pumping efficiency, defined as the ratio of the detected ISHE charge current to the absorbed microwave power. Our experimental results clearly show a significant differ- ence (up to factor of 150) between the different surface treatments. A sketch of the experimental setup is shown in Fig.1(a). The YIG samples of 2.1 µm and 4.1 µm thick- FIG. 1. (Color online) (a) Sketch of the sample and the setup geometry. (b) Illustration of spin pumping and inverse spin Hall effect. Details see text.2 FIG. 2. (Color online) (a) ISHE voltage as a function of the ap plied magnetic field H. Magnetic field dependence of (b) transmitted and reflected microwave power Ptrans,Prefland (c) absorbed microwave power Pabs. Applied microwave power Papplied= 10 mW, YIG thickness: 2.1 µm. ness and a size of 3 ×4 mm2were grown by liquid phase epitaxy on both sides of 500 µm thick gadolinium gallium garnet (GGG) substrates. Since the GGG substrate be- tween the two YIG layers is rather thick (500 µm), the second YIG layer has no influence on our studies. Afteraconventionalpre-cleaningstep, allsampleswere cleanedbyacetoneandisopropanolin anultrasonicbath. In order to provide the same cleanliness of all samples, the purity of each sample was monitored after this first step. Afterwards, the following surface treatments have been applied: •Cleaning in “piranha” etch, a mixture of H 2SO4 and H 2O2. This exothermic reaction at around 120◦Cisstronglyoxidizingand, thus, removesmost organic matter. Among the used surface treat- ments, “piranha” cleaning is the only one, which was performed outside the molecular beam epitaxy (MBE) chamber. Thus, we cannot exclude, that the samples are contaminated by microscopic dirt or a water film due to air exposure before the Pt deposition. •Heating at 200◦C for 30 min in order to remove water from the sample surfaces, performed in-situ. •Heating at 500◦C for 5 hours performed in-situin order to remove water from the sample surfaces. It might be that for this method lattice misfits in the YIG crystal are annealed as well.16,17 •In-situAr+plasma cleaning at energies of 50-100 eV for 10 min. SRIM simulations show that the usedenergiesarebelowthe thresholdfor sputtering and, thus, this cleaning method acts mechanically. •In-situO+/Ar+plasmacleaningatenergiesaround 50-100eVfor10min. Inadditiontothemechanical cleaning effect, O+oxidizes organic matter. The used surface processings are summarized in Tab. I. They fulfill different requirements: First of all, the sam- plesarecleanedremovingmicroscopicparticles,(organic) matter and water. In the case of the samples, which are heated, the crystalline YIG structure might be annealed as well. Another aspect of the used treatments mightbe the modification of the spin pinning condition (not part of the present studies; requires further deeper in- vestigations). After cleaning, the 10 nm thick Pt layer was grown by MBE at a pressure of 5 ×10−8mbar and a growth rate of 0.01 nm/s. It is important to note that the Pt film was deposited on each of the samples of one set simultaneously, ensuring identical growth conditions. The measurement of the spin pumping efficiency was performed in the following way. The samples were mag- netized in the film plane by an external magnet field H (seeFig. 1(a)). Themagnetizationprecessionwasexcited at a constant frequency of f= 6.8 GHz by applying mi- crowave signals of power Pappliedto a 600 µm wide Cu microstrip antenna. The Pt layer and the microstrip an- tenna were electrically isolated by a silicon oxide layer of 100µm thickness in order to avoid overcoupling of the YIG film with the antenna. While sweeping the exter- nal magnetic field the inverse spin Hall voltage UISHE (Fig.2(a)) as well as the microwave reflection and trans- mission (Fig. 2(b)) were recorded. The voltage UISHE was measured across the edges of the Pt layer perpendic- ular to the external magnetic field using a lock-in tech- nique. For this purpose the microwave amplitude was modulated with a frequency of 500 Hz. Changing the external magnetic field to the opposite direction results in an inverted voltage proving the ISHE nature of the observed signal.2,6,7The complicated absorption and re- flection spectra depicted in Fig. 2(b) are due to the in- terference of the electromagnetic signal in the microstrip line reflected from the YIG sample and the edges of the line.18This behavior does not influence our studies since we further use only the maximum of UISHEto calcu- late the spin pumping efficiency (see Fig. 2(c)). Know- ing the applied microwave power Pappliedand measuring microwave reflection Prefland transmission Ptrans(see Fig.2(b)) enables us to calculate the absorbed power asPabs=Papplied−(Ptrans+Prefl). The results are de- picted in Fig. 2(c).19At the FMR field HFMR≈170 mT, energy is transferred most effectively into the magnetic system and, thus, the microwave absorption is maximal (see Fig. 2(c)). In resonance condition, the angle of pre- cession is maximal and spin currents are most efficiently pumped from the YIG into the Pt layer (Fig. 1(b)) and transformed into an electric current by the ISHE.2,7Sub-3 sequently, we measure the maximal ISHE voltage UFMR ISHE atHFMR. The dependence of UFMR ISHEas a function of the applied microwave power Pappliedis shown in Fig. 3, left scale (illustrated is method 7, Tab. I, discussed below with a ratherhighenhancementfactorof104). Theinterfaceop- timization provides the possibility to observe UFMR ISHEover wide range of applied powers Papplied. We find a linear relationbetween UFMR ISHEandPappliedoverthe wholepower regionofnearlyfourordersofmagnitude. TheISHEvolt- ageUFMR ISHEincreases from 100 nV (for Papplied≈100µW) to approximately 500 µV (forPapplied≈500 mW). On the right scale in Fig. 3the absorbed microwave power Pappliedis shown as function of the applied microwave powerPabs.Pabsdepends also linearly on Papplied. In order to investigate the different cleaning methods described above we measure the spectra for three differ- ent microwave powers Pappliedof 1 mW, 10 mW and 100 mW. We investigate three sets of samples (2 sets at 2.1 µm, 1 set at 4.1 µm YIG thickness). Inordertocomparethesamplesofonesetweintroduce the spin pumping efficiency as η(Papplied,n) =UFMR ISHE R·PFMR abs, (1) wherenis the index number of the sample (i. e. the index number of the cleaning method, see Tab. I),UFMR ISHE isthe maximalISHE voltagein resonance HFMR,Risthe electric resistance of the Pt layer, Pappliedis the applied microwave power and PFMR absis the absorbed microwave power at HFMR. Further, we introduce the power and thickness inde- pendent parameter ǫby normalizing the efficiency of the n-th sample to the first sample of each set n= 1, which underwent only a simple cleaning process, as ǫ(n) =η(n,Papplied) η(n= 1,Papplied). (2) Theǫ-parameter is a measure for the enhancement of the spin pumping efficiency due to the surface treatment. For each of them we calculate the ǫ-values: three mi- crowave powers for each set. The general tendency is the same for all series of measurements. According to the literature21, the spin pumping efficiency should not de- pendent on the YIG thickness for the used samples due to their large thicknesses. We observe an ISHE voltage for the 4.1 µm set to be around 20 % of that of the 2.1 µm sets, which we associate with a better quality of the 2.1µm YIG film. Nevertheless, the general tendency of the enhancement ǫ(n) due to the used surface treatments is rather independent on the film thickness: the absolute value ofthe ǫ-parameterfor the 4.1 µm set is 80% ofthat of the 2.1 µm sets. In order to obtain an easily compa- rable measure for the spin pumping efficiency, we further introduce the mean value ¯ ǫof these ǫ-values. The stan- dard deviation is given by σ(¯ǫ) and is a measure for theFIG. 3. (Color online) Maximal inverse spin Hall effect in- duced voltage UFMR ISHEand absorbed microwave power Pabsat FMR as a function of the applied microwave power Papplied. YIG thickness: 2.1 µm, cleaning method 7, “piranha” etch and heating at 500◦C. reliability of the specific surface treatments. The results are summarized in Tab. I. Our investigations of the YIG/Pt interface optimiza- tion on spin pumping show the following trend. The con- ventional cleaning by acetone and isopropanol in an ul- trasonicbath(sample1)achievestheworstresults(¯ ǫ= 1, UISHE= 270 nV for Papplied= 10 mW). Surface clean- ing by “piranha” improves the efficiency by a factor of 14, but this method is not reliable (standard deviation ofσ(¯ǫ) = 131% is very large). Since this surface treat- ment takes place outside the MBE chamber and since the sample is in air contact after cleaning, the sample might be contaminated again by water and possibly by microscopic dirt before the Pt deposition. Heating the samples after “piranha” cleaning in the MBE chamber at 200◦C for 30 min (sample 3) removes mainly the wa- ter film from the surface and results in a 64 times higher efficiency. However, this cleaning method also does not guarantee a high ISHE voltage, which is reflected in the high standard deviation. Using an Ar+plasma (method 4) is more efficient (see Fig. 2) and particularly more re- liable. Method 4 acts mechanically and removes water as well as other dirt from the sample surface. The best results (¯ǫ= 152) are obtained for the O+/Ar+plasma (sample 5). The additional advantage of this surface treatment is the removal of organic matter. In order to check the reliability of the O+/Ar+plasma clean- ing (method 5), we substituted the “piranha” cleaning by heating the sample: even without “piranha” etching, but with heating at 200◦C and O+/Ar+plasma (method 6), we achieve a considerable spin pumping efficiency of ¯ǫ= 86. This is mainly attributed to the O+plasma. It is remarkablethat purely heating the sample at 500◦C for 5 hours (method 7) results in a comparable high efficiency of ¯ǫ= 104. The additional reason might be, that the temperature is sufficiently high to anneal crystal defects of the YIG samples.16,17 In order to determine the spin mixing conductance g↑↓ eff ofoursamples, we performed FMR measurementson one of the YIG samples with a pronounced mode structure4 samplen process ¯ ǫ σ(¯ǫ) [%]g↑↓ eff(×1019m−2) 1 simple cleaning 1 — 0.02 2 piranha 14 131 0.32 3 piranha + 200◦C 64 93 1.45 4 piranha + Ar+79 54 1.79 5 piranha + O+/Ar+152 61 3.43 6 200◦C + O+/Ar+86 52 1.94 7 piranha + 500◦C 104 40 2.35 TABLE I. ¯ ǫis the calculated mean value of the ǫ-parameter, which is the thickness and power independent spin pumping efficiency (Eq. 2), σ(¯ǫ) is the standard deviation, spin mixing conductance g↑↓ eff. (method 7, with and without Pt layer on the top) using a vector network analyzer.20The FMR linewidth ∆ H is related to the Gilbert damping parameter as15,21–23 α=γ∆H/2ω, where γis the gyromagnetic ratio and ω= 2πfis the microwave angular frequency. For the bare YIG sample we measure ∆ H0= 0.06 mT (corre- sponds to α0= 1.2×10−4at a frequency f= 6.8 GHz), whereas for the Pt coveredYIG sample ∆ HPt= 0.16mT (corresponds to αPt= 3.3×10−4, respectively). Thus, we obtain a change of the Gilbert damping constant ∆α= 2.1×10−4. The spin mixing conductance g↑↓ effis re- latedtothechangeoftheGilbertdamping∆ α=αPt−α0 as21–23 g↑↓ eff=4πMSdF gµB∆α. (3) TakingMS= 140 kA/m and dF= 2.1µm into account we obtain a spin mixing conductance of g↑↓ eff= 2.35×1019 m−2for this particular sample. Since the spin mixing conductance g↑↓ effis proportional to the spin pumping ef- ficiency and, thus, consequently to the enhancement pa- rameter¯ǫ, wecancalculate g↑↓ efffortheothersurfacetreat- ments. The results are summarized in Tab. I. As it is apparent from Tab. I, the spin mixing conduc- tance can be varied by treating the YIG surface before the Pt deposition in the range of two orders of magni- tude. The largest value of the spin mixing conductance is obtained for a combined surface treatment by “pi- ranha” etch and O+/Ar+plasma,g↑↓ eff= 3.43×1019m−2. The obtained values for g↑↓ effagree with the values re- ported in the literature.15,23–25Our maximal spin mix- ing conductance g↑↓ eff= 3.43×1019m−2is one order of magnitude larger than the one reported in Refs.15,23for YIG/Au ( g↑↓ eff=5×1018m−2) and even three orders of magnitude larger than the one estimated in Ref.2for YIG/Pt ( g↑↓ eff=3×1016m−2). On the other hand, our maximal value is still one order of magnitude smaller compared to the one reported in Ref.25for YIG/Pt (g↑↓ eff=4.8×1020m−2). In conclusion, we have shown a strong dependence ofthe spin pumping effect on the interface condition of YIG/Ptbilayerstructures. We improvedthe ISHE signal strengthbyafactorofmorethan150usingacombination of “piranha” etch and in-situO+/Ar+plasma treatment in comparison to standard ultrasonic cleaning. The com- bined cleaning by “piranha” etch and heating at 500◦C yields a comparable enhancement of the spin pumping efficiency (by a factor of 104). The spin mixing conduc- tances for the different surface treatments were calcu- lated. We find a maximal value of g↑↓ eff= 3.43×1019m−2. Since the voltage generated by the ISHE scales with the length of the Pt electrode, optimal interface conditions are extremely essential for the utilization of spin pump- ing and ISHE in micro-scaled devices. Our results are also important for studies on the reversed effects: the amplification26and excitation2of spin waves in YIG/Pt structures by a combination of the direct spin Hall and the spin-transfer torque effect.2,27 We thank E. Saitoh and K.Ando for helpful discus- sions. Financial support by Deutsche Forschungsgemein- schaft (CH 1037/1-1) and the Nano-Structuring Center, TU Kaiserslautern, for technical support, is gratefully acknowledged. 1I.ˇZuti´ c, J. Fabian, and S. Das Sarma, Rev. Mod. Phys. 76, 323 (2004). 2Y. Kajiwara, K. Harii, S. Takahashi, J. Ohe, K. Uchida, M. Mizuguchi , H. Umezawa, H. Kawai, K. Ando, K. Takanashi, S. Maekawa, and E. Saitoh, Nature 464, 262 (2010). 3A. A. Serga, A. V. Chumak, and B. Hillebrands, J. Phys. D: Appl. Phys. 43, 264002 (2010). 4Y. Tserkovnyak, A. Brataas, and G. E. W. Bauer, Phys. Rev. Lett.88, 117601 (2002). 5M. V. Costache, M. Sladkov, S. M. Watts, C. H. van der Waal, and B. J. van Wees, Phys. Rev. Lett. 97, 216603 (2006). 6J. E. Hirsch, Phys. Rev. Lett. 83, 1834 (1999). 7E. Saitoh, M. Ueda, H. Miyajima, and G. Tatara, Appl. Phys. Lett.88, 182509 (2006). 8X. Jia, K. Liu, K. Xia, and G.E.W. Bauer, Europhys. Lett. 96, 17005 (2011). 9M. B. Jungfleisch, A. V. Chumak, V. I. Vasyuchka, A. A. Serga, B.Obry, H.Schultheiss, P.A.Beck, A.D.Karenowska, E.Sait oh, and B. Hillebrands Appl. Phys. Lett. 99, 182512 (2011). 10A. V. Chumak, A. A. Serga, M. B. Jungfleisch, R. Neb, D. A. Bozhko, V. S. Tiberkevich, and B. Hillebrands, Appl. Phys. L ett. 100, 082405 (2012). 11C. W. Sandweg, Y. Kajiwara, A. V. Chumak, A. A. Serga, V. I. Vasyuchka, M. B. Jungfleisch, E. Saitoh, and B. Hillebrand s, Phys. Rev. Lett. 106, 216601 (2011). 12H. Kurebayashi, O. Dzyapko, V. E. Demidov, D. Fang, A. J. Ferguson, and S. O. Demokritov, Appl. Phys. Lett. 99, 162502 (2011). 13V. Castel, N. Vlietstra, J. Ben Youssef, and B.J. van Wees, Ap pl. Phys. Lett. 101, 132414 (2006). 14K. Ando, T. An, and E. Saitoh, Appl. Phys. Lett. 99, 092510 (2011). 15C. Burrowes, B. Heinrich, B. Kardasz, E.A. Montoya, E. Girt, Yiyan Sun, Young-Yeal Song, and M. Wu Appl. Phys. Lett. 101, 092403 (2012). 16O.G. Ramer and C.H. Wilts, phys. stat. sol. (b) 79, 313 (1977). 17Duk Yong Choi and Su Jin Chung, J. Cryst. Growth 191, 754 (1998). 18W. Barry, IEEE Trans. Microwave Theory Tech. MTT-34 , 1 (1986). 19In theUISHEsignal as well as in the microwave absorption spec- tra several modes are visible. They are slightly different fo r each5 sample and are identified as higher width modes and perpendic - ular standing thickness spin-wave modes. Since only the max i- mal voltage is used to determine the spin pumping efficiency, t he mode structures of the the different samples are of minor inte rest for the present study. 20S. S. Kalarickal, P. Krivosik, M. Wu, C. E. Patton, M. L. Schne i- der, P. Kabos, T. J. Silva, and J. P. Nibarger, J. Appl. Phys. 99, 093909 (2006). 21H. Nakayama, K .Ando, K. Harii, T. Yoshino, R. Takahashi, Y. Kajiwara, K. Uchida, and Y.Fujikawa, and E. Saitoh, Phys. Rev. B85, 144408 (2012). 22O. Mosendz, J.E. Pearson, F.Y. Fradin, G.E.W. Bauer, S.D. Bader, and A. Hoffmann, Phys. Rev. Lett. 104, 046601 (2010). 23B. Heinrich, C. Burrowes, E. Montoya, B. Kardasz, E. Girt,Young-Yeal Song, Yiyan Sun, and Mingzhong Wu, Phys. Rev. Lett.107, 066604 (2011). 24F.D. Czeschka, L. Dreher, M. S. Brandt, M. Weiler, M. Altham- mer, I.-M. Imort, G. Reiss, A. Thomas, W. Schoch, W. Limmer, H. Huebl, R. Gross, and S.T.B. Goennenwein, Phys. Rev. Lett. 107, 046601 (2011). 25S.M. Rezende, R.L. Rodrguez-Su´ arez, M.M. Soares, L.H. Vil ela- Le˜ ao, D. Ley Dom´ ınguez, and A. Azevedo, Appl. Phys. Lett. 102, 012402 (2013). 26Z. Wang, Y. Sun, M. Wu, V. Tiberkevich, and A. Slavin, Phys. Rev. Lett. 107, 146602 (2011). 27J. C. Slonczewski, J. Magn. Magn. Mater. 159, L1 (1996), and L. Berger, Phys. Rev. B 54, 9353 (1996).

2013-02-27

The dependence of the spin pumping efficiency and the spin mixing conductance on the surface processing of yttrium iron garnet (YIG) before the platinum (Pt) deposition has been investigated quantitatively. The ferromagnetic resonance driven spin pumping injects a spin polarized current into the Pt layer, which is transformed into an electromotive force by the inverse spin Hall effect. Our experiments show that the spin pumping effect indeed strongly depends on the YIG/Pt interface condition. We measure an enhancement of the inverse spin Hall voltage and the spin mixing conductance of more than two orders of magnitude with improved sample preparation.

Optimization of the yttrium iron garnet/platinum interface for spin pumping-based applications

1302.6697v1

Quantitative comparison of magnon transport experiments in three-terminal YIG/Pt nanostructures acquired via dc and ac detection techniques J. G uckelhorn,1, 2,a)T. Wimmer,1, 2S. Gepr ags,1H. Huebl,1, 2, 3R. Gross,1, 2, 3and M. Althammer1, 2,b) 1)Walther-Meiner-Institut, Bayerische Akademie der Wissenschaften, 85748 Garching, Germany 2)Physik-Department, Technische Universit at M unchen, 85748 Garching, Germany 3)Munich Center for Quantum Science and Technology (MCQST), Schellingstr. 4, D-80799 M unchen, Germany (Dated: August 5, 2020) All-electrical generation and detection of pure spin currents is a promising way towards controlling the diusive magnon transport in magnetically ordered insulators. We quantitatively compare two measurement schemes, which allow to measure the magnon spin transport in a three-terminal device based on a yttrium iron garnet thin lm. We demonstrate that the dc charge current method based on the current reversal technique and the ac charge current method utilizing rst and second harmonic lock-in detection can both eciently distinguish between electrically and thermally injected magnons. In addition, both measurement schemes allow to investigate the modulation of magnon transport induced by an additional dc charge current applied to the center modulator strip. However, while at low modulator charge current both schemes yield identical results, we nd clear dierences above a certain threshold current. This dierence originates from nonlinear eects of the modulator current on the magnon conductance. In the eld of spintronics, pure spin currents are promising for spin and information transport at low dis- sipation level. To this end, the ecient control of pure spin currents is an essential, but challenging task.1{4In magnetically ordered insulators, spin currents are carried by magnons, the elementary excitations of the spin sys- tem. These magnonic spin currents lead to interesting new device concepts for magnon-based information pro- cessing.5{8In this context, devices for magnon logic oper- ations mainly focus on coherent magnon transport. For instance, it has been shown that damping compensation via spin transfer torque is an ecient method to optimize coherent magnon propagation.9{12Furthermore, a logic majority gate13and the rst magnon transistor14using magnonic crystals15have been implemented. Recently, incoherent, thermally excited magnons have gained increasing interest as information carriers for logic operations. In bilayer systems consisting of magnetically ordered insulators (MOI) and heavy metals (HM) with strong spin orbit coupling, it has been shown that in- coherent magnons in the MOI can be excited electri- cally16{18as well as thermally16,19,20, which then can be detected electrically in the HM utilizing the inverse spin Hall eect (SHE).21{23Moreover, devices based on non-continuous HM electrodes have been used to show that a superposition of diusive magnon currents allow for the realization of a majority gate.24Later on, simi- lar device concepts were used to demonstrate the manip- ulation of magnon currents using a three-electrode ar- rangement in yttrium iron garnet (Y 3Fe5O12, YIG)/Pt bilayers.8,25{27In these experiments, a charge current is applied to a Pt strip (injector) injecting magnons into a)janine.gueckelhorn@wmi.badw.de b)matthias.althammer@wmi.badw.dethe YIG via the SHE and Joule heating (see Fig. 1(a)). These magnons are then electrically detected via the in- verse SHE as a voltage signal at a second Pt strip (de- tector). A charge current applied to a third Pt strip (modulator) placed between these two Pt strips allows to manipulate the magnon transport from injector to detec- tor.8The eect of the modulator in these experiments can be modeled as a change in the eective magnon conduc- tivity, which has to be distinguished from the expected change in the magnon transport signature due to spin Hall and spin Seebeck physics. In particular, the eective magnon resistance changes in a nonlinear fashion with the modulator current and shows a threshold behavior. Two main measurement schemes have been used to ac- cess the magnon transport properties, which are based on an ac8,16,25and a dc17,26,28stimulus applied to the injector. Although it is not obvious whether or not these techniques yield exactly the same result, a quantitative comparison is still missing. In this paper, we perform a quantitative comparison of the following measurement schemes: (i) a dc-detection technique utilizing the current reversal method29and (ii) an ac-readout technique based on lock-in detection. We corroborate that both techniques are quantitatively equivalent in the regime where the magnon resistance is weakly aected by the modulator current. In the nonlin- ear regime we nd that the two techniques are qualita- tively dierent which gives access to higher order terms originating from the injector current. As shown in Fig. 1(a), we investigate the magnon transport using a three-terminal YIG/Pt nanostructure.8 A charge current Iinjis applied to the Pt-injector, induc- ing a magnon accumulation in the YIG lm, both via the SHE generated spin accumulation and via Joule heating. The magnons diuse to the Pt-detector strip, where they induce a voltage Vdetvia the inverse SHE. A dc chargearXiv:2008.01416v1 [cond-mat.mes-hall] 4 Aug 20202 + -+ - modulator detectorinjectorPt YIGw 1w 2 dd w 2 M+ -Vdet(a) z xyHIinj φIdcmod (c)(b)+Iinj -Iinj0 Iinjsin(ωt) -Iinjcos(2ωt) Figure 1. (a) Sketch of the sample conguration with the electrical wiring scheme and the electrical connection scheme, and the coordinate system with the in-plane rotation angle 'of the applied magnetic eld 0H. (b), (c) Schematic de- pendence of the detector voltage Vdetas a function of time according to Eq. (1) for (b) the dc and (c) the ac technique. (b) For the dc technique, the current to the injector Iinjis stepwise varied from + IinjtoIinjand vice versa. (c) For the ac technique, Vdet acis shown for the rst (red) and sec- ond (blue) harmonic signal as well as for a constant oset detector voltage (green). The black line corresponds to their superposition. currentImod dcapplied to along the Pt-modulator allows to manipulate the magnon transport between injector and detector via a SHE induced spin accumulation and Joule heating eects. Following previous works8,25,28,30, we express the de- tector voltage as Vdet Iinj;Imod
=X i2finj;modg1X j=1Ri-det j Imod

 Iij: (1) Here,Ri-det j Imod
are the transport coecients describ- ing the conversion process at the YIG/Pt interface and the transport in the YIG layer. Note that we only ac- count for changes in Ri jviaImod. This assumption is only valid for small injector currents.8,25 For the dc-detection technique, we utilize an advanced current reversal scheme. We apply a dc charge current sequence + Iinj;0;Iinjto the injector, while a constant charge current Imod dc is applied to the modulator, and measure for each conguration the voltage Vdet dcat the detector, as sketched in Fig. 1(b). From these measure- ments, we can then dene VSHE dc=1 2 Vdet dc Iinj;Imod dc
Vdet dc Iinj;Imod dc
 =Rinj-det 1 Imod
Iinj+Rinj-det 3 Imod
Iinj3+:::(2) as the voltage due to the SHE induced magnons trans- ported from the injector to the detector assuming an odd symmetry with respect to Iinj. In similar fashion, we de- ne Vtherm dc =1 2 Vdet dc Iinj;Imod dc
+Vdet dc Iinj;Imod dc
2Vdet dc 0;Imod dc
 =Rinj-det 2 Imod
Iinj2+:::(3)as the voltage due to the thermally injected magnons assuming an even symmetry with respect to Iinj. This elaborate scheme allows us to disentangle the dc detec- tor voltages generated by ImodandIinj. Thus,VSHE dc andVtherm dc only contain contributions from Imodvia the transport coecients Rinj-det j Imod
. In case of the ac-readout technique, we simultaneously apply an ac charge current Iinj ac(t) =Iinjsin(!t) to the in- jector and a constant dc charge current Imod dcto the mod- ulator and record the rst and second harmonic signal of Vdet acvia lock-in detection (compare Fig. 1(c)). For the rst harmonic signal V1! acand a time interval T1=! we obtain: V1!=2 TZT 0sin(!t)Vdet ac Iinj ac(t);Imod dc
dt =Rinj-det 1 Imod
Iinj+3 4Rinj-det 3 Imod
Iinj3+:::(4) which corresponds to the SHE induced magnon transport signal. For the second harmonic signal V2! acwe obtain: V2! ac=2 TZT 0cos(2!t)Vdet ac Iinj ac(t);Imod dc
dt =1 2Rinj-det 2 Imod
Iinj2+:::(5) which corresponds to the thermally generated magnons via Joule heating in the injector. When measuring V2! ac one has to account for the 90phase shift of the signal with respect to the reference signal. Due to the lock-in technique, the rst and second harmonic signal only con- tain contributions from the magnon transport between injector and detector. If we now compare VSHE dc withV1! ac, we see that these two quantities should be identical if Rinj-det j = 0 forj2. Thus, a quantitative comparison of VSHE dc andV1! acen- ables us to obtain information on higher order SHE con- tributions. In contrast, the ratio V2! ac=Vtherm dc is constant and yields 1 =2 if only transport coecients up to the fth order ( j5) contribute. To conrm this model conjecture, we conducted magnon transport experiments in YIG/Pt heterostructures. For the experiment comparing the dc- and ac-detection techniques, we use a peak value of Iinj= 100 µA and in the case of lock-in detection a low frequency (7 :737 Hz) modulation of the ac charge current. The device consists of 5 nm thick Pt strips with an edge-to-edge distance of d= 200 nm and a modulator width of w1= 500 nm on a 11:4 nm thick YIG lm (see supplemental information for growth details). The injector and the detector have a width ofw2= 500 nm and a length of l2= 50 µm, while the length of the modulator is l1= 64 µm. To characterize the magnon transport in our device, we plot the voltage signals VSHE dc,Vtherm dc ,V1! ac,V2! ac as a function of the magnetic eld orientation '(cf. Fig. 1(a)) measured with a xed magnetic eld strength of0H= 50 mT at T= 280 K for various positive3 (b)ac AacAac (d)(a)dc (c)A1ω(-µ0H)A1ω(+µ0H) ASHE(-µ0H) AdcASHE(+µ0H) Adc Atherm(+µ0H) AdcAtherm(-µ0H) Adc AacA2ω(+µ0H)AacA2ω(-µ0H) Figure 2. Detector signals (a) VSHE dc, (b)V1! ac, (c)Vtherm dc , (d) V2! acplotted versus the magnetic eld orientation with con- stant magnitude 0H= 50 mT for various positive modulator currentsImod dc. ForImod dc>0, the magnon transport signal is signicantly increased at '=180and reduced at '= 0. For the SHE induced magnon transport signals the (a) dc de- tector signal VSHE dc and (b) the rst harmonic signal of the ac measurement technique V1! acare in perfect agreement. While the angle dependence of the thermal signals (c) Vtherm dc and (d)V2! acis in good agreement, their absolute amplitude values strongly dier. The voltage amplitudes ASHE dc,A1! ac,Atherm dc , A2! acare extracted from the angle dependence of the detector signals as shown by the vertical arrows. modulator currents Imod dc. We rst focus on VSHE dc and V1! acin Fig. 2(a) and (b). For Imod dc = 0 (black data points), we observe the distinctive cos2'modulation for magnon transport between the injector and detector for both measurement techniques with minima in VSHE dc and V1! acforHk^y('=180;0;180), corresponding to maxima in magnon transport between injector and detector.16,17ForImod dc>0, the magnon transport sig- nal is signicantly increased at '=180forVSHE dc as well asV1! ac. This enhancement can be explained as an increase in magnon conductivity due to a magnon accu- mulation underneath the modulator caused by the SHE induced magnon chemical potential and thermally gen- erated magnons due to Joule heating. This increase in magnon conductivity leads to a larger magnon transport signal at the detector and thus larger negative voltage in both measurement schemes. At '= 0, we obtain a decreased magnon transport signal for VSHE dc as well as V1! ac. This originates from the magnon depletion caused by the annihilation of magnons via the SHE. However, this depletion is counterbalanced by the thermally in- jected magnons arising due to Joule heating of the mod- ulator strip. Comparing the dc and ac case, not just the angle dependence is equivalent, but also the voltage amplitudes VSHE dc andV1! acare in agreement with the pre- dictions from our detector voltage model. Separate mea- surements on an additional sample yield identical results (see supplementary material). current density (1011 A/m²) (a) (b) (c)Figure 3. Extracted amplitudes (a) ASHE dcand (b)A1! acfor 0H= 60 mT (as indicated in Fig. 2) of the SHE injected magnon transport signal versus the dc charge current Imod dc. The curves and signal amplitudes show similar behavior for (a) the dc and (b) the ac scheme. The black dashed line is a t indicating the Imod dc+Imod dc2dependence in the low bias regime (jImod dcj0:55 mA). (c) Ratio of the extracted amplitudes ASHE dcandA1! ac. ForImod dc0:55 mA, the ratio shows a nearly constant behavior ( A1! ac=ASHE dc'0:98). For higher modulator current values, the ratio clearly deviates from 0.98. We now discuss the angle-dependent data obtained from the thermal signals Vtherm dc andV2! ac. In Fig. 2(c) and (d) we plot the angle-dependent thermal voltage sig- nals for the dc- and ac-detection technique for positive Imod dc, respectively. For Imod dc= 0 the measurements of the thermally induced magnons show the characteristic cos'modulation in agreement with previous work.16For Imod dc>0, we observe a signicant increase of the detector signalsVtherm dc andV2! acat'=180and a decrease at '= 0as already reported in Ref. 25. For Imod dc= 900 µA and 1000 µA, this dierence is signicantly increased. We attribute this enhancement and decrease of the signal to the same mechanisms as in the case for the SHE driven magnon transport ( VSHE dc andV1! ac). At'=180, the magnon conductance underneath the modulator is in- creased by the SHE and thermally injected magnons via Imod dc. At'= 0, the magnon depletion underneath the modulator is counterbalanced by the thermally injected magnons and only a small reduction in the signal ampli- tude is observed. Comparing dc and ac conguration, we observe that the thermally induced signals Vtherm dc and V2! acstrongly dier in their absolute amplitude values, as expected from our model. For a more elaborate quantitative comparison of the detected voltages in dc and ac measurements, we extract the signal amplitudes ASHE dc(0H) andA1! ac(0H) of the angle-dependent measurements, as indicated in Fig. 2, and plot them as a function of Imod dcfor a mag-4 netic eld magnitude of 0H= 60 mT in Fig. 3. We note that we use ASHE dcandA1! acin our analysis instead ofVSHE dc andV1! ac, since at'= 90the voltage mea- sured is close to 0 leading to signicant contributions of noise. At rst glance, the curves and the signal ampli- tudes show similar behavior for the dc (Fig. 3(a)) and ac (Fig. 3(b)) conguration. As reported in Refs. 8 and 25, the signal amplitudes can be modeled by a superpo- sition of a linear (SHE) and quadratic (Joule heating) dependence in the low bias regime ( jImod dcj0:55 mA). To illustrate this, we plot this linear and quadratic de- pendence as a black dashed line in Fig. 3(a) and (b). The t well reproduces the measured data points in the low bias regime. For larger currents ( Imod dc>0:55 mA) we observe a pronounced deviation from this behavior. This enhancement in magnon conductance is in agree- ment with our previous work, which we attribute to a zero eective damping state via SHE induced damping- like spin-orbit torque underneath the modulator.25To show that the extracted amplitudes as a function of the modulator current Imod dcfor the dc conguration is in accordance with the ac measurement technique, the ratioA1! ac=ASHE dc is plotted in Fig. 3(c). In the low and negative bias regime ( Imod dc0:55 mA) the ratio is nearly constant with A1! ac=ASHE dc'0:98. This value is close to 1, which our model predicts for only linear eects with Rinj-det j Imod dc
= 0 forj2. However, forImod dc>0:55 mA the ratio exhibits a clear devia- tion from 1. Following the arguments of our theoretical model, this deviation indicates that for Imod dc>0:55 mA Rinj-det j Imod dc
6= 0 (forj2), i.e. a deviation from the linearIinjdependence. We attribute this to a new regime established via the damping compensation underneath the modulator, re ecting a typical threshold behavior of nonlinear eects.28For negative eld polarity we extract a similar dependence of the ratio A1! ac=ASHE dcjust with a threshold for negative Imod dc(see supplementary material for analysis with varying 0H). In similar fashion, we extract the amplitudes Atherm dc (0H) andA2! ac(0H) of the thermally injected magnons as a function of Imod dcfor the same magnetic eld magnitude of 0H= 60 mT. The results are shown in Fig. 4(a) and (b) for the dc and ac conguration, re- spectively. The qualitative dependence on Imod dcis iden- tical forAtherm dc andA2! acfor all current ranges. In agree- ment with previous reports25, we nd a signicant kink inAtherm dc andA2! acabove a certain critical current value. To account for the dierences of the absolute ampli- tude values, we calculate the ratio A2! ac=Atherm dc , shown in Fig. 4(c). The ratio is nearly constant over the whole modulator current range and has a value of 0.5 within the experimental error for all measured magnetic eld mag- nitudes (see supplemental information for other 0H). The small deviation, most notably in the negative bias regime, may be explained by the low thermal signal am- plitude in our devices (yielding a worse signal-to-noise ratio) and dierences in thermal landscape due to a dif- current density (1011 A/m²) (a) (b) (c)Figure 4. Extracted amplitudes (a) Atherm dc and (b)A2! acfor 0H= 60 mT (as indicated in Fig. 2) of the thermally in- jected magnon transport signal for the dc and the ac scheme versus the dc charge current Imod dc. (c) Ratio of the extracted amplitudes for the ac and dc conguration. Over the whole modulator current range the ratio shows a nearly constant behavior (A2! ac=Atherm dc'0:5). ference in the average applied heating power for ac and dc measurements. Nevertheless, the thermally generated signals nicely agree with our simple model of the detector voltage signal. However, the quantitative comparison of the thermal signal is not suitable to detect higher order contributions. In summary, we compared two measurement tech- niques, both allowing for an all-electrical generation and detection of pure spin currents in MOI/HM heterostruc- tures. On the one hand, we employ a dc-detection technique, where we utilized a modied current reversal method to take into account the modulator in a three- terminal nanostructure and to dierentiate between SHE and thermally injected magnons arriving from the injec- tor at the detector. On the other hand, we used an ac- readout technique, where lock-in detection of the rst and second harmonic signal is utilized to distinguish between these two magnon contributions. We demonstrate that the dc and ac technique are both well suited to investi- gate incoherent magnon transport in these three-terminal structures. In addition, our results show that below a criticalImod dcthe detector voltage has contributions lin- ear and quadratic in Iinj. This especially manifests itself as a full quantitative agreement between VSHE dc andV1! ac, which allows to compare results obtained with dierent techniques with higher condence. For large modulator currents, deviations are observed, indicating a contribu- tion of higher order in Iinjto the detector voltage. This sheds new light onto this nonlinear contributions appear- ing above a certain threshold value (corresponding to the damping compensation regime in our previous work).255 SUPPLEMENTARY MATERIAL See supplementary material for details on the fabrica- tion process and the measurement techniques, separate measurements of an additional sample investigating the SHE injected magnons, angle-dependent measurements of the presented sample for negative led polarity, a study of the eld dependence of the extracted amplitudes of the electrically and thermally induced magnons for the dc- and the ac-detection technique and an investigation of the third harmonic voltage signal. ACKNOWLEDGMENTS We gratefully acknowledge nancial support from the Deutsche Forschungsgemeinschaft (DFG, German Re- search Foundation) under Germany's Excellence Strat- egy { EXC-2111 { 390814868 and project AL2110/2-1. DATA AVAILABILITY The data that support the ndings of this study are available from the corresponding author upon reasonable request. REFERENCES 1J. Ahopelto, G. Ardila, L. Baldi, F. Balestra, D. Belot, G. Fagas, S. D. Gendt, D. Demarchi, M. Fernandez-Bola~ nos, D. Holden, A. Ionescu, G. Meneghesso, A. Mocuta, M. Pfeer, R. Popp, E. Sangiorgi, and C. S. Torres, \NanoElectronics roadmap for europe: From nanodevices and innovative materials to system integration," Solid-State Electronics 155, 7{19 (2019). 2R. Jansen, \Silicon spintronics," Nature Materials 11, 400{408 (2012). 3A. Homann, \Pure spin-currents," physica status solidi (c) 4, 4236{4241 (2007). 4M. Coll, J. Fontcuberta, M. Althammer, M. Bibes, H. Boschker, A. Calleja, G. Cheng, M. Cuoco, R. Dittmann, B. Dkhil, I. E. Baggari], M. Fanciulli, I. Fina, E. Fortunato, C. Frontera, S. Fu- jita, V. Garcia, S. Goennenwein, C.-G. Granqvist, J. Grol- lier, R. Gross, A. Hagfeldt, G. Herranz, K. Hono, E. Houw- man, M. Huijben, A. 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2020-08-04

All-electrical generation and detection of pure spin currents is a promising way towards controlling the diffusive magnon transport in magnetically ordered insulators. We quantitatively compare two measurement schemes, which allow to measure the magnon spin transport in a three-terminal device based on a yttrium iron garnet thin film. We demonstrate that the dc charge current method based on the current reversal technique and the ac charge current method utilizing first and second harmonic lock-in detection can both efficiently distinguish between electrically and thermally injected magnons. In addition, both measurement schemes allow to investigate the modulation of magnon transport induced by an additional dc charge current applied to the center modulator strip. However, while at low modulator charge current both schemes yield identical results, we find clear differences above a certain threshold current. This difference originates from nonlinear effects of the modulator current on the magnon conductance.

Quantitative comparison of magnon transport experiments in three-terminal YIG/Pt nanostructures acquired via dc and ac detection techniques

2008.01416v1

Spin Waves and Spin Currents in Magnon -Phonon Composite Resonator Induced by Acoustic Waves of Various Polarizations S.G. Alekseeva,*, N.I. Polzikovaa, V.A . Luzanovb, S.A.Nikitova a Kotelnikov Institute of Radioengineering and Electronics RAS Mokhovaya Str. 11, Build. 7, Moscow 125009 Russian Federation b Fryazino branch Kotelnikov Institute of Radioengineering and Electronics RAS Vvedenskogo Squar. 1, Fryazino, Moscow region 141190 Russian Federation *E-mail: alekseev@cplire.ru Abstract. In this work, we present the results of a systematic experimental study of linear and parametric spin wave resonant excitation accompanied by spin currents (spin pumping) in a multifrequency composite bulk acoustic wave resonator with a ZnO -YIG-GGG -YIG/Pt structure. The features of magnetic dynamics excitation in YIG films due to magnetoelastic coupling with acoustic thickness modes of various polarizations are studied. Acoustic spin waves and spin pumping are detected by simultaneous frequency -field mappin g of the inverse spin Hall effect voltage and the resonant frequencies of thickness extensional modes. In the parametric range of frequencies and fields, acoustic spin pumping induced by both shear and longitudinal polarization modes was observed. Linear a coustic spin waves are excited only by shear thickness extensional modes because longitudinal acoustic waves do not couple with the magnetic subsystem in linear regime . Keywords: magnetoel astic interaction; spin waves; spin pumping, bulk acoustic waves; resonator; YIG; ZnO, HBAR . Introduction Magnon -phonon interactions determine the fundamental properties of magnetic materials and structures, such as relaxation processes, and are also of practical interest, for example, for low -energy consumption for spin waves (SW) and spin currents excitation [1 -5]. In composite heterostructures containing piezoelectric and ferro(ferri)magnetic layers, the excitation of so -called acoustic SW (ASW) and acoustic spin pumping (ASP) occurs due to a combi nation of magnetoelasticity and piezoelectric effect in various layers, not necessarily in direct contact. To generate ASW, both surface acoustic waves (AW) excited by interdigitated transducers [1, 4 -9] and volume AW, in particular, microwave modes in com posite High overtone Bulk Acoustic wave Resonator (HBAR) [3, 10, 11] are used. Currently, HBARs, along with surface AW resonators, have proven to be in great demand as sources of coherent phonons for fundamental and applied research [12-15]. In our previou s works, we studied phenomena associated with the interaction of coherent AW and SW in hybrid magnon -phonon HBARs with a layered structure: piezoelectric (ZnO) – ferrimagnetic (yttrium iron garnet - YIG) – dielectric substrate (gallium gadolinium garnet - GGG) – ferrimagnetic (YIG) – Pt (Fig.1) [16 -21]. In the structure involved, shear bulk acoustic thickness modes of high harmonics were excited using piezoelectric transducers in the gigahertz frequency range. A self -consistent theory was developed to descr ibe magnetoelastic phenomena in such structures [16 -18]. The acoustic excitation of both linear [16 -19] and parametric ASWs [20, 21] and their electrical detection via the effect of spin pumping and the inverse spin Hall effect (ISHE) were theoretically pr oved and experimentally demonstrated. Fig. 1. Schematic of hybrid HBAR In this work, we present the experimental study of linear and parametric ASWs excitation and the features of the spin pumping they create in the hybrid magnon -phonon HBAR due to thickness acoustic modes of various polarizations: transverse (shear) and longitudinal. As in our previous works, we use the method of acoustic resonator spectroscopy [22] in combination with the method of electrical detection of ISHE voltage. In particula r, the frequency -field ( f,H) dependences of the microwave signal complex reflection coefficient S11(f,H) from the transducer electrodes and the constant voltage UISHE(f,H) on a platinum strip are studied. 1. Methods Experimental hybrid HBARs (see Fig. 1) are fabricated based on ready - made structures consisting of a 500 μm thick (111) -oriented GGG substrate ( 3) and a 30 μm thick epitaxial YIG films ( 2), (4). The YIG films were doped with La and Ga. Piezoelectric transd ucer composed of ZnO film ( 1) sandwiched between two thin-film aluminum electrodes was deposited on one side of the structure by rf magnetron sputtering. The top and the bottom electrodes were patterned by photolithography and had an overlap with the apert ure a = 170 μm. The Pt film ( 5) was deposited onto the free film ( 4) and formed as a stripe. The HBAR technology and design are described in more detail in [18 -20, 22]. Electrical excitation and detection of bulk AWs of different polarizations occur due to the direct and reverse piezoelectric effect in a ZnO film with an inclined 𝑐⃗- axis [22, 23]. Depending on the magnitude of the applied magnetic field, ADSW excitation in YIG films due to magnetoelastic interaction takes place either at HBAR frequencies fn (linear regime) or at half frequencies fn/2 (parametric regime) when the threshold power is exceeded. The spin current from YIG into Pt 𝑗⃗𝑠, [24] created by ADSW, is converted into conductivity current by ISHE [25]. This results in a consta nt voltage detected at the ends of a platinum thin film strip 𝑈ISHE =−𝑎′(𝐸⃗⃗ISHE ∙𝑦⃗), 𝐸⃗⃗ISHE ∝−(𝑗⃗𝑠×𝑧⃗). (1) Here 𝐸⃗⃗ISHE is an electrostatic field, 𝑎′≈𝑎 is the length of the region in the y direction in which ADSW excitation takes place. Fig. 2. Frequency dependence of the microwave reflection coefficient modulus |S11(f)| in the absence of a magnetic field. The inset shows an enlarged fragment. The dips in the frequency response correspond to the resonant frequencies of the thickness modes of shear AW (S) and longitudinal AW (L); the intermodal distance of longitudinal modes is approximately twice as large as that of shear modes. 2. Results and discussion Figure 2 shows the frequency dependen ce of reflection coefficient modulus |S11(f)| of microwave signal from the piezoelectric transducer electrodes in the absence of a magnetic field. All the experiments were conducted at fixed power level 9 mW. To study the excitation features of ASW and ASP from acoustic modes of different polarizations, the frequency range corresponding to the inset in Fig. 2 was selected. In this range the transducer excites both longitudinal (L) and shear (S) modes with the same efficiency. We denote the frequencies of these modes as flL and fsS, where l and s are the overtone numbers of the thickness modes of the corresponding polarizations. Further studies are carried out in a tangential magnetic field H in the range (0 – 450 Oe). As will be shown below, this field range contains both linear and parametric regi mes of the ASW excitation [20]. Fig. 3. Frequency dependencies of | S11(f)| (a) and voltage UISHE(f) on Pt (b) at several magnetic fields. Curves: 1 – 60 Oe, 2 – 181 Oe, 3 – 238 Oe, 4 – 327 Oe, 5 – 352 Oe. Figure 3a shows the frequency dependences of the reflection coefficient (Fig. 3a) and the ISHE voltage UISHE (Fig. 3b) measured simultaneously at several magnetic fields. The measurements were carried out in a narrow frequency range, including closely located one longitudinal and one transverse AW modes (see inset in Fig. 2). As one can see from Fig. 3 a, the resonant frequency for the longitudinal mode flL(H) (1923.6 MHz at H=0) changes slightly with the field increase. The resonant frequency of the transverse mode fsS(H) (1923.3 MHz at Н=0) remains practically unchanged in weak fields and experiences a shift in the fields Н > 200 Oe. The shift increases as the field approaches the ferromagnetic resonance (FMR) region. Assuming fsS ≈ fFMR, where the FMR frequency is related to the magnetic field by the Kittel formula fFMR = γ[H(H +4πM0)]½ , (2) we find that HFMR ≈ 384 Oe. Here, γ =2.8 MHz/Oe, M0 - effective saturation magnetization. For doped YIG we use the value 4 πM0 = 845 Oe, established for an identical structure in [20]. The change in the positions of voltage maxima UISHE(f) upon excitation of the transverse mode demonstrates similar behavior in the fields H > 200 Oe, but significantly more diverse behavior at lower fields. Figure 3b clearly shows that the UISHE maximum splits into two. At the same time on the characteristic s |S11(f)| in Fig. 3a there is a mild feature: a minimum located at 80 kHz higher from the main one and corresponding to the splitted UISHE maxima mentioned above. Fig. 4. Frequency -field dependence of the voltage UISHE(f,H) (a). The dots show the minimums of the S11 reflectance. The field dependences of the SW spectra frequency limits (b). For detailed comparison of the behavior |S11(f, H)| and UISHE(f,H) let us present them on the same graph. For this, the 3D color map UISHE(f,H) (Fig. 4.a) is best suited, on which the minima | S11(f, H)|, (dots) are superimposed . Also let us consider the magnetic field dependencies mentioned above in accordance with the calculated dependences of the SW spectra frequency limits f = fH(H) = γH and f = fFMR(H) shown in Fig. 4 b. The horizontal lines mark the frequency fp=1.1923 GHz ≈ fs,lS,L, and its sub -harmonics fp /2 and fp /4. The critical fields marked in Fig. 4b are found from the relations НFMR = [(2fp /γ)2+(4πM0)2]½/2-2πM0 = 384 Oe , Hc1 = fp /(2γ) = 343 Oe, (3) Hc = [(fp /γ)2+(4πM0)2]½/2-2πM0 = 121 Oe, Hc2 = fp /(4γ) = 171.5 Oe. The linear excitation of ADSW results in the signal of UISHE(f, H) in the vicinity of the HFMR field. Additional non -resonant (i.e. frequency independent) contributions to the UISHE signal (Fig. 4 a) and to the decrease in the overall level of |S11(f)| at H ≈ HFMR are associated with inductive excitation of magnetic dynamics directly by the transduc er electrodes. Such mixed inductive and acoustic excitations, as well as the possibility of completely acoustic excitation of SW, were discussed in detail in [19]. The excitation of any parametric SW is possible if H < Hc1 = fp /(2γ). Therefore, in the field range Hc1 < H < HFMR ≈ HMER, only linear excitation of ADSW is possible due to the magnon – transverse phonon coupling. Here the field HMER is the field of magnetoelastic resonance, at which synchronism between SW and AW occurs, HMER(f)=HFMR+Hex, where Hex ~ 3 – 5 Oe is the field of inhomogeneous exchange [18]. It can be seen that in the linear field region there is a direct match between the voltage maxima position and the main resonant frequency of the shear mode. It can be noted also that both transverse AW modes induce voltage UISHE in the parametric region ( H < Hc1), and the signal maximum is located in the reg ion Hc < H < Hc2. The field Hc corresponds to the creation process of two parametric magnons with frequency fp/2 and zero momentum, and Hc2 corresponds to the upper limit on H for the possible decays of parametric magnons with the frequency fp/2 into two secondary parametric ones at a frequency fp/4. A more detailed discussion of the critical fields given in (3) see [19]. Let's consider the case of longitudinal mode. As can be seen from Fig. 4 a, the longitudinal AW mode does not affect UISHE in the linear regime, and in the parametric one its influence is limited by the fields H < Hc =121Oe. Note that a small UISHE signal is also detected in fields 130 < H < 280 Oe, but at excitation frequencies that do not correspond to either the L or S HBAR modes. This is clearly visible, for example, from a comparison of curves 2 in Fig. 3a and Fig. 3b. The reason for this response is not yet clear. Note that parametric spin pumping induced by bot h S and L modes at frequencies of about 2.4 GHz was also observed in [19]. Note that the presence of additional resonant frequencies in the HBAR spectrum is due to locality of the elastic oscillations excitation. The excitation region in the structure pla ne is determined by the transducer aperture with diameter a. Strictly speaking, these oscillations will propagate not only under the transducer, as shown in Fig. 1, but also outside it, carrying energy away from the excitation region in the form of plate L amb modes [26, 27]. The highest resonator quality factor is achieved when the so -called trapped -energy regime is realized. Namely, at a certain ratio of frequencies and geometric dimensions, there are no conditions outside the transducer region for propaga ting modes. In this case, the elastic energy remains localized in the transducer region with an energy distribution decreasing exponentially with distance from the electrode edge. In this region fundamentally trapped -energy high overtone thickness modes ar e quasi -uniform in plane. In addition to fundamental modes, one or more lateral standing modes may be exited. These modes, which are also trapped -energy, are located higher in frequency from the fundamental ones and are usually called spurious resonance [2 6]. In our case, at least a small spurious S -mode is observed near the main one at a frequency of 1923.3 MHz. It can be noted that the depths of the | S11| dips for the main and the spurious modes differ several times (Fig. 3a). However, the heights of the corresponding resonant peaks on the UISHE are comparable (Fig. 3 b, Fig. 4). Such inconsistency can be explained as follows. The ISHE voltage according to (1) depends on the EISHE field magnitude, which is obviously greater for the fundamental mode. As for the length of the spin pumping region a', it turns out to be larger for the spurious mode compared to the main one, for which a' = a, since spurious mod e is less localized near the electrode boundaries. In this way, partial compensation occurs for the acoustic energy attributable to the non - fundamental mode. Conclusion The electroacoustic excitation of magnetic dynamics in YIG films in a magnon -phonon bulk acoustic resonator has been studied. The regimes of linear and parametric spin waves and spin currents excitation due to thickness extensional modes with various polarizations have been studied. In the linear regime, spin dynamics in the YIG f ilms is excited only by transverse modes (both fundamental and spurious thickness overtones). 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2024-03-10

In this work, we present the results of a systematic experimental study of linear and parametric spin wave resonant excitation accompanied by spin currents (spin pumping) in a multifrequency composite bulk acoustic wave resonator with a ZnO-YIG-GGG-YIG/Pt structure. The features of magnetic dynamics excitation in YIG films due to magnetoelastic coupling with acoustic thickness modes of various polarizations are studied. Acoustic spin waves and spin pumping are detected by simultaneous frequency-field mapping of the inverse spin Hall effect voltage and the resonant frequencies of thickness extensional modes. In the parametric range of frequencies and fields, acoustic spin pumping induced by both shear and longitudinal polarization modes was observed. Linear acoustic spin waves are excited only by shear thickness extensional modes because longitudinal acoustic waves do not couple with the magnetic subsystem in linear regime.

Spin Waves and Spin Currents in Magnon-Phonon Composite Resonator Induced by Acoustic Waves of Various Polarizations

2403.06274v1

Frequency fluctuations of ferromagnetic resonances at milliKelvin temperatures Tim Wolz,1Luke McLellan,2Andre Schneider,1Alexander Stehli,1Jan David Brehm,1Hannes Rotzinger,1, 3 Alexey V. Ustinov,1, 4, 5and Martin Weides2 1)Institute of Physics, Karlsruhe Institute of Technology, 76131 Karlsruhe, Germany 2)James Watt School of Engineering, Electronics & Nanoscale Engineering Division, University of Glasgow, Glasgow G12 8QQ, United Kingdom 3)Institute for Quantum Materials and Technologies, Karlsruhe Institute of Technology, 76131 Karlsruhe, Germany 4)National University of Science and Technology MISIS, 119049 Moscow, Russia 5)Russian Quantum Center, 143025 Skolkovo, Moscow, Russia (*Electronic mail: martin.weides@glasgow.ac.uk) (Dated: 15 July 2021) Unwanted fluctuations over time, in short, noise, are detrimental to device performance, especially for quantum coher- ent circuits. Recent efforts have demonstrated routes to utilizing magnon systems for quantum technologies, which are based on interfacing single magnons to superconducting qubits. However, the coupling of several components often introduces additional noise to the system, degrading its coherence. Researching the temporal behavior can help to iden- tify the underlying noise sources, which is a vital step in increasing coherence times and the hybrid device performance. Yet, the frequency noise of the ferromagnetic resonance (FMR) has so far been unexplored. Here, we investigate such FMR frequency fluctuations of a YIG sphere down to mK-temperatures, and find them independent of temperature and drive power. This suggests that the measured frequency noise in YIG is dominated by so far undetermined noise sources, which properties are not consistent with the conventional model of two-level systems, despite their effect on the sample linewidth. Moreover, the functional form of the FMR frequency noise power spectral density (PSD) cannot be described by a simple power law. By employing time-series analysis, we find a closed function for the PSD that fits our observations. Our results underline the necessity of coherence improvements to magnon systems for useful applications in quantum magnonics. Fluctuations of the resonance frequency and other forms of noise can drastically hamper the performance of sensors, amplifiers, and information processing circuits. This is ac- curate at room temperature but particularly crucial for quan- tum devices, where environmental noise leads to decoher- ence. With the recent coupling of single magnons to su- perconducting qubits1–3and resonators4–6, research on hy- brid quantum magnonics7–9has emerged. There, the goal is a combination of quantum computing’s exponential speed- up with magnonics’10,11low-loss devices. First demonstra- tions of magnonic devices are, for example, a magnon based transistor12or a majority gate13, combining OR and AND logic. Moreover, with a radio frequency-to-light conversion based on magnons14–16, a possible direction towards a quan- tum internet exists, but also requires a coupling of several quantum systems. Such a coupling often gives rise to addi- tional loss channels and increased noise, which along with the short coherence times of magnons presents a major ob- stacle in quantum magnonics17. Yet, the influence and origin of magnonic noise is still largely an open question. Predom- inantly, phase noise has been considered in magnetic tunnel junction oscillators18,19, the amplitude noise in a magnonic waveguide20at room temperature, and theoretically the mag- netization noise of spins21–23, for instance. Frequency fluc- tuations of the most basic magnon mode, the ferromagnetic resonance (FMR), however, have eluded attention. Here, we experimentally observe such FMR frequency fluc- tuations with a focus on an yttrium-iron-garnet (YIG) sphere at mK temperatures and show that time-series analysis canyield additional information, especially when the noise fre- quency dependence of the fluctuations cannot be described by a simple power law. After an introduction to the measure- ment setup and the spectroscopic characterization of the YIG sample, we briefly recapitulate the concept of the power spec- tral density (PSD). Then, the results of the frequency noise measurements are presented and analyzed. After which, we compare the results to room temperature data and a different material, lithium ferrite (LiFe). Our experimental setup (Fig. 1 (a)) consists of a vector net- work analyzer (VNA) connected to the different magnetic me- dia via a strip-line in a notch-type configuration. For the mK temperature measurements, the sample, a YIG sphere with diameter d=0:2mm, is mounted in a solenoid coil in- side a dilution refrigerator. A VNA offers a straight forward procedure for frequency noise measurements. Sweeping the probe frequency allows for a characterization of the sample via its Si j(w)-matrix element, from which we extract the FMR linewidth. Then, to measure frequency fluctuations, we em- ploy the continuous wave mode of the VNA with probe fre- quency wp. Here, we record a time trace of the sample’s frequency response at one single point close to resonance (wpwr). Fluctuations in the phase arg S21can then be con- verted to resonance frequency fluctuations via the slope in the linear region of arg S21(w), see Fig. 1 (b) for a schematic overview and Supplementary Information A 1 for more de- tails. All measurements are performed and evaluated with the open-source measurement suite qkit28. We start with the spectroscopic characterization of our sam-arXiv:2107.06531v1 [cond-mat.mes-hall] 14 Jul 20212 Frequency sweep Circle fit Linewidth Continous wave mode Phase / frequency conversion Periodograms PSD YIG4K 50mKHEMT(a) (b) 10dB 10dB 20dB 10 dB 200mK FIG. 1. Experimental setup, measurement schemes, and sample characterization at mK temperatures. (a) The magnetic medium, a YIG sphere, is mounted over a micro strip line and placed inside a solenoid coil in a dilution refrigerator. The steady-state response is measured via a vector network analyzer. (b) The measurement schemes show how linewidth data is extracted from frequency sweeps, whereas the continuous wave mode allows for an estimation of the frequency noise power spectral density (PSD). (c,d) Amplitude jS21jand phase arg S21response of the ferromagnetic resonance, shown for input power at the sample of P=90dBm and temperature T=50mK (background corrected). Solid orange lines denote a circle fit24, which is used to determine the FMR linewidth. Linear region of the phase response yields the conversion from phase fluctuation to frequency fluctuations. (e) Internal linewidth extracted from circle fits shows a temperature and power dependence that was previously attributed to loss into a bath of two-level systems25–27. ple at mK temperatures. The FMR is tuned to wr=2p= 6:11GHz, corresponding to an external field 0H0:21T, where the sample is fully magnetized. Figures 1 (c,d) show the amplitude and phase of the background corrected com- plex S21frequency response. A circle fit24returns the inter- nal linewidth (HWHM) ki=wr=(2Qi), with Qias internal Q- factor. Varying power and temperature, we find a linewidth dependence that decreases with increasing power Pand tem- perature Tin accordance to previous reports, which attributed this effect to energy loss into a bath of two-level systems (TLS)25–27(see Fig. 1 (e)). Increasing temperature and power eliminates the loss channels into the TLS bath by equalizing the occupation numbers of excited and unexcited states of the TLS bath. In the standard tunneling model, this linewidth de- pendence is given by29 kTLSµtanh (¯hwr=kBT)p 1+P=Pc: (1) Pcdenotes the critical drive power, at which the Rabi drive rate exceeds the coherence of the TLS. For our sample Pccan be found in the range of 80dBm <Pc<70dBm. The cable loss is included and estimated to be 20 dB. These values correspond to photon numbers of 106to 107and are similar to previous results for magnon excitations in YIG26,27. We now focus on the noise measurements. A recorded time trace of a fluctuating parameter, in our case frequency fluc-tuation Df, can be difficult to interpret and, hence, the power spectral density S(f)of the underlying random process is esti- mated. Roughly speaking, the PSD represents the fluctuation strength for a given frequency interval. The Wiener Khinchin theorem30relates the autocorrelation function (ACF) of the measured time trace to the PSD via Fourier transform. Em- ploying the convolution theorem, one can calculate a so-called periodogram, an estimate of the PSD: SDf(f) =lim T!¥1 2TjF(Dfr(t))j2 1 N fsN å n=1Dfr;nei2pf ndt2 : (2) In the second line, we used the discrete Fourier transformation with Ndata points, sampling time dtand normalization by the sampling rate fs=1=dt. For clarity, a subscript rfor the fre- quency fluctuations Dfis added in these equation. To reduce the PSD’s variance, we utilize Welch’s method31, where the data is divided into multiple segments and the resulting peri- odograms are averaged. Different physical noise mechanisms can manifest in dis- tinct noise PSDs and are affected differently by external pa- rameters. The presence of TLS does not only lead to an increased energy loss but TLS near resonance are also re- sponsible for frequency fluctuations. However, these fluc- tuations can be covered by other, more dominating, noise3 /uni00000013 /uni00000018/uni00000013/uni00000013 /uni00000014/uni00000013/uni00000013/uni00000013 /uni00000014/uni00000018/uni00000013/uni00000013 /uni00000015/uni00000013/uni00000013/uni00000013 /uni00000037/uni0000004c/uni00000050/uni00000048/uni00000003/uni0000000b/uni00000056/uni0000000c/uni00000015/uni00000013/uni00000013 /uni00000013/uni00000015/uni00000013/uni00000013f/uni00000003/uni0000000b/uni0000004e/uni0000002b/uni0000005d/uni0000000c /uni0000000b/uni00000044/uni0000000c /uni00000030/uni00000048/uni00000044/uni00000056/uni00000058/uni00000055/uni00000048/uni00000050/uni00000048/uni00000051/uni00000057 /uni00000024/uni00000035/uni0000000b/uni00000016/uni0000000c/uni00000003/uni00000053/uni00000055/uni00000052/uni00000046/uni00000048/uni00000056/uni00000056 /uni00000014/uni00000013/uni00000016 /uni00000014/uni00000013/uni00000015 /uni00000014/uni00000013/uni00000014 /uni00000014/uni00000013/uni00000013/uni00000014/uni00000013/uni00000014 /uni00000029/uni00000055/uni00000048/uni00000054/uni00000058/uni00000048/uni00000051/uni00000046/uni0000005c/uni00000003/uni0000000b/uni0000002b/uni0000005d/uni0000000c/uni00000014/uni00000013/uni00000018/uni00000014/uni00000013/uni0000001a/uni00000014/uni00000013/uni0000001c/uni00000014/uni00000013/uni00000014/uni00000014/uni00000033/uni00000036/uni00000027/uni00000003Sf(f)/uni00000003(Hz2/Hz) /uni0000000b/uni00000045/uni0000000c /uni00000018/uni00000013/uni00000003/uni00000050/uni0000002e/uni0000000f/uni00000003/uni00000010/uni00000019/uni00000013/uni00000003/uni00000047/uni00000025/uni00000050 /uni00000018/uni00000013/uni00000003/uni00000050/uni0000002e/uni0000000f/uni00000003/uni00000010/uni0000001b/uni00000013/uni00000003/uni00000047/uni00000025/uni00000050 /uni0000001b/uni00000013/uni00000013/uni00000003/uni00000050/uni0000002e/uni0000000f/uni00000003/uni00000010/uni00000019/uni00000013/uni00000003/uni00000047/uni00000025/uni00000050/uni00000003 /uni00000026/uni00000058/uni00000055/uni00000055/uni00000048/uni00000051/uni00000057 /uni0000002b/uni00000028/uni00000030/uni00000037/uni0000000f/uni00000003/uni00000010/uni00000019/uni00000013/uni00000003/uni00000047/uni00000025/uni00000050/uni00000013 /uni00000014/uni00000013/uni00000013 /uni00000015/uni00000013/uni00000013 /uni0000002f/uni00000044/uni0000004a/uni00000013/uni00000011/uni00000013/uni00000013/uni00000011/uni00000018/uni00000014/uni00000011/uni00000013/uni00000024/uni00000026/uni00000029/uni0000000b/uni00000046/uni0000000c /uni00000013 /uni00000014/uni00000013 /uni0000002f/uni00000044/uni0000004a/uni00000013/uni00000011/uni00000013/uni00000014/uni00000011/uni00000013/uni00000033/uni00000024/uni00000026/uni00000029 /uni0000000b/uni00000047/uni0000000c /uni00000014/uni00000013/uni00000016 /uni00000014/uni00000013/uni00000015 /uni00000014/uni00000013/uni00000014 /uni00000014/uni00000013/uni00000013/uni00000014/uni00000013/uni00000014 /uni00000029/uni00000055/uni00000048/uni00000054/uni00000058/uni00000048/uni00000051/uni00000046/uni0000005c/uni00000003/uni0000000b/uni0000002b/uni0000005d/uni0000000c/uni00000014/uni00000013/uni00000018/uni00000014/uni00000013/uni0000001a/uni00000014/uni00000013/uni0000001c/uni00000014/uni00000013/uni00000014/uni00000014/uni00000033/uni00000036/uni00000027/uni00000003Sf(f)/uni00000003/uni0000000bHz2/Hz/uni0000000c /uni0000000b/uni00000048/uni0000000c /uni00000030/uni00000048/uni00000044/uni00000056/uni00000058/uni00000055/uni00000048/uni00000050/uni00000048/uni00000051/uni00000057 /uni00000024/uni00000035/uni0000000b/uni00000016/uni0000000c/uni00000003/uni0000000b/uni0000003a/uni00000048/uni0000004f/uni00000046/uni0000004b/uni0000000c /uni00000024/uni00000035/uni0000000b/uni00000016/uni0000000c/uni00000003/uni0000000b/uni00000046/uni0000004f/uni00000052/uni00000056/uni00000048/uni00000047/uni00000003/uni00000049/uni00000052/uni00000055/uni00000050/uni0000000c FIG. 2. Frequency fluctuations of a ferromagnetic resonance (FMR) at mK temperatures. (a) Measured time trace of the frequency fluctuations (P=60dBm ;T=50mK) compared to generated data, a realization of a third order autoregressive process AR(3). For better visibility the data are offset by 200kHz. Measured data is post averaged to 8 Hz, so that periodic signals are removed. (b) Power spectral density (PSD) of FMR frequency fluctuations for different input powers and temperature. Below 3 Hz, the sample noise PSD exceeds the amplitude of parasitic noise sources, such as the HEMT amplifier and the current source. No dependence on these external parameters can be observed in the low frequency range of the PSD. The functional form of the PSDs is close to a Lorentzian but shows a steeper decay at around 1 Hz. (c,d) Autocorrelation (ACF) and partial autocorrelation (PACF) of the time trace data displayed in (a). PACF only shows values significantly different from zero up to lag n=3, indicating an AR(3) process. Note the different scales of the x-axes. (e) A comparison of measured FMR noise data with an AR(3) process shows excellent agreement. For the generated data, the PSD is calculated via Welch’s method and for the closed form PSD data, Eq. (4) was employed with estimated coefficients. sources. TLS frequency fluctuations can be understood in the Jaynes-Cummings model, where the resonance frequency of the FMR receives a shift depending on the state of the TLS. Such fluctuations have been observed in superconduct- ing resonators32–36, revealing three main characteristics: First, the frequency dependence of the PSD shows the infamous 1 =f decay, which is explained29,37by TLS uniformly distributed in frequency space with coherence rates distributed according to P(k)µ1=k. Second, due to the saturation of a TLS bath with power, the amplitude Aof the noise PSD should scale accord- ing to Aµ(P=Pc)0:5, as shown in Ref. 34. Third, depend- ing on whether the TLS themselves interact with each other, the amplitude should either reduce for decreasing temperature (non-interacting) or increase (interacting)35. We can now ap- ply this knowledge to the results of our noise measurements, which are presented in Fig. 2. Panel (a) shows a recorded time trace of the FMR frequency fluctuations, which is used to evaluate the PSD, as displayed in Fig. 2 (b). We note that the observed frequency noise PSD is indeed higher than the am- plifier noise and the noise produced by the current source (see Supplementary Information A 1 c). We also observe a func- tional form of the PSD that does not fit a simple power law. Up to 1 Hz, it can be described by a Lorentzian function but then a steep decrease follows. To test the influence of externalparameters, we varied the temperature from 50 mK to 800 mK and swept the input power around the critical power from 60 dBm down to100 dBm. Three curves are shown as examples, see Supplementary Information A 2 for more data. The frequency noise PSDs all show an independence of tem- perature and power. The increased white noise part for the low power PSD arises from amplifier noise. Taking all these points together, we conclude that TLS as described by the standard tunneling model are not the most dominant noise source for frequency fluctuations in our magnetic system. The lack of a power dependence is the strongest argument. A comparison to superconducting resonators35supports this statement. There, the TLS noise PSD at 0 :1 Hz is three magnitudes lower than the observed FMR fluctuation PSD. Hence, despite showing a power or temperature dependent resonance linewidth, so far undetermined noise sources most likely mask the influence of TLS noise in the magnon system. As the measured PSD does not follow a simple power law, we search for a closed function that describes our data. For this purpose, we return back to the time trace and analyze it with a method closely related to maximum entropy spectral analysis38, and based on time series analysis. There, a basic model describing random data is the autoregressive (AR) pro-4 cess, defined as yt=et+p å i=1aiyti: (3) A random data point at time tis calculated via a weighted sum of the last pdata points plus a white noise term with a Gaussian probability density function N(s;m=0), where sandmare the standard deviation and mean value, respec- tively. The aiare free parameters and have to be estimated as well as the order pof the process. AR processes are ap- plicable if the influence of a single perturbation propagates via sums of exponential decays or damped oscillations. A fa- mous examples is the AR(1) process with ai=1, describing a random walk or Brownian motion. A reduction of a1re- sults in the damping of these fluctuations over time. See Sup- plementary Information A 3 a for more examples and higher order processes. To first test the applicability of an AR pro- cess to our data, we look at the ACF, which indeed shows an exponential-like decay (see Fig. 2 (c)) for the measured FMR frequency noise, and hence points towards an AR pro- cess. Next, we estimate the order of the process, as well as the values of our coefficients. Here, we make use of the par- tial autocorrelation function (PACF), which only returns the direct correlation between data points, i.e., the indirect in- fluence of data points lying between is switched off. Since per definition of the AR process, a direct influence only ex- ists up to order p, we count the time lags that show a value significantly different from zero, and find p=3 (Fig. 2 (c)), confirming the validity of the AR model for our data. The aicoefficients can then be calculated by employing the Yule- Walker equations39,40, which relate the ACF to the ai(Sup- plementary Information A 3 b). The estimated coefficients are a1=1:764;a2=1:079;a3=0:309;s=5:284kHz. Note that the order and subsequently the coefficients depend on the chosen sampling rate. To remove periodic signals and the 1 =f amplifier part, a digital post averaging to a sampling frequency of 8 Hz was performed, see Supplementary Information A 4 for different sampling rates. With the estimated values, we can generate a model time trace for comparison (Fig. 2 (a)) and calculate its PSD. Importantly, a closed form41for the PSD of an AR( p) process exists that depends on the aiparam- eters and the variance s2of the white noise part: S(f) =2s2dt1åp k=1akei2pkdt f: (4) Figure 2 (e) shows an excellent agreement of the measured PSD with both, the numerical simulation and the closed form. Furthermore, from the estimated parameters and the ACF, we can conclude that FMR frequency fluctuations are exponen- tially damped out over time without showing an oscillating behavior. The higher order of the AR process indicates sev- eral noise mechanisms that occur on different timescales22, and therefore require a weighted sum of the last pdata points. We also compare the previous results to room temperature measurements of YIG and LiFe. Two surprising results can be observed from the data in Fig. 3: First, focusing on YIG, we see that the frequency noise at low temperature is about /uni00000014/uni00000013/uni00000015 /uni00000014/uni00000013/uni00000014 /uni00000014/uni00000013/uni00000013 /uni00000029/uni00000055/uni00000048/uni00000054/uni00000058/uni00000048/uni00000051/uni00000046/uni0000005c/uni00000003/uni0000000b/uni0000002b/uni0000005d/uni0000000c/uni00000014/uni00000013/uni00000014/uni00000016 /uni00000014/uni00000013/uni00000014/uni00000014 /uni00000014/uni00000013/uni0000001c /uni00000014/uni00000013/uni0000001a /uni00000033/uni00000036/uni00000027/uni00000003Sy/uni00000003/uni0000000b1/Hz/uni0000000c/uni0000003c/uni0000002c/uni0000002a/uni0000000350/uni00000050/uni0000002e /uni0000003c/uni0000002c/uni0000002a/uni00000003/uni00000015/uni0000001c/uni00000013/uni0000002e /uni0000002f/uni0000004c/uni00000029/uni00000048/uni00000003/uni00000050/uni0000004c/uni00000051/uni0000004c/uni00000050/uni00000058/uni00000050 /uni0000002f/uni0000004c/uni00000029/uni00000048/uni00000003/uni0000004f/uni0000004c/uni00000051/uni00000048/uni00000044/uni00000055FIG. 3. Comparison of frequency fluctuations of YIG and LiFe at room temperature to the low temperature data. The amplitudes of the power spectral densities (PSD) are normalized by the resonance frequency Sy=SDf=f2r. LiFe exhibits a minimum in its field disper- sion, resulting in a magnetic field insensitivity. The PSD at this point in the dispersion are compared to a point in the linear regime. Differ- ent white noise baselines are due to phase frequency conversion and normalization. two magnitudes higher than at room temperature in the low frequency region and both curves exhibit different functional forms. The increased white noise level, compared to the low temperature measurement, can be attributed to the lower dy- namic range of a second VNA, employed in the room tem- perature setup. We note that these room temperature fluctu- ations could be caused by current noise, which we could not directly measure in this setup. Nevertheless, the higher ampli- tude at low temperature either suggests an extreme increase of noise with temperature in a region above 800 mK to room temperature or distinctly different noise mechanisms. Both possibilities emphasize that additional care has to be taken in the development of coherent quantum magnonic devices in the future. Second, we examine frequency noise of LiFe. For this material, a mode softening was observed yielding a min- imum in its dispersion42. At such a minimum, the resonance frequency is first-order insensitive to field fluctuations. The FMR of our sample exhibits a gradient of the dispersion that is almost zero (see Supplementary Information A 5). We per- form measurements in this region and in the linear dispersion regime. Despite the reduced field sensitivity, measurements at this insensitivity point show stronger fluctuations than in the linear dispersion regime. The fluctuations are also stronger than the low temperature noise of YIG. The strong frequency noise of LiFe around the insensitivity point therefore presents a considerable challenge for possible magnon based frequency precision applications43. In conclusion, we studied FMR frequency fluctuations at mK temperatures. The recorded PSDs do not show a simple power law and are also independent of temperature and in- put power, which indicates undetermined noise mechanisms stronger than the influence of TLS described by the the stan- dard tunneling model. We also presented a method to analyze noise data in the time domain, especially useful if a simple power law is not sufficient to describe the noise PSD. With5 this method and after post averaging of the data down to 8 Hz, we find an excellent agreement of the measured data with an AR(3) process, suggesting that several noise processes on dif- ferent time scales are at play. A comparison to room tem- perature measurements and LiFe has shown increased noise at low temperatures and, surprisingly, also a high noise PSD close to the field-insensitivity point of LiFe, underpinning the importance of improving magnon coherence for useful appli- cations. With this work, we hope to spark a broader interest into magnon decoherence research. ACKNOWLEDGMENTS We wish to acknowledge fruitful discussions with Jür- gen Lisenfeld, Khalil Zakeri, Dmytro Bohzko, and Mehrdad Elyasi. We acknowledge financial support from the for- mer Helmholtz International Research School for Teratronics (Tim Wolz), the Landesgraduiertenförderung (LGF) Baden- Württemberg (Alexander Stehli), the Carl-Zeiss-Foundation (Andre Schneider) and Studienstiftung des Deutschen V olkes (Jan David Brehm). This work was supported by the European Research Council (ERC) under the Grant Agreement 648011 (MW) and by the Ministry of Science and Higher Education of the Russian Federation in the framework of the State Pro- gram (Project No. 0718-2020-0025) (A VU). DATA AVAILABILITY STATEMENT The data that support the findings of this study are available from the corresponding author upon reasonable request. 1Y . Tabuchi, S. Ishino, A. Noguchi, T. Ishikawa, R. Yamazaki, K. Usami, and Y . Nakamura, “Coherent coupling between a ferromagnetic magnon and a superconducting qubit,” Science 349, 405–408 (2015). 2D. Lachance-Quirion, Y . Tabuchi, S. Ishino, A. Noguchi, T. Ishikawa, R. Yamazaki, and Y . Nakamura, “Resolving quanta of collective spin exci- tations in a millimeter-sized ferromagnet,” Science Advances 3, e1603150 (2017). 3D. Lachance-Quirion, S. P. Wolski, Y . Tabuchi, S. Kono, K. Usami, and Y . 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Tobar, “Cavity magnon polaritons with lithium ferrite and three-dimensional microwave resonators at millikelvin temperatures,” Physical Review B 97, 155129 (2018). 43G. Flower, M. Goryachev, J. Bourhill, and M. E. Tobar, “Experimental implementations of cavity-magnon systems: From ultra strong coupling to applications in precision measurement,” New Journal of Physics 21, 095004 (2019). 44E. Rubiola, “The Leeson effect - Phase noise in quasilinear oscillators,” arXiv:physics/0502143 (2005). 45R. Boudot and E. Rubiola, “Phase noise in RF and microwave amplifiers,” IEEE Transactions on Ultrasonics, Ferroelectrics and Frequency Control 59, 2613–2624 (2012). 46S. Seabold and J. Perktold, “Statsmodels: Econometric and statistical mod- eling with python,” in 9th Python in Science Conference (2010).7 Appendix A: Supplementary information 1. Experimental details a. Sample geometry In the low temperature experiment, the YIG sphere is placed over a 50 matched micro strip line such that the 110 axis is aligned parallel to the external field. The micro strip is made from a Rogers TMM10i copper cladded (35 µm) sub- strate with a thickness of 0 :64 mm. For the room temperature experiments, the external field is generated by two Helmholtz coils with an iron yoke. The sample is also placed over a micro strip made from a Rogers TMM10i substrate with the 110 axis along the external field. b. Phase frequency conversion Phase fluctuations can be converted to frequency fluctua- tions via the following formula: Dj= (2QL+2Qi)Dw w0: (A1) This equation represents the linearization of the phase roll-off around the resonance frequency. In our experiments, however, we fit the phase response around the resonance frequency by a linear function, and use the extracted parameters for the phase-frequency conversion. We also note that by having chosen sampling rates fs<200Hz all noise measurements are performed below the Leeson frequency wL=wr=(2QL), with QLas the loaded quality factor. This way only fre- quency fluctuations are observed and not instantaneous phase fluctuations44. c. Parasitic noise sources We identify two parasitic external noise sources in the setup: the HEMT amplifier, producing phase noise and the current source, with current fluctuations translating into fre- quency fluctuations of the FMR. Microwave amplifiers exhibit a low frequency 1 =fpart, which is independent of power, and a white noise part, scaling inversely scaling with input power45. To reduce the amplifier’s white noise, we prepared the sample in the under-coupled regime, so that most of the in- put power is transmitted. Yet according to Eq. (A1) the slope of the phase response flattens out and then again frequency noise of the sample will be low compared to the phase noise of the amplifier for a strongly under-coupled regime. As shown in Fig. 1 (c,d) the amplitude and phase signal is still strong enough and hence a good compromise was found. Addition- ally, the sample is shielded from HEMT noise by a circulator and an additional 10 dB attenuator before the HEMT, which is used to prevent compression of the HEMT due to the high input powers employed in the experiment. Band pass filters /uni00000014/uni00000013/uni00000018/uni00000014/uni00000013/uni0000001a/uni00000014/uni00000013/uni0000001c/uni00000014/uni00000013/uni00000014/uni00000014Sf(f)/uni00000003/uni0000000bHz2/Hz/uni0000000c T=40mK/uni0000000b/uni00000044/uni0000000c P/uni00000003/uni0000000b/uni00000047/uni00000025/uni00000050/uni0000000c /uni00000003/uni00000010/uni0000001a/uni00000013 /uni00000003/uni00000010/uni0000001b/uni00000013 /uni00000003/uni00000010/uni0000001c/uni00000013 /uni00000010/uni00000014/uni00000013/uni00000013 /uni00000014/uni00000013/uni00000018/uni00000014/uni00000013/uni0000001a/uni00000014/uni00000013/uni0000001c/uni00000014/uni00000013/uni00000014/uni00000014Sf(f)/uni00000003/uni0000000bHz2/Hz/uni0000000c P=80dBm /uni0000000b/uni00000045/uni0000000c T/uni00000003/uni0000000b/uni00000050/uni0000002e/uni0000000c /uni00000003/uni00000018/uni00000013 /uni00000015/uni00000013/uni00000013 /uni00000017/uni00000013/uni00000013 /uni00000019/uni00000013/uni00000013 /uni00000014/uni00000013/uni00000015 /uni00000014/uni00000013/uni00000014 /uni00000014/uni00000013/uni00000013/uni00000014/uni00000013/uni00000014 /uni00000029/uni00000055/uni00000048/uni00000054/uni00000058/uni00000048/uni00000051/uni00000046/uni0000005c/uni00000003/uni0000000b/uni0000002b/uni0000005d/uni0000000c/uni00000014/uni00000013/uni00000014/uni00000018 /uni00000014/uni00000013/uni00000014/uni00000016 /uni00000014/uni00000013/uni00000014/uni00000014 /uni00000014/uni00000013/uni0000001c Sy(f)(1/Hz)T=40mK P=60dBm /uni0000000b/uni00000046/uni0000000c r/2/uni00000003/uni0000000b/uni0000002a/uni0000002b/uni0000005d/uni0000000c /uni00000019/uni00000011/uni00000014/uni00000014 /uni00000017/uni00000011/uni00000016/uni00000015FIG. 4. Measured power spectral densities for different power, tem- perature and resonance frequencies. (a) Power sweep, at constant temperature T=40mK. Power is referenced to the sample input. No power dependence is visible. (b) Temperature sweep, at con- stant power P=80dBm. All recorded PSDs are temperature in- dependent. (c) Comparison of frequency fluctuations at two differ- ent resonance frequencies wr;1=6:11GHz (used in (a) and (b)) and wr;1=4:32GHz. The PSDs are normalized to their resonance fre- quencies. Differences are minimal and only visible around the kink at 1 Hz. (3–7 GHz), installed before and after the sample, reduce un- wanted external low frequency noise in the microwave lines. To determine the current fluctuations, we inserted a 1 re- sistor between current source and solenoid coil at room tem- perature. We then employed an FFT spectrum analyzer and measured the voltage drop at the resistor over an RC high pass with a cutoff frequency of fc=3102Hz to filter out the dc part, thereby circumventing the dynamic range limitation of the spectrum analyzer.8 /uni00000013 /uni00000015/uni00000013/uni00000013 /uni00000017/uni00000013/uni00000013 /uni00000037/uni0000004c/uni00000050/uni00000048/uni00000003/uni0000000b/uni00000056/uni00000048/uni00000046/uni0000000c/uni00000013/uni00000014/uni00000013/uni00000015/uni00000013 y/uni0000000b/uni00000044/uni0000000c /uni00000024/uni00000035/uni0000000b/uni00000014/uni0000000c /uni00000013 /uni00000015/uni00000013 /uni00000017/uni00000013 /uni00000037/uni0000004c/uni00000050/uni00000048/uni00000003/uni0000000b/uni00000056/uni00000048/uni00000046/uni0000000c/uni00000013/uni00000011/uni00000013/uni00000013/uni00000011/uni00000018/uni00000014/uni00000011/uni00000013 y /uni0000000b/uni00000045/uni0000000c /uni00000014/uni00000013/uni00000015 /uni00000014/uni00000013/uni00000014 /uni00000014/uni00000013/uni00000013 /uni00000029/uni00000055/uni00000048/uni00000054/uni00000058/uni00000048/uni00000051/uni00000046/uni0000005c/uni00000003/uni0000000b/uni0000002b/uni0000005d/uni0000000c/uni00000014/uni00000013/uni00000013/uni00000014/uni00000013/uni00000015/uni00000014/uni00000013/uni00000017/uni00000033/uni00000036/uni00000027/uni00000003/uni0000000b/uni00000014/uni00000012/uni0000002b/uni0000005d/uni0000000c/uni0000000b/uni00000046/uni0000000c /uni00000013 /uni00000015/uni00000013/uni00000013 /uni00000017/uni00000013/uni00000013 /uni00000037/uni0000004c/uni00000050/uni00000048/uni00000003/uni0000000b/uni00000056/uni00000048/uni00000046/uni0000000c/uni00000018 /uni00000013/uni00000018y/uni0000000b/uni00000047/uni0000000c /uni00000024/uni00000035/uni0000000b/uni00000015/uni0000000c /uni00000013 /uni00000015/uni00000013 /uni00000017/uni00000013 /uni00000037/uni0000004c/uni00000050/uni00000048/uni00000003/uni0000000b/uni00000056/uni00000048/uni00000046/uni0000000c/uni00000013/uni00000011/uni00000018 /uni00000013/uni00000011/uni00000013/uni00000013/uni00000011/uni00000018/uni00000014/uni00000011/uni00000013y /uni0000000b/uni00000048/uni0000000c /uni00000014/uni00000013/uni00000015 /uni00000014/uni00000013/uni00000014 /uni00000014/uni00000013/uni00000013 /uni00000029/uni00000055/uni00000048/uni00000054/uni00000058/uni00000048/uni00000051/uni00000046/uni0000005c/uni00000003/uni0000000b/uni0000002b/uni0000005d/uni0000000c/uni00000014/uni00000013/uni00000015 /uni00000014/uni00000013/uni00000014 /uni00000014/uni00000013/uni00000013/uni00000014/uni00000013/uni00000014/uni00000033/uni00000036/uni00000027/uni00000003/uni0000000b/uni00000014/uni00000012/uni0000002b/uni0000005d/uni0000000c/uni0000000b/uni00000049/uni0000000c FIG. 5. Examples of AR(1) and AR(2) processes. First row (a-c) shows two examples of the AR(1) process with parameters a1=1, repre- senting Brownian motion (brown line) and a1=0:92 depicting a damped random walk (blue line); second row (d-f) illustrates examples of the AR(2) process with parameters a1=0:6;a2=0:2 (red line) having an exponential ACF, and a1=0:9;a2=0:6 (purple line) featuring an oscillation around the mean value. From left to right, the panels display time traces, i.e., realizations of random process, propagations of a one-time shock, and the power spectral densities of the respective processes. 2. Additional power spectral density data In Fig. 4 (a-c) further frequency noise PSD data are shown, confirming the discussed noise independence of power and temperature. Data for panel (a) and (b) are taken at wr= 6:11GHz, and therefore the sample is fully magnetized. Over the measured range of the two external parameters, no dif- ference in the low-frequency part of the PSDs can be seen. Again, the increase in white noise in panel (a) is attributed to amplifier noise due to lower input power45. Figure 4 (c) displays a comparison of two different resonance frequencies, above and below the saturation magnetization. The data are normalized via the resonance frequencies for better compari- son. The low frequency parts are identical. Yet, a small dif- ference is visible around the kink at 1 Hz, suggesting a small influence of the sample magnetization. 3. Time series analysis a. Examples of AR processes For a better understanding of AR processes, we show ex- amples of first and second order AR processes with different parameters ai. The mathematical background can be found in a textbook by Box et al.41, for instance. Recall from Eq. 3, that in the AR process, white noise fluctuations are added to a weighted sum of the last pdata points. Examples of timetraces generated according to this equations are displayed, as well as the propagation of a one time-shock and their PSDs are shown in Fig. 5. As mentioned, in the main text, an AR(1) pro- cess with a1=1, represents the famous random walk or Brow- nian motion. In Fig. 5 (a), we can see how the summing of all white noise terms leads to a few big fluctuations over time. The summation can be also seen as the integration of white noise, giving an 1 =fterm in frequency space, which is subse- quently squared for the PSD and therefore yielding the 1 =f2 decay of so-called brown noise. Reducing a1filters out big fluctuations, since a one-time shock is exponentially damped over time (compare Fig. 5 (a,b)). This filtering is also visi- ble in the PSD (Fig. 5 (c)), having a Lorentzian form, which flattens out for low frequency compared to the pure random walk. Now, considering an AR(2) process and choosing the parameters accordingly, we see either the exponential damp- ing of the white noise fluctuations for a1=0:6;a2=0:2 or an oscillating behavior for a1=0:9;a2=0:6 (Fig. 5 (e)). Moreover, the differences can already be recognized in their time traces (Fig. 5 (d)), where the oscillating AR(2) process frequently crosses the mean value. In the PSD, Fig. 5 (f), the damped process has a form close to a Lorentzian, whereas the oscillating process shows a peak at its oscillating frequency. Increasing the order of the AR process shows a similar quali- tative behavior with oscillations and/or damped exponentials, but described by additional summands.9 b. Yule-Walker equations and partial autocorrelation In the main text, we stated the usefulness of the Yule- Walker equation (YWE) to estimate the specific values for the aiand the partial auto correlation function (PACF) to estimate the order of the AR process. Again, for more mathematical derivations, we refer to Box et al.41and present only the em- ployed procedure for our calculations. The YWE are a set of equations that relate the values of the ACF riat lag ito the coefficients aiof the AR process and are defined as follows: 0 BBBB@1 r1 r2::: r1 1 r1::: r2 r1 1::: ............ rp1rp2rp3:::1 CCCCA0 BBBB@a1 a2 a3 ... ap1 CCCCA=0 BBBB@r1 r2 r3 ... rp1 CCCCA: (A2) We see that for a specific order p, we obtain a set of pequa- tions, in which we can replace the riby their measured values and solve for the ai. Moreover, the PACF can also be calcu- lated with the YWE. Since the PACF only describes the direct correlation between data points and since there is no depen- dence for lag values n>pin the AR process, the coefficient ap, equaling the order of the AR process also represents the PACF value at lag p. This means one has to start with order p=1, take the measured r1, and calculate a1(which equals r1) for the first value in the PACF. Then pneeds to iteratively be increased and the procedure repeated. If an AR process is applicable the PACF will drop to zero after the first pvalues. The white noise part can be considered as zero order of the AR process and can also be incorporated into the Yule Walker equations as r0=p å i=1airi+s2=1: (A3) Hence, after the aiare determined, the variance of the Gaus- sian white noise process s2can be estimated. For the numeri- cal time-series analysis in this work, we employed the python statsmodel46package. 4. PACF dependence on sampling rate We showed the time series analysis for a post averaged sam- pling rate of 8 Hz in the main text. The sampling rate was chosen such that the steep decay in the PSD is still captured but the influence of the 1 =fHEMT noise and the periodicsignals, mainly 50 Hz current oscillations, are averaged out. Now, we consider different sampling rates below 8 Hz. Fig- ure 6 (a) shows the PACF for the first four lags depending on the sampling rate. We see that the third order becomes negli- gent below 2 Hz, where also the steep decay is averaged away. The PACF value at lag n=2 remains for even lower sampling rates, likely because of the slight curvature in the PSD leading to the knee at 1 Hz. Reducing the sampling rate even lower, the PSD becomes a simple Lorentzian and hence only the PACF at lag n=1 is of importance. Values at higher lags are within the grey shaded region denoting the 95 % confidence interval and are hence not significant anymore. Figure 6 (b) emphasizes this point by showing the PACF for several lag values at the lowest evaluated sampling rate. 5. Room temperature characterization Figure 7 (a) shows the dispersion relation of the Kittel mode and its gradient (b) for LiFe at room temperature. Goryachev et al. observed a minimum in the dispersion due to a mode softening42. There the resonance frequency is first-order in- sensitive to fluctuations in the external field. For our sample, the FMR dispersion exhibits a flat region over roughly 15 mT. Due to the high linewidth, k17MHz (HWHM) and there- fore the small slope of the phase response, frequency noise of LiFe could not be observed at low temperature. It was masked by the HEMT phase noise. We note that the linewidth at this minimum is higher than in the linear region ( k13MHz) and also that the sample is not fully magnetized. /uni00000014/uni00000013/uni00000014 /uni00000014/uni00000013/uni00000013/uni00000014/uni00000013/uni00000014 /uni00000036/uni00000044/uni00000050/uni00000053/uni0000004f/uni0000004c/uni00000051/uni0000004a/uni00000003/uni00000055/uni00000044/uni00000057/uni00000048/uni00000003/uni0000000b/uni0000002b/uni0000005d/uni0000000c/uni00000013/uni00000011/uni00000018 /uni00000013/uni00000011/uni00000013/uni00000013/uni00000011/uni00000018/uni00000014/uni00000011/uni00000013/uni00000033/uni00000044/uni00000055/uni00000057/uni0000004c/uni00000044/uni0000004f/uni00000003/uni00000024/uni00000026/uni00000029 /uni0000000b/uni00000044/uni0000000c n=1 n=2n=3 n=4 /uni00000013 /uni00000014/uni00000013 /uni0000002f/uni00000044/uni0000004a/uni00000003n /uni0000000b/uni00000045/uni0000000c /uni00000036/uni00000044/uni00000050/uni00000053/uni0000004f/uni0000004c/uni00000051/uni0000004a/uni00000003/uni00000055/uni00000044/uni00000057/uni00000048 /uni00000013/uni00000011/uni00000013/uni00000016/uni00000016/uni00000003/uni0000002b/uni0000005d FIG. 6. Dependence of the partial autocorrelation (PACF) on the post-processing sample rate. (a) PACF for lag n=1 to 4. (b) PACF at lowest sampling rate for different lags. Only first lag shows a value significantly different from zero, as depicted by the grey region, the 95 % confidence interval.10 /uni00000014/uni00000017/uni00000013 /uni00000014/uni00000019/uni00000013 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 /uni00000014/uni00000017/uni00000013 /uni00000014/uni00000019/uni00000013 /uni00000014/uni0000001b/uni00000013 /uni00000028/uni0000005b/uni00000057/uni00000048/uni00000055/uni00000051/uni00000044/uni0000004f/uni00000003/uni00000049/uni0000004c/uni00000048/uni0000004f/uni00000047/uni00000003/uni0000000b/uni00000050/uni00000037/uni0000000c/uni00000017 /uni00000015 /uni00000013/uni00000015/uni00000017/uni0000002a/uni00000055/uni00000044/uni00000047/uni0000004c/uni00000048/uni00000051/uni00000057/uni00000003/uni0000000b/uni0000002a/uni0000002b/uni0000005d/uni00000012/uni00000037/uni0000000c /uni0000000b/uni00000045/uni0000000c /uni00000013/uni00000011/uni00000019/uni00000016 /uni00000014/uni00000011/uni00000014/uni00000017 |S21| FIG. 7. Dispersion relation and gradient of LiFe. (a) A flat region between 150 mT and 165 mT is visible in the dispersion relation, at- tributed to a mode softening42and making the ferromagnetic reso- nance less susceptible to field fluctuations. The arrow indicates the bias point at which the fluctuation measurements were performed. Values higher than one in the Smatrix element are due to the back- ground correction in combination with an impedance mismatch in the system. (b) Gradient of the dispersion spectrum, numerically calcu- lated. Points are the extracted FMR frequencies with a median filter (solid line) as a guide to the eye.

2021-07-14

Unwanted fluctuations over time, in short, noise, are detrimental to device performance, especially for quantum coherent circuits. Recent efforts have demonstrated routes to utilizing magnon systems for quantum technologies, which are based on interfacing single magnons to superconducting qubits. However, the coupling of several components often introduces additional noise to the system, degrading its coherence. Researching the temporal behavior can help to identify the underlying noise sources, which is a vital step in increasing coherence times and the hybrid device performance. Yet, the frequency noise of the ferromagnetic resonance (FMR) has so far been unexplored. Here, we investigate such FMR frequency fluctuations of a YIG sphere down to mK-temperatures, and find them independent of temperature and drive power. This suggests that the measured frequency noise in YIG is dominated by so far undetermined noise sources, which properties are not consistent with the conventional model of two-level systems, despite their effect on the sample linewidth. Moreover, the functional form of the FMR frequency noise power spectral density (PSD) cannot be described by a simple power law. By employing time-series analysis, we find a closed function for the PSD that fits our observations. Our results underline the necessity of coherence improvements to magnon systems for useful applications in quantum magnonics.

Frequency fluctuations of ferromagnetic resonances at milliKelvin temperatures

2107.06531v1

Spin transport across antiferromagnets induced by the spin Seebeck eect Joel Cramer1;2, Ulrike Ritzmann3;4, Bo-Wen Dong1;2;5, Samridh Jaiswal1;6, Zhiyong Qiu7, Eiji Saitoh7;8;9;10, Ulrich Nowak;4, Mathias Kl aui?;1;2 1Institute of Physics, Johannes Gutenberg-University Mainz, 55099 Mainz, Germany 2Graduate School of Excellence Materials Science in Mainz (MAINZ), Mainz, 55128 Mainz, Germany 3Department of Physics and Astronomy, Uppsala University, 751 05 Uppsala, Sweden 4Department of Physics, University of Konstanz, 78457 Konstanz, Germany 5School of Materials Science and Engineering, University of Science and Technology Beijing, 100083 Beijing, People's Republic of China 6Singulus Technologies AG, 63796 Kahl am Main, Germany 7Advanced Institute for Materials Research, Tohoku University, Sendai 980-8577, Japan 8Institute for Materials Research, Tohoku University, Sendai 980-8577, Japan 9Center for Spintronics Research Network, Tohoku University, Sendai 980-8577, Japan 10Advanced Science Research Center, Japan Atomic Energy Agency, Tokai 319-1195, Japan E-mail:ulrich.nowak@uni-konstanz.de ,?klaeui@uni-mainz.de Abstract. For prospective spintronics devices based on the propagation of pure spin currents, antiferromagnets are an interesting class of materials that potentially entail a number of advantages as compared to ferromagnets. Here, we present a detailed theoretical study of magnonic spin current transport in ferromagnetic-antiferromagnetic multilayers by using atomistic spin dynamics simulations. The relevant length scales of magnonic spin transport in antiferromagnets are determined. We demonstrate the transfer of angular momentum from a ferromagnet into an antiferromagnet due to the excitation of only one magnon branch in the antiferromagnet. As an experimental system, we ascertain the transport across an antiferromagnet in YIG jIr20Mn80jPt heterostructures. We determine the spin transport signals for spin currents generated in the YIG by the spin Seebeck eect and compare to measurements of the spin Hall magnetoresistance in the heterostructure stack. By means of temperature-dependent and thickness-dependent measurements, we deduce conclusions on the spin transport mechanism across IrMn and furthermore correlate it to its paramagnetic-antiferromagnetic phase transition.arXiv:1803.03416v1 [cond-mat.mtrl-sci] 9 Mar 2018Spin transport across antiferromagnets induced by the spin Seebeck eect 2 1. Introduction For the development of next-generation, energy- ecient spintronic devices for information transmis- sion, processing, and storage, the investigation of pure spin currents has attracted great interest during re- cent years. In contrast to spin-polarized charge cur- rents, with a broad spectrum of applications in current spintronics schemes (e.g. spin-transfer-torque operated magnetic tunnel junctions [1]), pure spin currents ex- clusively transfer angular momentum and have no net charge ow. While in normal metals that exhibit the spin Hall eect [2] pure spin currents are realized by charge currents of opposite spin-polarization owing in opposite directions, magnetically ordered systems provide a further spin transport channel via magnonic (spin wave) excitations with no moving charges [3]. Aside from information transfer and data handling, pure spin currents have furthermore proven as a useful tool to investigate magnetic material properties. Spin Hall magnetoresistance (SMR) [4] measurements, for instance, allow to probe the orientation of magnetic sublattice moments in complex magnetic oxides [5, 6], which are otherwise not accessible using common char- acterization methods, e.g. SQUID magnetometry. With respect to magnonic spin current propaga- tion, insulating ferromagnets (FM) pose an interesting medium and therefore caught notable renewed atten- tion in recent years. As compared to metallic systems, insulators prevent spin transfer mediated by charge motion and consequently do not exhibit Joule heat- ing losses within the insulator. The most prominent representative of this material class is the yttrium iron garnet Y 3Fe5O12(YIG) [7], since it reveals excellent insulating properties and extremely low Gilbert damp- ing [8]. In single crystalline YIG, magnon propagation lengths in the range of several micrometer have been re- ported [9{12]. More recently, however, due to potential advantages over ferromagnets antiferromagnets (AFM) have gained increased interest considering spintronics applications [13]. In AFMs, neighboring magnetic mo- ments are ordered alternatingly, such that the macro- scopic moment Mof the solid vanishes. As a result, adjacent AFM devices do not exhibit mutual interac- tion due to the lack of stray elds and furthermore are insensitive to external magnetic eld perturbations. It has been shown that insulating AFMs are able to exhibit thermal magnon currents induced by the spin Seebeck eect (SSE) [11, 14, 15] when driveninto the spin- op state [16{18]. Magnon propagation across AFM thin lms has been investigated in FM/AFM/HM heterostructures both experimentally [19{24] and theoretically [25{27]. Since the excitation frequency of antiferromagnetic magnons usually lies in the range of several THz, they cannot be excited by optical or current electrical methods. Therefore, spin currents are generated in the FM layer, pumped into the AFM, and eventually detected in the HM by means of the inverse spin Hall eect (ISHE). The change of the ISHE voltage signal when measured as a function of AFM thickness or temperature eventually then allows one to infer information on the AFM magnon propagation properties. Here, we put forward an analytical model describing the details of magnon propagation in antiferromagnets. We demonstrate an exponential spatial decay of AFM magnons in insulators, which is in line with experimental observations. Despite the high speed of antiferromagnetic magnons, their range is limited due to a very short life time. Though the propagation length reveals a clear maximum just above the energy gap, it is signicantly smaller as compared to ferromagnetic systems. Our analytical work is well in agreement with the results obtained for atomistic spin dynamics simulations. Moreover, we present angular momentum transfer due to magnon propagation from a FM into an AFM. We identify two dierent regimes: Below the frequency gap, evanescent modes with a very strong spatial decay are excited within the AFM. Above the frequency gap, antiferromagnetic magnons are excited that propagate on a longer range within the AFM. On the experimental side, we investigate spin cur- rent transmission across the metallic AFM Ir 20Mn80 (IrMn) in YIG/IrMn/Pt trilayers to identify po- tentially dominant spin transmission channels (elec- tronic vs. magnonic). We perform both SSE and SMR measurements and compare the temperature- and thickness-dependent signal amplitudes obtained to examine whether genuine spin transport across IrMn or interface exchange coupling phenomena are observed. It was shown before that the thickness-dependent antiferromagnetic-paramagnetic phase transition of IrMn thin lms can be probed by means of temperature-dependent ferromagnetic reso- nance spin pumping measurements [28]. In trilayers of Ni81Fe19/Cu/IrMn, the Gilbert damping constant of Ni81Fe19exhibits an enhancement near TN eel, revealingSpin transport across antiferromagnets induced by the spin Seebeck eect 3 increased spin sink properties of the IrMn layer for the pumped spin current due to spin uctuations. As simi- lar observations were made for systems including insu- lating AFMs [22{24], this implies a signicant coupling of the spin current to the antiferromagnetic ordering parameter in IrMn. Consequently, this method allows one to indirectly gain insight into the magnetic prop- erties of IrMn. While no direct information about spin propagation was previously obtained, we here compare dierent layer stacks to identify the spin transport con- tribution to the signal. 2. Analytical model of magnon propagation in ferromagnets and antiferromagnets We start the development of the theoretical model by discussing spin transport in FMs and AFMs individually and the length scales involved. For that purpose, we consider a simple cubic lattice with lattice constant a. In the Hamiltonian, we include exchange interaction of nearest neighbors with exchange constant Jand an anisotropy leading to an easy axis in x-direction with anisotropy constant dx. The Hamiltonian is then given by H=X hijiJijSiSj+X idx(Sx i)2. (1) We perform atomistic spin dynamics simulations [29] as well as analytical calculations based on the Landau- Lifshitz-Gilbert (LLG) equation, _S= s(1 +2)SH  s(1 +2)S(SH). (2) This equation of motion describes the precession of normalized magnetic moments Saround their eective eldH=@H=@Sand relaxation depending on the damping constant . denotes the gyromagnetic ratio andsis the magnetic moment. In Ref. [30], the propagation length of magnons was investigated for FM systems. By linearizing the LLG equation and assuming Sex, the coupled equations of motions were solved. The imaginary part of the eigenvalue denes the magnon frequency h!FM= 2dx+JX (1cos(qa)) , (3) wheredenotes the cartesian components. The real part describes the lifetime = 1=(!). A magnon accumulation was dened as the transferred magnetic moment that scales with
mP q1=2A2 q, where Aqis the spin wave amplitude. Considering a spin wave propagating only in z-direction, q=qzez, the propagation length was dened as the decay of the dx= 0 .1|J|dx= 0 .05|J|dx= 0 .01|J|dx= 0.001 |J| frequency ¯hω/Jmagnon propagation length ξ/a 6 5 4 3 2 1 035 30 25 20 15 10 5 0Figure 1. Frequency dependent magnon propagation length in an AFM for various anisotropy constants dxand a damping constant of = 0:01. Numerical data, depicted as data points, are in agreement with the analytical model, which is shown as corresponding continuous lines. magnon accumulation
mand was obtained via the lifetimeand the group velocity vz=@!=@qz. The result for the propagation length was FM(!) =vz 2=Ja 2h!r 1 1h!2dx J2 . (4) The propagation length has a maximum close to the frequency gap and decays with increasing frequency. For low damping and low anisotropies, the propagation length of low frequency magnons is in the range of up to a fewm. These results explain a saturation eect of the SSE in YIG and a suppression eect due to large external magnetic elds [11,31]. Here, we now develop the analogous model for AFMs. We consider a similar system and choose J <0. This system consists of two sublattices A and B. To describe magnon excitations, we linearize the LLG equation for each sublattice and assume Sx i;A1 and Sx i;B1, as well as a small damping constant 1. The considered AFM has two magnon branches. A magnon describes a collective precession of magnetic moments in both sublattices, but with unequal amplitudes. The ratio of the amplitudes of the two sublattices is wave-vector dependent and it is reversed for the two magnon branches. Therefore, magnons of opposite branches carry opposite angular momentum. Moreover, magnetic moments precess either all clockwise or counterclockwise within the two dierent magnon branches. In the absence of an external magnetic eld, the magnon branches are degenerate and their dispersion relation is given by h!AFM =s 2dx+ 6jJj
24J2X cos(qa)
2. (5) In contrast to FMs, AFMs have a large frequency gap of h!0p 24dxjJj. Due to degeneracy, magnons from both branches are excited thermallySpin transport across antiferromagnets induced by the spin Seebeck eect 4 with equal probablity and no magnetitization occurs at constant temperatures. The total magnetization is also compensated in linear temperature gradients. It was shown that around a temperature step no net spin transfer occurs in AFMs, although a magnon current appears [32]. Despite the fact that in an isolated AFM no net spin current occurs, the length scale of magnon propagation in AFMs is interesting to study. Thermally induced magnons do not transfer angular momentum, but they still transfer heat and are the origin of thermally driven domain wall motion in AFMs [33, 34]. Moreover, external magnetic elds lift the degeneracy. It has been shown, that thermally activated spin currents appear due to the SSE [17,18]. The lifetime of AFM magnons is given by the real part of the eigenvalue and one obtains =h (2dx+ 6jJj)
. (6) The resulting lifetime is shorter than in FMs and independent of the magnon frequency. We obtain for the frequency-dependent magnon propagation length (!) =ajJjp H2 0(h!)2 H0h!s 1p H2 0(h!)2 2jJj22 , (7) where we use the abbreviation H0= 2dx+ 6jJj. We simulate the decay of magnons in an AFM with 88512 magnetic moments. To excite monochromatic spin waves with a group velocity only inz-direction, we attach an additional layer in the x-y-plane, in which all magnetic moments precess homogeneously with frequency !. The magnetic moments of the two sublattices are aligned in oppposite directions and their precession has a phase shift of 180 degrees. Due to exchange interaction, this layer couples to the system and monochromatic spin waves enter. By tting the exponential decay of the spin wave amplitudes, we calculate their propagation length. The results from the analytical formula as well as from numerical simulations are shown in Fig. 1. The propagation length increases strongly just above the frequency gap !0until a maximum value is reached and then decreases with further increasing frequency. The maximum values are much shorter than in FMs. Despite the higher velocity for magnons at a frequency close to gap, their range is still small due to their short lifetime. As shown in the gure, the analytical formula describes the general behavior of the propagation length. However, for high frequencies deviations between analytical calculation and numerical simulation appear due to the limited cross section in the simulations. Note that in contrast dx= 0.001 |J|dx= 0 .01|J|dx= 0 .05|J|dx= 0 .1|J| damping constant αpropagation length ξ/a 1 0.1 0.01 0.0011000 100 10 1Figure 2. Magnon propagation length as a function of the damping constant for dierent anisotropy constants dx. The numerical data are shown as data points and the continuous lines represent the maximum value of the analytical one-dimensional model. to FMs, the dispersion relation of AFMs depends on the spatial dimension of the lattice. Similar to our previous studies on FMs [30], we study the length scale of thermally triggered magnon propagation using a temperature step to excite the magnons. We simulate 8 8512 magnetic moments and apply a temperature step along the z-axis from kBT1= 0:1jJjtokBT2= 0. We t the decay of the magnon accumulation in both sublattices and compare the resulting length scale with the maximum propagation length from equation 7. Figure 2 shows the results from numerical simulation as well as from the analytical model. For high damping values, the analytical formula deviates since we neglected 2- terms in the derivation. But both methods give similar results for low damping values. In contrast to low frequency magnons in FMs, which can propagate over severalm, the AFM magnons have a much shorter range in the nm-regime. 3. Magnon transfer in ferromagnet-antiferromagnet-heterostructures To compare to experimental work, we study the excitation of spin waves in hetero- structures consisting of a FM and an AFM layer. We excite a monochromatic spin wave in the FM and study the transfer of angular momentum into the AFM. We perform simulations with a FM system with 8 8256 magnetic moments and additionally an AFM layer of the same size attached to it. For simplication, we use a layered AFM by considering antiferromagnetic exchange interaction only in z-direction, JFM = Jx AFM =Jy AFM =Jz AFM. The exchange interaction at the interface is given by JIF=JFM. The monochromatic spin wave is excited by a homogenous precession of the magnetic moments with a given frequency!at the 0th layer of the FM.Spin transport across antiferromagnets induced by the spin Seebeck eect 5 ¯hω= 0.5J¯hω= 0.1J space position z/amagnetization comp. |mx| AFM FM 60 40 20 0 −20 −401 0.995 0.99 0.985 Figure 3. Absolute value of the x-component of the magnetization for spin wave propagation from a FM ( z < 0) to AFM (z >0) layer for an evanescent mode ( h!= 0:1J) and a normal mode ( h!= 0:5J). The dots (triangles) in the AFM- regime show mxfor sublattice A (B). Dependent on the frequency of the spin wave, two dierent regimes for the spin wave propagation within the AFM appear. For frequencies below the gap of the dispersion relation of the AFM, the signal decays exponentially with distance to the interface. These are evanescent modes [27]. Spin waves with frequencies above the gap excite a spin wave of the same frequency within the antiferromagnet. Note that in this quasi one-dimensional AFM, the dispersion relation is given by h!=q 2dx+ 2jJj
24J2 cos(qza)
2. (8) The frequency gap in this case is  h!0p 8dxjJj. In Fig. 3, we show the x-component (easy axis) of the magnetization for two dierent examples. The red curves show an evanescent mode where no precession of they- andz- components of the magnetization in the AFM is observed and the signal disappears on a very short length scale. The blue curves represent a normal mode in the AFM, where the spin wave propagates within the AFM with the same frequency as in the FM. They- andz- components of the magnetization of the single sublattices show precession due to AFM spin wave propagation, whereas the x-component of the magnetization in both sublattices decays exponentially within the magnon propagation length. The orientation of the magnetization of the FM determines the sense of the rotation of the magnetic moments as well as the transferred angular momentum in the FM. Therefore, only one of the two magnon branches is excited and due to the dierent amplitudes of the two sublattices, angular momentum is transferred. The oscilattion of the x-component within the FM layer illustrates interference of the incoming spin wave with a strongly re ected wave at the interface and only a small ratio of the signalis transferred in both cases into the AFM. Note that both spin waves in the ferromagnet have been excited with the same initial amplitude at z=256a. The higher frequency has a much shorter propagation length in the FM and, therefore, its amplitude at the interface is signicantly smaller. Nevertheless, with larger distances to the interface, the normal AFM magnon causes a larger signal than the evanescent mode. The chosen frequency is close to the gap and the propagation length is several nm. Here, we demonstrate the propagation of spin waves for a single monochromatic wave. For temper- ature gradients inducing the SSE a broad frequency spectra would be excited in the ferromagnetic layer. Due to the larger propagation length at low frequen- cies within the FM, these frequencies play an impor- tant role in the SSE in YIG [31]. Due to the high frequency gap of antiferromagnets, mainly evanescent modes should be excited. The transferred spin cur- rent should decay exponentially within distances in the range of a few nm. 4. Experimental investigation of spin current transmission across a metallic antiferromagnet Having established the theory of spin transport in and across AFMs using pure magnonic spin currents, we next investigate spin transport experimentally in a combination of ferromagnetic, antiferromagnetic and heavy metal layers. To begin with, let us compare the results of the theory to experimental ndings for systems including insulating AFMs, where the spin current can only be carried by magnons. The extensive literature [19{24] shows that indeed an exponential decay of the signal is found with increasing thickness of the AFM. So qualitatively, in these systems the theoretical description seems to hold and is apt to describe the spin transport mechanism. As a next step, we probe here experimentally the spin current transport in conducting AFMs. In systems including the latter, the spin current can be transported by magnons as described above, but additionally also by charge- based spin currents. To check if charge-mediated transport of spin information occurs in addition to the magnonic spin currents described above, we performed temperature-dependent spin transmission experiments in a stack including the metallic AFM Ir 20Mn80(IrMn) using YIG/IrMn/Pt trilayers. In the experiment, spin currents are either triggered by the spin Seebeck eect [11, 14, 15] or via the spin Hall eect using spin Hall magetoresistance measurements [4]. As a rst dierence to insulating AFMs, one has to take into account the fact that in addition to Pt, which is widely usedSpin transport across antiferromagnets induced by the spin Seebeck eect 6 as a model material for ISHE based experiments, IrMn itself as well exhibits a spin Hall eect [35]. Therefore, in order to understand this more complex system, one needs to study not just the trilayer YIG/IrMn/Pt but also the individual combinations YIG/IrMn and YIG/Pt. Initially, single crystalline YIG is grown epitaxially on (111)-oriented Gd 3Ga5O12 (GGG) substrates by liquid-phase-epitaxy with a lm thickness of 5 µm. Onto GGG/YIG samples of size 2 mm6 mm0:5 mm, IrMn/Pt bilayers with varying IrMn thickness but constant Pt thickness (dIrMn = 0:8;1:3 nm,dPt= 5 nm) are deposited via magnetron sputtering. Furthermore, YIG/Pt( dPt= 5 nm) and YIG/IrMn ( dIrMn = 1:3 nm) reference samples are fabricated for comparison. The temperature-dependent SSE measurements are performed in a cryostat with a variable temperature insert (5 KT300 K), employing the conventional longitudinal conguration [11, 36]. By sandwiching the samples in between a top resistive heater and a bottom temperature sensor, an out-of-plane ( z direction) temperature gradient is generated, which induces the thermal spin current in the YIG layer. Base temperature and temperature gradient are determined via the resistance change of heater and sensor. An external magnetic eld His applied in-plane along the sample short edge ( ydirection), such that a detectable ISHE voltage drop in the long axis of the sample ( x direction) appears. The SSE voltage VSSEis extracted from the dierence between the ISHE voltages obtained for positive and negative magnetic eld divided by 2. To account for the dierent lm resistivities, the SSE currentISSE=VSSE=Ris considered in the following. The temperature-dependent SMR measurements are carried out in a superconducting vector cryostat that allows to align the magnetic eld in all directions. The SMR ratio is extracted from angular-dependent resistance measurements, in which the magnetic eld His rotated in the yz-plane and a sin2'yzresistance change [low (high) resistance for Hin-plane (out-of- plane)] is observed. To ensure that the magnetization follows the applied eld direction, the eld strength is xed to a value of 0H= 0:8 T, which is much larger than the coercivity of the YIG. In the following, we start by describing the experimentally determined spin signals as a function of temperature. Then, in a second step we discuss the results of the dierent measurements and the implications for the spin transport that we can deduce. First, we show in Fig. 4 the measured SSE current amplitude divided by the temperature dierence between sample top and bottom as a function of temperature for the stacks investigated. For enhanced readability, the data obtained for the samples with and without a Pt top layer are presented separately 024681012141618 Pt IrMn (0.8nm)/Pt IrMn (1.3nm)/Pt 50 100 150 200 250 300 T(K)0.20.30.40.50.6 IrMn (1.3nm)a b(nA K-1) ISSE/ΔT (nA K-1) ISSE/Δ TFigure 4. Detected spin Seebeck current as a function of temperature for (a) YIG/Pt or YIG/IrMn/Pt and (b) YIG/IrMn bi- and tri-layers. in Fig. 4a and Fig. 4b. The YIG/Pt only sample (red circles) exhibits a clear signal maximum near T= 90 K, whereas broad, at maxima are observed at dierent temperatures for the samples with the additional IrMn interlayer. For the samples with IrMn layers, the detected SSE signal amplitudes become signicantly suppressed at low temperatures below the maxima [Tcrit(dIrMn = 0:8 nm)150 K,Tcrit(dIrMn = 1:3 nm)200 K]. We nd at low temperatures, where the IrMn orders antiferromagnetically, that the insertion of IrMn generally yields a thickness- dependent signal reduction, which is in line with the theory described above. However, at higher temperatures ( T200 K), where the IrMn is likely in the paramagnetic phase, a larger Isse=
Tamplitude is observed for YIG/IrMn (0 :8 nm)/Pt as compared to the YIG/Pt sample. This behavior clearly goes beyond the theoretical description put forward above, since there only the AFM phase is considered. Possible origins of this behavior include an enhanced eective spin-mixing conductance of the YIG/IrMn interface as compared to the YIG/Pt interface [37, 38]. While of interest, this aspect is however not the focus of this work and further studies are necessary to understand this, which go beyond the scope of the current work. Finally, comparing the samples with and without Pt capping layers, we see that the temperature dependence of Issefor YIG/IrMn (1 :3 nm) in Fig. 4b exhibits, similar to YIG/Pt, a clear signal maximum nearT= 120 K, but with a signicantly reduced signal amplitude. Next, we compare the results of SSE measure-Spin transport across antiferromagnets induced by the spin Seebeck eect 7 IrMn (0.8nm)/Pt IrMn (1.3nm)/Pt T(K)50 100 150 200 250 300024681012 0.0000.0050.0100.0150.020 SMR (%)(nA K-1) ISSE/Δ T Figure 5. Comparison between temperature-dependent SSE (closed symbols) and SMR (open symbols) amplitudes for YIG/IrMn (0 :8 nm)/Pt (blue squares) and YIG/IrMn (1:3 nm)/Pt (green diamonds). ments with the results of the SMR measurements to un- derstand and dierentiate between interface and spin transport eects. The temperature-dependent SMR amplitudes obtained by the angular-dependent mea- surements are shown in Fig. 5 (open symbols), directly compared to the ISHE current amplitude (closed sym- bols). Apart from a small dierence in the amplitude ratio, both SMR and SSE feature similar temperature- dependent proles with an overlapping, strong signal suppression that sets in at low temperatures. In the following, we discuss the results above to understand the measured signals and the dierent contributions. To deduce information about the spin current transmission details across IrMn, we analyze and compare the dierent data sets obtained for the dierent sample stacks individually: Firstly, we discuss the temperature-dependent generation and detection of magnon spin currents. For that we consider the bilayers of YIG/Pt and YIG/IrMn, which do not involve spin current transmission across the full IrMn layer. In YIG/Pt, as shown in Fig. 4a, the detected spin Seebeck current exhibits a distinct amplitude maximum near T= 90 K, which was explained before as a consequence of an increasing magnon propagation length in YIG with decreasing temperature, counteracted by a reduced occupation of magnon states due to lower thermal energy [12]. However, rather than being a pure bulk eect of the FM, the position of the signal maximum also depends on the employed ISHE detection layer [12,39], implying a spectral-dependent transmission of magnons across the YIG/metal interface. YIG/IrMn (Fig. 4b) shows a qualitatively similar behavior as compared to YIG/Pt but with a shifted peak position near T= 120 K, which can be explained from the dierent magnon mode transmissions for YIG/Pt and YIG/IrMn as discussed for dierent detection layers in the literature [12,39]. Next, we discuss the spin current transport andto understand its properties, we compare the stacks YIG/IrMn/Pt and YIG/IrMn. The large dierence in the SSE signal amplitude for YIG/IrMn and YIG/IrMn/Pt can be easily understood considering material properties such as a smaller spin Hall angle (IrMn SH0:8Pt SH[35]), a shorter spin diusion length (IrMn sf = 0:7 nm vs.Pt sf= 2 nm [40, 41]) as well as a higher lm resistivity ( IrMn=Pt0:15 [35]) of IrMn as compared to Pt. We now look closely at the comparison between YIG/IrMn (1 :3 nm) (purple diamond, Fig. 4b) and YIG/IrMn (1 :3 nm)/Pt (green diamond, Fig. 4a). Given the much lower signal amplitude of YIG/IrMn as compared to YIG/IrMn/Pt and furthermore the much lower resistance of the Pt, it is clear that in the YIG/IrMn/Pt sample the signal contribution from the ISHE voltage generation in the IrMn is negligible. Thus, we can interpret the YIG/IrMn/Pt signal as the pure signal of the spin current transmitted from the YIG across the IrMn into the Pt, where due to the ISHE it is converted into the measured voltage. Comparing the temperature dependences, we nd in YIG/IrMn (1 :3 nm) a clear signal maximum nearT= 120 K, while in YIG/IrMn (1 :3 nm)/Pt at temperatures below 150 K the signal is strongly attenuated. To explain this key feature of the strong attenuation, we go through all the processes to identify the origin: (i) We have established from the YIG/Pt system measurements that the spin current generated in the YIG and the detection in the Pt are large below 150 K (Fig. 4a). (ii) From the YIG/IrMn system, we know that the spin transport across the YIG/IrMn interface below 150 K is large (Fig. 4b). Hence, what remains to explain the attenuation of the signal below 150 K in the YIG/IrMn/Pt system is the spin transport across the IrMn, which apparently is suppressed below 150 K. The transmission of the spin current can be of both electronic and magnonic nature, with the temperature dependence of ISSE=
Tin YIG/IrMn/Pt implying that the dominating contribution to the spin transport is strongly suppressed at low temperatures. Hence, we need to understand whether the magnonic or the electronic spin current dominates. From the fact that the signal in the YIG/IrMn system is still large below 150 K, we deduce that the charge- based spin currents in the IrMn, which are necessary for the ISHE so they can be converted into a charge current signal, are also still large at temperatures below 150 K. The observed strong attenuation of the measured signal in the YIG/IrMn/Pt system thus must stem from the magnonic spin current transport across the IrMn layer. Finally and importantly this is then also in line with the theory put forward above, where a short spin transport length is found for antiferromagnetically ordered systems.Spin transport across antiferromagnets induced by the spin Seebeck eect 8 0 50 100 150 200 250 300 T(K)0100200300400500600Hex(Oe) 01234567 ISSE Hex (nA K-1) ISSE/Δ T Figure 6. (a) Exchange-bias anisotropy eld detected in SiO2/IrMn (1:3 nm)/CoFe (2 nm) (blue circles) and (b) spin Seebeck current measured for YIG/IrMn (1 :3 nm)/Pt (green diamonds) as a function of temperature. To further reinforce this interpretation of a po- tential relation of our experimental ndings with the phase transition between the antiferromag- netic and the paramagnetic phase, we performed temperature-dependent magnetometry measurements on a SiO 2/IrMn (1:3 nm)/CoFe (2 nm) reference sam- ple. This reference sample is necessary to identify the transition temperature as the very large thickness of the used YIG lms does not allow one to observe exchange-bias in the YIG/IrMn/Pt samples used for the transport experiments. From the magnetometry data, the additional exchange anisotropy eld of the IrMn lm exerted on the CoFe layer is extracted as a function of temperature, see Fig. 6. The exchange-bias eld vanishes at the so-called blocking temperature TB80 K, which in thin lms usually is found to be smaller than TN eel[42]. While the absolute value needs to be taken with care, however, considering the compo- sitional dierences of the investigated samples, the N eel temperature of the YIG/IrMn (1 :3 nm)/Pt stack is ex- pected to be below 150 K. One observes that above TB, ISSEstarts to increase signicantly in the correspond- ing sample, which we identify as a further indication for a correlation between the signal suppression and the AFM phase transition of the IrMn lm. Above the N eel temperature, the magnonic spin current can be transported by short-range correlations [43], while belowTN eelthe AFM magnon gap (see. Eq. 8) in IrMn opens up and increases when further decreasing the temperature. According to the physical processes depicted in Fig. 3, this signies a transition from spin angular momentum transfer via precessing spin waves to evanescent waves at low temperatures, which can explain the strong suppression of ISSE=
Tdue to the strong decay of the evanescent waves. Therefore, from all the indications, we conclude that the spin current is at least partially transported by AFM magnonic spin currents in the IrMn layer. This conclusion is further corroborated by recent studies by Saglam et al. [44], who report on two transportregimes in Ni 80Fe20/FeMn/W systems with varying FeMn thickness. In the short-range regime (small thickness), spin propagation is dominated by electronic transport, whereas in the long-range regime (larger thickness) magnonic excitations yield the leading spin transport channel. Note that FeMn exhibits a larger spin-diusion length as IrMn [40]. Furthermore, in the experiment by Saglam et al. the spin current is emitted by the Ni 80Fe20FMR mode excited at f= 9 GHz, whereas in SSE experiments thermal magnons up to the THz regime are present. The correlation between the AFM order in IrMn and its spin current propagation properties becomes furthermore apparent when considering the trilayer samples with varying IrMn thickness. Whereas the thickness-dependent reduction of ISSE=
Tis to be un- derstood as a result of spin diusion (either electronic and magnonic), the thickness-dependent critical tem- perature for signal suppression is a direct indication of the paramagnetic-antiferromagnetic phase transition. In agreement with the ndings by Frangou et al. [28], who report an increasing TN eelwith increasing IrMn thickness, the signal suppression for thicker IrMn sets in at higher temperatures. Finally, the comparison of SSE and SMR amplitudes reveals very good agreement (Fig. 5), showing in particular coinciding low-temperature behavior, despite the conceptional dierences of the underlying eects. The SMR includes strong interface eects, considering that the pure spin current induced in a heavy metal due to the SHE interacts with the surface spins of an adjacent magnetic layer [4], which results in a spin-orientation-dependent lm resistance. The SSE, on the other hand, includes the conversion of bulk magnon spin currents into electronic spin currents and eventually charge currents by the ISHE. Taking into account the dierences of thickness, conductivity and spin Hall angle of Pt and IrMn, one can assume that in the SMR experiment the SHE spin current is mainly generated in the Pt layer. The observed angular dependence of the resistance change corresponds to a positive SMR that appears in systems in which the spin currents interact with the surface magnetization of FMs. For AFMs, on the other hand, the SMR follows the N eel order parameter and a negative SMR is observed [45{47]. Therefore, we conclude that for the SMR signal measured, the spin current that is generated in the Pt transmits across the IrMn and interacts with the YIG surface magnetization (absorption/re ection). Potential negative SMR contributions may appear at magnetic elds of sucient strength to align and rotate the N eel order parameter in IrMn, which is not the case here. Assuming the validity of the aforementioned magnonic spin transport mechanism in IrMn, theSpin transport across antiferromagnets induced by the spin Seebeck eect 9 coinciding temperature dependences of SSE and SMR amplitudes imply a strong coupling of the electronic spin current in Pt to the order parameter in IrMn at the IrMn/Pt interface and a dominating contribution of the spin transport across the IrMn layer for the temperature dependence. 5. Summary In conclusion, we have studied both theoretically and experimentally the propagation of pure spin currents in antiferromagnetic systems. While in insulating AFMs spin information transmission is exclusively provided by magnonic excitations, metallic AFMs as well can exhibit charge-mediated spin currents. AFM magnons exhibit a high-frequency gap. Despite the high velocity of antiferromagnetic magnons close to the frequency gap, the analytical model of magnonic transport shows that AFM magnons decay on much shorter distances, due to a shorter and frequency- independent lifetime. Using atomistic spin dynamics simulations, we demonstrate the propagation of spin waves from a FM to an AFM and show that short range evanescent modes are excited below the frequency gap, whereas normal modes with a longer propagation length are excited above the frequency gap. Beyond theoretical considerations, we furthermore investigate spin transmission across the metallic AFM IrMn by temperature-dependent SSE and SMR measurements in YIG/IrMn, YIG/Pt and YIG/IrMn/Pt heterostructures. From a systematic comparison of the obtained results, we conclude that the spin currents are at least partially mediated by AFM magnons. At low temperatures, where IrMn orders antiferromagnetically, the detected spin signals in YIG/IrMn/Pt transmitted across the IrMn become strongly suppressed, whereas in YIG/IrMn a notable signal induced by solely an electronic spin current is still detected. This is explained by the AFM magnon gap in IrMn to open up, such that the spin current is transported by evanescent waves that exhibit a strong decay over the lm thickness. Furthermore, the critical temperature, at which the suppression sets in, increases with increasing IrMn thickness as expected for a thickness-dependent phase transition temperature. Eventually, the coinciding temperature dependences observed for SSE and SMR suggest strong interaction of the electronic spin current in Pt towards the order parameter in the AFM IrMn. Acknowledgements The authors would like to thank the Deutsche Forschungsgemeinschaft (DFG) for nancial support (SPP 1538 Spin Caloric Transport", SFB767 inKonstanz and SFB TRR173 in Mainz), the Graduate School of Excellence Materials Science in Mainz (DFG/GSC 266), the EU projects (IFOX FP7-NMP3- LA-2012246102, INSPIN FP7-ICT-2013-X 612759), ERATO "Spin Quantum Rectication Project" (No. JPMJER1402) from JST, Japan, Grant-in-Aid for Scientic Research on Innovative Area "Nano Spin Conversion Science" (No. JP26103005) and Grant-in- Aid for young scientists (B) (No. JP17K14331) from JSPS KAKENHI, Japan. References [1] Ikeda S, Miura K, Yamamoto H, Mizunuma K, Gan H, Endo M, Kanai S, Hayakawa J, Matsukura F and Ohno H 2010 Nat. Mater. 9721{724 [2] Sinova J, Valenzuela S O, Wunderlich J, Back C H and Jungwirth T 2015 Rev. Mod. Phys. 871213{1260 [3] Kruglyak V, Demokritov S and Grundler D 2010 J.Phys. D:Appl. 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2018-03-09

For prospective spintronics devices based on the propagation of pure spin currents, antiferromagnets are an interesting class of materials that potentially entail a number of advantages as compared to ferromagnets. Here, we present a detailed theoretical study of magnonic spin current transport in ferromagnetic-antiferromagnetic multilayers by using atomistic spin dynamics simulations. The relevant length scales of magnonic spin transport in antiferromagnets are determined. We demonstrate the transfer of angular momentum from a ferromagnet into an antiferromagnet due to the excitation of only one magnon branch in the antiferromagnet. As an experimental system, we ascertain the transport across an antiferromagnet in YIG$|$Ir$_{20}$Mn$_{80}|$Pt heterostructures. We determine the spin transport signals for spin currents generated in the YIG by the spin Seebeck effect and compare to measurements of the spin Hall magnetoresistance in the heterostructure stack. By means of temperature-dependent and thickness-dependent measurements, we deduce conclusions on the spin transport mechanism across IrMn and furthermore correlate it to its paramagnetic-antiferromagnetic phase transition.

Spin transport across antiferromagnets induced by the spin Seebeck effect

1803.03416v1

Tailoring magnetic insulator proximity effects in graphene : First-principles calculations A. Hallal1;2;3, F. Ibrahim1;2;3, H. X. Yang1;2;3, S. Roche4;5and M. Chshiev1;2;3 1Univ. Grenoble Alpes, INAC-SPINTEC, F-38000 Grenoble, France 2CEA, INAC-SPINTEC, F-38000 Grenoble, France 3CNRS, SPINTEC, F-38000 Grenoble, France 4Catalan Institute of Nanoscience and Nanotechnology (ICN2), CSIC and The Barcelona Institute of Science and Technology, Campus UAB, Bellaterra, 08193 Barcelona, Spain 5ICREA-Institució Catalana de Recerca i Estudis Avançats, 08010 Barcelona, Spain Abstract. We report a systematic first-principles investigation of the influence of different magnetic insulators on the magnetic proximity effect induced in graphene. Four different magnetic insulators are considered: two ferromagnetic europium chalcogenides namely EuO and EuS and two ferrimagnetic insulators yttrium iron garnet (YIG) and cobalt ferrite (CFO). The obtained exchange-splitting varies from tens to hundreds of meV. We also find an electron doping induced by YIG and europium chalcogenides substrates, that shift the Fermi level up to 0.78 eV and 1.3 eV respectively, whereas hole doping up to 0.5 eV is generated by CFO. Furthermore, we study the variation of the extracted exchange and tight binding parameters as a function of the EuO and EuS thicknesses. We show that those parameters are robust to thickness variation such that a single monolayer of magnetic insulator can induce a large magnetic proximity effect on graphene. Those findings pave the way towards possible engineering of graphene spin-gating by proximity effect especially in view of recent experiments advancement. PACS numbers: 75.70.Ak, 73.22.Pr, 75.70.Cn, 75.70.Tj, 72.80.Vp, 85.75.-darXiv:1610.09554v1 [cond-mat.mtrl-sci] 29 Oct 2016Tailoring magnetic insulator proximity effects in graphene 2 1. Introduction Graphene spintronics is one of the most promising directions of innovation for two- dimensionalmaterials,openingnewprospectsforinformationtechnologies[1,2]. Besides its exceptional electrical, thermal, and mechanical properties [3, 4], two-dimensional graphene possesses a unique electronic band structure of massless Dirac fermions with a very long spin-diffusion lengths up to room temperature owing to its weak intrinsic spin- orbit coupling [5, 6, 7, 8, 9, 10, 11, 12, 13]. Accordingly graphene stands as a potential spin-channel material. However, a fundamental challenge lies in the development of external ways to control the propagation of spin currents at room temperature, in view of designing spin logics devices. Since carbon is non-magnetic, a significant effort is focused on injecting spins and inducing magnetism in graphene. Magnetism in graphene can be induced and controlled throughmaterialdesignordefectsandseveralmethodshavebeenproposedtomagnetize graphene [14, 15]. For instance, edge magnetism has been shown to develop in a few nanometers wide graphene nanoribbons for certain edge geometries [16, 17], or the hole structure of graphene nanomesh [18] was also theoretically proposed to offer robust and room temperature magnetic states able to affect spin transport [19, 20]. Much attention isalsopaidtotailorspin-polarizedcurrentsandmagnetoresistancesignalsbyintentional defects, ordepositingatoms[21,22,23,24,25,26,27]ormolecules[28,29,30]. Recently, the production of spin-polarized currents and magnetoresistance signals by growing grapheneonmagneticsubstrates, suchasYIG,hasraisedalotofinterest[31,32,33,34]. However, the conductivity mismatch is an important factor that influences the spin injection from magnetic metallic substrates into graphene restricting, thus, the design of novel types of spin switches. Therefore, the use of magnetic insulators (MIs) has attracted much interest as an alternative route to induce magnetism in graphene via the exchange-proximity interaction. Prior theoretical study of proximity effects of a ferromagnetic insulator (EuO) on graphene reported a large spin polarization of porbitals together with a large exchange-splitting band gap [35]. However, the drawback of using EuO is its low Currie Temperature (T C) and the predicted strong electron doping (about 1.4 eV). Thus, many theoretical works have been dedicated to investigate different MIs with higher TCand weaker doping which is crucial for practical spintronic devices. Additionally, theoreticalinvestigationsofgrapheneinproximityoftopological[36,37]andmultiferroic insulator [38] have found large exchange-splitting up to 300 meV. Recently, it has been proposed to insert 2D insulators (e.g. hBN) between graphene and the ferromagnetic material to induce exchange splitting [39]. In this case, the doping of graphene and exchange coupling strengh can be tuned by varying the number of hBN layers. On the other hand, recent experiments on YIG/Gr [31, 32, 33, 40] and EuS/Gr [41] demonstrated a large exchange-coupling between MI and graphene. Namely, a large magnetic exchange field up to 14 T is found in case of EuS on graphene with a potential of reaching hundreds of Tesla. However, EuS has even lower T Ccompared to that ofTailoring magnetic insulator proximity effects in graphene 3 Table 1. Computational and Structural details for the four investigated systems, effective Hubbard term, the bulk lattice parameter, the lattice mismatch between the MIs and graphene and the Curie temperature of each magnetic insulators. Structure Package Potential Ueff(eV) Latt. param.(Å) Mismatch(%) Tc(K) EuO SIESTA LDA+U Eu- f7.6 O-p3.4 5.18 0.8 77 EuS SIESTA LDA+U Eu- f6.3 and Eu- d4.4 5.92 -1.76 16.5 Y3Fe5O12SIESTA LDA+U Fe- d2.7 12.49 2.5 550 CoFe 2O4VASP GGA+U Fe- d3.61 Co-d3.61 8.46 -3.6 793 EuO. For YIG/Gr some experiments show a very large exchange-coupling of the order of tens of meV [31] while others reported smaller values of 0.2 T [32, 33] or 1 T [34]. Such discrepancy between the two reports might be due to different absorption strengths between graphene layer and YIG. In this Letter, using first-principles calculations we explore how the nature of the magnetic insulator affects the features of the magnetic proximity effect induced in graphene. Four cases of different MIs are studied: europium oxide (EuO), europium sulfite (EuS), cobalt ferrite CoFe 2O4(CFO) as well as yttrium iron garnet Y 3Fe5O12 (YIG). The exchange-proximity parameters are obtained from the electronic-band structure of graphene calculated in each case. We obtain electron doping for all cases except the CFO where the Dirac point lies above the Fermi level at 0.5 eV. The magnetic proximity effect results in a large exchange-splitting band gap of a few tens of meV. The presence of spin-dependent band gap around Dirac point is clear in all cases, except for cobalt ferrite where no bandgap is formed. In addition, we report systematic studies of electronic band structure of graphene as a function of EuO and EuS thickness where we show that the exchange-splitting gaps are robust to MI thickness variation. These findings pave the way towards possible engineering of graphene spin-gating by proximity effect especially in view of aforementioned recent experiments on EuS and YIG on top of graphene. 2. Methodolgy The Vienna ab initio simulation package (VASP) [42, 43, 44] is used for structure optimization, where the electron-core interactions are described by the projector augmented wave method for the potentials [45], and the exchange correlation energy is calculated within the generalized gradient approximation (GGA) of the Perdew-Burke- Ernzerhof form [46, 47]. The cutoff energies for the plane wave basis set used to expand the Kohn-Sham orbitals are 500 eV for all calculations. Structural relaxations and total energy calculations are performed ensuring that the Hellmann-Feynman forces acting on ions are less than 102for all studied structure. Except for YIG, due to its large supercell, relaxation is done using SIESTA code [48], where the exchange correlation energy is calculated within the local density approximation (LDA) [49, 50].Tailoring magnetic insulator proximity effects in graphene 4 Figure 1. (Coloronline)Sideviewandtopviewofthecalculatedcrystallinestructures for graphene on top of (a) EuO film (b) CoFe 2O4(C) EuS (d) Y 3Fe5O12. All the calculated structure are passivated with with hydrogen atoms. Since Eu is a heavy element with atomic number 63and its outer shell ( 4f76s2) contains 4felectrons, the GGA and LDA approaches fail to describe the strongly correlated localized 4felectrons and predicts a metallic ground state for the europium chalcogenides, whereas a clear band gap is observed in experiments. Similarly, GGA and LDA fail to describe the electronic interaction in Mott insulator such as iron oxides or cobalt oxides. Such a deficiency of these approaches is expected in correlated systems as transition metal oxides. Thereby, to account for the strong on-site Coulomb repulsion among the localized 3d(4f) electrons in YIG and CFO (EuO and EuS) we have introduced a Hubbarad-U parameter as described by the authors of Ref. [51, 52] for SIESTA and as described and implemented in the VASP code. The LDA+U and GGA+U represented by the Hubbard-like term Uand the exchange term J, which led to an improvement of the ground state properties such as the energy band gap and the spin magnetic moments in the MIs. The Ueff=UJvalue used for each system is summarized in Table 1, and in addition to Uefffor Eu-fin EuO, the LDA in EuS is also corrected by adding Ueffterm to the Eu- dorbitals following Ref. [53]. In all cases investigated, the density of states of bulk MIs are calculated and compared to those obtained using the VASP package and a good agreement is found between the two approaches using the same U parameters. The two investigated EuO and EuS compounds have a ferromagnetic ground state with a rocksalt structure with lattice parameters of 5.18 Å and 5.92 Å, respectively. Lattice structure and lattice mismatch between graphene and EuO are described in detail in Ref. [35]. It is found that a 3 3 unit cell of graphene can fit easily on a a 2 2 EuO (111) surface unit cell with a lattice about 7.33 Å and with a lattice mismatch of about 0.8%. For EuS, the bulk lattice parameter is quite larger than that of bulk EuO.Tailoring magnetic insulator proximity effects in graphene 5 Nevertheless, graphene can still fit on a EuSp 3p 3(111) substrate with a lattice mismatch in order of 1.8%. Due to this difference in the lattice parameter between EuO and EuS, a different graphene absorption on top of the surface occurs as seen in Figure 1 [(a) and (c)]. In both cases the supercell is composed of six bilayer of europium chalcogenides with graphene in top of Eu termination, which is the energetically most stable configuration. Next, we consider the lattice mismatch between graphene and YIG. Their lattice parameters are 2.46 Å and about 12.49 Å, respectively. as shown in Figure 1(d), the 11 unit cell of YIG (111) substrate with a lattice constant of about 17.66 Å can fit on the 77 graphene unit cell, with a lattice mismatch of about 2.5%. The resulting supercell is composed of six YIG trilayers and a graphene layer placed on top. For CFO the bulk lattice parameter is 8.46 Å and again along the (111) direction a 5 5 graphene unit cell can fit on 1 1 CFO(111) substrate with a lattice mismatch of about 3.6%. The supercell in this case is composed of six trilayer of CFO with graphene in top of Fe atoms (cf. Figure 1(b)). In all the cases, the bottom surface is passivated with hydrogen atoms in order to avoid the bottom surface effect on graphene and the vacuum region is chosen to be larger than 14 Å. The lattice structure of graphene/MIs are presented in figure Figure 1 with a vertical distance between Eu and C layers around 2.57 Å and 2.52 Å for EuO and EuS, respectively. For graphene/YIG and graphene/CFO, due to the large lattice mismatch, the graphene lattice is corrugated with corrugation height in order of 0.6 Å and 0.15 Å for YIG and CFO, respectively. The average vertical distance between Fe and C atoms is close to 2.7 Å for both YIG/graphene and CFO/graphene. This strong corrugation presents in graphene may affect its electronic band structure as shown previously for graphene on top of MgO substrate [54]. Finally, using the SIESTA package and the optimized structures of graphene on MIs shown in Figure 1, we calculate the electronic structure of the systems with LDA+U for the exchange correlation functional (c.f. Table 1). The self-consistent calculations are performed with an energy cutoff of 600 Ry and with a 4 41 K-point grid for EuO and EuS and 331 for YIG. A linear combination of numerical atomic orbitals with a double-polarized basis set is used for the small structures and and a single- for the larger ones. For graphene on CFO, the the electronic structure is calculated using GGA+U as implemented in VASP package with a 3 31 K-point grid. 3. Results and Discussion 3.1. Electronic structure of graphene in proximity of MIs Graphene honeycomb structure comprising two equivalent carbon sublattices AandBis responsible for the fact that charge carriers are described by massless Dirac excitations. Of particular importance for the physics of graphene are the two Dirac points KandK0 at the inequivalent corners of the graphene Brillouin zone (BZ). In the vicinity of these two points, the electronic structure of graphene is characterized by a linear dispersionTailoring magnetic insulator proximity effects in graphene 6 Figure 2. (Color online) Band structures of graphene on (a) EuO (b) CoFe 2O4(c) EuS (d) Y 3Fe5O12. Blue (Green) and red (black) represent spin up and spin down bands of graphene (magnetic insulator), respectively. EuO case is taken from Ref. [35] relation with a Dirac point separating the valence and conduction bands with a zero band gap as follows: H1(q) =vF
q (1) whereqisthemomentummeasuredrelativetotheDiracpointand vFrepresentsthe Fermivelocitywhichdoesnotdependontheenergyormomentum[3]. ThegaplessDirac cones atKandK0are protected by time-reversal and inversion symmetry. Since Dirac points are separated in the BZ, small perturbations cannot lift this valley degeneracy. Once graphene is in proximity of a substrate, AandBsublattices feel different chemical environment which leads to the inversion symmetry breaking between KandK0and giving rise to a band gap. This can be modeled by the following Hamiltonian describing the graphene’s linear dispersion relation in proximity of magnetic insulator: H2(q) =vF
q1s+1sz+
s 2z1s (2) whereandsare the Pauli matrices that act on sublattice and spin, respectively. The second term represents the exchange coupling induced by the magnetic moment of magnetic atoms, with being the strength of exchange spin-splitting of the hole or electron. The last term results from the fact that graphene sublattices AandBare now feeling different potential which might result in a spin-dependent band gap opening atTailoring magnetic insulator proximity effects in graphene 7 Table 2. Extracted energy gaps and exchange parameters of graphene/MIs structures at Dirac point compared with parameters for graphene in proximity of EuO heterostructure reported in Ref. [55]. E Gis the band-gap of the Dirac cone.
"and
#are the spin-up and spin-down gap, respectively. The spin-splitting of the electron and hole bands at the Dirac cone are eandh, respectively. E Dis the Dirac cone doping with respect to Fermi level. Structure E G(meV)
"(meV)
#(meV)e(meV)h(meV) E D(eV) EuO/Gr/EuO(1BL) aligned[55] 127 309 344 182 217 -2.8 EuO/Gr/EuO(1BL) misaligned[55] -38 137 182 211 220 -2.8 GR/EuO(6BL) 50 134 98 84 48 -1.37 GR/EuS(6BL) 160 192 160 23 -10 -1.3 Gr/Y 3Fe5O12 1 116 52 -52 -115 -0.78 Gr/CoFe 2O4 -37 12 8 -45 -49 +0.49 the Dirac point and
sis the spin-dependent staggered sublattice potential. A Rashba spin orbit coupling term can also be added to the Hamiltonian and can be represented by 2(^s)at the left side of Equation (2). Let us now discuss the electronic band structures of graphene in proximity to MIs as shown in Figure 2. For graphene on top of europium chalcogenides a 33unit cell is used and due to zone folding of grapheneŠs BZ, both KandK0valleys get mapped to the point [35]. Therefore for EuO and EuS, the Dirac cone of graphene becomes located at the point instead of Kone’s. The linear dispersion of the graphene band structure is modified with a band gap opening at the Dirac point. More interestingly, thisdegeneracyliftingattheDiracpointisspindependentasdemonstratedforEuO[35]. The spin-dependent band gaps found in the EuO/graphene are about 134 and 98 meV for spin up and spin down states, respectively (see Figure 2(a)). Here, however, we fit the band structure parameters according to Hamiltonian given by Equation (2) to which the exchange splitting gaps of 84 and 48 meV are added for electrons and holes, respectively. ReplacingEuObyEuSincreasesdrasticallythebandgapopeningasshown inFigure2(c). Thespin-dependentbandgapsinthiscaseareabout192(resp. 160meV) for spin up (resp. spin down) states. However, the spin splitting is strongly reduced to 23 (resp. -10 meV) for electrons (resp. holes). This difference between EuO and EuS results from the fact that all 3 Eu atoms in EuS case are sitting in a hollow site of graphene hexagon while for EuO, the atoms belong to the bridge site and to the hollow site as shown in Figure 1(a) and (c). Recently, Su et al [55] reported that while Eu atom sitting at the hollow site of graphene hexagon is described by an inter-valley scattering term in the induced proximity Hamiltonian, Eu atoms at the bridge site reduces the graphene lattice symmetry and can be represented by a valley pseudospin Zeeman term inx-direction in sublattice space that shifts slightly the Dirac cones from the point. Let us now discuss the proximity effects induced by yttrium garnet (YIG) and cobalt ferrite (CFO) oxides. In Figure 2(d) we present the electronic bands of the YIG/Graphene structure where we see that the proximity of YIG induces a band gapTailoring magnetic insulator proximity effects in graphene 8 opening around the Dirac point. Furthermore, due to the interaction between graphene and the magnetic substrate, the spin-degeneracy around Dirac point is lifted. The spin- dependent band gaps found in the YIG/graphene are 116 and 52 meV for spin up and spin down states, respectively. The spin splittings estimated from the band structure are found to be about -52 and -115 meV for electrons and holes, respectively. Due to its strong interaction with the magnetic insulator, graphene becomes heavily doped and the Dirac Cone is shifted below the Fermi level as seen in Figure1 (b). Interestingly, the band structure presented in Figure 1(b) shows that graphene on top of YIG has a half metallic behavior. The spin-up Dirac cone lies in the middle of the spin-down gap and vice versa. For the CFO/graphene case the induced band gap opening around the Dirac point is absent (see Figure 2(b)). This is due to the quite large interlayer distance between the ferrimagnetic insulator and graphene and due to the physisorption interaction which does not perturb the inversion symmetry of the two Dirac points. Nevertheless, due to the interaction between graphene and the magnetic substrate, the spin-degeneracy around Dirac point is lifted and spin-dependent band gaps are still induced in this case and found about 12 and 8 meV for spin up and spin down states, respectively. The strength of the exchange-splitting estimated from the band structure is -45 and -49 meV for electrons and holes, respectively. Due to the weak interaction with the magnetic insulator graphene becomes slightly doped and the Dirac Cone is shifted above the Fermi level as seen in Figure 2(b). The extracted energy band gaps and exchange-splitting values at Dirac point induced in graphene by the proximity of magnetic insulators are summarized in Table 2 with EG,
"and
#representing the energy band gap and the spin-dependent gaps for spin-up and spin-down, respectively. The spin splitting of the electron and hole bands are denoted as eandh. Finally, E Dindicates how large is the Dirac point doping with respect to Fermi energy. In Table 2 the positive value of E Gindicates a band gap between conduction and valence band, whereas negative value indicates a spin resolved band overlap as seen in CFO case shown in Figure 2(b). Spin-splittings are defined by spindependentenergydifferencesatDiracpointwithnegativevalueindicatingthatspin- up bands are lower in energy than that of spin-down bands. The extracted values are compared with that aligned and misaligned EuO heterostructure with graphene between two EuO layers reported in Ref. [55]. As illustrated in Table 2, doping graphene with EuO will push further the Dirac point below the Fermi level and makes impossible to harvest the graphene linear dispersion in practical electronic devices. To overcome the problem of strong doping one can deposit on the top side of the structure a material which can hole dope graphene. For instance, we propose that CFO deposited on the top side of europium chalcogenides/graphene or even YIG/graphene will bring Dirac cone closer to Fermi level and the exchange-splitting parameter induced by proximity effect, in such a heterostructure, is expected to double to be in the range of hundreds of meV.Moreover, thistypeofasymmetricheterostructurewillbreakthein-planeinversion symmetry of the graphene layer and might give rise to topological properties such as quantum anomalous Hall effect [55].Tailoring magnetic insulator proximity effects in graphene 9 Figure 3. Thickness variation of the exchange-coupling parameters presented in table 2 for the graphene in proximity of chalcogenides EuO and EuS. 3.2. Thickness variation effect on the graphene exchange parameter in proximity of europium chalcogenides Finally, let us check the robustness of aforementioned results by exploring the variation of the energy band gaps and proximity exchange-splitting in graphene at Dirac point as a function of MIs thickness. As seen in Figure 3, all the plotted values tends to saturate above a thickness of 3 bilayers indicating that already 3 or 4 bilayers of MIs are sufficient to mimic the bulk effect. The results also indicate that already MIs as thin as 1 bilayer of europium chalcogenides can induce large proximity effect in graphene. For instance, the spin-splitting of the electron and hole bands at the Dirac cone in the case of one bilayer of EuS (EuO) are found about 120 and 80 meV (55 and 5 meV), respectively. As EuS (EuO) thickness increases, both spin-splitting values decreases (increases) to reach the bulk value shown in Table 2. As for spin-dependent band gaps
"and
#, both decreases as a function of MI thickness with variation of spin-down and spin-up band gaps being less dramatic in the case of EuS compared to that for EuO. Since the induced magnetism in graphene due to proximity of europium chalcogenides arises from graphene hybridization with polarized Eu- 4fstate right below the Fermi level [35], the observed variation at low thicknesses is related to the variation of the energy level of these Eu- 4fstates. 4. Conclusion In summary, using first-principles calculations we investigated proximity effects induced in graphene by magnetic insulators. Four different MIs have been considered: two ferromagnetic europium chalcogenides and two ferrimagnetic insulators yttrium iron garnet and cobalt ferrite. In all cases, we find that the exchange-splitting varies inTailoring magnetic insulator proximity effects in graphene 10 the range of tens to hundreds meV. While Dirac cone is negatively doped for europium chalcogenides and YIG, it is found to be positively doped for CFO substrate. In order to bring the Dirac cone closer to the charge neutrality point, we propose to deposit on the topsideofthenegativelydopedstructureamaterialwhichcanpositivelydopegraphene, such as CFO. In such a heterostructure the exchange-coupling parameter induced by proximity effect is expected to be doubled. Moreover, we explored the variation of the extracted magnetic exchange parameters as a function of europium chalcogenides thicknesses. This analysis show that the extracted parameters are robust to thickness variationandonemonolayerofmagneticinsulatorcaninducealargemagneticproximity effectongraphene. Thesefindingspavethewaytowardspossibleengineeringofgraphene spin-gating by proximity effect especially in view of recent experiments advancement. Acknowledgments This project has received funding from the European Union’s Horizon 2020 research and innovation programme under grant agreement No. 696656 (Graphene Flagship). S. R. acknowledges Funding from the Spanish Ministry of Economy and Competitiveness and the European Regional Development Fund (Project No. FIS2015-67767-P (MINECO/FEDER)),theSecretariadeUniversidadeseInvestigacióndelDepartamento de Economía y Conocimiento de la Generalidad de Cataluña, and the Severo Ochoa Program (MINECO, Grant No. SEV-2013-0295). References [1] S.Roche, J.Åkerman, B.Beschoten, J.-C.Charlier, M.Chshiev, S.P.Dash, B.Dlubak, J.Fabian, A. Fert, M. Guimarães, F. Guinea, I. Grigorieva, C. Schönenberger, P. Seneor, C. Stampfer, S. O. Valenzuela, X. Waintal and B. van Wees, 2D Materials 2, 030202 (2015) [2] W. Han, R. K. Kawakami, M. Gmitra and J. Fabian, Nature Nanotechnology 9, 794 (2014). [3] A. H. Castro Neto, F. Guinea, N. M. R. Peres, K. S. Novoselov, and A. K. Geim, Rev. Mod. 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2016-10-29

We report a systematic first-principles investigation of the influence of different magnetic insulators on the magnetic proximity effect induced in graphene. Four different magnetic insulators are considered: two ferromagnetic europium chalcogenides namely EuO and EuS and two ferrimagnetic insulators yttrium iron garnet (YIG) and cobalt ferrite (CFO). The obtained exchange-splitting varies from tens to hundreds of meV. We also find an electron doping induced by YIG and europium chalcogenides substrates, that shift the Fermi level up to 0.78 eV and 1.3 eV respectively, whereas hole doping up to 0.5 eV is generated by CFO. Furthermore, we study the variation of the extracted exchange and tight binding parameters as a function of the EuO and EuS thicknesses. We show that those parameters are robust to thickness variation such that a single monolayer of magnetic insulator can induce a large magnetic proximity effect on graphene. Those findings pave the way towards possible engineering of graphene spin-gating by proximity effect especially in view of recent experiments advancement.

Tailoring magnetic insulator proximity effects in graphene: First-principles calculations

1610.09554v1

Eect of magnon decays on parametrically pumped magnons Viktor Hahnand Peter Kopietz Institut f ur Theoretische Physik, Universit at Frankfurt, Max-von-Laue Strasse 1, 60438 Frankfurt, Germany (Dated: February 19, 2021) We investigate the in uence of magnon decays on the non-equilibrium dynamics of parametrically excited magnons in the magnetic insulator yttrium-iron garnet (YIG). Our investigations are moti- vated by a recent experiment by Noack et al. [Phys. Status Solidi B 256, 1900121 (2019)] where an enhancement of the spin pumping eect in YIG was observed near the magnetic eld strength where magnon decays via con uence of magnons becomes kinematically possible. To explain the experimental ndings, we have derived and solved kinetic equations for the non-equilibrium magnon distribution. The eect of magnon decays is taken into account microscopically via collision in- tegrals derived from interaction vertices involving three powers of magnon operators. Our results agree quantitatively with the experimental data. I. INTRODUCTION In a recent experiment1the parametric excitation of magnons in the magnetic insulator yttrium-iron garnet (YIG) was investigated by coupling an oscillating mi- crowave eld into the system and measuring the magnon density via the inverse spin-Hall eect.2This eect, which converts a spin current into an electric eld per- pendicular to the directions of the spin current and the spin polarization, is caused by the relativistic spin-orbit interactions that are also responsible for the direct spin- Hall eect.3In solids this eect is enhanced due to the strong potential of atomic nuclei.4In the experiment1 a thin YIG lm was exposed to an oscillating magnetic eldH(t) =H0ez+H1cos(!0t)ez, where the static part H0ezforces the macroscopic magnetization to be aligned along thez-axisez, while the oscillating part with am- plitudeH1H0drives the magnons in the sample out of equilibrium. Noack et al.1observed that the spin- pumping eect was enhanced for certain values of the static eld H0, and that the magnon density in the sta- tionary non-equilibrium state displayed peaks or dips for those values of H0where magnon decays due to the con- uence of two parametrically excited magnons with iden- tical energy and momentum becomes kinematically pos- sible. Recall that magnon decays due to the con uence and the reverse splitting process conserve the total en- ergy and momentum of the magnons involved in these scattering processes.5,6 In this work we provide a quantitative microscopic ex- planation for the experimental observations of Ref. [1]. It turns out that therefore a proper understanding of magnon damping under non-equilibrium conditions in YIG is crucial. We therefore construct a kinetic theory of pumped magnon gases including microscopi- cally derived collision integrals describing the relevant dissipative eects. While theoretical investigations of pumped magnon gases in magnetic insulators have a long history5{24in all works published so far the eect of collisions on the non-equilibrium magnon dynamics was considered only phenomenologically by introducing (by hand) a relaxation rate into the kinetic equations for themagnon distribution functions. Although the relevant microscopic collision integrals have been derived within the Born approximation in Ref. [11], to our knowledge a microscopic treatment of the eect of magnon colli- sions on the non-equilibrium dynamics of magnons is still missing in the literature. An alternative method to in- vestigate the dynamics of pumped magnons in YIG is based on the numerical solution of the stochastic non- Markovian Landau-Lifshitz-Gilbert equation with a mi- croscopically derived noise and dissipation kernel.25The approach based on kinetic equations adopted here has the advantage that it allows us to identify the experi- mentally relevant con uent scattering processes directly in the collision integral. Still, the resulting non-linear integro-dierential equations are very complicated and can only be solved numerically. Moreover, the deriva- tion of the collision integrals starting from an eective spin Hamiltonian for YIG is a demanding technical prob- lem because the distribution function of the magnon gas in YIG with external pumping has an o-diagonal com- ponent so that we have to deal with various types of anomalous cubic interaction vertices. While in principle the collision integrals can be derived diagrammatically using the Keldysh formalism,26to keep track of all terms contributing to the collision integrals we have found it more convenient to use an unconventional method de- veloped in Ref. [27] based on a systematic expansion of the collision integrals in terms of connected equal-time correlation functions. The rest of this article is organized as follows. In Sec. II we introduce the eective Hamiltonian describ- ing pumped magnons in YIG which is the starting point for our investigations. In Sec. III we derive collision- less kinetic equations for the magnon distribution func- tions in YIG. We also discuss the usual phenomenological strategy of introducing dissipative eects into the colli- sionless kinetic equations, derive the resulting stationary non-equilibrium distributions for YIG, and show that the experimental results of Noack et al.1cannot be explained within this approximation. In Sec. IV we derive the col- lision integrals containing the cubic vertices using an ex- pansion in powers of connected equal-time correlations.27arXiv:2012.07870v2 [cond-mat.stat-mech] 18 Feb 20212 Our numerical results for the stationary non-equilibrium solution including the eects of the cubic vertices are pre- sented in Sec. V. Finally, in Sec. VI we summarize our results and present our conclusions. To make this work self-contained we have added three appendices with tech- nical details. In Appendix A we outline the derivation of the Hamiltonian of pumped magnons in YIG follow- ing mainly Refs. [28,29]. In Appendix B we review the method of deriving kinetic equations via an expansion in terms of connected equal-time correlations developed by Fricke,27and in Appendix C we give the explicit expres- sions for the relevant collision integrals for YIG obtained with this method. II. HAMILTONIAN FOR PUMPED MAGNONS IN YIG In the experimental setup of Ref. [1] a thin stripe of YIG is exposed to an oscillating microwave eld in the parallel pumping geometry where the oscillating compo- nent of the magnetic eld is parallel to its static com- ponent. At the energy scales of interest the magnon dy- namics can be described by the following time-dependent eective Hamiltonian,5,6,19,20,28{31 H(t) =1 2X ijX h Jij+D iji S iS j [h0+h1cos (!0t)]X iSz i; (2.1) where the indices i;jlabel theNsites of a cubic lat- tice and;denote the three spin components x;y;z of the spin operators S i. The nearest neighbor exchange couplings connecting lattice sites riandrjare denoted byJij, whileD ijdenotes the matrix elements of the dipolar tensor dened in Eq. (A1) of Appendix A. The last term in Eq. (2.1) represents the coupling of the spins to a static magnetic eld H0and a time-dependent mi- crowave magnetic eld H1oscillating with frequency !0, whereh0=H0andh1=H1are the corresponding Zeeman energies. The geometry of the system and our choice of the coordinate system is shown in Fig. 1. The Hamiltonian (2.1) can be bosonized using the Holstein- Primako transformation32as described in Appendix A. We expand the resulting bosonized Hamiltonian in pow- ers of the inverse spin quantum number 1 =S, H(t) =H0(t) +H2(t) +H3+H4+O(S1=2);(2.2) whereHncontainsnpowers of the boson operators. Ex- plicit expressions for the terms in the expansion (2.2) are given in Refs. [20,24,28,29] and are reproduced in Ap- pendix A. It is convenient to use a canonical (Bogoliubov) transformation to diagonalize the time-independent part ofH2(t), which then assumes the form given in Eq. (A10). For our purpose it is sucient to further simplify H2(t) by dropping all non-resonant terms which are explicitly wx yzd = aNH0k θkFIG. 1: Sketch of a long YIG stripe oriented along the z- axis with width winy-direction and thickness d=aNinx- direction. Here ais the lattice spacing and Nis the number of lattice sites in x-direction. In this work we consider wavevec- torskin they-z-plane with kbeing the angle between kand the static magnetic eld magnetic eld H0=H0ez. time-dependent in the rotating reference frame dened by the canonical transformation (2.9) below.20,24,29In this approximation H2(t) =X k kay kak+1 2Vkei!0tay kay k +1 2V kei!0takak ;(2.3) whereakanday kannihilate and create magnons with momentum kand energy k. For small kthe magnon energy can be approximated by28,33,34 k=q h0+k2+ (1fk)
sin2k [h0+k2+fk
]; (2.4) while the pumping energy Vkcan be written as Vk=h1
4k fk+ (1fk) sin2k : (2.5) Here,is the exchange stiness of long-wavelength magnons,28the dipolar energy scale
=42S a3(2.6) is determined by the eective magnetic moment and the eective spin S[see Eq. (A18)], and the form factor fkfor a thin stripe of YIG shown in Fig. 1 is given by28,33 fk=1ejkjd jkjd; (2.7) wheredis the thickness of the YIG stripe. We parametrize the in-plane wavevector as k=kyey+kzez=jkj(sinkey+ coskez);(2.8)3 wherekis the angle between the wavevector kand the static magnetic eld H0ezas shown in Fig. 1. The explicit time-dependence of the quadratic part of the Hamiltonian (2.3) can be removed via a canonical transformation to the rotating reference frame, ~ak= ei!0 2tak;~ay k= ei!0 2tay k: (2.9) The quadratic part of the Hamiltonian then be- comes20,24,29 ~H2=X k Ek~ay k~ak+Vk 2~ay k~ay k+V k 2~ak~ak ;(2.10)where Ek=k!0=2 (2.11) is the shifted magnon energy in the rotating reference frame. It turns out that in this frame the cubic and the quartic parts of the magnon Hamiltonian acquire an explicit time-dependence. Explicitly, after Bogoliubov transformation and transformation the cubic and quar- tic part of the magnon Hamiltonian are in the rotating reference frame of the form ~H3(t) =1p NX k1;k2;k3k1+k2+k3;01 2aaa 1;2;3ei!0t=2~ay 1~a2~a3+1 2aaa 1;2;3ei!0t=2~ay 1~ay 2~a3 +1 3!aaa 1;2;3e3i!0t=2~a1~a2~a3+1 3!aaa 1;2;3e3i!0t=2~ay 1~ay 2~ay 3 ; (2.12) ~H4(t) =1 NX k1;k2;k3;k4k1+k2+k3+k4;0" 1 (2!)2aaaa 1;2;3;4~ay 1~ay 2~a3~a4+1 3!ei!0taaaa 1;2;3;4~ay 1~a2~a3~a4 +1 3!ei!0taaaa 1;2;3;4~ay 1~ay 2~ay 3~a4+1 4!e2i!0taaaa 1;2;3;4~a1~a2~a3~a4+1 4!e2i!0taaaa 1;2;3;4~ay 1~ay 2~ay 3~ay 4 ;(2.13) where we have introduced the short notation ki!ifor the momentum labels. In Eqs. (A14) and (A15) of Ap- pendix A we explicitly give the rather cumbersome ex- pressions for the vertices appearing in Eqs. (2.12) and (2.13). At the rst sight it seems that within the rotating- wave approximation we should drop all oscillating terms in Eqs. (2.12) and (2.13). However, as will be shown in Sec. IV, the collision integrals originating from the cu- bic part ~H3(t) of the Hamiltonian contain products of two cubic vertices, so that some of the time-dependent factors in Eq. (2.12) cancel in the collision integrals and at this point we do not neglect the oscillating terms in Eq. (2.12). We conclude this section with a cautionary remark about the validity of the spin Hamiltonian (2.1) which describes only the lowest (acoustic) branch of the magnon spectrum. Since YIG is a ferrimagnetic insulator with a rather large number of spins per unit cell, the magnon spectrum has also several high-energy (optical) branches5 which are not taken into account via the spin Hamilto- nian (2.1). It turns out, however, that in thermal equi- librium at room temperature these optical magnons have a much lower occupancy than the low-energy magnons, so that at the energy scales probed in the experiment1 we can safely neglect the optical magnons. In principle we cannot exclude the possibility that non-equilibrium scattering processes lead to a signicant population of the optical magnons. In fact, a recent calculation of the inverse spin-Hall voltage and the spin Seebeck ef-fect in YIG by Barker and Bauer35suggests that optical magnons can signicantly contribute to spin transport. On the other hand, in Ref. [35] is is also shown that the inclusion of the optical magnons does not qualitatively change the predicted inverse spin-Hall voltage. Since in the present work we do not attempt to calculate the ab- solute size of the inverse spin-Hall voltage but consider only the magnon density (which is expected to be pro- portional to the inverse spin-Hall voltage), for our pur- pose it is sucient to work with the eective low-energy spin Hamiltonian (2.1). The high-energy magnon bands can at least partially be taken into account by consid- ering the parameters in Eq. (2.1) as eective quantities which include renormalization eects due to the optical magnon bands. This argument is further strengthened by the fact that the Hamiltonian (2.1) correctly describes the dynamics of non-equilibrium magnon condensation in YIG.25 III. COLLISIONLESS KINETIC EQUATIONS AND S-THEORY WITH PHENOMENOLOGICAL DAMPING Before deriving in Sec. IV kinetic equations for the distribution functions of magnons in YIG including the relevant collision integrals, it is instructive to consider rst the collisionless limit. As recently pointed out in Ref. [24], for a complete description of the non-4 equilibrium time-evolution of the magnon distribution in YIG, we should take into account that in the presence of a time-dependent microwave eld the magnon annihilation operators can have a nite expectation value exhibiting a non-trivial dynamics. In the rotating reference frame we dene ~ k(t) =h~ak(t)i=ei!0t=2hak(t)i=ei!0t=2 k(t);(3.1) where the time-evolution is in the Heisenberg picture and h:::idenotes to the non-equilibrium statistical average. In addition, we should consider the time-evolution of the connected diagonal- and o-diagonal distribution func- tions, nc k(t) =hay k(t)ak(t)i=h~ay k(t)~ak(t)i;(3.2) ~pc k(t) =h~ak(t)~ak(t)i= ei!0tpk(t); (3.3) whereak(t) =ak(t)hak(t)i=ak(t) k(t). Note that the phase factors ei!0t=2generated by the trans- formation to the rotating reference frame cancel in the diagonal distribution function nc k(t). A. Collisionless kinetic equations The equations of motion for the distribution functions can be derived from the Heisenberg equations of motion for the operators in the rotating reference frame, i@t~ak= ~ak;~H(t) ; (3.4a) i@t~ay k= ~ay k;~H(t) : (3.4b) To begin with, let us approximate the magnon Hamil- tonian by its quadratic part ~H2neglecting all magnon- magnon interactions. In this approximation,24 @tnc k+i Vk(~pc k)V k~pc k = 0; (3.5a) @t~pc k+ 2iEk~pc k+iVk[2nc k+ 1] = 0; (3.5b) @t~ k+iEk~ k+iVk~  k= 0: (3.5c) Unfortunately, these equations do not provide a satisfac- tory description of the experimental results of Ref.[1]. In particular, in the strong pumping regime jVkj>jEkj these equations predict an exponential growth of the magnon distributions,9,10,20whereas experimentally one observes a saturation for suciently long times. To de- scribe this saturation we have to take magnon-magnon interactions into account. This can be done by employing a time-dependent self-consistent Hartree-Fock approxi- mation, which in this context is called S-theory.6{8,17,19 The kinetic equations (3.5) are then replaced by non- linear integro-dierential equations, which in the rotating reference frame take again the form24 @tnc k+ih ~Vk(~pc k)~V k~pc ki = 0; (3.6a) @t~pc k+ 2i~Ek~pc k+i~Vk[2nc k+ 1] = 0; (3.6b) @t~ k+i~Ek~ k+i~Vk~  k= 0; (3.6c)where the renormalized magnon energy ~Ekand the renor- malized pumping energy ~Vkdepend on the distribution functions as follows, ~Ek=Ek+1 NX qTk;q nc q+~ q2 ; (3.7a) ~Vk=Vk+1 2NX qSk;q ~pc q+~ q~ q :(3.7b) HereTk;qandSk;qare dened via the following ma- trix elements of magnon-magnon interaction vertices in Eq. (2.13), Tk;q= aaaa k;q;q;k; (3.8a) Sk;q= aaaa k;k;q;q: (3.8b) Note that in Eq. (3.7) we have dropped oscillating terms arising from the vertices of ~H4(t) in Eq. (2.13) involving time-dependent factors of ei!0tande2i!0t, which is consistent within the rotating-wave approximation. B. Stationary non-equilibrium distribution with phenomenological damping In the experiment by Noack et al.1the magnetic-eld dependence of the magnon distribution in a stationary non-equilibrium state of a YIG sample subject to an os- cillating microwave eld is measured. Let us now try to explain this experiment using a simple modication of the collisionless kinetic equations (3.6) where we in- troduce (by hand) a phenomenological damping rate k. Note that without such a damping rate the solutions of the collisionless kinetic equations never reach a stationary non-equilibrium state.24In the rotating reference frame the equations of motion for the magnon operators includ- ing the phenomenological damping kare @t~ak(t) = (iEk k) ~akiVk~ay k;(3.9a) @t~ay k(t) = (iEk k) ~ay k+iV k~ak: (3.9b) In Refs. [7,8] it was argued that the damping selects the pair of magnon modes with momentum kthat is char- acterized by the smallest damping to be the only signif- icantly occupied modes, so that the dynamics of these modes is eectively decoupled from the other modes. Moreover, it is argued that, if initially other magnon modes are signicantly occupied as well, after suciently long times only this single pair of magnon modes will survive. This argument justies the approximation of re- placing the integrals dening the renormalized energies in Eq. (3.7) by a single term where the loop momentum qis equal the external momentum k, ~EkEk+1 NTk;k nc k+~ k2 ; (3.10a) ~VkVk+1 2NSk;k ~pc k+~ k~ k :(3.10b)5 800 900 1000 1100 120000.020.040.060.080.1 H0 / Oens / N7dB 8dB 10dB 12dB FIG. 2: Dependence of the magnon density ns=N =P kns k=Non the external magnetic eld strength H0in the stationary non-equilibrium state within S-theory given by Eq. (3.11a) for dierent pumping strengths. The maxi- mum ofVkwas chosen to be larger than the relaxation rate k= 2:19103GHz. To describe the experiment of Noack et al1we have performed our calculations for a thin YIG lm with thickness d= 22:8m (corresponding to N= 18422) subject to a microwave eld with frequency !0= 13:857 GHz. Neglecting the expectation values of the magnon opera- tors, Zakharov et al.7,8nd that the stationary solution of the collisionless kinetic equations (3.6) with additional damping is given by ns k=Np V2 k 2 kEk Tk;k+1 2Sk;k; (3.11a) ~ps k=ns k; (3.11b) provided the pumping is strong enough to compensate the losses due to damping, jVkj>j kj: (3.12) We shall refer to Eq. (3.11) as the stationary solution within S-theory. Taking explicitly the expectation values of the magnon operators in Eq. (3.10) into account yields the same result,24,36 nc k+~ k2 =ns k; (3.13a) ~pc k+~ 2 k=ns k: (3.13b) In Fig. 2 we plot the stationary magnon density ns=P kns kwithin S-theory obtained from Eq. (3.11) as a function of the external magnetic eld assuming a con- stant phenomenological relaxation rate k= 2:08 103GHz. For comparision, we reproduce in Fig. 3 the experimental results for the inverse spin-Hall eect volt- age from Fig. 4 a) of Ref. [1], which is expected to be pro- portional to the density of pumped magnons. Obviously, 800 900 1000 1100 1200010203040 H0 / Oe VISH / µ V7dB 8dB 10dB 12dBFIG. 3: Experimental results for the inverse spin-Hall voltage VISHreproduced from Fig. 4 a) of Ref. [1]. in a certain range of magnetic elds the experimental data exhibit characteristic features which are missed by S-theory, which explains only the average linear growth of the observed magnon density with increasing magnetic eld. An obvious reason for the failure of S-theory is that the phenomenological damping introduced by hand nei- ther takes into account the kinematic constraints nor the microscopic magnon dynamics responsible for the dissi- pative eects which are essential for the emergence of a stationary non-equilibrium state in the pumped magnon gas. For a satisfactory explanation of the experimental data1reproduced in the lower part of Fig. 3 we should therefore use kinetic equations with microscopically de- rived collision integrals describing the relevant scattering processes. The collisionless kinetic equations (3.5) are then replaced by @tnc k(t) +ih ~Vk(t) (~pc k(t))~V k(t)~pc k(t)i =In k(t); (3.14a) @t~pc k(t) + 2i~Ek(t)~pc k(t) +i~Vk(t) [2nc k(t) + 1] =Ip k(t); (3.14b) @t~ k(t) +i~Ek(t)~ k(t) +i~Vk(t)~  k(t) =I k(t); (3.14c) where all interactions beyond S-theory are taken into ac- count via three types of collision integrals In k(t),Ip k(t), andI k(t). These collision integrals should be derived from the Hamiltonian (2.2), including the cubic part ~H3(t) which determines the damping to leading order in the small parameter 1 =S. In spite of many decades of theoretical research on pumped magnon gases,5{24a com- plete derivation of the relevant collision integrals In k(t), Ip k(t), andI k(t) and the subsequent numerical solution of the resulting kinetic equations cannot be found in the literature. In the rest of this work we will solve this6 technically very complicated problem using an unconven- tional approach to non-equilibrium many-body systems developed by J. Fricke27which we review in Appendix B. Before deriving in the following section explicit micro- scopic expressions for the collision integrals in Eq. (3.14) let us generalize the construction of a stationary solu- tion with phenomenological damping discussed above by assuming that the collision integrals are of the form In k(t) = n knk(t); (3.15a) Ip k(t) = p k~pk(t); (3.15b) where n kand p kare assumed to be constant in time and independent of the magnon distribution functions. For simplicity we assume that the expectation values of the magnon operators are negligible and set I k(t) = 0. In this case the stationary non-equilibrium solution of Eq. (3.14) can easily be obtained analytically. The imag- inary part of p kcan be grouped together with the renor- malized magnon energy ~Ekand we therefore modify the expression for the renormalized magnon energy as fol- lows, ~Ek=Ek1 2Im p k+1 NX qTk;qnq(t): (3.16) ForjVkj>1 4 n kRe p kthe stationary non-equilibrium so- lution of Eq. (3.14) is then given by ns k=s Re p k n kj~pkj; (3.17a) ~ps k= s 1 n kRe p k 4V2 k+is n kRe p k 4V2 k! j~pkj; (3.17b) j~ps kj=Nq V2 k1 4 n kRe p kjEkjp n k=Re p k Tk;k+1 2Sk;k: (3.17c) Note that for1 2 n k=1 2 p k kwe recover the sta- tionary solution within conventional S-theory7,8given in Eqs. (3.11). Contrary to the case without collision inte- grals, the result for non-vanishing expectation values ~ k diers as the collision integrals cannot be written in the form n k(nk+j~ kj2). IV. COLLISION INTEGRALS In this section we present a microscopic derivation of the collision integrals In k(t),Ip k(t), andI k(t) appearing in the kinetic equations (3.14). Given the fact that for YIG the eective spin S14 is rather large,28we work to leading order in 1 =Swhere only the cubic part ~H3(t) of the Hamiltonian in Eq. (2.12) has to be taken intoaccount. The assumption that the experimentally ob- served ne structure of the inverse spin-Hall signal shown in Fig. 3 can be explained with the help of the scatter- ing processes described by the cubic vertices contained in~H3(t) is also supported by the fact that the peaks and dips of the observed signal as a function of the magnetic eld agree with the points where the splitting processes (in which one magnon is absorbed and two magnons are emitted) and the con uence processes (in which two magnons are absorbed and one magnon is emitted) de- scribed by the vertices in ~H3(t) become kinematically possible.1Note that a nite cubic part ~H3(t) of the magnon Hamiltonian arises entirely from dipole-dipole interactions. The corresponding scattering processes con- serve energy and momentum, but do not conserve the number of magnons.37As we do not expect magnon- phonon interactions, magnon-defect interactions, and in- teractions with thermal optical magnons to be responsi- ble for the eect observed in the experiment1we neglect these interactions. In principle, the collision integrals can be derived us- ing the Keldysh formalism.26However the Keldysh for- malism has the disadvantage that it produces two-time correlations, whereas in our case we are only interested in equal-time correlations. Although the reduction of two-time correlations to equal-time correlations can be achieved by means of standard methods such as the gen- eralized Kadano-Baym-Ansatz,38in view of the com- plexity of the collision integrals for YIG we nd it more ecient to use a method involving only equal-time corre- lations at every step of the calculation. We therefore use the method developed by J. Fricke,27which allows us to to derive directly a hierarchy of coupled kinetic equations for equal-time correlations and provides us with a system- atic scheme for decoupling the correlations for arbitrary order. To make this work self-contained, in Appendix B we outline the main features of this method. A. Collision integrals due to cubic interaction vertices Consider rst the diagonal collision integral In k(t) ap- pearing in the kinetic equation (3.14a) for the connected partnc k(t) of the diagonal magnon distribution. Us- ing the method developed in Ref. [27] (which we review in Appendix B) and omitting for simplicity the time- arguments, we nd In k(t) =ip NP qh 1 2aaa k;q;kqei!0t=2h~ay q~ay kq~akicc:c: +(aaa q;qk;k)ei!0t=2h~ay q~aqk~akicc:c:i ; (4.1) where we have used momentum conservation to carry out one of the summations. Here h~ay q~ay kq~akicand h~ay q~aqk~akicare connected equal-time correlations in-7 k kq k-qk kq q-k FIG. 4: Diagrammatic representation of contributions to the collision integral In k(t) given in (4.1) which determine the time-evolution of the connected diagonal distribution func- tionnc k(t). For simplicity we do not draw the two conjugated diagrams obtained by ipping the direction of each arrow cor- responding to the complex conjugated terms in Eq. (4.1). The symbols have the following meaning: Outgoing arrows represent creation operators, incoming arrows represent an- nihilation operators, and the black dots represent external or interaction vertices. The left diagram contains two external vertices and the interaction vertex aaa k;q;kq; the empty circle (correlation bubble) represents the correlation h~ay q~ay kq~akic. As the lines between the correlation bubble and the interac- tion vertex in the left diagram form a pair of equivalent lines we have to insert a prefactor of 1 =2 in front of the rst vertex in Eq. (4.1). The right diagram contains the vertex aaa k;qk;q and the correlation h~ay q~aqk~akic.volving three magnon operators. In the graphical rep- resentation of Eq. (4.1) shown Fig. 4 these correlations are represented by empty circles with three external legs (correlation bubbles). Note that the diagrams shown in Fig. 4 dier from Feynman diagrams as they represent contributions to the dierential equations for the corre- lations at a xed time. Next, we express the three-point correlations in Eq. (4.1) in terms of the four-point corre- lations using the equation of motion. As a representative example, let us consider the correlation h~ay q~ay kq~akicin the rst term on the right-hand side of Eq. (4.1) and ex- plicitly evaluate only the diagram shown in Fig. 5. The other terms entering the equation of motion correspond- ing to the remaining diagrams have the same form and are represented by the dots in Eqs. (4.2){4.4) below. The calculations leading to the collision integrals are analo- gous for all terms. The equation of motion implies d dt+i(kqkq) h~ay q~ay kq~akic=ip NX q01 2 aaa k;q0;kq0
ei!0t=2h~ay q~ay kq~aq0~akq0ic+::: : (4.2) Integrating Eq. (4.2) over the time we obtain h~ay q~ay kq~akic=ip NX q02 41 2tZ t0dt0ei(kqkq)(tt0) aaa k;q0;kq0
ei!0t=2h~ay q~ay kq~aq0~akq0ic+:::3 5: (4.3) Finally, substituting Eq. (4.3) into Eq. (4.1) we obtain In k(t) =1 NX q;q02 41 2tZ t0dt0cos [(kqkq) (tt0)] aaa k;q;kq aaa k;q0;kq0
h~ay q~ay kq~aq0~akq0ic+:::3 5 t0!1!2 NX q;q01 2(kqkq) aaa k;q;kq aaa k;q0;kq0
h~ay q~ay kq~aq0~akq0ic+::: ; (4.4) where in the last step we have taken the limit t0!1 and the dots denote the contributions of the other di- agrams. The other terms entering this equation repre- sented by the dots are of the same form. Note that the terms with two annihilation operators or two creation op- erators within the two-particle correlations are complex. Therefore, there appears an exponential function with imaginary valued argument instead of the cosine func- tion leading in the thermodynamic limit to a term of the same form as in Eq. (4.4) without the factor of two. In this way all terms entering the equation of motion forthe one-particle distribution functions can be obtained from the diagrams. A complete list of all diagrams con- tributing to the equation of motion of the three-point correlationsh~ay q~ay kq~akicandh~ay q~aqk~akicis shown in Fig. 18 of Appendix C. The approach outlined above can also be used to ob- tain the o-diagonal collision integral Ip k(t) in the kinetic equation (3.14b) for the o-diagonal distribution function ~pc k(t). In this case there are only two diagrams containing the relevant vertices shown in Fig. 6. The corresponding8 kq k-qq' kq k-q kq k-qqk k-q qk k-q k-qk qk-qk q k-qk qqk k-q k qk-qq' -q' q' q' -q' k-q+q' q-k-q' -q' k-q+q' q'k-q' q'-k k-q' q'-q q-q' q-q' -q' FIG. 5: One of the diagrams contributing to the equation of motion of the three-point correlation h~ay q~ay kq~akic. This diagram, which corresponds to the term explicitly written out in Eq. (4.2), contains the interaction vertex aaa q0;kq0;kand the four-point correlation h~ay q~ay kq~aq0~akq0ic. As the lines between the correlation bubble and the interaction vertex are a pair of equivalent lines this diagram should be weighted by an extra factor of 1 =2. k -kq k-qk -kq q-k FIG. 6: The two diagrams contributing to the time-evolution of the o-diagonal distribution function pk=h~ak~aki. The diagrams correspond to the two terms on the right-hand side of Eq. (4.5). The left diagram contains the interaction ver- tex aaa q;kq;kand the correlation h~aq~aqk~akic. The left diagram should be multiplied by a factor of 1 =2 because the lines between the correlation bubble and the vertex are equiv- alent. The right diagram contains the vertex aaa qk;q;kand the correlationh~ay q~aqk~akic. expression for the o-diagonal collision integral is Ip k(t) =i1p NX qh 1 2 aaa k;q;kq ei!0t=2h~aq~aqk~akic +aaa qk;q;kei!0t=2h~ay q~aqk~akici : (4.5) The correlationh~aq~aqk~akicin the rst term leads for large times to a delta function of the form (k+q+kq). Keeping in mind that the magnon dispersionkis positive for all momenta, this term does not contribute to the o-diagonal collision integral Ip k(t) for large times. The diagrams contributing to the corre- lationh~ay q~aqk~akichave already been discussed in the context of the diagonal collision integral In k(t), see Fig. 18 in Appendix C. Finally, the collision integral I k(t) enter- ing the kinetic equation (3.14) for the expectation values ~ kof the magnon operators vanishes, I k(t) = 0; (4.6) because there is no diagram contributing to the time- evolution of ~ k(t) that is quadratic in the three-point vertices.B. Decoupling of the equations of motion for the connected correlations So far, we have expressed the contributions to the col- lision integrals involving the various types of three-point vertices in terms of connected four-point correlations. The next step is to decouple the hierarchy of equations of motion by replacing the connected four-point correla- tions by one-point and connected two-point correlations. Keeping in mind that the only non-vanishing distribution functions are nc k, ~pc k, and ~ kwe nd h~ay k~ak~ay q~aqic=1!h~ay k~akih~ay q~aqi+ 2!h~ay k~akih~ay qih~aqi +2!h~ay kih~akih~ay q~aqi3!h~ay kih~akih~ay qih~aqi =nc knc q+nc kj qj2+nc qj kj2 3j kj2j qj2: (4.7) Analogous expressions can be written down for the other four-point correlations, so that the collision integrals can be expressed in terms of the two types of two-point cor- relationsnc k(t) and ~pc k(t) and the non-equilibrium expec- tation values k(t) of the magnon operators. It is conve- nient to decompose the collision integrals as Ip k=Ip k;inIp k;out; (4.8a) Ip k=Ip k;inIp k;out; (4.8b) whereIk;inis the in-scattering or arrival term, and Ik;out is the out-scattering or departure term. The explicit ex- pressions for the various contributions to the collision in- tegrals are rather cumbersome and are given in Eqs. (C1) - (C4) of Appendix C. Within the rotating-wave approx- imation the fast oscillating terms containing factors of ei!0tshould be neglected to be consistent with a simi- lar approximation in the renormalized magnon dispersion ~Ekand the pumping energy ~Vk. V. EXPLANATION OF THE MAGNETIC FIELD DEPENDENCE OF THE INVERSE SPIN-HALL SIGNAL IN YIG Having derived explicit expressions for the collision in- tegralsIn k(t) andIp k(t) we can now construct stationary solutions of the kinetic equations (3.14) and determine the non-equilibrium magnon distribution which is pro- portional to the inverse spin-Hall signal observed in the experiment.1As discussed in Sec. III B, in order to under- stand the magnetic eld dependence of the inverse spin- Hall signal we need a microscopic understanding of the momentum-dependent magnon damping. In this section we rst calculate the magnon damping in thermal equi- librium which we need in the subsequent calculation of the collision integrals. We then present an approximate solution of the kinetic equations (3.14) with microscopic collision integrals derived in Sec. IV and obtain excellent agreement with the experiment.19 A. Magnon damping in thermal equilibrium In thermal equilibrium with temperature Tthe nor- mal magnon distribution is given by the Bose-Einstein distribution nk=1 ek=T1: (5.1) The magnon damping in equilibrium can then be ob- tained from the imaginary part of the magnon self-energy obtained within the imaginary-time (Matsubara) formal- ism. Alternatively, the magnon damping n kin equilib- rium can be obtained by writing the departure term of the collision integral as In k;out= n knk; (5.2) where for simplicity we consider only the normal (diag- onal) part In k;outof the collision integral. To simplify the explicit evaluation of the damping n klet us assume that the momentum kis suciently large so that we can neglect the eect of dipole-dipole interactions on the magnon dispersion. In this regime the long-wavelength magnon dispersion is determined by the exchange inter- action, k=q A2 kjBkj2jAkj=h0+k2; (5.3) with exchange stiness =JSa2: (5.4) In the expressions for the magnon dispersion given in Appendix A [see Eqs. (A9c) and (A11)] we can then set Bk= 0 andVk= 0. According to Ref. [28], for the eec- tive exchange energy in YIG is J1:29K, the eective spin isS14:2, and the lattice constant is a12:376A. The Bogoliubov transformation from Holstein-Primako bosonsbkto magnon operators akis then not neces- sary so that we may identify the corresponding vertices, aaa k1;k2;k3= bbb k1;k2;k3. Moreover, in the regime where the magnon dispersion is dominated by the exchange energy we may neglect the diagonal elements of the dipolar ten- sorD kdened in Eq. (A19). In the geometry shown in Fig. 1 the only non-zero elements of the dipolar tensor are thenDyz k=Dzy k, see Eq. (A19d). This greatly sim- plies all quantities appearing in the kinetic equations for the magnon distribution. To get a rough estimate for the order of magnitude of the damping, let us also ne- glect the contributions from the o-diagonal distribution pk(t) and the expectation values kof the magnon oper- ators to the collision integral In k;outin Eq. (5.2). In this approximation we obtain n k= n k;con+ n k;split; (5.5) where the contribution from the con uent process is n k;con= NX q(kkqq) jaaa k;q;kqj2[nq+nkq+ 1];(5.6)(a) (b) FIG. 7: Feynman diagrams representing the contributions to the magnon self-energy which generate (a) the con uent and (b) the splitting contributions to the magnon damping given in Eqs. (5.6) and (5.7). Here the arrows represent the magnon propagators and the dots represent the cubic interaction ver- tices. and the contribution from the splitting process is n k;split =2 NX q(k+qkq) jaaa q;k;qkj2[nqknq]: (5.7) Note that these expressions can also be obtained di- rectly from the diagonal part of the imaginary frequency magnon self-energy ( k;i!) via analytic continuation, n k=Im(k;k+i0+): (5.8) The Feynman diagrams for the self-energy corrections associated with the con uence and the splitting processes are shown in Fig. 7. For vanishing wavevector k= 0 the con uent contribution has been carefully evaluated by Chernyshev.39Here we are only interested in the range of wavevectors kwhere the magnon dispersion is dominated by the exchange energy so that it can be approximated by k=h0+k2. Keeping in mind that in our geometry the only non-vanishing matrix elements of the dipolar tensor areDyz k=Dzy kand using Eq. (A19d) we nd that the relevant cubic interaction vertex in Eqs. (5.6) and (5.7) is given by aaa k1;k2;k3= bbb k1;k2;k3=r S 2 Dzy k2+Dyz k3

p 2Sk2yk2z k2 2+k3yk3z k2 3 ;(5.9) where the energy scale
associated with the dipolar in- teraction is dened in Eq. (2.6). Since the experiment1 has been performed at room temperature which is large compared with the typical magnon energies, we may approximate the equilibrium magnon distribution in Eqs. (5.6) and (5.7) by a Rayleigh-Jeans distribution, nqT=q; n kqT=kq: (5.10) Shifting the integration variable q=q0+k=2 in Eq. (5.6), we obtain for the ratio of the con uent magnon damping to the magnon energy at high temperatures, n k;con k=T 8J
h0S2 (jkj)Fcon(k=);(5.11)10 where the threshold momentum is dened by 2= 2h0=; (5.12)and the dimensionless function Fcon(p) is dened via the following integral Fcon(p) =Z2 0d' 21 h 1 +1 2 p+^q'p p212ih 1 +1 2 p^q'p p212i 2 64 py+p p21 cos' pz+p p21 sin'  p+^q'p p212+ pyp p21 cos' pzp p21 sin'  p^q'p p2123 752 ;(5.13) 1 1.5 2 2.5 3 3.5 400.00050.0010.00150.0020.0025 py Fcon(py, 0) FIG. 8: Numerical evaluation of the function Fcon(py;0) dened in Eq. (5.13) as a function of py=ky=. For large py we nd that Fcon(py;0)/1=p4 y. where ^q'=eycos'+ezsin'. At the threshold momen- tumk=^kthis reduces to Fcon(^k) =16 9^k2 y^k2z: (5.14) A numerical evaluation of Fcon(py;pz= 0) is shown in Fig. 8. A rough estimate for the order of magnitude of the con uent magnon damping for YIG at room temperature is given by the prefactor in Eq. (5.11), which yields1 n k;con kT 16J
h0S2 =290K 161:29K1750G 1000G142 = 14(0:125)20:22: (5.15) This indicates that at room temperature the damping due to magnon con uence can be substantial. Next, consider the contribution from the splitting pro- cess to the magnon damping in equilibrium representedby the diagram (b) in Fig. 7. With the same approxima- tions as above we obtain n k;split k= 2Ta2Zd2q (2)2(h02k
q) qq+k 
2 2Sh ^ky^kz+ ^qy^qzi2 : (5.16) Setting for simplicity kz= 0, we see that the -function enforcesqy=q0 y=h0=(2ky). The condition jq0 yj=a then reduces to jkyj> h 0a=(2) =2a=(4). With a1, it is clear that the splitting contribution to the magnon damping has a much lower threshold than the con uent contribution. Using the quadratic approx- imation (5.3) for the magnon dispersion and the deni- tion (5.12) of we nd that for parametrically pumped magnons with k=!0=2 the conditionjkj>is satised for h0<!0 6; (5.17) where the upper bound !0=6 coincides with the mag- netic eld strength below which the con uent damping process is kinematically possible. On the other hand, for h0> ! 0=6 the damping is dominated by the splitting processes. Unfortunately, the approximations made in this sec- tion are only valid for small pumping energy jVkj, whereas the experiment1has been performed in the regime of parametric instability where jVkj>jEkj. Therefore we expect that the estimates for the magnon damping in this subsection are not relevant for the ex- periment of Ref. [1]. This is also conrmed by the linear magnetic eld dependence of the damping due to the con uent and the splitting processes in thermal equi- librium shown in Fig. 9, which can be obtained by nu- merically evaluating Eqs. (5.6) and (5.7). In Fig. 10 we show the corresponding magnon density obtained by in- serting this damping into the expression (3.11a) for the magnon distribution predicted by S-theory. Obviously, the magnetic-eld dependence is linear in a wide range of elds and shows a small discontinuity at H082011 800 900 1000 1100 120005e-061e-051.5e-052e-05 H0 / Oe γn / T γ n con / T γ n split / T FIG. 9: Magnetic eld dependence of the magnon damping in thermal equilibrium due to the con uence and the split- ting processes. The plotted damping rates n conand n splitare obtained from Eqs. (5.11) and (5.16) by averaging over all momenta ksatisfyingk=!0. For the calculation we have assumed a lm thickness of d= 22:8m and a pumping fre- quency!0= 13:857 GHz. 800 900 1000 1100 12000.020.030.040.050.060.070.08 H0 / Oen / N7dB 8dB 10dB 12dB FIG. 10: Magnetic eld dependence of the stationry magnon density within S-theory given by Eq. (3.11a) for the same parameters as in Fig. 9. The continuous lines are obtained assuming a constant magnon damping k= 2:19103GHz, while the dashed lines are obtained by substituting the equi- librium magnon damping shown in Fig. 9 into Eq. (3.11a). Oe where the condition (5.17) is violated. By compar- ing Fig. 10 with the experimental result for the inverse spin-Hall voltage shown in Fig. 3, we conclude that by inserting the equilibrium magnon damping into the S-theory result for the stationary magnon density of the pumped magnon gas we cannot explain the experimental results. B. Solution of the kinetic equations with microscopic collision integrals In this section we show that the experimental results can be explained when the eect of collisions on the sta- tionary distribution of the pumped magnon gas is taken into account microscopically within a non-equilibrium many-body approach where we approximately solve the kinetic equations (3.14) with collision integrals given in Appendix C. As it stands, this system of non-linear integro-dierential equations is very complicated and we have not been able to solve it directly. Fortunately, we have found an approximation strategy which is su- ciently simple to allow for a numerical solution of the ki- netic equations while it still contains the relevant physical processes which determine the detailed form of the exper- imentally observed inverse spin-Hall signal. Our strategy is to divide the magnons into the following two groups corresponding to dierent regimes in momentum space and dierent energy windows: 1.Parametric magnons are directly excited by the os- cillating microwave eld via parametric resonance. From S-theory7,8,24we know that only magnons in a small area of the momentum space near the res- onance surface dened by k=!0=2 are generated by the parametric pumping so that it is justied to assume that all parametric magnons fulll the resonance condition k=!0=2. 2.Secondary magnons are created by con uence pro- cess of two parametric magnons. As a consequence, their energy k=!0is twice as large as the energy of parametric magnons. Assuming that the non-equilibrium magnon dynamics is dominated by these two groups of magnons, we can ap- proximate the distribution of all other magnons in the collision integrals by the thermal equilibrium distribu- tion. These approximations signicantly simplify the col- lision integrals as the arguments of the delta functions only vanish if two of the energies correspond to paramet- ric magnons and the other one to secondary magnons. The complexity of evaluating the collision integrals nu- merically is then greatly reduced. Neglecting the expec- tation values of the magnon operators we nd from the general expressions for the collision integrals given in Ap- pendix C that the collision integrals associated with the two dierent magnon groups can be written as12 In(1) k =2 NX q q=!0 qk=!0=2aaa q;k;qk2 n(2) qh 1 +n(1) qki n(1) kh n(1) qkn(2) qi +Reh aaa q;k;qk
aaa qk;k;q(~p(2) q)~p(1) qki ; (5.18a) In(2) k =2 NX q q=!0=2 kq=!0=21 2aaa k;q;qk2 n(1) qn(1) kqn(2) qh 1 +n(1) q+n(1) kqi Reh aaa k;q;kq aaa q;kq;k
~p(2) k(~p(1) q)+ aaa k;q;qk
aaa kq;k;q~p(2) k(~p(1) kq)i ;(5.18b) Ip(1) k=2 NX q q=!0 qk=!0=2 aaa qk;k;q aaa k;qk;q
~p(1) qk n(2) qn(1) k aaa qk;k;q2~p(1) k 1 +n(1) qk1 2n(2) q~p(2) q ; (5.18c) Ip(2) k= 0; (5.18d) wheren(1) kand ~p(1) krefer to the magnon distribution func- tions of parametric magnons and n(2) kand ~p(2) krefer to the secondary magnon group. When summing over the loop momentum q, we have to implement the conditions k=qk=!0=2,q=!0in the collision integrals of the parametric magnon group, and the conditions k=!0 andq=kq=!0=2 for the secondary magnon group. When all of these conditions can be fullled simultane- ously, there is only one possible combination of wavevec- tors so that only a single term contributes to the sums in Eq. (5.18). In order to calculate the collision integrals nu- merically we thus have to nd the specic combination of wavevectors that fulll momentum and energy conserva- tion. Then, we interpolate linearly between the magnon distribution functions dened on a nite grid in momen- tum space and evaluate the expressions (5.18). It is also possible that for certain parameters the conservation laws cannot be fullled, so that the collision integrals vanish in our approximation. All other magnons which do not belong to the above two groups are assumed to be in thermal equilibrium where the stationary distributions are given by the Bose-Einstein distribution (5.1) with T= 290 K. We take the contribution of these equilibrium magnons to the damping of the non-equilibrium magnons into account using the equilibrium damping rates derived in Sec. V A. To obtain a self-consistent solution of the kinetic equa- tions (3.14) with collision integrals given by Eq. (5.18) we use the following iterative procedure: Initially, we com- pletely neglect the collision integrals and use the station- ary distribution (3.17) of the kinetic equations with phe- nomenological damping n k= p k= 0= 2:87103GHz to construct the initial seed for the iteration. We thensubstitute the resulting stationary distribution back into our microscopic expressions (5.18) for the collision inte- grals and calculate a new estimate for the collision inte- grals. Next, we use the result to re-calculate a rened estimate for the stationary solution of the kinetic equa- tions (3.17). To obtain new values for non-equilibrium damping rates n(1) kand p(1) kwe assume that the terms proportional to n(1) kand ~p(1) kdominate the collision in- tegrals and estimate n(1) kand p(1) kbyIn(1) k=n(1) kand Ip(1) k=~p(1) k. The result is again substituted into the right- hand side of the collision integrals (5.18) and the proce- dure is iterated again. Gradually, we obtain corrections to the initial estimate of the magnon distribution in the stationary non-equilibrium state. To control the conver- gence of this algorithm we estimate the error by evaluat- ing the derivatives @tnkand@t~pkgiven by the equations of motion (3.14) and summing up the absolute values for every magnon mode. This expression should vanish if our estimates for the magnon distributions approach the ex- act stationary solutions. If this estimated error tends to zero during the iteration, our algorithm has produced a self-consistent stationary solution of the kinetic equations (3.14). Note that the vanishing of the o-diagonal colli- sion integral Ip(2) kassociated with the secondary magnons implies that the stationary solution of the kinetic equa- tion (3.17a) has the property that n(2) kvanishes indepen- dently of the value of ~ p(2) k. In Fig. 11 we show our numerical results for a YIG lm with thickness d= 22:8m (corresponding to N= 18423) in a microwave eld with frequency !0= 13:857GHz for four dierent pumping strengths between 7dB and 12dB,13 800 900 1000 1100 120000.0050.010.0150.020.025 H0/ Oe n / N7dB 8dB 10dB 12dB FIG. 11: The magnon density obtained by the procedure de- scribed in Sec. V B for a thin YIG lm of thickness d= 22:8m and!0= 13:857 GHz is plotted over the external eld strengthH0for four dierent pumping strengths. The pa- rameter for the pumping strength is hVk1 0ik1with the average taken over all momenta k1of parametric magnons. Our theoretical result shown in this gure should be compared with the experimental results by Noack et al.1reproduced in Fig. 3. where the parameter controlling the pumping strength is hVk1 0ik1with the average taken over all momenta k1of parametric magnons. The magnon density shown in Fig. 11 is approximated by taking the sum over all magnon modes used for the calculations, ns=NX i=1ns i; (5.19) where the momentum dependence of the magnon distri- bution functions are parameterized by the angle i=ki of the in-plane wavevectors dened in Eq. (2.8) and we useN= 40 angles of equal size in the interval [0 ;=2]. The wavevectors k1andk2of parametric and secondary magnons for a given angle iare calculated by solving the equations k1=!0=2 andk2=!0numerically for k1andk2with magnon dispersion given by Eq. (2.4). Apart from a small oset in the overall eld strength by about 50 Oe, the main features of the experimentally ob- served line-shape of the inverse spin-Hall signal shown in Fig. 3 are reproduced remarkably well by our calcu- lation. Recall that S-theory with phenomenological con- stant damping cannot explain this line-shape. In partic- ular, the experimentally observed dip around H01050 Oe for small pumping strength which evolves into a peak at the same eld for larger pumping strength is repro- duced by our method. Note, however, that in the exper- iment these features appear at a slightly lower eld of H01000 Oe. A possible explanation for this discrep- ancy in the overall eld strength is the in uence of cu- bic crystallographic and uni-axial anisotropy elds whichcan modify the saturation magnetization. It is therefore plausible that the experimentally relevant value of the saturation magnetization diers from the value of 1750 G assumed in our calculation which can explain the 50 Oe shift in the position of the peaks and dips in the upper and lower part of Fig. 11 . To show that dip and the peak are related to the con- uent magnon damping, we have plotted in Fig. 12 the cumulative damping rates n=PN i=1 n iand Re p=PN i=1Re p ifor the stationary non-equilibrium state we have obtained from our kinetic equations. Obviously, the (a) 800 900 1000 1100 120000.020.040.060.080.1 H0/ Oe γn/ GHz (b) 800 900 1000 1100 120000.050.10.150.2 H0/ Oe Re γp/ GHz FIG. 12: The damping dened by Eqs. (3.15) in the stationary non-equilibrium state shown in Fig. 11 is plotted over the external eld strength H0for the same parameter values as in Fig. 11. peaks in the cumulative magnon damping are observed at the same magnetic eld strength where the enhancement of the magnon density takes place. Not all magnon modes show these enhancements. The distribution functions for most of the magnon modes still increase linearly with the external eld strength and only a few magnon modes aroundk40have peaks between H0= 1050 Oe and H0= 1100 Oe. It is interesting to compare the order of magnitude14 of the cumulative non-equilibrium damping nshown in the upper panel of Fig. 12 with the established value of the Gilbert damping used in phenomenologi- cal approaches for YIG.40{42Usually the momentum- dependent damping kis parameterized in terms of a dimensionless damping parameter = k=(2k), where kis the magnon dispersion.40According to Refs. [41,42] for thermal acoustic magnons in YIG the typical value ofis for small wavevectors of order 104. On the other hand, our cumulative non-equilibrium damping n in the upper panel of Fig. 12 is typically of order 0 :02 GHz, which yields a dimensionless damping parameter 1:4103. We conclude that the non-equilibrium damping obtained within our microscopic approach is roughly an order of magnitude larger than the accepted phenomenological value of the equilibrium damping of thermal magnons in YIG. The rather complicated dependence of the non- equilibrium magnon density on the external magnetic eld shown in Fig. 12) cannot be reproduced within con- ventional S-theory where the microscopic collision inte- grals are replaced by a phenomenological relaxation rate. In the relevant parameter regime, S-theory predicts a lin- ear dependence of the magnon density on the external eld strength as shown in Fig. 2. Note also that within S-theory the damping is assumed to be strong so that only magnon modes near the maximum of the pumping energyVkatk= 90are signicantly occupied. In fact, the magnon modes which we have identied to be re- sponsible for the observed peaks and dips are assumed to be suppressed by the phenomenological damping in S-theory. Thus, it is evident that the experimentally ob- served structures in the non-equilibrium magnon density are caused by the con uence and splitting decay pro- cesses; the kinematic constraints controlling these pro- cesses are fully taken into account in our collision inte- grals which couple pairs of parametric magnons at spe- cial wavevectors depending on the external eld strength. The mathematical structure of the equations of motion is complicated and leads to peak structures appearing in the collision integrals at certain eld strengths. This in turn gives rise to similar structures in the eld-dependent magnon density close to magnetic elds where con uent magnon decay is kinematically possible. VI. SUMMARY AND CONCLUSIONS In this work we have derived and solved kinetic equa- tions for pumped magnons in YIG with collision integrals discribing dissipative eects associated with magnon de- cays. The collisionless limit of these equations has re- cently been discussed in Ref. [24]. However, to explain recent experimentel data1for the magnetic eld depen- dence of the inverse spin-Hall voltage in the stationary non-equilibrium state of pumped magnons in YIG a mi- croscopic understanding of magnon decays is crucial. We have derived the relevant collision integrals due to cu-bic interaction vertices using a systematic expansion in powers of connected equal-time correlations.27We have obtained the collision integrals for the diagonal and o- diagonal distribution functions containing terms which are linear and quadratic in the magnon distribution func- tions as well as the expectation values of the magnon op- erators. In previous works these collision integrals were not taken into account due to their complexity or were only derived within Born approximation11and evaluated in thermal equilibrium. We have found a way to numerically solve the result- ing kinetic equations within an approximation where only two groups of magnons are asumed to be driven out of equilibrium: parametric magnons that are generated by the pumping, and secondary magnons that are involved in con uence and splitting processes described by the microscopic collision integrals. We have explicitly con- structed the stationary non-equilibrium solution of the kinetic equations for the pumped magnon gas. Our results show in a large parameter regime a roughly linear magnetic eld dependence of the magnon density, in agreement with previous results obtained within a col- lisionless kinetic theory. However, near the magnetic eld strength where magnon decays (con uence and splitting processes) become kinematically allowed, we have ob- tained peak and dip structures in the magnon density, in good agreement with the experiment by Noack et al.1 where the non-equilibrium magnon density has been mea- sured via the inverse spin-Hall eect. ACKNOWLEDGEMENTS We are grateful to A. A. Serga for his comments on the manuscript. We also thank A. A. Serga and T. Noack for helping us to prepare Fig. 3 and for their permission to present the experimental data of Ref. [1] in this gure. APPENDIX A: HAMILTONIAN FOR PUMPED MAGNONS IN YIG Here we derive the magnon Hamiltonian for the para- metrically pumped magnon gas in YIG following mainly Ref. [28]. We start from the eective spin Hamiltonian for YIG5,6,19,20,28{31given in Eq. (2.1). The exchange couplingsJijassume the value J1:29 K for all pairs of nearest neighbor spins located at lattices sites riand rj, and the dipolar tensor is28,37 D ij= (1ij)2 jrijj3h 3^r ij^r iji ; (A1) whereis the magnetic moment of the spins, rij= rirj, and ^rij=rij=jrijj. After Holstein-Primako transformation32and expansion in powers of 1 =Sthe spin Hamiltonian is mapped onto an eective boson Hamilto- nian of the form (2.2) where the terms Hican be ex- pressed in terms of Holstein-Primako bosons biandby i.15 The zeroth order contribution H0(t) can be dropped as it does not contain any boson operators. Transforming to momentum space, bi=1p NX keik
ribk; (A2) whereNis the total number of lattice sites, the contribu- tions to the Hamiltonian up to fourth order in the bosons can be written as29 H2(t) =X k Akby kbk+Bk 2 by kby k+bkbk +h1cos (!0t)X kby kbk; (A3a) H3=1p NX k1;k2;k3k1+k2+k3;01 2!h bbb 1;2;3by 1b2b3 +bbb 1;2;3by 1by 2b3i ; (A3b) H4=1 NX k1;:::;k4k1+


+k4;0" 1 (2!)2bbbb 1;2;3;4by 1by 2b3b4 +1 3!bbbb 1;2;3;4by 1b2b3b4+1 3!bbbb 1;2;3;4by 1by 2by 3b4 : (A3c) The vertices in (A3a)-(A3c) can be expressed in terms of the Fourier transforms of the exchange and dipolar couplings, Jk=X ieik
rijJij; (A4a) D k=X ieik
rijD ij: (A4b) The coecients AkandBkin Eq.(A3a) are Ak=h0+S(J0Jk) +S Dzz 01 2(Dxx k+Dyy k) ; (A5a) Bk=S 2[Dxx k2iDxy kDyy k]; (A5b) while the cubic vertices depend only on the dipolar tensor as follows, bbb 1;2;3=r S 2 Dzy k2iDzx k2+Dzy k3iDzx k3 +1 2(Dzy 0iDzx 0) ; (A6a) bbb 1;2;3= bbb 3;2;1 ; (A6b)and the quartic vertices are bbbb 1;2;3;4=1 2[Jk1+k3+Jk2+k3+Jk1+k4+Jk2+k4 +Dzz k1+k3+Dzz k2+k3+Dzz k1+k4+Dzz k2+k4 4X i=1 Jki2Dzz ki
# ; (A7a) bbbb 1;2;3;4=1 4 Dxx k22iDxy k2Dyy k2+Dxx k32iDxy k3Dyy k3 +Dxx k42iDxy k4Dyy k4 ; (A7b) bbbb 1;2;3;4= bbbb 4;1;2;3 : (A7c) Next, we diagonalize the time-independent part of H2(t) by introducing magnon annihilation and creation opera- torsakanday kvia the Bogoliubov transformation, bk by k = ukvk v kukak ay k ; (A8) where uk=r Ak+"k 2"k; (A9a) vk=Bk jBkjr Ak"k 2"k; (A9b) "k=q A2 kjBkj2: (A9c) In terms of the magnon operators the time-dependent term in Eq. (A3a) leads to o-diagonal terms, so that the total quadratic Hamiltonian reads,29 H2(t) =X k "kay kak+"kAk 2 +h1cos (!0t)Ak "kay kak"kAk 2"k +X kh Vkcos (!0t)ay kay k+V kcos (!0t)akaki ; (A10) with pumping energy Vk=h1Bk 2"k: (A11) Expressing also the cubic and quartic parts of the Hamil- tonian in terms of magnon operators we obtain2916 H3=1p NX k1;k2;k3k1+k2+k3;01 2aaa 1;2;3ay 1a2a3+1 2aaa 1;2;3ay 1ay 2a3+1 3!aaa 1;2;3a1a2a3+1 3!aaa 1;2;3ay 1ay 2ay 3 ; (A12) H4=1 NX k1;k2;k3;k4k1+k2+k3+k4;0" 1 (2!)2aaaa 1;2;3;4ay 1ay 2a3a4+1 3!aaaa 1;2;3;4ay 1a2a3a4 +1 3!aaaa 1;2;3;4ay 1ay 2ay 3a4+1 4!aaaa 1;2;3;4a1a2a3a4+1 4!aaaa 1;2;3;4ay 1ay 2ay 3ay 4 ; (A13) with cubic vertices given by aaa 1;2;3=bbb 1;2;3v1u2u3bbb 2;1;3v2u1u3bbb 3;1;2v3u1u3+ bbb 1;2;3v1v2u3+ bbb 2;3;1v2v3u1+ bbb 1;3;2v1v3u2;(A14a) aaa 1;2;3= bbb 1;2;3u1u2u3+ bbb 2;1;3v1v2u3+ bbb 3;1;2v1v3u2bbb 3;2;1v3v2v1bbb 1;2;3v2u1u3bbb 1;3;2v3u1u2;(A14b) aaa 1;2;3= aaa 3;2;1
; (A14c) aaa 1;2;3= aaa 1;2;3
; (A14d) and quartic vertices aaaa 1;2;3;4= bbbb 1;2;3;4u1u2v3v4+ bbbb 1;3;2;4u1u3v2v4+ bbbb 1;4;2;3u1u4v2v3+ bbbb 2;3;1;4u2u3v1v4 +bbbb 2;4;1;3u2u4v1v3+ bbbb 3;4;1;2u3u4v1v2 bbbb 4;1;2;3u1u2u3v4bbbb 3;1;2;4u1u2u4v3bbbb 2;1;3;4u1u3u4v2bbbb 1;2;3;4u2u3u4v1 bbbb 2;3;4;1u1v2v3v4bbbb 1;3;4;2u2v1v3v4bbbb 1;2;4;3u3v1v2v4bbbb 1;2;3;4u4v1v2v3; (A15a) aaaa 1;2;3;4=bbbb 2;1;3;4u2v1v3v4bbbb 3;1;2;4u3v1v2v4bbbb 4;1;2;3u4v1v2v3bbbb 2;3;1;4u2u3u1v4 bbbb 2;4;1;3u2u4u1v3bbbb 3;4;1;2u3u4u1v2 +bbbb 1;2;3;4u1u2u3u4+ bbbb 4;3;2;1u3u2v1v4+ bbbb 3;4;2;1u4u2v1v3+ bbbb 2;4;3;1u4u3v1v2 +bbbb 1;2;3;4u4u1v2v3+ bbbb 1;2;4;3u3u1v2v4+ bbbb 1;3;4;2u2u1v3v4+ bbbb 4;3;2;1v4v2v3v1; (A15b) aaaa 1;2;3;4= bbbb 1;2;3;4u1u2u3u4+ bbbb 1;3;4;2u1u4v3v2+ bbbb 1;4;3;2u1u3v4v2+ bbbb 2;3;4;1u2u4v3v1 +bbbb 2;4;3;1u2u3v4v1+ bbbb 3;4;2;1v1v2v3v4 bbbb 4;3;2;1u3v2v1v4bbbb 3;4;2;1u4v2v1v3bbbb 2;3;4;1u2u3u4v1bbbb 1;3;4;2u1u3u4v2 bbbb 2;3;4;1u2v3v4v1bbbb 1;3;4;2u1v3v4v2bbbb 1;2;4;3u1u2u3v4bbbb 1;2;3;4u1u2u4v3; (A15c) aaaa 1;2;3;4= aaaa 1;2;3;4; (A15d) aaaa 1;2;3;4= aaaa 4;3;2;1
: (A15e) Finally, let us give simplied expressions for the Fourier transforms JkandD kfor the geometry shown in Fig. 1 which reduce the complexity of the coecients Akand Bkand the higher-order vertices. For the energy scales probed in the experiment1it is sucient to retain only the lowest magnon band, so that we can derive the dis- persion from an eective in-plane Hamiltonian. The sim- plest approximation for the lowest transverse mode is the uniform mode approximation where we approximate the transverse modes by plane waves.28This approach is valid if the thickness dof the YIG lm is small comparedto the extensions in y- andz-direction. Then we nd Ak=h0+JS[42 cos (kya)2 cos (kza)] S 2(Dxx k+Dyy k) +
3; (A16) Bk=S 2(Dxx kDyy k); (A17) where
=42S a3(A18)17 is the dipolar energy and the Fourier transformed ele- ments of the dipolar tensor are28 Dxx k=42 a31 3fk ; (A19a) Dyy k=42 a31 3(1fk) sin2k ;(A19b) Dzz k=42 a31 3(1fk) cos2k ;(A19c) Dyz k=Dzy k=22 a3sin (2k); (A19d) Dxy k=Dyx k= 0: (A19e) The form factor fkis given in Eq. (2.7). For in-plane wavevectors Dyz k=Dzy kis the only non-zero o-diagonal matrix element of the dipolar tensor. Within these ap- proximations the expressions for the magnon energy k and the pumping energy Vkreduce to Eqs.(2.4) and (2.5) of the main text. APPENDIX B: EXPANSION IN POWERS OF CONNECTED CORRELATIONS In this appendix we review the method of deriving ki- netic equations in terms of connected equal-time correla- tions developed by J. Fricke in Ref. [27]. In the follow- ing we refer to this method as the Fricke approach . In Sec. IV we have used this method to derive the leading contributions of the cubic interaction vertices to the col- lision integrals appearing in the kinetic equations (3.14). While it is also possible to use the Keldysh formalism26 for this task, the Fricke approach is more ecient for our purpose because it produces directly a hierarchy of coupled kinetic equations involving only equal-time cor- relations and provides us with a systematic decoupling scheme for correlations of arbitrary order. Note also that the Fricke approach generates an expansion of the colli- sion integrals in powers of connected equal-time correla- tions and is therefore very convenient for including the eect of time-dependent non-Gaussian correlations in the non-equilibrium dynamics; in contrast, the Keldysh for- malism relies on the perturbative expansion in terms of single-particle Green functions. 1. Equations of motion Consider the bosonic many-body system with second quantized Hamiltonian Hwhich may explicitly depend on time. In the Heisenberg picture the time-dependence of an operator A(t) is given by the Heisenberg equation of motion, id dtA(t) = [A(t);H]: (B1)The expectation value of A(t) is given by hAit= Tr [0A(t)]; (B2) where the density matrix 0species a mixture of states at the initial time t0. The time-dependence of the expec- tation value is described by id dthAit=h[A;H ]it: (B3) WritingH=Ht 0+V, where the one-particle part Ht 0 contains the terms that are quadratic in the bosonic op- erators and Vdescribes interactions, we obtain id dthAith A;Ht 0 it=h[A;V]it: (B4) The contribution of the one-particle Hamiltonian Ht 0to the time-evolution of the system is easy to handle. In order to derive the contribution of the right hand side of Eq. (B4) containing the interaction Hamiltonian V, it is useful to introduce connected correlations. 2. Connected correlations In order to express expectation values of an arbitrary set of bosonic operators at the same time in terms of connected equal-time correlations we introduce the clus- ter expansion. Following again Ref. [27], let us consider a set of bosonic operators Bilabeled by a set of integers i2N. The explicit expressions for the connected cor- relations contain sums over all partitions Pof an index setIdened as the set of all non-empty disjoint sub- setsJofIwithS J2PJ=I. Furthermore, we dene BIBi1


Bikas the product of all operators with in- dicesi1;:::;ik, wherei1<:::<ikandI=fi1;:::;ikg. In our case, the Biare bosonic eld operators, i.e. lin- ear combinations of bosonic creation operators by jand annihilation operators bj. They obey the commutation relations  bi;bj = 0; bi;by j =ij: (B5) Since the operators do not commute in general, we keep track of their ordering by requireing that the indices ik within the sets Iare ordered as denoted above.27 The connected correlations h:::iccan be dened recur- sively as follows,43 hBIi=X P2PIY J2PhBJic; (B6) wherePIrefers to the set of all partitions of I. With the help of Eq. (B6) we can write n-point correlation func- tions as sums over all partitions Pof the index set I with each summand being the product of all correlations of the subsets J2P. Note that the correlations preserve18 the ordering of the indices in the sets J. It is also pos- sible to obtain an explicit expression for the connected correlations,27,43 hBIic=X P2PI(1)#P1(#P1)!Y J2PhBJi;(B7) where #Idenotes to the cardinality of the set Iwhich we will refer to as the order of the correlations. For example, the connected correlations up to third order are,27 hB1ic=hB1i; (B8a) hB1B2ic=hB1B2ihB1ihB2i; (B8b) hB1B2B3ic=hB1B2B3ihB1B2ihB3ihB1ihB2B3i hB1B3ihB2i+ 2hB1ihB2ihB3i:(B8c) The commutation relations (B5) imply that correla- tions with a permuted sequence of eld operators dier. Using b1b2c= b1b2 b1 b2 it follows that the con- nected one-particle correlations are hbibjic=hbjbiic; (B9a) hbiby jic=ij+hby jbiic: (B9b) On the other hand, in correlations of order greater than two the eld operators permute trivially,27 h


bibj


ic=h


bjbi


ic; (B10a) h


biby j


ic=h


by jbi


ic: (B10b) We note that only connected correlations of order n= 2 obey a non-trivial commutation relation and are thus a special case. For this reason, we will refer to one- particle connected correlations as contractions. As we will see later, contractions play an important role for this method. A proof of Eqs. (B10a) and (B10b) can be found in Refs. [27,43]. The denition of the cluster expansion can also be ex- tended to fermionic eld operators in such a way that we obtain analogous equations. Then the correlations of ordern6= 2 anti-commute27and hence sign rules have to be included. In this work we are only interested in bosonic operators. 3. Linked-cluster theorem We are interested in the time-evolution of n-point func- tionshBIit=hB1


Bkit, whereBiare linear combina- tions of bosonic eld operators with i2I. First, we simplify the interaction Hamiltonian Vin Eq. (B4) by assuming the form V=BK. This can be justied by the fact that the equation of motion (B4) is a linear combi- nation of the BK. The linked-cluster theorem discussed in this appendix still holds for the full interaction Hamil- tonian with the form of V=P KvKBK. The equation of motion of the expectation value hBIi is given by27 id dthBIit=h[BI;V]it=hBIBKBKBIit;(B11)where we have chosen IandKto be disjoint without loss of generality. As the connected correlation obey non- trivial commutation relations in general, we have to keep track of the sequence of indices of the operators inside the expectation values in Eq. (B11). Therefore we dene I+Kas the setI[Kwith the order relation given by the order relations of IandKrespectively and the condition i < k8i2I; k2K. Note that the sets I+Kand K+Iare identical; their order relation diers though. ForJ2I+Kwe dene ~Jas the identical set Jbut with order relation of K+I, following Ref. [27]. It can be shown that there is a linked-cluster-theorem for the equation of motion of the connected correlations which is given by27 id dthBIic t=X P2Pc I;K0 @Y J2PhBJic tY ~J2PhB~Jic t1 A;(B12) wherePc I;Kis the set of all connected diagrams and is dened by Pc I;KfP2PI+Kj8J2P:J\K6=;g: (B13) The right-hand side of Eq. (B12) can be further simpli- ed. We have seen in the above section that only contrac- tions which are one-particle connected correlations obey non-trivial commutation relations. Therefore the correla- tionshBJic tandhB~Jic tdier only for contraction and are identical for connected correlations of order n6= 2. Ob- viously, the term within the brackets on the right-hand side of Eq. (B12) is non-zero only if it contains at least one contraction, so that in a diagrammatic representa- tion (see below) only diagrams that contain contractions of external vertices with the interaction vertex contribute to the equation of motion of connected correlations. 4. Diagrams The diagrams introduced here dier from Feynman di- agrams because they represent dierential equations for the correlations. As a consequence, each diagrams con- tain only one interaction vertex. Moeover, each diagrams describe the time-evolution of a particular correlation at timetso that there is no time or energy integration involved.27Also, we introduce a new graphical symbol, the correlation bubble44representing the time-dependent correlations. Let us now introduce the graphical elements of the dia- grams. External vertices (Fig. 13) represent annihilation or creation operators. At least one external vertex is con- tracted with an interaction vertex (Fig. 14) which repre- sents the interaction associated with a certain matrix el- ement. Contractions (Fig. 15) are connected one-particle correlations. They are represented by their own graphi- cal element as they play a special role for this method. Connected correlations of order n6= 2 are represented19 k' k1 1 FIG. 13: External vertices represent annihilation operators with wavevector k0 1or creation operators with wavevector k1. j i j i1 s1 r FIG. 14: Interaction vertices describe the interactions. They are connected with matrix elements vi1;:::;ir;j1;:::;js. by correlation bubbles (Fig. 16).27As there is not nec- essarily a conservation of particle numbers for bosons, the number of incoming lines can dier from the number of outgoing lines for interaction vertices and correlation bubbles. Now we explain the rules for obtaining the time deriva- tive of thenpoint functionhbk1


bksby k0r


by k0 1itdue to interactions from the diagrams, where n=r+s. The cluster expansion of h[bk1


bksby k0r


by k0 1;V]itleads to all possible diagrams where vertices are connected with contractions and correlation bubbles. The resulting dia- grams contain r+sexternal vertices and only one inter- action vertex.27The fact that there is only one interac- tion vertex simplies the diagrammatic rules; there are no rules regarding time-ordering. We start with the rule for the prefactor. From Eq. (B4) we get a factor of ( i). Furthermore, we can write the interaction Hamiltonian in the form V=1 r!s!X vi1;:::;ir;i0 1;:::;i0sby i1


by isbi0r


bi0 1:(B14)Usually the interaction matrix elements vi1;:::;ir;i0 1;:::;i0sfulll symmetry properties, causing the prefactor of 1=(r!s!) to drop out because permutating annihilation and creation operators in the interaction term gives the same contribution. However, there exists an exception: if two lines connected to a correlation bubble point into the same direction, permutating the operators yields the k' k1 1 FIG. 15: The contraction by k0 1bk1ic. Contractions are con- nected correlations of order two. If the order is larger than two they are correlation bubbles (see Fig.16). k' k k' k1 r1 s FIG. 16: Correlation bubbles represent connected correla- tions, in this case hbk1


bksby k0r


by k0 1ic t. Note that the order of the connected correlation has to be larger than two. Oth- erwise it is a contraction (see Fig.15). same graph and thus the prefactor remains.27 The diagrammatic expansion of the equation of mo- tion for the n=r+s-point function has the following structure, d dt+i k1+


+ksk0 1


k0r
 hbk1


bksby k0r


by k0 1it=iX diagrams1 2neX i1;:::;ir j1;:::;jsvi1;:::;ir;j1;:::;jsXdiagram; (B15) whereXdiagram is the collision term containing the con- tractions and correlations and nedenotes to the number of equivalent pairs of lines. By comparing Eq. (B15) with the general structure (B12) of the linked cluster expan- sion we notice that the collision term contains the dif-ference of the partitions of hbk1


bksby k0r


by k0 1Vitand the partitions of hVbk1


bksby k0r


by k0 1it. As connected correlations of order n6= 2 obey trivial commutation re- lations, there will only be a dierence of these two terms due to contractions. As a result, the collision term is a20 j i1 1 jr isk kk' k'1 1 r s } to correlations FIG. 17: Schematic diagram showing the interaction vertex. In totalr+slines connect the interaction vertex with external vertices. The other lines go to correlation bubbles. product of a several factors: First of all, Xdiagram con- tains all correlations which are denoted by correlation bubbles. Furthermore, there are contributions from con- tractions. Contractions starting and ending at the in- teraction vertex give a normal-ordered contribution in the formhby kibkjic t, contractions between external ver-tices give an anti-normal-ordered contribution of the form hbkiby k0 jic t. Finally, there is a contribution from the re- maining contractions connecting the external vertices with the interaction vertex. Labeling the diagram as shown in Fig. 17, this contribution has the form27 h hbk1by i1ic t


hbksby isic t hby k0 1bj1ic t


 hby k0rbjric t  hby i1bk1ic t


 hby isbksic t hbj1by k0 1ic t


hbjrby k0ric ti : (B16) Diagrams without at least one contraction connecting an external vertex and an interaction vertex can be omit- ted, because the contributions of these diagrams vanish due to the fact that only contractions obey non-trivial commutation relations. Also, we only consider connected diagrams as unconnected diagrams do not contribute to the time-evolution of the correlations according to the linked-cluster theorem (B12). APPENDIX C: COLLISION INTEGRALS FOR PUMPED MAGNONS IN YIG Here we use the general formalism outlined in Appendix B to derive the collision integrals due to the cubic interaction vertices in the kinetic equations (3.14) describing the pumped magnon gas in YIG. The diagrams contributing to the correlationshay qay kqakicandhay qaqkakicare shown in Fig. 18. Recall that these diagrams are dierent from Feynman diagrams as they describe the time-evolution of correlations. Therefore they only contain one interaction vertex and there is no time or energy integration associated with the diagrams. The diagrams shown in Fig. 18 represent contributions to the connected three-point correlations which determine the collision integrals as described in Sec. IV, see Eqs. (4.1) and (4.5). For the collision integrals associated with the diagonal distribution function we obtain for the arrival term In k;in=2 NX q( 1 2("k"q"kq)aaa k;q;kq2nc qnc kq+("k+"kq"q)aaa q;k;qk2nc q 1 +nc qk
+("k"q"kq) aaa k;q;kqX q0 1;q0 2q0 1+q0 2k;0 aaa q0 1;q0 2;kei!0th~ay q~aqk~ay q0 2~aq0 1ic +1 2 aaa k;q0 1;q0 2 h~ay q~aqk~aq0 1~aq0 2ic +(k+qkq) aaa q;k;qk
X q0 1;q0 2q0 1+q0 2k;01 2 aaa k;q0 1;q0 2 ei!0th~ay q~ay kq~aq0 1~aq0 2ic +aaa q0 1;k;q0 2h~ay q~ay kq~aq0 1~ay q0 2ic) ; (C1)21 kq k-qq' kq k-q kq k-qqk k-q qk k-q k-qk qk-qk q k-qk qqk k-q k qk-qq' -q' q' q' -q' k-q+q' q-k-q' -q' k-q+q' q'k-q' q'-k k-q' q'-q q-q' q-q' -q' kq q-kkq q-k kq q-kqk q-k qk q-k q-kk qq-kk q q-kk qqk q-k q q-kkq' q'-kq' k-q' -q' k-q'q' q-q' q' q'-q-q' q-q' k-q+q' q'q-k-q' -q' q-k-q' q' FIG. 18: Diagrammatic representation of all terms contributing to the time-evolution of the correlation hay qay kqakic(left set of diagrams) and to the correlation hay qaqkakic(right set of diagrams). and for the departure term In k;out=2 NX q( 1 2("k"q"kq)aaa k;q;kq2nc k 1 +nc q+nc kq
+("k+"qk"q)aaa q;k;qk2nc k nc qknc q
+("k"q"kq) aaa k;q;kqX q0 1;q0 2q0 1+q0 2k;01 2aaa q;q0 1;q0 2ei!0th~ak~aqk~ay q0 1~ay q0 2ic + aaa q0 1;q0 2;q h~ak~aqk~ay q0 1~aq0 2icaaa q0 2;q0 1;qkei!0th~ak~ay q~ay q0 1~aq0 2ic 1 2 aaa qk;q0 1;q0 2 h~ak~ay q~aq0 1~aq0 2ic +("k+"qk"q) aaa q;k;qk
X q0 1;q0 2q0 1+q0 2k;0 aaa q0 1;q0 2;q ei!0th~ak~ay kq~ay q0 1~aq0 2ic +1 2aaa q;q0 1;q0 2h~ak~ay kq~ay q0 1~ay q0 2ic+1 2aaa kq;q0 1;q0 2h~ak~ay q~ay q0 1~ay q0 2ic +1 2 aaa q0 2;q0 1;kq ei!0th~ak~ay q~aq0 1~aq0 2ic+ aaa q0 1;q0 2;kq ei!0th~ak~ay q~ay q0 1~aq0 2ic) :(C2)22 For the collision integrals of the o-diagonal distribution function we obtain for the arrival term Ip k;in=2 NX q( ("k+"qk"q) aaa qk;q;kaaa q;qk;knc q 1 +nc qk
+X q0 1;q0 2("k+"qk"q)k+q0 1q0 2;0aaa qk;q;k aaa q0 1;q0 2;kei!0th~ay q~aqk~aq0 1~ay q0 2ic +1 2 aaa k;q0 1;q0 2 h~ay q~aqk~aq0 1~aq0 2ic) ;(C3) and for the departure term Ip k;out=2 NX q( ("k+"qk"q) aaa qk;q;kaaa q;qk;knc k nc qknc q +X q0 1;q0 2("k+"qk"q)k+q0 1q0 2;0aaa qk;q;k1 2aaa q;q0 1;q0 2ei!0th~ak~aqk~ay q0 1~ay q0 2ic + aaa q0 1;q0 2;q h~ak~aqk~ay q0 1~aq0 2icaaa q0 2;q0 1;qkei!0th~ay q~ak~ay q0 1~aq0 2ic 1 2 aaa qk;q0 1;q0 2 h~ay q~ak~aq0 1~aq0 2ic) :(C4) Electronic address: hahn@itp.uni-frankfurt.de 1T. B. Noack, V. I. Vasyuchka, D. A. Bozhko, B. Heinz, P. Frey, D. V. Slobodianiuk, O. V. Prokopenko, G. A. Melkov, P. Kopietz, B. Hillebrands, and A. A. Serga, Enhancement of the Spin Pumping Eect by Magnon Con uence Process in YIG/Pt Bilayers , Phys. Status Solidi B 256, 1900121 (2019). 2J. E. Hirsch, Spin Hall Eect , Phys. Rev. Lett. 83, 1834 (1999). 3K. Ando, S. Takahashi, J. Ieda, Y. Kajiwara, H. Nakayama, T. Yoshino, K. Harii, Y. Fujikawa, M. Mat- suo, S. Maekawa, and E. Saitoh, Inverse spin-Hall eect induced by spin pumping in metallic system , J. Appl. Phys. 109, 103913 (2011). 4N. Nagaosa, Spin Currents in Semiconductors, Metals, and Insulators , J. Phys. Soc. Jpn. 77, 031010 (2008). 5V. Cherepanov, I. Kolokolov, and V. S. L'vov, The saga of YIG: spectra, thermodynamics, interaction and relaxation of magnons in a complex magnet , Phys. Rept. 229, 81 (1993). 6V. S. L'vov, Wave Turbulence Under Parametric Excita- tions , (Springer, Berlin, 1994). 7V. E. Zakharov, V. S. L'vov, and S. S. Starobinets, Sta- tionary nonlinear theory of parametric excitation of waves , Zh. Eksp. Teor. Fiz. 59, 1200 (1970) [Sov. Phys. JETP 32, 656 (1971)]. 8V. E. Zakharov, V. S. L'vov, and S. S. Starobinets, Usp. Fiz. Nauk 114, 609 (1974) [ Spin-wave turbulence beyond the parametric excitation threshold , Sov. Phys. Usp. 17, 896 (1975)]. 9H. Suhl, The theory of ferromagnetic resonance at highsignal powers , J. Phys. Chem. Solids 1, 209 (1957). 10E. Schl omann, J. J. Green, and U. Milano, Recent Devel- opments in Ferromagnetic Resonance at High Power Lev- els, J. Appl. Phys. 31, 386S (1960); E. Schl omann and R. I. Joseph, Instability of Spin Waves and Magnetostatic Modes in a Microwave Magnetic Field Applied Parallel to the dc Field ,ibid.32, 1006 (1961); E. Schl omann and J. J. Green, Spin-Wave Growth Under Parallel Pumping ,ibid. 34, 1291 (1963). 11I. A. Vinikovetskii, A. M. Frishman, and V. M. Tsukernik, Kinetic equation for a system of parametrically excited spin waves , Zh. Eksp. Teor. Fiz. 76, 2110 (1979) [Sov. Phys. JETP 49, 1067 (1979)]. 12C. B. Araujo, Quantum-statistical theory of the nonlin- ear excitation of magnons in parallel pumping experiments , Phys. Rev. B 10, 3961 (1974). 13V. M. Tsukernik and R. P. Yankelevich, Stationary distri- bution of magnons following parametric excitation in fer- romagnetic substance , Zh. Eksp. Teor. Fiz. 68, 2116 (1975) [Sov. Phys. JETP 41, 1059 (1976)]. 14A. V. Lavrinenko, V. S. L'vov, G. A. Melkov, and V. B. Cherepanov, "Kinetic" instability of a strongly nonequi- librium system of spin waves and tunable radiation of a ferrite , Zh. Eksp. Teor. Fiz. 81, 1022 (1981) [Sov. Phys. JETP 54, 542 (1981)]. 15A. A. Zvyagin, V. Ya. Serebryannyi, A. M. Frishman, and V. M. Tsukernik, Dynamics of spin waves under paramet- ric excitation by a stepped periodic eld of arbitrary ampli- tude, Fiz. Nizk. Temp. 8, 1205 (1982) [Sov. J. Low Temp. Phys. 8, 612 (1982)]. 16A. A. Zvyagin and V. M. Tsukernik, A change in equilib-23 rium conguration of magnetic system during parametric excitation , Fiz. Nizk. Temp. 11, 88 (1985) [Sov. J. Low Temp. Phys. 11, 47 (1985)]. 17S. P. Lim and D. L. Huber, Microscopic theory of spin- wave instabilities in parallel-pumped easy-plane ferromag- nets, Phys. Rev. B 37, 5426 (1988); Possible mechanism for limiting the number of modes in spin-wave instabilities in parallel pumping ,ibid.41, 9283 (1990). 18Yu. D. Kalafati and V. L. Safonov, Thermodynamic approach in the theory of paramagnetic resonance of magnons , Zh. Eksp. Teor. Fiz. 95, 2009 (1989) [Sov. Phys. JETP 68, 1162 (1989)]. 19S. M. Rezende, Theory of microwave superradiance from a Bose-Einstein condensate of magnons , Phys. Rev. B 79, 060410(R) (2009); Theory of coherence in Bose-Einstein condensation phenomena in a microwave-driven interact- ing magnon gas , Phys. Rev. B 79, 174411 (2009). 20T. Kloss, A. Kreisel, and P. Kopietz, Parametric pump- ing and kinetics of magnons in dipolar ferromagnets , Phys. Rev. B 81, 104308 (2010). 21A. A. Zvyagin, Re-distribution (condensation) of magnons in a ferromagnet under pumping , Fiz. Nizk. Temp. 33, 1248 (2007) [Sov. J. Low Temp. Phys. 33, 948 (2007)]. 22V. L. Safonov, Nonequilibrium Magnons , (Wiley-VCH, Weinheim, Germany, 2013). 23D. V. Slobodianiuk and O. V. Prokopenko, Kinetics of Strongly Nonequilibrium Magnon Gas Leading to Bose- Einstein Condensation , J. Nano- Electron. Phys. 9, 03033 (2017). 24V. Hahn and P. Kopietz, Collisionless kinetic theory for parametrically pumped magnons , Eur. Phys. J. B 93 , 132 (2020). 25A. R uckriegel and P. Kopietz, Rayleigh-Jeans condensation of pumped magnons in thin lm ferromagnets , Phys. Rev. Lett. 115, 157203 (2015). 26See, for example, A. Kamenev, Field Theory of Non- Equibrium Systems , (Cambridge University Press, Cam- bridge, 2011). 27J. Fricke, Transport Equations Including Many-Particle Correlations for an Arbitrary Quantum System: A Gen- eral Formalism , Ann. Phys. 252, 479 (1996); see also J. Fricke, Transportgleichungen f ur quantenmechanische Vielteilchensystems , (Cuvillier Verlag, G ottingen, 1996). 28A. Kreisel, F. Sauli, L. Bartosch, and P. Kopietz, Micro- scopic spin-wave theory for yttrium-iron garnet lms , Eur. Phys. J. B 71, 59 (2009). 29J. Hick, F. Sauli, A. Kreisel, and P. Kopietz, Bose-Einstein condensation at nite momentum and magnon condensa- tion in thin lm ferromagnets , Eur. Phys. J. B 78, 429 (2010). 30S. M. Rezende, F. M. de Aguiar, and A. Azevedo, Magnonexcitation by spin-polarized direct currents in magnetic nanostructures , Phys. Rev. B 73, 094402 (2006). 31A. R uckriegel, P. Kopietz, D. A. Bozhko, A. A. Serga, and B. Hillebrands, Magnetoelastic modes and lifetime of magnons in thin yttrium iron garnet lms , Phys. Rev. B 89, 184413 (2014). 32T. Holstein and H. Primako, Field Dependence of the Intrinsic Domain Magnetization of a Ferromagnet , Phys. Rev.58, 1098 (1940). 33B. A. Kalinikos and A. N. Slavin, Theory of dipole- exchange spin wave spectrum for ferromagnetic lms with mixed exchange boundary conditions , J. Phys. C 19, 7013 (1986). 34I. S. Tupitsyn, P. C. E. Stamp, and A. L. Burin, Stability of Bose-Einstein Condensates of Hot Magnons in Yttrium Iron Garnet Films , Phys. Rev. Lett. 100, 257202 (2008). 35J. Barker and G. E. W. Bauer, Thermal Spin Dynamics of Yttrium Iron Garnet , Phys. Rev. Lett. 117, 217201 (2016). 36Eq. (3.13) suggests that there is an ambiguity in the choice of the partition of the connected part nc kand the contri- butionj~ kj2from the expectation values of the magnon operators which is eventually removed by the microscopic collision integrals. 37R. N. Costa Filho, M.G. Cottam, and G. A. Farias, Micro- scopic theory of dipole-exchange spin waves in ferromag- netic lms: Linear and nonlinear processes , Phys. Rev. B 62, 6545 (2000). 38P. Lipavsk y, V. Spi cka and B. Velick y, Generalized Kadano-Baym ansatz for deriving quantum transport equations , Phys. Rev. B 34, 6933 (1986). 39A. L. Chernyshev, Field dependence of magnon decay in yttrium iron garnet thin lms , Phys. Rev. B 86, 060401(R) (2012). 40S. A. Bender, R. A. Duine, A. Brataas, and Y. Tserkovnyak, Dynamic phase diagram of dc-pumped magnon condensates , Phys. Rev. B 90, 094409 (2014). 41S. Homan, K. Sato, and Y. Tserkovnyak, Landau-Lifshitz theory of the longitudinal spin Seebeck eect , Phys. Rev. B 88, 064408 (2013). 42L. J. Cornelissen, K. J. H. Peters, G. E. W. Bauer, R. A. Duine, and B. J. van Wees, Magnon spin transport driven by the magnon chemical potential in a magnetic insulator , Phys. Rev. B 94, 014412 (2016). 43K. Baumann and G. C. Hegerfeldt, A Noncommutative Marcinkiewicz Theorem , Publications of the Research In- stitute for Mathematical Sciences, Kyoto University, Vol. 21, No. 1 (1985). 44H. Schoeller, A New Transport Equation for Single-Time Green's Functions in an Arbitrary Quantum System. Gen- eral Formalism , Ann. Phys. 229, 273 (1994).

2020-12-14

We investigate the influence of magnon decays on the non-equilibrium dynamics of parametrically excited magnons in the magnetic insulator yttrium-iron garnet (YIG). Our investigations are motivated by a recent experiment by Noack et al. [Phys. Status Solidi B 256, 1900121 (2019)] where an enhancement of the spin pumping effect in YIG was observed near the magnetic field strength where magnon decays via confluence of magnons becomes kinematically possible. To explain the experimental findings, we have derived and solved kinetic equations for the non-equilibrium magnon distribution. The effect of magnon decays is taken into account microscopically via collision integrals derived from interaction vertices involving three powers of magnon operators. Our results agree quantitatively with the experimental data.

Effect of magnon decays on parametrically pumped magnons

2012.07870v2

Note Microwave cavity tuned with liquid metal and its application to Electron Paramagnetic Resonance C.S.Gallo1, E.Berto2, C.Braggio2;3, F.Calaon3, G.Carugno2;3, N.Crescini1;3, A.Ortolan1, G.Ruoso1, M.Tessaro3 1INFN, Laboratori Nazionali di Legnaro, Viale dell'Universit a 2, 35020 Legnaro, Padova, Italy 2Dipartimento di Fisica e Astronomia, Via Marzolo 8, 35131 Padova, Italy 3INFN, Sezione di Padova, Via Marzolo 8, 35131 Padova, Italy E-mail: carmelo.gallo@lnl.infn.it Submitted to: Measurement Science and Technology Keywords : microwave cavity, tunable resonance, liquid metal, Electron Paramagnetic Resonance, GaInSn Abstract. This note presents a method to tune the resonant frequency f0of a rectangular microwave cavity. This is achieved using a liquid metal, GaInSn, to decrease the volume of the cavity. It is possible to shift f0by lling the cavity with this alloy, in order to reduce the relative distance between the internal walls. The resulting modes have resonant frequencies in the range 7 8 GHz. The capability of the system of producing an Electron Paramagnetic Resonance (EPR) measurement has been tested by placing a 1 mm diameter Yttrium Iron Garnet (YIG) sphere inside the cavity, and producing a strong coupling between the cavity resonance and Kittel mode. This work shows the possibility to tune a resonant system in the GHz range, which can be useful for several applications.arXiv:1804.03443v1 [physics.ins-det] 10 Apr 2018Microwave cavity tuned with liquid metal 2 Resonant cavities are typically completely enclosed by conducting walls that can contain oscillating electromagnetic elds. The resonant frequency f0of a mode in a rectangular cavity depends on the distances between the internal surfaces of the walls. Imposing the boundary conditions on the electromagnetic eld trapped inside the cavity, it is possible to obtain an analytic expression of the resonant frequency fm;n;l 0 for the m;n;l mode. If we call a;b;d the dimensions of a cavity lled with vacuum, this calculation yields fm;n;l 0 =c2 2rm a2 +n b2 +l d2 ; (1) wherecis the speed of light in vacuum [1]. Eq.(1) states that a cavity resonates at frequencies which are determined by the dimensions of the resonant cavity: as the cavity dimensions increase, the resonant frequencies decrease, and vice versa. Thus a reduction of one of this distances necessarily results in increased resonant frequencies of the modes, allowing a tuning of the cavity within certain ranges. To shift the resonance frequency of a chosen mode we change one single dimension by lling the cavity with a liquid metal. It is to be noticed that this is not the only way to shift the resonance frequency of a mode, for example it is possible to insert dielectric materials into the cavity [2, 3, 4, 5, 6, 7, 8]. The copper cavity used in this work has dimensions ( a= 30 mm)(b= 10 mm)(d= 60 mm); we aim to shift the resonance frequency of a Transverse Magnetic mode (TM102), whose resonance frequency f0is 7.093 GHz. The cavity partially lled with liquid metal is shown in Fig.(1). Figure 1. Left: the cavity. Right : half view of the cavity with liquid metal. Let us call a0,f0andai,fithe coordinate in the vertical direction and resonant frequencies of the empty and lled cavity, respectively. We tune the resonant frequency of the selected mode injecting dierent volumes of a liquid metal. We have selected GaInSn (liquid metal at room temperature), which is an eutectic mixture of the metals gallium, indium and tin. We control the volume of the injected metal using a syringeMicrowave cavity tuned with liquid metal 3 connected to the cavity by a copper tube as shown in Fig.(2). The dimensions of copper tube are not important, in our experiment we used a copper tube 130 mm long and with 1.78 mm of diameter. Figure 2. Picture of the microwave cavity connected to the GaInSn syringe by means of a copper tube soldered to the bottom of the cavity itself. The measures were made at room temperature. The measures consist of i= 1;2;3;4;5 steps. For every ithstep the volume of liquid metal inside the cavity is increased of 1 cm3. In each step, we measured the scattering parameters of the system using a Network Analyzer, which is connected to the microwave cavity by two loop antennas (see Fig.(2)). For each variation of the cavity volume, we simulated the system by CST Microwave Studio to nd the resonant frequencies, and the electromagnetic eld of the TM102 cavity mode. In Fig.(3) we report an example of the elds in the cavity without liquid metal. While in Fig.(4) we show the measurements of the module of the transmission coecient (S12) at dierent volume of liquid metal, taken with the Network Analyzer.Microwave cavity tuned with liquid metal 4 Figure 3. Simulation of the TM102 cavity mode elds: the top of the gure represents the magnetic eld, while the bottom part is the electric eld. Figure 4. Module of Transmission Coecient S12 of TM102 mode at dierent volume of liquid metal. Now we were able to compare the frequencies measured with the simulated ones. The results are reported in Tab.(1).Microwave cavity tuned with liquid metal 5 Table 1. Results of the dierent measurements and simulations. The columns represents the cm3of metal injected into the cavity, the residual volume of the partially lled cavity, the simulated ( fth i) and the measured ( fi) resonance frequencies, and the corresponding measured loaded Qfactor. [cm3]aibd[mm3]fth i[GHz] fi[GHz] Q(L) i 0 30.01060 7.066 7.093 2728 1 28.31060 7.282 7.333 2156 2 26.71060 7.516 7.580 2106 3 25.01060 7.805 7.806 2296 4 23.31060 8.146 8.159 2039 As expected, the resonant frequencies change in good agreement with the simulations and the loaded quality factors Q(L) iof the modes does not dier drastically from those of the empty cavity. We have thus shown that it is possible to tune the resonance frequencies of a cavity lled with a liquid metal. Now we demonstrate that the system can be used for a physical application like an Electron Paramagnetic Resonance (EPR) experiment. In this way we tested the coupling of the cavity with a ferromagnetic resonance of a magnetic material placed inside. The cavity is equipped with a magnetized sample of volume VSand then placed inside an electromagnet that generates a static magnetic eld B0. In a ferromagnetic material, electron spins tend to align parallel to the external magnetizing eld B0and the Larmor frequency of the ferromagnetic resonance reads fL= B0 2, where is the electron gyromagnetic ratio. If the total number of spin of the sample is sucient, when the Larmor frequency and the cavity resonance coincide, the resonant mode of the system splits in two separate modes (hybridization). The mode separation is given by the total coupling strength g=g0pnSVS, wherenSis the spin density of the material, VSis the volume of the material, g0= q 0}!L VCis the single spin coupling, VCis the volume of the cavity and !L= 2fL[9]. To measure the hybridization, with the loop antennas, in addition to the eld B0, an RF eld B1is applied to the sample in a direction orthogonal to B0. We placed a 1 mm diameter Yttrium Iron Garnet (YIG) sphere in the center of the cavity, where the RF magnetic eld B1is maximum (see Fig.(3)). This material has very high spin density nS= 2
1028m3. The volume of the material is VS=4 3r3= 0:52
109m3, so the total expected coupling strength is g= 3:24
109g0. We also performed 5 measures, labelled with j, for the system cavity plus YIG. For everyj= 1;2;3;4;5 step the volume of liquid metal inside the cavity has been increased of 1 cm3. We measured the scattering parameters of the system using the Network Analyzer. The results of measurements are reported in Tab.(2), and in Fig.(5) we show the measurements of the module of the transmission coecient (S12) of hybrid system at dierent volume of liquid metal.Microwave cavity tuned with liquid metal 6 Table 2. Results of the dierent measurements in the strong coupling regime. The columns represents the cm3of metal injected into the cavity, the applied magnetic eld, the lower ( f< i) and higher ( f> i) hybrid resonances, and the respective Q factors. [cm3]Bz[T]f< i[GHz] Q(L)< i f> i[GHz] Q(L)> i 0 0.253 7.071 4159 7.117 4448 1 0.262 7.315 3325 7.353 3501 2 0.270 7.561 3979 7.599 2923 3 0.279 7.778 3709 7.824 3556 4 0.291 8.141 3540 8.174 3027 Figure 5. Module of Transmission Coecient S12 of hybrid resonances at dierent volume of liquid metal. If we consider, for example, the cavity without liquid metal, the single spin coupling isg0= 0:016 Hz, so the total coupling strength is g= 5:18
107Hz. The total coupling strength measured is g= 4:60
107Hz, which are comparable within about 10%. The system works as expected, and for all the dierent levels of liquid metal we were able to obtain hybridization. This demonstrates the capability of our tunable resonant system of performing EPR measurement. In conclusion, in this note we introduce a new method to tune the frequencies of the modes of cavities using liquid metal, and how it can be exploited in an EPR application. This process has been veried in the 7 8 GHz frequency range, however we can sweep over dierent frequency ranges by changing the geometry of the empty cavity. Acknowledgments We wish to thank doctor Antonio Barbon for helpful discussions concerning the experiment.Microwave cavity tuned with liquid metal 7 References [1] Pozar D 2004 Microwave Engineering (Wiley) ISBN 9780471448785 [2] Liu X, Katehi L P B, Chappell W J and Peroulis D 2010 Journal of Microelectromechanical Systems 19 774-784 ISSN 1057-7157 [3] Stefanini R, Chatras M, Pothier A, Orlianges J C and Blondy P 2009 High q tunable cavity using dielectric less rf-mems varactors 2009 European Microwave Integrated Circuits Conference (EuMIC) pp 391-394 [4] Perigaud A, Pacaud D, Delhote N, Tantot O, Bila S, Verdeyme S and Estagerie L 2016 Frequency- tunable microwave-frequency wave lter with a dielectric resonator including at least one element that rotates uS Patent 9,343,791 [5] E K 1969 Tunable microwave cavity using a piezoelectric device uS Patent 3,471,811 [6] C Carvalho N, Fan Y and Tobar M 2016 Review of Scientic Instruments 87 094702 [7] Sakaguchi J, Gilg H, Hayano R, Ishikawa T, Suzuki K, Widmann E, Yamaguchi H, Caspers F, Eades J, Hori M, Barna D, Horvth D, Juhsz B, Torii H and Yamazaki T 2004 Nuclear Instruments and Methods in Physics Research Section A: Accelerators, Spectrometers, Detectors and Associated Equipment 533 598 { 611 ISSN 0168-9002 URL http://www.sciencedirect.com/science/article/pii/S0168900204014639 [8] Carter P S 1961 IRE Transactions on Microwave Theory and Techniques 9 252{260 ISSN 0097- 2002 [9] Tabuchi Y, Ishino S, Ishikawa T, Yamazaki R, Usami K and Nakamura Y 2014 Phys. Rev. Lett. 113(8) 083603

2018-04-10

This note presents a method to tune the resonant frequency $f_{0}$ of a rectangular microwave cavity. This is achieved using a liquid metal, GaInSn, to decrease the volume of the cavity. It is possible to shift $f_{0}$ by filling the cavity with this alloy, in order to reduce the relative distance between the internal walls. The resulting modes have resonant frequencies in the range $7\div8\,$GHz. The capability of the system of producing an Electron Paramagnetic Resonance (EPR) measurement has been tested by placing a 1 mm diameter Yttrium Iron Garnet (YIG) sphere inside the cavity, and producing a strong coupling between the cavity resonance and Kittel mode. This work shows the possibility to tune a resonant system in the GHz range, which can be useful for several applications.

Microwave cavity tuned with liquid metal and its application to Electron Paramagnetic Resonance

1804.03443v1

1Submitted for publication in Physical Review B January 29, 2009 Theory of coherence in Bose -Einstein condensation phenomena in a microwave driven interacting magnon gas Sergio M. Rezende Departamento de Física, Univ ersidade Federal de Pernambuco, Recife, PE 50670 -901, Brazil Strong experimental evidences of the formation of quasi -equilibrium Bose -Einstein condensation (BEC) of magnons at room temperature in a film of yttrium iron garnet (YIG) excited by microwave r adiation have been recently reported. Here we present a theory for the magnon gas driven by a microwave field far out of equilibrium showing that the nonlinear magnetic interactions create cooperative mechanisms for the onset of a phase transition leading to the spontaneous generation of quantum coherence and magnetic dynamic order in a macroscopic scale. The theory provides rigorous support for the formation of a BEC of magnons in a YIG film magnetized in the plane. We show that the system develops cohere nce only when the microwave driving power exceeds a threshold value and that the theoretical result for the intensity of the Brillouin light scattering from the BEC as a function of power agrees with the experimental data. The theory also explains quantita tively experimental measurements of microwave emission from the uniform mode generated by the confluence of BEC magnon pairs in a YIG film when the driving power exceeds a critical value. PACS numbers: 75.30.Ds, 03.75.Nt, 05.30.Jp I. Introduction In a recent series of papers Demokritov and co -workers have reported remarkable experimental evidence of the formation of Bose -Einstein condensation (BEC) and related phenomena in a magnon gas driven by microwave radiation [1 -6]. Bose -Einstein condensation, a phenomenon that occurs when a macroscopic number of bosons occupies the lowest available quantum e nergy level [7 ], has only been unequivocally observed in a few physical systems, such as superflu ids [7 ], excitons and biexcitons in semiconductors [8,9], atomic gases [10 ] and certai n classes of quantum magnets [11 ]. BEC phenomena usually takes place by cooling the system to very low temperatures. The room temperature experiments reported in [1 -6] ha ve ingeniously materialized earlier proposals for producing Bose -Einstein condensation of magnons [ 12,13] and demonstrated powerful techniques for observing its unique properties. The experiments were done at room temperature in epitaxial crystalline films of yttrium -iron garnet (YIG) magnetized by an applied in -plane field. In these films the combined effects of the exchange and magnetic dipolar interactions among the spins produce a dispersion relation (frequency ωk versus wavevector k) for magnons propag ating at angles with the field smaller than a critical value that has a minimum 0k at 0k ~ 105 cm-1. In bulk samples the dispersion relation has the usual parabolic shape with a minimum at k = 0, where the density of states vanishes. In films the energy minimum away from the Brillouin zone center produces a peak in the density of states at 0k, providing an important condition for the formation of the condensate. 2The experiments reported in [1 -6] employ a microwave magnetic field with pumping frequency pf= 8.1 GHz applied parallel to the static field in the so -called parallel pumping process [ 14,15] to drive magnons in YIG films magnetized in the plane. In some of the latest experiments reported [4,5], short microwave pulses (30 ns) are used to create a hot magnon gas, allowing its evolution to be observed with time resolved Brillouin light scattering (BLS). Several important features are observed with increas ing microwave pow er. Initially, when the power exceeds a first threshold value, there is a large increase in the population of the parametric magnons with frequency in a narrow range around 2/pf = 4.05 GHz. Then the energy of these primary magnons redis tributes in about 50 ns through modes with lower frequencies down to the minimum frequency 2/0 min k f = 2.9 GHz (for H = 1.0 kOe) as a result of magnon interactions that conserve the number of magnons. This produces a hot magnon gas that remai ns decoupled from the lattice for over 200 ns due to the long spin -lattice relaxation time. The BLS spectrum in this time span reflects the shape of the magnon density of states weighted by the appropriate the rmal distribution exhibiting a peak at the freq uency minf. Thereafter this peak decays exponentially in time in the range of several hundred ns due to the thermalization with the crystal lattice. However if the microwave power exceeds a second threshold value, much larger than the on e for parallel pumping, two striking features are observed, namely, the decay rate of the BLS peak at minf doubles in value while its intensity increases by two orders of magnitude. The behavior of the BLS peak was attributed to a cha nge in the magnon state from incoherent to coherent, indicating the formation of a room -temperature BEC of magnons [4,5]. Coherence of photon fields has a formal quantum treatment developed by Roy J. G lauber over four decades ago [16 ]. Coherent magnon states, introduced in analogy with the photon states also have a formal quantum treatment [ 17,18]. In this paper we show that an interacting magnon system in a YIG film driven by microwave radiation develops a spontaneous coherent state with properties that expla in the main features of the experimental observations. Since the coherent state corresponds to a quantum macrosc opic wavefunction, the theory provides rigorous support for the existence of Bose -Einstein condensation of magnons in the experiments of [1 -6]. Note that r ecently it has been argued [19 ] that the intermagnon interactions in a YIG film magnetized in the plane prevents the conditions for stabilization of the BEC. Contrary to the conclusions of [19 ], we show that the magnon -magnon interactions play a n essential role in the formation of the BEC at room -temperature in a YIG film driven by microwave radiation as in the experiments of Demokritov and co -workers [1 -6]. In another recent paper of the same group, Dzyapko et al. [6] show that if the applied in-plane static field has a value such that the frequency of the 0k magnon is 0= 20k, a microwave radiation signal is generated by 0k magnons created by pairs of BEC magno ns 0 0,k k through a three -magnon confluent process. The 0k value is necessary for emission because the wavenumber of electromagnetic radiation with frequency 1.5 GHz, as in the experiments [6], is f k /2 0.3 cm-1. In an earlier paper [20] we have show n that the 0k magnons created by the BEC are coherent magnons states, corresponding to a nearly uniform magnetization precessing with frequency 0 and generating a microwave signal. The microwave emission from the collective action of the spins is identified with superradiance. Here we present other aspects of the theoretical model for this phenomenon and show that the predicted radiated signal power agrees with the ex perimental data [6]. The paper is organized as follows. In Sec. II we discuss the nature of the spin -wave modes in thin films based on the results of earlier work by several authors, in order to establish the background for the remainder of the paper. Sec. III is devoted to a review of the properties of coherent magnons states. In Sec. IV we discuss the excitation of spin waves in films by the parallel pumping technique. Sec. V is dev oted to the proposed cooperative mechanism for the formation of the BEC of magnons. In Sec. VI we show that the states resulting from the cooperative action have quantum coherence. In Sec. VII we show th at the results of the model agree with experimental data for the intensity of the Brillouin light scattering from the BEC and f or the microwave emission from the uniform mode resulting from the coalescence of a pair of BEC magnons. Sec. VIII summarizes the main results. 3II. Spin -wave modes in thin films Since the pioneering work of Damon and Eshbach [21] the theory of spin waves in ferromagnetic films has been studied and reviewed by many authors [22 -31]. The theory of Damon and Eshbach (DE) was developed for waves with very small wavenumbers k that have energies with negligible contribution from the exchange interaction between t he spins. They used a semi -classical approach for the equation of motion for the magnetization in which the magnetic dipolar field plays a dominant role. This field w as obtained with the so -called magnetostatic approximation valid for wavenumbers much larg er than the values for the electromagnetic field ( k ~1 cm-1) so DE coined the term magnetostatic waves to the resulting wave solutions. Later several authors included the exchange interaction in the equations o f motion and in the boundary conditions, used various approaches and approximations to find the normal modes and introduced other names to the waves, such as dipole -exchange waves. Actually they are all simply spin waves, pictured classically by the view of the spins precessing about the equilibrium d irection with a phase that varies along the direction of propagation. Various results have been successfully applied to explain experimental observations in thin slabs or films of YIG and other low loss ferrite materi als as well as in ultrathin films of fe rromagnetic metals. In this section we present the background information on th e normal spin -wave modes in a thin ferromagnetic film necessary for the discussion of the theory of the interacting spin waves. Initially we employ the DE theory extended to inc lude exchange in order to obtain exact dispersion relations for waves in films corresponding to the nearly uniform transverse mode. Then we develop a quantum model based on the second quantization of the spin excitations involving magnon creation and annih ilation operators which is the most convenient approach to treat interactions. Consider an unbounded flat ferromagnetic film with thickness d magnetized in the plane by a static magnetic field H. We use a coordinate Cartesian system with the x and z coordi nates in the plane of the film, zˆ along the field direction. Anisotropy is neglected since it is very small in YIG so that the magnetization M in equilibrium lies along zˆ and one can write y x z my mx MzM ˆ ˆ ˆ . The DE approach consists of solving the Landau -Lifshitz equations of motion for the small -signal time -varying components of the magnetization xm and ym under the action of the magnetic dipol ar field they create added to the static field H [21]. Furthermore it is assumed that xm and ym are described by waves with frequency ωk and wavevector k propagating in the film x-z plane and a standing wave pattern in the perpendicular direction. The corresponding dipolar field dh can be obtained from Maxwell’s equations in the magnetostatic approximation 0dhx , which allows expressing the field in terms of a magnetic potential as dh . The equation for follows from 0) 4 (. m hd and its solutions are subjected to the electromagnetic boundary conditions involving the internal and external fields on the two surfaces of the film. One then obtains a transcendental equation relating the frequency ωk with the wavevector components [21,27 -29], 01 sin ) 1( ) cot()() 1(22 2 2 2/1 k ydk (1) where k is the angle between the wavevecto r k in the plane and the z-direction, ky is the wavenumber characterizing the mode pattern in the direction normal to the film and the other parameters are rel ated to the frequency by 2 2 k HM H , 2 2 k HM k and (2) 1sin 12 k , (3) where HH , MM 4 and = gB/ is the gyromagnetic ratio (2.8 GHz/kOe for YIG). Note that the components of the wavevector k in the plane enter in (1) -(3) through k xkksin and k zkkcos . 4From the equation for the potential in the film one can see [21,29] that the transverse wavenumber ky is related to the wavenumber k in the plane by k ky2/1)( . (4) It follows that for each pair of values of kx, kz , or equivalently k, θk, Equation (1) has several solutions for the frequency ωk, each corresponding to a different tranverse mode pattern characterized by a discrete ky. From (2) -(4) it is clear that ky can be real or imaginary, depending on the range of frequency. Real value s of ky correspond to the so -called volume magnetostatic modes, for which the magnetization components have a dependence on the transverse coordinate y of the type cos kyy, sin kyy. Imaginary values of ky correspond to the surface modes, which have an expo nential dependence on y decaying away from one of the film surfaces. The surface modes have a unique property of being non -reciprocal, in the sense that the wave associated with one surface propagates only in one direction but not in the oppo site [21,22,27 -29]. From (2) -(4) it can be shown [27 -29] that for each frequency there is a critical angle of propagation θkc above which δ becomes positive so that ky is imaginary and the mode is a surface wave, 2/12 2 ) ( sin HH k kc . (5) For typical numbers appropriate to the experiments of [1 -6] with YIG films, H = 1.0 kOe, M4 = 1.76 kG, 2/k f = 4.0 GHz, the critical angle is θkc = 50.26o. For the specific case of the surface wave with θk = 90o, Equation (1) has a simple solution with an explicit dependence of the frequency on the wavevector given by [29], ) 1(412 2 2 2 kd M M H H k e . (6) The introduction of the exchange interaction complicates considerably the problem of finding the spin-wave normal modes in films. Fi rst of all one can see that in films with thickness on the order of 1 µm or less, the exchange introduces a sizeable separation in the frequencies of the volume modes with different transverse patterns because ky ~ ny π /d and the exchange energy varies with the square of ky. The exact solution of the wave equations must in volve the matching of mixed electromagnetic and exchange boundary conditions [24 -26]. A nearly exact expression for the frequency of the lowest lying exchange branch can be obtained by simply introducing the exchange interaction as an effective field in Equations (1) -(4) which is added to the applied field, so that the parameter ωH becomes, ) (2kDHH , (7) where BgaSJ D / 22 is the exchange stiffness, J being the nearest neighbor exchange constant and a the lattice parameter of the film. The dispersion relations obtained by solving numerically Equ ations (1) -(4), with ωH as in (7), are shown by the solid lines in Figure 1 for several angles θk in two YIG films with thickness d = 0.1 µm and 5 µm, using H = 1.0 kOe, M4 = 1.76 kG, and D = 2 x 10-9 Oe.cm2. The main feature of the dis persion curves is that for propagation angles below certain values the frequency exhibits a minimum at a k value that depends on the thickness. This is a consequence of the fact that the frequency initially decreases with increasing k due to the role of th e dipolar energy but then at larger values of k it changes slope due to the effect of exchange as in (7). In the quantum approach which will be used to treat interactions we use a Hamiltonian in the form, )('int 0 tH H HH , (8) where H0 is the unperturbed Hamiltonian that describes free magnons, Hint represents the nonlinear magnetic interactions and )('tH represents the external microwave driving. The magnetic Hamiltonian can be written as H = HZ + Hexc + Hdip, represen ting respectively, the Zeeman, exchange, and dipolar contributions. We treat the quantized excitations of the magnetic system with the approach of Holste in- Primakoff [32 -35], which consists of three transformations that allow the spin operators to be expresse d in terms of boson operators that create or destroy magnons. In the first transformation the components of the local spin operator are related to the creation and annihilation operators of spin deviation at site j, denoted respectively by jaand ja, which satisfy the boson commutation rules ij j iaa],[ and 0],[j iaa . Using a 5 coordinate system with zˆ along the equilibrium direction of the spins, def ining y jx j j iSSSand y jx j j iSSS, where the factor i is the imaginary unit, not to be confused with the subscript denoting lattice site i, it can be shown that the relations that satisfy the commutation rules for the spin components and the boson operators are [32 -34] jj j j a SaaS S2/1 2/1) 21()2( , (9a) 2/1 2/1)21( )2(Saaa S Sj j j j , (9b) j jz j aaSS , (9c) where S is the spin and j j j aan is the operator for the number of spin deviations at site j. One of the main advantages of this approach is that the nonlinear interactions are treated analytically by expa nding the square root in (9a) and (9b) in Taylor series. We use only the first two terms of the expansion, so that )4/ ()2(2/1S aaa a S Sj j j j j (10a) and )4/ ()2(2/1S aaa a S Sj j j j j . (10b) In order to find the normal modes of the system we use the linear approximation, whereby only the first terms in (9c) and (10) are kept, i.e., j j a S S2/1)2(, j j a S S2/1)2( , and SSz j. With these transformations one can express the magnetic Hamiltonian in a quadratic form containing only lattice sums of products of two boson operators. The second step is to introduce a transformation from the localized field operators to collective boso n operators kaand ka using the Fourier transform kkrki j aeNa . 2/11 (11) where N is the number of spins in the system. The condition that the new collective operators satisfy the boson commutation rules ', '],[kk k kaa and 0],['k kaa , requires that the transformation coefficients satisfy the usual orthonormality relations. The contributions from the Zeeman and exchange energies to the Hamiltonian H0 with quadratic form in boson operat ors can be shown to be [32 -35] kk k exc Z aakDH H H ) (2 . (12) 10310410510623456789Figure 1 (a) 40o 20o (a) H = 1.0 kOe d = 0.1 m k = 90o k = 0 Frequency f = / 2 (GHz) Wavenumber k (cm-1)10110210310410510623456789Figure 1 (b) f = 4.05 GHz60o 50o 40o 20o (b) H = 1.0 kOe d = 5.0 m k = 90o k = 0 Frequency f = / 2 (GHz) Wavenumber k (cm-1) Figure 1: Dispersion relations for spin waves propagating at various angles with the in -plane applied field H = 1.0 kOe in a YIG film with thickness (a) 0.1 m and (b) 5 m. The curves with full lines represent the calculation with the DE theory inclu ding exchange, Equation (1) while the dotted lines represent the calculation with the approximate theory Equation (26) for θk = 0, 20o and 40o. 6 The contribution of the dipolar energy to the Hamiltonian can be obtained with approximations valid for the nearly uniform transverse mode, which corresponds to the lowest lying exchan ge branch with ky ~ 0. Following [30 -31] we neglect the variation of the magnetization on the transverse coordinate and work with the averages over y, 2/ 2/, , , );,(1);( );,(d dyx yx yx dytzx mdtr mtzx m . (13) The magnetic potential created by the spat ial variation of the small -signal transverse components of the magnetization is written in the form [30,31], krki keyVzyx . 2/1)(1),,( (14) where V is the volume of the film and k and r denote the wavevector and the position vector in the plane. The Fourier transform of the potential )(yk can be obtained from the solution of m. 42 derived from Maxwell’s equations subject to the electromagnetic boundary conditions at 2/d y [31], )(1)(sinh 4)( ]1)( cosh [4)(2/ 22/kmkky e kmkkky ei yykd xx kd k (15) where the Fourier components of the magnetization appearing in (15) can be expressed in terms of the collective boson operators using the relation yx B yx SVN g m, , )/( in (10) and (11), ) ()2( )(2/1 k k x aaVNSkm , (16a) ) ()2( )(2/1 k k y aaVNSi km . (16b) The small -signal transverse components of the dipolar field can be obtained from the magnetic potential with dh so that the contribution of the dipolar energy to the magnetic Hamiltonian can be calculated with ) (21dip y ydip x x dip hm hmdzdydx H . (17) The integration in (17) can be performed without difficulty by expressing the magnetization and the dipolar field in terms of their Fourier transforms and using the orthon ormality relations. One can show that, kk k k k k k k k k k dip ch aaF F aaF F M H }.. ] sin) 1([{21] sin) 1([ 22 2 . (18) With (12) and (18) one can write the total Hamiltonian for the free magnon system as, kk k k k k k k k k aaB aaB aaA H* * 021 21 (19) where ] 2 sin) 1( 2 [2 2 k k k k FM F M DkH A , (20a) ] 2 sin) 1( 2[2 k k k k FM F M B , (20b) kd e Fkd k /) 1( . (20c) In order to diagonalize the quadratic Hamiltonian it is necessary to introduce new collective boson operators kcand kc satisfying the commutation rules ' '],[kk k kcc and 0],['k kcc , related to kaand kathrough the Bogoliubov transformation [32-34] ` k k k k k cvcua , (21a) k k k k k cv cua* , (21b) 7where 12 2k kvu , as appropriate for a unitary transformation. The coefficients of this transformation must be such that the quadratic Hamiltonian acquires the diagonal form kk k kcc H 0 , (22) because this leads to the Heisenberg equation of moti on k k kkci Hci dtdc ],[1 0 . (23) This equation has stationary solutions of the form tkie which assures that kc is the operator for the normal -mode excitations of the magnetic system. Hence kc and kc are the creation and annihilation operators for magnons. It can be shown [32-34] that the coefficients of the transformations (21) are given 2/1)2( kk k kAu , (24a) and 2/1 2/1 2)2( )1 ( kk k k kAu v , (24b) where the sign of vk in (24b) is the opposite one of the parameter Bk and the frequency k of the eigenmodes is 2/122) (k k k B A . (25) Using the expressions for the parameters in (20) we obtain from (25) an explicit equ ation for the dependence of the spin -wave frequency on the wavevector in the plane, ] 4 [] sin) 1( 4 [2 2 2 2 2 k k k k FM DkH F M DkH (26) This equation is the same as the one obtained for the lowest lying branch of the “dipole -exchange” modes with more rigorous treatment o f the exchange interaction [24 -26]. It also agrees with the results of [30,31] in the limit kd << 1. The dispersion curves shown by the dotted lines in Figure 1 are obtained with (26). The agreement with the Damon -Eshabch result extended to include exchang e is quite good for any angle θk in the YIG film with d = 0.1 µm. In the case of the film with d = 5 µm the agreement is good for θk < 50o. The results in Figure 1 shows that the second quantization approach just presented describes quite well the nearly uniform transverse spin -wave mo de in films. To conclude this section we express the components of the magnetization vector operators in terms of the magnon creation and annihilation operators using the relations with the spin operators and the transformations (10), (11) and (21), ) )( ()2()(* . 2/1 k k k k krki x ccvu eNSMrm , (27a) ) )( ()2()(* . 2/1 k k k k krki y ccvueNSMi rm , (27b) With these equations one can calculate the expectation values of the magnetization components for any spin excitation in films expressed in terms of the magnon states. III- Coherent m agnon states If the nonlinear interactions are neglected, the spin -wave excitations with wavevector k and frequency k described by magnon creation and annihilation operators kcand kcform a sy stem of independent harmonic oscillators, governed by the unperturbed Hamiltonian kk k kcc H 0 . The eigenstates kn of this 8Hamiltonian which are also eigenstates of the number operator k k k ccn can be obtaine d by applying integral powers of the creation operator to the vacuum, 0])!/()[(2/1 kkn k k n c n , (28) where the vacuum state is defined by the condition 00kc . These stationary states describe systems with a precisely defined number of magnons kn and uncertain phase. They form a complete orthonormal set which can be used as a basis for the expansion of any state of spin excitation. They are used in nea rly all quantum treatments of thermodynamic properties, relaxat ion mechanisms, and other phenomena involving magnons. However, as can be seen from the expressions in (27), they have zero expectation value for the small -signal transverse magnetization operators xm and ym and th us do not have a macroscopic wavefunction. In order to establish a correspondence between classical and quantum spin waves one should use the concept of coherent magnon states [ 17,18], defined in analogy to the coherent photon states introduced by Glauber [16]. A coherent magnon state is the eigenket of the circularly polarized magnetization operator y xim m m. It can be written as the direct product of single -mode coherent states, defined as the eigenstates of the annihilation operator, k k k kc , (29) where the eigenvalue k is a complex number. Although the coherent states are not eigenstates of the unperturbed Hamiltonian and as such do not have a well defined number of magnons, they have nonzero expectation values for the magnetization mwith a well defined phase. Here we review a few important properties of the coherent states. First we recall that they can be expanded in terms of the eigenst ates of the unperturbed Hamiltonian [16-18], k knkkn kk k n n e2/122/1)!/()( . (30) The probability of finding kn magnons in the coherent state k obtained directly from (30) is given 2 2 2)!/ ( )(k kkn k k k k coh en n n . (31) This function is a Poisson distr ibution [1 6] that exhibits a peak at the expectation value of the occupation number operator 2 k kn in the coherent state. It can be shown that coherent states are not orthogonal to one another, but they form a complete set, so that they c onstitute a basis for the expansion of an arbitrary state. The distribution (31) is very different from the one prevailing in systems in thermal equilib rium, which cannot be described by pure quantum states. Instead they are described by a mixture in which one can find any number of magnons kn with energy k. The average number of magnons with energy k in thermal equilibrium at a temperature T is given by the Bose -Einstein distribution 11 /TBkkken (32) where Bk is the Boltzmann constant. The probability of finding kn magnons with energy k in the mixture describing the thermal equilibrium with the average value (32) can be shown to be [1 6] 1) 1()()( kn kkn k k thnnn . (33) Note that for large kn Equation (33) approaches the exponential function ) exp(kn . To stress the difference between the coherent state and the mixture descr ibing the thermal equilibrium we show in Figure 2 the distributions (31) and (33) corresponding to 50kn . 9 Figure2: Distributions of magnons in a system in thermal equilibrium and in a coherent state with 50kn . Another important property of a coherent state is that it can be generated by the application of a displacement operator to the vacuum [1 6-18], 0)(k kD , (34a) where ) exp()(* k k k k k c c D . (34b) In order to stud y the coherence properties of a magnon system, it is convenient to use the density matrix operator ρ and its representation as a statistical mixture of coherent states, k k k k d P 2)( , (35) where )(kP is a probability density, called P representation, satisfying the normalization condition 1 )(2k kd P and ) (Im) (Re2 k k k d d d . As shown by Glauber [1 6], if ρ corresponds to a coherent state, )(kP is a Dirac δ -function. On the other hand, if ρ represents a thermal Bose -Einstein distribution, )(kP will be a Gaussian function. To conclude this section it is important to obtain the expectation values of the components of the magnetization operators for a single coherent state with eigenvalue ) exp(k k k i . Using (29) in the expressions (27) it is straightfor ward to show that ) .(cos) ()2/(),( 2/1 k k k k k x t rk vuNSMtrm , (36a) ) .(sin) ()2/(),( 2/1 k k k k k y t rk vuNSMtrm . (36b) The transverse components of the magnetization in (36) together with zMzˆ correspond to the classical view of a spin wave, namely, the magnetizati on precesses around the equilibrium direction with a phase that varies along the direction of propagation and with an ellipticity given by kk k k kk k yx BA vuvu mm ) ( maxmax . (37) Note that the elliptical precession of the transverse magnetization with freq uency k results in an oscillation of the z-component with frequency k2. As is well known it is this fact that makes possible to excite spin -waves with a microwave field parallel to the static field. 0 20 40 60 80 1000.000.020.040.060.08Figure 2 <ncoh> = 50 <n th> = 50Magnon distribution Number of magnons 10 IV- Microw ave excitation of spin waves Spin waves can be nonlinear excited in a magnetic material by means of several techniques employing microwave radiation, with the microwave magnetic field applied either perpendicular or parallel to t he static field. The exci tation is provided by the oscillation in the coupling parameter between two or more magnon modes, so the processes are called parametric. As in other nonlinear processes, the excitatio n occurs when the driving field exceeds a certain threshold value which depends on the rate at which the magnon mode relaxes to the heat bath. In the parallel pumping process the driving Hamiltonian in (8) follows from the Zeeman interaction of the microwave pumping field ) cos(ˆ t hzpwith the magnetic system. One ca n express the Zeeman interaction in terms of the magnon operators using (9c), (11) and (21) and keeping only terms that conserve energy and show that the driving Hamiltonian for a ferromagnetic film is given by kk ktpi k ch cceh tH ..2)(', (38a) where k k k k M k k k F F vu 4/] sin) 1([2 (38b) represents the coupling of the pumping field h (frequency p) with the kk , magnon pair with frequency k equal or close to 2/p . Note that for a thick film, or a large wavevector, or a combination of both such that kd >> 1 Equation (20c) gives Fk << 1. In this case the coupling coefficient approaches the value for bulk samples k k M k 4/ sin2 . This is maximum for waves propagating perpend icularly to the field since they have the largest ellipticity and vanishes for waves with k along the field. However in films with kd on the order of 1 or less Fk is finite and the parallel pumping field can drive waves with any value of k. This is what happens in the case of the experiments in [1 -6] with H = 1.0 kOe. As seen in Figure 1 (b) in a YIG film with d = 5 µm magnons with frequency 4.05 GHz and k = 0 can have two values for k, approximately 2 x 103 cm-1 and 5 x 105 cm-1. The first value corresponds to 1kd and 6.0kF and the second to 250kd and 0~kF . This means that magnons with frequency 4.05 GHz and k ~ 2 x 103 cm-1 with k = 0 have a finite ellipticity and can be parallel -pumped. In fact, as can be seen in Figure 1 (b), for H = 1.0 kOe only waves with k in the range from 0 to about 50o can be pumped at 2/pf = 4.05 GHz. It turns out that as k increases with fixed frequency the wavevector k increases so Fk decreases. In a film with thickness d = 5 µm this approximately compensates the increase in the k2sin term so that the factor k kvu which determines the parallel -pumping coupling remains about 0.2 in the whole range of k, 0 -50o. The Heisenberg equation of motion for the operators kcand kc with the Hamiltonian )('0 tH HH given by (22) and (38) can be easily solved assuming that the pumping field is applied at t = 0 to g ive the evolution of the expectation value of the number of magnons, t k kke n tn2)0( )( , (39a) where k k k k h 2/12 2] )([ , (39b) )0(kn is assumed to be the thermal number of magnons, 2/p k k is the de tuning from the frequency of maximum pumping strength and k is the magnon relaxation rate which was introduced phenomenologically in the equations of motion. Equations (39) express the well known effect of the parallel -pumping excita tion. Magnon pairs with frequency k equal or close to 2/p and wavevectors kk , determined by the dispersion relation 11 are driven parametrically and their population grow exponentially when the fie ld amplitude exceeds a critical value hc, given by the condition 0k in (39b), k k k ch /) (2/1 2 2 . (40) The large increase in the magnon population enhances the nonlinear interactions causing a reaction that limits its grow th. Due to energy and momentum conservation the important mechanism in this process is the four -magnon interaction, which can be represented by a Hamiltonian of the form [32 - 36] ' ' ',' ' ' ' 21 )4((k k k k kkkk k k k k kk ccccT ccccS H ) , (41) where the interaction coefficients are determined mainly by the dipolar and exchange energies. For the k- values relevant in the experiments [1 -6] the contribution from the exchange energy is negligible compared to the dipolar [33]. The four -magnon dipolar Hamiltonian can be obtained from (17) using for m and diph the first and second terms of the expansions in (10), following procedures similar to those in Sec. II and keeping only terms with two creation and two annihilation magnon operators. The result has several terms with coefficients containing the form factor Fk in (20c) and products of the parameters uk and vk in (24), as given in Ref. [ 19]. It turns out that for the conditions of the experiments, Fk << 1, uk ~ 1 and vk << 1, so that the coefficien ts in (41) are given approximately by NS T SM kk kk / 2 2' ' . Using the Hamiltonian (8) with (41) as the interaction term one can write the Heisenberg equations for the operators kcand kc from which several quanti ties of interest can be obtained. One of them is the correlation function k defined by [36], tki ki k k k k een cc 2 , (42) where kn is the magnon number operator and k the phase between the states of the pair. From the equation of motion for k it can be shown that for h > hc, in steady -state [36 -38] )4(2/1 2 2 2] ) [( Vhnk k k ss k , (43) where NS T S VM kk kk / 4 2)4( . (44) It can also be shown that the phase k varies from 2/ to as h increases from hc to infinity. In the range of pumping power of the experiments [1 -6] 2/ ~k . By using methods of quantum statistical mechanics and t he probability density defined in (35) it has been demonstrated that the magnon pairs excited by parallel -pumping are in coherent magnon states but this is so only when the four -magnon interaction is taken into account [38]. Equation (43) shows that magn on pairs with frequency within a certain range around 2/p are pumped by the microwave field when its amplitude exceeds a critical value given by kk k ch2/1 2 2) ( . (45) Note that the population of the parametric magnons i s maximum for 2/p k and for the allowed k that maximizes k. The modes with 2/p k are excited when the field amplitude h is larger than a critical value k k ch/ . In the reported experiments the minimum hc corresponds to a critical power pc in the range of 100 µW to 1 mW determined by the experimental geometry and the spin -lattice relaxation rate in YIG, ~SL 2 x 106 s-1 [1-6]. However, when very short microwave pulses are used, much higher power levels are required to reduce the rise time and to build up large magnon populations. In this case i t is the larger magnetic relaxation rate, SL m 25~ 5 x 107 s-1 in the experiments [4,5], t hat must be overcome by 12 the driving. So one can define a critical field SL m c k m c h h / /1 for driving magnons with short pulses. Using the fact that the driving microwave power p is proportional to h2, we can write from (43) an expression for the ste ady-state number of parametric magnons with frequency 2/p k as a function of power, mc c ss kVp ppn/ 2]/) [( )4(2/1 1 1 (46) where 2 1 )/(SL m c cp p . Using numbers appropriate for the experiments [4 -5], pc = 100 µW, m= (1/20 ns) = 5 x 107 s-1, pc1 = 0.0625 W, M NSV 4)4( = 1.24 x 1011 s-1, for a driving power p = 4 W, Equation (46) gives for the normalized number of parametric magnons NS n ss k/ = 1.6 x 10-3. The number of magnons pumped by the microwave field is actually larger than this because many modes with frequency in the vicinity of 2/p are also driven. From (46) one can write an approximate equation for the total number of magnons pumped into the system as 2/1 1 1 ]/) [(c c H p p p pp nrn (47a) where M m m H NS V n 8/ 2/)4( (47b) and pr is a factor that represents the number of pumped modes weighted by a factor relative to the number of magnons of the mode with maximum coupling. V. Mod el for Bose -Einstein condensation in the microwave driven interacting magnons In the experiments of [1 -6] magnon pairs are parametrically driven by parallel -pumping in a YIG film at large numbers compared to the thermal values. The population of these pri mary magnons with frequency equal or close to 2/p is quickly redistributed over a broad frequency range down to the minimum frequency 2/0 min k f . This redistribution is caused by four -magnon scattering events which conserve the total number of magnons so that a quasi -equilibrium hot magnon gas is formed. Since the spin -lattice relaxation time in YIG is much longer than the intermagnon decay time, the hot magnon gas remains practically decoupled from the lattice for several h undred ns with an essentially constant number of magnons. In this situation the occupation number of the system is given by the Bose -Einstein distribution 11),,( /) ( TBk BEeT n (48) where is the associated chemical potential. As is well known [ 7] in systems with constant number of particles it is (48) and not (32) that determines the distribution of the number of bosons with ener gy at a given temperature T, provided the system is in equilibrium and there is no interaction between the bosons. The experiments of [1 -5] were done with 8.1 GHz microwave pumping in two types of pulsed regimes and the properties of the pumped magnon system were measured with time -resolved Brillouin light scattering. In the first one long pulses o f duration 1 s were employed to ensure that quasi - equilibrium was established in the hot magnon gas while still decoupled from the lattice. This made possible the observation of the full thermal equilibrium spectr a between fmin and the parametric magnon frequency of 4.05 GHz as a function of the microwave pumping power. The authors of [1 -5] argue that without external driving the magnons are in thermal equilibrium with the lattice and have uncertain number so that = 0. If a microwave driving is applied a nd the power exceeds the threshold for parallel pumping the total number of particles in the magnon gas increases and can be expressed as 13 dT n D NBE tot ),,()( (49) where )(D is the magnon density of states and the integral in (49) is carried out over the whole range of magnon frequencies. Clearly as the microwave power is raised the total number of magnons increases s o that the temperature and the chemical potential increase. Using (49) and the similar equation for th e energy of the system it is possible to determine the values of and T for a given Ntot. In the experiments with long pulses [1 -3] the BLS spectra could be fitted with the spectral density function ),,()( T n DBE , allowing the determination of and T for each power value. At a high enough power the chemical potential reaches the energy corresponding to fmin resulting in an overpopulation of magnons with that frequency relative to the theoretical fit. It was then necessary to add a singularity at fmin to fit the spectrum [2]. This was interpreted as a signature of the Bose -Einstein condensation of magnons, namely: when the number of magnons reaches a critical value defined by the condition min2fc the gas is spontaneously divided in two part s, one with the magnons distributed according to (48) and another one with the magnons accumulated in the state of minimum energy. The experiments with short microwave pulses (30 ns) [4,5] allowed the observation of the dynamics of the redistribution of e nergy from the primary magnons to the modes in the broader energy range and the formation of the strong BLS peak at fmin. The behavior of the peak intensity and of the relaxation to the lattice with increasing microwave pumping power revealed that above a critical power level the magnons accumulated at the bottom of the spectrum develop a spontaneous emergence of coherence. The coherence of the BEC was further confirmed in experiments showing the microwave emission from the k = 0 mode generated by the coal ition of a pair of BEC magnons when the applied field has a value for which its frequency is 2 fmin [6]. While the thermodynamic interpretation of the experiments in [1 -5] is quite satisfactory and explains qualitatively several observed features, it fails in providing quantitative results to compare with data and, most serious, it does not explain the obser ved spontaneous emergence of quantum coherence in the BEC of magnons. This is not surprising because a system of free noninteracting magnons cannot poss ibly evolve spontaneously from quantum states describing thermal magnons, represented by the distribution (33), to coherent magnons states corresponding to (31). The theory presented in this section shows that the cooperative action of the magnon gas throu gh the four -magnon interaction can provide the mechanism for the observed spontaneous emergence of quantum coherence in the BEC. The theory relies in part on some assumptions based on the experimental observations and on some approximations to allow an ana lytical treatment of the problem. The ultimate justification for the assumptions and approximations is the good agreement of the theoretical results with the experimental data for the BLS intensity and for the emitted microwave signal as a function of the microwave pumping power presented in th e next section . We consider that with microwave pumping the magnon system can then be decomposed in two sub-systems, one with frequency above 2/p in thermal equilibrium with the lattice at room te mperature and another one with frequency in the range 2/0 p k in quasi -equilibrium at a higher temperature T. The second sub -system, which we call the magnon reservoir, is characterized by an occupation number given by the Bose -Einstein dis tribution with its own temperature and chemical potential. We also assume that after the hot magnon reservoir is formed by the redistribution of the primary magnons, the corr elation between the phases of the magnon pairs lasts for a time that can be as lar ge as m/4 , which is about 100 ns in the experiments [1 -6]. This is a sufficient time for the four -magnon interaction to come into play for establishing a cooperative phenomenon to drive a specific k mode. The effective driving Hamiltoni an for this process is obtained from Equation (41) by taking averages of pairs of destruction operators for reservoir magnons to form correlation functions as defined in (42), Rkk ktRkiRki Rk Rkk ch cc eenS tH .. )('2 21 . (50) Equation (50) has a form that resembles t he Hamiltonian (38) for parallel pumping, revealing that under appropriate conditions magnon pairs can be pumped out of equilibrium in the gas. To treat (50) we note 14 that since the number of the magnons pn pumped into the system is muc h larger than the number of thermal magnons in the range 2/0 p k one can write for the magnon reservoir dT n D nBE p ),,()( , (51) where pn is related to the power as in (47). Of course the calculation of the population in each state of the reservoir as a function of power is a formidable task. So we use some approximations to treat the problem analytically . Consider that the population of the primary magnons is distributed among the RN modes Rk in the magnon reservoir, so that with (47) we can write an expression for the average population of modes Rk as a function of pumping power p, 2/1 1 1 ]/) [(c c H R p pp nr n , (52a) where R pNrr / . (52b) If all the reservoir states had the same magnon number the sum in Rk in (50) would reproduce the density of states D(ω). Actually the number of magnons in each state Rk depends on its energy as given by (48) and can be written approximately as R kR BE Rk n f n )( , where )(kR BEf is a function proportional to (48) with a normalizati on constant so that its average over the frequency range of the reservoir modes is unity, BE BE BE C n f /)( )( , (53a) dn CBE RBE1 , (53b) 0 2/k p R being the frequency range of the reservoir modes. Thus the relev ant quantity for determining the frequency dependence of the coefficient in the Hamiltonian (50) is the density of st ates weighted by the normalized Bose -Einstein distribution, )()( )( BEf D G . (54) Note that )(BEf and )(G also vary with and T but we omit them in the functions to simplify the notation. Figure 3 shows plots of (54) for several values of and the corresponding T for a 5 m thick YIG film. The density of states was calculated numerically using the approximate dispersion relation (26) by counting the number of states with z z z x x x L n kL n k /2 ,/2 having frequenc ies in discrete intervals 2x 1.0 MHz in the range 2/ 0p . The value of were chosen so that their differences to 0k are the same as the ones used in [3] to fit the measured BLS spectra with varying microwave power. The corresponding values of T were estimated by the fits to the BLS spectra in [3]. The dimensions used to calc ulate the density of states are Lx = Lz = 2 mm. As expected )(G has a peak at the minimum frequency that becomes sharper the chemical potential rises and approaches the minimum energy. The consequence of this is that as the microwave pu mping power increases and TkB k /) (0 becomes very small the peak in )(G dominates the coefficient in (50) revealing that it is possible to establish a cooperative action of the modes with frequency Rk close to 0k so as to drive magnon pairs nonlinearly as in the parallel pumping process. Considering that the pumping is effective for frequencies Rk in the range m k0 , the sum over Rk in (50) can be replaced by m kD)(0 so that one can write an effective Hamiltonian for driving 0 0,k k magnon pairs as, .. )()(' 0 00 2ch cc e h t Hk ktk i eff eff , (55a) where 2/ )( )()4( 0 R m k eff nV Gi h (55b) 15 Figure 3: )(G as a function of frequency for spin waves in a 5 m thick YIG film in a field H = 1.0 kOe with the following parameters: = 0, T = 300 K (lowest values at fmin = 2.898 GHz); / h = 2.718 GHz, T = 900 K; / h = 2.868 GHz, T = 120 0 K ( h is Plank’s constant). represents an effective field proportional to the average number of magnons Rn in the reservoir. Note that the factor –i in (55b) arises from the phase between pairs that is approximately 2/ in the range of power of interest. From the analysis in Sec. IV one can see that there is a critical number of reser voir modes above which they act cooperatively to pump the 0 0,k k magnons parametrically. The condition m effh)( gives the critical average number of reservoir magnons )( /20 )4( k c GV n . (56) Since the Hamiltonian (55) has the same form as (38), the population of the 0k mode driven by the effective field and saturated b y the effect of the four -magnon interaction is calculated in the same manner as done for the direct parallel -pumping process. Thus from (43) with 0k we have )4(2/1 22 02] )([ Vh nm eff k . (57) Using (47b), (55b) and (56) in Equati on (57) one can write the population of the 0k mode in terms of the average reservoir number Rn, 2/1 2 2 0 ) (c R cH k nnnnn . (58) Alternatively 0kn can be written in terms of the pumping p ower using (52) and (56) in (58), 2/1 1 2 2 0] (/) [(c c c H k p p pp n n , (59) where Hn is given by (47b) and }])( [/161{2 0 1 2 k m c c Gr p p (60) is another threshold power level pc2 >> pc1. Note that with (52) and (60) the effective driv ing field (55b) can be expressed in terms of power as 2/1 1 2 2 ] (/) [( )(c c c m eff p p pp i h . (61) Notice that since )(0kG depends on and consequently on the power, the value of µ that enters in (56) and (60) is the one for 2cpp. Equations (58) and (59) are valid only for c Rnn or equivalently 2.5 3.0 3.5 4.0 4.50.05.0x1051.0x1061.5x106Figure 3 G () (number / MHz) Frequency f (GHz) 16 Figure 4: Variation with microwave pumping power of the normalized reservoir average magnon number and of the BEC magnon population . 2cpp and they represent the first important result of this paper. For nR < nc, or p < pc2 the population of the 0k mode is that of thermal equilibrium with the reservoir given by )(0 0 k BE R k fn n . (62) However, for c Rnn or 2cpp the population of mode 0k is pumped -up out of equilibrium as a result of a spontaneous cooperative action of the reservoir modes. As it will be shown in the next section the 0k mode with population given by (57) -(59) above the threshold is in a coherent magnon state. This means that when the average reservoir magnon number reaches the critical value (56) the magnon gas separat es in two parts, one in thermal equilib rium with the reservoir having frequencies in a wide range and one with a higher magnon number in a narrow range around the minimum frequency. This is one of the characteristic features of a Bose -Einstein condensate. We now have the necessary elements to i nterpret the behavior of the magnon system with increasing microwave pumping power. First we note that in the interacting magnon gas the formation o f the BEC occurs at a value of the chemical potential that is close but not equal to the minimum energ y 0k. This is so because as the microwave power increases and approaches 0k, the average reservoir number reaches the critical value (56) corresponding to a small but finite ) (0k . The value of the chemical potential satisfying (56) can be identified as the critical value c for the formation of the BEC. Using (48), (53), (54) and (60) one can obtain the following relation between c and the critical power pc2 2/1 1 1 20 0 ]/) [(4)( c c c BEB k m c k p p pCTk Dr , (63) where we have considered that TkB c k /) (0 << 1 to use the binomial expansion of the exponential function in (48). Of course Equation (63) is not an explicit expression for the critical chemical po tential in terms of pc2 because CBE and also the effect ive temperature T vary with . Equation (63) is important to demonstrate that the difference ) (0 c k is finite in the interacting magnon gas. As the microwave power increases above pc1 the average reservoir magnon number nR increase s contin uously as given by (52). The variation of nR with p is shown in Figure 4. Correspondingly the chemical potential increases with power and reaches the critical value c when p reaches pc2, giving rise to the nonlinear driving of the 0k mode. This process leads to a sharp increase in the magnon population at the state with minimum frequency 0k characteristic of the condensation of bosons . Thus the population 0kn will henceforth be called condensat e or BEC magnon number . For 2cpp the chemical potential locks at the value c so that the dependence of effh)( on power is entirely contained in (61). Since the four -magnon interaction that produces the cooperative actio n conserves the number of magnons, as p increases further the number of 2.5 3.0 3.5 4.0 4.5 5.00.01.0x10-42.0x10-41.0x10-31.5x10-32.0x10-3Figure 4 nk0 nR/ rMagnon number / NS Pumping power p (W) 17 magnons in the reservoir stays constant and the additional magnons originating from the primary magn ons end up at the condensate state. Figure 4 also shows the variation with power of the BEC number 0kn for 2cpp. VI. Quantum coherence of the Bose -Einstein condensate In order to study the coherence properties of the 0k mode pumped above threshold one has to use methods of st atistical mechanics appropriate for boson systems interacting with a heat -bath. We follow here the same procedure used to study the direct parallel pumping process [3 8]. The first step is to represent the magnon reservoir and its interactions with a specif ic k mode by a Hamiltonian that allows a full description of the thermal and driving processes for the interacting magnon system, RS R eff H Ht H H HH )(')4( 0 , (64) where the first three terms are given respectively by (22), (41) and (55), RkkR kR kR R cc H (65) is the Hamiltonian for the magnon reservoir, assumed to be a system with large thermal capacity and in thermal equilibrium and k kR kRk Rkkk kR kRk RS cc cc H, ,* , (66) represents a linear interaction between the magnons k and the heat reservoi r. Note that (66) also has its origin in the four -magnon interaction which provides the main mechanism for the intermagnon relaxation. Using the Heisenberg equation for the magnon operators for a mode k in the vicinity of k0 with the total Hamiltonian (64) and assuming that nk = n-k we obtain, )( )( ) 2 (0 2 )4( tF c e hicnVi idtdc k ktki eff k k m kk (67) where 2 ,)(kRk k m D , (68a) tkRi kR kRkRk k ec i tF, )( , (68b) represent respectively the magnetic relaxation rate expressed in terms of the interaction between ma gnon k and the heat reservoir and a Langevin random force with correlators of Markoffian systems type [3 8-40]. Using Equation (67) and the corresponding one for the operator kc, transforming them to the representation of coherent magnon states k and working with variables in a rotating frame ktki k k k et c )( we obtain an equation of motion for coherent state eigenvalue with k = k0, )( )( )(4)( 2)( 4 2 )4(22 2 )4(tSt tVh V dtt d k k km eff mk (69a) where tki k meff tki k k etFhi etFtS )()()( )(* . (69b) Equations (69) contain all the information carried by the equations of motion for the magnon operators. It is a typical nonlinear Langevin equation which appears in Brownian motion studies and laser theory [3 9,40]. It shows that the magnon modes with amplitud e kare driven thermally by the hot magnon 18 reservoir and also by an effective driving field. The solutions of (69) confirm the previous analysi s. For negative values of the driving term ] )([22 m effh the magnon amplitudes are essentially the ones of the thermal reservoir. For positive values they grow exponentially and are limited by the effect of the four- magnon interactions. Above the threshold condition the steady -state solution of (69) gives for the number of magnons 2 k kn an expression identical to (57). The final step to obtain information about the coherence of the excited mode is to find an equation for the probability density )(kP, defined in (35), that is stochastically equivalent t o the Langevin equation. Using ) exp(k k k i a we obtain a Fokke r-Plank equation in the form [38 ], ) (1) (1) ([)(1 '22 22 4 kP x xPxxxPxxAxxtP , (70) where t n n tH m k3/1 2 3 2 0 )/ (' , (71a) k k H a nn x6/1 02) /2( (71b) represent normalized time and m agnon amplitude and the parameter A is given by, 2 03/2 02 )4(2/1 22 3/2 02)2(2] )([ )2(k k Hm eff k Hnnn Vh nnA . (71c) Note that A can alternatively be written in terms of the average reservoir number Rn or the power p as, ]1)/[(])(2[2 3/2 0 c R R k BEHnnn fnA , (72a) ) () (] [])(2[ 1 22 3/1 11 3/2 0 c cc cc k BE p ppp ppp frA . (72b) Application of Equation (70) to describe the full dynamics of the pulsed experiments [1 -6] must consider that the factors relating t’ to t and x to ka, as well as the parameter A, are all time dependent . However, for typical numbers appropriate for the experiments, t’ ~ t x 2x106 s-1, so that the dynamics of the pulsed experiments is relatively slow in the renormalized time scale. Thus in a first approximat ion we assume that all parameters are constant a nd obtain the stationary solution of (70) independent of kin the form, ) (exp )(6 61 2 21x xA CxP . (73) where C is a normalization constant such that the integral of P(x) in the range of x from zero to infinity is equal to unity. No te that for obtaining (73) all integration constants were set to zero to satisfy this condition. Figure 5 shows plots of P(x) for four values of the parameter A, -1, 0, 80 and 250. In choosing the positive values we have considered parameters which enter i n (72a) and (72b) appropriate for the experiments [1 -6]: cp = 100 mW, 1cp = 0.0625 W and 2cp = 2.8 W; )(0k BEfr ~ 8 x10-7 obtained from the fit of theory to the BLS data as shown in the next Section. With these numbers we obtain A = 250 for c Rnn/ = 1.0 23, or equivalently 2/cpp = 1.047 . Equation (72a) shows that for reservoir average populations below the critical number, c Rnn, the parameter A is negative. In this case the function P(x) in (73) behaves as a Gaussian distribution, characteristic of systems in thermal equilibrium and described by incoherent magnon states [1 6]. On the other hand for c Rnn, or 2cpp, A > 0 and the stationary state consists of two components, a coherent one convoluted with a much smaller fluctuation with Gaussian distribution. Since the variance of P(x) is 19 Figure 5: Probability density characteristic of a microwave driven interacting magnon system for several values of the parameter A: Negative values correspond to c Rnn or for 2cpp ; A = 0 corresponds to the threshold; A = 80 and 250 correspond to c Rnn/ = 1.008 and 1.023, or to 2/cpp = 1.015 and 1.047. proportional do A-1, for A >> 1 the function P(x) becomes a delta -like distribution, characteristi c of a coherent magnon state [1 6]. Figure 5 shows that in the conditions of the experime nts P(x) becomes a delta -like function at power levels just above the critical value. Note that only in the presence of the f our- magnon interaction the magnon state driven collectively by the reservo ir modes is a coherent state [38 ]. Note also that P(x) has a peak at 4/1 0Ax , so that it represents a coherent state with an average number of magnons given by 2/1 2 0Ax . From (71b) and (71c) we see that this corresponds to a magnon number 2 0a which is precisely the value 0kn given by (58) and (59). This means that the magnon 0k driven cooperatively by the reservoir modes is a quantum coherent state. This is the second and most import ant result of this paper since the coher ence implies a macroscopic wavefunction satisfying an essential condition for the condensate. The theoretical interpretation of the observations of Demokritov and co -workers [1 -6] is now clear. After the reservoir of hot magnons with population Rn is formed as a result of the fast redistribution of the energy of the primary parallel -pumped magnons, the modes with frequency Rkclose to 0k act together to drive the mode 0k. However only if the microwave power is above a critical value 2cp, Rn exceeds cn and the system spontaneously develops a coherent state with frequency 0k. According to (36) the small -signal dynamic magnetization is proportional to the amplitude of the coherent state, 2/1 0 0 k kn a m. Thus one can write from (59) that for 2cpp the dynamic magnetization scales with microwave power as 4/1 2) (cpp m, characteristic of a second -order phase transition. The spontaneous emergence of quantum coherence [41 ] caused by a phase -transition and the associated magnetic dynamic order in a macroscopic scale, constitute rigorous theoretical support for the for mation of Bose -Einstein condensation of magnons at room temperature, as claimed by Demokritov and co -workers [1 -6]. VII. Comparison with experimental data In this section we apply the model for the formation of the BEC of magnons developed to treat an interacting magnon gas driven by microwave radiation in a YIG film. We compare the results of the theory with the data obtained by Demokritov and co -workers [1 -6] using two very different techniques, Brillouin light scattering form the magnon condensate a nd microwave emission from the uniform mode driven by BEC magnon pairs. In both cases the theory developed here allows the calculation of quanti ties of interest as a function of microwave pumping power to compare with data. 0 1 2 3 4 5 6 7 80.00.10.20.30.40.50.6Figure 5 A = 250 A = 80 A = -1 A = 0Probability density P (x) Normalized magnon amplitude x 20 a- Intensity of the Brillouin Light Scattering In the experiments of [4,5] with short microwave pulse driving the coherence properties of the excited magnons states emerge clearly in the behavior of the intensity of the BLS peak at minf. As argued in [4,5], for i ncoherent scatterers the BLS intensity is proportional to their number, whereas for coherent scatterers it is proportional to the number squared. Thus, in order to compare theory with data we e xpress the BLS intensity in terms of the microwave power p in two regimes: for 2cpp the number of magnons with frequency fmin is the thermal number 0kn given by (52) and (62); for 2cpp the condensate is characterized by a coherent magnon state with number given by (59). Consider that the relevant number of scatterers is the number of magnons per spin site. Using (52) and (62) we obtain for 2cpp, 2/1 11 00) ()()( ) ( cc H k BEk inc ppp NnfrbNnb I , (74) and with (59) we have for 2cpp, ) ()( ) ( 1 22 2 2 0 c cc H k coh p ppp NnbNnb I , (75) where b is a scale factor proportional to the magneto -optical constant and involves electromagnetic, magnetic and geometrical quantities . Figure 6 shows a fit of (74) and (75) to data, using 2/1 1 1 ) (cincppc I and ) (2 2 ccohppc I , with c1 = 6.7, c2 = 370.0 and 2cp = 2.8 W. Using (47b), (74) and (75) one can obtain a relation that allows the calculation of the factor )(0k BEfr at the critical chemical potential from the fitting parameter s, 22/1 1 21 08)( cc Mm k BEpp ccfr , (76) from which we obtain )(0k BEfr = 8.1 x 10-7. Before discussing the implications of this result it is interesting to compare its value with the one obtained directly from the measured 2cp = 2.8 W. Using this value and 1cp = 0.0625 W in (52) and (60) we obtain 6.0 )()(0 0 m k k BE D fr . Considering )(0kD 105 / MHz, calculated numerically as described earlier, and 2/m 8 MHz, we find )(0k BEfr = 7.4 x 10-7, which is very close to the value obtained from (76). Figure 6: Fit of the theoretical result (solid line) for the BLS intensity as a function of microwave pumping power to the experimental data (symbols) of Demokritov and co -workers [4,5]. 2 3 4 5 6 7101102103 Microwave power p (W)Figure 6BLS Intensity (a.u.) 21 To obtain a value for )(0k BEf at the critical chemical potential we use the definition (53a) and consider that the difference between the minimum energy 0k and µc is, in frequency units, in the range (10 – 20) MHz. The normalization constant CBE is calculated by the integration of (53b) in the frequency range (2.9 – 4.05) GHz using the binomial expansion of the exponential in (48) and assuming T = 103 K. We obtain CBE = (0.8 – 0.9) x 105 and )(0k BEf (10 – 25) for the range of µc above. With these value s we have an order of magnitude estimate for 710~/R pNrr . Considering the number of reservoir states 9 810 10~RN obtained numerically we find for the pumping factor 21010~pr . This is quite sm all compared to the value 4 310 10~pr calculated numerically by counting the states with frequency in the range m p2/ for the conditions of the experiments. We attribute this discrepancy to one or a combination of the followin g reasons: a flaw in the theoretical model; a failure of Equation (26) in reproducing the correct slopes of the dispersion curves near the frequency minima introducing considerable error in the calculation of density of states; a large number of magnons is lost on the way to the region of minimum frequency in the process of redistribution of the primary magnon population. b- Microwave emission from the BEC of magnons As observed by Dzyapko et al. [6], if the static field applied to a microwave pu mped YIG film has a value such that the frequency of the 0k magnon is 0= 20k, a microwave signal is emitted with frequency 0. They interpret this radiation as due to 0k magnons created by pairs of BEC magnons 0 0,k k through a three -magnon confluent process. The 0k value is necessary for emission because the wavenumber of electromagnetic radiation with frequency 1.5 GHz, as in the experiments [6], is f k /2 0.3 cm-1. Figure 7 illustrates the three -magnon confluent process in the dispersion relation for modes propagating along the field in a 5 µm thick YIG film for H = 520 Oe , which is the field value for 0= 20k. As w e have sh own earlier [20] , the 0k magnons created by the BEC are coherent magnons states . Thus they correspond to a nearly uniform magnetization precessing with frequency 0 that emits electromagnetic radiation with this fr equency [42 -44]. To calculate the power emitted by the uniform mode as a function of the microwave pumping power we need to study the process by which this mode is driven by the BEC magnon pairs. Consider a Hamiltonian as in (8) in which the magnon interac tion include three -and four -magnon contributions , )(')4( )3( 0 t H H H HHeff , (77) where )('t Heff is the effective Hamiltonian for driving 0 0,k k magnon pairs given by (55) and (61) and the Hamiltonian for th e three -magnon confluence process is [32 -34] ..0 0 0 )3()3(ch cccV Hk k , (78) where the vertex of the interaction for small wavevectors is dominated by the dipolar interaction be tween the spins S and is given approximately by 2/1 )3( )2/(SN VM . To study the process by which the pairs of BEC coherent magnons 0 0,k k are produced and then generate k ~ 0 modes we use t he Hamiltonian (77) to obtain the Heisenberg equations of motion for the magnon operators, 22 Figure 7: Dispersion relation for magnons propagating along the field H = 520 Oe applied in the plane of a YIG film with thickness 5 m with illustration of the 3 -magnon coalescence process that generates a k = 0 from a pair of BEC magnons. 0 0 )3( 0 0 )4( 0 00) (k kccVicnVi idtdc , (79) 00 2 0 )3( 0 0 )4( 0 00] )( [ ) 2 (ktki eff k k k kkc e h cVi cnVi idtdc , (80) where the relaxation was introduced phenomenologically. We consider that all states involved are coherent magnon states as demonstrated earlier and work with the corresponding eigenvalues k. In addition we assume that there are 0kp pair modes with wavevectors close to 0 0,k k to drive the k ~ 0 modes and that the resonance condition is satisfied 0 02k , determined by the value of the applied field H. The equations of motion for the eigenvalue 0 and the correlation function 0 0 0 k k k in a frame rotating with frequency 0 become, 0 )3( 0 0 0 )4( 00) (k kVip niVdtd , (81) 0 0 0 )3( 0 0 )4( 00])( / [2 ) 2 (2k eff k k k kkn h p Vi nVidtd . (82) Note that in (82 ) the term representing the coupling with the k = 0 mode is divided by the number of modes 0kp assumed in the driving because 0k represents only one pair -mode 0k. The coupling term in (82) represents a reaction of the k = 0 mode that influences the behavior of the BEC modes. In steady -state 0 /dtd (81) leads to, 0 0 )4( 0)3( 0 0 kk niVVip . (83) This result , valid for the resonanc e condition 0 02k , shows that the BEC magnon pairs drive the uniform mode as an effective microwave magnetic field by means of the three -magnon interaction. Note that there is no threshold condition in this process, BEC magnon pairs with any value of 0kn will create k = 10-710-510-310-110110310510712345Figure 7 k00 -k0k0Frequency f = / 2 (GHz) Wavenumber k (cm-1) 23 0 magnons. This is in contrast to the so -called subsidiary resonance instability process in which the three - magnon splitting process occurs only if the microwave field exceeds a critical value [34,37,45] . The presence of the term 0 )4(niV in the denominator due to the four -magnon interaction represents a detuning from the resonance condition due to the renormalization of the k = 0 mode frequency. In fact, this term is responsible for the sa turation in the growth of the k = 0 mode amplitude with microwave pumping power observed experimentally [6]. In order to compare theory with data we have solved numerically the coupled equations (8 1) and (8 2) with their real and imaginary parts to find th e steady -state values of the magnon populations 0n and 0kn for each value of the pumping power. The calculations were done considering that the relaxation of all modes involved is dominated by the magnetic interacti ons, m k0 0 . We also use normalized variables and parameters: SNnnk k / ' , t tm' , m SN V V 2/)( '2/1 )3( )3( , m SNV V 2/)( ')4( )4( and m eff eff h h /)( )'( . With M4 1.76 kG and m 5x107 s-1 we have 0.219')3(V and 0. 1240')4(V . Figure 8 shows the variation with microwave power of the normalized steady -state values of the populations of the uniform mode n0 and the BEC mode nk0 multiplied by the fa ctor pk0. Notice that they are both nonzero only for pumping power above the threshold value. The total power radiated by the uniform magnetization precessing about the static field with frequency 0 is given by [44 ], ) (322 2 34 02 2 y xm mcNP (84) where N is the number of spins in the region of emission, is the volume of the spin unit cell, c is the speed of light and mx and my are the small -signal components of the transverse magnetization. In (84 ) we have written the volume of the sample as NV to stress the dependence of the radiated power on the square of the number of spins. This characterizes superradiance, a term introduced in 1954 by Dicke [46] to designate the type of spontaneou s emission of radiation from an assembly of N atoms that has as an intensity proportional to N 2 instead of N as in usual situations. This emission requires some kind of quantum coherence in the atomic states, a topic which became understood many years aft er Dicke’s paper was published. The observation of macroscopic superradiance of microwaves in ferromagnetic resonance in YIG was achieved only in the 1970s [43]. The recent experiments of Dzyapko et al. [6] constitute the first observation of superradiance originating from a Bose -Einstein condensate. Figure 8: Variation with microwave pumping power of the normalized steady -state magnon numbers of the uniform mode n0 and the BEC mode nk0 (multiplied by the factor pk0 = 5 x 104). 4.0 4.5 5.0 5.5 6.0 6.5 7.00.00.20.40.60.81.01.21.41.61.82.0Figure 8 n0 nk0 x pk0Magnon number / NS Pumping power p 24 Figure 9: Microwave emission signal power vs pumping power. Symbols represent the experimental data of Dzyapko et al. [6] and the solid line is the fit with theory. Since the microwave signal power is a fraction of the total radiated power given by (84), we use the expression 0'nCps to fit the data of Dzyapko et al. [6]. In Figure 9 the symbols represent the data of [6] and the solid line represents the theoretical fit with using C = 13.2 W, 0kp= 5 x 103 and 2cp= 4.45 W. The fit is quite good but it is important to check if the values of the fitting parameters bear conn ection to reality. A good estimate for the number of BEC modes that drive the k = 0 magnon is obtained by counting the modes with frequency in the range 2/0 0 m k k and with k in the z-direction of the static field, z z Lnk / , where n is an integer and zL the sample length. The result obtained numerically wit h the dispersion relation (26) is 20 x 103. The value of 0kp obtained from the fitting is somewhat smaller than this, which is expected since it represents the number of modes weighted by the number of magnons of the mode k relative to the maximum number at k0 . To calculate the emitted microwa ve signal we use in Equation (84 ) the expressions for the magnetization components of a coherent state (37) obtaining, 0 32 4 02 'ncM VP . (85) Using in (8 5) 20x 3.0 GHz, M = 300 G, c = 3 x 1010 cm/s and an estimated emission volume V = 1 mm x 0.5 mm x 5 m = 2.5 x 10-6 cm3, we obtain for the factor of 0'n in (8 5) approximately 400 W. This is two orders of magnitude larger than the value of C obtained from the fit of theory to experiment, which is quite reasonable considering that the measured signal power is only a very smal l fraction of the t otal radiated power given by (84 ). It is important to note that if (8 1) and (8 2) are solved considering 10kp , the calculated 0'n is smaller than the value obtained with 0kp = 5 x 103 by a factor 107. This means that with 10kp the total emitted power calc ulated with (85 ) would be smaller than the measured signal power by a factor 105, which is completely unrealistic. Note that this model is also consistent with the 6 MHz linewidth of the microwave emission spectrum observed in [6]. This value was considered too large by the authors of [6] who expected a linewidth one order of magnitude smaller corresponding to the spin -lattice relaxation rate. In fact the linewidth is close to the value determined by the magnetic relaxation rate, 2/m = 8 MHz, which in our theory d ominates decay process. 4.0 4.5 5.0 5.5 6.0 6.5 7.00123456Figure 9Signal power ps Pumping power p 25 VIII - Summary In conclusion, we have shown that in a magnon system in a YIG film driven by microwave radiation far out of equilibrium, the four -magnon interactions acting on the reservoir modes with frequencies close to the m inimum in the dispersion relation create the conditions for the spontaneous generation of coherent magnon states. As the microwave power p is increased and exceeds a critical value 2cp, the magnetic quantum states change from incoherent to coherent magnon states. Correspondingly, the small -signal magnetization changes from zero to 4/1 2) (cpp m for 2cpp. Since the magnetization represents the order parameter of the dynamic magnetic system, this characterizes a true second order phase transition with critical exponent 1/4 . The spontaneous em ergence of quantum coherence [41 ] caused by a phase - transition and the associated magnetic dynamic order in a macroscopic scale, constitute rigorous theoretical support for the formation of Bose -Einstein condensation of magnons at room temperature, as claimed by Demokritov and co -workers [1 -6]. We have also shown that the nearly uniform mode generated by Bose -Einstein condensate (BEC) magnon pairs emits superradiance as a re sult of the cooperative action of the spins. The theory explains quantitatively recent experimental observations of Dzyapko et al. [6] of microwave emission when the driving power exceeds a critical value. The theoretical results fit very well the data for the emitted signal power versus microwave pumping power with realistic parameters. The author would like to thank Professor Roberto Luzzi of UNICAMP for calling our attention to the recent challenges of BEC of magnons and Professor Sergej Demokritov of University of Muenster for providing important information on the experiments. The author is also very grateful to Professor Cid B. de Araújo for many stimulating discussions and for the Ministry of Science and Technology for supporting this work. REFER ENCES [1] S.O. Demokritov, V.E. Demidov, O. Dzyapko, G.A. Melkov, A.A. Serga, B. Hillebrands, and A.N. Slavin, Nature 443, 430 (2006). [2] V.E. Demidov, O. Dzyapko, S.O. Demokritov, G.A. Melkov, and A.N. Slavin, Phys. Rev. Lett. 99, 037205 (2007). [3] O. D zyapko, V.E. Demidov, S.O. Demokritov, G.A. Melkov, and A.N. Slavin, New J. Phys. 9, 64 (2007). [4] V.E. Demidov, O. Dzyapko, S.O. Demokritov, G.A. Melkov, and A.N. Slavin, Phys. Rev. Lett. 100, 047205 (2008). [5] S.O. Demokritov, V.E. Demidov, O. Dzyapko, G.A. Melkov, and A.N. Slavin, New J. Phys. 10, 045029 (2008). [6] O. Dzyapko, V.E. Demidov, S.O. Demokritov, G.A. Melkov, and V.L. Safonov, Appl. Phys. Lett. 92, 162510 (2008). [7] See for example, R.P.Feynman, Statistical Mechanics: a Set of Lectures (W.A. Benjamin, Reading, Massachusetts, 1972); L.D. Landau and E.M. Lifshitz, Statistical Physics (Pergamon Press, Oxford, 1980) [8] See for example, S. A. Moskalenko and D.W. Snoke, Bose Einstein Condensation of Excitons and Biexcitons (Cambridge Univ. Pres s, Cambrige, 2000). [9] J. Kasprzak et al., Nature 443, 409 (2008). [10] See for example, Bose Einstein Condensation in Atomic Gases , Proceedings of the International School of Physics “Enrico Fermi”, edited by M. Inguscio, S. Stringari, and C.E. Wieman ( IOS Press, Amsterdam, 1999). [11] L. Yin, J.S. Xia, V.S. Zapf, N.S. Sullivan, and A. Paduan -Filho, Phys. Rev. Lett. 101, 187205 (2008). [12] Yu. D. Kalafati and V.L. Safonov, Zh. Eksp. Teor. Fiz. 95, 2009 (1989); Sov. Phys. JETP 68, 1162 (1989). 26 [13] G.A. Melkov, V.L.Safonov, A.Yu. Taranenko, and S.V. Sholom, J. Mag. Mag. Mat. 132, 180 (1994). [14] F.R. Morgenthaler, J. Appl. Phys. 31, 95S (1960). [15] E. Schlömann, J.J. Green, and V. Milano, J. Appl. Phys. 31, 386S (1960). [16] R.J. Glauber, Phys. Rev. 131, 2766 (1963). [17] S.M. Rezende and N. Zagury, Phys. Lett. A 29, 47 (1969). [18] N. Zagury and S.M. Rezende, Phys. Rev. B 4, 201 (1971). [19] I.S. Tupitsyn, P.C.E Stamp, and A.L. Burin, Phys. Rev. Lett. 100, 257202 (2008). [20] S.M. Rezende, Phys. Rev. B (to be published). [21] R.W. Damon and J.R. Eshbach, J. Phys. Chem. Solids 19, 308 (1961). [22] M. Sparks, Phys. Rev. B 1, 3831 (1970). [23] T. Wolfram and R.E. DeWames, Phys. Rev. B 4, 3125 (1971). [24] B.A. Kalinikos, IEE Proc. (London) 127, (H1), 4 (198 0). [25] B.A. Kalinikos and A.N. Slavin, J. Phys. C 19, 7013 (1986). [26] M.G. Cottam and A.N. Slavin, in Linear and Nonlinear Spin Waves in Magnetic Films and Superlattices , Ed. M.G. Cottam (World Scientific, Singapore, 1994), Chapter 1. [27] D.D. Stancil , Theory of Magnetostatic Waves (Springer Verlag, New York, 1993). [28] P. Kabos and V.S. Stalmachov, Magnetostatic Waves and Their Applications (Chapman and Hall, London, 1994). [29] M.J. Hurben and C.E. Patton, J. Mag. Mag. Mat. 139, 263 (1995). [30] R.E . Arias, and D.L. Mills, Phys. Rev. B 60, 7395 (1999). [31] P. Landeros, R.E. Arias, and D.L. Mills, Phys. Rev. B 77, 214405 (2008). [32] M. Sparks, Ferromagnetic Relaxation Theory (McGraw -Hill, New York, 1964). [33] A.I. Akhiezer, V.G. Bar’yakhtar, and S. V. Peletminskii, Spin Waves (North -Holland, Amsterdam, 1968). [34] R.M. White, Quantum Theory of Magnetism , 3rd Edition (Springer -Verlag, Berlin 2007). [35] S.M. Rezende, F.M. de Aguiar, and A. Azevedo, Phys. Rev. B 73, 094402 (2006). [36] V.E. Zakharov, V.S. L’vov, and S.S. Starobinets, Usp. Fiz. Nauk. 114, 609 (1974) [Sov. Phys. Usp. 17, 896 (1975)]. [37] S.M. Rezende and F.M. de Aguiar, Proc. IEEE 78, 893 (1990). [38] Cid B. de Araújo, Phys. Rev. B 10, 3961 (1974). [39] H.Haken, Rev. Mod. Phys. 47, 67 ( 1975). [40] P. Meystre and M. Sargent III, Elements of Quantum Optics (Springer -Verlag, Berlin, 1992). [41] D. Snoke, Nature (London) 443, 403 (2006). [42] N. Bloembergen and R.V. Pound, Phys. Rev. 95, 8 (1954). [43] E. Montarroyos and S.M. Rezende, Solid St. Comm. 19, 795 (1976). [44] S. Chaudhuri anf F. Keffer, J. Phys. Chem. Solids 45, 47 (1984). [45] Cid B. de Araújo and S.M. Rezende, Phys. Rev. B 9, 3074 (1974). [46] R.H. Dicke, Phys. Rev. 93, 99 (1954).

2009-02-18

Strong experimental evidences of the formation of quasi-equilibrium Bose-Einstein condensation (BEC) of magnons at room temperature in a film of yttrium iron garnet (YIG) excited by microwave radiation have been recently reported. Here we present a theory for the magnon gas driven by a microwave field far out of equilibrium showing that the nonlinear magnetic interactions create cooperative mechanisms for the onset of a phase transition leading to the spontaneous generation of quantum coherence and magnetic dynamic order in a macroscopic scale. The theory provides rigorous support for the formation of a BEC of magnons in a YIG film magnetized in the plane. We show that the system develops coherence only when the microwave driving power exceeds a threshold value and that the theoretical result for the intensity of the Brillouin light scattering from the BEC as a function of power agrees with the experimental data. The theory also explains quantitatively experimental measurements of microwave emission from the uniform mode generated by the confluence of BEC magnon pairs in a YIG film when the driving power exceeds a critical value.

Theory of coherence in Bose-Einstein condensation phenomena in a microwave driven interacting magnon gas

0902.3138v1

1 Voltage -Controlled Magnon Transistor via Tunning Interfacial Exchange Coupling Y. Z. Wang#, T. Y. Zhang#, J. Dong, P. Chen, C. H. Wan*, G. Q. Yu, X. F. Han* 1Beijing National Laboratory for Condensed Matter Physics, Institute of Physics, University of Chinese Academy of Sciences, Chinese Academy of Sciences, Beijing 100190, China 2Center of Materials Science and Optoelectronics Engineering, University of Chinese Academy of Sciences, Beijing 100049, China 3Songshan Lake Materials L aboratory, Dongguan, Guangdong 523808, China *Email: xfhan@iphy.ac.cn ; wancaihua@iphy.ac.cn Abstract : Magnon transistors that can effectively regulate magnon transport by an electric field are desired for magnonics which aims to provide a Joule -heating free alternative to the conventional electronics owing to t he electric neutrality of magnons (the key carriers of spin -angular momenta in the magnonics). However, also due to their electric neutrality, magnons have no access to directly interact with an electric field and it is thus difficult to manipulate magnon transport by voltages straightforwardly. Here, we demonstrated a gate voltage ( 𝑉g) applied on a nonmagnetic metal/magnetic insulator (NM/MI) interface that bended the energy band of the MI and then modulated the possibility for conduction electrons in the NM to tunnel into the MI can consequently enhance or weaken the spin -magnon conversion efficiency at the interface. A voltage -controlled magnon transistor based on the magnon -mediated electric current drag (MECD) effect in a Pt/Y 3Fe5O12 (YIG)/Pt sandwich was then experimentally realized with 𝑉g modulating the magnitude of the MECD signal. The obtained efficiency (the change ratio between the MECD voltage at ±𝑉g) reached 10% /(MV/cm) at 300 K. This prototype of magnon transistor offers an effective scheme to control magnon transport by a gate voltage. Magnons as the collective excitation of a magnetically ordered lattice possess both spin -angular momenta and phases but no charges [1], born an ideal information carrier for the Joule -heating - free electronics [2,3] . In order to efficiently manipulate magnon transport, magnon transistors, as an elementary brick for magnonics, are long -desired. Despite of great achievements in efficiently exciting [4-10], propagating [11-13] and detecting [14-18] magnons, the ele ctric neutrality of magnons sets a high level of difficulty in controlling magnon transport by electric fields. 2 Several magnon gating methods have been realized. The YIG/Au/YIG/Pt magnon valves [19] and YIG/NiO/YIG/Pt magnon junctions [20-22] are gated by an external magnetic field (H). Large (small) spin -Seebeck voltage was output by setting the two YIG layers into the parallel (antiparallel) state by H currently, though spin -orbit torques (SOT) are potentially used to gate the magnon valves/junctions in t he future [23]. Another magnon -spin valve YIG/CoO/Co was also H- gateable [24]. Its spin pumping voltage depends on the H-controlled parallel/antiparallel states between YIG and Co. Another method is gating current . In the magnon transistor consisting of three Pt stripes on top of a YIG film [25], a charge current in the leftmost Pt stripe excites a magnon current in YIG via (i) the spin Hall effect (SHE) in Pt and (ii) the interfacial s-d coupling at the Pt/YIG interface. The as -induced magnon current diffu ses toward the rightmost Pt stripe where the inverse process occurs, resulting in a detectable voltage. This phenomenon, featured by the nonlocal electric induction across a MI with the help of magnons, is named as the magnon - mediated electric current drag (MECD) effect [4,6] . A gating current flowing in the middle Pt strip changes magnon density of YIG in the gate region and consequently modifies the MECD efficiency. Gate voltage is advantageous in energy consumption. However, due to no direct coupling of magnons with any electric fields, magnon transistors inherently controlled by Vg are still missing. Inspired by the model of Chen et al. [26,27] , we realize the spin mixing conductance ( 𝐺↑↓) at a NM/MI interface relies sensitively on the interfacial s-d exchange coupling. Here, we proposed a voltage -gated magnon transistor (Fig.1 (a)) where Vg across the NM/MI interface tilts downward (upward) the energy band of the MI (Fig.1 (b)), decreases (increases) the probability of electrons penetrating into the MI (Fig.1 (c)), thus weakens (strengthens) the spin -magnon conversion efficiency at the interfa ce and consequently changes the magnon excitation efficiency in the magnon transistor. We extended the model by including the Vg-induced band bending of MI via the Hamiltonian 𝐻MI=𝑝22𝑚⁄ +𝑉0+Г𝐒·𝛔+𝑒𝑧𝑉g𝑡⁄ (1) , where 𝑉0 is the energy barrier at the interface; Г𝐒·𝛔 describes the s-d coupling of electron spins 𝛔 in NM with localized moments 𝐒 in MI; 𝑒𝑧𝑉g𝑡⁄ describes the conduction band bending by Vg and t is the MI thickness (calculation details in Supplementary Materials). The predicted V0 and 3 electric field E (𝐸=𝑉g𝑡⁄) dependence of the real part of 𝐺↑↓ (Gr) was plotted in Fig.1 (d) (taking Fermi energy of NM 𝜀=5 eV and s-d coupling strength Г=0.5 eV for the Pt/YIG interface [4]). The typical Gr-E curves at three V0 values (Fig.1 (e)) suggest the positive Vg can efficiently increase Gr and vice versa. The spin -magnon convertance at the NM/MI interface is proportional to Gr [28] and thus the magnon current excited in MI can be modified by Vg as experimentally shown below. Fig.1. Mechanism of voltage -gated magnon transistor. (a) Schematics of the voltage -gated magnon transistor. A spin current is generated by the spin Hall effect (SHE) in the bottom (B) -NM, which induces imbalanced spin accumulation ( 𝜇𝑠) at the B -NM/MI interface. Due to the s-d exchange coupling at the interface, 𝜇𝑠 relaxes by annihilating (generating) magnons in MI as 𝜇𝑠 has parallel (antiparallel) polarization to the magnetization of MI. The excited magnon current was thus manipulated by Vg: positive (negative) Vg increase s (decreases) its magnitude. (b) Schematics of potential profile near the B -NM/MI interface under positive Vg. (c) Schematics of probability | φ|2 at the B -NM/MI interface under positive and negative Vg. (d) The predicted V0 and E dependence of Gr (the colo r scale bar in units of e2/ℏa2). (e) The E-dependence of Gr under V0=5.625, 5.675 and 5.725 eV extracted from Fig.1 (d). The Vg-controlled magnon transistor was then experimentally achieved in a Pt(10)/YIG(80)/Pt(5 nm) sandwich (details in Method and Supplementary Material) where Vg across the YIG was able to tune the MECD effect. The measured voltage V along the top (T) -Pt electr ode follows the 4 coming 3 characteristics: (1) the angular dependence of 𝑉=𝑉drag cos2𝜃 (𝜃 the angle between spin polarization 𝛔 and magnetization M, Fig.S6 (a)), (2) the linear dependence of 𝑉drag on the input current ( 𝐼in) along the B -Pt electrode (Fig.S6 (b,c)) and (3) the 𝑇5/2 temperature -dependence (Fig.S6 (d)), all coinciding with Ref.6&7 [7,8] . These features confirmed the MECD nature of the measured voltage . The insulating property of YIG was also checked by 𝐼leak −𝑉g curves (Fig.S3 (b)) with the leakage current 𝐼leak. 𝐼leak was independent on H, assuring the irrelevance of the observed H-dependent V with 𝐼leak (Fig.S4). Fig.2. Voltage -controlled MECD effect. (a) The γ-dependence of ∆𝑉 with H rotated in the yoz plane and 𝐼in= 5 mA under 𝑉g=−5,0 and +5 V. The open circles (solid lines) are the experimental data (fitted curves by ∆𝑉=𝑉drag cos2γ). (b) The 𝐼in-dependences of 𝑉drag under different 𝑉g and their linear fittings. (c) The 𝑉g- dependence of magnon drag parameter α. Error bars for Device 1 and 2 are from the standard deviations of the linear fittings of the 𝑉drag −𝐼in relation and the ∆𝑉=𝑉drag cos2γ fittings , respectively. The red line is the hyperbolic tangent fitting of the 𝛼- Vg curve. (d) The γ-dependence of the difference in ∆𝑉 between 𝑉g=±5 V. The 𝑉g-controllability of the MECD effect is clearly shown in Fig.2. The MECD magnitude was noticeably enhanced (weakened) under 𝑉g=+5 V (−5 V) (Fig.2 (a)), which was further confirmed by the slope change of the 𝑉drag −𝐼in curves (Fig.2 (b)). The magnon drag parameter 5 𝛼 was then calculated by 𝑉drag 𝑅T−Pt=𝛼𝐼𝑖𝑛. The 𝑉g-dependence of the extracted α (see method) (Fig.2 (c)) showed a clear change as 𝑉g= [−2 V,+2 V] and nearly saturated beyond the region. The maximum 𝛼 tunability by 𝑉g ( 𝛼(𝑉g>+2𝑉)−𝛼(𝑉g<−2𝑉) 𝛼(𝑉g<−2𝑉)) reached ~ 5% with 𝛼(𝑉g> +2𝑉)~1.71×10−5 and 𝛼(𝑉g<−2𝑉)~1.63×10−5. The 𝛼-controllability by Vg was also repeated in another Device 2 . In order to trace the trend of the Vg-induced change in 𝛼, we fitted the 𝛼-Vg curve by a hyperbolic tangent function 𝛼=𝑎+𝑏tanh (𝑐𝑉g) as shown by the red line in Fig.2(c). Note that this fitting only mathematically impacts with |𝑏𝑎⁄| and c reflecting the magnitude and saturation speed of the Vg-tunability , respectively . Here, for the 𝛼-Vg curve |𝑏𝑎⁄|=0.019 and c=-0.55 V-1. In the following, we reveal the origin of the 𝑉g-tunability over the MECD effect. First, the 𝑉g- dependence of the MECD effect cannot be caused by any magnon coupling possibilities with the leakage current since Ileak increased divergently with the increa se in |𝑉g| but 𝛼 nearly saturated above ±2 V. Second, the resistance of T -Pt directly changed by Vg was negligibly small (<0.008%, Fig.S8), also impossible to cause such significant change ~5% in the MCD signal. Third, though negligibly small in garnets [29-31], the interfacial Dzyaloshinsky -Moriya interaction (DMI) may introduce an additional magnon -drift velocity 𝐯DMI =𝐳̂×𝐦̂2𝛾 𝑀𝑠𝐷 to influence magnon transport with 𝐳̂ the interfacial normal, 𝐦̂ (𝑀𝑠) the magnetization direction (saturated magnetization), 𝛾 the gyromagnetic ratio and 𝐷 a Vg-changeable parameter quantifying the DMI [32-34]. However, this DMI mechanism, if any, would bring about a 360o period in the yoz rotation owing to the 𝐦̂- dependence of 𝐯DMI. In stark contrast, the Δ𝑉+5 𝑉−Δ𝑉−5 𝑉 vs 𝛾 curve (Fig.2 d) shows a cos2 𝛾 symmetry (180o period), thus ruling out the DMI origin of the 𝑉g controllability. To be more specific, the MECD effect can be explicitly expressed as below [4,6] : 𝐣eT−Pt∝𝜃SHtop𝜃SHbottom𝐺Ss−m𝐺Sm−s𝛔×(𝐌×𝐣eB−Pt) (2) here, 𝐣eT−Pt (𝐣eB−Pt) is the induced (input) charge current density along the T -Pt (B -Pt) electrode, 𝜃SHtop(bottom ) is the spin Hall angle of the top (bottom) Pt electrode, 𝐺Ss−m(𝐺Sm−s) is the effective spin-magnon (magnon -spin) convertance at the B -Pt/YIG (YIG/T -Pt) inter face, 𝛔 is the spin polarization perpendicular to 𝐣eT−Pt and M is the YIG magnetization. Ruling out the above 3 6 reasons, the MECD voltage can still be potentially manipulated by 𝑉g in the following scenarios: (1) 𝑉g-induced changes in the effective mag netization of YIG, (2) the spin Hall angles ( 𝜃SH) of Pt or (3) the spin -magnon conversion efficiency across the B -Pt/YIG or YIG/T -Pt interfaces. Hereafter, we experimentally check their possibilities one -by-one. Fig.3. Schematics setups for (a) spin pumping measurement where the spin pumping voltage ( 𝑉SP) was picked up along the B -Pt stripe with H perpendicular to the stripe and 𝑉g applied across the sandwich and for (e) SMR measurement where the resistance change Δ𝑅B of the B -Pt stripe was measured with H rotated in the yoz plane. (b) The H-dependence of the normalized 𝑉SP(𝐻)/𝑉SPmax under different rf frequencies ( 𝑓) and 𝑉g=-3.9, 0 and +3.9 V. (c) The 𝐻-dependence of 𝑉SP at 𝑓=5 GHz and 𝑉g=−3.9,0,+3.9 V. (Error bars from standard deviation by fitting 𝑉SP−𝐻 curves with the Lorentzian function.) (d) The 𝛾-dependences of the Δ𝑅B (open circles) and their ∆𝑅B=Δ𝑅SMR cos2γ fittings. (f) The resonance field ( 𝐻r) dependence of f under 𝑉g= −3.9,0 and +3.9 V (open circles) and their Kittle fittings. The 𝑉g-dependence of (g) the peak value of 𝑉SP− 𝐻 curve ( 𝑉SPpeak) under 𝑓=5 GHz and (h) the SMR ratio. (Error bars from standard deviation of the ∆𝑅B= Δ𝑅SMR cos2γ fittings.) Red lines in Fig.3(c &d) are the hyperbolic tangent fitting of the 𝑉SPpeak-Vg and SMR ratio - Vg curve s, respectively. To investigate the 𝑉g-dependence of Ms, we conducted spin pumping experiments ( experimental details in Method). The spin pumping voltage VSP picked up in the B -Pt electrode at various Vg is exhibited in Fig.3 (a). The H-dependences of a normalized VSP at different f and 𝑉g show no noticeable change s (variation<0.3%) in the resonance field ( 𝐻r) (Fig.3 (b)) and the overlapped Kittle fittings manifested no changes in the magnetization and anisotropy of YIG under Vg. Interestingly, the magnitude of 𝑉SPpeak changed by Vg (Fig.3 (c)). The tunability defined by 7 𝑉SPpeak(𝑉g=+3.9 𝑉)−𝑉SPpeak(𝑉g=−3.9 𝑉) 𝑉SPpeak(𝑉g=−3.9 𝑉) was also ~5 %. Moreover, the 𝑉SPpeak-Vg tendency seemed similar to the Vdrag-Vg relation , with |𝑏𝑎⁄|=0.021 and c=-0.54 V-1 extracted from the hyperbolic tangent fitting . We also tested VSP along the T -Pt stripe, which had ideally identical Hr but opposite polarity with the B -Pt stripe (Fig.S7 (a)). However, 𝑉SPpeak was not changed by Vg for the T -Pt detector (Fig.S7 (c)). Since spin currents were both pumped out from the sandwiched YIG, the different Vg- controllability on VSP for the B -Pt and the T -Pt detectors strongly hinted an interfacial gating origin instead of any bulk YIG reasons. The following Vg-dependent spin H all magnetoresistance (SMR) effect also supported this interfacial claim. Since SMR originates from spin -transfer at interfaces and shunted by a thick Pt layer, we fabricated another Pt(4)/YIG(80)/Pt(5 nm) sandwich. Its Δ RB-Pt-γ relation at various Vg and the summarized Vg-dependence of the SMR ratio are shown in Fig.3 (d,h). The similar coefficient s of |𝑏𝑎⁄|=0.022 and c=-0.51 V-1 were obtained from the hyperbolic tangent fitting (the red line in Fig.3(h)), illustrat ing the Vg-tunability on the SMR ratio also followed the similar trend as the Vg-dependence of 𝛼 and 𝑉SPpeak. The SMR effect in the T -Pt stripe was independent on Vg (Fig.S7 (b,d)). Fig.4. (a) The γ-dependence of Δ V(Vg=5V) -ΔV(Vg=-5V) under different T. (b) The T-dependence of the difference in the magnon drag parameter Δ α=α(Vg=5 V) -α(Vg=-5 V) between Vg=± 5 V. The solid lines are obtained by fitting data using ∆𝛼=𝐴𝑒−∆𝐸kB𝑇 ⁄. (c) The calculated Vg-dependence of Gr by taking redistributed voltage on the contact r esistance ( Rcontact ) into consideration . The red line is the hyperbolic tangent fitting result. After the above analysis we have narrowed possibility for the Vg-controlled MCD effect to (1) a Vg-changeable spin Hall angle in B -Pt or (2) a Vg-controllable spin -magnon conversion efficiency across the B -Pt/YIG interface. If the bulk spin Hall angle was modulated by Vg, we would not expect a substantial difference between the B -Pt and T -Pt stripes si nce they were both textured in 8 the (111) orientation (Fig.S5). The Vg-independent resistivity of B -Pt (Fig.S8) did not support a Vg-modulated spin Hall angle of the B -Pt as well [35]. We further measured the Vg-controlled MECD effect at different T. The Vg-tunability over the MECD effect was strongly depended on T from 240 K to 300 K (Fig.4 (a)). The difference in α under Vg=± 5 V increased by a factor of 3.5 (from 0.6× 10-7 at 240 K to 2.1× 10-7 at 300 K) (Fig.4 (b)). This strong T-dependence cannot favor the possibility of a Vg-controlled intrinsic spin Hall conductivity ( 𝜎SHint) since the electronic structure of Pt varies little with T. Nevertheless, the strong T-dependence can be naturally obtained as following. According to the spin -mixing conductance model across a NM/MI interface [4,6,26,27,36] , the spin -torque -transfer efficiency and the spin - magnon convertance both depend on the s-d exchange coupling strengt h and thus probability of electrons penetrating into the insulating YIG as evanescent states. The probability certainly depends on the interface barrier (thus Vg) and also T since T determines the kinetic energy of electrons in YIG. Supposing (1) ± 5 V gati ng leads to the similar band bending at different T and (2) the classic thermal activation theory holds, we would expect an exponential T-dependence (Arrhenius law [37]) for the MECD coefficient. Fig.4 (b) shows the fitting well matched the experimental dat a and the caused difference in the effective tunneling barrier by ± Vg reached 0.13 eV. Since the spin -mixing conductance depended on the s-d coupling in the same way as the spin - magnon convertance, the SMR shared the same Vg-dependence as the MECD effect naturally. Band bending at interfaces relies on charged defect density which pins the Fermi level and influences bending degree, which probably accounts for the observation that a smoother and well - crystallized B -Pt/YIG interface (evidenced by a sharper el ectron diffraction pattern at this region) contributed to the Vg-controllability. Now the above experimental data persuade us to attribute the Vg-controlled MECD effect to the Vg-induced changes in the spin -magnon conductance across the B -Pt/YIG interface. However, the measured Vg-α deviated from the theoretical prediction by the saturation trend at large Vg. We attribute this deviation to the redistributed voltage on the contact resistance ( Rcontact ) since Ileak increases divergently with Vg. In practice, we rewrote the Hamiltonian in YIG 𝐻MI=𝑝22𝑚⁄ + 𝑉0+Г𝐒·𝛔−𝑒𝑧𝑉g−𝐼leak ∙𝑅contact 𝑡, considering the voltage dropped on Rcontact . The calculated Gr-Vg relation (Fig.4 (c)) using parameters for Pt/YIG: Fermi energy 𝜀=5 eV, 𝑉0=5.5 eV, Г=0.5 eV [4] and 𝑅contact =15 MΩ agrees well with experiment. Gr increased (decreased) with positive 9 (negative) Vg and saturated at 𝑉g≈±2 V. The calculated Gr change by Vg saturated at 13%/(MV/cm), also in a quantitative agreement with the experiment value ~10%/(MV/cm). The calculated result can also be well fitted with the hyperbolic tangent function with |𝑏𝑎⁄|=0.041 and c=-0.45 V-1, which was the reason why we had used the hyperbolic tangent fitting to mathematically trace the Vg-dependences of α, 𝑉SPpeak and SMR ratio. In summary we have experimentally demonstrated a field -effect magnon transistor based on the MECD effect in the P t/YIG/Pt sandwich. With the voltage -induced band bending of YIG, the energy profile of the B -Pt/YIG interfacial barrier and consequently its spin -magnon convertance was modulated. In this sense, the MECD effect was directly modulated by the gate voltage. O ur finding promises direct modulation of spin -magnon conversion by electric fields, which shows a feasible pathway toward electrically controllable magnonics. References : [1] F. Bloch, Z. Angew. Phys. 61, 206 (1930). [2] A. V. Chumak, A. A. Serga, and B. Hillebrands, Nat. Commun. 5, 4700 (2014). [3] A. V. Chumak, V. I. Vasyuchka, A. A. Serga, and B. Hillebrands, Nat. Phys. 11, 453 (2015). [4] S. S. L. Zhang and S. Zhang, Phys. Rev. Lett. 109, 096603 (2012). [5] S. S. L. Zhang and S. Zhang, Phys. Rev. B 86, 214424 (2012). [6] S. M. Rezende, R. L. Rodrí guez -Suá rez, R. O. Cunha, A. R. Rodrigues, F. L. A. Machado, G. A. Fonseca Guerra, J. C. Lopez Ortiz, a nd A. Azevedo, Phys. Rev. B 89, 014416 (2014). [7] H. Wu, C. H. Wan, X. Zhang, Z. H. Yuan, Q. T. Zhang, J. Y. Qin, H. X. Wei, X. F. Han, and S. Zhang, Phys. Rev. B 93, 060403 (2016). [8] J. Li, Y. Xu, M. Aldosary, C. Tang, Z. Lin, S. Zhang, R. Lake, and J. Shi, Na t. Commun. 7, 10858 (2016). [9] K. Uchida, S. Takahashi, K. Harii, J. Ieda, W. Koshibae, K. Ando, S. Maekawa, and E. Saitoh, Nature 455, 778 (2008). [10] O. Mosendz, J. E. Pearson, F. Y. Fradin, G. E. W. Bauer, S. D. Bader, and A. Hoffmann, Phys. Rev. Lett. 104, 046601 (2010). [11] L. J. Cornelissen, J. Liu, R. A. Duine, J. B. Youssef, and B. J. van Wees, Nat. Phys. 11, 1022 (2015). [12] L. J. Cornelissen and B. J. van Wees, Phys. Rev. B 93, 020403 (2016). [13] L. J. Cornelissen, J. Shan, and B. J. van Wees, Phys. Rev. B 94, 18040 2 (2016). [14] A. V. Chumak, A. A. Serga, M. B. Jungfleisch, R. Neb, D. A. Bozhko, V. S. Tiberkevich, and B. Hillebrands, Appl. Phys. Lett. 100, 082405 (2012). [15] Y. Onose, T. Ideue, H. Katsura, Y. Shiomi, N. Nagaosa, and Y. Tokura, Science 329, 297 (2010). [16] G. P. Zhang, W. Hü bner, G. Lefkidis, Y. Bai, and T. F. George, Nat. Phys. 5, 499 (2009). [17] S. Murakami and A. Okamoto, J. Phys. Soc. Jpn. 86, 011010 (2016). [18] G. G. Siu, C. M. Lee, and Y. Liu, Phys. Rev. B 64, 094421 (2001). [19] H. Wu, L. Huang, C. Fang, B. S. Yang, C. H. Wan, G. Q. Yu, J. F. Feng, H. X. Wei, and X. F. Han, Phys. Rev. Lett. 120, 097205 (2018). [20] C. Y. Guo et al. , Phys. Rev. B 98, 134426 (2018). [21] C. Y. Guo et al. , Nat. Electron. 3, 304 (2020). [22] Z. R. Yan, C. H. Wan, and X. F. Han, Phys. Rev. Appl. 14, 044053 (2020). 10 [23] C. Y. Guo, C. H. Wan, M. K. Zhao, H. Wu, C. Fang, Z. R. Yan, J. F. Feng, H. F. Liu, and X. F. Han, Appl. Phys. Lett. 114, 192409 (2019). [24] J. Cramer et al. , Nat. Commun. 9, 1089 (2018). [25] L. J. Cornelissen, J. Liu, B. J. van Wees, and R. A. Duine, Phys. Rev. Lett. 120, 097702 (2018). [26] W. Chen, M. Sigrist, J. Sinova, and D. Manske, Phys. Rev. Lett. 115, 217203 (2015). [27] W. Chen, M. Sigrist, and D. Manske, Phys. Rev. B 94, 104412 (2016). [28] L. J. Cornelissen, K. J. H. Peters, G. E. W. Bauer, R. A. Duine, and B. J. van Wees, Phys. Rev. B 94, 014412 (2016). [29] C. O. Avci, E. Rosenberg, L. Caretta, F. Bü ttner, M. Mann, C. Marcus, D. Bono, C. A. Ross, and G. S. D. Beach, Nat. Nanotechnol. 14, 561 (2019). [30] S. Vé lez et al. , Nat. Commun. 10, 4750 (2019). [31] S. Ding et al. , Phys. Rev. B 100, 100406 (2019). [32] H. Wang et al. , Phys. Rev. Lett. 124, 027203 (2020). [33] H. T. Nembach, J. M. Shaw, M. Weiler, E. Jué , and T. J. Silva, Nat. Phys. 11, 825 (2015). [34] J.-H. Moon, S. -M. Seo, K. -J. Lee, K. -W. Kim, J. Ryu, H. -W. Lee, R. D. McMichael, and M. D. Stiles, Phys. Rev. B 88, 184404 (2013). [35] S. Dushenko, M. Hokazono, K. Nakamu ra, Y. Ando, T. Shinjo, and M. Shiraishi, Nat. Commun. 9, 3118 (2018). [36] S. Ok, W. Chen, M. Sigrist, and D. Manske, J. Phys.: Condens. Matter 29, 075802 (2016). [37] A. K. Galwey and M. E. Brown, Thermochim. Acta 386, 91 (2002). Acknowledgements : This work was fi nancial supported by the National Key Research and Development Program of China [MOST Grant No. 2022YFA1402800], the National Natural Science Foundation of China [NSFC, Grant No. 51831012, 12134017], and partially supported by the Strategic Priority Resear ch Program (B) of Chinese Academy of Sciences [CAS Grant No. XDB33000000, Youth Innovation Promotion Association of CAS (2020008)]. Contributions : X.F.H. led and was involved in all aspects of the project. Y.Z.W., J. D. and P. C. deposited stacks and fabri cated devices. Y.Z.W. and C.H.W. conducted magnetic and transport property measurement. T.Y.Z., C.H.W. and Y.Z.W. contributed to modelling and theoretical analysis. C.H.W., Y.Z.W., T. Y. Z., G. Q. Y. and X.F.H. wrote the paper. X.F.H. and C.H.W. supervised and designed the experiments. All the authors contributed to data mining and analysis. Conflict of Interests : The authors declare no competing interests. Methods : Sample Preparation: The Si/SiO 2//Pt(10)/Y 3Fe5O12 (YIG)(80)/Pt(5 nm) heterostructures are deposited by ultrahigh vacuum magnetron sputtering system (ULVAC -MPS -400-HC7) with a base pressure ＜5× 10-6 Pa. The bottom Pt Hall bar (B -Pt) with dimensions of 20×200 μm2 was first fabricated on substrates 11 by standard photolithography, followed by deposition of 80 nm YIG film. After deposition, a high-temperature annealing was carried out in an oxygen atmosphere to improve the crystalline quality of both YIG and Pt/YIG interface. Finally, another round of deposition and photolithography was carried out to fabricate top Pt Hall bar (T -Pt) with same dimensions. The terminal of T -Pt and B -Pt Hall bars are designed away from each other allow two Hall bars being input and detected independently. For the spin pumping device, B -Pt and T -Pt are fabricated into two independent stripes with dimensions of 10×360 μm2 by the above mentioned method. And an 80 nm Au co -planar wave guide (CPW) was deposited afterwards. The two Pt strips are placed in the gap of the CPW. Measurement of Magnetic Property: The M -H hysteresis was measured with a vibrating sample magnetometer (VSM, MicroSense EZ - 9) with field applied parallel to the film plane (IP curve) or perpendicular to film plane (OOP curve). Measurement of Transport Property: All the magnon mediate curr ent drag (MCD) and spin magnetoresistance (SMR) test were carried out in a physical property measurement system (PPMS -9 T, Quantum design) with magnetic field up to 9 T and temperatures down to 1.8 K. During measurements, the input current was supplied by a Keithley 2400 source -meter while a Keithley 2182 nanovoltmeter detected the corresponding voltage. The gate voltage was provided by another Keithley 2400 across the Hall channel of T -Pt and B -Pt (grounded) Hall bars. The magnetic field was fixed at 1 T a nd sample rotated in xoy, xoz or yoz plane. For angular dependent of MCD signal measurements, the input current ( Iin) was applied in the long axis of B -Pt Hall bar and the voltage signal (Δ V) was picked up alone the long axis of T -Pt Hall bar. Then the ma gnitude of MCD voltage Vdrag under certain Iin was obtained by fitting the ∆𝑉−𝛾 curves measured at different 𝐼in with ∆𝑉=𝑉drag cos2𝛾. The magnon drag parameter was then calculated by 𝛼≡𝑉drag 𝐼in𝑅T−Pt, where 𝑅T−Pt is the resistance of T -Pt electrode. 12 And for SMR test the resistance of the B -Pt (T -Pt) electrode was measured by four -terminal method, and the angular dependence of change in RPt (ΔRPt-γ) was well fitted by ∆𝑅Pt= 𝑅SMR cos2𝛾 and the SMR ratio was thus ob tained by |𝑅SMR 𝑅Pt⁄ |. The spin pumping test was carried out at room temperature in a home -build electromagnet with magnetic up to ~ 3500 Oe. A signal generator (ROHDE&SCHWARZ SMB 100A) supplies a microwave signal modulated with a 1.172 kHz signal to CPW and the voltage signal was picked up by a lock -in amplifier (Stanford SR830), while external magnetic field H applied perpendicular to the direction that spin pumping voltage was picked up. To minimize interference, 𝑉g was provided by dry batteries during sp in pumping measurements. Data availability : The data that support the findings of this study are available from the corresponding author upon reasonable request.

2023-01-13

Magnon transistors that can effectively regulate magnon transport by an electric field are desired for magnonics which aims to provide a Joule-heating free alternative to the conventional electronics owing to the electric neutrality of magnons (the key carriers of spin-angular momenta in the magnonics). However, also due to their electric neutrality, magnons have no access to directly interact with an electric field and it is thus difficult to manipulate magnon transport by voltages straightforwardly. Here, we demonstrated a gate voltage ($V_{\rm g}$) applied on a nonmagnetic metal/magnetic insulator (NM/MI) interface that bended the energy band of the MI and then modulated the possibility for conduction electrons in the NM to tunnel into the MI can consequently enhance or weaken the spin-magnon conversion efficiency at the interface. A voltage-controlled magnon transistor based on the magnon-mediated electric current drag (MECD) effect in a Pt/Y$_{\rm 3}$Fe$_{\rm 5}$O$_{\rm 12}$ (YIG)/Pt sandwich was then experimentally realized with $V_{\rm g}$ modulating the magnitude of the MECD signal. The obtained efficiency (the change ratio between the MECD voltage at $\pm V_{\rm g}$) reached 10%/(MV/cm) at 300 K. This prototype of magnon transistor offers an effective scheme to control magnon transport by a gate voltage.

Voltage-Controlled Magnon Transistor via Tunning Interfacial Exchange Coupling

2301.05592v1

Theory for electrical detection of the magnon Hall eect induced by dipolar interactions Pieter M. Gunnink,1,Rembert A. Duine,1, 2and Andreas R uckriegel3 1Institute for Theoretical Physics and Center for Extreme Matter and Emergent Phenomena, Utrecht University, Leuvenlaan 4, 3584 CE Utrecht, The Netherlands 2Department of Applied Physics, Eindhoven University of Technology, P.O. Box 513, 5600 MB Eindhoven, The Netherlands 3Institut f ur Theoretische Physik, Universit at Frankfurt, Max-von-Laue Strasse 1, 60438 Frankfurt, Germany (Dated: April 26, 2021) We derive the anomalous Hall contributions arising from dipolar interactions to diusive spin transport in magnetic insulators. Magnons, the carriers of angular momentum in these systems, are shown to have a non-zero Berry curvature, resulting in a measurable Hall eect. For yttrium iron garnet (YIG) thin lms we calculate both the anomalous and magnon spin conductivities. We show that for a magnetic eld perpendicular to the lm the anomalous Hall conductivity is nite. This results in a non-zero Hall signal, which can be measured experimentally using Permalloy strips arranged like a Hall bar on top of the YIG thin lm. We show that electrical detection and injection of spin is possible, by solving the resulting diusion-relaxation equation for a Hall bar. We predict the experimentally measurable Hall coecient for a range of temperatures and magnetic eld strengths. Most strikingly, we show that there is a sign change of the Hall coecient associated with increasing the thickness of the lm. I. INTRODUCTION One of the earliest successes of the concepts of geom- etry and topology in condensed matter was the explana- tion of the anomalous Hall eect in terms of the Berry phase. The anomalous Hall eect was therefore a step- ping stone for further understanding of geometrical and topological eects, such as the quantum Hall eect [1]. Since it is a geometrical eect, the anomalous Hall eect is not restricted to electronic systems. Indeed, it has also been observed for other types of carriers, such as phonons and photons [2{4]. Since spin waves, or magnons, are the carriers of angular momentum in ferromagnets, the question thus naturally arises if a magnon analogue of the anomalous Hall eect can also exist. Continuing the analogy with the anomalous Hall eect, the magnon Hall eect could lead to further understanding of topology in magnonic systems. Previously, a thermal magnon Hall eect has been proposed, where magnons are the heat carriers. First predicted for chiral quantum magnets [5], it was subse- quently observed in Lu 2V2O7[6, 7]. In these systems the chiral nature of the spin waves provides the time- reversal symmetry breaking that is necessary for a nite anomalous Hall response. For forward volume magneto- static spin waves in a thin-lm ferromagnet a thermal magnon Hall eect has also been proposed [8, 9], where the dipole-dipole interaction provides the required sym- metry breaking. A transverse thermal Hall conductivity has also been calculated for this system [10], but has not yet been measured experimentally. This is most likely p.m.gunnink@uu.nldue to the small transverse thermal conductivities pre- dicted for the most commonly used insulating ferromag- net, yittrium iron garnet (YIG) [6]. Moreover, phonons also contribute to the thermal Hall eect, and it might therefore be hard to disentangle the contributions of the two heat carriers. An eort has been made by Tanabe et al.[11] to excite spin waves using a coplanar waveguide and measure the temperature gradient perpendicular to the propagation direction. However, they were only able to measure a transverse temperature gradient in the un- saturated regime, which can therefore not directly be at- tributed to magnons. Recent advances have shown that it is possible to elec- trically inject and detect spin waves using metallic leads [12]. This has opened the way to electrically measure the magnon Hall eect. However, a complete picture of the interaction between the electrical detection and the Hall eect is still lacking. Electrical detection via metal strips can signicantly modify magnon transport prop- erties [13], and it is not clear if a nite magnon Hall response can still survive. In this work we therefore de- velop a theory for the electrical detection of the magnon Hall eect in order to determine if the magnon Hall eect can be measured electrically. We numerically calculate the Hall response, using the diusion-relaxation equation for magnons in a Hall bar geometry, as depicted in Fig. 1. In order to determine the magnitude of the expected Hall response two con- tributing factors need to be calculated: (1) the magnon spin and anomalous conductivities and (2) the bound- ary conditions which incorporate the electrical detection. We numerically calculate these using a microscopic de- scription. Starting from the Keldysh quantum kinetic equations [14], we derive the equation of motion of the magnon distribution function to leading order in a semi-arXiv:2104.11304v1 [cond-mat.mes-hall] 22 Apr 20212 0.5 0.00.51.01.52.02.53.03.54.0 FIG. 1. The Hall bar with electrical injection and detection of spin currents using Permalloy strips on top of YIG. Spin cur- rent is injected by the Py strip 1, and is detected by the strips 2, 3 and 4. The colorscale shows the diusion of the magnon chemical potential throughout the lm, obtained by solving the diusion-relaxation equations as described in Sec. III B. The Hall bar has size MM, and the Py detectors and de- tectors have size LaLb, whereLbLa. The magnetic eld is oriented out of plane, as shown in Fig. 2, where also the interface between the YIG and the Py is shown in more detail. classical expansion in gradients. This allows us to sepa- rate the spin diusion and anomalous Hall contributions to the spin current. This work is ordered as follows. We rst discuss the specic Hall geometry required to measure a nite magnon Hall eect in Sec. II. Next, in order to deter- mine the magnitude of the magnon Hall eect we derive the equations of motion for the spin density in Sec. III. We also show how the equations of motion have to be modied if a metallic lead is interfaced with the system, in order to detect or inject spins. From the equation of motion we derive a diusion-relaxation equation, which fully describes the magnon diusion and relaxation in the Hall bar geometry, including boundary conditions. In Sec. IV we show how the conductivities and damping can be numerically evaluated and we discuss results for a typical thin lm of YIG. In Sec. V we solve the diusion- relaxation equation numerically and present our results for a YIG Hall bar, where spin waves are injected and de- tected electrically. A summary and conclusion are given in Sec. VI. In the appendices A-E we give a more de- tailed derivation of the quantum kinetic equations for general bosonic systems, and more details regarding the diusion-relaxation equation and the Hamiltonian. FIG. 2. The considered geometry, with the magnetic eld pointing slightly o the ^ z-axis, as explained in the main text. The Py strip on top of the YIG has a charge current Irunning parallel to the lm, which induces a spin current Jssuch that there is an accumulation of spin at the interface between the YIG and the Py. II. SETUP First, we discuss the experimental setup necessary to measure a magnon Hall eect electrically. We consider a Hall bar geometry, as shown in Fig. 1. There are four ter- minals, formed by metal strips on top of a YIG thin lm. The strips act as injectors and detectors of spin currents. Magnons are injected at terminal 1 and diuse through the lm. They are then detected at terminals 2, 3 and 4. By comparing the detected currents at terminals 2 and 4 a Hall signal can be measured. Note that in electronic Hall experiments terminal 3 is necessary in order for a current to ow, but in our case we have only included it for completeness. The Berry curvature is only non-zero if either time- reversal or inversion symmetry is broken [15]. Breaking these symmetries can be achieved by applying a mag- netic eld perpendicular to the plane, which leads to for- ward volume modes, as was previously suggested by Mat- sumoto and Murakami [8]. Conventionally, one would use the spin Hall eect (SHE) in the metal strips to excite magnons in the YIG lm [12]. However, the polarization of the spin current induced by the SHE is always in-plane [16] and can therefore not excite forward-volume modes in the YIG lm. Instead, we propose to use ferromagnetic Permalloy (Py) strips. If a charge current ows through the Py strip, the anomalous spin Hall eect (ASHE) in- duces a spin current polarized parallel to the magnetiza- tion of the Py strip, as shown in Fig. 2. For suciently large external magnetic elds the magnetization of the Py strips and the YIG will both be aligned to the exter- nal eld. This spin current can therefore excite magnons in the YIG lm. However, the spatial direction of the spin current is JsIM, where IandMare the charge current and magnetization respectively [17, 18]. Therefore, if the magnetic eld is oriented along the ^ z direction and the charge current ows along the ^ ydi- rection the spin current ows along the ^ xdirection. In3 other words, the spin current in the Py strip ows par- allel to the YIG lm and can therefore not enter it to excite magnons. However, one can tilt the magnetic eld slightly o-axis, i.e. o the ^ z-axis, as depicted in Fig. 2. The spin current induced by the ASHE then gains an out-of-plane component and is able to excite magnons in the YIG [18]. At the detectors the opposite process, the inverse ASHE, converts a spin current in a measurable charge current. III. METHOD In this section we consider the microscopic Hamilto- nian for a thin lm of YIG and derive the equations of motion for the spin density. The formalism that we use, however, is completely general and can be applied to any bosonic Hamiltonian with anomalous coecients. We consider a thin lm of YIG, with Nlayers, of thick- nessd=Na, with a magnetic eld perpendicular to the lm. We include both the dipole-dipole and exchange interaction, which gives us a full description of the spin wave dynamics. We apply a Holstein-Primako trans- formation to the Hamiltonian, retain terms up to second order, and Fourier-transform along the x;ydirections. We can then write the quadratic part of the Hamiltonian as Hk=X k by kbk
AkBk By kAkbk by k ; (1) where by k= (by k(z1);:::;by k(zN)) are the creation opera- tors for magnons with the two-dimensional wave vector kandAkandBkareNNmatrices with Nthe number of internal degrees of freedom within a unit cell, which is in our case equivalent to the number of layers. More details are found in Appendix E. We evaluate the dipole- dipole interaction using the Ewald summation method [19]. This allows us to accurately compute the magnon spectrum, even at long wavelengths, where conventional summing methods are slow [20], but where we do expect the Berry curvature to be large [9]. From the anomalous coecients Bk, which are due to the dipole-dipole inter- action, it is clear that spin is not conserved. The dipolar interactions couple the magnons the the lattice, which therefore acts as a spin sink and/or source. We note that the anomalous coecients in the Hamil- tonian create a squeezed magnon state, which is not an eigenstate of the spin in the z-direction [21]. Between a metallic lead and the magnetic system there is thus an interface of a squeezed (the YIG) and a spin state with denite spin in the z-direction (the metallic lead). This leads to corrections to the spin current over the interface, which we show in more detail in Sec. III A. In a bosonic system with anomalous coecients, the Bogoliubov-de Gennes (BdG) Hamiltonian Hkis diago- nalized by a para-unitary transformation [22], such thatTy kHkTk=Ek;Ty kTk=; (2) whereEk= diag E1 k;:::;EN k;E1 k;:::;EN k ,= diag [1;:::;1;1;:::;1] andTkis a para-unitary trans- formation matrix of size 2 N2N. Note that we only haveNdistinct bands, since the bands nandn+Nare related to each other via the para-unitary structure. In order to derive the equations of motion we perform the gradient expansion of the Hamiltonian. We rst de- ne the Berry connection (suppressing the k-label from here onwards) A=iTy(@kT); (3) where2(x;y). Numerically, we calculate the Berry connection using the component-wise form A nm=i Ty(@kH)T nm EnnmEm; n6=m; (4) wheren;m = 1;:::;2N. This form also makes it clear that the Berry connection increases close to band crossings. From the Berry connection we dene the Berry curva- ture for the n-th band as  n= @kA@kA
nn =i AAAA
nn(5) The Berry curvature satises the sum ruleP n  n= 0, wherenis summed over all 2 Nbands. We note that these denitions for the Berry phase and curvature are equivalent to those given by Shindou et al. [23], who were the rst to consider the topology of magnons, and also to those of Lein and Sato [24], who showed rigorously that the concept of the Berry phase can be applied to BdG-type Hamiltonians. Now we are able to derive the equations of motions for general bosonic systems with non-zero anomalous co- ecients. As noted, this is applicable to the magnons described here, but also for other bosonic systems, such as phonons and photons [4, 25], where geometrical ef- fects are also known. We start from the quantum kinetic equations in the Keldysh formalism, which are derived by performing a Wigner transformation and expanding the gradients up to rst order [14]. Moreover, we assume damped quasiparticles in (local) thermal equilibrium. We have relegated the details of this calculation to Appendix A and will only state the equation of motion for the spin densitysz(r;t) here, which is given by @tsz+r
Js= sm; (6) where we have only kept terms up to rst order in the magnon chemical potential m. Here, sdescribes the relaxation rate of the magnons. The spin current Jsis written component-wise as J s=s@rm+H sX "@rm; (7)4 wheresis the magnon spin conductivity, H sis the Hall conductivity and "is the two-dimensional Levi-Civita symbol. The Berry curvature only aects the magnon Hall conductivity H s, and bands with a greater Berry curvature contribute to a larger Hall conductivity. From the Keldysh formalism the coecients s;H sand scan be calculated using the microscopic Hamiltonian, by in- tegrating the relevant quantities over the entire Brillouin zone. We show the details of this calculation in Appendix E. We consider a clean system in the low-temperature limit, such that the dominant damping source is the Gilbert damping [26]. Moreover, we disregard heat trans- port, since long-range magnon transport is dominated by the magnon chemical potential [27]. The complete magnon dynamics are thus given by Eq. (6), where we calculate the transport coecients us- ing the microscopic Hamiltonian. We therefore do not have to rely on tting parameters. A. Metallic lead In order to model the electrical detection and injection, we consider a metallic lead interfaced with the YIG lm, as shown in Fig. 2. As a result of this interface the equa- tions of motion have to be modied, such that we have at the interface between the magnet and the metallic lead that @tsz(r;t) +r
Js= sm+Am+Be+C; (8) whereeis the electron spin accumulation in the lead. We show the detailed derivation of this correction and the coecients A;B andCin Appendix C. The correc- tionAm, withA > 0, describes the relaxation of the magnons into the metallic lead. Beis the injection of spin driven by the chemical potential in the metallic lead. The constant Cis related to the fact that the magnons are squeezed, whereas the spins in the metallic lead are not squeezed. The main correction is a constant injec- tion of angular momentum into the YIG, even with zero chemical potential in the Py lead, which is a characteris- tic feature of elliptic magnonic systems [28]. The source of this spin current is the lattice, which couples to the magnons via the dipole-dipole interaction. The constant Cis therefore zero in the absence of dipolar interactions. There are also corrections due to dipolar interactions to the constants AandB, which are of less importance. In absence of these corrections we would have A=B, such that the spin current is zero when e=m[27]. With the metallic lead modelled, we now have all the necessary parts for a full description of the dynamics of magnons in a Hall bar. B. Diusion-relaxation We now write down the full diusion-relaxation equa- tion, which we solve numerically to give the full descrip-tion of the Hall bar, including electrical injection and detection. Since the Hall conductivities enter through antisymmetric terms in the current, see Eq. (7), these drop out in the nal diusion-relaxation equation, which becomes sr2m= sm: (9) The Hall conductivities only appear in the expressions for the boundary conditions, where we require that the normal component of the current vanishes, i.e. that Js
^n= 0 at the edges of the lm if there is no metallic lead present, where ^nis the normal vector to the boundary. To measure a nite Hall response we consider a Hall bar setup, as shown in Fig. 1. The Hall response can then be measured between terminals 2 and 4. As far as we are aware, there are no analytical solutions for such a geometry. We therefore numerically solve the diusion- relaxation equation, Eq. (9). Specically, we solve the diusion-relaxation equation on the square 0xMand 0yM, where the diusion is given by Eq. (9). We use a Finite Element Method, with a symmetric square grid, implemented in the FreeFEM++ software [29]. At the open boundaries we require that Js
^n= 0. At the injector and detectors we have the boundary condition Js
^n=Jint s(m), where the interface current Jint sis a function of the magnon chemical potential at the interface int mand includes the contributions A;B andCas discussed in Sec. III A. We give the full form of Jint sin Appendix D. We then dene the total spin current injected or detected at Py strip i asIi=R @SiJs
^nds, where@Siis the interface between the Py and the YIG. IV. HALL ANGLE AND DIFFUSION LENGTH With the full description of the transport coecients complete, we now numerically evaluate these using the microscopic Hamiltonian. We have relegated the deriva- tion of these coecients to Appendix B. The parameters used in this work are shown in Table I. We only consider the low-temperature regime T <2 K, since at higher tem- peratures we expect other damping mechanisms besides the Gilbert damping to play a role. Moreover, one might expect the ferrimagnetic branches in the YIG dispersion relation to be relevant at room temperature [30], which are not captured in our model. First, we show the results for the spin diusion length, `m=p s=s, for a lm of thickness N= 75 in Fig. 3. The diusion length peaks for low temperatures, and converges to a constant value in the high temperature regime. This can be explained by the energy dependence of the Gilbert damping: for low temperature only the lowest energy bands contribute, which have the lowest Gilbert damping, since the damping is proportional to energy. The drop-o of the diusion length at low tem- perature and high magnetic eld is explained by the fact that the temperature is not high enough to occupy the5 0.0 0.5 1.0 1.5 2.0 T (K)102103lm(µm)N=75 1800 2000 2200 2400 2600 2800 3000H (Oe) FIG. 3. The spin diusion length lmfor a thin lm of YIG with thickness N=75, for varying magnetic eld strength. The corresponding Hall angle is shown in Fig. 4a. TABLE I. Parameters for YIG used in the numerical calcu- lations in this work. Note that Sfollows from S=Msa3=, where= 2Bis the magnetic moment of the spins, with B the Bohr magneton. We are not aware of any values of the parameters eandIFfor a YIGjPy interface and have there- fore assumed values that are equivalent to the YIG jPlatinum interface. Since the injection and detection is described in lin- ear response, their exact values do not aect the nal results. Quantity Value a 12:376/RingA [31] S 14.2 4Ms 1750 G [32] J 1:60 K [19] G 104[33] IF 102[33] e 8V [27] rst band, and there is thus no transport possible. At ele- vated temperatures we compare the spin diusion length to a simple model that only considers the lowest exchange band of YIG, from which the spin diusion length is es- timated as lm4p J=3kBTMs2 G[27]. We expect this approximation to be only valid for relatively high tem- peratures, where the higher exchange bands are occu- pied, and for thicker lms. We therefore compare this approximation with our calculations at T= 2 K and nd thatlm35µm, whereas our numerical model found lm= 55 µm forN= 150 andH= 1800 Oe. Moreover, as is evident from Fig. 3, our numerically calculated diu- sion length also scales as 1 =p T. For dierent thicknesses (not shown here) the behaviour and order of magnitude of the spin diusion length is similar. Next, we consider the Hall angle, H=H s=s. We compare two lms with thicknesses N= 75 andN= 150 in Fig. 4. It is clear that the Hall angle peaks for small temperature, and tends to a lower constant value for higher temperature. The complete drop-o at T= 0 is explained by the fact that there are no magnons ther-mally excited at zero temperature. In order to further explain these results we rst need to focus on the Berry curvature for these thin lms, since the Berry curvature is directly related to the Hall conductivity in this system. We therefore show the Berry curvature yz nof then-th band in Fig. 5 for these two lms. We can see that the Berry curvature is largest for the lowest band, which we therefore expect to dominate transport. Furthermore, in the dipolar regime, at small wavevectors, the Berry curvature is largest. This explains the temperature de- pendence of Hwe observe in Fig. 4. At low tempera- tures the dipolar magnons dominate transport, and they have a large Berry curvature. Furthermore, the exchange bands naturally have a larger contribution to transport than the dipolar magnons (not shown here). As the tem- perature increases, the ratio between the exchange and dipolar magnons shift towards the exchange magnons, in- creasing the magnon spin conductivity, but not the Hall conductivity. For the lm with thickness N= 150, shown in Fig. 4b, the Hall angle is negative for low magnetic eld. Here the shaded region indicates the error from integrating the Berry curvature nover the Brillouin zone. The larger errors can be explained from the behaviour of the Berry curvature close to band crossings, as shown in Fig. 5b. The Berry curvatures grows at band crossings|but never diverges, since none of the bands are ever degenerate. This can also be seen from Eq. (4), where it is clear that the Berry connection matrix and therefore the Berry cur- vature of the band nis inversely proportional to the en- ergy gap. Integrating such a function is numerically very costly, and we only reach the precision as indicated by the shaded region. The avoided band crossings in the dispersion, which lead to an increased Berry curvature, are only present for thicker lms ( N&150). The results for the Berry curvature can directly be compared to the Berry curvature as obtained by Okamoto and Murakami [34], who showed the same behaviour as we have shown for theN= 150 lm, with an enhanced Berry curvature at the band crossings and a negative Berry curvature for some of the higher bands. The negative Hall angle can be explained from the neg- ative Berry curvature, which is present for N= 150, but not for N= 75, as was shown in Fig. 5. This sign switch of the Hall angle is similar to what was observed by Hirschberger et al. [35] in measuring the thermal Hall eect in a Kagome magnet. For the forward volume modes, the magnetic eld acts as a way to introduce a nite energy shift of the bands. This can be used to explain the behaviour of the spin diusion length as shown in Fig. 3. A higher magnetic eld reduces the diusion length, since by shifting all the bands the magnetic eld changes which bands are occu- pied and therefore contribute. For the Hall angle, H, the magnetic eld dependence is more complicated, at least for smaller magnetic elds. As a function of mag- netic eld strength, the Hall angle rises rapidly, until it peaks for a eld of strength 2400 Oe, after which6 0.0 0.5 1.0 1.5 2.0 T (K)0.00.51.01.5θH×10−7 (a) N=75 0.0 0.5 1.0 1.5 2.0 T (K)−2024θH×10−6 (b) N=150 1800 2000 2200 2400 2600 2800 3000H (Oe) FIG. 4. The Hall angle H=H s=sfor two dierent lm thicknesses, (a) N= 75 and (b) N= 150. The shaded area indicates the error, which results from a slowly converging integral over the Brillouin zone. 102103104105106 k(cm−1)0.00.51.01.52.0E (GHz)(a) N=75 102103104105106 k(cm−1)0.00.51.01.52.0E (GHz)(b) N=150 10−2100102104Ωn(a.u.) −105−104−103−102−101−100100101102103104105106Ωn(a.u.) FIG. 5. The Berry curvature yz nper band for the forward-volume modes of a thin lm with (a) N= 75 and (b) N= 150 layers, and a magnetic eld strength H= 1800 Oe. Note the more complicated Berry curvature structure for N= 150, which is not present for the N= 75 thin lm and is due to the band crossings. We also note that the Berry curvature is negative for certain bands for N= 150, but for none for N= 75. it drops again. For higher elds, the magnetic eld es- sentially shifts the ratio between which type of magnons contribute at a given energy: the exchange or the dipolar magnons. This does not explain the low magnetic eld behaviour though, since we expect this behaviour to be (roughly) linear. Further research is needed to under- stand this in more detail. Since we have determined that thickness plays a role in the Hall eect of YIG, we also show the results for a xed magnetic eld, with increasing thickness in Fig. 6. It can clearly be observed that the Hall angle increases for thicker lms. However, one should be aware that this is still assuming that there is no diusive transport along the lm normal, i.e. the spin diusion length is larger than the lm thickness. The spin diusion length for YIG thin lms at the temperature range considered here has not yet been measured, but for T= 30 K itis roughly 5 µm [36], which would make our description valid for thin lms up to N= 5000. We have now calculated the transport coecients s;H sand s. Not discussed in the main text are the coecients A;B andCthat govern spin injection at the metallic lead interface, which we show in App. C. Next, we solve the diusion-relaxation equation, in order to de- termine if the magnon Hall eect can be measured elec- trically. V. DIFFUSION IN THE HALL BAR Experimentally, the main observable is the dierence between the spin currents detected by terminals 2 and 4. We dene a Hall coecient as the signal dierence7 0.0 0.5 1.0 1.5 2.0 T(K)10−1010−910−810−710−6θHN 150 100 75 50 25 FIG. 6. The Hall angle HforH= 2600 Oe, as a function of temperature and for varying thicknesses. We were not able to numerically calculate Hfor thicker lms, so it is not clear if the Hall angle will continue to increase. 0.0 0.5 1.0 1.5 2.0 T (K)02468∆I×10−8 1800 2000 2200 2400 2600 2800 3000H (Oe) FIG. 7. The Hall coecient
I, which follows from the nu- merical solution to the diusion-relaxation equation for a Hall bar geometry. The thickness of the lm is N= 75 and this can therefore be directly compared to the Hall angle Hin Fig. 4a. From this comparison it is clear that a Hall response can be measured, and that His a direct predictor of
I. between detectors 2 and 4,
I=I2I4 I2+I4: (10) In order to conrm that a non-zero Hall angle Hre- sults in a nite
Iwe numerically solve the diusion- relaxation equation. We choose M= 8µm,La= 3µm andLb= 0:1µm, which are the same dimensions used by Daset al. [18] to measure the planar Hall eect in YIG. The distribution of the chemical potential for a typical system is shown in Fig. 1. The chemical potential dif- fuses through the lm and gets picked up by the three detectors. Note that the dierence between the currents picked up by detectors 2 and 4, i.e.
I, is too small to be visible on the color scale of Fig. 1. We then calculate the Hall coecient
IforN= 75 and show the results in Fig.7. These results can be com- pared to the Hall angle, H, in Fig. 4a. From this com-parison it is clear that the Hall angle His directly related to the Hall coecient
I. We see little to no eect from the magnon relaxation, since the spin diusion length is much longer than the size of the Hall bar. Most im- portantly, there are no (large) corrections from interface eects due to the electrical injection and detection. This is also the case for dierent thicknesses. We therefore conclude that the magnon Hall eect can in principle be measured electrically in a Hall bar geometry. VI. CONCLUSION AND DISCUSSION We have derived and calculated the anomalous Hall conductivity for magnons in a thin lm of YIG, us- ing a microscopic model. Furthermore, we have shown that a non-zero anomalous Hall conductivity results in a measurable signal in a Hall bar setup and can be mea- sured electrically. The magnon Hall eect has previ- ously only been measured thermally in materials with a Dzyaloshinskii-Moriya spin-orbit interaction [6], but with a Hall bar setup as discussed here this magnon Hall eect could also be measured electrically in YIG. Using realistic parameters we have calculated the size of the expected Hall angle, and its dependency on tem- perature and magnetic eld. Moreover, we have shown that for thicker lms of YIG, there is a sign change in the Hall angle as a function of the magnetic eld, which would be a strong experimental indicator of the magnon Hall eect. The presented method can be applied to any bosonic system with anomalous coecients to determine anoma- lous transport properties. In fact, the physical origin of the anomalous transport properties discussed here are the dipole-dipole interactions, which are universally present in any magnetic system. As such, this method can be applied to a wide range of magnetic materials. In order to measure this eect it is possible to use the fact that the sign of the Hall angle switches as the eld is reversed. Therefore, by comparing measurements with opposite eld, the anomalous contributions can be iso- lated. This is especially useful since the spin diusion and relaxation means that the distance between the in- jector at lead 1 and the detectors at leads 2 and 4 is critical. As was shown by Takahashi and Nagaosa [37] and Okamoto et al. [38], for magnetoelastic waves the Berry curvature is enhanced at the crossing of the magnon and phonon branches. This could therefore serve to further enhance the magnon Hall eect discussed here. The in- clusion of magnon-phonon coupling on our formalism is left for future work. ACKNOWLEDGMENTS R.D. is member of the D-ITP consortium, a program of the Dutch Organization for Scientic Research (NWO)8 that is funded by the Dutch Ministry of Education, Cul- ture and Science (OCW). This project has received fund- ing from the European Research Council (ERC) under the European Union's Horizon 2020 research and inno- vation programme (grant agreement No. 725509). This work is part of the research programme of the Founda- tion for Fundamental Research on Matter (FOM), which is part of the Netherlands Organization for Scientic Re- search (NWO). We thank Timo Kuschel for discussions and Ruben Meijs for doing his thesis work on this sub- ject. Appendix A: Quantum Kinetic Equations In this appendix we derive the equation of motion for the spin density of a bosonic Hamiltonian. We start from the quantum kinetic equations:  ^^H ^GK=^K^GA+^R^GK; (A1) ^GK ^^H =^GR^K+^GK^A; (A2) where hats indicate matrices in space and time, ^=(rr0)(tt0)i~@t0,^R=A=Kare the re- tarded/advanced/Keldysh self-energies, ^GR=A=Kare the retarded/advanced/Keldysh Green's functions and = diag [1;:::;1;1;:::;1]. We apply a Wigner transforma- tion, dened as A(r;t;p;") =Z dr0Z dt0 ^A r+r0 2;t+t0 2;rr0 2;tt0 2 ei(k
r0!t0) and expand up to rst order in ~, such that we have (suppressing all labels from here on)  "HR+i~ 2@t +i 2(rpH)
rr GK=KGA; (A3) GK "HAi~ 2 @t i 2 rr
(rpH) =GRK; (A4)where we assume that the Hamiltonian does not depend explicitly on position or time, i.e. H(r;t;k;!) =H(k) and have used arrows to indicate to which function the derivative applies, if there are ambiguities. Furthermore, we dene a covariant derivative as DkETy(@kH)T=@kE+iEAiEA:(A5) We introduce the transformed Green's functions gR=A=K=T1GR=A=K Ty
1and self-energies R=A=K=TyR=A=KTand assume damped quasipar- ticles in (local) thermal equilibrium, such that R=A(k;!) =i[mm(k;!) + mr(k;!)] ; (A6) K(r;k;!) =2imm(k;!)Fn(r;!) 2imr(k;!)Fm=0 n (r;!);(A7) where  mn(k;!) =mn( (k;!) 1nN; (k;!)N+ 1n2N; with2fmr;mmgrepresenting the magnon relaxation processes (which do not conserve spin) and magnon- magnon interactions (which conserve spin) respectively. The distribution function is dened as Fmn(r;!) =mn( fB(r;!) 1nN; fB(r;!)N+ 1n2N; wherefB= coth ~!m 2kBT is the symmetrized Bose-Einstein distribution. The distribution function Fm=0 n (r;!) describes the relaxation of magnons to the lattice. For brevity, we write n(k;!) = mr n(k;!) + mm n(k;!). The retarded and advanced Green's functions are then given by gR=A=nm n(~!in)En: (A8) For the Keldysh Green function we rst solve the diago- nal component of the distribution function, fni~ 2gK nn, using the dierence between Eqs. (A1) and (A2), such that @tfn+X @rj n=2mr n ~" fn~n (~!nEn)2+ 2nFm=0 n# 2mm n ~" fn~n (~!nEn)2+ 2nFn# ; (A9) where the current density j n=n ~(@kEn)fn+ni 4X m6=n (DkE)nmgK mn+gK nm(DkE)mn ; (A10)9 has contributions from the o-diagonal components. We now assume that there is local thermal equilibrium, and thus that the local distribution function fncan be described by small corrections fnon top of the thermal equilibrium. This is possible because the spin conserving processes (represented by mm) are much faster than the non-spin-conserving processes (represented by mr). Thus, we disregard the Fm=0 n term in Eq. (A9) and make the ansatz fn=~n (~!nEn)2+ 2nFn+fn; (A11) wherefnis at least one order higher in gradients. In a steady state (such that @tfn= 0) we further note that from Eq. (A9) it is clear that X @rj n=2n ~ fn+mr n (~!nEn)2+ 2n FnFm=0 n
! : (A12) This can then be solved up to rst order in gradients by inserting the ansatz, Eq. (A11), into the current density, Eq. (A10), and using the fact that gK nmis one order higher in gradients and can thus be discarded. Then we nd fn=n~ 2(@kEn)1 (~!nEn)2+ 2n(@rFn)~mr n (~!nEn)2+ 2n FnFm=0 n
: (A13) In order to nd gK nmwe consider the sum of Eqs. (A1) and (A2) and nd for m6=nthat [2~!nEnmEm+i(nm)]gK nm=i 2X X l n(DkE)nl @rgK lm
m @rgK nl
(DkE)lm :(A14) It is convenient to proceed in the quasiparticle limit (lim n!0+), where lim !0+fn=(!nEn=~)Fn(!) +fn: (A15) We now use the fact that gK nmis one order higher in gradients than fn, and as such can write gK nm=1 ~i mEnnEmX @r(DkE)nmh m(!nEn=~)Fn(!) +n(!mEm=~)Fm(!)i ; (m6=n);(A16) where we have used the diagonal components fnto rewrite Eq. (A14), only keeping terms up to rst order in gradients. Using the denition of the covariant derivative in Eq. (A5) we now write the current as j n=n(@kEn)(!nEn=~) Fnmr n n FnFm=0 n
 1 2n~X (@kEn) @kEn
(!nEn=~)@rFn +i 4~X m6=nX (nmEmEn) A mnA nmA mnA nm
@r[n~(!mEm=~)Fm+m~(!nEn=~)Fn]; (A17) such that we now have a full description of the equation of motion, Eq. (A9) for the distribution function of the magnons. Note that the rst term in Eq. (A17) will be zero if integrated over, due to inversion symmetry. We continue with the spin density, which is dened as sz(r;t) =i~ 4Trh ^GKi ; =i~ 4Zddk (2)dZd! 2Tr TyTgK ; (A18) such that @tsz(r;t) =1 2Zddk (2)dZd! 2X n TyT
nn"X @rj n+ 2mr ~ fn~n (~!nEn)2+ 2nFm=0 n!# ; (A19) where we have only kept terms up to rst order in gradients. Since the processes described by mmalways conserve spin and because we assume them to approximately conserve momentum, we furthermore disregard all terms related to mm, such that n= mr n. Its inclusion up to this point was however necessary, since without it a local thermal equilibrium cannot be properly dened and a current density cannot be expressed in terms of the magnon chemical potential.10 Appendix B: Coecients From here on, we assume Gilbert damping for the magnon relaxation process, such that mr(k;!) = 2G~![26], whereGis the bulk Gilbert damping parameter. With the generic equation of motion, Eq. (A19), we now derive the equation of motion up to linear order in the magnon chemical potential, giving @tsz(r;t) +X @rJ s= sm; (B1) whereJ s=@rm+P @rm, with =1 32~GkBTZd2k (2)2X n TyT
nn(@kEn)2 EncschEn 2kBT2 ; (B2) =1 32kBT~Zd2k (2)2X n;m;m6=n n TyT
nn+m TyT
mm
(nmEmEn)  mcschEn 2kBT2 ; (B3) s=1 2kBTZd2k (2)2Zd! 2X n TyT
nn(2G~!)2 (~!nEn)2+ (2G~!)2cschn~! 2kBT2 : (B4) Here we have disregarded the mmterm, since magnon- magnon scattering preserves momentum and should therefore not contribute to the magnon spin conductivity . We then have s=xx=yyandH s=xy, since the system is rotationally invariant. In order to calculate these coeencts we diagonalize the Hamiltonian Hwith a paraunitary matrix T, which also gives the energies E. Moreover, we construct @kH, such that we calculate the Berry phase and subsequently the Berry curvature using Eq. (4). These terms are further shown in Appendix E. We can then integrate the coecients H s;sand sover the entire Brillouin zone, where we use the translation invariance to employ the one-dimensional Gauss{Kronrod quadrature formula, which also gives an error estimate. These results are shown in Sec. IV. Appendix C: Metallic lead We now consider how the equation of motion for the spin density has to be modied if a metallic lead is inter-faced to the ferromagnet. Attaching a metallic lead, the self-energies are modied such that R=A=K= R=A=K bulk+ R=A=K IF , with R=A IF(r;t;k;!) =iIF(~!e); (C1) K IF(r;t;k;!) = 2R IFFe(!); (C2) where Fe(r;!) =nm8 < :cothh ~!e 2kBTi 1nN; cothh ~!e 2kBTi N+ 1n2N; (C3) andIFis the interfacial Gilbert damping. The equation of motion for the spin density, Eq. (A19), is then modied to@tsz+r
Js= s+ IF s, where IF s=1 4Zddk (2)dZd! 2Tr TyK IFTgATyK IFTgR+TyR IFTgKTyA IFTgK : (C4) Noting that, up to lowest order in the interfacial coupling, the Green's functions gR=A=Kare unchanged by the interfacial self-energies, we can further write this as (in the quasiparticle limit) IF s=IF 2~Zddk (2)dTr Ty(Ee)TF(E)Ty(Ee)Fe(E)T
 : (C5) We again keep only terms linear in mande, such that we can write IF s=Am+Be+C, with11 A=IF 4~kBTZd2k (2)2Tr" TyETcschE 2kBT2# ; (C6) B=IF 2~Zd2k (2)2Tr" TyTcothE 2kBT 1 2kBTEcschE 2kBT2 + cothE 2kBT# ; (C7) C=IF 2~Zd2k (2)2Tr TyETE
cothE 2kBT : (C8) Appendix D: Boundary conditions With the equation of motion for the spin density com- pletely determined, we can now consider the boundary condition for the spin density in the Hall bar geome- try. For the metal strips we assume a thin strip, where LaLband the long side Lbinterfaces the Hall bar, as shown in Fig. 1. Then the detector can be described by Eq. (8), with the boundary condition that the current at its interface with the main region is continuous. Thus we have, Z @SidsJs
^n=Z SidS[(s+A)m+Be+C];(D1) whereSiis the area of detector i. Note thate for the detectorse= 0. We now Taylor expand the chemi- cal potential in the detector strip perpendicular to the interface, and integrate over the short side of the strip, keeping only terms linear in La, which gives the bound- ary condition Z @SidsJs
^n=LaZ @Sids[(s+A)m+Be+C]; (D2) where we have required that Js
^n= 0 at the other three sides of the detector. The boundary condition can now be identied as Js
^n=Jint s(m); (D3) where Jint s(m) =La[(s+A)m+Be+C]: (D4) Appendix E: Hamiltonian In order to determine the dynamics of the magnons in the YIG, we describe this system using the Heisenberg spin Hamiltonian [39] H=1 2X ijJijSi
SjHe
X iSi 1 2X ij;i6=j2 jRijj3h 3 Si
^Rij Sj
^Rij Si
Sji ; (E1)where the sums are over the lattice sites Ri, with Rij= RiRjand ^Rij=Rij=jRijj. We only consider nearest neighbour exchange interactions, so Jij=Jfor near- est neighbours and 0 otherwise. Here = 2Bis the magnetic moment of the spins, with B=e~=(2mec) the Bohr magneton. Heis the external magnetic eld, which we take strong enough to fully saturate the ferromagnet. We apply the Holstein Primako transformation up to quadratic order, S+ i=p 2Sbi;S i=p 2Sby i;Sz i=Sby ibi(E2) to the Heisenberg spin Hamiltonian, Eq. (E1), and ap- ply the Fourier transformation in the xy-plane, intro- ducing k= (kx;ky). The coordinate system used is summarized in Fig. 2 in the main text. We can now write the quadratic part of the Hamiltonian in the basis (bk(z1);:::;b k(zN);by k(z1);:::;by k(zN))Tas Hk=AkBk By kAk ; (E3) where the amplitude factors are Ak(zij) =X rijeik
rA(zizj;r); =ij" h+SX nDzz 0(zin)# S 2[Dyy k(zij) +Dxx k(zij)] +SJk(zij);(E4) Bk(zij) =X rijeik
rB(zizj;r); =S 2[Dxx k(zij)Dyy k(zij) +iDxy k(zij)];(E5) where Jk(zij) =J[ij(6j1jN 2 cos(kxa)2 cos(kya))ij+1ij1];(E6) rij= (xij;yij) andD k(zij) describes the dipole-dipole interaction. For the Berry curvature we need to calculate @kHk, where2(x;y). This is given by @kHk=@kAk@kBk @kBy k@kAk ; (E7)12 where @kAk(zij) =S 2[@kDyy k(zij) +@kDxx k(zij)] + 2SJasin(k a); (E8) @kBk(zij) =S 2[@kDxx k(zij)@kDyy k(zij) +i@kDxy k(zij)]; (E9) For the dipolar sums we apply the Ewald summation method, as previously developed by Kreisel et al. [19], and nd Dzz k(zij) =2 a2X g8p" 3pep2q2jk+gjf(p;q) 42 3r "5 X r jrijj23z2 ij
cos (kxxij) cos (kyyij)'3=2(jrijj2"); (E10) Dyy k(zij) =2 a2X g4p" 3pep2q2(ky+gy)2 jk+gjf(p;q) 42 3r "5 X r jrijj23y2 ij
cos (kxxij) cos (kyyij)'3=2(jrijj2"); (E11) Dxy k(zij) =2 a2X g(ky+gy)(kx+gx) jk+gjf(p;q) 4"5=22 pX ryijxijsin(kxxij) sin(kyyij)'3=2(jrijj2"); (E12) where '3=2(x) =ex3 + 2x 2x2+3pErfc (px) 4x5=2(E13) andq=zijp",p=jk+gj=(2p") andf(p;q) =e2pqErfc(pq) +e2pqErfc(p+q). The sums are either over the real space lattice or the reciprocal lattice, where the reciprocal lattice vectors are gx= 2m,gy= 2n,fm;ng2Z. "determines the ratio between the reciprocal and real sums. We choose "=a2, such that 2 pq1 and exp[2pq] converges quickly. Note that Dxx kfrom the symmetry Dyy k=Dxx k(kx!ky;ky!kx). Taking the derivatives w.r.t.13 kxandkywe nd @kyDzz k(zij) =2 a2X g16pp" 3pep2q2@p @ky+p2p"@f @ky+ky+gy 2p"pf(p;q) +42 3r "5 X ryij jrijj23z2 ij
cos (kxxij) sin (kyyij)'3=2(jrijj2"); (E14) @kyDyy k(zij) =2 a2X g 8pp" 3pep2q2@p @ky+(ky+gy)2 jk+gj@f @ky+2(ky+gy)jk+gj2(ky+gy)3 jk+gj3f(p;q)! +42 3r "5 X ryij jrijj23y2 ij
cos (kxxij) sin (kyyij)'3=2(jrijj2"); (E15) @kxDyy k(zij) =2 a2X g8pp" 3pep2q2@p @kx(ky+gy)2 jk+gj@f @kx+(ky+gy)2(kx+gx) jk+gj3f(p;q) +42 3r "5 X rzij jrijj23y2 ij
cos (kyyij) sin (kzzij)'3=2(jrijj2"); (E16) @kyDxy k(zij) =2 a2X g(ky+gy)(kx+gx) jk+gj@f @ky+(kx+gx)jk+gj2(ky+gy)2(kx+gx) jk+gj3f(p;q) 4"5=22 pX ry2 ijxijsin(kxxij) cos(kyyij)'3=2(jrijj2"); (E17) where @p @k=k+g 4"p; (E18) @f @k= 2qe2pqErfc(p+q)2qe2pqErfc(pq)4pep2q2k+g 4p"(E19) and the remaining terms follow from symmetry, by swapping ky$kx. [1] N. Nagaosa, J. Sinova, S. Onoda, A. H. MacDonald, and N. P. Ong, Anomalous Hall eect, Reviews of Modern Physics 82, 1539 (2010). [2] T. Qin, J. Zhou, and J. Shi, Berry curvature and the phonon Hall eect, Physical Review B 86, 104305 (2012). [3] C. Strohm, G. L. J. A. Rikken, and P. Wyder, Phe- nomenological Evidence for the Phonon Hall Eect, Physical Review Letters 95, 155901 (2005). [4] M. Onoda, S. Murakami, and N. Nagaosa, Hall Eect of Light, Physical Review Letters 93, 083901 (2004). [5] H. Katsura, N. Nagaosa, and P. A. Lee, Theory of the Thermal Hall Eect in Quantum Magnets, Physical Re- view Letters 104, 066403 (2010). [6] Y. Onose, T. 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J. van Wees, and I. J. Vera- Marun, Ecient Injection and Detection of Out-of-Plane Spins via the Anomalous Spin Hall Eect in Permalloy Nanowires, Nano Letters 18, 5633 (2018). [19] A. Kreisel, F. Sauli, L. Bartosch, and P. Kopietz, Mi- croscopic spin-wave theory for yttrium-iron garnet lms, The European Physical Journal B 71, 59 (2009). [20] R. N. Costa Filho, M. G. Cottam, and G. A. Farias, Mi- croscopic theory of dipole-exchange spin waves in ferro- magnetic lms: Linear and nonlinear processes, Physical Review B 62, 6545 (2000). [21] A. Kamra, W. Belzig, and A. Brataas, Magnon-squeezing as a niche of quantum magnonics, Applied Physics Let- ters117, 090501 (2020). [22] J. H. P. Colpa, Diagonalization of the quadratic boson hamiltonian, Physica A: Statistical Mechanics and its Applications 93, 327 (1978). [23] R. Shindou, R. Matsumoto, S. Murakami, and J.-i. Ohe, Topological chiral magnonic edge mode in a magnonic crystal, Physical Review B 87, 174427 (2013). [24] M. Lein and K. 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2021-04-22

We derive the anomalous Hall contributions arising from dipolar interactions to diffusive spin transport in magnetic insulators. Magnons, the carriers of angular momentum in these systems, are shown to have a non-zero Berry curvature, resulting in a measurable Hall effect. For yttrium iron garnet (YIG) thin films we calculate both the anomalous and magnon spin conductivities. We show that for a magnetic field perpendicular to the film the anomalous Hall conductivity is finite. This results in a non-zero Hall signal, which can be measured experimentally using Permalloy strips arranged like a Hall bar on top of the YIG thin film. We show that electrical detection and injection of spin is possible, by solving the resulting diffusion-relaxation equation for a Hall bar. We predict the experimentally measurable Hall coefficient for a range of temperatures and magnetic field strengths. Most strikingly, we show that there is a sign change of the Hall coefficient associated with increasing the thickness of the film.

Theory for electrical detection of the magnon Hall effect induced by dipolar interactions

2104.11304v1

Giant Real-time Strain-Induced Anisotropy Field Tuning in Suspended Yttrium Iron Garnet Thin Films Renyuan Wang1†, Sudhanshu Tiwari2*, Yiyang Feng2, Sen Dai2, and Sunil A. Bhave2 1FAST LabsTM, BAE Systems, Inc. 65 Spit Brook Road, Nashua, NH 03087, USA 2Elmore Family School of Electrical and Computer Engineering, Purdue University 1205 West State St., West Lafayette, IN 47907, USA Yttrium Iron Garnet based tunable magnetostatic wave and spin wave devices are poised to revolutionize the fields of Magnonics, Spintronics, Microwave devices, and quantum information science. The magnetic bias required for operating and tuning these devices is traditionally achieved through large power-hungry electromagnets, which significantly restraints the integration scalability, energy efficiency and individual resonator addressability. While controlling the magnetism of YIG mediated through its magnetostrictive/magnetoelastic interaction would address this constraint and enable novel strain/stress coupled magnetostatic wave (MSW) and spin wave (SW) devices, effective real-time strain-induced magnetism change in YIG remains elusive due to its weak magnetoelastic coupling efficiency and substrate clamping effect. We demonstrate a heterogeneous YIG-on-Si MSW resonator with a suspended thin-film device structure, which allows significant straining of YIG to generate giant magnetism change in YIG. By straining the YIG thin-film in real-time up to 1.06%, we show, for the first time, a 1.837 GHz frequency-strain tuning in MSW/SW resonators, which is equivalent to an effective strain-induced magnetocrystalline anisotropy field of 642 Oe. This is significantly higher than the previous state-of-the-art of 0.27 GHz of strain tuning in YIG. The unprecedented strain tunability of these YIG resonators paves the way for novel energy-efficient integrated on-chip solutions for tunable microwave, photonic, magnonic, and spintronic devices. Single crystal Yttrium iron garnet (YIG, Y 3Fe5O12) exhibits the lowest magnon damping among known magnetic materials1. Fueled by the material’s many interesting cross- physical-domain coupling properties, there have been many explorations into using YIG to realize devices such as magneto-optic devices for cryogenic applications2, non- reciprocal optical and RF/microwave devices3–5, tunable RF/microwave devices6–8, hybrid quantum circuits for quantum information processing9–13, spintronic devices14–17, and devices for realizing room temperature Bose-Einstein condensates18. In most of these applications, the magnetic bias needed for operating and/or tuning of the YIG device was achieved through a bulky and power hungry electromagnet, which significantly restraints the integration and energy- efficiency scalability. Electrical control of the magnetism of YIG through its magnetoelastic interaction, for example, through piezoelectric actuation, would enable beyond CMOS scalable energy-efficient devices for high efficiency magnetoelectric energy harvesting19, ultra-low power non- volatile memory20, electrically small magnetoelectric Figure 1: (a) A 3D rendering of the YoS strain tuning device consisting of a YIG MSW resonator suspended over two movable Si shuttles, which are connected to a fixed Si frame by 28 thin Si springs. (b) The measurement setup used for characterization of the strain tunability of the device. (c) An optical image of the suspended YIG resonator and (d) A microscope photo of the strain-tuned resonator device in (a) consisting of a YIG thin-film resonator ion-sliced from a bulk single crystal YIG subsrate, suspended on a bulk micromachined silicon frame. (e) Measured impedance spectra of the device under various levels of applied strain in the YIG resonator. (f) Obtained strain vs frequency response from the measured spectrum as shown in (e). antenna21, spin-state manipulation in quantum sensing and quantum information processing9,22–27, and novel spintronic and tunable devices28–38. However, effective real-time strain or stress mediated control of magnetism in YIG has remained intractable due to considerably small magneto-elastic coefficients compared to other magnetoelastic materials such as terfenol-D and FeGaB39–41. This results in the requirement of a large amount of strain to generate a useful strain-induced magnetocrystalline anisotropy field42. This is exacerbated by the fact that high quality YIG thin-films are typically grown on lattice matched crystalline bulk substrates42,43 that are incompatible with modern microfabrication technologies for etching. Consequently, most reported literature on strain control of YIG devices utilize solid mount YIG, which prevents transduction of large amounts of stress/strain in YIG due to substrate clamping28,36,41,44. In addition, intimate heterogeneous integration of YIG with piezoelectric and ferroelectric materials is needed to facilitate electrical control of strain/stress transduction, which has been proven challenging to achieve45,46. State-of-the-art piezoelectric materials can only achieve 0.3% ~ 0.6% of maximum unloaded strain47 and under loaded conditions, this is not sufficient to generate a useful amount of strain (therefore strain-induced magnetocrystalline anisotropy field) in YIG with a solid-mounted device structure. To address these challenges and enable strong mechanically mediated control of magnetism in YIG, we demonstrate heterogeneous suspended thin-film YIG-on-Si (YoS) magnetostatic wave resonators (Figure 1). The resonators were fabricated on a YoS material platform (Figure 2), with 2.19 µm thick YIG ion-sliced from bulk single crystal substrate, bonded to an oxidized Si wafer using a gold to gold thermocompression bonding process (see Methods). While highly anisotropic etching of YIG is challenging to achieve with conventional wet etching or reactive ion etching techniques, we developed an anisotropic ion mill etching process to pattern thick (>2 µm) YIG thin-film device structures, which is beneficial for minimizing spin-wave scattering. The devices consist of suspended ion-sliced YIG thin film spin-wave resonators anchored on two silicon shuttles connected to the Si substrate through Si spring beams, which are formed by deep reactive ion etching (DRIE) of silicon. The suspended region of YIG has dimensions of 260 µm x 500 µm. Since the YIG is free-standing by removing the substrate, the device structure has significantly lower stiffness compared to the solid mount devices. This allows a large amount of mechanical strain to be transduced in to the free- standing YIG membrane. By actuating the Si shuttles using linear translation actuators, we transduce up to 1.06% of tensile strain in the suspended YIG thin film, which leads to a 1.837GHz of real-time strain induced frequency tuning. This Figure 2: Ion-sliced and transferred YIG film on Si material platform: (a) laser interferometry measurement of ion-sliced YIG film on Si substrate; (b) cross-section schematic of the layer stack (from top to bottom: 2.19 µm YIG, 10 nm Ti, 60 nm Au, 10 nm Ti, 2 µm wet thermal oxide, 500 µm Si, 2 µm wet thermal oxide); (c) microscope photo of ion-sliced YIG film on Si, which was ion-sliced from a quarter of a 5mm diameter bulk single crystal YIG substrate; (d) SEM image of anisotropic etching profile of ion-mill patterning of YIG thin-film, etched through the gold bottom electrode stopping on the thermal oxide layer . corresponds to a record strain-induced magnetocrystalline anisotropy field of 642 Oe28,36,41,44,48 (Methods). The measurement setup used for strain tunability is described in Figure 1. To strain the suspended YIG film in real-time, we actuate the silicon shuttles in the length direction by a linear translation stage (see Methods), and the actuation force is measured by a 6-axis load-cell. Meanwhile, a magnetic bias field from a 3D projection electromagnet is applied to the thin film. This allows us to experimentally characterize the frequency tuning by strain-induced anisotropy field of all three types of MSWs that can exist in the YIG thin-film. The magnetocrystalline anisotropy field of YIG in the (111) plane exhibits a 6-fold symmetry with respect to the crystal basis. However, it is relatively weak compared to the strain-induced anisotropy field relevant to this work. Therefore, without any loss of generality, we orient the length direction of our devices perpendicular to the crystal [1 -1 0] direction, which is determined by XRD before the first level of photolithography. On the other hand, the strain induced anisotropy exhibits a 2-fold symmetry with respect to the angle formed by the uniaxial stress and the saturation magnetization (which is always along the direction of applied DC magnetic bias). Therefore, we study the strain induced anisotropy field frequency tuning in three scenarios: when the static magnetic bias is aligned i) perpendicular and ii) parallel to the length direction of the device within the YIG thin-film plane, and iii) perpendicular to the film plane, while tensile straining the thin-film in the length direction. When the static magnetic bias is in the thin-film plane and perpendicular to the length direction of the device, the resonator operates predominantly in the magnetostatic backward volume wave (MSBVW) mode due to the symmetry of the transducer (Figure 3), where the wave is excited through the out-of-plane component of the H field Figure 3: Measured strain-induced frequency tuning of suspended YIG thin-film MSW resonator operating in magnetostatic forward volume wave, magnetostatic backward volume wave, and magnetostatic surface wave configurations. Due to the anisotropic contribution of the magnetoelastic energy to the total free energy of magnetization, the frequency tuning efficiency for the three different configurations exhibits vast difference: (a) impedance of the thin-film YIG resonator operating in MSBVW configuration under strain- induced frequency tuning, inset annotates direction and magnitude of static magnetic bias and direction of strain; (b) resonant frequency vs. strain of the resonator operating in MSBVW configuration, showing a tuning efficiency of 160.3GHz/strain from linear regression fitting; (c) impedance of the same device operating in MSSW configuration under strain-induced frequency tuning, inset annotates direction and magnitude of static magnetic bias and direction of strain; (d) resonant frequency vs. strain of the resonator operating in MSSW configuration, showing a tuning efficiency of -290.8GHz/strain from linear regression fitting; (e) impedance of the same device operating in MSFVW configuration under strain-induced frequency tuning, inset annotates direction and magnitude of static magnetic bias and direction of strain; (f) resonant frequency vs. strain of the resonator operating in MSFVW configuration, showing a tuning efficiency of 162.5GHz/strain from linear regression fitting. from the RF current flowing through the transducer fingers. The wave travels predominantly along the width direction of the suspended YIG film, parallel to the external applied magnetic bias. Figure 1 shows the measured impedance under a static magnetic bias of 526 Oe, while the suspended YIG thin film is strained in real-time by a linear translation stage from 0.06% to 1.06% of strain. Strong spurious modes exist on the lower frequency side of the main resonance. This is consistent with the fact that MSBVW exhibits anomalous dispersion. The measured resonant frequency of the main mode is tuned from 2.734 GHz to 4.571 GHz, resulting in a 1.837 GHz of tuning range with a tuning efficiency of 183.7 GHz/strain. This is consistent with our numerical model (see Methods). The 1.837 GHz of tuning is equivalent to an effective strain-induced magnetocrystalline anisotropy field of 642 Oe. Figure 3a also shows the impedance spectrum of a later iteration of the same design operating in the same configuration. This device is fabricated using an optimized process where the resputtered materials during ion-mill are removed by phosphoric acid at a temperature of 70°C for 25 minutes. This modified process produces a cleaner etching profile and results in sharper resonances in the spectrum. The measurement results shown in Figure 3a are with an in-plane bias of 1202 Oe. When no stress is applied, the resonant frequency is ~5.2 GHz. This is consistent with the theoretical value predicted for a 2.19 µm thick YIG film assuming a saturation magnetization of 1760 Oe (Methods), which indicates that the saturation magnetization of the suspended YIG film is consistent with the saturation magnetization of bulk single crystal YIG. When the uniaxial strain is varied from 0.1% to 0.28%, the resonant frequency is tuned from 5.35 GHz to 5.65 GHz, resulting in a tuning efficiency of 160.3 GHz/strain. As we further increase the external static magnetic bias, we notice a reduction in strain-induced frequency tuning efficiency as summarized in Figure 4. Intuitively, the total free energy at high bias is dominated by the Zeeman energy leading to a diminishing effect on the effective bias from change of magnetoelastic energy by straining. 400 600 800 1000 1200 140050100150200250300Slope (GHz/strain) Bias (Oe) Figure 4: Strain tuning efficiency of MSBVW under different magnitudes of externally applied magnetic bias, showing decreasing tuning efficiency with increasing magnitude of externally applied static bias as the Zeeman energy dominates at high bias. Figure 3c shows the impedance of the same device when a 969 Oe of external magnetic bias is applied parallel to the length direction of the thin film. In this configuration, the transducer predominantly couples to the magnetostatic surface waves (MSSW) through the in-plane component of the RF H-field. The wave travels predominantly along the width direction of the suspended thin-film, perpendicular to the externally applied magnetic bias. As surface wave exhibits normal dispersion, spurious models are predominantly on the higher frequency side of the main resonance. Consistent with our theoretical model, the resonant frequency of the device decreases as the suspended YIG is tensile strained, which is opposite to the MSBVW configuration. The tuning efficiency of MSSW is ~290.8 GHz/strain (Figure 3d), which is approximately 1.8 times higher than that of the MSBVW under a similar static magnetic bias. Finally, Figure 3e shows the measured strain-induced frequency tuning for the magnetostatic forward volume waves (MSFVW), where the magnetic bias is out of plane with a magnitude of 2670 Oe, and the wave propagates in the thin-film’s width direction. In this configuration, the MSFVWs are excited through the in- plane component of the RF H-field from the transducer fingers. The tuning efficiency is 162.5 GHz/strain from linear fitting (Figure 3f). This is similar to that from the MSBVW configuration, and consistent with our numerical model. Our results demonstrate a material and device platform that enables opportunities to explore abundant physics and engineering applications of the YIG material system. Combining a suspended YIG thin-film MSW resonator structure on a heterogeneous YoS material platform, we realized giant magneto-mechanical interactions in YIG, resulting in a record 1.837 GHz of real-time strain induced frequency tuning. Leveraging this device structure, we investigated the interaction of uniaxial tensile strain with the frequency tuning behavior of all three types of magnetostatic waves. An immediate advantage of this platform becomes apparent when we consider that the strain is a normalized quantity; hence, scaling the actuator dimension of the piezoelectric transducer appropriately can result in a much larger strain in the suspended YIG than the intrinsic strain capability of the piezoelectric material. In combination with our heterogeneous YIG-on-Si material platform, this would allow potential future integration of voltage-controlled, efficient, piezoelectric strain transduction using CMOS circuits. Acknowledgment : The YIG on silicon substrate material platform was developed at FAST LabsTM, BAE systems. Microfabrication including ion-milling of YIG, and back-side deep silicon etching was performed at the Birck Nanotechnology Center, Purdue University. Measurements were performed at Seng-Liang Wang Hall at Purdue University. The authors greatly appreciate the help and support from Prof. Michael Capano and Prof. Pavan Nukala on X-ray diffraction measurements. The views, opinions, and/or findings expressed are those of the authors and should not be interpreted as representing the official views or policies of the Department of Defense or the U.S. Government. This work was supported in part by the Air Force Research Laboratory (AFRL) and the Defense Advanced Research Projects Agency (DARPA). Authors Contributions : R.W. invented device concept and design, completed modeling, and developed YoS material platform. Y.F. and S.D. performed short-loop fabrication runs to identify YOI release recipes and YIG material properties. S.T. worked closely with R.W. to update the design for testability and high yield, and micromachined the suspended YoS resonators. S.T. built measurement setup and conducted resonator measurements. R.W. and S.T. analyzed the experimental data. R.W. and S.T wrote the manuscript with input from others †rw364@cornell.edu *tiwari40@purdue.edu [1] Cullity, B. D., Cullity, B. D. & Graham, C. D. 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Philips Res. Rep. 10, 113–130 (1955). [52] Spin Waves: Theory and Applications . (Springer US, Boston, MA, 2009). doi:10.1007/978-0-387-77865-5. [53] Kittel, C. On the Theory of Ferromagnetic Resonance Absorption. Phys. Rev. 73, 155–161 (1948). [54] Pearson, R. F. Magnetocrystalline Anisotropy of Rare‐Earth Iron Garnets. Journal of Applied Physics 33, 1236–1242 (1962). [55] The Measurement of Magnetostriction Constants by Means of Ferrimagnetic Resonance - IOPscience. https://iopscience.iop.org/article/10.1143/JJAP.11.1303/meta. METHODS I. Magnetostatic wave dynamics MSW are lattice waves, where the lattice consists of magnetic dipole precessions. Under a strong DC torque exerting bias, a localized RF disturbance sets the dipoles into precession, which then propagates through the material. Collectively, they form a propagating precession wave. In the classical regime (where wavelength is much longer than the exchange coupling length), the wave dynamics are governed by the Landau-Lifshitz-Gilbert (LLG) equation49 coupled with Maxwell’s equations under magneto-quasi-static approximation. The LLG equation governs the magnetic dipole precession dynamics, where M is the magnetic dipole moment per unit volume, and Heff is the summation of any effective fields that can exert torque on the magnetic dipoles, The second term on the right-hand side of the equation accounts for the relaxation effect, suggested by T. L. Gilbert. 𝜕𝑴 𝜕𝑡= −𝛾𝑴×𝑯 𝒆𝒇𝒇+𝛼 𝑀௦𝑴×𝜕𝑴 𝜕𝑡 As the wavelength approaches the characteristic exchange interaction length λex, the exchange interaction among nearby electron spins become significant and gives rise to an extra torque exerting term ( 𝑯𝒆𝒙= 𝜆௫∇ଶ𝑴) in Heff. And magnetic dipole precession wave in such regime are referred to as spin waves. To facilitate the study of stress induced anisotropy field, we avoid the complexity arising from spin wave interactions by operating in the classical MSW regime in this work. Figure 1. Types of magnetostatic waves that can be supported by thin-film ferro/ferromagnetic materials. It is well known that a ferro/ferromagnetic thin film structure, in general, supports three types of MSW due to the 2D confinement of the film. Namely, there are magnetostatic forward volume waves, magnetostatic backward volume waves, and magnetostatic surface waves. The particular type of waves that can be supported and excited in a thin-film structure depends on the configuration of the directions of the static magnetic bias, the wave-vector, and the RF excitation field, which are annotated in Figure 2. Our MSW transducer is designed to couple to all wave types depending on the static bias configuration to facilitate the study of stress induced anisotropy field tuning, albeit the coupling efficiencies are different for different types of waves. Figure 2: HFSS simulation of small signal RF response (impedance) of the suspended YIG resonator devices operating in the MSBVW, MSSW, and MSFVW configurations: (a) simulated impedance of the resonator operating in the MSBVW configuration under 1250Oe static magnetic bias; (b) simulated impedance of the resonator operating in the MSSW configuration under 1250Oe static magnetic bias; (c) simulated impedance of the resonator operating in the MSFVW configuration under 1786Oe out of plan static magnetic bias. In addition, it has been shown that the dispersion relations for the lowest order MSWs can be written as50 MSFVW: 𝜔ଶ= 𝜔ቂ𝜔+𝜔ெቀ1−ଵିషೖ ௗቁቃ (1) MSBVW: 𝜔ଶ= 𝜔ቂ𝜔+𝜔ெቀଵିషೖ ௗቁቃ (2) MSSW: 𝜔ଶ= 𝜔(𝜔+𝜔ெ)+ఠಾమ ସ(1−𝑒ିଶௗ) (3) where ω0 = µ0γHeffDC, ωM = µ0γMs, and ω and k are respectively the frequency and wave-vector of the MSFW (parallel to film), d is the film thickness, γ is the gyromagnetic ratio, µ0 is the vacuum permeability, Ms is the magnitude of the saturation magnetization. For our devices, we operate in the regime where kd << 1 (thin-film approximation), so that the tuning efficiency of MSW resonances approaches that of uniform precession mode. We model the small signal RF behavior of our devices using ANSYS HFSS, as the physics is captured by Maxwell’s equations and the permeability tensor from linearized LLG equation. Figure 2a shows the simulated resonator impedance as a function of frequency for the device in main manuscript operating in the MSBVW configuration. The internal magnetic bias is 1250Oe in the plane of the YIG thin film and perpendicular to the device length direction. Spurious modes exists on the lower frequency side of the main resonance, consistent with our measurements. Figure 2b shows the simulated impedance of the same device operating in the MSSW configuration with an internal magnetic bias of 1250Oe, while Figure 2c shows the impedance of the device operating in the MSFVW wave with an out-of-plane internal magnetic bias of 1786Oe. II. Tuning of Resonator Frequency through Strain Induced Anisotropy Field Figure 3: Decomposition of the total free energy into its individual components: (a) total free energy surface only due to the magnetocrystalline anisotropy; (b) total free energy surface only due to the Zeeman energy; (c) total free energy surface only due to the demagnetization field; (d) total free energy surface only due to the magnetoelastic effect. It has been shown that the uniform magnetic dipole precession frequency of a ferromagnetic body, hence the effective magnetic bias, can be calculated as51 (4) where θ and φ are the polar coordinates, γ is the gyromagnetic ratio, µ0 is the vacuum permeability, 𝐻 is the effective magnetic bias, and E is the total free energy of the magnetic dipole moments. Many fields that can exert torque on the dipole moments contribute to the total free energy. As it is assumed that a strong static bias is always present during the operation of our devices, and the wavelength considered here is much longer than the exchange-coupling length in YIG52, we do not consider the exchange field in our formulation. Here, we consider the energy from static magnetic bias (Zeeman energy1), demagnetization field53, magnetocrystalline anisotropy field54, and stress induced magnetocrystalline anisotropy field (magnetoelastic energy42,55), and they can be written as below 𝐸 = −𝜇 𝑀ௌ∙𝐻ா௫௧ (5) 𝐸 = 𝜇𝑀ௌ∙𝐻/2 (6) 𝐸= 𝐾+𝐾ଵ(𝛼ଵଶ𝛼ଶଶ+𝛼ଶଶ𝛼ଷଶ+𝛼ଷଶ𝛼ଵଶ)+𝐾ଶ(𝛼ଵଶ𝛼ଶଶ𝛼ଷଶ)+⋯ (7) 𝐸ொ= −3𝜆 (𝛼ଵଶ𝜎ଵଵ+𝛼ଶଶ𝜎ଶଶ+𝛼ଷଶ𝜎ଷଷ)/2−3𝜆 ଵଵଵ(𝛼ଵ𝛼ଶ𝜎ଵଶ+𝛼ଶ𝛼ଷ𝜎ଶଷ+𝛼ଷ𝛼ଵ𝜎ଷଵ) (8) where Ms is the saturation magnetization, HExt is the externally applied static magnetic bias, HDemag is the demagnetization field (shape anisotropy field), Ki are the crystalline anisotropy constants, αi are the directional consines of the saturation 𝜔=𝜇𝛾𝐻=𝛾 𝑀௦sin𝜃 ඨ𝜕ଶ𝐸 𝜕𝜃ଶ𝜕ଶ𝐸 𝜕𝜑ଶ−ቆ𝜕ଶ𝐸 𝜕𝜃𝜕𝜑ቇଶ magnetization with respect to the crystal axis, λijk are the magnetostriction constants, and σij are the components of the stress tensor. To model the strain-induced anisotropy field, we start with calculating the total free energy. For example, Figure 4b shows the total free energy surface of a (001) YIG thin film, with a 3500Oe external static magnetic bias and a 1% uniaxial strain both applied in the [1 -1 0] direction, where the magnitude of the vector connecting the origin to the points on the surface denotes the total free energy. The free energy surface exhibits a global minimum in the [1 -1 0] direction, which is consistent with fact that the strong static bias dominates that potential energy so that the saturation magnetization is aligned in [1 -1 0]. Figure 3 shows the decomposition of the free energies into their individual components caused by only the magnetocrystalline anisotropy energy (Figure 3a), Zeeman energy (Figure 3b), demagnetization energy (Figure 3c), and strain-induced crystalline anisotropy energy (Figure 3d). As anticipated, the free energy surface, due only to the magnetocrystalline anisotropy exhibits six maximum energy lobes pointing to the [001] equivalent directions, and 8 minimum energy points in the [111] equivalent directions. This is consistent with the fact that the [001] directions are the hard axis of YIG while [111] are the soft axis of YIG. Meanwhile, the free energy Figure 4: Numerical calculation of effective bias from strain-induced anisotropy field: (a) total free energy when the material is stress free; (b) total free energy when the thin-film is subjected to 1% of uniaxial strain in the [1 -1 0] direction; (c) calculated frequency surface from the total free energy surface in (a); calculated frequency surface from the total free energy surface in (b). surface due only to the demagnetization field are minimized in the (001) plane, consistent with that the shape anisotropy of a ferromagnetic thin film tends to align the magnetization within the thin film. The Zeeman energy surface exhibits a global minimum at the [1 -1 0] direction, which is consistent with that the external applied magnetic bias is in the [1 -1 0] direction. Using eq. 4, we then numerically calculate the uniform precession resonant frequency (Figure 4d) and the effective magnetic bias, which results in a resonant frequency of 9.66GHz or equivalently an effective bias of 2681Oe calculated using Kittel’s formula53 assuming a saturation magnetization of 1760 Oe for YIG. It is worth noting that the resonant frequency surface only has physical meaning in the direction of the global minimum of the free energy surface. Comparing to the exact same configuration without any strain in YIG (Figure 4a, c), the resonant frequency is 12.11GHz or equivalently an effective bias of 3534Oe. The minor difference in the effective bias calculated from Kittel’s formula versus the 3500Oe model input was due to the effect of crystalline anisotropy of YIG. As indicated by eq. (5) ~ (8), the effective bias depends on the direction of the Ms as well as the stress tensor with respect to the crystal axis. In Figure 5, we calculate the stress-induced frequency change of a (111) YIG thin-film under 2000 Oe of in- plane static magnetic bias, when an in-plane uniaxial stress of 1% is applied. The simulation is performed for different directions of uniaxial stress with respect to the crystal basis and for different directions of static magnetic bias with respect to the direction of the uniaxial stress. As shown in the figure, the strain-induced frequency tuning exhibits a 2-fold symmetry with respect to the direction of the bias. When the bias is parallel to the uniaxial stress, the frequency tuning is -2.35GHz at 1% strain comparing to no strain. Meanwhile, when the bias is perpendicular to the uniaxial stress, the frequency tuning is positive 1.2GHz. In addition, the frequency change shows a very weak 6-fold symmetry with respect to the direction of the uniaxial stress, which is due to the weak interaction between the strain-induced anisotropy and the magnetocrystalline anisotropy. Figure 5: Strain-induced frequency tuning in a (111) YIG thin film under an in-plane uniaxial strain of 1% and an in-plane magnetic bias of 2000Oe. Figure 6: Strain-induced frequency tuning for MSBVW under different externally applied magnetic bias. Figure 6 shows the calculated strain-induced frequency tuning under different externally applied magnetic bias. Here, the uniaxial stress is parallel to the YIG [1 -1 0] direction and the static bias is perpendicular to the [1 -1 0] direction, which corresponds to the MSBVW case in our measurements. At an external bias of 434Oe, the frequency change at 1.1% strain is 1.95GHz. This is consistent with our measurement in Figure 1, where a 1.02% strain led to 1.837GHz of frequency tuning. The frequency tuning efficiency from straining exhibits weak dependence on the magnitude of externally applied magnetic bias. The tuning efficiency at 434Oe, 934Oe, and 1434Oe are 177GHz/strain, 151GHz/strain, and 139GHz/strain, respectively. In addition, Figure 7 shows the frequency tuning as a function of strain for different wave configuration under different externally applied magnetic bias. The orange curve shows tuning corresponding to the MSBVW case, similar to Figure 6 but at an external bias of 1202Oe. The tuning efficiency is 148GHz/strain. Meanwhile, the yellow curve is for the case where the uniaxial stress is parallel to the YIG [1 -1 0] direction and the static bias is also parallel to the [1 -1 0] direction with a static bias of 969Oe, corresponding to the MSSW case in our measurements. The tuning efficiency is 294GHz/strain, which matches with our measurements well. The simulated tuning efficiency of MSSW configuration at this bias is about 2 times of that of the MSBVW case, while the measured tuning efficiency for the MSSW case is about 1.8 times of the MSBVW. In addition, the blue curve is for the case where the uniaxial stress is parallel to the YIG [1 -1 0] direction and the static bias is normal to the thin-film with a static bias of 2670Oe, corresponding to the MSFVW case in our measurements. The tuning efficiency is about 12% lower than that of the MSBVW case in the model. In comparison, our measurements indicated that the tuning efficiencies are similar. We attribute these numerical mismatches between model and measurements to the inaccuracy of the material parameters used in the model as well as the fact that the stress in the suspended YIG thin-film is not strictly uniaxial. 00.002 0.004 0.006 0.008 0.01 0.012 Strain2000300040005000600070008000 External Bias = 434Oe External Bias = 934Oe External Bias = 1434Oe Figure 7: Strain-induced frequency tuning as a function of strain induced by uniaxial stress for static bias perpendicular and parallel to the uniaxial stress. III. Fabrication process flow Figure 8: Microfabrication process flow of the suspended YIG thin-film devices. The (a) Starting YoS material stack, (b) Optical image and corresponding cross-sectional schematic after patterning of YIG layer, (c) Optical image and corresponding cross-sectional schematic after patterned deposition of Ti/Au layers which serve as transducers. (d) Cross-sectional schematic and an optical image of the sample after backside DRIE of Si layer. (e) Cross-sectional schematic and an optical image of the sample after etching of SiO 2, and bonding metallic layers. The sample is on a carier wafer for steps (d-e). (f) Optical image and a scross-sectional schematic after demounting of the sample from the carier wafer. The fabrication process (Figure 8) begins with a thin-film YoS substrate (Main manuscript), which is fabricated by an ion- slicing, and thin-film transferred process. During the ion-slicing process, a bulk single crystal YIG grown by floating zone method is irradiated with 1MeV He+ ions. This creates a damaged layer centered at ~3 µm blow the YIG surface. Next, the YIG is flipped bonded on a high resistivity Si wafer with 2 µm thick thermal oxides on both sides by gold to gold compression bonding process. The bonding layer consists of 10nm of Ti adhesion layer and 30nm of Au bonding layer on both the Si and the YIG. This is followed by an anneal process that slice off the YIG film from the plane damaged by the high energy helium ions. A phosphoric acid etch at 65C with 85% concentration selectively removes the ion-implantation damaged surface layer on the sliced-off YIG, which results in a YIG thin-film of around 2.4 µm thickness (Figure 8a). The high Si substrate resistivity minimize RF loss for the final resonator device. As the bonding strength of YIG on the Si wafer is crucial for the straining of the suspended YIG thin-film, the bonding strength has been tested by die shearing test indicating a >100MPa bonding shear strength. In addition, samples have been thermally cycled to over 450℃, and IR imaging of the bonding interface showed no signs of delamination. As the thermal expansion coefficients of YIG and Si are 10.4ppm/℃ and 2.6ppm/℃, the peak shear thermal stress at the bonding interface from the thermal cycling exceeds 300MPa, indicating extremely high bonding quality. The microfabrication process flow after on the ion sliced YIG-on-Si wafer is shows in Fig. 8. The cross-sectional schematics are accompanied by an optical image of the sample at each step. The optical from topside of the wafer are situated over the cross-sectional schematic while the bottom side optical images are situated under the schematic. After slicing off, the YIG films (>2 µm) are patterned through ion milling utilizing an AJA International, Inc. system. An optimizes photoresist mask is used to avoid burning of photo-resist during this thick YIG etching (Figure 8b). This ion milling process causes some sputtering of etched material on the vertical sidewalls. To remove this resputtered material, the sample is immersed in phosphoric acid at a temperature of 70 °C for 25 minutes. Consequently, this combined etching procedure results in an almost vertical sidewall profile. Additionally, the phosphoric acid soak reduces the thickness of the YIG film by 100 nm. During the ion milling process for YIG etching, an over-etch is carried out to etch the Au/Ti layers, which were used for bonding the YIG to the Si wafer. Consequently, the buried SiO 2 layer is exposed. This 2 μm SiO 2 layer is removed using reactive ion etching (RIE). Next, a top electrode consisting of 10nm Ti adhesion layer and 300nm of Au is defined over the YIG by lift-off (Figure 8c) process, which serves as the magnetostatic wave transducer. After the front-side process, the YoS substrate is flip-mounted on a carrier wafer for back-side processing. To protect the top side of the sample, a photoresist layer is spin coated, and a 100 nm layer of Aluminum (Al) is deposited. The Al layer serves to prevent the formation of a difficult-to-clean mixture of photoresist and crystal-bond 555, which is utilized for bonding the sample to a carrier wafer. Next, deep reactive ion etching (DRIE) is used to etch through the bulk of the Si wafer. The thermal oxide on the backside of the Si wafer is patterned by RIE using photoresist masks, which serves as a hardmask for the DRIE process. The buried thermal oxide between the bonding metal and Si serves as the etch stop for the DRIE (Figure 8d), which is exposed after the DRIE etch. The buried SiO 2 layer is then partially etched using reactive ion etching. The remaining SiO 2 layer and bonding metal layers (Ti/Au/Ti) are removed using wet chemical processes (Figure 8e). The removal of the gold layer is accomplished using a standard Au etchant, while BOE is employed for the removal of the SiO 2 and Ti layers. Throughout this entire process, the sample remains attached to the carrier wafer. Subsequently, the sample is demounted by immersing the carrier wafer in hot water. Following the demounting, the sample is soaked in acetone to lift off the protective Al layer. As a result of this process, a suspended YIG film with MSW transducers on top is obtained (Figure 8f). IV. Measurements The strain tuning measurement requires force-displacement measurement simultaneously with RF measurement under a magnetic bias field. In order to achieve this feature, a custom test setup was designed and assembled. The measurement setup consists of a linear translation stage (X-LDM060C from Zaber Technologies Inc) to provide necessary movements, thereby enabling strain of YIG film, a 6-axis load cell (MC3A from Advanced Mechanical Technology, Inc.) to completely characterize the forces and moments generated during the movement of the linear stage, a projection magnet (Model 5201 from GMW Associates) and a typical RF measurement setup (i.e., VNA and GSG RF probes along with a microscope). Two machined aluminum plates, one each, are mounted on the linear stage and the load cell. These plates are machined with a M1.4 screw whole where a partially screwed M1.4 screw is fixed. The sample is mounted between the linear stage and load cell by aligning the holes in the Si with the screws. This allows the stretching of the Si plates, which in turn transfers stresses to the YIG thin film. The measurement setup is schematically shown in Figure 9. The equivalent spring model of this measurement is shown in Figure 10. Both the load cell and the linear stage have their own stiffnesses as denoted by K LC and K LS, which are connected in series with the device where the suspended YIG film and the straight Si springs are in parallel combination. When the linear stage is actuated, the complete spring assembly comes under tension, and a fraction of the total force is transferred to the YIG film. After each stage movement the force reading is allowed to stabilize, and the corresponding RF response of the device is saved using the VNA. A constant bias field is maintained using the projection magnet, and measurements are taken for incremental displacement intervals. This process is repeated for a combination of bias fields in all three axes. The spring model can be further simplified by using an effective spring of stiffness, K C which, accounts for K LS and K LC in series. To measure this stage assembly stiffness K C, a rigid Si block with the same mounting hole but without any spring is fabricated. A force vs. displacement response is measured after mounting this Si block on the measurement setup. Since the Si block can be assumed rigid for this measurement, the slope of this curve yields the stiffness of the stage assembly. For accurate characterization of the spring constant of the Si springs, the force displacement measurement is carried out on a device with ruptured YIG film. From this measurement, the combined stiffness of stage assembly and Si springs (K Si and K C in series) is obtained. Since K C has already been calculated, K Si can be easily calculated. From the different stiffness measurements the fraction of total force that is experienced by the YIG film is calculated. This force is used to calculate the strain in the YIG film for each measurement point. Figure 9: Setup for testing the suspended thin-film YIG devices. Figure 10: The equivalent spring connection model of the full characterization setup.

2024-05-22

Yttrium Iron Garnet based tunable magnetostatic wave and spin wave devices are poised to revolutionize the fields of Magnonics, Spintronics, Microwave devices, and quantum information science. The magnetic bias required for operating and tuning these devices is traditionally achieved through large power-hungry electromagnets, which significantly restraints the integration scalability, energy efficiency and individual resonator addressability. While controlling the magnetism of YIG mediated through its magnetostrictive/magnetoelastic interaction would address this constraint and enable novel strain/stress coupled magnetostatic wave (MSW) and spin wave (SW) devices, effective real-time strain-induced magnetism change in YIG remains elusive due to its weak magnetoelastic coupling efficiency and substrate clamping effect. We demonstrate a heterogeneous YIG-on-Si MSW resonator with a suspended thin-film device structure, which allows significant straining of YIG to generate giant magnetism change in YIG. By straining the YIG thin-film in real-time up to 1.06%, we show, for the first time, a 1.837 GHz frequency-strain tuning in MSW/SW resonators, which is equivalent to an effective strain-induced magnetocrystalline anisotropy field of 642 Oe. This is significantly higher than the previous state-of-the-art of 0.27 GHz of strain tuning in YIG. The unprecedented strain tunability of these YIG resonators paves the way for novel energy-efficient integrated on-chip solutions for tunable microwave, photonic, magnonic, and spintronic devices.

Giant Real-time Strain-Induced Anisotropy Field Tuning in Suspended Yttrium Iron Garnet Thin Films

2405.13303v1

arXiv:1702.03119v1 [cond-mat.mes-hall] 10 Feb 2017Relative weight of the inverse spin Hall and spin rectificati on effects for metallic Py,Fe/Pt and insulating YIG/Pt bilayers estimate d by angular dependent spin pumping measurements S. Keller,1J. Greser,1M. R. Schweizer,1A. Conca,1B. Hillebrands,1and E. Th. Papaioannou1 Fachbereich Physik and Landesforschungszentrum OPTIMAS, Technische Universit¨ at Kaiserslautern, Erwin-Schr¨ odinger-Str. 56, 67663 Kaiserslautern, Germa ny (Dated: 13 February 2017) We quantify the relative weight of inverse spin Hall and spin rectificat ion effects occurring in RF-sputtered polycrystalline permalloy, molecular beam epitaxy-grown epitaxial iro n and liquid phase epitaxy-grown yttrium-iron-garnet bilayer systems with different capping materia ls. To distinguish the spin rectification signal from the inverse spin Hall voltage the external magnetic field is rotated in-plane to take advantage of the different angular dependencies of the prevailing effects. We pro ve that in permalloy anisotropic magne- toresistance is the dominant source for spin rectification while in epit axial iron the anomalous Hall effect has an also comparable strength. The rectification in yttrium-iron-gar net/platinum bilayers reveals an angular dependence imitating the one seen for anisotropic magnetoresista nce caused by spin Hall magnetoresistance. Spintronic bilayers composed of a ferromagnetic (FM) and a nonmagnetic (NM) layer with large spin-orbit- interaction are promising devices for the spin-to-charge conversion for future applications. At ferromagnetic res- onance (FMR) the spin pumping (SP) effect allows for the injection of a pure spin current from the FM into the NM layer1. There, the spin current is converted into a charge current by the inverse spin Hall effect (ISHE)2. A wide range of metallic, semiconducting or insulating ferro-3and ferrimagnets4and NM5materials, like Au, Pd, Ta, W, and Pt, have been investigated up to this point. In metallic FM layers, an overlapping additional effect take place, the so called spin rectification (SR) ef- fect, which hinders the access to the pure ISHE signal. Different approachesforseparationhavebeen thoroughly investigated6–8. Thickness variations of the FM and the NM layers show different dependencies for ISHE and SR, but require a lot of effort for producing whole sample series or wedged microstructures. Another method is a sweep of the excitation frequency, which cannot be ap- plied to all experimental setups and requires a careful calibration of the microwave transmission properties of the setup. Also the minimization of the electrical mi- crowave field at the location of the sample by using a microwavecavity is possible, but in most cases only fixed frequencies can be applied. The rotation of the magneti- zation angle by rotatingthe external magnetic field in- or out-of-plane is one of the most common and practicable methods, with out-of-plane rotation normally requiring larger magnetic fields for thin films7,8. Here, we quan- tify with the help of the in-plane angular dependent spin pumping measurements the ISHE and the SR contribu- tions, mainly anisotropic magnetoresistance (AMR) and anomalous Hall effect (AHE). Therefore, bilayers com- posed of magnetic (Fe, Py, and YIG) and non-magnetic (Pt, Al and MgO) materials have been used. Capping layers with significant spin Hall angle Θ SH(Pt) should show a large ISHE, while materials with small Θ SH(Al) and insulating materials (MgO) should not. All bilayer FIG. 1. Experimental setup and coordinate system: x,yand zare the lab fixed coordinates. The external field /vectorHrotates in-plane, while the angle Θ His defined as the angle between zand/vectorH. The bilayer films are lying in the xandzplane andyis the out-of-plane coordinate. The exciting stripline antenna is parallel to zand is generating an in-plane dynamic magnetic field hx, an out-of-plane field hyand also a dynamic electrical field ez, which induces an electrical current jzin the samples in zdirection. Eddy currents jeddypotentially can flow transverse to the microwave electrical field in xdirection. The electrical contacts for measuring the DC voltage are ei- ther transverse ( Vx) or parallel ( Vz) to the stripline antenna. samples with metallic FM (Py/Al, Py/Pt, Fe/MgO and Fe/Pt) have the dimensions of (10 ×10) mm2, while the YIG/Pt sample is of smaller dimensions (2 ×3) mm2. We first address the measurements on polycrystalline Py/Pt and Py/Al bilayers, that is, with presence and ab- sence of ISHE voltage, respectively. We will use the data ofthis model system to illustrate the angulardependence of the measured signal and the analysis method used2 FIG. 2. Theoretical in-plane magnetization angular depen- dencies of spin rectification effects and ISHE with contacts transverse to the microwave antenna ( xdirection) and differ- ent dynamic magnetic field geometries, adapted from Harder et. al.8. ΘHis the magnetic field angle (defined in Fig. 1), ALandADare the amplitudes of the effects contributing to the symmetric voltage (L: Lorentzian) and the antisymmetri c voltage (D: Dispersive). to separate the different contributions. Second, we will presentthedataforepitaxialFe/PtandFe/MgOsamples and we will apply again the same analysis method com- paring the weights of the different contributions with the Pycase. Finally, resultsinYIG/Ptbilayersarepresented to compare the situation for a system with an insulating magnetic layer where no AMR or AHE can be present. Prior to concluding, we will present some important re- marks about the validity and limitations of the analysis method based on angular measurements. In the experiment for a fixed excitation frequency and external field angle, the external field amplitude is swept. The voltage measured by lock-in-amplification technique exhibits peaks consisting of symmetric and antisymmet- ric componentswhich arefitted by the followingequation for each individual external magnetic field sweep5: Vmeas(H) =Vsym(∆H)2 (H−HFMR)2+(∆H)2 +Vasym−2∆H(H−HFMR) (H−HFMR)2+(∆H)2,(1) whereVsymandVasymaretheamplitudeofthesymmetric and antisymmetric components, respectively. ∆ His the FIG. 3. Angular dependent spin pumping measurements of Py(12nm)/Al(10nm) (top graph) and Py(12nm)/Pt(10nm) (bottom graph) at 13 GHz excitation frequency with contacts transverse to the direction of the stripline antenna. Black and orange arrows are highlighting the side-maxima/minima ori g- inating from AMR. linewidth, His the applied magnetic field, and HFMRis the corresponding FMR field value. While the SP/ISHE effect contributes only to Vsym, the SR effects contribute to both voltage amplitudes. The relative contribution of AMR to VsymandVasymand AHE to VsymandVasym is determined by the phase difference between the dy- namic magnetization /vector m(t) and the microwave electrical field induced AC current /vectorj(t) inside the FM layer. This phase difference is not easily accessible5and the relative contribution of AMR does not necessarily have to be the same as the one of AHE8. To fit the measured voltage amplitudes it is needed to calculate the angular depen- dencies of SP/ISHE and SR (a detailed derivation can be found in7,8). The symmetric aswell asthe antisymmetric voltage will then be fitted. First let us consider the coordinate system (see Fig. 1), wherexandzare the in-plane and ythe out-of-plane lab fixed coordinates, Θ His the angle between the ex- ternal magnetic field /vectorHand the zaxis. The electrical3 contacts are either in x(transverse to the stripline an- tenna) or z(parallel to the stripline antenna) direction. jzinduced by the microwave electrical field ezandjeddy inxdirection (explained later) are the in-plane current components. The dynamic magnetic microwave fields hx (in-plane) and hy(out-of-plane) are determined by the microwave stripline antenna. At first the model of the measurements, where the DC voltage is measured in xdirection (transverse to the an- tenna, shown in Fig. 1), is discussed: For this measure- ment configuration significant values for jzandhx(in- plane dynamic magnetic field component), and smaller values for hy(out-of-plane dynamic magnetic field com- ponent), which is estimated a magnitude smaller than the in-plane field components, are considered. The the- oretical angular dependencies of the underlying effects are graphically shown in Fig. 27,8. It can be recog- nized that in-plane excited AHE is similar to in-plane excited ISHE bearing only one maximum and one mini- mum, but with different slopes at zero crossing. In-plane excited AMR is showing three maxima/minima where one is of higher amplitude (referred to as main maxi- mum/minimum in the following) and two of smaller am- plitude (referred to as side maxima/minima). Out-of- plane excited AMR is showing two maxima/minima with equal amplitude. Out-of-plane AHE will generate a con- stant offset and out-of-plane ISHE has an identical shape as in-plane AHE and can therefore not be distinguished fromit. As tobe shownlaterin the measurementsforthe Py samples, an additional AMR effect also takes place. This AMR effect is shownin Fig. 2in blue and is the only one antisymmetric around 0◦. This AMR scales with an electrical current jxperpendicular to the microwave in- duced currents and with an out-of-plane microwave field component hyand is affiliated to eddy currents9. To fit the experimental data all considered effects are linear su- perimposed: Vx sym=Vhx ISHEcos3(ΘH)+Vhy,hx ISHE,AHEcos(ΘH) + Vhy AHE+Vhx,jz AMRcos(2Θ H)cos(Θ H) + Vhy,jz AMRcos(2Θ H)+Vhy,jeddy AMRsin(2Θ H). Vx asym=Vhx AHEcos(ΘH)+Vhy AHE+ Vhx,jz AMRcos(2Θ H)cos(Θ H) + Vhy,jz AMRcos(2Θ H)+Vhy,jeddy AMRsin(2Θ H).(2) Equations 2were then used to fit the angular depen- dent spin pumping measurements shown in Fig. 3and4 for the of VsymandVasymof Py/Al, Py/Pt, Fe/MgO and Fe/Pt bilayers. The voltage amplitudes from the fits of the Py and Fe sample measurements have been summa- rized in Table Ifor comparison. After familiarizing with the angular dependencies of the ISHE and SR effects the measurements for the Py bi- layers are now discussed: In Fig. 3we see for the Py/Al FIG. 4. Angular dependent spin pumping measurements of Fe(12nm)/MgO(10nm) (top graph) and Fe(12nm)/Pt(10nm) (bottom graph) at 13 GHz excitation frequency with contacts transverse to the direction of the stripline antenna. sample that the signal is mainly consisting of AMR in the symmetric as well as in the antisymmetric ampli- tude, since the signals exhibit pronounced side-maxima (arrows). The antisymmetricvoltageamplitudeofPy/Pt has almost identical shape as the one of Py/Al. For both samples the AMR to AHE ratio of the antisymmet- ric voltage is approximately 1 to 4 (see Table I). Py/Al and Py/Pt also show that their side-maxima (arrows) are having not the same amplitudes. This is correlated to AMR caused by eddy currents with an out-of-plane dynamic magnetic field component (see Table I). The measurements of Fe/MgO and Fe/Pt can be seen in Fig.4. Since epitaxial Fe has a strong magneto- crystalline anisotropy the magnetization will in general not be aligned to the external magnetic field due to the anisotropy fields. The ISHE and SR effects are, however, only dependent on the angle of magnetization Θ M. For this reason, an additional rescalingof the angle axis is re- quired. A numerical analysis has been performed where ΘMhas been calculated for the data measured at Θ H. For this K1/Ms(K1: cubic anisotropy constant, Ms:4 Vsym/VasymSample Vhx ISHE(µV)Vhy,hx ISHE,AHE†(µV)Vhy AHE(µV)Vhx,jz AMR(µV)Vhy,jz AMR(µV)Vhy,jeddy AMR (µV) VsymPy/Al 0 1.43±0.04 0.15±0.025.63±0.06 0 -1.68±0.03 Py/Pt amb. amb. 0.02 ±0.02 amb. 0 -0.61±0.02 Fe/MgO 0 6.85±0.12 -0.01±0.055.12±0.150.18±0.09 0 Fe/Pt amb. amb. 0.12 ±0.05 amb. -0.25 ±0.08 0 VasymPy/Al 0 -1.49±0.05 0.03±0.03-5.95±0.07 0 1.02±0.04 Py/Pt 0 -0.61±0.03 0.01±0.01-2.36±0.04 0 0.44±0.02 Fe/MgO 0 4.07±0.15 0.07±0.06-6.20±0.190.18±0.10 0 Fe/Pt 0 3.55±0.09 0.01±0.04-7.88±0.110.13±0.05 0 TABLE I. Results of the angular spin pumping measurements: s ymmetric and antisymmetric voltage amplitudes of Py/Al, Py/Pt, Fe/MgO and Fe/Pt. Items marked with amb. are ambiguou s (see text). The voltage of the effects mainly contributing are marked in bold. The voltage marked with†corresponds to the term ∝cos(Θ H), which is comprised of in-plane AHE and out-of-plane ISHE in the symmetrical voltage and only of in- plane AHE in the antisymmetric voltage (to be seen in Fig. 1). Absolute values between samples are not comparable because of different excitation frequencies. saturation magnetization) has been extracted from the dependence of HFMRon the frequency (Kittel fit10). For RF-sputtered polycrystalline samples which are isotropic ΘHand Θ Mare identical. However, in the case of epi- taxial Fe they can differ more than 10◦. After the angle rescaling the angular dependent mea- surementsoftheFe bilayerscanbe discussed: In thesebi- layers (Fig. 4) in-plane excited AHE seems to be equally prominentasin-planeexcited AMR, ascanbe recognized from the lack of side-maxima and also from the voltage amplitudesofthefits shownin Table I. Thisisamaindif- ference to the Py case where AMR is strictly dominant. The AHE excited by the out-of-plane dynamic magnetic field is rather small (as it also is in the case for Py). The difference is not due to the difference in growth (poly- or single-crystalline) of the FM layers. For instance, recent results on also polycrystalline CoFeB/Pt and CoFeB/Ta layers show that there AHE is the only dominant effect while AMR is almost negligible11. The weight of the different spin rectification contribution is reflecting only the strength of the different effects (AHE, AMR) in the ferromagnetic material. The different capping materials (insulator, respectively metal) is changing the relative contribution of the SR effects onto Vasym. Additionally the epitaxial Fe samples, especially Fe/MgO, show char- acteristic features around angles, where the external field is oriented equidistant between the magnetic hard (e.g. 45◦) and easy axis (e.g. 0◦) of Fe. This is due to the intrinsic magnetic anisotropy influencing the angular de- pendencies of ISHE and SR. In addition, to compare with the measurements of the bilayers with metallic FM, a bilayer with an insulator magnetic material was measured with the same setup. For this YIG(100 nm)/Pt(10nm), where AMR and AHE are suppressed, was chosen and the results are shown in Fig.5(top graph) and Table II. A surprisingly non- vanishing antisymmetric voltage with angular dependen- cies similar to AMR and AHE can be seen. The sym- metric voltage amplitude seems to be consisting mainly of an ISHE contribution and of a contribution ∝cos(ΘH) which can be either in-plane ISHE or out-of-plane AHE, as shown in Fig. 2. In order to understand this behav- FIG. 5. Angular dependent spin pumping measurements of YIG(100nm)/Pt(10nm) at 6.4 GHz excitation frequency with contacts transverse (top graph) and parallel (bottom graph ) to the direction of the stripline antenna. ior we performed a second measurement with electrical contacts parallel to the stripline antenna ( z-direction), measurements shown in Fig. 5(bottom graph). In this contact geometry the ISHE and SR effects have differ- ent angular dependencies as shown in Fig. 6. Here in-5 Vsym/VasymContacts Vhx ISHE(µV)Vhy ISHE(µV)Vhx AHE(µV)Vhy AHE(µV)Vhx,jz AMR(µV)Vhy,jz AMR(µV) Vsymtransverse amb. -1.35⋆amb. -0.02 ±0.01 amb. -0.04 ±0.02 parallel -1.48†-1.82±0.03-0.19±0.040.05±0.02 0†-0.09±0.02 Vasymtransverse 0 0 0.53±0.030.00±0.010.20±0.030.03±0.02 parallel 0 0 0.11 ±0.03-0.01±0.020.65±0.04-0.08±0.03 TABLE II. Results of the angular spin pumping measurements: symmetric and antisymmetric voltage amplitudes of YIG/Pt with contacts transverse and parallel to the microwave ante nna. Values marked with * are ambiguous (see text). The volta ge of the effects mainly contributing are marked in bold. The out -of-plane ISHE voltage with transverse contacts marked wit h⋆ has the same shape as the AHE and could easily be confused with it, but the comparison with the measurement with parallel contacts confirms it as an ISHE voltage. In-plane ISHE in the p arallel contacts case marked with†cannot be distinguished from in-plane AMR. In-plane AMR and AHE in the symmetrical vo ltage are estimated small since out-of-plane AMR and AHE are also small despite of relatively high out-of-plane exci tation fields. plane excited ISHE and AMR have the same angular de- pendence, but the out-of-plane excited ISHE exhibits an unique cos(Θ H) dependence. To fit the measured data with contacts parallel to the antenna following equations has been used: Vz sym=Vhx ISHE,AMRsin(2Θ H)cos(Θ H) + Vhy ISHEsin(ΘH)+Vhx AHEcos(ΘH) + Vhy AHE+Vhy,jz AMRsin(2Θ H). Vz asym=Vhx AHEcos(ΘH)+Vhy AHE+ Vhx AMRsin(2Θ H)cos(Θ H) + Vhy,jz AMRsin(2Θ H).(3) LookingatTable IItheout-of-planeISHEcontribution Vhy ISHEin transverse contacts ( −1.82µV) has almost the same value as the one of parallel contacts ( −1.35µV). The discrepancy is roughly reflecting the difference in sample width (2 mm) and length (3 mm) where the DC contacts have been applied to. The term ∝cos(ΘH) used for the fit of Vsymin Fig.5(top graph) is therefore con- firmed as out-of-plane ISHE contribution. To understand the enhancement of the out-of-plane magnetic microwave field component it is needed to con- sider that the size of the YIG/Pt sample is much smaller than the Fe and Py samples, its dimensions being closer to the width of the stripline antenna. This changes the distribution of the microwave fields and therefore en- larges the out-of-plane field components to the extent that they can be comparable in magnitude to the in- plane fields. In Fig. 5(bottom graph) we also see a non-vanishing antisymmetric voltage with AMR-like de- pendence. Other authors also reported rectification ef- fects in YIG/Pt bilayers in spin pumping experiments at room temperature caused by spin Hall magnetoresis- tance (SMR)12–14. SMR is a rectification effect occuring in bilayers consisting of a FM insulator and a NM layer, where a spin current induced by spin Hall effect (SHE) forms a spin accumulation at the interface. When the magnetization is aligned parallel to the polarization of the accumulation, fewer spin currents can enter the FM FIG. 6. Theoretical in-plane magnetization angular depen- dencies of spin rectification effects and ISHE with contacts parallel to the microwave antenna and different dynamic ex- ternalmagnetic fieldgeometries, adaptedfrom Harderet.al .8. ΘHis the magnetic field angle (defined in Fig. 1),ALand ADare denoting the amplitudes of the effects contributing to the symmetric voltage (L: Lorentzian) and the antisymmetri c voltage (D: Dispersive). layerandspinback-flowinducesanadditionalchargecur- rent by ISHE reducing the resistivity of the NM layer. This in-plane angular dependent change in resistivity in- duces effects similar to AMR and AHE. Additionally the magnetic proximity effect can also contribute to spin rec- tification in YIG/Pt bilayers15,16: a ferromagnetic layer in contact to Pt can induce a finite magnetic moment in Pt near the interface because of the high paramagnetic susceptibility of Pt. This thin ferromagnetic Pt film can also exhibit spin rectification by itself with the same an-6 gular dependence. This is also true in metallic systems but therethe spinrectificationgeneratedbytheFM layer is dominating. Summarizing this section we have shown that the symmetric voltage of YIG/Pt is mainly consist- ing of ISHE contributions and the antisymmetric voltage is indeed small but not negligible and consisting of SMR induced rectification. Furthermore, the results from the analysis of the YIG/Pt can be used to interpret the ISHE contribution in Py/Pt, seen in Fig. 3: There the side-maxima of the symmetric amplitude are stronger pronounced than the ones of Py/Al. This is due to the reduction of the am- plitude of the main-maximum of Vsymof Py/Pt. The reason for this is the opposite sign of the ISHE to the SR contributions. As shown in Table IIISHE from the YIG/Pt measurements shows a negative sign. The sign ofthe ISHE voltageis determined by the sign ofspin Hall angle, thedirectionofthespinpolarizationandthedirec- tion of the spin current. They are all the same for both Py/Pt and YIG/Pt, therefore, the voltages generated by ISHE in Py/Pt and YIG/Pt should have the same sign. In Tables IandIIsome values of the fits have not been shown, indicated by “amb.”. An intrinsic limita- tion of the analysis procedure is present when rotating the external magnetic field in-plane and investigating in- plane excited effects. According to Equation 4an ambi- guity exists with the main in-plane contributions of this measurement configuration: ISHE, AMR and AHE are mathematically linearly dependent. Therefore, the abso- lute values obtained from the fits for the Py/Pt, Fe/Pt and YIG/Pt from Vsymmay not be relevant. Neverthe- less, the overall angular dependence and the Vasymdata, where no ambiguity is present, support the interpreta- tions shown in this paper. cos(2Θ H)cos(Θ H) =[2cos2(ΘH)−1]cos(Θ H) =2cos3(ΘH)−cos(ΘH).(4) In summary, we have shown that the spin rectification effect does scale differently in Fe, Py and YIG bilayer systems, as summarized in Table IandII: While AMR is more pronounced than AHE in RF magnetron sputtered Py, AHE seems to be equal in magnitude for epitaxial Fe systems. Spin rectification with an angular depen- dence similar to AMR is appearing in the antisymmetric Lorentzianshape in nanometer thin YIG/Pt bilayerfilms originating from the spin Hall magnetoresistance. The symmetric signal of YIG/Pt is mainly consisting of equal ISHE contributions excited by in- and out-of-plane dy- namic magnetic fields. In epitaxial Fe systems the effects due to non-collinearitybetween the external field and themagnetization needs to be taken into account. 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Rezende, Spin pumping and anisotropic mag- netoresistance voltages in magnetic bilayers: Theory and e xper- iment, Phys. Rev. B 83, 144402 (2011). 6E.Th. Papaioannou, P. Fuhrmann, M.B. Jungfleisch, T. Br¨ ach er, P. Pirro, V. Lauer, J. L¨ osch, B. Hillebrands, Optimizing the spin- pumping induced inverse spin Hall voltage by crystal growth in Fe/Pt bilayers , Appl. Phys. Lett. 103, 162401 (2013). 7R. Iguchi, and E. Saitoh, Measurement of spin pumping voltage separated from extrinsic microwave effects , arXiv:1607.04716v1 (2016). 8M. Harder, Y. Gui, and C.-M. Hu, Electrical detec- tion of magnetization dynamics via spin rectification effect s, arXiv:1605.00710v1 (2016). 9V. Flovik, and E. Wahlstrœm, Eddy current interactions in a Ferromagnet-Normal metal bilayer structure, and its impac t on ferromagnetic resonance lineshapes , J. Appl. Phys. 117143902 (2015). 10C. Kittel, On the Theory of Ferromagnetic Resonance Absorp- tion, Phys. Rev. 73, 155 (1948). 11A. Conca, B. Heinz, M. R. 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2017-02-10

We quantify the relative weight of inverse spin Hall and spin rectification effects occurring in RF-sputtered polycrystalline permalloy, molecular beam epitaxy-grown epitaxial iron and liquid phase epitaxy-grown yttrium-iron-garnet bilayer systems with different capping materials. To distinguish the spin rectification signal from the inverse spin Hall voltage the external magnetic field is rotated in-plane to take advantage of the different angular dependencies of the prevailing effects. We prove that in permalloy anisotropic magnetoresistance is the dominant source for spin rectification while in epitaxial iron the anomalous Hall effect has an also comparable strength. The rectification in yttrium-iron-garnet/platinum bilayers reveals an angular dependence imitating the one seen for anisotropic magnetoresistance caused by spin Hall magnetoresistance.

Relative weight of the inverse spin Hall and spin rectification effects for metallic Py,Fe/Pt and insulating YIG/Pt bilayers estimated by angular dependent spin pumping measurements

1702.03119v1

1 Giant Spin Seebeck Effect through an Interface Organic Semiconductor V. Kalappattil1, R. Geng2, R. Das1, H. Luong2, M. Pham2, T. Nguyen2, A. Popescu1, L.M. Woods1, M. Kläui3, H. Srikanth1, and M.H. Phan1 1 Department of Physics, University of South Florida, Tampa, Florida 33620, USA 2 Department of Physics and Astronomy, University of Georgia, Athens, GA 30602, USA 3 Institute of Physics, Johannes Gutenberg University Mainz, 55128 Mainz, Germany Interfacing an organic semiconductor C 60 with a non -magnetic metallic thin film (Cu or Pt) has creat ed a novel heterostructure that is ferromagnetic at ambient temperature , while its interface with a magnetic metal (Fe or Co) can tune the anisotropic magnetic surface property of the material . Here, we demonstrate that sandwiching C60 in between a magnetic insulator ( Y3Fe5O12: YIG) and a non -magnetic , strong spin -orbit metal (Pt) promotes highly efficient spin current transport via the thermally driven spin Seebeck effect (SSE). Experiment s and first principles calculations consistently show that t he presence of C 60 reduces significantly the conductivity mismatch between YIG and Pt and the surface perpendicular magnetic anisotropy of YIG , giving rise to enhanced spin mixing conductance across YIG/C60/Pt interface s. As a result, a 600% increase in the SSE voltage (VLSSE) has been realized in YIG/C 60/Pt relative to YIG/Pt. Temperature -dependent SSE voltage measurements on YIG/C 60/Pt with varying C 60 layer thicknesses also show an exponential increa se in VLSSE at low temperatures below 200 K, resembling the temperature evolution of spin diffusion length of C 60. Our study emphasizes the important roles of the magnetic anisotropy and the spin diffusion length of the intermediate layer in 2 the SSE in YIG/C 60/Pt structures, providing a new pathway for developing novel spin - caloric materials. Generating pure spin current s has been the main challenge for realizing highly efficient spintronics devices.1 Notable effects under study for attaining pure spin current s are the spin Hall effect (SHE )2, spin pumping mechanism3,4, and the spin Seebeck effect (SSE).5,6 In the last two approaches, a pure spin current is generated in a ferromagnetic (FM) material ( which could be a metal7, insulator8, or semiconductor9) and converted into a voltage drop via the inverse spin Hall effect (ISHE) in a nonmagnetic metal (NM) possessing strong spin -orbit coupling. In case of the SSE, the spin current Js can be expressed as10 𝐽𝑠=𝐺 2𝜋𝛾ℏ 𝑀𝑠𝑉𝑎𝐾𝑏Δ𝑇, (1) where G is the spin mixing conductance, 𝛾 is the gyromagnetic ratio, Va is the magnetic coherence volume, Ms is the saturation magnetization, and Δ𝑇 is the temperature difference betwe en magnons in FM and electrons in NM. One of the key parameters essential for an efficient spin transport across a FM/NM interface is having a large spin mixing conductance , which directly correlates with a large spin current, as evident from Eq. (1).10 According to Eq. (1), increasing G is essential to transport large spin current s from FM to NM. Since the first observation of SSE in the ferromagnetic insulator Y3Fe5O2 (YIG) in 2010 ,8 this material has become one of the most intensively studied systems for fundamental understanding of the underlying spin transport and for prospective spin caloritronic applications , among others .5,6 Significant efforts have been devoted towards improv ing the value of G and hence the SSE voltage by utilizing YIG/Pt interface s and adding various thin intermediate layer s, such as Cu, NiO, CoO, Fe 70Cu30, NiFe, or Py.11-15 Inserting a non -magnetic layer of Cu, for 3 example, between YIG and Pt has been reported to decrease the SSE signal11, while the addition of an ultrath in magnetic layer of Fe 70Cu3014 or NiFe15 between YIG and Pt has increased the magnetic moment density or G at their interface, leading to an overall improvement in the SSE voltage . Recently, Lin et al. reported a significant enhancement of the SSE in YIG/ M/Pt (M = NiO or CoO) heterostructures,12 demonstrating the important role played by an additional antiferromagnetic layer on the spin current transport. Despite the enhancement of SSE reported in these heterostructures, the responsible underlying mechanis ms associated with the role of the intermediate layer on the SSE have remained an open question.10-15 It is generally accepted that spin transport through a material is limited by its spin diffusion length 𝜆. While YIG has a large 𝜆 (~10 μm), a small 𝜆 value of Pt (~ 2 nm ) has been reported .16 Consequently, undesired effects, such as large surface roughness of YIG17and/or large perpendicular surface magnetic anisotropy of YIG18 causing strong spin scattering, could suppress considerably the spin current injection into the Pt layer. The large conductivity mismatch between YIG and Pt could also decrease considerably the efficient spin transport in YIG/Pt.19 It is therefore desirable to seek a n intermediate material that can reduce both the perpendic ular surface magnetic anisotropy of YIG and the conductivity mismatch between the YIG and Pt layers . In this regard, an organic semiconductor (OSC) such as C 60 buckyballs can be considered as a promising candidate material due to its low spin-orbit coupling that essentially results in weak spin scattering and consequently a large spin diffusion length ( 𝜆 can be of the order of several hundred nanometers).20,21 Additionally, t he C60 buckyballs are semiconducting in nature 22, thus when sandwiched between YIG and Pt layers, the conductivity mismatch between them is likely to be reduced. Recent studies have also revealed that C60 hybridizes 4 strongly with metal lic substrates , which leads to inducing ferromagnetic order at the surface layer of a non -magnetic metal , such as Cu or Pt23, or tuning the surface magnetism (e.g. surface magnetic anisotropy) of a magnetic metal , such as Fe or Co .24 These findings lead us to propose a new approach for improving the SSE in ferromagnetic insulator/metal sy stems such as YIG/Pt by adding a thin intermediate layer of an organic semiconductor such as C 60 and forming a novel heterostructure of YIG/C 60/Pt, as shown in Fig. 1 . In this Letter, we synthesize YIG/C 60/Pt systems containing buckyball layers of various thickness es and report the first comprehensive experimental demonstration of thermally generated pure spin current s in the systems through the longitudinal SSE (LSSE) . Our experiments show that the spin current in the YIG/C 60/Pt heterostructures is si gnificantly enhanced as compared to YIG/Pt structures. This enhancement can be as large as 600% and it is dependent on temperature and the thickness of the C 60 layer. We demonstrate that the long spin diffusion length of C 60 has a much pronounced positive impact on the SSE signal . The exponential temperature dependence of both the SSE voltage and 𝜆 suggests that 𝜆 scales with the SSE voltage in magnitude. Our first principles calculations based on density functional theory (DFT) corroborate the experimental measurements and provide further insight into the role of C 60 on the transport and magnetic properties of the YIG/C 60/Pt heterostructure. Effect of C 60 on the bulk and surface magnetic properties of YIG : First, we show how the coating of C 60 modifies the bulk and surface magnetic properties of YIG by means of magnetometry and radio -frequency (RF) transverse susceptibility. Figure 2a shows the magnetic hysteresis ( M-H) loops taken at 300 K for YIG, YIG/Pt and YIG/C 60(5 nm)/Pt structures. All three plots superimpose d on each other , indicate that it is not possible to use magnetometry to identify the magnetic difference among these samples. In a recent study, we have demonstrated 5 the excellent capacity of using our RF transverse susceptibility (TS) technique to probe the surface perpendicular magnetic anisotropy (PMA) field and its temperature evolution in YIG, providing the first experimental evidence for a strong effect of the surface PMA on VLSSE.25 Note that TS uses a self-resonant tunnel diode oscillator with a resonant frequency of ~12 MHz . The sensitivity of 10 Hz has been reached and validated by us over the years as a highly efficient tool for precisely measuring magnetic anisotropy in a wide range of magnetic mat erials.25-29 In the present study, we have performed detailed TS experiments on YIG, YIG/Pt, and YIG/C 60/Pt structures. Figure 2b shows typical TS curves at 200 K for these samples as dc magnetic field was swept from positive saturation to negative saturation. As the field was swept from positive to negative saturation, the first peak corresponds to the bulk magnetocrystalline anisotropy field (HK) and the second peak corresponds to the surface/interface magnetic anisotropy field (HKS) of the system. The unique TS method of measuring magnetic anisotropy provides the advantage of measuring surface and bulk anisotropy separately . Since SSE is a surface/interface -related phenomen on, we are interested in HKS and its temperature evolution, as it controls surface magnetization and hence LSSE voltage . Figure 2c shows t he temperature dependence of HKS for the YIG, YIG/Pt, and YIG/C 60/Pt samples. As expected, HKS (T) for YIG shows a peak around 75 K, which has been attributed to the rotation of surface spins away from the perpendicular easy-axis direction.25 It is worth noting in Fig. 2c that while the coating of Pt on the surface of YIG considerably increases HKS in YIG/Pt, as compared to YIG, the coating of C 60 on the surface of YIG drastically decreases HKS in YIG/C 60/Pt. The decrease in HKS in YIG/C 60/Pt can be due to the hybridization between the d z2 orbital of Fe and p orbitals of C atoms.24 As HKS inversely scales with VLSSE,25 the decrease in HKS of YIG due to C 60 interface is expected to increase VLSSE in YIG/C 60/Pt relative to YIG/Pt. 6 To provide an in -depth understanding of the interfacial properties of the YIG/C 60/Pt system in regards to the experimental measurements, we perform first principles simulation based on DFT for the atomic and electronic structure properties of the studied system . Details of the computational approach are given in the Methods section. Due to the limitations imposed by the complexity of the studied composite and the potentially large number of atoms in the supercell, we take advantage of the fact that the majority of the magnetic moment within the YIG unit cell is highly localized to the Fe sites30, and thus model the YIG layer by a Fe layer . The periodically distributed C 60 molecules between Fe and Pt layers, each composed of three monolayers, are shown in Fig. 3a, where some characteristic distances are also denoted. To get a better idea of the atomic bonds of C-Pt and C -Fe, a schematic accentuating the atomic locations directly above and below of the buckyball are shown in Fig. 3b together with some relative displacements with respect to the horizontal planes fixed by the rest of the atoms in the Fe and Pt layers. It is found that, in the relaxed configuration, the C 60 molecule is chemisorbed with one hexagonal C face atop of one Pt atom, which gets pushed below the initial layer position by 𝛿𝑃𝑡(1)≈0.4 Å, while the neighboring Pt atoms are pulled above by 𝛿𝑃𝑡(2)≈0.2 Å. The Fe atoms in the immediate vicinity of the C 60 cage are also displaced by 𝛿𝐹𝑒≈0.1 Å forming an armchair. The C atoms in the hexagonal face adjacent to the Pt layer form chemical bonds with a length of 𝑑𝑃𝑡−𝐶≈2.2 Å, while those in the hexagon close to the Fe layer form bonds with inequivalent lengths of 𝛿𝐹𝑒−𝐶(1)≈2 Å and 𝛿𝐹𝑒−𝐶(2)≈2.25 Å, respectively. The distance between the C 60 molecules is 𝑑𝐶60−𝐶60≈4.13 Å and it is larger than the overall separation of 3.13 Å of the buckyball cryst al31, which ensures a minimal interaction between adjacent buckyballs . All structural parameters are summarized in Table 1 , where results including the effects of the van der Waals interaction via the DFT -D3 approach are also shown. Our calculations indicate that 7 most of the characteristic distances do not change significantly, although 𝐷 is reduced by 0.01 Å, while 𝛿𝑃𝑡(1) and 𝛿𝑃𝑡(2) are reduced by 0.04 Å and 0.03 Å, respective ly, upon taking the van der Waals dispersion into account. The calculated average magnetic moments per layer are also shown in Fig. 3b. The C atoms in the hexagonal face close to the Fe layer acquire an antiparallel average magnetic moment of about 0.01 μ B. The intercalation of the C 60 molecule decreases the magnetic moments of the interfacial Fe atoms in the immediate vicinity of the C 60 cage to 2.27 μ B, while it reduces to zero the proximity induced magnetization of the Pt atoms in the first layer near the C 60 cage. This suggests that there is a reduction of the interfacial PMA, which is consistent with our measurements of HKS in YIG/C 60/Pt, and it is attributed to the hybridiza tion of the Fe dz2 orbitals with C p z orbitals24. We also calculate the electronic density of states (DOS), where the spin -resolved results are given in Fig. 3c, while the total density of states is shown in Fig. 3d. In addition to DOS for the Pt/C 60/Fe, we also present the obtained DOS for the Pt/Fe for comparison. The Fe/Pt structure is for med by removing the C 60 molecules and allowing the adjacent layers of Pt and Fe to relax and bond. It is interesting to note that while the Pt/Fe system exhibits a spi n-polarized DOS near the position of the Fermi level, with the minority spins having the dominant contribution to the transport, for the Pt/C 60/Fe structure both spins contribute almost equally to the DOS at the Fermi level ( Fig. 3c). This ultimately leads to a reduction in the conductivity mismatch between the Pt and Fe layers as a result of the C 60 interface. This situation is further clarified by the total DOS in Fig. 3d, which shows a significant enhancement of the conduction states near 𝐸𝐹 for the Pt/C 60/Fe heterostructure as compared to the Pt/Fe system. In fact, it is found that DOS at 𝐸𝐹 for Pt/C 60/Fe is about 600% larger than DOS at 𝐸𝐹 for Pt/Fe, which further 8 corroborates the giant SSE enhancement in our experiments. Even though the synthes ized samples involve layers with different C 60 thicknesses, we note that the individual buckyball has an energy gap between its highest occupied molecular orbital and the lowest unoccupied molecular orbital, which is very similar to the semiconducting gaps of a linear chain of C 60 molecules32, thus the characteristic DOS behavior is due to the interface effects with the Pt and Fe layers and they are expected to be preserved regardless of the thickness of the C 60 layer. Thus the simulated structure as depict ed in Fig. 3a is expected to be a good representative of the measured samples. Effect of C 60 on the LSSE voltage in the YIG/C 60/Pt structure : Figure s 4a and b show the LSSE voltage (VLSSE) vs. magnetic field (H) curves for YIG/C 60/Pt samples with varying C 60 thicknesses ( tC60 = 0, 5, 10, 30, and 50 nm) at two representative temperatures of 140 and 300 K , for a temperature gradient of T = 2 K . It can be seen in this figure that in the low field region ( H ≤ 0.3 kOe), VLSSE is relatively small (almost zero) and remains almost unchanged with increasing the magnetic field. This low field anomal y has been attributed to the presence of the surface PMA of YIG.18,25 It is worth mentioning in the present case that even after the introduction of the C60 layer between the YIG and Pt layers , the anomalous low field VLSSE (H) behavior is still persistent , underlin ing the same mechanism for LSSE voltage generation in YIG/ C60/Pt. Saturated VLSSE has been calculated as the average of positive and negative peak voltage s. At 300 K, YIG/Pt with no C 60 layer has produced VLSSE of 110 nV (Fig. 4a). When a 5nm C 60 film was introduced between the YIG and Pt interfaces, VLSSE increased to 190 nV. The enhancement of VLSSE due to the C 60 intermediate layer bec omes more prominent at low temperature. At 140 K, VLSSE increases from 70 to 660 nV with the insertion of the 5 nm C 60 thin film (Fig. 4b). As can be summarized in Fig. 4c, when the thickness of C 60 is increased from 5 to 9 50 nm, VLSSE decrease s sharply first and then gradually . At 300 K, YIG/C 60/Pt samples with tC60 = 10, 30, and 50 nm show smaller values of VLSSE as compared to YIG/Pt. At 140 K, however, the opposite trend is observed. For the thickest C 60 layer (50 nm), the VLSEE value of YIG/C 60/Pt is still greater than that of YIG/Pt. To elucidate th e observed phenomenon , we have studied in detail the temperature evolution of VLSEE in YIG/Pt, YIG/C 60(5 nm)/Pt, YIG/C 60(10 nm)/Pt, YIG/C 60(30 nm)/Pt, and YIG/C 60(50 nm)/Pt . All measurements were performed from 300 to 140 K , and the results are shown in Fig. 5a. It should be recalled that for YIG/Pt (with no C 60 layer) as the temperature was decreased , VLSEE decreas ed with a slope change around 170 K , and this slope change has been attributed to the effective magnetic anisotropy change in YIG , due to spin reorientation transition .25 However , all YIG/ C60/Pt samples have shown an opposite temperature dependence of VLSEE; VLSEE remain s almost constant up to 20 0 K but below which it starts increasing exponentially (Fig. 5a). To better visualize this, the LSSE signal normalized to the signal at 140 K is shown in Fig. 5b. The e xponential rise of VLSEE below 200 K is evident from this figure, for all C 60-coated samples. All t his suggests a dominant effect of C 60 deposition on the LSSE signal in the YIG/C 60/Pt system s. It has been experimentally shown that the spin diffusion length of C 60 possesses an exponential increase with lowering temperature just below 200 K when the film thickness is below 60 nm .33 This logically relates the temperature dependence of VLSEE to that of the spin diffusion length of C 60. In other words, the strong increase of VLSEE with a temperature below 200 K can be attributed to the strong temperature dependence of the spin diffusion length of C 60 in this temperature region. To verify this , the VLSEE (T) data has been fitted to an exponential function that can be used to describe the temperature dependence of the spin diffusion length of 10 C60, and a n example of this fit is shown in the inset of Fig. 5b. This result indicate s that t he long spin diffusion length of C 60 has indeed played a crucial role in promoting spin transport in YIG/C 60/Pt. To quantitatively explain the C 60 thickness -depende nt LSSE behavior in the YIG/C 60/Pt systems, we have adapted the model proposed for YIG/Pt by Lin et al.12 Since the exchange spin transport mechanism is dominant in the organic material C60, the spin current can be written in the following form: 𝐽𝑠𝑃𝑡/𝐶60/𝑌𝐼𝐺 =(𝑘∇𝑇𝑒−(𝑥 𝜆𝑃𝑡) ) 1+𝐺𝑌𝐼𝐺 (1 𝐺𝑃𝑡 𝐶60+1 𝐺𝑃𝑡) 1 cosh(𝑡𝑐60 𝜆𝑐60)+𝐺𝑐60(1 𝐺𝑃𝑡+1 𝐺𝐶60 𝑌𝐼𝐺)(sinh(𝑡𝑐60 𝜆𝑐60)) , (2) where k is the spin current coefficient, ∇𝑇 is the temperature gradient, G is the spin current conductance, 𝜆 is the spin diffusion length of the corresponding material, tC60 is the thickness of the C 60 layer. Since it is difficult to obtain spin current magnitude from LSSE measurements, we have considered the ratio of the spi n currents in both cases for our comparison purpose, 𝐽𝑠𝑃𝑡/𝐶60/𝑌𝐼𝐺 𝐽𝑠𝑃𝑡/𝑌𝐼𝐺 = ( 1+((𝐺𝑃𝑡 𝐶60 𝐺𝑃𝑡 𝑌𝐼𝐺)−1)𝐺𝑃𝑡 𝐺𝑃𝑡 𝐶60+𝐺𝑃𝑡 ) 1 cosh(𝑡𝑐60 𝜆𝑐60)+𝐺𝑐60(1 𝐺𝑃𝑡+1 𝐺𝐶60 𝑌𝐼𝐺)(sinh(𝑡𝑐60 𝜆𝑐60)) (3) In the spin-wave approximation ,12,34 𝐺𝑃𝑡 𝑌𝐼𝐺 is proportional to ( T/TC)3/2 , where TC is the Curie temperature of YIG. Since TC of YIG is very high (~560 K), spin c urrent injection should 11 increase as a thin layer C 60 is sandwiched in between YIG and Pt. This can explain the increased VLSEE for YIG/C 60(5nm)/Pt relative to YIG/Pt. It was previously reported that the insertion of a thin antiferromagnetic NiO layer (~1 nm ) between YIG and Pt increase d VLSEE.12 However, since 𝜆 of NiO is relatively small (~1-2 nm ), the hyperbolic function in th e denominator of Eq. (3) increases, result ing in reducti on of the spin current , as the NiO thickness is increased . In our case, as the thickness of the C 60 layer is increased, both hyperbolic terms in Eq. (3) increase, resulting in an exponential decrease of VLSEE. At the same time, reduction in HKS in YIG/C 60/Pt due to hybridization between the d z2 orbital of Fe and C atoms , which is evident from our TS studies and DFT calculations, would result in a net increase of spin moments at the YIG surface.24 Also , it has recently been shown that the Stoner criteria for magnetism can be beaten in C60 by the metal -molecule interface and can induce a magnetic moment in the metal surface .23 Theoretical studies have show n that increase in surface magnetic moment density increases the spin mixing conductance.14 This explains our observation of the enhanced VLSSE at room temperature in YIG/ 5nm C60/Pt for both single crystal and thin film38 of YIG as compared to YIG/Pt, when 𝜆c60 is relatively small at room temperature (~12 nm) . The decrease in VLSSE with increasing the C 60 thickness ( tC60) for the YIG/C 60/Pt systems can be decribed by the relation VLSSE e-t/.39 Fiting the C 60 thickness -dependent VLSSE data of YIG/5nm C 60/Pt at 300 K to this equation has yielded 𝜆c60 ~11±2 nm, which is similar to th at reported for the Ni 80Fe20/C60/Pt system (𝜆c60 ~13±2 nm at 300 K ) using the spin pumping method.39 This indicates that the C 60 spin current arriving at the C 60/Pt interface is proportional to e-t/. In conclusion, we have demonstrated a new, effective approach for enhancing the LSSE in a ferromagnetic insulator/metal system like YIG/Pt by adding a thin, intermediate layer of high spin diffusion length organic semiconductor like C 60. We have shown that the presence of 12 C60 reduces significantly the conductivity mismatch between YIG and Pt and the surface magnetic anisotropy of YIG , giving rise to the enhanced spin mixing conductance and hence the enhanced LSSE. Results from our first principles simulations demonstrate that the density of carriers at the Fermi level is much enhanced upon inclusion of the interface C 60 layer, which is also accompanied by a reduced magnetic anisotropy. The LSSE of YIG/C 60/Pt strongly depends on the spin diffusion le ngth of the intermediate layer C60; the temperature dependence of LSSE resembles that of the spin diffusion length of C 60. Our study provides a pathway for designing novel hybrid materials with prospective applications in spin caloritronics and other multi functional devices. Methods Sample characterization. Single crystal YIG was purchased from Crystal Systems Corporation, Hokuto, Yamanashi, Japan , which was grown using Floating zone method along (111) direction. Various layer s of C 60 were deposited on top of the YIG surface , using the therm al evaporation method with the evaporation rate of 0.2 Å/s at the base pressure of 2 x 10-7 torr. Figure 1b shows the SEM cross -section and EDX color map image of the 50 nm thick C 60 deposited YIG slab. From the SEM image, it can be concluded that the C 60 was evenly deposited on the surface of the YIG single crystal slab. Measurements. Longitudinal spin Seebeck voltage m easurements were performed on a YIG single crystal of dimension 6 mm 3 mm 1 mm (length width thickness) . A platinum strip of 6 mm 1 mm 15 nm was deposited on YIG using DC sputtering. The sputtering chamber was evacuated to a base pressure of 5 × 10−6 Torr and Argon pressure of 7 mT during the deposition. DC current and volt age used for deposition we re 50 mA and ~ 350 V, respectively. The schematic of the LSEE measurement set -up is shown in Fig. 1a. For LSSE measurements 13 YIG/Pt w as sandwiched between two copper plates. A Peltier module wa s attached to the bottom plate and top plate temper ature was controlled through m olybdenum screws attached to the cryogenic system. A temperature gradient of approximately 2 K was achieved by applying 3 A current to the Peltier module. K -type thermocouples were used to monitor the temperat ure of the top and bottom pla tes. After stepping the system t emperature and Peltier module current, measurements were performed after 2 h of stabilization time . The SSE voltage was recorded as the magnetic field wa s swept between positive and negative saturation of YIG, using a Keithley 2182 Nano voltmeter . Transverse susceptibility (TS) measurements were performed using a self -resonant tunnel diode oscillator with a resonant frequency of 12 MHz and sensitivity in resolving frequency shift on the order of 10 Hz.25,26 The tunnel diode oscillator is integrated with an insert that plugs into a commercial Physical Properties Measurement System (PPMS, Quantum Design), which is used to apply dc magnetic fields (up to ±7 T) as well as provide the measurement tempe rature range (10 K < T < 300 K). In the experiment, the sample is placed in an inductive coil, which is part of an ultrastable, self -resonant tunnel -diode oscillator in which a perturbing small RF field ( HAC ≈ 10 Oe) is applied perpendicular to the DC field. The coil with the sample is inserted into the PPMS chamber which can be varied the temperature from 10 K to 350 K in an applied field up to 7 T. Computational Methods . The first principles simulations are performed using the local spin density appro ximation to the density functional theory (DFT) as implemented in the Quantum ESPRESSO package 35. We use ultrasoft pseudopotentials with a kinetic energy cutoff of 320 eV. The exchange -correlation is treated within the Perdew -Burke -Ernzerhof (PBE) general ized gradient approximation36. For the calculations, we construct a supercell consisting of equally 14 spaced C 60 buckyballs sandwiched between a Fe layer composed of three bcc Fe (001) monolayers and a Pt layer composed of three fcc Pt (001) monolayers. In total the supercell consists of 156 atoms with 48 Pt atoms, 48 Fe atoms and 60 C atoms. The reciprocal space is sampled with a uniform Monkhorst -Pack 4 x 4 x 1 mesh. The outermost Fe and Pt monolayers are kept fixed during the relaxation, and all the other atoms are allowed to relax until the change in energy is less than 10-5 eV and the forces acting on atoms are less than 0.02 eV/A. We also included the vdW -D3 dispersion correction37 in the calculations. Acknowledgments Research at USF was supported by the Army Research Office through Grant No. W911NF -15-1- 0626 (Spin -thermo -transport studies) and by the U.S. Department of Energy, Office of Basic Energy Sciences, Division of Materials Sciences and Engineering un der Award No. DE -FG02 - 07ER46438 (Magnetic st udies) . LW acknowledges support from the US Department of Energy, Office of Basic Energy Sciences , under Grant No. DE-FG02 -06ER46297. The use of the University of South Florida Research Computing facilities are also acknowledged. TN acknowledges support fr om the STYLENQUAZA LLC. DBA VICOSTONE USA . Author contributions V.K., R.G. and R.D. had equal contributions to the work. M.H.P. and T.N. developed the initial concept. V.K., R.D., R.G., T.N., and M.H.P. designed the study. YIG/C 60 samples were fabricated by R. G. and M.P. YIG/C 60/Pt samples were fabricated by V. K. and R.D. Structural and m agnetic characterization , and spin Seebeck effect measurements were performed and analyzed by V.K. , and R.D. A.P. and L.M. W. performed DFT calculations and simulations. All 15 authors discussed the results and wrote the manuscript. M.H.P. and H.S. jointly led the research project. Additional information Competing financial interests: The authors declare no competing financial int erests. Corresponding auth ors: phanm@usf.edu (M.H.P ); ngtho@uga.edu (N.D.T. ); sharihar@usf.edu (H.S.) 16 References 1Wolf, S. A. et al. 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Nature 524, 69–73 (2015) 24Bairagi, K. et al. Tuning the Magnetic Anisotropy at a Molecule -Metal Interface. Phys. Rev. Lett. 114, 247203 (2015) 25Kalappattil , V. et al., Roles of bulk and surface magnetic anisotropy on the longitudinal spin Seebeck effect of Pt/YIG. Sci. Rep., 7 (1) (2017) 26Srikanth , H., Wiggins & Rees J. H. Radio -frequency impedance measurements using a tunnel - diode oscillator technique. Review of Scientific Instruments 70, 3097 (1999) 18 27Frey N. A. et al. Magnetic anisotropy in epitaxial CrO 2 and CrO 2/Cr 2O3 bilayer thin films. Phys. Rev. B 74, 024420 (2006). 28Woods G. T. et al. Observation of charge ordering and the ferromagnetic phase transition in single crystal LSMO using rf transverse susceptibility. J. Appl. Phys . 97, 10C104 (2005). 29Frey N. A. et al. Transverse susceptibility as a probe of the magnetocrystalline anisotropy - driven phase transition in Pr 0.5Sr0.5CoO 3, Phys. Rev. B 83, 024406 (2011) 30Xie, L . S., Jin, G . X., He, L., Bauer, G ., Barker, J ., Xia, K. First-principles study of exchange interactions of yttrium iron garnet. Phys. Rev. B 95, 014423 (2017) 31Krätschmer, W., Lamb, L. D., Fostiropoulos, K. & Huffman, D. R. Solid C 60: a new form of carbon . Nature 347, 354 –358 (1990) 32Belavin et al, Stability, electronic structure and reactivity of the polymerized fullerite forms, J. Phys. and Chem. Of Solids, 61, 1901 (2000) 33Nguyen, T. D., Wang, F., Li, X. G., Ehrenfreund, E. & Vardeny , Z. V. Spin diffusion in fullerene -based devices: Morphology effect. Phys. Rev. B 87, 075205 (2013) 34Zhang, S. S. L. & Zhang, S. Spin convertance at magnetic interfaces. Phys. Rev. B. 86, 214424 (2012) . 35Paolo, G. et al. QUANTUM ESPRESSO: A modular and open -source software project for quantum simulations of materials . J. Phys. Condens. Matter 21, 395502 (2009). 36Perdew, J. P., Burke, K. & Ernzerhof, M. Generalized gradient approximation made simple . Phys. Rev. Lett. 77, 3865 (1996) 37 Grimme, S., Antony, J., Ehrlich, S. & Krieg, H. A consistent and accurate ab initio parametrization of density functional dispersion correction (DFT -D) for the 94 elements H-Pu. J. Chem. Phys. 132, 154104 (2010) 38 Das R., Kalappattil V., Geng R., Luong H., Pham M., Nguyen T., Liu T., Wu M.Z., Phan M.H., and Srikanth H., Enhanced room -temp erature spin Seebeck effect in a YIG/C 60/Pt layered heterostructure , AIP Advances 8, 055906 (2018 ) 19 39 Liu, H.L., Wang, J.Y., Groesbeck, M., Pan, X., Zhang, C., Valy Vardeny, Z. Studies of spin related processes in fullerence C 60 devices. J. Mater. Chem. C 6, 3621 (2018) 20 Figure captions Figure 1 (a) Measurement geometry and spin transport through the YIG/C 60/Pt layer. Cross - sectional SEM and EDX color map images of the 50 nm thick C 60 deposited YIG slab. Figure 2 (a) Magnetic hysteresis ( M-H) loops taken at 300 K for YIG, YIG/Pt, and YIG/C 60/Pt structures; (b) Transverse susceptibly spectra taken at 200 K for YIG, YIG/Pt , and YIG/C 60/Pt structures ; and (c) Temperature dependence of surface/interface perpendicular magnetic anisotropy field ( HSK) for YIG , YIG/Pt , and YIG/C 60/Pt structures . Figure 3 (a) The Pt/C 60/Fe heterostructure, where Pt and Fe atoms are indicated with grey and red colors and C atoms are in yellow. Characteristic interatomic distances are also denoted. (b) Schematic representation of the Pt/C 60/Fe relaxed structure, showing the buckling (enhanced to help visualization) of the innermost Pt and Fe layers due to the intercalation of the C 60 molecules. Some characteristic vertical displacements are also shown. The numbers on the right represent the calculated average magnetic moments, in Bohr magneton, per layer. The considered C layers in are composed of the C hexagons closest to the metallic surfaces. (c) Spin resolved DOS for the Pt and Fe layers, with and without the C 60 molecule. (d) Total DOS for the same structures as in (c). Figure 4 LSSE voltag e vs. magnetic field curves taken at (a) 300 K and ( b) 140 K for YIG/C 60/Pt with different thickness es of C 60. Figure 5 (a) C60 thickness dependence of the L SSE signal at 300 K and 140 K; (b) Temperature dependence of L SSE voltage for YIG/C 60/Pt with different thickness es of C 60; (c) The normalized value of LSSE for different C60 thickness es of YIG/C 60/Pt. Inset of (c) shows the fit 21 for the 5 nm C 60 thickness. The temperature dependence of spin diffusion length determined from our SSE method and the MR method .33 22 Table 1 Relaxed structural parameters of the Fe/C60/Pt layer shown in Fig s 3a and 3b. Specifically, 𝐷, 𝑑𝐹𝑒−𝐶(1,2), , 𝑑𝑃𝑡−𝐶, 𝑑𝐶60−𝐶60, 𝛿𝑃𝑡(1,2), , and 𝛿𝐹𝑒 (in Å ) represent the separation between the Fe and Pt layers, the lengths of the covalent bonds formed between the Fe and C atoms, between Pt and C atoms, the distance between the C 60 molecules, and the displacements of the Pt and Fe atoms located immediately in the vicinity of the C 60 molecule, respectively. The numbers in parenthesis represent the same distances calculated by including the vdW correctio n. 𝐷 (Å) 𝑑𝐹𝑒−𝐶(1)(Å) 𝑑𝐹𝑒−𝐶(2)(Å) 𝑑𝑃𝑡−𝐶(Å) 𝑑𝐶60−𝐶60(Å) 𝛿𝑃𝑡(1)(Å) 𝛿𝑃𝑡(2)(Å) 𝛿𝐹𝑒(Å) 10.33 (10.31) 2.021 (2.02) 2.25 (2.25) 2.21 (2.198) 4.13 (4.13) 0.44 (0.4) 0.22 (0.18) 0.11 (0.11) 23 Figure 1 24 Figure 2 25 Figure 3 26 Figure 4 27 Figure 5

2019-05-11

Interfacing an organic semiconductor C60 with a non-magnetic metallic thin film (Cu or Pt) has created a novel heterostructure that is ferromagnetic at ambient temperature, while its interface with a magnetic metal (Fe or Co) can tune the anisotropic magnetic surface property of the material. Here, we demonstrate that sandwiching C60 in between a magnetic insulator (Y3Fe5O12: YIG) and a non-magnetic, strong spin-orbit metal (Pt) promotes highly efficient spin current transport via the thermally driven spin Seebeck effect (SSE). Experiments and first principles calculations consistently show that the presence of C60 reduces significantly the conductivity mismatch between YIG and Pt and the surface perpendicular magnetic anisotropy of YIG, giving rise to enhanced spin mixing conductance across YIG/C60/Pt interfaces. As a result, a 600% increase in the SSE voltage (VLSSE) has been realized in YIG/C60/Pt relative to YIG/Pt. Temperature-dependent SSE voltage measurements on YIG/C60/Pt with varying C60 layer thicknesses also show an exponential increase in VLSSE at low temperatures below 200 K, resembling the temperature evolution of spin diffusion length of C60. Our study emphasizes the important roles of the magnetic anisotropy and the spin diffusion length of the intermediate layer in the SSE in YIG/C60/Pt structures, providing a new pathway for developing novel spin-caloric materials.

Giant Spin Seebeck Effect through an Interface Organic Semiconductor

1905.04555v1

Spin-torque oscillation in a magnetic insulator probed by a single-spin sensor H. Zhang1,2, †, M.J.H. Ku1,2, †, F. Casola1,2, C.H. Du2, T. van der Sar2, ‡, M.C. Onbasli3,4, C.A. Ross3, Y. Tserkovnyak5, A. Yacoby2,6, R.L. Walsworth1,2,7,* 1Harvard-Smithsonian Center for Astrophysics, 60 Garden Street, Cambridge, MA 02138, USA. 2Department of Physics, Harvard University, 17 Oxford Street, Cambridge, MA 02138, USA. 3Department of Materials Science and Engineering, Massachusetts Institute of Technology, 77 Massachusetts Avenue, Cambridge, MA 02139, USA. 4Koç University, Department of Electrical and Electronics Engineering, Sarıyer, 34450 Istanbul, Turkey. 5Department of Physics and Astronomy, University of California, Los Angeles, 475 Portola Plaza, Los Angeles, CA 90095, USA. 6John A. Paulson School of Engineering and Applied Sciences, Harvard University, Cambridge, MA 02138, USA. 7Center for Brain Science, Harvard University, Cambridge, Massachusetts 02138, USA. †These authors contributed equally to this work. ‡Present address: Kavli Institute of Nanoscience, Delft University of Technology, 2628CJ Delft, Netherlands. *e-mail: rwalsworth@cfa.harvard.edu Coherent, self-sustained oscillation of magnetization in spin-torque oscillators (STOs)1,2,3,4 is a promising source for on-chip, nanoscale generation of microwave magnetic fields. Such fields could be used for local excitation of spin-wave resonances, control of spin qubits, and studies of paramagnetic resonance. However, local characterization of fields emitted by an STO has remained an outstanding challenge. Here, we use the spin of a single nitrogen-vacancy (NV) defect in diamond to probe the magnetic fields generated by an STO in a microbar of ferromagnetic insulator yttrium-iron-garnet (YIG). The combined spectral resolution and sensitivity of the NV sensor allows us to resolve multiple spin-wave modes and characterize their damping. When damping is decreased sufficiently via spin injection, the modes auto-oscillate, as indicated by a strongly reduced linewidth, a diverging magnetic power spectral density, and synchronization of the STO frequency to an external microwave source. These results open the way for quantitative, nanoscale mapping of the microwave signals generated by STOs, as well as harnessing STOs as local probes of mesoscopic spin systems. Spin-torque oscillators (STOs) have been proposed as on-chip sources of spin waves2, as nanoscale microwave generators2,3,4, and as building blocks in neural networks2,3,5. While optical methods6,7,8 such as Brillouin light scattering and magneto-optical Kerr effect microscopy have been used to spatially investigate the magnetization dynamics of STOs, the detection of on-chip STO microwave fields with a probe that allows nanoscale spatial imaging and MHz spectral resolution has remained an outstanding challenge. Recently, the electron spin associated with the nitrogen-vacancy (NV) defect in diamond has emerged as a sensitive magnetic-field sensor9 that allows nanometer-scale spatial resolution10,11, sub-Hz spectral resolution12, and excellent magnetic-field sensitivity13. Here, we use NV magnetometry to study the local magnetic fields generated by an STO in a YIG microbar that can be driven into auto-oscillation by a spin-current injected via the spin-Hall effect in a platinum thin film8,14,15. To locally detect the STO magnetic fields, we position a diamond nanobeam containing an individually addressable NV sensor at ~100 nm from a Pt/YIG hybrid microstructure (Fig. 1a). Au electrical leads supply the DC current Idc to the Pt wire for spin injection. A nearby Au stripline (Fig. 1b) delivers microwave signals for both control of the NV spin state and for microwave-driving of spin-wave modes in the YIG bar. Further information on NV physics relevant to sensing of magnetic fields can be found in refs. 9-13,16-18, and details of the fabrication processes can be found in the Methods and in Supplementary Information 1. We start by probing the spin-wave spectrum of the YIG micro-magnet using microwave excitation. We sweep the frequency of a microwave drive field and use the NV spin sensor to detect changes in the stray static magnetic field due to changes in the YIG magnetization upon exciting a spin-wave resonance16. Fig. 1c depicts the sensing sequence, and Fig. 1d shows the spectrum at zero DC current. We observe multiple spectral peaks, the most prominent of which persists as the strongest mode throughout the entire sweep range of the external magnetic field Bext. We attribute this peak to the spatially homogeneous (n=1) ferromagnetic resonance (FMR) of the YIG bar as it couples efficiently to our microwave drive field. The centre frequency of this mode vs Bext follows a Kittel-like dependence (Fig. 1e). Next, we demonstrate control of spin-wave resonance damping by injecting a spin current into the YIG via the Pt contact. As illustrated in Fig. 2a, the effect of a spin current on the dynamics of the magnetization can be described as a spin-orbit torque acting on the YIG magnetization19. Depending on the relative orientation between the injected spins and the equilibrium magnetization, the spin-orbit torque can either reduce or enhance the effective damping of precessional motion of the magnetization vector (see Supplementary Information 3). We observe such damping modification by measuring the response of the YIG stray field to the microwave drive as a function of Idc. In Fig. 2b, we show example FMR spectra of the YIG stray field 𝛥𝐵||, normalized by the microwave drive amplitude b1, as a function of the drive frequency at several Idc. A strong dependence of the peak amplitude on Idc indicates that damping is reduced (enhanced) for positive (negative) Idc. In addition, we observe a shift in the resonance frequency as a function of Idc (Fig. 2b) that is well described by a second-order polynomial8. We attribute the quadratic (symmetric) part of this polynomial to Joule heating, while the linear part may be attributed to a combination of effects such as an Oersted field generated by the current in the Pt, and a change in the YIG magnetization caused by the spin-orbit torque (see Supplementary Information 4). When the intrinsic magnetic damping is compensated by the anti-damping torque exerted by the injected spin current, we expect an increase in the rate of change of the FMR peak amplitude as a function of the microwave drive power. Figure 2c plots the on-resonance peak amplitude of 𝛥𝐵||, extracted from data such as those in Fig. 2b, as a function of the microwave drive power 𝑏!! at different Idc. We observe two distinct regimes: for Idc ≲ 4 mA, the peak amplitude increases approximately linearly over a large range of microwave drive power, while for Idc≳4 mA, the signal increases sharply and then saturates. The distinction between these two regimes is more evident in Fig. 2d, which plots the inverse of the initial slopes of 𝛥𝐵|| vs 𝑏!! (i.e., 𝑑𝑏!!𝑑𝛥𝐵||) as a function of Idc. The trend of the inverse on-resonance slopes quantifies the evolution of the effective damping coefficient 𝛼!"", which monotonically changes with 1𝜒!!!∝𝑏!𝑚!~𝑑𝑏!!𝑑Δ𝐵||, where 𝜒" is the imaginary part of the YIG bar’s magnetic susceptibility, and m is the oscillation amplitude of the transverse magnetization20. In Fig. 2d we observe 𝑑𝑏!!𝑑Δ𝐵|| decreases as a function of Idc (diagonal dashed line) until a certain threshold, after which it plateaus (horizontal dashed line). We interpret the plateau to be due to auto-oscillation of a spin-torque oscillator (STO), with the crossing of the dashed lines corresponding to the STO onset threshold current ~ 3.8 mA. Note that this threshold current value agrees well with an independent estimate obtained from Fig. 4a. To study auto-oscillation of the spin-wave modes in the YIG bar, we characterize the power spectral density of the magnetic-noise generated by the modes as a function of Idc and in the absence of microwave excitation. We use the NV spin as a field-tunable spectrometer via a technique known as NV spin relaxometry16, where the NV spin relaxation rates Γ are measured to quantify the magnetic-noise power spectral density 𝐵!(𝜔) at the NV 𝑚!=0↔±1 transition frequencies 𝜔±=2π(Dgs±γBext) via the relation 𝛤±=!!!𝐵!(𝜔±). Here, γ=2.8 MHz/G is the NV gyromagnetic ratio, Dgs =2.87 GHz is the NV zero-field splitting, and Bext is the external static magnetic field aligned with the NV axis. To characterize NV spin relaxation, we prepare the NV spin in the mS=0 state and determine the spin-relaxation rate 𝛤 in the presence of YIG bar magnetic noise by measuring the spin-dependent photoluminescence 𝑃𝐿𝜏,𝜔=𝑃𝐿(𝜏=0)𝑒!!(!)! after a hold time τ (Fig. 3a). In Fig. 3b, we show an example NV spin relaxometry experiment at Idc=5.8 mA, where we measure the PL at a fixed τ as we vary the external magnetic field that sweeps the lower transition frequency 𝜔! over several spin-wave resonances. In this situation, the rate 𝛤≈𝛤!; Γ+ is negligible as the density of thermal magnons is suppressed at energy 𝜔! which is far detuned from spin-wave resonances. The magnetic field fluctuations produced by the different YIG spin-wave modes increase the NV spin relaxation rate, which results in decreased NV PL21,22. By performing this measurement at multiple Idc, we map the noise spectrum of spin-waves as a function of Bext and Idc (Fig. 3c). The red/blue stars denote the locations of prominent spin-wave resonances (obtained by fitting the peak centres at each Idc), which we call STO1 and STO2. We identify STO1 as the spatially homogeneous (n=1) FMR and STO2 as a higher order (n=2) spin wave mode (Supplementary Information 5). The spectral resolution of the NV sensor allows us to zoom in closely on the regions where the two spin-waves approach each other (Figs. 3d and 3e), and observe what seems to be a mode anti-crossing, hinting at hybridization of the spin-wave modes due to mode interactions. Micromagnetic simulations elucidate the nature of the modes and point to the possibility of mode mixing (Supplementary Information 5). We estimate a mode coupling strength of about 10 MHz (Supplementary Information 6), which is larger than the linewidth of the individual modes (see Fig. 4b). To quantitatively study the power spectral density of the magnetic field noise generated by STO1 and STO2, we tune the NV transition frequency into resonance with the modes by adjusting Bext and extract the NV spin relaxation rate Γ as a function of Idc (Fig. 4a). As we change Idc, we observe a dramatic increase in the STO magnetic-noise power spectral density of up to three orders of magnitude, a key signature of auto-oscillation23. The inset of Fig. 4a maps 1/Γ as a function of Idc, and we use the intersection of a linear fit of 1/Γ at low current with 1/Γ = 0 to indicate the onset (threshold) of auto-oscillation, following the relation24,25 1/𝛤∝1𝑝∝1−𝐼!"𝐼!!, where p is the peak power spectral density emitted by the STO and Ith is the threshold current. For STO1, we estimate Ith1 ≈ 3.5 mA, close to the estimate made above from the stray-field magnetometry measurements in Fig. 3d. For STO2, we obtain a higher threshold current Ith2 ≈ 4.4 mA. A strong correlation between linewidth reduction (Fig. 4b) and divergence of magnetic fluctuations (Fig. 4a) is consistent with Landau-Lifshitz-Gilbert phenomenology: that is, the spin-orbit torque reduces the damping torque and the associated STO linewidth. As we increase Idc further, we surprisingly observe a reduction in the power spectral density accompanied by linewidth broadening and the appearance of a higher-order STO (STO* as shown in Fig. 3b). This may imply that strong spin injection introduces an additional magnon decay channel and the magnetic system is approaching a re-thermalization scenario26, though we leave a study of this phenomenon to future work. Finally, we demonstrate that the Pt/YIG STO can be synchronized with an external microwave source, as observed previously for STOs in metallic ferromagnets26. We use the measurement scheme shown in Fig. 5a, which is an NV spin relaxometry measurement with an added microwave drive field. We sweep the frequency fMW of this drive field around the free-running STO frequency. By monitoring the magnetic-noise power spectral density at the NV transition frequency, we observe locking of the STO over a frequency interval Δfs (Fig. 5b). Figures 5c and 5d show that the locking interval increases approximately linearly with the drive amplitude b1 as expected for frequency-locked oscillators. We observe an increase of the synchronization bandwidth for larger Idc (Fig. 5d). Frequency locking to an external microwave source can be used to quickly tune the STO in and out of resonance with the control frequency of a target system. In summary, we used the spin of a single NV center in diamond as a nanoscale magnetic sensor to measure the local magnetic fields generated by spin-torque oscillators (STOs) driven by spin-current in a Pt/YIG hybrid microstructure. We demonstrated STO auto-oscillation15 in this magnetic insulator using three independent methods: suppression of the effective damping torque, divergence of the power spectral density at the STO frequency, and STO synchronization to an external microwave source. High spectral resolution is a key capability of NV sensing and can be further improved below ~1 Hz by, for example, quantum interpolation27, synchronized readout12, or modest cooling17. Thus in future work, NV sensors should be able to provide access to the sub-Hz regime of advanced research on STOs3, as well as nanoscale spatial characterization of STO-generated magnetic fields10,11. Spatial mapping at such length scales would provide access to locations of large STO magnetic-field intensity, with the potential to use an STO to drive magnetic excitations in other systems of interest, such as spin waveguides28 and spin qubits29. Finally, studies of spin-torque oscillation may provide insight into phenomena such as magnon thermodynamics30, strongly-correlated many body physics18, and control over magnetic phase transitions31. Methods Hybrid Pt/YIG device preparations. Fabrication of the Pt/YIG device starts with a 17-nm YIG film epitaxially grown on a (111) orientation GGG substrate using pulsed laser deposition32. A 10 nm layer of Platinum (Pt) is sputtered on top of the YIG film, which is first cleaned by an Ar+ plasma at a pressure below 5×10-8 Torr to ensure good Pt purity. The Pt/YIG stripe is defined by electron-beam lithography (Elionix F125, 125 kV) with a PMMA (495A2, ~ 30 nm)/HSQ (XR-1541-006, ~250 nm and FOX-16, ~ 500 nm) resist stack, followed by developing in 25% TMAH. Ar+ ion milling is used to transfer the pattern onto the substrate and form the Pt/YIG hybrid microstructure. Finally, leads for DC current and microwave driving are defined by electron-beam lithography and e-beam evaporation techniques. See Supplementary Information 1 for further details. Experimental setup and nanobeam fabrication. The experimental setup is based on a home-built laser scanning confocal microscope, which has been described previously11. As part of the experimental sensing platform, we pattern bulk diamond containing NVs into a nanobeam structure33 and place it close to the sample of interest, with a single NV sensor within about 100 nm of the Pt/YIG microstructure to access the sample’s relatively weak localized fields. References 1. Chumak, A. V., Vasyuchka, V. I., Serga, A. A. & Hillebrands, B. Magnon spintronics. Nat. Phys. 11, 453–461 (2015). 2. Locatelli, N., Cros, V. & Grollier, J. Spin-torque building blocks. Nat. Mater. 13, 11–20 (2014). 3. Chen, T. et al. Spin-Torque and Spin-Hall Nano-Oscillators. Proc. IEEE 104, 1919–1945 (2016). 4. Demidov, V. E. et al. Magnetization oscillations and waves driven by pure spin currents. Phys. Rep. 673, 1-31 (2017). 5. Torrejon, J. et al. Neuromorphic computing with nanoscale spintronic oscillators. Nature 547, 428–431 (2017) 6. Demokritov, S. O. et al. Bose-Einstein condensation of quasi-equilibrium magnons at room temperature under pumping. Nature 443, 430–433 (2006). 7. 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USA 114, 2149-2153 (2017). 28. Collet, M. et al. Spin-wave propagation in ultra-thin YIG based waveguides. Appl. Phys. Lett. 110, 092408 (2017). 29. Sutton, B. & Datta, S. Manipulating quantum information with spin torque. Sci. Reports 5, 17912 (2015). 30. Safranski, C. et al. Spin caloritronic nano-oscillator. Nat. Commun. 8, 117 (2017). 31. Giamarchi, T., Rüegg, C. & Tchernyshyov, O. Bose-Einstein Condensation in Magnetic Insulators. Nat. Phys. 4, 198 (2008). 32. Lang, M. et al. Proximity Induced High-Temperature Magnetic Order in Topological Insulator - Ferrimagnetic Insulator Heterostructure. Nano Lett. 14, 3459 (2014). 33. Burek, M. J. et al. Free-standing mechanical and photonic nanostructures in single-crystal diamond. Nano Lett. 12, 6084 (2012). Acknowledgements The authors acknowledge the provision of diamond samples by Element 6, assistance with nanobeam fabrication from M. Warner and M. Burek, the use of a setup for nanobeam transfer from P. Kim, use of the ion mill facility in the J. Moodera lab, and experimental assistance from K. Arai, M. Han, and J.-C. Jaskula. This material is based upon work supported by, or in part by, the United States Army Research Laboratory and the United States Army Research Office under Contract/Grants No. W911NF1510548 and No. W911NF1110400. A.Y. acknowledges support from the Army Research Office under Grant Number W911NF-17-1-0023. The views and conclusions contained in this document are those of the authors and should not be interpreted as representing the official policies, either expressed or implied, of the Army Research Office or the U.S. Government. The U.S. Government is authorized to reproduce and distribute reprints for Government purposes notwithstanding any copyright notation herein. Work at the Massachusetts Institute of Technology was supported by the Solid-State Solar-Thermal Energy Conversion Center (S3TEC), an Energy Frontier Research Center funded by DOE, Office of Science, BES under award no. DE-SC0001299/DE-FG02-09ER46577. Work at the University of California, Los Angeles, is supported by the U.S. Department of Energy (DOE), Office of Basic Energy Sciences (BES) under award no. DE-SC0012190. F.C. acknowledges support from the Swiss National Science Foundation grant no. P300P2-158417. This research is also funded in part by the Gordon and Betty Moore Foundation’s EPiQS Initiative through Grant GBMF4531, the STC Center for Integrated Quantum Materials, NSF Grant No. DMR-1231319, and by the National Science Foundation under Grant No. EFMA-1542807. This work was performed in part at the Center for Nanoscale Systems (CNS), a member of the National Nanotechnology Coordinated Infrastructure Network (NNCI), which is supported by the National Science Foundation under NSF award no. 1541959. CNS is part of Harvard University. Author contributions H.Z. and M.J.H.K. contributed equally to this work. H.Z, M.J.H.K, A.Y., and R.L.W. conceived the project. R.L.W. and A.Y. supervised the project. H.Z. and M.J.H.K developed measurement protocols, built the experimental setup for NV measurement, performed the measurements, and analyzed the data. H.Z. fabricated the Pt/YIG device and performed micromagnetic simulations. F.C. and H.Z. developed the nanobeam platform and fabricated the nanobeams. F. C. helped with micromagnetic simulations. M.C.O. and C.A.R. provided the YIG sample. H.Z., M.J.H.K., T.v.d.S., F.C., C.H.D., Y.T., A.Y., and R.L.W. contributed to the interpretation. H.Z., M.J.H.K., T.v.d.S., C.H.D., and R.L.W. wrote the manuscript with the help from all co-authors. Competing interests The authors declare no competing financial interests Materials & Correspondence. Correspondence to R. L. Walsworth (rwalsworth@cfa.harvard.edu). Figures Figure 1 | Local probing of a Pt/YIG spin-torque oscillator using a single nitrogen-vacancy (NV) sensor spin in diamond. a, Schematic illustration of the device: a diamond nanobeam containing a single NV spin is positioned ~100 nm from a hybrid Pt/YIG structure (2.5×9 µm2). The Pt (YIG) thickness is 10 nm (17 nm). Au wires provide electrical contact to the Pt film. b, False-colored scanning electron micrograph of the device (before positioning the diamond nanobeam). c, Measurement sequence for stray-field magnetometry of spin-wave resonances. A green laser pulse initializes the NV spin. The first microwave π/2 pulse prepares an NV spin superposition, followed by a spin-echo sequence with two π pulses. A spin-wave (FMR) drive is applied during the central 2τ period. A change in the YIG stray field ΔΒ|| imparts a phase ϕ=γΔΒ||2τ on the NV spin state, where γ=2.8 MHz/G is the NV gyromagnetic ratio. The final π/2 pulse converts this phase to an NV spin population difference, which is read out via spin-dependent photoluminescence. A free precession time τ ≈ 5.5 µs is used for such stray-field magnetometry. d, Example of YIG spin-wave resonances measured with the pulse sequence in c, at applied static magnetic field Bext = 337 G aligned with the NV axis. Plotted is the NV-measured stray static magnetic field along the NV axis, ΔB|| , as a function of the spin-wave drive frequency. The signal is normalized by b!!, which is proportional the spin-wave drive power. b1 is independently measured on-chip using the same NV sensor via tuning Bext to bring the m!=0↔−1 transition on resonance with the drive field and measuring the NV Rabi frequency16. Blue dots: data. Red line: double Gaussian fit, yielding FWHM = 8.5(6) MHz for the dominant mode attributed to the spatially homogeneous (n=1) ferromagnetic resonance (FMR) of the YIG bar. e, Green dots: Magnetic-field (Bext) dependence of the fundamental spin-wave resonance frequency extracted from fits to measurements such as shown in d. Blue line: fit reveals characteristic Kittel-like behavior of FMR. Black lines: NV transition frequencies corresponding to the m!=0↔±1 transitions. NV-spin manipulation pulses are applied on the m!=0↔+1 transition. Figure 2 | Controlling spin-wave damping using electrically controlled spin injection. a, Sketch of the magnetization dynamics of the Pt/YIG device under the influence of a spin current. The electrical current (Je) injects a spin current (Js) into the YIG, leading to a spin-orbit torque (labeled SOT) that either reduces or enhances magnetic damping depending on the relative orientation between the injected spins and the magnetization M. DT denotes the (intrinsic) damping torque. b, NV-measured, microwave-driven spin-wave resonance spectra in the YIG as a function of the DC current Idc through the Pt. (Measurement sequence shown in Fig. 1c.) Blue traces: normalized change in YIG stray field ∆B||b!! as a function of microwave driving frequency for Idc = 5, 4, 3, 2, 1, 0, -2, -5 mA. Red lines: double Gaussian fit to data. Green dots: centre frequency of the fundamental spin-wave mode vs. Idc. Black curve: parabolic fit to green dots. c, On-resonance ∆B|| as a function of microwave driving power b!! for different values of Idc. Black, red, and pink dots correspond to Idc = 4.5, 4.75, and 5 mA, for which the initial slopes (d∆B||db!!) have no discernable difference. Blue and green squares correspond to Idc = 3 and 3.5 mA, for which initial slopes are significantly smaller. d, Plot of the inverse of the initial slopes, i.e., db!!d∆B||, as a function of Idc. Diagonal and horizontal dashed lines serve as eye guide to illustrate that there exists a current threshold as onset of an auto-oscillating spin torque oscillator (STO). Figure 3 | Spin-wave noise spectroscopy via NV spin relaxometry. a, NV spin relaxometry measurement sequence. The NV spin is initialized into m!=0 by a green laser pulse and let to relax for a time τ, after which the spin population is characterized via the spin-dependent PL during a laser readout pulse. The current Idc enhances or reduces spin-wave damping in the YIG bar and alters the power spectral density of the magnetic-field noise accordingly. Noise that is resonant with an NV transition frequency causes NV spin relaxation. b, Example NV spin relaxometry measurement at Idc=5.8 mA and τ=5 µs. By tuning the magnetic field Bext, the frequency of the m!=0↔−1 transition is swept over three spin-wave (SW) modes in the YIG, whose field-noise causes strong NV spin relaxation and thus dips in the normalized PL signal. c, Performing the measurement shown in panel b for different Idc yields a 2D plot of PL vs Idc and Bext that displays the presence and dispersion of spin-torque oscillators (STOs) in the system. Different delay times τ of 150 µs, 50 µs, 15 µs, 5 µs, and 3 µs are used for the different Idc ranges of [-5 mA:0 mA], [0.2 mA:1.8 mA], [2 mA:3 mA], [3.2 mA:5 mA], and [5.2 mA:6 mA], respectively. Top horizontal axis shows the m!=0↔−1 transition frequency at corresponding Bext. Blue stars indicate fits of peak centres for the first resonance on the left-hand-side (STO1), while red stars are fits of peak centres for the second (STO2). These two STOs are also indicated in panel b. Note that an additional oscillator (data points are orange in color and designated as STO* in panel b) appears when Idc = 5.8 mA and persists for higher current. Inset illustrates mode spatial distribution of STO1 and STO2 along width of Pt/YIG microstructure (W). d&e. Zoomed-in, high-resolution views of c, where spin-wave modes are observed to approach each other. Figure 4 | Detection of spin-torque auto-oscillation by NV spin relaxometry. a, NV spin relaxation rate (Γ) is measured at the current and magnetic field values indicated by the blue and red stars in Fig. 3c, where the spin-torque oscillators (STOs) are resonant with the m!=0↔−1 transition frequency. At each of these current and magnetic field values, we sweep τ, perform an NV spin relaxometry measurement sequence (Fig. 3a), and extract the exponential decay time constant Γ. ΓSTO1 (red) and ΓSTO2 (blue) are plotted as a function of Idc. The dramatic order-of-magnitude increase of the relaxation rate above Idc~3 mA indicates spin-torque induced auto-oscillation of the STOs. Inset shows 1/Γ vs Idc for both STOs. Linear fits at low current (Idc<4 mA) intersect with T1 = 0 at Ith1=3.5 mA and Ith2=4.4 mA, which we define as the auto-oscillation threshold currents. b, Measured STO linewidth ∆B as a function of Idc for STO1 (blue dots) and STO2 (red dots). The vertical axis on the right gives the linewidth in frequency (MHz), calculated from ∆B using the Kittel relation at Bext ~ 250 G and Idc = 0. Figure 5 | Locking STO frequency to an external microwave (MW) source. a, NV spin relaxometry measurement sequence as in Fig. 3a, with added MW drive. For synchronization measurement, Bext is tuned such that the NV m!=0↔−1 transition coincides with STO resonance. b, Measured NV photoluminescence (PL) as a function of the MW drive frequency fMW, at Idc = 5.2 mA, Bext = 292 G, and MW drive amplitude b1 = 1.5 G. When the MW drive is resonant with the NV transition frequency, a dip in the PL is observed because the driving depletes the ms=0 population. Over a frequency interval Δfs the STO can be locked to the MW drive and thus detuned from the NV transition, thereby decreasing the NV spin relaxation and correspondingly increasing the measured PL. When the MW drive frequency is detuned beyond the locking interval (i.e., synchronization bandwidth), the STO remains resonant with the NV transition frequency, leading to strong NV-spin relaxation and a corresponding reduced PL. (See Supplementary Information 8 for detailed data analysis.) c, 2D map of PL vs fMW and MW drive amplitude b1. The synchronization bandwidth increases linearly with b1. d, Synchronization bandwidth vs b1 at different Idc (4, 5.2, and 5.6 mA).

2018-10-16

Coherent, self-sustained oscillation of magnetization in spin-torque oscillators (STOs) is a promising source for on-chip, nanoscale generation of microwave magnetic fields. Such fields could be used for local excitation of spin-wave resonances, control of spin qubits, and studies of paramagnetic resonance. However, local characterization of fields emitted by an STO has remained an outstanding challenge. Here, we use the spin of a single nitrogen-vacancy (NV) defect in diamond to probe the magnetic fields generated by an STO in a microbar of ferromagnetic insulator yttrium-iron-garnet (YIG). The combined spectral resolution and sensitivity of the NV sensor allows us to resolve multiple spin-wave modes and characterize their damping. When damping is decreased sufficiently via spin injection, the modes auto-oscillate, as indicated by a strongly reduced linewidth, a diverging magnetic power spectral density, and synchronization of the STO frequency to an external microwave source. These results open the way for quantitative, nanoscale mapping of the microwave signals generated by STOs, as well as harnessing STOs as local probes of mesoscopic spin systems.

Spin-torque oscillation in a magnetic insulator probed by a single-spin sensor

1810.07306v1

1 Atomic -Scale Structure and Chemistry of YIG/GGG Interface Mengchao Liu1, Lichuan Jin2, Jingmin Zhang1, Qinghui yang2, Huaiwu Zhang2, Peng Gao1,3,4,a), Dapeng Yu4,5,6 1Electron Microscopy Laboratory, School of Physics, Peking University, Beijing, 100871, China 2State Key Laboratory of Electronic Thin Films and Integrated Devices, University of Electronic Science and Technology of China, Chengdu 610054, China 3International Center for Quantum Materials, School of Physics, Peking University, Beijing, 100871, China 4Collaborative Innovation Centre of Quantum Matter, Beijing 100871, China. 5State Key Laboratory for Mesoscopic Physics, School of Physics, Peking University, Beijing 100871, People’s Republic of China 6Institute for Quantum Science and Engineering and Depar tment of Physics, South University of Science and Technology of China, Shenzhen 518055, People’s Republic of China Authors to whom correspondence should be addressed: a)p -gao@pku.edu.cn; Keywords YIG, interface, STEM, atomic structure, EELS Abstract: Y3Fe5O12 (YIG) is a promising candidate for spin wave devices. In the thin film devices, the interface between YIG and substrate may play important roles in determining the device properties. Here, we use spherical aberration -corrected scanning electron micro scopy and spectroscopy to study the atomic arrangement, chemistry and electronic structure of the YIG/Gd 3Ga5O12 (GGG) interface. We find that the chemical bonding of the interface is FeO -GdGaO and the interface remains sharp in both atomic and electronic s tructures. These results provide necessary information for understanding the properties of interface and also for atomistic calculation. 2 Spin waves (magnons) that have a large group velocity up to a few tens of μm/ns and a frequency in the gigahertz/tera hertz range,[1-4] are promising for the application of information transport and processing,[5-10] as the conventional semiconductor devices are approaching their limitation.[11] One promising candidate material for the spin -wave devices[14-16] is yttrium iron garnet (Y 3Fe5O12, YIG),[12-20] which has the smallest relaxation parameter, high Curie temperature, excellent chemical stability[5, 21 -23] and a very low damping coefficient and thus allows the magnons to propagate over several centimeters in distanc e.[24-28] For the large scale magnonic circuits integration, YIG is usually required to be in the form of thin film with smooth interface and thickness in nanometer scale in order to be compatible with conventional silicon technology.[16, 28 -30] In fact, t he energy consumption can also be effectively reduced in the thin film YIG devices.[13, 31 ] Particularly, the nanometer -thick YIG film is highly desirable for construction of spin wave nonreciprocity logic devices and voltage switched magnetism. However, w hen the thickness of YIG film decreases, the effects of interface between YIG and the substrate are expected to become pronounced or even may completely dominate the properties of the entire devices. Therefore, it’s of great significance to study the atomi c structure, chemistry and electronic structure of interface of YIG thin film. In this paper, we employ aberration -corrected scanning transmission electron microscopy (AC-STEM) and spectroscopy to study the YIG film on the gadolinium gallium garnet (Gd 3Ga5O12, GGG) substrate.[27] The recent advancements of AC -STEM imaging enable us to directly visualize the atomic bonding at the interface. In addition, combining atomically resolved imaging and spectroscopy such as energy -dispersive X -ray spectroscopy (EDS) and electron energy loss spectroscopy (EELS) in the STEM mode allows us simultaneously to determine the elemental 3 distribution and electronic structures of the heterostructure. By combining these state -of-the-art electron microscopy and spectroscopy techni ques, we reveal the interfacial bonding of YIG/GGG is FeO-GdGaO. No significant elemental diffusion is observed at the interface. The EELS measurements show that the electronic structures of interfacial Fe remain the same with that in the interior film. Su ch atomically sharped interface in both chemistry and electronic structures indicates it is possible to fabricate ultrathin YIG film for future nanodevices for which no intrinsic interfacial zone exists at the YIG/GGG interface. Th e detailed structure info rmation also provides necessary information for future atomistic simulation of the interface. A cross -sectional atomically resolved high angle annular dark filed (HAADF) image of YIG is presented in Figure 1 (a) with the atomic model being overlapped . The red arrows mark the interface of YIG and GGG. Since the HAADF image is Z-contrast (atomic number) image, in which the contrast directly reflects the atomic number of the element, the darker side of the image is YIG and the brighter side is GGG. It can be n oticed that O is invisible in the HAADF image . The HAADF image shows perfectly epitaxial growth and the interface is atomically sharp . The overlapped atomic model highlights the atom positions , which will be discussed below . The crystal structure of YIG is cubic with a dimension 12.376 Å in unit cell and houses 80 atoms. In each unit cell, there are twenty Fe3+ ions occupying two different sites. Among of them, 8 Fe3+ ions occupy octahedral sites and 12 Fe3+ ions with opposite magnetic moment occupy tetrahe dral sites.[5] YIG and GGG have the same garnet structure. The mismatch between YIG and GGG is smaller than 0.05%.[32] This makes the high quality and defect -free unstressed film fabrication possible. In addition, for the best matching, we also dope YIG by lanthanum lightly.[7] Therefore, no dislocations are observed at the interfaces for all these YIG thin films. The atomically resolved EDS maps of YIG are shown in Figure 1(b) -(e), which are element Fe, 4 Y , O and Fe along with Y respectively. The atomic mo del on the EDS map in Fig ure 1(e) further highlights the locations of Fe and Y atom columns. Figure 1(f)-(i) show the distribution of elements Fe, Y , Ga, Gd of the YIG/GGG interface , the yellow arrows mark the interface and the scale bar in these figures i s 1 nm . These EDS maps are acquired at the same area as shown in Fig ure 1(a). The yellow arr ows mark the interface position based on the Z -contrast image . There are Fe atoms diffuse across the interface into GGG, while the Y , Gd and Ga remain sharp edges a t the interface from the EDS mappings . For the YIG grown on GGG (111) substrate, there are two possible interfacial bonding between them, as shown in Figure 2 . Along the [111] direction, there are two types of atom planes of garnet structure, which we cal l A and B atom plane respectively ( see the detail s in the supporting information ). B atom plan e in YIG (GGG) consists of Fe, Y (Ga, Gd) and O atoms while A atom plane in YIG (GGG) consists of Fe (Ga) and O atoms only. The atom planes arrange in ABAB… order inside the crystal. Therefore, t he interfacial bonding should be either FeO -GdGaO or YFeO -GaO. Based on the atomically resolved EDS maps, the bonding at the interface of YIG/GGG is identified to be A/B type, i.e., FeO-GdGaO bonding. The schematic illustra tion of interfacial bonding is overlaid with HAADF image in Figure 1 (a). The detailed structure information of the interface viewing from another two zone axis directions is included in the supporting information. The counts of elemental distribution from the EDS maps are averaged along the interface and depicted in Figure 2(e), which shows the width of the interfacial region is ~1.4 nm. The counts of Fe near the int erface is higher than th ose of Y compared to that in the interior film, due to the interfaci al bonding of FeO-GGG and slight Fe diffusion . We measured 18 EDS maps from different locations in different TEM specimens , and the frequency distribution histogram shown in Figure 2(f) indicates the width of the transition area is equal to the width of 1. 9 unit (2.3 nm). However, it 5 should be noted that the practical interfacial region should be even thinner due to the presence of delocalization effects from the EDS measurement. Therefore, we conclude that no significant interdiffusion takes place at the i nterface. To reveal the local electronic structure of the YIG/GGG interface, core -loss electron energy loss spectroscopy (EELS) experiments are carried out on the Titan Cubed Themis G2 300 aberration -corrected transmission electron microscope with the Gat an EnfiniumTMER (Model 977) spectrometer. Figure 3(a) is a STEM image of the YIG/GGG interface along ] direction. The big green rectangle highlights the location s where the EEL spectra were recorded with a spatial step of 4.5 Å . The O -K edge a nd Fe -L2,3 edge of the spectra are shown in Figure 3(b). As marked by the dashed line, the peak of Fe -L2,3 edge do es not show any detectable shift when the probe move s across the interface. Furthermore, the intensity ratio of L 3 to L 2 is sensitive to the e lectronic structures of Fe , too. The ratio is calculated in Figure 3(c) marked by star s which show no distinguishable change either. Since the energy of Fe-L2,3 edge is sensitive to the Fe valence, no peak shift or ratio change indicates the interfacial Fe remains the same nature with that in the film.[32-35] The integration of L 3 and L 2 (marked by rhombus) is shown in Fig ure 3(c), from which we can obtain the width of the transition area is 1.8 nm, which is consistent with the EDS measurements . The na ture of the interface usually plays important role s in the properties for thin film devices. Particularly for those devices with nanometer scale, the interface properties could be dominated. By combining atomically resolved imag e and EDS results , we reveal that at the interface the FeO atom plane of YIG bonds with GdGaO atom plane of GGG. Slight Fe diffusion in the GGG is also observed. The EELS measurements show that the electronic structures of Fe remain unchanged at the interface . The atomically sharped interface in structure and electronic structures may indicate 6 there are no intrinsic interfacial effects for YIG thin film devices . The finding of a tomic arrangement of interface structure provides necessary information for the future atomistic simulation such as density functional theory calculations. Supp orting Information Supporting Information is available from the Wiley Online Library or from the author. Acknowledgements The authors greatly acknowledge the helpful discussion from Prof. Xiaoyan Z hong and Prof. Jing Zhu from Tsinghua University, and Prof. Jia Li from Peking University. This work was supported by the National Key R&D Program of China (2016YFA0300804), National Natural Science Foundation of China (51672007, 51502007), the National Pr ogram for Thousand Young Talents of China and “2011 Program” Peking -Tsinghua -IOP Collaborative Innovation Center of Quantum Matter. 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Saitoh, Nature 2010 , 464, 262. [24] C. W. Sandweg, Y . Kajiwara, A. V . Chumak, A. A. Serga, V . I. Vasyuchka, M. B. Jungfleisch, E. Saitoh, B. Hillebrands, Phys. Rev. Lett. 2011 , 106, 216601. [25] B. Heinrich, C. Burrowes, E. Montoya, B. Kardasz, E. Girt, Y . -Y . Song, Y . Sun, M. Wu, Phys. Rev. Lett. 2011 , 107, 066604. [26] H. L. Wang, C. H. Du, Y . Pu, R. Adur, P. C. Hammel, F. Y . Yang, Phys. Rev. B 2013 , 88, 100406. 9 [27] E. A. Giess, J. D. Kuptsis, E. A. D. White, J. Cryst. Growth 1972 , 16, 36. [28] C. Dubs, O. Surzhenko, R. Linke , A. Danilewsky, U. Brueckner, J. Dellith, J. Phys. D. Appl. Phys. 2017 , 50, 204005. [29] Y . Sun, Y . -Y . Song, H. Chang, M. Kabatek, M. Jantz, W. Schneider, M. Wu, H. Schultheiss, A. Hoffmann, Appl. Phys. Lett. 2012 , 101, 152405. [30] M. Balinskiy, S. Ojh a, H. Chiang, M. Ranjbar, C. A. Ross, A. Khitun, J. Appl. Phys. 2017 , 122, 123904. [31] O. d'Allivy Kelly, A. Anane, R. Bernard, J. Ben Youssef, C. Hahn, A. H. Molpeceres, C. Carretero, E. Jacquet, C. Deranlot, P. Bortolotti, R. Lebourgeois, J. C. Mage, G . de Loubens, O. Klein, V . Cros, A. Fert, Appl. Phys. Lett. 2013 , 103, 082408. [32] D. Song, L. Ma, S. Zhou, J. Zhu , Appl. Phys. Lett. 2015 , 107, 042401 . [33] P. A. van Aken, B. Liebscher, Phys. Chem. Miner. 2002 , 29, 188. [34] R. F. Egerton, Electron e nergy -loss spectroscopy in the electron microscope , Vol. 30, Springer , US, 2011 , p. 94. [35] L. A. J. Garvie, P. R. Buseck, Nature 1998 , 396, 667. 10 Figures and caption Figure 1. Atomic structure of YIG/GGG interface. (a) Atomically resolved STEM image of a YIG/GGG interface along the [10 ] direction. The red arrows mark the interface. The left side is YIG which appears dark contrast in the HAADF image. (b) -(e) Atomically resolved EDS maps of (b) element Fe, (c) element Y , (d) element O, and (e) overlap of element Fe and Y in YIG. The atomic arrangement model is overlapped on Figure 1(e). (f) -(i) Atomically resolved EDS maps of interface. (f) element Fe, (g) element Y , (h) element Ga and (i) element Gd. The yellow arrows mark the interface. 11 Figure 2. The atomic arrangement of the interface of YIG and GGG. (a) Two alternative atom planes of YIG along [111] direction. The oxygen is invisible for clarity. (b) A/B type bonding model at the interface between YIG and GGG. This mode is in good agreem ent with experimental data. (c) Two alternative atom planes of GGG along [111] direction. The oxygen is invisible for clarity. (d) B/A type bonding model at the interface between YIG and GGG. (e) The EDS results of the interface. D marks the width o f the transition area of the YIG/GGG interface. (f) The frequency distribution of D of altogether 18 EDS results. 12 Figure 3. EELS measurements of the YIG/GGG interface. (a) A STEM image of the YIG/GGG interface along [10 ] direction. The ar ea selected to get EELS spectr a is marked by the big green rectangle (consists of many small rectangles with different colors). (b) The averaged elemental line profile across the YIG/GGG interface. The spectr a presented by colored lines correspond to those rectangles with the same color in (a). (c) The Fe L 2,3 white line ratio and sum across the YIG/GGG interface. The sum indicates the location of interface; while the ratio remain unchanged at the interface indicates no distinguished chemical shift in Fe at th e interface. 13 Supporting information The red arrows mark the same atom plane of two different zone axis directions which are [ 21] in Figure S1 (a) and [3 ] in Figure S1(b), though the atoms surrounding them seems to be quite differ ent from each other. It illustrates that the atom planes which consists Fe atoms only are all equivalent. Also , the atom planes which consist of Fe and Y atom are of the same kind. Two cross sectional atomically resolved STEM images of YIG/GGG interfac e with different viewing directions are shown in Figure S2 (a) and Figure S3 (a), and the corresponding FFT patterns are presented in Figure S2(b) and Figure S3(b) respectively. The simulations of the electron diffraction in Figure S2(c) and Figure S3(c) con firm the directions of the zone axis are [ 21] for Figure S2(a) and [11 ] for Figure S3(a), respectively. The atomistic models of these two zone axes are overlapped on the STEM images. As marked by A and B plane in Figure S2(a) and Figure S3(a), it is clear that there are two kinds of atom planes which are parallel to the (111) interface. B atom plane of YIG (GGG ) consists of Fe, Y (Gd, Ga) and O atoms , while A atom plane of YIG (GGG) consist s of Fe (Ga) and O atoms only . The atom pl anes arrange in ABAB… order inside the crystal. Figure S2(d) -(g) and Figure S3(d) -(g) show the EDS results and the yellow arrows in these figures mark the interface. It can be noted that only Fe atoms diffuse across the interface for both [ 21] and [11 ] directions. This also confirms the point of view in the main text that the YIG/GGG interface belongs to A/B type as shown in Figure 2(c). The atomic structure of interface data from different zone axis directions means observing the sample from different viewing directions which are all parallel to the (111) interface. They present different information of the sample and support each other. 14 Figure S1. (a) The atomic model of YIG along [ 21] zone axis direction. The oxygen is invisible for clarity. (b) The atomic arrangement model of YIG along [3 ] zone axis direction. The oxygen is invisible for clarity. The red arrows mark the same atom plane of the two zone axis direction s. 15 Figure S2. (a) Atomically resolved STEM image with viewing direction of [ 21] direction with atomic model and labeled atom planes on it. The red arrows mark the interface. The yellow arrows mark the two kinds of atom plane of YIG. (b) The Fourier transfor mation pattern. (c) The simulation of electron diffraction. (d)-(g) EDS maps of (d)element Fe, (e) element Y, (f) element Ga and (g) element Gd. T he scale bar is 1 nm. The yellow arrows mark the interface. 16 Figure S3. (a) Atomically resolved STEM image with viewing direction of [ ] direction with atomic model and labeled atom planes on it. The red arrows mark the interface. The yellow arrows mark the two kinds of atom plane of YIG. (b) The Fourier transformation pattern. (c) The simu lation of electron diffraction. (d)-(g) EDS maps of (d)element Fe, (e) element Y , (f) element Ga and (f) element Gd. T he scale bar is 1 nm. The yellow arrows mark the interface.

2018-03-10

Y3Fe5O12 (YIG) is a promising candidate for spin wave devices. In the thin film devices, the interface between YIG and substrate may play important roles in determining the device properties. Here, we use spherical aberration-corrected scanning electron microscopy and spectroscopy to study the atomic arrangement, chemistry and electronic structure of the YIG/Gd3Ga5O12 (GGG) interface. We find that the chemical bonding of the interface is FeO-GdGaO and the interface remains sharp in both atomic and electronic structures. These results provide necessary information for understanding the properties of interface and also for atomistic calculation.

Atomic-scale structure and chemistry of YIG/GGG Interface

1803.03799v1

arXiv:1305.3117v2 [cond-mat.mes-hall] 1 Jul 2013Exchange magnetic field torques in YIG/Pt bilayers observed by the spin-Hall magnetoresistance N. Vlietstra,1J. Shan,1V. Castel,1J. Ben Youssef,2G. E. W. Bauer,3,4and B. J. van Wees1 1)Physics of Nanodevices, Zernike Institute for Advanced Mate rials, University of Groningen, Groningen, The Netherlands 2)Laboratoire de Magn´ etisme de Bretagne, CNRS, Universit´ e de Bretagne Occidentale, Brest, France 3)Kavli Institute of NanoScience, Delft University of Techno logy, Delft, The Netherlands 4)Institute for Materials Research and WPI-AIMR, Tohoku Univ ersity, Sendai, Japan (Dated: 23 July 2018) The effective field torque of an yttrium-iron-garnet film on the spin a ccumulation in an attached Pt film is measured by the spin-Hall magnetoresistance (SMR). As a result , the magnetization direction of a ferro- magnetic insulating layer can be measured electrically. Experimental transverse and longitudinal resistances are well described by the theoretical model of SMR in terms of the d irect and inverse spin-Hall effect, for different Pt thicknesses [3, 4, 8 and 35nm]. Adopting a spin-Hall angle of PtθSH= 0.08, we obtain the spin diffusion length of Pt ( λ= 1.1±0.3nm) as well as the real ( Gr= (7±3)×1014Ω−1m−2) and imaginary part (Gi= (5±3)×1013Ω−1m−2) of the spin-mixing conductance and their ratio ( Gr/Gi= 16±4). Keywords: spin-Hall magnetoresistance, yttrium iron garnet, YI G, spin-mixing conductance, effective field torque In spintronics, interfaces between magnets and nor- mal metals are important for the creation and detec- tion of spin currents, which is governed by the difference of the electric conductance for spin up and spin down electrons.1–3Another important interaction between the electron spins in the magnetic layer and those in the normal metal, that are polarized perpendicular to the magnetization direction, is governed by the spin-mixing conductance G↑↓,4which is composed of a real part and an imaginary part ( G↑↓=Gr+iGi).Gris associ- ated with the “in-plane” or “Slonczewski” torque along /vector m×/vector µ×/vector m,5–7where/vector mis the direction of the magnetiza- tion of the ferromagnetic layer and /vector µis the polarization ofthe spin accumulationat the interface. Gidescribes an exchangemagneticfieldthat causesprecessionofthespin accumulation around /vector m. This “effective-field” torque as- sociated with Gipoints towards /vector µ×/vector m. While several experiments succeeded in measuring Gr,3,4,7–10Giis difficult to determine experimentally, mainly because it is usually an order of magnitude smallerthan Gr.4The recentlydiscoveredspin-Hallmag- netoresistance (SMR)11–14offers the unique possibility to measure Gifor an interface of a normal metal and a magnetic insulator by exposing it to out-of-plane mag- netic fields. Althammer et al.15carried out a quan- titative study of the SMR of Yttrium Iron Garnet (YIG)/Platinum (Pt) bilayers. They obtained an esti- mate of Gi= 1.1×1013Ω−1m−2by extrapolating the high field Hall resistances to zero magnetic field.16 In this paper, we report experiments in which the con- tribution of GrandGican be controlled the magnetiza- tion direction of the YIG layer by an external magnetic field. Thereby either GrorGican be made to dominate the SMR. By fitting the experimental data by the the- oretical model for the SMR,11the magnitude of Gr,Giand the spin diffusion length λin Pt are determined. For SMR measurements, Pt Hall bars with thicknesses of 3, 4, 8 and 35nm were deposited on YIG by dc sputtering.12Simultaneously, a referencesample wasfab- ricated on a Si/SiO 2substrate. The length and width of the Hall bars are 800 µm and 100 µm, respectively. The YIG has a thickness of 200nm and is grown by liquid phase epitaxy on a single crystal Gd 3Ga4O12(GGG) substrate.17The magnetization of the YIG has an easy- plane anisotropy, with an in-plane coercive field of only 0.06mT. To saturate the magnetization of the YIG in the out-of-planedirectionafieldabovethesaturationfield Bs (µ0Ms= 0.176T)17has to be applied. All measurements are carried out at room temperature. The magnetization of the YIG is controlled by sweep- ing the out-of-plane applied magnetic field with a small intended in-plane component (see insets of Fig. 1(a,b)). Fig. 1(a) shows out-of-plane magnetic field sweeps for various directions of the in-plane component of B(and thusM), while measuring the transverse resistance (us- ing a current I= 1mA). Above the saturation field (B > B s), alinearmagneticfield dependence isobserved, that can be partly ascribed to the ordinary Hall effect, but its slope is slightly larger, which suggests the pres- ence of another effect (discussed below). Furthermore, extrapolation of the linear regime for the positive and negative saturated fields to B= 0mT, reveals an off- set between both regimes, that, as shown below, can be ascribed to Gi. When Bis smaller than the satu- ration field, the observed signal strongly depends on the angleαbetween the direction of the charge current Je and the in-plane component of the magnetic field. This α-dependence is not observed for B > B s. The maxi- mum/minimum magnitude of the peak/dip observed in the non-saturated regime exactly follows the SMR be-2 (b)(a) B V Jez - yx α z - yxV B Jeα-800 -600 -400 -200 0 200 400 600 800-80-60-40-20020406080 -800 -600 -400 -200 0 200 400 600 800-1.2-0.8-0.40.0-800 -400 0 400 800-20-1001020 Pt [3nm] RT [mΩ ] B [mT]α = 00 450 900 1350 RL-R0 [Ω] Pt [3nm] B [mT]α = 00 450 900 RT [mΩ] B [mT]α = 900 FIG. 1. (a) Transverse and (b) longitudinal resistance of Pt [3nm] on YIG under an applied out-of-plane magnetic field. αis the angle between Jeand the small in-plane component of the applied magnetic field. The insets show the configu- ration of the measurements, as well as a separate plot of the transverse resistance for α= 90◦, where the contribution of Giis most prominent. R0is the high-field resistance of the Pt film, here 1695Ω. haviour for in-plane magnetic fields.12,13By increasing the magnetic field strength, the magnetization is tilted out of the plane and less charge current is generated by theinversespin-Halleffectinthetransverse(andalsolon- gitudinal) direction, resulting in a decrease of the SMR signal. The sharp peak observed around zero applied field can be explained by the reorientation of Min the film plane when Bis swept through the coercive field of the YIG. The corresponding measurements of the longitudinal resistance are shown in Fig. 1(b) (For currents I= 1−100µA). In this configuration, the signal for B > B s does not show a field dependence nor an offset between positive and negative field regimes when linearly extrap- olated to zero field. The observed features for the transverse (Fig. 1(a)) as well as the longitudinal (Fig. 1(b)) resistance can be described by the following equations11 ρT= ∆ρ1mxmy+∆ρ2mz+(∆ρHall+∆ρadd)Bz(1) ρL=ρ+∆ρ0+∆ρ1(1−m2 y) (2) whereρTandρLare the transverse and longitudinal re- sistivity, respectively. ρis the electrical resistivity of thePt. ∆ρHallBzdescribes the change in resistivity caused by the ordinaryHall effect and ∆ ρaddBzis the additional resistivity change on top of ∆ ρHallBz, as observed for saturated magnetic fields.18Bzis the magnetic field in thez-direction. mx,myandmzare the components of the magnetization in the x-,y- andz-direction, re- spectively, defined by mx= cosαcosβ,my= sinαcosβ andmz= sinβ, where αis the in-plane angle be- tween the applied field BandJe, andβis the angle by whichMis tilted out of the plane. For an applied field in thez-direction, from the Stoner-Wohlfarth Model,19 β= arcsin B/Bs. ∆ρ0, ∆ρ1and ∆ρ2are resistivity changes as defined below11 ∆ρ0 ρ=−θ2 SH2λ dNtanhdN 2λ(3) ∆ρ1 ρ=θ2 SHλ dNRe/parenleftBigg 2λG↑↓tanh2dN 2λ σ+2λG↑↓cothdN λ/parenrightBigg (4) ∆ρ2 ρ=−θ2 SHλ dNIm/parenleftBigg 2λG↑↓tanh2dN 2λ σ+2λG↑↓cothdN λ/parenrightBigg (5) whereθSH,λ,dN,G↑↓andσare the spin-Hall angle, the spin relaxation length, the Pt thickness, the spin-mixing conductance ( G↑↓=Gr+iGi) and the bulk conductivity, respectively. From Eq. (1), Giis most dominant in the transverse configurationwhen the product mxmyvanishes (∆ ρ2is a function of Gi). This is the case for α= 0◦andα= 90◦, as is shown in Fig. 1(a). As mzscales linearly with B, the term ∆ ρ2mz, contributes an additional linear depen- dence for B < B sthat causes an offset between resis- -800 -600 -400 -200 0 200 400 600 800-20-1001020304050607080 RT [mΩ] B [mT]Pt thickness [nm] 3 4 8 35 α= 4500 0.1 0.2 0.3 0.4-16-12-8-40 SiO2/ Pt YIG / Pt∆RT/∆B [ µΩ/mT] 1/d [nm-1] FIG. 2. Out-of-plane magnetic field sweeps on YIG/Pt for different Pt thicknesses [3, 4, 8 and 35nm], fixing α= 45◦. In the saturated regime ( B > B s), linear behaviour is ob- served. The inset shows the measured slope ∆ RT/∆Bin the saturated regimes (red dots). The expected (black line) and measured (black dots) curves display the slopes for the ordi - nary Hall effect on a SiO 2/Pt sample. The red dotted line is a guide for the eye.3 tances for positive and negative saturation fields. This behaviour is clearly observed in the inset of Fig. 1(a), where the measurement for α= 90◦is separately shown. Forα= 45◦(135◦), the product mxmyis maximized (minimized) and a maximum (minimum) change in re- sistance is observed. These measurements were repeated for a set of sam- ples with different Pt thicknesses [3, 4, 8 and 35nm]. Results of the thickness dependent transverse resistance are shown in Fig. 2. For α= 45◦, at which both GrandGicontribute to a maximum SMR signal, a clear thickness dependence is observed at all field val- ues. The thickness dependence of the slope ∆ RT/∆B at saturation fields is shown in the inset of Fig. 2, where the red dots represent the experiments. The black line (dots) shows the expected (observed) slope from the ordinary Hall effect (measured on a SiO 2/Pt sam- ple) given by the equation (∆ RT/∆B)Hall=RH/dN, whereRH=−0.23×10−10m3/C is the Hall coefficient of Pt.20∆RT/∆Bfor YIG/Pt behaves distinctively dif- ferent. When decreasing the Pt thickness, ∆ RT/∆Bof YIG/Pt increases faster than expected from the ordinary Hall effect. This discrepancy cannot be explained by the present theory for the SMR and may thus indicate a dif- ferent proximity effect. The red dotted line in the inset of Fig. 2 is a guide for the eye and represents the term ∆ρHall+∆ρaddin Eq. (1). The SMR, including the resistance offset obtained by linear extrapolation of the high field regimes, is only sig- nificant for the thin Pt layers [3, 4 and 8nm]. The thick Pt layer [35nm] shows no (or very small) SMR. Using Eqs. (1) and (2), all experimental data can be fitted simultaneously by the adjustable parameters θSH, λ,GrandGi.ρ= 1/σfollows from the measured resistances R0for each Pt thickness given in the cap- tion of Fig. 3. The quality of the fit is demonstrated by Fig. 3(a)-(f) for θSH= 0.08,λ= 1.2nm,Gr= 4.4×1014Ω−1m−2andGi= 2.8×1013Ω−1m−2. The measurementsareverywelldescribed bythe SMR theory (Eqs. (1) and (2)), for all Pt-thicknesses and magnetic field strength and direction. However, due to the cor- relation between the fitting parameters, similarly good fitting results can be obtained by other combinations of θSH,λ,GrandGi, notwithstanding the good signal-to- noise-ratio of the experimental data. We therefore fixed the Hall angle at θSH= 0.08, which is within the range 0.06 to 0.11 obtained from the fitting and consistent with results published by several groups.12,21–24By fixing θSH the quality of the fits is not reduced, but the accuracy of the parameter estimations improves significantly. By Fig. 4 it is observed that a strong correlation exists be- tween both GrandGi, andλ, whereas the ratio Gr/Gi does not significantly change (see inset Fig. 4). A good fit cannot be obtained for λ >1.4nm. For λ <0.8nm the error bars become very large and for λ <0.4nm a good fit can no longer be obtained. Inspecting Fig. 4 we favour λ= 1.1±0.3nm,Gr= (7±3)×1014Ω−1m−2 andGi= (5±3)×1013Ω−1m−2, where the higher val-(a) (d) (f)(e) (c)(b) -100102030 RT [mΩ ] -800-600-400-200 0 200 400 600 800-4-2024 RT [mΩ ] B [mT]-20020406080 Pt [8nm]Pt [3nm] Pt [4nm] Pt [4nm]Pt [3nm] RT [mΩ ]α = 450 α = 900 Pt [8nm]-1200-900-600-3000 RL-R0 [mΩ ] α = 00 α = 450 α = 900 -800-600-400-200 0 200 400 600 800-60-40-200 RL-R0 [mΩ ] B [mT]-500-400-300-200-1000 RL-R0 [mΩ ] FIG. 3. Theory Eqs. (1,2) (solid lines) fitted to (a)-(c) tran s- verse and (d)-(f)longitudinal observed resistances (open sym- bols) for different αand Pt thicknesses 3, 4 and 8nm, respec- tively, using θSH= 0.08,λ= 1.2nm,Gr= 4.4×1014Ω−1m−2 andGi= 2.8×1013Ω−1m−2.R0is the high-field longitudinal resistance of the Pt film of 1695Ω, 930Ω and 290Ω for the 3, 4 and 8nm Pt thickness, respectively. 0.8 0.9 1.0 1.1 1.2 1.3 1.4024681012 0.8 0.9 1.0 1.1 1.2 1.3 1.4812162024 Gr (x1014 ) Gi (x1013 )Gr,i [ Ω-1m-2] λ [nm]θSH = 0.08 G r / G i λ [nm] FIG. 4. Obtained magnitude and uncertainties of GrandGi (Gr/Giin the inset) as a function of λ, forθSH= 0.08. ues ofGrandGicorrespond to smaller λ. The ratio Gr/Gi= 16±4 does not depend on λ. Insummary,byemployingtheSMR,includingthecon- tribution of the imaginary part of the spin-mixing con- ductance, it is possible to fully determine the magne- tization direction of an insulating ferromagnetic layer, by purely electrical measurements. The experimental data are described well by the spin-diffusion model of the SMR, for all investigated Pt thicknesses and mag- netic configurations. By fixing θSH= 0.08, we find the parameters λ= 1.1±0.3nm,Gr= (7±3)×1014Ω−1m−2, Gi= (5±3)×1013Ω−1m−2andGr/Gi= 16±4 for YIG/Pt bilayer structures.4 We would like to acknowledge B. Wolfs, M. de Roosz and J. G. Holstein for technical assistance. This work is part of the research program of the Foundation for Fundamental Research on Matter (FOM), EU-ICT-7 ”MACALO” and DFG Priority Programme 1538 ”Spin- Caloric Transport” (BA 2954/1-1) and is supported by NanoNextNL, a micro and nanotechnology consortium of the Government of the Netherlands and 130 partners, by NanoLab NL and the Zernike Institute for Advanced Materials. 1C. Burrowes, B. Heinrich, B. Kardasz, E. A. Montoya, E. Girt, Y. Sun, Y.-Y. Song, and M. Wu, Applied Physics Letters 100, 092403 (2012). 2Y. Tserkovnyak, A. Brataas, and G. E. W. Bauer, Phys. Rev. B 66, 224403 (2002). 3Y. Kajiwara, S. Takahashi, S. Maekawa, and E. Saitoh, Magnet - ics, IEEE Transactions on 47, 1591 (2011). 4K. Xia, P. J. Kelly, G. E. W. Bauer, A. Brataas, and I. Turek, Phys. Rev. B 65, 220401 (2002). 5D. Ralph and M. Stiles, Journal of Magnetism and Magnetic Materials 320, 1190 (2008). 6Z. Wang, Y. Sun, Y.-Y. Song, M. Wu, H. Schultheiß, J. E. Pearson, and A. Hoffmann, Applied Physics Letters 99, 162511 (2011). 7Y. Kajiwara, K. Harii, S. Takahashi, J. Ohe, K. Uchida, M. Mizuguchi, H. Umezawa, H. Kawai, K. Ando, K. Takanashi, S. Maekawa, and E. Saitoh, Nature (London) 464, 262 (2010). 8X. Jia, K. Liu, K. Xia, and G. E. W. Bauer, EPL (Europhysics Letters) 96, 17005 (2011). 9F. D. Czeschka, L. Dreher, M. S. Brandt, M. Weiler, M. Altham- mer, I.-M. Imort, G. Reiss, A. Thomas, W. Schoch, W. Limmer, H. Huebl, R. Gross, and S. T. B. Goennenwein, Phys. Rev. Lett. 107, 046601 (2011). 10V. Castel, N. Vlietstra, J. Ben Youssef, and B. J. van Wees, Applied Physics Letters 101, 132414 (2012). 11Y.-T.Chen, S. Takahashi, H.Nakayama, M.Althammer, S. T. B. Goennenwein, E. Saitoh, and G. E. W. Bauer, Phys. Rev. B 87, 144411 (2013).12N. Vlietstra, J. Shan, V. Castel, B. J. van Wees, and J. Ben Youssef, Phys. Rev. B 87, 184421 (2013). 13H. Nakayama, M. Althammer, Y.-T. Chen, K. Uchida, Y. Kaji- wara, D. Kikuchi, T. Ohtani, S. Gepr¨ ags, M. Opel, S. Takahas hi, R. Gross, G. E. W. Bauer, S. T. B. Goennenwein, and E. Saitoh, Phys. Rev. Lett. 110, 206601 (2013). 14C. Hahn, G. de Loubens, O. Klein, M. Viret, V. V. Naletov, and J. Ben Youssef, Phys. Rev. B 87, 174417 (2013). 15M. Althammer, S. Meyer, H. Nakayama, M. Schreier, S. Alt- mannshofer, M. Weiler, H. Huebl, S. Gepr¨ ags, M. Opel, R. Gro ss, D. Meier, C. Klewe, T. Kuschel, J.-M. Schmalhorst, G. Reiss, L. Shen, A. Gupta, Y.-T. Chen, G. E. W. Bauer, E. Saitoh, and S. T. B. Goennenwein, Phys. Rev. B 87, 224401 (2013). 16The authors of Ref.15obtained Giby adding the saturation mag- netization to the applied magnetic field to obtain the total m ag- netic field in the Pt. In our opinion the saturation magnetiza tion should not be included, which leads to a different zero-field e x- trapolation resulting in Gi= 1.7×1013Ω−1m−2, which is more close to the uncertainty interval of our results. 17V. Castel, N. Vlietstra, B. J. van Wees, and J. Ben Youssef, Phys. Rev. B 86, 134419 (2012). 18From the measurements for α= 90◦, shown in the inset of Fig. 1(a) and in Figs. 3(a)-(c), we deduce that also in the non - saturated regime this additional effect likely scales linea rly with B. The dominant linear effect observed in the non-saturated regime is attributed to Gi. The remaining linear signal is ex- plained by the sum of the ordinary hall effect and the addition al term as defined in Eq. (1). 19E. Stoner and E. Wohlfarth, IEEE Transactions on Magnetics 27, 3475 (1991). 20C.M.Hurd, The Hall Effect in Metals and Alloys (Plenum Press, New York, 1972). 21L. Liu, R. A. Buhrman, and D. C. Ralph, arXiv:1111.3702v3 [cond-mat.mes-hall]. 22L. Liu, T. Moriyama, D. C. Ralph, and R. A. Buhrman, Phys. Rev. Lett. 106, 036601 (2011). 23K. Ando, S. Takahashi, K. Harii, K. Sasage, J. Ieda, S. Maekaw a, and E. Saitoh, Phys. Rev. Lett. 101, 036601 (2008). 24A. Azevedo, L. H. Vilela-Le˜ ao, R. L. Rodr´ ıguez-Su´ arez, A . F. Lacerda Santos, and S. M. Rezende, Phys. Rev. B 83, 144402 (2011).

2013-05-14

The effective field torque of an yttrium-iron-garnet film on the spin accumulation in an attached Pt film is measured by the spin-Hall magnetoresistance (SMR). As a result, the magnetization direction of a ferromagnetic insulating layer can be measured electrically. Experimental transverse and longitudinal resistances are well described by the theoretical model of SMR in terms of the direct and inverse spin-Hall effect, for different Pt thicknesses [3, 4, 8 and 35nm]. Adopting a spin-Hall angle of Pt $\theta_{SH}=0.08$, we obtain the spin diffusion length of Pt ($\lambda=1.1\pm0.3$nm) as well as the real ($G_r=(7\pm3)\times10^{14}\Omega^{-1}$m$^{-2}$) and imaginary part ($G_i=(5\pm3)\times10^{13}\Omega^{-1}$m$^{-2}$) of the spin-mixing conductance and their ratio ($G_r/G_i=16\pm4$).

Exchange magnetic field torques in YIG/Pt bilayers observed by the spin-Hall magnetoresistance

1305.3117v2

Detection of DC currents and resistance measurements in longitudinal spin Seebeck eect experiments on Pt/YIG and Pt/NFO Daniel Meier,1,a)Timo Kuschel,1Sibylle Meyer,2Sebastian T.B. Goennenwein,2Liming Shen,3Arunava Gupta,3 Jan-Michael Schmalhorst,1and G unter Reiss1 1)Center for Spinelectronic Materials and Devices, Department of Physics, Bielefeld University, Universit atsstrae 25, 33615 Bielefeld, Germany 2)Walther-Meissner-Institut, Bayerische Akademie der Wissenschaften, Walther-Meissner-Strasse 8, 85748 Garching, Germany 3)Center for Materials for Information Technology, University of Alabama, Tuscaloosa, Alabama 35487, USA (Dated: 10 June 2021) In this work we investigated thin lms of the ferrimagnetic insulators Y 3Fe5O12and NiFe 2O4capped with thin Pt layers in terms of the longitudinal spin Seebeck eect (LSSE). The electric response detected in the Pt layer under an out-of-plane temperature gradient can be interpreted as a pure spin current converted into a charge current via the inverse spin Hall eect. Typically, the transverse voltage is the quantity investigated in LSSE measurements (in the range of V). Here, we present the directly detected DC current (in the range of nA) as an alternative quantity. Furthermore, we investigate the resistance of the Pt layer in the LSSE conguration. We found an in uence of the test current on the resistance. The typical shape of the LSSE curve varies for increasing test currents. In the recent years, the spin Seebeck eect (SSE), the thermal generation of a pure spin current, has been attracted much attention in spintronics1and has opened the branch of spin caloritronics.2The rst observation on thin Ni 81Fe19 (permalloy - Py) lms3in the now called transverse conguration (TSSE) could not be reproduced by many groups.4{7 Additionally, unintended charge transport phenomena like the anomalous Nernst or planar Nernst eect appeared in most attempts to investigate the TSSE which disguised the voltage measured. These unintended Nernst eects are eected by temperature gradients in unintended directions7or small parasitic magnetic elds.8 Magnetic insulators like Y 3Fe5O12(yttrium iron garnet - YIG) or NiFe 2O4(nickel ferrite - NFO) seem to be a more promising material class for TSSE investigations.9The lack of free charge carriers suppress the appearance of any thermally driven charge current phenomena. However, it could be shown very recently that the TSSE on YIG and NFO could also not be reproduced.10Nevertheless, a pure spin current generation could be observed in spite of that. This was accomplished by a spin Seebeck eect in the longitudinal conguration (LSSE).11Here, the spin current is generated longitudinal to the temperature gradient which is applied perpendicular to the spin detector/ferromagnet bilayer system. The LSSE on ferromagnetic or ferrimagnetic insulators is now a well established phenomena and could be reproduced in many groups.12{16The generally used quantity which is presented in all of the given publications is the voltage which arises in the spin detector material transverse to the generated spin current due to the inverse spin Hall eect (ISHE). The ISHE describes the conversion of a spin current into a charge current due to spin dependent scattering of the electrons in a heavy metal.17Generally, the voltage is used as the electrical response quantity that is measured in ISHE experiments. Recently, Omori et al. have investigated the detection of the converted charge current in lateral spin valve structures by means of the ISHE.18The direct detection of the charge current generated in the spin detector material in LSSE experiments will be presented in this work. Another transport phenomena which is connected with Pt/magnetic insulator bilayers is the recently observed spin Hall magnetoresistance (SMR).19,20Here, an interplay of the spin Hall eect and the ISHE leads to a magnetore- sistance eect when an electrical current ows through the Pt lm. Most of the given literature show eld rotation measurements to distinguish between the SMR and the anisotropic magnetoresistance. The latter eect could appear if the Pt is spin polarized at the interface by the ferromagnetic or ferrimagnetic material due to a magnetic proximity eect.12This is still under discussion for the investigated Pt/YIG systems21,22, but could be excluded for Pt/NFO by x-ray resonant magnetic re ectivity23,24and for Pt/CoFe 2O4by x-ray magnetic circular dichroism very recently.25 The SMR can also emerge as symmetric peaks in magnetic eld loop measurements due to magnetic anisotropies. In this work we present measurements on Pt/YIG and Pt/NFO in the LSSE conguration (Fig. 1 (f)). The electric response detected in the Pt layer is shown as a function of the external magnetic eld Hfor various angles with respect to the x-direction. Here, we present the voltage as the typically used quantity for reference and compare this with the directly detected DC current as an alternative quantity. Furthermore, we show how the resistance a)Electronic mail: dmeier@physik.uni-bielefeld.de; www.spinelectronics.dearXiv:1601.00304v1 [cond-mat.mtrl-sci] 3 Jan 20162 varies during the LSSE measurement. We could observe dierent behaviour when the test current is increased. Large test currents lead to a dominant SMR which suppresses the appearance of the LSSE. Therefore, we will show that the LSSE and the SMR can be created simultaneously with dierent sources. While the LSSE is generated by the temperature gradient, the SMR is produced by a charge current through the Pt. Furthermore, we used the same contacts for the applied current and the voltage measurement. Slightly dierent experiments are reported by Schreier et al.26and Vlietstra et al.27about simultaneous LSSE/SMR measurements with the same source. ThetY IG= 60 nm thick YIG lm investigated in this work was deposited on 0.5 mm thick yttrium aluminium garnet (Y3Al5O12) (111)-oriented single crystal substrates with 5 mm 2 mm in dimension by pulsed laser deposition from a stoichiometric polycrystalline target.20The KrF excimer laser had a wavelength of 248 nm, a repetition rate of 10 Hz and an energy density of 2 J/cm2. The YIG lm was capped by a 2 nm thin Pt lm deposited by e-beam evaporation. ThetNFO = 1m thick NFO lm was deposited by direct liquid injection-chemical vapour deposition on 0.5 mm thick MgAl 2O4(100)-oriented substrates with 8 mm 5 mm in dimension.28The NFO lm was cleaned with ethanol in an ultrasonic bath after a vacuum break and was capped by a 10 nm thin Pt lm deposited by dc magnetron sputtering. The LSSE measurements were performed in a vacuum chamber with a base pressure of 1
106mbar. The samples were clamped between two copper blocks with a piece of 0.5 mm thick sapphire substrate for electrical isolation between the Pt and the top copper block. The temperature gradient through the sample stack was established by heating the top copper block by Joule heating. Two 25 m thin aluminium bonding wires at the sample edges measured the electrical response transverse to the temperature gradient and perpendicularly to the external magnetic eld which was applied in the sample plane. For Pt/YIG and Pt/NFO the electrical contacts had a distance of about 4 mm and 7 mm, respectively. The measurement conguration and all observed responses are consistent if a rigorous sign check is applied.16 -8-4048 voltage V (µV) -200 -100 0100 200 magnetic field H (Oe)!T = 35K 0° 80° 90° 100° -10-50510 current I (nA) -200 -100 0100 200 magnetic field H (Oe)!T = 35K 0° 80° 90° 100° -60-3003060 current I (nA) -400 -200 0200 400 magnetic field H (Oe) 2.8 K 11.6 K 15.4 K 21.4 K!T-6-3036 voltage V (µV) -400 -200 0200 400 magnetic field H (Oe) 2.8 K 10.6 K 15.5 K 19.4 K!T9 6 3 0voltage Vsat (µV) 25 20 15 105 0 temperature di fference T (K)90 60 30 0 current Isat (nA)Pt/YIG Pt/NFO (a) (b)(c) (e) (d)(f) MAO[YAG] NFO[YIG] PtV ∆ T H a xyz z NSHi(+)Lo(-) FIG. 1. (a) The transverse voltage Vis shown as a function of the external magnetic eld Hfor various angles with respect to the x-direction measured on Pt/YIG. The temperature dierence between the top and bottom is
T= 35K. (b) The transverse current Imeasured at the Pt/YIG bilayer is shown as a function of Hfor various angles . (c) The voltage Vshown as a function of Hfor various temperature dierences
Ton Pt/NFO. (d) The current Imeasured at the Pt/NFO bilayer plotted against H. (e) The saturated voltage Vsatand saturated current Isatshown as a function of
Tfor Pt/YIG. (f) The measurement conguration with the temperature gradient rTz, the magnetic eld vector Hand the connections as well as polarity of the electrical measurement. In the rst LSSE measurements shown in Fig. 1 (a) the voltage Von the Pt/YIG bilayer was obtained as a function of the external magnetic eld Hwith a xed temperature dierence
T= 35Kfor various angles of the magnetic eld vector with respect to the x-direction. The voltage in saturation is about 6 :8Vfor= 0which decreases for angles up to = 90where it reaches zero due to the cross product of the ISHE given by E/JS, with the electric eld E, the spin current JSand the spin-polarization vector of the electrons in the Pt. For angles above 90the voltage in saturation changes its sign. For magnetic eld values Haround the coercive eld of the YIG the voltage also switches its sign. This can be seen for all angles . However, for angles around 90there are two peaks around the coercive eld. These peaks results from a switching behaviour for materials with a magnetic anisotropy3 which was investigated recently.15While Kehlberger et al. could observe an antisymmetric switching with respect to H we observe a symmetric switching. This can be a result of a symmetric reversal process of the magnetization vector which rotates between 0and 180passing the 90direction for both hysteresis branches and never rotating over the 270direction. Additionally, this symmetric curve can be reminiscent to a magnetoresistive switching in a manner like the SMR and will be part of future investigations. In Fig. 1 (b) the current Imeasured at the Pt contacts is plotted as a function of H. The current generated by the ISHE conversion in the Pt shows the same angle dependency of the saturated value compared to the voltage. The current in saturation decreases for angles between 0and 90until it vanishes. This shows the same behaviour expected for the LSSE via the ISHE. The peaks in the voltage for angles near 90(Fig. 1 (a)) are vaguely perceptible in the current due to the dierent measurement accuracy. In the additional system Pt/NFO the current Iand the voltage Vgenerated by the LSSE were studied. A detailed LSSE investigation for this system was previously reported.14In Fig. 1 (c) Vis shown as a function of Hfor various temperature dierences
T. The magnitude of these curves is shown in Fig. 1 (e) which shows the typical proportionality expected for the LSSE. This could be conrmed for the current Idirectly measured at the Pt lm (Fig. 1 (d)) for slightly divergent temperature dierences. However, the linearity between Iand
Tbecomes obvious in Fig. 1 (e) in spite of the poorer accuracy. Therefore, the current Ishows the expected LSSE behaviour for the
Tproportionality as well as the angle dependency regarding the ISHE exemplarily shown for each material system, Pt/YIG and Pt/NFO. This makes it an equivalent quantity for LSSE investigations. antisym. partsym. part714.25714.00713.75713.50R (Ω)100 µA exp. data -200-1000100200magnetic field H (Oe)antisym.sym. partpart715.50715.00714.50714.00713.50R (Ω)exp. data 10 µA antisym. partsym. part713.80713.75713.70R (Ω)-200-1000100200magnetic field H (Oe)1000 µA exp. data(a)(d)(b)(e)(c)(f)1.20.80.40.0∆R (Ω)100806040200inverse current 1/IT (1/mA)(g) (h)-8-4048voltage V (µV)-200-1000100200magnetic field H (Oe) V I•R FIG. 2. The resistance Rmeasured on Pt/YIG transverse to the applied external magnetic eld Hfor
T= 35Kwith dierent test currents ITin (a) 10A, (b) 100A and (c) 1000 A. The experimental data was separated mathematically into the antisymmetric and symmetric part (d), (e) and (f) with respect to H. (g) The directly measured LSSE voltage compared to the product of the measured DC current and the residual resistance of the Pt lm. (h) The margin of the saturated values in the antisymmetric parts
Rplotted against the inverse test current 1 =IT. In further investigations, we measured the resistance of the bilayers for
T= 35 K. The measurement is performed as a voltage measurement with dierent test currents applied. In Fig. 2 (a), (b) and (c) the resistance Ris shown as a function of Hfor three dierent test currents (10 A, 100A and 1000 A). For low test currents, e.g., 10 A the resistance is antisymmetric with respect to Hand shows a hysteretical behaviour. The experimental data can be separated mathematically into a complete antisymmetric and symmetric part which is shown in Figs. 2 (d), (e) and (f). Since the experimental data are nearly completely antisymmetric for IT= 10A the symmetric part shows only a mean resistance without any magnetic eld dependent behaviour (Fig. 2 (d)). For larger test currents, however, the mathematical separation shows a switching behaviour with two peaks around the coercive elds of the YIG lm which is symmetric with H(Fig. 2 (e)). The magnitude of the antisymmetric part, i.e., the dierence of the saturated voltages for positive and negative magnetic elds
R, decreases. When the test current is further increased the symmetric eect is more dominant in the experimental data compared to the antisymmetric contribution. Here, the antisymmetric part shows a very low magnitude (Fig. 2 (f)) and the symmetric part is more dominant even in the experimental data. The LSSE contribution normalized to the used test current is always the same. This can be shown by the proportionality between
Rand the inverse test current 1/ ITin Fig. 2 (h). When the the measured ISHE4 current (Fig. 1 (a)) is multiplied by the residual resistance of the Pt lm the obtained curve is similar to the previously measured ISHE voltage (Fig. 2 (g)). Recently, Schreier et al. have shown that the temperature gradient can also be established by heating the top of the sample with a large current through the Pt layer which is the spin detector at the same time.26The Joule heating of a large d.c. current (about 10 mA) transverse to the voltage measurement generated the LSSE which is antisymmetric with the external magnetic eld H. Furthermore, the current generated a SMR represented by large symmetric peaks. Both could be separated by taking the dierence of two measurements with the reversed current applied at the Pt. For the test currents used in this work there is no additional heating which would be manifest in a deviation of the linearity between
Rand 1=ITin Fig. 2 (h). Very recently, Vlietstra et al. have extended these investigations by using a.c. currents in a similar range of the absolute value compared to Schreier et al. They measured the rst- and second-harmonic voltage in order to separate the SMR and LSSE contribution which are generated by the same current source.27 In conclusion, we have shown that the direct measurement of the d.c. current is an equivalent quantity in LSSE experiments which shows the same properties of the saturated values compared to the commonly used voltage. Furthermore, the resistance was measured in the LSSE conguration by applying dierent test currents. The same switching behaviour expected for the LSSE could be obtained for low enough test currents applied. However, large enough test currents can aect the result and create an additional contribution given by the SMR. The variation of the test current can obscure the real interpretation of magnetoresistive experiments which become extremely worthwhile for the sample systems used in LSSE experiments. The authors thank Michael Schreier for valuable discussions and gratefully acknowledge nancial support by the EMRP JRP EXL04 SpinCal and the Deutsche Forschungsgemeinschaft (DFG) within the priority programme SpinCaT (KU 3271/1-1 and RE 1052/24-2). 1A. Homann and S. D. Bader, Phys. Rev. Applied 4, 047001 (2015). 2G. E. W. Bauer, E. Saitoh, and B. J. van Wees, Nat. Mater. 11, 391 (2012). 3K. Uchida, S. Takahashi, K. Harii, J. Ieda, W. Koshibae, K. Ando, S. Maekawa, and E. Saitoh, Nature 455, 778 (2008). 4S. Huang, W. Wang, S. Lee, J. Kwo, and C. Chien, Phys. Rev. Lett. 107, 216604 (2011). 5A. Avery, M. Pufall, and B. Zink, Phys. Rev. Lett. 109, 196602 (2012). 6M. Schmid, S. Srichandan, D. Meier, T. Kuschel, J. M. Schmalhorst, M. Vogel, G. Reiss, C. Strunk, and C. H. Back, Phys. Rev. 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Huebl, S. Gepr ags, M. Opel, R. Gross, D. Meier, C. Klewe, T. Kuschel, J.-M. Schmalhorst, G. Reiss, L. Shen, A. Gupta, Y.-T. Chen, G. E. W. Bauer, E. Saitoh, and S. T. B. Goennenwein, Phys. Rev. B 87, 224401 (2013). 21Y. Lu, Y. Choi, C. Ortega, X. Cheng, J. Cai, S. Huang, L. Sun, and C. Chien, Phys. Rev. Lett. 110, 147207 (2013). 22S. Gepr ags, S. Meyer, S. Altmannshofer, M. Opel, F. Wilhelm, A. Rogalev, R. Gross, and S. T. B. Goennenwein, Appl. Phys. Lett. 101, 262407 (2012). 23T. Kuschel, C. Klewe, J. M. Schmalhorst, F. Bertram, O. Kuschel, T. Schemme, J. Wollschl ager, S. Francoual, J. Strempfer, A. Gupta, M. Meinert, G. G otz, D. Meier, and G. Reiss, Phys. Rev. Lett. 115, 097401 (2015). 24T. Kuschel, C. Klewe, P. Bougiatioti, O. Kuschel, J. Wollschl ager, L. Bouchenoire, S. Brown, J.-M. Schmalhorst, D. Meier, and G. Reiss, submitted for publication (2015). 25M. Valvidares, N. Dix, M. Isasa, K. Ollefs, F. Wilhelm, A. Rogalev, F. S anchez, E. Pellegrin, A. Bedoya-Pinto, P. 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2016-01-03

In this work we investigated thin films of the ferrimagnetic insulators YIG and NFO capped with thin Pt layers in terms of the longitudinal spin Seebeck effect (LSSE). The electric response detected in the Pt layer under an out-of-plane temperature gradient can be interpreted as a pure spin current converted into a charge current via the inverse spin Hall effect. Typically, the transverse voltage is the quantity investigated in LSSE measurements (in the range of \mu V). Here, we present the directly detected DC current (in the range of nA) as an alternative quantity. Furthermore, we investigate the resistance of the Pt layer in the LSSE configuration. We found an influence of the test current on the resistance. The typical shape of the LSSE curve varies for increasing test currents.

Detection of DC currents and resistance measurements in longitudinal spin Seebeck effect experiments on Pt/YIG and Pt/NFO

1601.00304v1

Modulation of pure spin currents with a ferromagnetic insulator Estitxu Villamor,1Miren Isasa,1Sa¨ul V ´elez,1Amilcar Bedoya-Pinto,1Paolo Vavassori,1, 2Luis E. Hueso,1, 2F. Sebasti ´an Bergeret,3, 4and F `elix Casanova1, 2 1CIC nanoGUNE, 20018 Donostia-San Sebastian, Basque Country, Spain 2IKERBASQUE, Basque Foundation of Science, 48011 Bilbao, Basque Country, Spain 3Centro de F ´ısica de Materiales (CFM-MPC) Centro Mixto CSIC-UPV/EHU, 20018 Donostia-San Sebastian, Basque Country, Spain 4Donostia International Physics Center (DIPC), 20018 Donostia-San Sebastian, Basque Country, Spain We propose and demonstrate spin manipulation by magnetically controlled modulation of pure spin currents in cobalt/copper lateral spin valves, fabricated on top of the magnetic insulator Y 3Fe5O12(YIG). The direction of the YIG magnetization can be controlled by a small magnetic field. We observe a clear modulation of the non-local resistance as a function of the orientation of the YIG magnetization with respect to the polarization of the spin current. Such a modulation can only be explained by assuming a finite spin-mixing conductance at the Cu/YIG interface, as it follows from the solution of the spin-diffusion equation. These results open a new path towards the development of spin logics. Spintronics is a rapidly growing field that aims at using and manipulating not only the charge, but also the spin of the elec- tron, which could lead to faster data processing speed, non- volatility and lower electrical power consumption as com- pared to conventional electronics [1]. Sophisticated applica- tions such as hard-disk read heads and magnetic random ac- cess memory (MRAM) have been introduced in the last two decades. Further progress could be achieved with pure spin currents, which are an essential ingredient in an envisioned spin-only circuit that would integrate logics and memory [2]. The most basic unit in such a concept is the spin analog to the transistor, in which the manipulation of pure spin currents is crucial. The original proposal by Datta and Das [3], which is also applica- ble to pure spin currents [4], suggested a spin manipulation that would arise from the spin precession due to the spin-orbit interaction modulated by an electric field (Rashba coupling). However, a fundamental limitation appears here, because the best materials for spin transport are those showing the lowest spin-orbit interaction and, therefore, there has been no success in electrically manipulating the spins and propagating them at the same environment, with few exceptions [4]. Alternative ways to control pure spin currents are thus desirable. One could take advantage of the spin- mixing conductance concept [5, 6] at nonmagnetic metal (NM)/ferromagnetic insulator (FMI) interfaces, which gov- erns the interaction between the spin currents present at the NM and the magnetization of the FMI. This concept is at the basis of new spin-dependent phenomena, including spin pumping [6–12], spin Seebeck effect [6, 13], and spin Hall magnetoresistance (SMR) [6, 14–18]. In these cases, a NM with large spin-orbit coupling is required to convert the in- volved spin currents into charge currents via the inverse Spin Hall effect [19]. In this Rapid Communication, we demonstrate an alterna- tive way of modulating pure spin currents based on magnetic, instead of electric, gating. To that end, we use lateral spin valves (LSVs). These devices allow an electrical injection and detection of pure spin currents in a NM channel by us-ing ferromagnetic (FM) electrodes in a nonlocal configuration [20–29]. The LSVs have been fabricated on top of a FMI, in order to enable the magnetic gating of the pure spin currents. The basic idea is depicted in Fig. 1: when the spin polariza- tion ( s) has the same direction as the magnetization ( M) of the FMI, the spin current reaching the detector will not vary with respect to the case where no FMI is used [Fig. 1(a)]. How- ever, when sandMare noncollinear, part of the spin current will be absorbed by Mvia spin-transfer torque [30–32], lead- ing to maximum spin absorption for perpemdicular Mands [Fig. 1(b)]. By using LSVs, we are able to extract the spin- mixing conductance of NM/FMI interfaces in the absence of charge currents, which otherwise could lead to spurious ef- fects, as suggested by some authors [33, 34]. Furthermore, the use of NM metals with low atomic number, employed in LSVs, rules out spin-orbit interaction effects that might exist for other systems, such as Pt/YIG [35]. We chose Y 3Fe5O12(YIG) [36] as a magnetic gate be- cause it is ferromagnetically soft and has a negligible mag- netic anisotropy. Mas a function of the applied in-plane mag- netic field ( H) measured by a vibrating sample magnetometer (VSM) saturates at 100 Oe [Fig. 2(a)], allowing control ofMabove this field. Cobalt (Co)/copper (Cu) LSVs were fabricated on top of YIG by two-step electron-beam lithogra- phy, ultrahigh-vacuum evaporation, and a lift-off process [Fig. 2(b)] [37]. Ar-ion milling was performed prior to the Cu de- position in order to remove resist leftovers [37]. To overcome the low spin injection of Co when using transparent interfaces [21–23], an oxide layer was created at the Co/Cu interface by letting Co oxidize after milling and before Cu deposition. The presence of an interface resistance, estimated to be RI5W, is known to enhance the spin injection efficiency [24, 25]. The LSVs were bridged by the same Cu channel, with thickness t100 nm, width w200 nm, and different edge-to-edge distances ( L) between the FM electrodes [37]. All measurements were performed using a ”dc reversal” technique [27] in a liquid-He cryostat with an applied mag- netic field Hat a temperature of 150 K. The sample can be rotated in plane in order to change the direction of H, whicharXiv:1404.2311v2 [cond-mat.mes-hall] 18 Feb 20152 !"#$!"$%"$"$&'$!"$($(b) FMI x y z NM I V- V+ !"#$!"$%"$ "$&'$!"$($(a) FMI x y z NM V- V+ I FM FM FM FM FIG. 1. (Color online) Scheme of the device used to modulate a pure spin current with magnetic gating. It consists of a ferromag- netic (FM)/ nonmagnetic (NM) lateral spin valve on top of a ferro- magnetic insulator (FMI). The nonlocal measurement configuration is shown. The x,yandzaxes are indicated as used in the text. (a) When the magnetization of the FMI ( M) and the polarization ( s) of the injected pure spin current ( js) are parallel, there will be no spin absorption. (b) When Mandsare perpendicular, the spin absorption will be maximum. is given by the angle adefined in Fig. 2(b). The nonlocal volt- ageVNLmeasured at the detector, normalized to the injected current I, is defined as the nonlocal resistance RNL=VNL=I [Fig. 2(b) shows a measurement scheme]. First, in order to check the standard performance of the LSV , the direction of H was fixed parallel to the FM electrodes ( a=0) and its value was swept from positive to negative, and vice versa, while RNLwas measured. This is plotted in Fig. 2(c), where RNL changes from positive to negative when the relative magneti- zation of the FM electrodes changes from parallel (P) to an- tiparallel (AP) by sweeping H. This measurement is an unam- biguous demonstration that a pure spin current is transported along the Cu channel [20–29]. It is worth noting that the rel- ative magnetization of the Co electrodes changes at H400 Oe, far above the saturation field of YIG ( 100 Oe). This detail is important for the performance of the next measure- ment, which consists in measuring RNLwhile fixing the value ofHand sweeping a. As shown in Fig. 2(d), this was done for both the P and AP configurations of the Co electrodes, which can be chosen with the proper magnetic field history. In this case, Hwas fixed to 250 Oe [see the dots in Fig. 2(c)], which is large enough to control Mof YIG but not to rotate the magnetization of the Co electrodes, as confirmed by magneto- optic Kerr effect (MOKE) microscopy [37, 38]. As intended, Fig. 2(d) shows a clear modulation of the measured RNL(i.e., FIG. 2. (Color online) (a) Magnetization of YIG ( M) as a function of the applied in-plane magnetic field Hmeasured at 150 K. (b) Colored scanning electron microscopy (SEM) image of a LSV . The nonlocal measurement configuration, materials, direction of Hand its angle a with respect to the FM electrodes are shown. (c) Nonlocal resistance (RNL) measured at 150 K as a function of Hwitha=0for a LSV with a separation distance between Co electrodes of L=1:6mm. The solid (dashed) line indicates the decreasing (increasing) sweep of H. A constant background of 0 :14 mWis subtracted from the data. Blue and red dots correspond to the value of RNLat the parallel (P) and antiparallel (AP) configurations of the Co electrodes, respectively, at H=250 Oe. (d) RNLas a function of a, measured for both the P and AP configurations, at 150 K with H=250 Oe for the same LSV . a modulation of the spin current) when Mof YIG is rotated in plane, clearly demonstrating a direct magnetic gating to a pure spin current. The reflection symmetry between the P and AP modulations again rules out the possibility of a relative tilting between the magnetization of Co electrodes [39]. In addition, the measurements were repeated in a control sam- ple, fabricated on a SiO2 substrate, in order to exclude any other possible artifacts [37]. The total change in RNL, caused by the spin absorption at the Cu/YIG interface, is defined as the nonlocal modulation dRNL=RNL(a=0)RNL(a=90)(tagged in Fig. 3). This figure contains the same data from Fig. 2(d), although, for the sake of clarity, P and AP configurations are plotted separately. In this case, for an Lof 1:6mm,dRNLhas a magnitude of  0:025 m W. We can define the factor b=dRNL=RNL(a=0) as an analog of a magnetoresistance, which gives a measure of the efficiency of the magnetic gating. Here, b=8:33% is obtained for the LSV with L=1:6mm, whereas b=2:96% forL=570 nm, showing that longer channels provide more efficient modulations. In order to quantify the observed modulation of RNL, we3 FIG. 3. (Color online) Nonlocal resistance (black solid squares) as a function of the angle abetween the FM electrodes and the applied magnetic field H, measured for the parallel (a) and antiparallel (b) configuration, at 150 K and H=250 Oe for a LSV with a separation distance of L=1:6mm. The red solid line corresponds to the fit of the data to Eq. 2. The blue dashed line corresponds to Eq. 2 in the absence of the spin-mixing conductance of the FMI/NM interface. The nonlocal modulation dRNLis tagged. solve the spin-diffusion equation [20, 21, 24] in the NM chan- nel, Ñ2~ms=~ms l2+1 l2m~msˆn; (1) where ~msis the spin accumulation at the NM metal and the vector refers to the spin-polarization direction. lis the spin- diffusion length of the NM and lm=q D¯h 2mBjBjis the magnetic length determined by the amplitude of the magnetic field Bˆn ( ˆnis the unit vector giving its direction). The last term in Eq. 1 describes the well-known spin precession due to the applied field [40, 41]. Bis proportional to Hand, for Cu, we can approximate Bm0H.Dis the electronic diffusion constant of the NM, and mBis the Bohr magneton. Assuming tl, we can integrate Eq. 1 in the zdirection and use the Brataas- Nazarov-Bauer boundary condition at the NM/FMI interface [5]. From the solution one can obtain an expression for the nonlocal resistance at the FM detector that reads [37, 42, 43] RNL=P2 IRN 2 cos2aeL=l+sin2aRel1 leL=l1 ;(2) which is only valid in the high interface resistance limit, i.e., ifRIRN.PIis the spin polarization of the FM/NM interface at both the FM injector and detector, RN=rl=wtis the spin resistance of the NM, and ris its electrical resistivity. The length l1is defined as l1=lr 1+2rGrl2 t+i l lm2; (3) FIG. 4. (Color online) Representation (solid lines) of the bfac- tor, based on Eq. 2 for an applied magnetic field H=250 Oe, as a function of (a) the distance ( L) between FM electrodes, (b) the thickness ( t) of the NM channel, and (c) the spin-mixing conduc- tance per unit area ( Gr) of the NM/FMI interface. The parame- ters used for the simulation are: (a) l=522 nm, r=2:1mWcm, Gr=51011W1m2, and t=100 nm. (b) l=522 nm, r= 2:1mWcm,Gr=51011W1m2, and L=1:6mm. (c) l=522 nm,r=2:1mWcm,L=1:6mm, and t=100 nm. where Gris the real part of the spin-mixing conductance per unit area [5] of the FMI/NM interface. We have dis- regarded the imaginary part of the spin-mixing conductance in accordance with Refs. [14, 32]. Notice that for a=0, theRNLfor the case without FMI [24, 28, 29] is recovered: RNL=P2 IRN 2eL=l. At a=90we obtain a similar expres- sion for RNLas in the a=0case, but with a reduced spin- diffusion length Re (l1):RNL=P2 IRN 2Re l1 leL=l1 . Equa- tion 3 shows that two quantities renormalize the spin-diffusion length: the spin-mixing conductance by means of the real term 2rGrl2=t, and the imaginary Hanle term i(l=lm)2originat- ing from the applied field. While the former leads to a reduc- tion of ldue to the torque exerted by the NM/FMI interface to the spins [30, 32], the latter causes, in addition, the precession of the spins when sandHare noncollinear [40]. At a first glance, one might think that the Hanle term could be enough to explain the observed modulation of RNLas a function of a. However, as shown in Fig. 3, a field of 250 Oe in the absence of Grleads to a modulation of RNL(blue dashed line) which is one order of magnitude smaller than the measured one. This is experimentally confirmed in the control sample performed on top of SiO 2[37]. Increasing H would eventually lead to a Hanle effect of the same order as theGreffect. Nevertheless, our experiment is limited to low magnetic fields ( H<400 Oe), to avoid the magnetization of the Co electrodes being affected by the direction of H, and thus the Hanle term will not be dominant. Considering both the Grand Hanle terms, Eq. 2 accurately fits the measured RNL(Fig. 3), reproducing the observed mod- ulation of the spin current. Note also that Eq. 2 reproduces the reflection symmetry between the P and AP configurations, be- cause the product P2 Ihas opposite sign for each configuration. The fact that the modulation is observed in a pure spin current in a metal such as Cu excludes any proximity effect as the ori-4 gin of the modulation [33, 34], confirming the validity of the Grconcept. There are two fitting parameters: PIandGr, whereas w,t andLare known geometrical parameters, and r(2.1mWcm) andl(522 nm) are obtained from resistance measurements andRNLmeasurements as a function of L[37]. From the fitting for the LSV with L=1:6mm (Fig. 3), we obtained PI=0:1280:001 and Gr= (4:280:06) 1011W1m2for the P state [Fig. 3(a)], and PI=0:129 0:001 and Gr= (5:630:07)1011W1m2for the AP state [Fig. (b)], which are almost identical for both magnetic con- figurations. Therefore, the value of Grobtained for this par- ticular Lis(4:960:68)1011W1m2. The same fitting was performed for the LSV with L=570 nm, where it was also possible to measure RNLas a function of a, obtaining PI=0:1230:001 and Gr= (2:820:66)1011W1m2. Since Gris extracted separately for each device, this trans- fers the unavoidable device-to-device dispersion (spin trans- port is very sensitive to any minor defect) into the value of Gr. The difference, which is less than a factor of 2, can thus be considered to be small, taking into account that, in order to observe a relevant variation in b, a much larger change in Gris needed [Fig. 4(c)]. Whereas PIis within the range reported in similar systems [22, 28, 29], Gris sub- stantially smaller than the values obtained for Pt/YIG (rang- ing from 1 :21012to 6:21014W1m2) [? ? ], Ta/YIG (4:31013W1m2) [16], and Au/YIG (between 3 :51013 and 1 :91014W1m2) [10, 11] either by SMR or spin pumping experiments. There is a possibility of underestimating Grif the assump- tion for Eq. 2, RIRN, is not fulfilled. For b8%, Gr would increase by a factor of 2, to81011W1m2, by considering transparent interfaces [37], which is still low com- pared to other NM/YIG interfaces. Another possible reason for the low Grvalue could be the Ar-ion milling performed before the Cu deposition [12] or the YIG surface quality. We rule this out by performing a control experiment in Pt/YIG where we obtain Gr=3:341013W1m2from SMR mea- surements [37, 44, 45]. Particularities of the grain structure and the growth condition of the evaporated Cu on YIG could also lead to an effective reduction of Grat the interface. Alter- natively, the spin-orbit interaction effects that might exist for Pt/YIG, Au/YIG or Ta/YIG [35] could lead to an overestima- tion of the obtained Grin these systems. Such effects are un- likely in Cu/YIG. It is worth noting that the Grof a NM/YIG interface, for a NM with a negligible spin-orbit coupling, was not experimentally measured before due to the need of the inverse Spin Hall effect (and thus a high spin-orbit coupling metal) in the experiments made so far [6–16]. Finally, a representation of b, based on Eq. 2, is plotted in Fig. 4 as a function of different parameters ( L,tandGr) which can be controlled in order to improve the efficiency of the magnetic gating. The values of the different parameters used for the representation are listed in the caption and corre- spond to realistic values taken from our devices. bincreases linearly with the length ( L) between the FM electrodes, reach-ing30% for L=5mm [Fig. 4(a)]. When the spin current flows over a longer distance, the spin scattering and absorp- tion caused by the NM/FMI interface will be enhanced (i.e., bwill be larger). This is in agreement with our experimen- tal results discussed above. However, there is an experimen- tal limit, since the nonlocal signal decays exponentially and will be negligible when Ll[23, 26]. By decreasing the thickness ( t) of the Cu channel, bincreases asymptotically when tapproaches 0 [Fig. 4(b)]. In this case, by decreasing t, the relative contribution of the NM/FMI interface to the spin- flip scattering processes increases, enhancing b. For instance, when t20 nm, balready increases to 50%. However, the decrease of lwith t[26], which has not been taken into ac- count for the representation, will lower b. The most effective way of improving bseems to be increasing Gr[Fig. 4(c)]. By increasing it by two orders of magnitude, i.e., for a Grof the order that Pt/YIG systems have, breaches almost up to a 100%, which would lead to a perfect magnetic gating of the pure spin currents. This seems feasible by improving the in- terface between Cu and YIG or by using another NM material with a high spin-mixing interface conductance with YIG. To conclude, we present an approach to control and ma- nipulate spins in a solid state device, by means of a magnetic gating of pure spin currents in Co/Cu LSV devices on top of YIG. A modulation of the pure spin current is observed as a function of the relative orientation between the magnetization of the FMI and the polarization of the spin current. Such mod- ulation is explained by solving the spin-diffusion equation and considering the spin-mixing conductance at the NM/FMI in- terface. The accuracy between the measured data and the ex- pected modulation provides an effective way of studying the NM/FMI interface. From our results, a spin-mixing conduc- tance of Gr41011W1m2is obtained for the Cu/YIG interface. An increase of this value will enhance the efficiency of the magnetic gating. This can be achieved by carefully tun- ing the fabrication parameters. Our experiment paves the way for different manners of spin manipulation, bringing closer pure spin currents and logic circuits. ACKNOWLEDGEMENTS We thank Professor Joaqu ´ın Fern ´andez-Rossier for fruit- ful discussions. This work was supported by the Euro- pean Commission under the Marie Curie Actions (256470- ITAMOSCINOM), NMP Project (263104-HINTS) and the European Research Council (257654-SPINTROS), by the Spanish MINECO under Projects No. MAT2012-37638, No. MAT2012-36844, and No. FIS2011- 28851-C02-02, and by the Basque Government under Project No. PI2012-47 and UPV/EHU Project No. IT-756-13. E.V . and M.I. thank the Basque Government for support through a Ph.D. fellowship (Grants No. BFI-2010-163 and No. BFI-2011-106). 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Vila, and H. Jaffr`es, Phys. Rev. Lett. 112, 106602 (2014).6 Modulation of pure spin currents with a ferromagnetic insulator SUPPLEMENTAL MATERIAL Estitxu Villamor, Miren Isasa, Sa ¨ul V´elez, Amilcar Bedoya-Pinto, Paolo Vavassori, Luis E. Hueso, F. Sebasti ´an Bergeret, and F `elix Casanova S1. Experimental details Lateral spin valves (LSVs) were fabricated by a two-step electron-beam lithography, ultra-high-vacuum evaporation and lift- off process (see Fig. 2(b) from the main text for a SEM image of the device). Since yttrium iron garnet (YIG) on gadolinium gallium garnet (GGG) is an insulating substrate, a thin gold (Au) layer of 2.5 nm was sputtered on top of the PMMA resist before each lithography step. This prevents the charging of the substrate during the e-beam exposure, which otherwise would distort the pattern. After the e-beam exposure and before developing the patterned resist, the Au layer was removed with Au etchant. In the first lithography step, FM electrodes were patterned and cobalt (Co) was electron-beam evaporated with a base pressure 1109mbar. In the second lithography step, the NM channel was patterned and copper (Cu) was thermally evaporated with a base pressure1109mbar. Ar-ion milling was performed prior to the Cu deposition in order to remove resist left-overs; the parameters used for the Ar-ion milling are an Ar flow of 15 standard cubic centimeters per minute, an acceleration voltage of 50 V , a beam current of 50 mA, and a beam voltage of 300 V for 30 s, as described in Ref. [1]. To overcome the low spin injection of Co when using transparent interfaces [1, 2], an oxide layer was created at the Co/Cu interface by letting the Co oxidize after the milling and before the Cu deposition. The presence of an interface resistance is known to enhance the spin injection efficiency [3]. An interface resistance RI5Wis estimated in this case. Both Co electrodes have a thickness of 35 nm and different widths (115 nm and 175 nm) to obtain different switching fields by means of shape anisotropy. Three LSVs were fabricated, bridged by the same Cu wire (of width w200 nm and thickness t100 nm) with edge-to-edge distances between the Co electrodes of L= 250 nm, 570 nm and 1600 nm. Such configuration allows the measurement of the four-point resistance Rof the wire as a function of L(where the electrodes belong to the same LSV or to two contiguous LSVs). By performing a linear fitting of Ras a function of L(see lower inset in Fig. S1) and knowing the dimensions of the Cu wire, it is straightforward to obtain its electrical resistivity, which has the value ofr=2:1mWcm at a temperature of 150 K. The non-local resistance RNL=VNL=I, i.e. the non-local voltage Vmeasured at the detector normalized to the value of the injected current I, is measured as a function of the applied magnetic field Hfor the three LSVs. Note that the LSV with L= 250 nm broke after measuring RNLas a function of Hata=0and, for this reason, the results of the RNLmeasurement as a function of aare not included in the main text. RNLchanges from positive to negative when the relative magnetization of the FM electrodes changes from parallel (P) to antiparallel (AP) by sweeping H(see upper inset of Fig. S1). The difference between positive and negative RNLis defined as the spin signal ( DRNL), which, for LSVs with a high interface resistance, can be expressed as [3–6]: DRNL=P2 IRNeL=l; (S1) where PIis the spin polarization of the Co/Cu interface, RN=rl=wtis the spin resistance of Cu and lis the spin diffusion length of Cu. Figure S1 shows the measured DRNLas a function of L(black squares), which is fitted to Eq. (S1) (red solid line) in order to obtain PIandl. The obtained values at 150 K are PI=0:180:01 and l=52225 nm. Even though PI is within the range of values that are observed in literature in similar systems [2, 4–7], it is slightly higher than the PIvalues obtained in the main text for the fitting of Eq. 2. This is due to the dispersion of the interface quality between different LSVs. The device with L=250 nm has a higher PI, which enhances the averaged PIobtained from the fitting of Eq. (S1). The value obtained for lis similar but slightly lower than our previous values obtained in Py/Cu LSVs on top of Si/SiO 2measured at 150 K (l=68015 nm) [1]. This could be due to the different growth of Cu on top of YIG as compared to SiO 2.7 FIG. S1. Spin signal as a function of the separation distance between Co electrodes ( L) for three Co/Cu LSVs, measured at 150 K with the applied field ( H) parallel to the electrodes (black squares). Red line is a fit to Eq. (S1). Upper inset: Non-local resistance measured as a function of Hfor the LSV with L=570 nm. A constant background of 0.38 m Wis subtracted from the data. Solid (dashed) line indicates the decreasing (increasing) sweep of H. Spin signal DRNLis tagged in the plot. Lower inset: Four-point resistance as a function of L(black squares). Red line is a linear fit. S2. Control experiment: MOKE measurements to rule out magnetization rotation in Co electrodes A tilting of the magnetization direction in the Co electrodes during the measurement of RNLas a function of the angle a between the Co electrodes ( ydirection, see Fig. 1 from the main text) and the direction of the applied magnetic field H, could in principle be invoked to explain the observed modulation of RNLwithabecause RNLµcosq, where q(a)is the relative angle between the magnetizations of both electrodes. This tilting could be caused by the torque exerted directly by the applied magnetic field or/and by a coupling between the Co electrodes and the YIG substrate. A modulation of 8%, such as the observed one, could correspond to q23. Even if the reflection symmetry between the RNLmodulation observed in the parallel and anti-parallel magnetization states of the Co electrodes is sufficient to rule out such explanation (as stated in the main text), to further exclude a magnetization rotation of the electrodes, MOKE microscopy measurements were performed at room temperature directly on the same sample used for the magnetic gating experiment. The capability of our MOKE microscope to measure the field induced magnetization reorientation of ultra-small ferromagnetic nanostructures was demonstrated earlier [8]. The MOKE measurements were performed on top of the widest electrode, which is the one whose magnetization can rotate more easily due to shape anisotropy. Figure S2 shows hysteresis loops of the Co electrode (red circles) and of the YIG (black squares), i.e.the projection of the magnetization in the ydirection, My, is measured as a function of the magnetic field applied in the ydirection, Hy, and normalized to the saturation magnetization, Ms. In both cases,the MOKE signal was acquired from a subset of the pixels of the CCD detector that corresponds to an area on the sample surface equal to 100 800 nm2[8]. The coercive field of the Co electrode is 500 Oe, in agreement with the RNLmeasurements as a function of Hshown in Fig. 1(c) from the main text. For the YIG substrate, magnetic saturation around 100 Oe is observed, in agreement with the VSM measurements shown in Fig 1(a) from the main text. To check for a possible rotation of the magnetization of the Co electrode, its My=Mswas measured while the direction of the magnetic field H, which had a fixed intensity of 250 Oe, was rotated by avaried from 0 to 360. Figure S3 shows My=Msof the Co electrode and the YIG substrate. Whereas the magnetization of YIG coherently rotates with the direction ofH(My=Msµcosa), given that His largely exceeding the saturation field of YIG, the magnetization of the Co electrode is constant for every a. Based on the signal-to-noise ratio of our measurements, the smallest detectable change in My=Ms corresponds to a rotation of Msof less than 5. Therefore, from our measurements, we can directly conclude that the rotation of the Co magnetization, if any, is less than 5, which could only explain a variation of less than 0.4% in RNL, well below the experimentally observed 8% .8 -800-4000 4 008 00-1.0-0.50.00.51.0 Co YIG My /MsH y [Oe] FIG. S2. Projection of the magnetization in the ydirection, My, of the YIG (black squares and line) and of the Co electrodes (red circles and line) normalized to the saturation magnetizations, Ms, measured as a function of the magnetic field applied in the ydirection, Hy. Measurements are performed at 300 K. 09 01 802 703 60-1.0-0.50.00.51.01.5H α y(250 Oe) YIG Co stripeMy/Msα FIG. S3. Projection of the magnetization in the ydirection, My, of the YIG (black squares) and of the Co electrodes (red circles) normalized to the saturation magnetizations, Ms, measured as a function of the angle abetween the direction of the Co electrodes ( y) and the applied magnetic field, H=250 Oe. Measurements are performed at 300 K.9 S3. Control experiment: Non-local resistance measurements on top of a silicon oxide substrate Even though a possible rotation of the FM electrodes, which could explain a variation in the non-local resistance as the one we observe, was excluded with the previous control experiment (section S2), an additional control experiment was performed in order to rule out any other possible artifact. With this purpose, the main experiment was repeated in a LSV fabricated on top of SiO 2instead of YIG. Fig S4(b) shows RNLmeasured at 150 K, applying a magnetic field of 250 Oe, as a function of afor both the parallel (P) and antiparallel (AP) magnetizations of the FM electrodes in a LSV with L=750 nm. Apparently, no periodic modulation of RNLis observed. However, by taking a closer view (Figs. S4(c) and S4(d)), one can guess a periodic modulation of the order of the noise, which behaves as the one observed in the main experiment, including a reflection symmetry between the P and AP case. The observed modulation corresponds to a 1%, which is the estimated value for the Hanle effect at this L, and is certainly smaller than the b=2:96% and b=8:33% observed in the main experiment for L=570 nm and L=1:6mm, respectively. This confirms experimentally that, for these values of LandH, the Hanle contribution is much smaller than the modulation due to spin-mixing conductance, and also excludes any other possible artifact. FIG. S4. (a) Non-local resistance ( RNL) measured at 150 K as a function of Hwitha=0for a LSV fabricated on top of SiO 2with L=750 nm. (b) RNLas a function of ameasured for both the (c) P and (d) AP configuration, at 150 K with H=250 Oe for the same LSV .10 S4. Control experiment: Spin Hall magnetoresistance measurements in a Pt/YIG sample In order to see if the low Grvalue obtained for Cu/YIG interfaces originates from the quality of the YIG substrate or from any effect that might be induced at the YIG substrate for the Ar-ion milling process (see section S1), we fabricated a Pt/YIG control sample and tested it within the Spin Hall magnetoresistance (SMR) framework. The SMR arises from the simultaneous effect of the spin Hall effect and the inverse spin Hall effect in the Pt layer in combination with the interaction of the generated spin current with the magnetization of the YIG surface. Depending on the relative orientation between the spin polarization vector and the direction of the magnetic moment of the YIG surface, the spin current might be absorbed via spin-transfer torque resulting in a modulation of the resistance of the Pt layer, which is fundamentally related to the spin-mixing conductance Grof the Pt/YIG interface [9, 10]. With this purpose, a 7-nm-thick Pt Hall bar (with a width W=100mm and a length L=800mm) was sputtered on top of a YIG substrate grown as the one used for the fabrication of the LSV . Prior to the Pt deposition, the YIG surface was subjected to the same Ar-ion milling process (see section S1). Angular dependent magnetoresistance (ADMR) measurements were performed by rotating a fixed Halong the three main rotation planes of the system: XY , YZ and XZ, with the corresponding angles a,band g. A large enough His applied to ensure the magnetization of the YIG substrate follows the direction of the applied magnetic field. The resistance was measured using both longitudinal ( RL) and transverse ( RT) configurations. Figure S5(a) shows a sketch of the resulting device with the definition of the axes and the transverse configuration. As expected from the SMR theory [9, 10]: (i) no ADMR is observed in RL(g), (ii) a large modulation is observed in RL(a)andRL(b), with the same amplitude and a cos2 dependence, and ( iii)RT(a)shows a sin a
cosadependence, with the same amplitude as in RL(a)but with a L=Wfactor. As illustrative example, the transverse resistance RT(a)obtained for H= 1 kOe and 150 K is plotted in Fig. S5(b). FIG. S5. (a) Sketch of the Pt Hall bar on YIG. Charge current ( JC) and applied external magnetic field ( H), measurement configuration, axes and the angle ( a) between HandJCare indicated. (b) Transverse resistance ( RT) measured as a function of a. A small spurious baseline resistance RT0was subtracted. According to the SMR theory, the amplitude of the observed magnetoresistance is related to the microscopic properties of the Pt layer by [9, 10]: Dr r=q2 SH2rl2 tGrtanh2t 2l 1+2rlGrcotht l; (S2) where qSHis the spin Hall angle, lis the spin diffusion length, ris the electrical resistivity, tis the thickness of the Pt and Gr is the real part of the spin-mixing conductance per unit area of the Pt/YIG interface. In our case, with a measured longitudinal resistance of RL=281:5Wat 150 K, one can determine r=24:7mWcm and the SMR signal Dr=r=5:48105. Knowing the values of qSHandlin Pt, one can extract the Grvalue of the Pt/YIG interface using Eq. S2. These values cannot be inferred from our measurements, but can be obtained from literature. Despite there is a big dispersion of the given values for qSHandl [9, 11, 12], we will use the ones recently reported in Ref. [12] ( qSH= 0.056 and l= 3.4 nm), since they are highly consistent within different methods used to determine them. A Gr=3:41013W1m2is thus obtained for our Pt/YIG interface, which is in agreement with the previously reported range of values [9, 13–18]. We can take this result as a proof of the good quality of the YIG substrates used in the present experiments and as an indication that the Ar-ion milling process in the LSV experiment is not at the origin of the low Grobtained.11 S5. Theory In order to model the experimental results, we consider the geometry shown in Fig. 1 from the main text. It consists of a normal (NM) layer, Cu in our case, deposited on top of a ferromagnetic insulator (FMI), YIG in this case. At x=0 there is a ferromagnetic (FM) electrode, Co in our case, that injects a charge current Ithat flows in x<0 direction. Coming from a FM, this current induces a spin accumulation ma s(adenotes the spin polarization direction) in the NM layer. In the absence of spin-orbit coupling, a spin current density in the NM ja k(kdenotes the flow direction) is then originated by the gradient of the spin accumulation ma s ja k=1 2er¶kma s; (S3) where ris the electrical resistivity of the NM and ethe absolute value of the electron charge. In a normal metal, the mean free path lis smaller than other characteristic lengths, and therefore the spin accumulation is determined by the Bloch equation with an added diffusion term [19, 20], which, in the steady state, has the simple form: Ñ2~ms=~ms l2+1 l2m~msˆn: (S4) Here ~ms= (mx s;my s;mz s),ldenotes the spin diffusion length which is related to the diffusion coefficient Dand the spin-flip relaxation time ts fbyl=pDts f, and lm=p D¯h=2mBBis the magnetic length determined by the amplitude of the applied magnetic field Bˆn( ˆnis a unit vector along the magnetic field direction). Alternatively, Eq. (S4) can be derived from the Keldysh Green’s function formalism [21, 22]. In the case of an intrinsic spin-orbit coupling, due to an inversion asymmetry, Eqs. (S3- S4) have the same form if one substitutes the gradient by a SU(2) covariant derivative [21]. In the case of extrinsic spin-orbit coupling, due to random impurities, these equations acquire some extra terms [22]. However, spin-orbit effects are negligible in accordance in our NM, Cu. We assume that the system is invariant in ydirection and therefore the spin accumula- tion only depends on xandz:ms(x;z). In order to solve the diffusion equation (S4) for the spin accumulation one needs proper boundary conditions. At the upper interface of the NM with the vacuum the spin current should vanish: ¶zmsjz=t=0; (S5) where tis the thickness of the NM. We are assuming z=0 at the NM/FMI interface, and z=tat the NM/vacuum interface. At the interface with the FMI we use the Brataas-Nazarov-Bauer boundary condition [23]: ¶z~msjz=0=2r[Grˆm(ˆm~ms)+Giˆm~ms]; (S6) where ˆ mis a unit vector along the magnetization of the FMI, and Gmix=Gr+iGiis the the complex spin-mixing interface conductance per unit area [23]. In the experiment the thickness tof the NM layer is smaller than the characteristic scale of variation of ms(l) and therefore we can integrate Eq. (S4) over zassuming that msdoes not depend on z. By performing this integration and using Eqs. (S5-S6) we obtain the following (1D) equation for ~ms: ¶2 xx~ms=~ms l2+1 l2m+1 l2 i ~msˆm1 l2rˆm(ˆm~ms); (S7) where we have considered an in-plane magnetization of the FMI, ˆ m= (sina;cosa;0), and defined l2 r=2rGr=tandl2 i= 2rGi=t. The latter term acts as an effective magnetic field parallel to the magnetization of the FMI which is assumed to be parallel to the applied magnetic field. Equation (S7) describes the spatial dependence of the spin accumulation in a thin FM layer in contact with a FMI. It consists of three coupled linear second order differential equations. In order to solve it we have to write the boundary conditions corresponding to the experimental situation: at x=0 an electric current Iis injected. This induces at x=0 a spin current equal to PII, where PIis the spin polarization of the FM/Cu interface. At a distance Lfrom the injection point there is a detector. The spin accumulation and its derivative are continuous in the NM layer. The boundary conditions for ~ms(x)at the injector and detector12 are obtained from the spin current continuity and read: PIIˆy=l eRN¶x~msjx=0l eRN¶x~msjx=0+ (S8) 0=l eRN¶x~msjx=Ll eRN¶x~msjx=L+; (S9) where the spin current at both sides of the FM injector (detector) is considered. RN=rl=wtis defined as the spin resistance, where wis the width of the NM channel. The FM injector is polarized in ydirection (whose unit vector is ˆ y) due to shape anisotropy, and, thus, the injected spin current as well. In order to obtain the boundary conditions Eqs. (S8-S9), a high interface resistance ( RI) was considered at the interfaces between the NM and the FM injector ( x=0) and between the NM and the FM detector ( x=L) [3], i.e. RIRN. IfRIis of the order of RN, a spin current that might flow back into the FM electrodes [20, 24] has to be taken into account. In the case considered above, it is rather straightforward to solve Eq. (S7) with the conditions Eqs. (S8-S9), in order to obtain the spin accumulation in all three spin polarization directions: mx s(x) =PIIeR Ncosasina ex=l+Rel1 lex=l1 ; (S10) my s(x) =PIIeR N cos2aex=l+sin2aRel1 lex=l1 ; (S11) mz s(x) =PIIeR NsinaIml1 lex=l1 ; (S12) where the characteristic length in the second exponential is defined as l1=lp1+g(S13) withg=gr+igi,gr=l2=l2 randgi=l2(1=l2 m+1=l2 i). It is interesting to note that, even if the injected spin current is polarized in the ydirection, a spin accumulation is created with the spins polarized in the xdirection, due to both the torque exerted by Mat the NM/FMI interface and the spin precession caused by the magnetic field perpendicular to the spin polarization, and in the zdirection only due to the spin precession caused by the magnetic field perpendicular to the spin polarization. Since the magnetization of the injector and detector are in ydirection, only my scan be detected at x=L. From Eq. (S11) we can determine the non-local resistance ( RNL) defined in terms of the non-local voltage VNLmeasured at the detector [3, 24]: RNL=VNL I=PImy s(L) 2eI; (S14) where we assume that the polarization at the detector contact is the same as at the injector. By inserting Eq. (S11) into this last expression we finally obtain RNL=P2 IRN 2 cos2aeL=l+sin2aRel1 leL=l1 : (S15) It follows that, in the absence of the spin-mixing conductance and if the magnetic field is in the xdirection ( i.e.a=90), the expression is identical to the one obtained in Ref. [20] in which the Hanle effect was studied. Eq. (S15) is a general expression that describes the non-local resistance in the NM/FMI structure and takes into account both the effect of the external applied field and the spin-mixing conductance describing the magnetic interactions at the interface. We have fitted our measurements of the RNL(a)dependence with Eq. (S15) by neglecting the imaginary part of the spin-mixing conductance which, according to first-principle calculations [25] and our discussion below, seems to be a good approximation. As we can see in Fig. 3 from the main text, the effect of the applied field on the RNL(a)modulation is small in comparison to the one induced by Gr. This demonstrates that the modulation observed can only be explained by the effect of the spin-mixing conductance. According to our estimations, Gr41011W1m2. This value is in principle smaller than the value reported in13 previous works [9, 13–18, 26, 27]. This discrepancy is discussed in the main text. FIG. S6. Non-local resistance measured as a function of the angle abetween the spin polarization and the applied magnetic field H. Two measurements have been done for H=250 Oe and H=350 Oe, with identical results. The experimental results shown in the main text have been obtained for an applied magnetic field of 250 Oe. Measurements have been performed also at 350 Oe with almost identical results (see Fig. S6). From theory, the difference in RNLfor these two fields is of the order of the measurement noise and, thus, not detectable in principle. For such values, (l=lm)21. But also (l=lr)2(and presumably also (l=li)2) are very small according to our estimation of Gr. For a Gr41011W1m2, and with the parameters of the used LSVs, (l=lr)2=0:037 is obtained. Therefore, one can go analytically one step further by treating the parameter gin Eq. (S13) perturbatively. Instead of focusing in RNLlet us analyze the amplitude of the effect when varying a(the field direction), as shown in Fig. 2(b) in the main text. We introduce the dimensionless parameter bdefined as b=1RNL(a=90) RNL(a=0): (S16) In the limit g1 we obtain up to lowest order in g bgr 2L+l l=l(L+l)rGr t; (S17) while the lowest correction in giis of second order and therefore negligible in this approach. If we insert here the value for Gr41011W1m2, we obtain an amplitude of the effect b9:3% (b4:8%) for the LSV with L=1600 nm ( L=570 nm), which is in good agreement with the experimentally obtained ones. As explained above, all the previous results have been obtained assuming the high RIlimit ( RIRN). However, if one allows for an arbitrary value of RI=RN, one should take into account the possible back flow of spin current in Eqs. (S8-S9). An expression for RNLwith arbitrary RIin a perpendicular magnetic field (without FMI) has been presented in Ref. [20]. In such a case only the lengths landlmenter in Eq. (S7). After inspection of the latter equation, it turns out that the general expression forRNLderived in Ref. [20] is also valid in the presence of the FMI layer, if one substitutes the magnetic length by l1of Eq. (S13). This result can be used to determine the parameter busing Eq. (S16). In Fig. S7 we show the dependence of bas a function of Grin both the RIRNandRI=0 cases. We see that for the value obtained from our measurements ( b8%) Gr is slightly larger (by a factor of 2) in the transparent case. We can conclude from this that the actual value of Grlies between 41011W1m2and 81011W1m2. It is also worth noticing that, according to Fig. S7, in the hypothetical case that Gris of the order of 10131014W1m2a full modulation ( b=100%) can be achieved. This means that RNLcan be switched between a finite value and 0 by switching the field from a=0toa=90, respectively.14 FIG. S7. bfactor as a function of Grfor the RI=0 and RIRNcases.15 [1] E. Villamor, M. Isasa, L. E. Hueso, and F. Casanova, Phys. Rev. B 87, 094417 (2013); E. Villamor, M. Isasa, L. E. Hueso, and F. Casanova, Phys. Rev. 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2014-04-08

We propose and demonstrate spin manipulation by magnetically controlled modulation of pure spin currents in cobalt/copper lateral spin valves, fabricated on top of the magnetic insulator Y$_3$Fe$_5$O$_{12}$ (YIG). The direction of the YIG magnetization can be controlled by a small magnetic field. We observe a clear modulation of the non-local resistance as a function of the orientation of the YIG magnetization with respect to the polarization of the spin current. Such a modulation can only be explained by assuming a finite spin-mixing conductance at the Cu/YIG interface, as it follows from the solution of the spin-diffusion equation. These results open a new path towards the development of spin logics.

Modulation of pure spin currents with a ferromagnetic insulator

1404.2311v2

Measurements of the exchange stiness of YIG lms by microwave resonance techniques S Klingler1, A V Chumak1, T Mewes2, B Khodadadi2, C Mewes2, C Dubs3, O Surzhenko3, B Hillebrands1, A Conca1 1Fachbereich Physik and Landesforschungszentrum OPTIMAS, Technische Universit at Kaiserslautern, 67663 Kaiserslautern, Germany 2Department of Physics and Astronomy, MINT Center, University of Alabama, Tuscaloosa, Alabama 35487, USA 3INNOVENT e.V. Technologieentwicklung, 07745 Jena, Germany E-mail: klingler@physik.uni-kl.de Abstract. Measurements of the exchange stiness Dand the exchange constant A of Yttrium Iron Garnet (YIG) lms are presented. The YIG lms with thicknesses from 0.9m to 2.6m were investigated with a microwave setup in a wide frequency range from 5 to 40 GHz. The measurements were performed when the external static magnetic eld was applied in-plane and out-of-plane. The method of Schreiber and Frait [1], based on the analysis of the perpendicular standing spin wave (PSSW) mode frequency dependence on the applied out-of-plane magnetic eld, was used to obtain the exchange stiness D. This method was modied to avoid the in uence of internal magnetic elds during the determination of the exchange stiness. Furthermore, the method was adapted for in-plane measurements as well. The results obtained using all methods are compared and values of Dbetween (5:180:01)
1017T
m2and (5:340:02)
1017T
m2were obtained for dierent thicknesses. From this the exchange constant was calculated to be A= (3:650:38) pJ/m.arXiv:1408.5772v1 [cond-mat.mtrl-sci] 25 Aug 20142 1. Introduction In order to employ the degree of freedom of the spin in future information technology, materials with low Gilbert damping and long spin-wave propagation distances are needed for data transport. Yttrium Iron Garnet (YIG) is a material which fullls the aforesaid requirements. New technologies employing YIG are being developed and new physical phenomena were investigated. Logic operations with spin waves in YIG waveguides [2, 3, 4], data-buering elements [5] and magnon transistors [6] are only a few examples for the latest technology progress. Especially, YIG lms of nanometer thickness [7, 8, 9, 10, 11, 12] are of large importance since they allow for the realization of nano- and microstructures [6, 8, 13, 14] and an enhancement of spin-transfer-torque related eects [12, 15]. In this context the material parameters of YIG are of crucial importance for its application potential. In a magnetic system, the exchange interaction contributes strongly to the energy of the system. From a classical point of view, this interaction is responsible for the parallel alignment of adjacent spins, thus, it strongly in uences the spin-wave characteristics. The strength of the exchange interaction is given by the exchange stiness D, but the existing approaches for its measurement are often in uenced by internal magnetic elds depending consequently on crystal anisotropies and the saturation magnetization. Thus, methods are required for the exact determination of the exchange stiness without the uncertainties added by the aforementioned parameters. Here, such a method is presented and compared to the results obtained by commonly used data evaluation methods. Firstly, the classical approach of Schreiber and Frait [1] is used for the determination of the exchange stiness when the external static magnetic eld is applied out-of-plane. Secondly, the method is modied to avoid any in uence of the anisotropy elds and the saturation magnetization in order to achieve highly-precise values for D. Thirdly, the method of Schreiber and Frait is adapted and used for in-plane measurements. All values ofDobtained by using the dierent methods are compared and, then, values of the exchange constant Aare calculated for our YIG samples. 2. Theory The precessional motion of the magnetization in an eective magnetic eld is described by the Landau-Lifshitz and Gilbert equation [16]. The eective magnetic eld depends on various parameters, such as the applied static and time dependent magnetic elds (0H0and0h(t), respectively), anisotropy elds (the cubic anisotropy eld 0Hc, the uniaxial out-of-plane 0Hu?and in-plane 0Hukanisotropy elds), as well as the exchange eld 0Hex=Dk2which describes the exchange interaction in the investigated material. Here, D=2A MSis the exchange stiness, Athe exchange constant and MSthe saturation magnetization. The wavevector kis the wavevector of perpendicular standing spin-waves which is quantized over the sample thickness. Under the assumption of perfect pinning of the spins at the sample surface kis dened by k=n=d [1], wheren3 Table 1. Parameters of the studied YIG samples. The average growth rate was calculated from the thickness dand the deposition time which was 5 min for all YIG lms. SampleThickness Growth rate Lattice mist d(m)(m/min)
a?=aGGG(104) E1 2:590:01 0.52 +5 :330:07 E2 1:590:02 0.32 +7 :680:02 E3 0:9030:003 0.18 +8 :720:03 is the mode number. The case n= 0 corresponds to the classical case of ferromagnetic resonance. The resonant precession frequency for cubic crystals is presented in reference [17]. For the case when the static magnetic eld is applied in-plane one obtains !k j j2 =2 0(H0+Hex) (H0+Hex+MSHu?Hc): (1) This equation is valid if the magnetization of the sample points along the h110i-axis of the crystal. If the static magnetic eld is applied out-of-plane the frequency is given by !? j j=0 H0+HexMS+Hu?4 3Hc+Huk : (2) Here,!?and!kare the applied microwave frequencies, is the gyromagnetic ratio and 0H0is the applied static magnetic eld. 3. Samples and Experimental Setup The YIG lms were grown by liquid phase epitaxy (LPE) on (111)-oriented Gadolinium Gallium Garnet (GGG) substrates. Due to the dierence in the lattice parameters of Czochralski-grown GGG (a GGG = 12:383A) and pure YIG (a YIG = 12:376A) [18] the lms exhibit a room temperature lattice mist
a?=aGGGaYIGwhich results in strained epitaxial lms. This strain is one of the main factors dening the uniaxial anisotropy elds 0Hu?and0Huk. In the case of LPE growth of garnet lms incorporation of lead ions from the PbO solvent plays an important role in adjusting the lm mist [19]. Therefore, the mist essentially depends on growth parameters (growth temperature, growth rate, etc.). For this reason, the growth rate was varied to obtain (Y1xPbx)3Fe5O12lms (0:005x0:015 [20]) with reduced lattice mists. In Tab. 1 important material parameters of the samples are shown. It can be seen that the lattice mist increases with decreasing growth rate. The lm thickness dwas measured by a prism coupler technique, and the YIG/GGG lattice mist values were determined by X-ray diraction. Then, the samples were cut in sizes of 3 3 mm2for microwave studies.4 For measuring the exchange stiness, a waveguide microwave resonance setup was used. An electromagnet is used to apply external elds up to 0Hdc<1650 mT0:1 mT, where a low-amplitude ( 0Hac= 0:1 mT) rf-frequency ( f1 MHz) modulation eld is used by a lock-in amplier as reference signal. The scan of a Lorentzian absorption peak with the modulation eld results in an output voltage which has the form of the derivative of the original signal. A microwave eld with a power of 10 dBm is applied in a wide frequency range from 5 GHz to 40 GHz with a rotatable coplanar waveguide (CPW) so that the angle between the eld and the sample surface can be varied from 0 to 360. For the in-plane measurements the external magnetic eld is applied along the edges of the sample which is positioned in the middle of the CPW. In all measurements the frequency is xed and the eld is swept. 4. Determination of exchange stiness 4.1. Method of Schreiber and Frait A typical dependence of the lock-in signal on the applied static eld from the out-of-plane measurements is shown in Fig. 1(a). The ferromagnetic resonance ( n= 0) can be found at the highest eld values, whereas the thickness modes are located at lower eld values. In any case mainly resonances with an even mode number are observed. This eect can be understood with the assumption of \perfect pinning", since in this case, only the even modes absorb energy from the homogeneous antenna eld [21]. The experimental observation of odd PSSWs can be caused by small microwave inhomogeneities across the lm thickness. In the classical approach of Schreiber and Frait the exchange stiness is determined in the out-of-plane conguration using Eq. (2). Here, the anisotropy elds and the saturation magnetization are absorbed in the eective magnetization Me;?. Then, the resonance eld for a certain frequency is dened by: 0Hres 0(n) =0Me;?+!? j jD2 d2n2, where (3) Me;?=MSHu?+4 3HcHuk (4) In a plot where the resonance eld is drawn over the square of the mode number n2, the exchange stiness can be extracted with the slope of a linear function, where the y-intercept delivers information about the eective magnetization. The presence of resonances with odd mode numbers introduces some ambiguity regarding the identication of the modes. However, the mode intensity together with the n2- dependence of the resonance eld shift enables a consistent identication, as can be seen in Fig 1(b). Here, the resonance elds of sample E2 are shown for dierent frequencies. The slopes of the linear functions are the same for all measurements and they-intercepts are dierent due to the use of dierent excitation frequencies. In the performed measurements no deviations from the linear functions were detected which5 850 860 870 880Lock-in signal (arb. units)30 20 10 0 -10 -20 -30 -40odd mode numbereven mode numberFMR x5030 GHz 25 GHz 20 GHz 15 GHz 10 GHz 7.5 GHz Square of mode number ² n20 0 40 60 80 100 120 140Resonance field µ (0.1 T)0 0Hresa) b) 456789101112 External field µ (mT)0 0H Figure 1. (a) Example spectrum for sample E2 in the out-of-plane conguration at 20 GHz. The spectrum on the left hand side of the dashed lines is magnied by a factor of 50. (b) Line positions found in the spectrum of sample E2 in out-of-plane conguration. It is obvious that the line positions follow a linear function in dependence ofn2. SinceDis a shared tting parameter, it is the same for every frequency. The dashed lines show the positions of the resonances with even mode numbers. Table 2. Results for the YIG samples with dierent thicknesses. The shown errors are the statistical tting errors. The values in the column with the out-of-planey measurement are obtained using the original method of Schreiber and Frait. The values in the column with the out-of-planemeasurements are obtained based on the dierence between the resonance eld of higher modes and the ferromagnetic resonance eld. D(1017T
m2) or 109(erg/G
cm) Sample out-of-planeyout-of-planein-plane Schreiber and Frait E1 5 :330:09 5 :180:01 5:290:04 E2 5 :320:09 5 :340:02 5:300:02 E3 5 :290:05 5 :310:02 5:400:02 would occur for small ndue in uence of the surface anisotropy. Thus, the assumption of perfect pinning is justied. From the slope the exchange stiness values from all samples are extracted, which are presented in the left column of Tab. 2. The average value for samples E1-E3 is D= (5:320:07)1017T
m2. With the shown method, the slope and the eective magnetization are optimized together during the tting process, i.e. the residuum is minimized. The optimization of both parameters at the same time leads to a mutual in uence of the parameters. This eect is clearly visible in the size of the error bars, if compared to the modied method which is presented in the next section.6 30 GHz 25 GHz 20 GHz 15 GHz 9.29 GHz 5 GHz 20 0 40 60 80 100 Square of mode number ² nExchange field µ (mT)0 exH70 60 50 40 30 20 10 0 Figure 2. The plotted exchange elds of sample E3 are obtained building the dierence between the resonance elds of the higher modes ( n6= 0) and the ferromagnetic resonance ( n= 0) eld. The exchange elds only depend on the square of the mode number and not on the frequency. 4.2. Modied method of Schreiber and Frait As shown before, the method of Schreiber and Frait requires several parameters to be taken into account in order to obtain the exchange stiness. Here, a method which is completely independent on assumptions for the anisotropy elds and the saturation magnetization is presented. For this the ferromagnetic resonance eld 0Hres 0(0) is subtracted from the resonance elds of the higher modes 0Hres 0(n6= 0) in order to determine the exchange eld 0Hexof the thickness modes. Since the resonance eld of the ferromagnetic resonance contains all information about the anisotropy elds and MS, as can be seen in Eq. (4), the exchange eld only depends on D: 0Hex=0Hres 0(n)0Hres 0(0) =D2 d2n2(5) In Fig. 2 the exchange elds are shown as a function of n2. One can see that the measured exchange elds for dierent frequencies collapse in a point for each mode number. This indicates that the exchange elds are independent on any external parameter. Furthermore, all collapsed data points lie on a linear function with Hex(0) = 0. This data can now be analyzed using a simple linear t with no oset, i.e. only one tting parameter is used. Thus, any mutual in uence of parameters is avoided which is the reason for a signicantly reduced statistical error. The results are shown in the middle column of Tab. 2. All values are in the same range as obtained with the former method. However, it is visible that the exchange stiness of sample E1 is signicantly smaller that the others. This dierence can be understood by a larger saturation magnetization for sample E1 than for samples E2 and E3 as shown below. In comparison to the method of Schreiber and Frait, the error is decreased by a factor of up to 9 due to the avoided in uence of the eective magnetization during the data evaluation. This allows for the7 even mode number FMR x130 odd mode number 620.0 622.5 625.0 627.5 External field µ (mT)0 0HLock-in signal (arb. units)40 30 20 10 0 -10 -20a) b) Square of mode number ² n5 0 10 15 20 25 30 350510152025 Exchange field µ (mT)0 exH Figure 3. (a) Example spectrum for the YIG sample E2 in in-plane conguration at 20 GHz. The rst two resonances overlap in such a way that a multiple resonance t had to be used to extract the linewidth and position of both resonances. The spectrum on the left hand side of the dashed lines is magnied by a factor of 130. (b) The exchange elds of the sample E3 follow a linear function of the square of the mode number. The slope of the function is proportional to the exchange constant. Each point in the graph stands for the exchange eld of one resonance. Even modes are marked with dashed lines. identication of the exchange stiness with a high accuracy. 4.3. Method for in-plane measurements Classically the method of Schreiber and Frait is used for the determination of the exchange stiness in out-of-plane conguration. Here, the method is adapted for the use in in-plane conguration. A sample spectrum of the in-plane measurements is shown in Fig. 3(a). It is slightly modied in comparison to the out-of-plane spectrum. The resonances are shifted to smaller eld values due to decreased demagnetizing eects. In the in-plane case, the former methods cannot be used for data evaluation since !kis not linearly dependent on the static eld in Eq. (1). However, the former procedure can be applied to the pure exchange eld of the PSSWs. For this we propose the following steps. Firstly, Eq. (1) must be rewritten in a way which is convenient for the tting process: !k=j j0q (H0+Hex) H0+Hex+Me;k
: (6) Here, the dierent eld contributions, including the saturation magnetization, are summarized in Me;k=MSHu?Hc. Now the eective magnetization is obtained by tting the n= 0-mode (FMR mode). Secondly, the same equation is used to obtain the exchange eld 0Hexof the higher PSSW modes ( n6= 0). For this, the obtained value of Me;kis used as a constant during8 Table 3. Exchange constants of the YIG samples are shown for dierent methods. The proposed modied method of Schreiber and Frait (middle column) with excluded in uence of anisotropies and MSgives the best agreement for dierent samples. A(1012J/m) or (107erg/cm) Sample out-of-planeyout-of-planein-plane Schreiber and Frait E1 3 :640:40 3 :650:38 3:710:39 E2 3 :640:43 3 :650:38 3:630:38 E3 3 :760:44 3 :660:37 3:730:40 the data evaluation process. The resulting exchange elds 0Hexof the PSSW modes depend on the square of a mode number which is unknown at rst. As in the case of the out-of-plane measurements there is an ambiguity regarding the identication of the mode number nfor each observed mode. The identication procedure of the mode numbers is shown next. Thirdly, the exchange elds of the modes are varied manually over the presumed mode numbers (see Fig. 3(b)). As rst indicator, the peak height of the resonance can be used to determine whether a mode is an even mode or not. To proove the mode numbering a graphical feedback can be obtained by plotting 0Hex(n) overn2, where a wrong mode numbering would be directly visible. If the exchange eld follows a linear function, as shown in Fig. 3(b), the exchange stiness is given by the slope of this function. For all three samples the values of the exchange stinesses are shown in the right column of Tab. 2. They are in the same range with the other methods which supports the value of this method. However, the data evaluation is much more complicated and the systematic uncertainties are increased in comparison to the other methods. 5. Determination of the exchange constant For the determination of the exchange constant A=DM S=2 of our YIG samples, the saturation magnetization MSmust be known. Vibrating sample magnetometry (VSM) was used to dene MSand values of 141 kA/m, 136 kA/m and 137 kA/m for samples E1, E2 and E3, respectively, are found with an accuracy of 10 %. The large error is due to the error in volume determination of the YIG lms. The results obtained for Ausing dierent methods of the denition of Dare shown in Tab. 3. One can see that all values agree within the error bars. However, only the proposed out-of-plane method gives practically the same value for all samples. This is due to the increased accuracy in the denition of the exchange stiness. The exchange constant of the YIG lms is determined to be (3 :650:38) pJ/m, which is the average value of the out-of- planemethod. The presented result is in good agreement with the values obtained by other groups [22]. Finally, one can state that the YIG lms have the same material9 characteristics independent on the lattice mismatch and thickness, which speaks for the high quality of the YIG lms (see Tab. 1). 6. Conclusion Dierent methods were developed and compared to estimate the exchange stiness D from the microwave absorption spectra. Firstly, the method of Schreiber and Frait [1] was used to estimate Dof the out-of-plane magnetized sample. The method was shown to be in uenced by anisotropy elds and the saturation magnetization choice. Therefore, the exchange stinesses Dwere accompanied with appreciable errors which resulted in dierent values for the exchange constant A. This problem was solved with the proposed method by avoiding additional t parameters including anisotropy elds by preliminary extraction of the pure exchange eld contributions [see Eq. (5)]. The method was demonstrated to give more accurate results which is the reason for the similar values of the exchange constant A. The modied method is recommended for determination of the exchange stiness in general. Finally, the former method of was adapted for the in-plane conguration. The in-plane estimates of the exchange stiness Dwere found to agree well with those obtained in the out-of-plane conguration. However, because of the nonlinear dependence of the PSSW mode frequency versus n2, an evaluation of the in-plane measurement data was more complicated and resulted in a similar spread of the exchange constant Aas in the original method of Schreiber and Frait [1]. Finally, it was also proven that the exchange constant in thin YIG lms remain nearly independent of the YIG/GGG lattice mist and a value of A= (3:650:38) pJ/m was extracted. 7. Acknowledgements We thank the Nano Structuring Center in Kaiserslautern for technical support. Part of this work was supported by NSF-CAREER Grant #0952929 and by EU-FET grant InSpin 612759. The measurements were performed during the annual MINT Summer Internship Program of the University of Alabama. [1] Schreiber F and Frait Z 1996 Phys. Rev. B 546473 [2] Khitun A, Bao M and Wang K L 2008 IEEE Transactions on Magnetism 442141 [3] Schneider T, Serga A A, Leven B, Hillebrands B, Stamps R L and Kostylev M P 2008 Appl. Phys. Lett.92022505 [4] Klingler S, Pirro P, Br acher T, Leven B, Hillebrands B, Chumak A V 2014 Design of a spin-wave majority gate employing mode selection arXiv:1408.3235 [cond-mat.mtrl-sci] [5] Chumak A V, Vasyuchka V I, Serga A A, Kostylev M P, Tiberkich V S and Hillebrands B 2012 Phys. Rev. Lett. 108257207 [6] Chumak A V, Serga A A and Hillebrands B 2014 Nat. Commun. 54700 [7] Jung eisch M B, Chumak A V, Kehlberger A, Lauer V, Kim D H, Onbasli M C, Ross C A, Kl aui M and Hillebrands B 2013 Thickness and power dependence of the spin-pumping eect in Y3Fe5O12/Pt heterostructures measured by the inverse spin Hall eect arXiv:1308.3787 [cond- mat.mes-hall].10 [8] Pirro P, Br acher T, Chumak A V, L agel B, Dubs C, Surzhenko O, G ornert P, Leven B and Hillebrands B 2014 Appl. Phys. Lett. 104012402 [9] d'Allivy Kelly O, Anane A, Bernard R, Ben Youssef J, Hahn C, Molpeceres A H, Carretero C, Jacquet E, Deranlot C, Bortolotti P, Lebourgeois R, Mage J C, de Loubens G, Klein O, Cros V and Fert A 2013 Appl. Phys. Lett. 103082408 [10] Liu T, Chang H, Vlaminck V, Sun Y, Kabatek M, Homann A, Deng L and Wu M 2014 J. Appl. Phys. 11517A501 [11] Sun Y, Chang H, Kabatek M, Song Y-Y, Wang Z, Jantz M, Schneider W, Wu M, Montoya E, Kardasz B, Heinrich B, te Velthuis S G E, Schultheiss H and Homann A 2013 Phys. Rev. Lett. 111106601 [12] Hahn C, de Loubens G, Klein O, Viret M, Naletov V V and Ben Youssef J 2013 Phys. Rev. B 87 174417 [13] Hahn C, Naletov V V, de Loubens G, Klein O, d'Allivy Kelly O, Anane A, Bernard R, Jacquet E, Bortolotti P, Cros V, Prieto J L and Mu~ noz M 2014 Appl. Phys. Lett. 104152410 [14] Ciubotaru F, Chumak A V, Grigoryeva N Y, Serga A A and Hillebrands B 2012 J. Phys. D: Appl. Phys. 45255002 [15] Kajiwara Y, Harii K, Takahashi S, Ohe J, Uchida K, Mizuguchi M, Umezawa H, Kawai H, Ando K, Takanashi K, Maekawa S and Saitoh E 2010 Nature 464262 [16] Landau L and Lifshitz E 1935 Phys. Z. Sowjetunion 16914 [17] Bobkov V B, Zavuslyak I V and Romanyuk V F 2003 J. Commun. Technol. & El. 48196 [18] Wang H, Du C, Hammel C and Yang F 2013 Strain-tunable magnetocrystalline anisotropy in epitaxial Y 3Fe5O12thin lms arXiv:1311.0238 [cond-mat.mtrl-sci] [19] Hergt R, Pfeier H, G ornert P, Wendt M, Keszei B and Vandlik J 1987 Phys. Stat. Sol. (a) 104 769 [20] Pb contents in formular units were estimated from the lattice mist by comparison with literature data [23]. [21] Kittel 1958 Phys. Rev. 1101295 [22] Hoekstra B. 1978 Spin wave resonsance studies of inhomogeneous La,Ga:YIG epitaxial lms , Dissertation [23] Sure S 1995, Herstellung magnetischer Granatlme durch Fl ussigphasen-Epitaxie f ur Anwendungen in der Integrierten Optik , Dissertation

2014-08-25

Measurements of the exchange stiffness $D$ and the exchange constant $A$ of Yttrium Iron Garnet (YIG) films are presented. The YIG films with thicknesses from 0.9 $\mu$m to 2.6 $\mu$m were investigated with a microwave setup in a wide frequency range from 5 to 40 GHz. The measurements were performed when the external static magnetic field was applied in-plane and out-of-plane. The method of Schreiber and Frait, based on the analysis of the perpendicular standing spin wave (PSSW) mode frequency dependence on the applied out-of-plane magnetic field, was used to obtain the exchange stiffness $D$. This method was modified to avoid the influence of internal magnetic fields during the determination of the exchange stiffness. Furthermore, the method was adapted for in-plane measurements as well. The results obtained using all methods are compared and values of $D$ between $(5.18\pm0.01) \cdot 10^{-17}$T$\cdot$m$^2$ and $(5.34\pm0.02) \cdot 10^{-17}$ T$\cdot$m$^2$ were obtained for different thicknesses. From this the exchange constant was calculated to be $A=(3.65 \pm 0.38)~$pJ/m.

Measurements of the exchange stiffness of YIG films by microwave resonance techniques

1408.5772v1

Electrical control of spin mixing conductance in a Y 3Fe5O12/Platinum bilayer Ledong Wang, Zhijian Lu, Jianshu Xue, Peng Shi, Yufeng Tian, Yanxue Chen, Shishen Yan,and Lihui Baiy School of Physics, State Key Laboratory of Crystal Materials, Shandong University, 27 Shandanan Road, Jinan, 250100 China Michael Harder Department of Physics, Kwantlen Polytechnic University, 12666 72 Avenue, Surrey, BC V3W 2M8 Canada (Dated: 2019/03/15/00:38:20) We report a tunable spin mixing conductance, up to 22%, in a Y 3Fe5O12/Platinum (YIG/Pt) bilayer. This control is achieved by applying a gate voltage with an ionic gate technique, which exhibits a gate-dependent ferromagnetic resonance line width. Furthermore, we observed a gate- dependent spin pumping and spin Hall angle in the Pt layer, which is also tunable up to 13.6%. This work experimentally demonstrates spin current control through spin pumping and a gate voltage in a YIG/Pt bilayer, demonstrating the crucial role of the interfacial charge density for the spin transport properties in magnetic insulator/heavy metal bilayers. I. INTRODUCTION Spin currents in Y 3Fe5O12/Platinum (YIG/Pt) bilay- ers have attracted much attention in the past decade due to the unique spin current transport properties in mag- netic insulators and spin charge conversion in heavy met- als [1{14]. Mechanisms including spin pumping at the bi- layer interface [15], spin diusion [16] and the inverse spin Hall eect in heavy metals [17], have been established to interpret the generation, transfer and conversion of spin current [18]. This understanding has advanced the devel- opment of spintronic devices, as evidenced by, for exam- ple, nano oscillators. However spintronic devices based on spin pumping require additional control beyond the basic generation and detection of spin current [19]. In this regard improvement of spin current transport at the interface of the YIG/Pt bilayer is the kernel toward the application of spin current due to spin pumping. The eciency with which the spin current crosses a bilayer interface is characterized by the spin mixing con- ductance,g"#, which is a constant for a given sample and is usually measured by comparing the ferromagnetic res- onance (FMR) line width in a bilayer device to that of a bare YIG layer. Some attempts have been made to change this interfacial spin transport. For example, a thin NiO [20, 21] or CrO [22] layer inserted at the inter- face has been reported to enhance or suppress the spin current, respectively. Additionally, Pt alloyed with Al or Au was demonstrated to enhance the spin transfer torque at the interface [23]. Importantly, such previous works hint that the conductivity in the Pt layer may change the boundary at the interface, hence changing the spin transport properties as well as the spin mixing conduc- tance. To control the spin transport properties of bilayer sam- ples a gate voltage can be used to control interfacial shishenyan@sdu.edu.cn ylhbai@sdu.edu.cnboundary conditions. Interestingly gate voltage tech- niques, originally developed to control the carrier den- sity in semiconductors, have recently been applied to thin metal lms [24]. In this context the addition of an ionic gate has been shown to enlarge the eld eect in metallic lms due to the huge number of ions accumulated at the interface under certain bias voltage conditions [25]. As a result it has been reported that the carrier density and anomalous Hall eect in Pt may be modulated [26, 27]. (a) (b) (c) VG xyz GateGate JsJs YIGPt YIG Pt/s32/s33/s32/s34 /s32/s33/s32/s32 /s35/s32/s33/s32/s34Rxy (Ω) /s35/s36/s32/s36 µ0H (T) VG = 2 V VG = 0 V VG = -2 V FIG. 1. (a) Spin pumping measurement setup using a YIG/Pt bilayer with a gate voltage VGapplied using an ionic gate tech- nique. An external magnetic eld Hwas applied along the y-axis perpendicular to the measurement direction ( x-axis). (b) A Hall measurement in a 3-nm-thick Pt lm indicates that the carrier density was modulated by the gate voltage. (c) Sketch of the carrier density change due to the gate volt- age, which changes the boundary conditions at the YIG/Pt interface, in uencing the spin current.arXiv:1903.05865v1 [cond-mat.mes-hall] 14 Mar 20192 This brings about the interesting possibility to develop additional control techniques to manipulate spin trans- port at the YIG/Pt interface [28]. In this work we applied a gate voltage to a YIG/Pt bilayer using an ionic gate technique to modulate the charge accumulation at the bilayer interface. We exper- imentally observed that the FMR line width is both en- hanced and suppressed depending on the polarity of the gate voltage. By compare the FMR line width in the bi- layer to that of the bare YIG thin lm, we found a gate tunable spin mixing conductance in the YIG/Pt bilayer. Additionally, the observation of a shift in the FMR reso- nance eld towards both high and low elds, depending on the polarity of the gate voltage, allows us to rule out the Joule heating eect. This shift in the FMR resonance eld indicates that the eective eld of the magnetization was changed by the gate voltage which may be induced by charge accumulation at the interface. Using the Landau- Lifshitz-Gilbert equation and spin pumping theory, we evaluated the spin current and the spin Hall eect in Pt, nding a strong gate voltage dependence. Thus we have experimentally demonstrated control of the spin current transport at a bilayer interface, which will be useful for understanding spin transfer at the interfaces of magnetic insulator and heavy metal bilayers. II. EXPERIMENTAL METHODS To fabricate the YIG/Pt bilayer a 20-nm thick YIG layer was rst deposited on a GGG substrate using pulsed laser deposition. The YIG had a saturation magnetiza- tion of0Ms= 0.175 T and a Gilbert damping of Y IG = 0.00049. A 2.5-nm-thick Pt layer was then deposited on top of the YIG using magneto sputtering in which the base pressure and the sputtering pressure were 2 105 Pa and 0.68 Pa, respectively. The Gilbert damping of the bilayer was Y IG=Pt = 0.00157 which is 3 times larger than that of the bare YIG lm. The lateral dimensions of the bilayer were 5 mm 2.5 mm, and a Hall bar was fabricated using lithography techniques for reference and Hall measurements. To measure the spin current signal due to spin pump- ing the YIG/Pt bilayer was driven to FMR by microwaves applied through a coplanar waveguide beneath the sam- ple. The microwave output power was 158 mW and the modulation frequency of the microwave power (used for lock-in measurements) was 8.33 kHz. An external mag- netic eld Hwas applied in-plane and perpendicular to the Pt strip to enhance the spin pumping signal. The spin pumping voltage was measured along the x-axis of the Pt layer using a lock-in technique while sweeping the magnetic eld Hat room temperature. An ionic gate, composed of a composite solid elec- trolyte, was placed on top of the Pt layer and a gate voltageVGwas applied between a contact (labelled as Gate) on top of the solid electrolyte, and the Pt layer. The composite solid electrolyte used for gating was pre-pared using 81 wt% of acrylic resin, 14.6 wt% of succi- nonitrile, and 4.4 wt% of cesium perchlorate. As shown in Fig. 1 (b) the sheet carrier density in the 3-nm-thick Pt layer was found to be 1.8 1017cm2when no gate voltage was applied, VG= 0 V, assuming the single band relationRH= 1=ne(eis the electron charge and nis the carrier density). This is comparable to the reported results for Pt [26]. However, as indicated by the squares and triangles for VG= 2 V and -2 V respectively, the Hall resistance, and hence the carrier density, changes signi- cantly when a gate voltage is applied. For a 3-nm-thick Pt thin lm it is expected that the boundary at the in- terface of the YIG and Pt layer is strongly dependent on the carrier density as well as the spin orbit interaction in the Pt layer as shown in Fig. 1 (c). Thus the change in boundary conditions due to charge accumulation at the YIG/Pt interface will change the interfacial spin trans- port properties, which was experimentally observed in the FMR measurement. In our work we dened a pos- itive gate voltage when the carrier density was reduced and vice versa. As a reference the FMR absorption in a bare YIG lm was also measured as a function of the gate voltage[29]. In this case the voltage induced FMR changes were found to be small and ignorable. III. RESULTS AND DISCUSSION A. VG-dependent spin mixing conductance Figure 2 (a) shows the spin pumping voltage measured in the YIG/Pt bilayer for a variable gate voltage VG, with the FMR absorption signal in the YIG bare layer plotted for reference. Here the FMR spectra has been shifted by the resonance eld H0and normalized to the max- imum signal amplitude, in order to highlight the FMR line width change due to the gate voltage. The signi- cant broadening of the YIG/Pt line width, as compared to the FMR line width in the bare YIG layer, has been experimentally observed and theoretically studied pre- viously [15, 30]. Here we also observe that the bilayer line width is dependent on the applied gate voltage VG. Figure 2 (b) shows the FMR line width
H(half-width- half-maximum) in both the YIG and YIG/Pt samples. The Gilbert damping, determined from the gradient of the line width as a function of the microwave frequency, is enhanced in the YIG/Pt bilayer compared to that in YIG. Clearly the gate voltage applied to the YIG/Pt bi- layer suppresses and enhances the Gilbert damping of the bilayer. The inhomogeneous broadening
H0is around 0.1 mT which is one order smaller than the line width
Hat 6.8 GHz in both the YIG/Pt and YIG samples. Furthermore
H0was barely aected by the gate volt- age. Therefore it is a good approximation to subtract the Gilbert damping directly using the line with
Hat 6.8 GHz. Such a line width change as a function of VG is summarized in Fig. 2 (c), where the dashed horizontal line indicates the FMR line width of the bare YIG. In a3 /s32/s33/s34 /s35/s33/s34 /s36/s33/s34µ0ΔH (mT) /s37/s38 /s37/s35 /s34 /s35 /s38 VG (V)YIG/Pt YIGη = − 0.038ω/2π = 6.8 GHz(a) /s35/s33/s39 /s35/s33/s34 /s36/s33/s39/s32↑↓ (×1018 m-2) /s37/s38 /s37/s35 /s34 /s35 /s38 VG (V)/s38 /s35 /s34µ0ΔH (mT) /s36/s34/s39/s34 ω/2π (GHz)VG -4 V 0 V 4 V YIG(b) (c) (d)/s36 /s34Normalized VSP /s37/s38/s34/s38 µ0(H-H0) (mT)-4 V 0 V 4 Vω/2π= 6.8 GHz VG YIG FIG. 2. Spin pumping voltage in the YIG/Pt bilayer, normal- ized by the maximum amplitude to highlight the line width changes induced by the gate voltage VG. The FMR absorption signal (squares) from a bare YIG layer is used as a reference. (b) The FMR line width as a function of microwave frequency depends on the gate voltage. (c) The FMR line width
H is plotted as a function of the gate voltage VGand compared to that in bare YIG. The solid line indicates the tting results to Eq. (1). Here the microwave frequency is 6.8 GHz. (d) Spin mixing conductance is subtracted and predicted based on Eqs. (2) and (1). rst order approximation we assume that the line width
His in uenced by the gate voltage VG, according to
H=
H0+! Y IG=Pt (1 +VG): (1) Here
H0is the frequency independent inhomoge- neous broadening, the gyromagnetic ratio = 2028 GHz/T, the Land e factor g= 2,Bis the Bohr mag- neton,!is microwave angular frequency, Y IG=Pt is the Gilbert damping of the YIG/Pt bilayer and is a pro- portionality factor that characterizes the in uence of the gate voltage on the Gilbert damping. The units of  areV1. Here we nd = -0.038V1by tting theline width of the YIG/Pt bilayer to Eq. (1). Such a VG-dependent line width was also observed for various dierent frequencies (not shown here). The spin mixing conductance g"#was experimentally evaluated by com- paring the FMR line width
HY IG=Pt in the YIG/Pt bilayer to the FMR line width
HY IG in the bare YIG layer [30], g"#=4Ms tY IG gB!(
HY IG=Pt
HY IG);(2) whereMsis the saturation magnetization and tY IG is the thickness of the YIG layer. By comparing the line width of FMR in the YIG/Pt bilayer to that in the YIG thin lm according to Eq. (2), one can evaluate the spin mixing conductance g"#as summarized in Fig. 2 (d). We nd that g"#is roughly linearly-dependent on the gate voltage VGas indicated by the solid line, which can be predicted by Eq. (2) using = -0.038V1. This indicates that we have experimen- tally manipulated the YIG/Pt interfacial spin transport properties, which is a key issue concerning spin injection in the spintronics community. We note that here the spin mixing conductance is an eective value since spin back ow will play a role in the 2.5-nm-thick Pt [31, 32]. B. Spin current manipulated by VG Since the spin mixing conductance is VGtunable we may expect that the spin current due to spin pumping is also controlled by the gate voltage. As evidenced by the broadened line width, the gate voltage enhanced spin mixing conductance is the key source of additional FMR damping. This leads to a reduced FMR amplitude and thus a (1/)2decrease in the spin current pumped by FMR. Therefore even though the enhanced spin mixing conductance leads to a large spin current transparency at the bilayer interface, the observed spin current amplitude is reduced. Figure 3 (a) shows the spin pumping voltage at dierent values of VG. The amplitude of VSPis en- hanced by applying a positive voltage and suppressed by a negative voltage, with a total change up to 46.3% as shown in Fig. 3 (b). We also carefully examined the re- sistance change of the Pt layer as a function of VG, which shows a much smaller change ( 3.4%) as highlighted in Fig. 3 (c). Fig. 3 (d) displays the VG-dependent charge currentJC;SP which has been generated from the spin current through the inverse spin Hall eect of the Pt layer. The gate voltage dependence of the charge current can be evaluated by considering the Polder tensor for FMR in the YIG layer, the spin mixing conductance at the interface, the spin diusion into the Pt layer, and the inverse spin Hall eect in the Pt layer. For a given mi- crowave frequency and power the spin current produced by the FMR and injected into the Pt layer at the inter- face will have the form js;0/g"#=2 Y IG=Ptand therefore the charge current may be written as, JC;SP =kSHg"#=2 Y IG=Pt: (3)4 /s32 /s33 /s34 /s35VSP (µV) /s34/s36/s35 /s34/s37/s35 /s34/s32/s35 µ0H (mT)VG = 4 V VG = 0 V VG = -4 V /s32/s38/s36 /s33/s38/s36 /s34/s38/s36L /s39/s37/s39/s33/s35/s33/s37 VG (V)/s37/s35/s35 /s32/s35/s35 /s33/s35/s35R (Ω) /s39/s37/s39/s33/s35/s33/s37 VG (V) /s34/s38/s33 /s34/s38/s35 /s35/s38/s40θSH / θSH,0 /s39/s37/s35/s37 VG (V)/s40/s38/s35 /s41/s38/s36 /s36/s38/s35JC,SP (nA) /s39/s37 /s39/s33 /s35 /s33 /s37 VG (V) VG-independent θSH VG-dependent θSH ζ = 0.034(a) (b) (c) (d) FIG. 3. (a) Spin pumping voltages VSPat various VG, demon- strating a gate voltage dependence. The spin pumping voltage VSPand the resistance of the YIG/Pt bilayer are plotted in (b) and (c) respectively as a function of the gate voltage VG. (d) The charge current due to spin pumping JC;SP was com- pared to the predictions with and without the VG-dependent spin Hall angle. The inset displays the spin Hall angle as a function of the gate voltage. Here, the microwave frequency is 6.8 GHz. Herekis aVG-independent constant which depends on the microwave frequency and power, saturation mag- netization of the YIG and the spin current diusion properties of the Pt layer. Based on this analysis we can predict the VG-dependent charge current using Eq. (3) as shown by the dashed line in Fig. 3 (d), where kSH= 7:641024nA
m2. Although the predicted dashed line does have a VG-dependence (due to the VG- dependent spin mixing conductance), it does not match the experimental observation. This indicates that for a given spin Hall angle, although the spin mixing conduc- tance was enhanced, less spin current was produced than that required to generate the necessary charge current. Therefore, in order to compare with the observed JC;SP it is reasonable to assume that the spin Hall angle, SH in Eq. (3), is also VG-dependent, /s32 /s33 /s34 /s35VSP (µV) /s34/s36/s37 /s34/s32/s38 /s34/s32/s34 µ0H (mT)/s34 /s35 FMR absorption (a.u.)ω/2π = 6.8 GHzVG 4 V0 V-4 V YIGYIG/Pt /s34/s36/s39 /s34/s36/s35 /s34/s32/s39µ0H0 (mT) /s40/s36 /s40/s33 /s35 /s33 /s36 VG (V)YIG YIG/PtYIG/Pt(a) (b)FIG. 4. (a) The FMR resonance eld H0was shifted to the high eld and low eld sides for positive and negative gate voltages, respectively. The FMR absorption in bare YIG is displayed for comparison. (b) The FMR resonance eld H0 as a function of the gate voltage is summarized and compared to that in a bare YIG layer. Here, the microwave frequency was 6.8 GHz. SH=SH;0(1 +VG): (4) Heredenes the gate voltage dependence of the spin Hall angle and has units of V1andSH;0is the spin Hall angle for VG= 0 V. By combining Eqs. (3) and (4), we predict the charge current due to spin pumping as shown by the solid line in Fig. 3 (b), which matches well with the experimental data. In this calculation kSH;0= 7:581023nA
m2, which is the same as kSHused for Eq. (3) (dashed line), and = 0.034V1indicating that the spin Hall angle in the Pt layer is also tunable by the gate voltage. Therefore, we nd a spin Hall angle which can be tuned by up to 13.6% as shown by the inset in Fig. 3 (d). This tendency is opposite the behaviour observed for the VG-dependent spin mixing conductance. C. VG-dependent anisotropy eld An analogous electrically tunable anomalous Hall eect in Pt was previously reported using an ionic gate technique [26] and may involve similar underlying physics, related to a strongly charge density dependent spin orbit interaction. However in previous results the gate voltage in uence on the resistance and Hall eect was irreversible, whereas the spin mixing conductance control in our work can be repeated many times and may5 therefore be utilized to control spin current transport in future spintronic devices. We have performed spin pumping measurements in YIG(20 nm)/Pt( tnm) for a variety of Pt thickness t[29]. Compared to the large VG- tunable eect in the 2.5-nm-thick-Pt sample discussed above, the spin pumping signal was greatly reduced in a 4-nm-thick-Pt sample and barely observable in a 10-nm- thick-Pt sample. These results further indicate that the eective spin mixing conductance change is due to charge accumulation at the YIG/Pt interface, which is greatly enhanced in the thin Pt samples. The physics underlying the tunable spin mixing con- ductance and spin Hall angle is the change in carrier density due to the gate voltage, which will also induce a boundary change at the bilayer interface as we high- lighted in Fig. 1 (c). Interestingly this also appears to induce an anisotropy change in the YIG/Pt bilayer as a function of the gate voltage. In Fig. 4 (a) we have plot- ted the FMR signals in the bare YIG layer and in the YIG/Pt bilayer with dierent gate voltages while the ex- ternal magnetic eld was applied in the lm plane. The vertical dashed lines highlight the resonance positions. The large 6.8 mT shift to low elds which is observed in the bilayer device, compared to the bare YIG, is due to the in uence of the Pt layer on the boundary conditions of the bare YIG surface. When we apply a gate volt- age the FMR resonance eld H0shifts to higher elds by 2.06 mT for VG= 4.0 V and to lower elds by 2.07 mT forVG= -4.0 V. The VG-dependent H0is summa- rized in panel (b) and shows nearly a linear dependence. A more complex anisotropy change was observed in dif- ferent samples and the mechanism is still an open ques-tion for future work, but bears an interesting analogy to the electrical eld induced anisotropy changes in systems such as CoFeB/MgO [24] and CoO/Co [33, 34]. IV. CONCLUSION In summary, we report the modulation of the charge carrier density at the interface of a Y 3Fe5O12/Platinum (YIG/Pt) bilayer using an ionic gate technique. We elec- trically detected the ferromagnetic resonance (FMR) at variable gate voltages, observing three major features: (1) The line width of FMR is controlled by a gate volt- age, which indicates that the spin mixing conductance in the bilayer can be tuned; (2) The voltage amplitude of spin pumping is strongly dependent on the gate volt- age. To model the voltage change we found that the spin Hall angle in Pt should be a function of the gate voltage; (3) The anisotropy change indicates that the boundary conditions at the interface of the bilayer are changed by the gate voltage. Thus, we experimentally demonstrated control of spin current due to spin pumping in a YIG/Pt bilayer using a gate voltage. This observation may be used to better understand spin transfer at magnetic in- sulator/heavy metal interfaces. V. 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2019-03-14

We report a tunable spin mixing conductance, up to $\pm 22\%$, in a Y${}_{3}$Fe${}_{5}$O${}_{12}$/Platinum (YIG/Pt) bilayer.This control is achieved by applying a gate voltage with an ionic gate technique, which exhibits a gate-dependent ferromagnetic resonance line width. Furthermore, we observed a gate-dependent spin pumping and spin Hall angle in the Pt layer, which is also tunable up to $\pm$ 13.6\%. This work experimentally demonstrates spin current control through spin pumping and a gate voltage in a YIG/Pt bilayer, demonstrating the crucial role of the interfacial charge density for the spin transport properties in magnetic insulator/heavy metal bilayers.

Electrical control of spin mixing conductance in a Y$_3$Fe$_5$O$_{12}$/Platinum bilayer

1903.05865v1

Global biasing using a Hardware-based artificial Zeeman term in Spinwave Ising Machines Victor H. Gonz ´alez1, Artem Litvinenko1, Roman Khymyn1, and Johan ˚Akerman1 1Department of Physics, University of Gothenburg, Gothenburg, 41296, Sweden, victor.gonzalez@physics.gu.se and johan.akerman@physics.gu.se A spinwave Ising machine (SWIM) is a newly proposed type of time-multiplexed hardware solver for combinatorial optimization that employs feedback coupling and phase sensitive amplification to map an Ising Hamiltonian into phase-binarized propagating spin-wave RF pulses in an Yttrium-Iron-Garnet (YIG) film. In this work, we increase the mathematical complexity of the SWIM by adding a global Zeeman term to a 4-spin MAX-CUT Hamiltonian using a continuous external electrical signal with the same frequency as the spin pulses and phase locked with with one of the two possible states. We are able to induce ferromagnetic ordering in both directions of the spin states despite antiferromagnetic pairwise coupling. Embedding a planar antiferromagnetic spin system in a magnetic field has been proven to increase the complexity of the graph associated to its Hamiltonian and thus this straightforward implementation helps explore higher degrees of complexity in this evolving solver. Index Terms —combinatorial optimization problems, Ising machines, spinwaves, unconventional computing, physical computing, spinwaves. IN THE LANDSCAPE of physical computation schemes, Ising machines (IM) have attracted considerable attention and investment over the last decade, in both academic and industrial research [1]–[6], for their applicability to combi- natorial optimization problems, potential for scalability and progressive increase in mathematical complexity. In this work, we contribute to the latter as we implement a global bias to artificial spin states in a spinwave Ising machine (SWIM). A SWIM is a newly proposed [2] time-multiplexed hard- ware solver circuit for problems in the NP (nonpolynomial time) complexity class that employs feedback coupling and phase sensitive amplification to map an Ising Hamiltonian into spinwave (SW) RF pulses propagating in an Yttrium- Iron-Garnet (YIG) film. Spinwaves are suitable for Ising machines because of their GHz oscillation frequencies, which permits the development of multiphysical systems using cheap and efficient off-the-shelf microwave components for signal processing [7], [8] and amplification which results in a small circuitry footprint and high per-spin power efficiency. In this work, we use an external continuous microwave signal to im- plement a global biasing to propagating spinwave RF artificial spin states. Exploring the complexity limits of this circuit implementation thus holds significant interest for technical and commercial applications. An IM operates as an mapping of an objective function into the Ising Hamiltonian of a device composed of an array of N binarized physical units referred to as spins si=±1: H=−1 2NX i=1NX j=1Jijsisj−NX i=1hisi (1) The objective function associated with the NP problem to solve is encoded into the pairwise coupling Jijand external Zeeman bias hisuch that ground state of the system represents the solution to it. Combinatorial problems of practical use, such as the traveling salesman and knapsack with integer PSC1C2 PSA ωrefCoupling delay VAS1 � LNAC3S2 GGGYIGωbiasFig. 1. Zeeman-biased spinwave Ising machine. PSA and LNA stand for phase-sensitive and low noise amplifiers, respectively. The propagating RF pulses have frequency of 3.13 GHz. The sign of the coupling is controlled by the total phase accumulation in the coupling delay and the Zeeman field amplitude and sign is controlled by the amplitude and phase of the injected signal ωbias. weights, have been shown to be encodable if one can add constraints to the degrees of freedom of the system [9]. Adding a global bias (i.e. the hiis identical for all si) to an antiferromagnetic planar graph has been shown as a straight- forward way to increase its mathematical complexity [10]. Implementation of a global Zeeman term using software has also shown to improve the stability and performance of time- multiplexed IMs with frustated lattices [11]. Thus, exploring hardware-based alternative schemes can lend versatility to small scale low-power devices.arXiv:2308.07718v1 [cond-mat.mes-hall] 15 Aug 20230 π 02π πPulse phasePhase (rad) 3.11 3.15 ωrefωrefPower spectral density (dBm/Hz) Frequency (GHz)(a) (b) Time (ns)50 100 150 200 250 300-15 dBm h<0 h=0 h>0 -100-500-100-500h<0 h=0 h>0 (c) Pulse amplitude (V)h<0 h=0 h>024 18APSA (dB)Fig. 2. Influence of Zeeman term at different signs of magnetic field. (a) Measured potential landscape shift of the artificial spin states. The sign of his given by the phase between ωrefandωbias. The total phase sensitive amplification of the circuit APSA favors either 0 or πdepending on hand thus changes the magnetic ordering of the spins. (b) Power spectral densities (PSD) as a functions of frequency for different hsigns. We observe that the modulation harmonics can be used to identify the type of solution achieved, with both ferromagnetic states having the same spectra. (c) Time traces of the RF pulses colored with their respective instantaneous phase for different signs of h. The non-zero signals have an amplitude of -15 dBm. ωbias allows us to change the ordering and direction of the artificial spin state. The circuit of the SWIM with a hardware implemented Zeeman term is shown in Fig. 1. The SWIM’s construction of the artificial spin state relies in phase-sensitive amplification (PSA) of a reference signal ωref =3.13 GHz. The PSA binarizes the phase of the signal by amplifying only its in- phase ( ϕ= 0) and out-of-phase ( ϕ=π) components. The signal is then simultaneously injected into a YIG waveguide and a coaxial delay line using coupler C2. The electrical signal excites spin waves within waveguide that propagate at a much slower speed, allowing the coupling delay to re-inject a shifted signal using coupler C1. Switch 1 (S1) then pulses the signal and a low noise amplifier (LNA) compensates propagation losses as the cycle starts again. The resulting time-multiplexed pulse train is our artificial spin state, where each pulse is an artificial spin, with their electrical amplitudes representing the norm of siand their individual phases representing its sign. The coupling delay’s length is such that every pulse interferes (or couples) with the previous nearest neighbor. Phase shifter PS ensures that the coupling term Ji,i+1is antiferromagnetic and variable attenuator V A controls its strength. The Zeeman term is implemented with external signal ωbiasapplied to the propagating pulses after PSA. The role of ωbiasis to unbalance the potential landscape of the artificial spin states and favor one phase over the other. Fig.2(a) shows the changes in PSA amplitude ( APSA) for different signs of h. The effective sign of the Zeeman term is given by its relative phase with respect to the reference signal, with negative hbeing in-phase and positive out-of- phase. The effective magnitude of his given by the signals’ amplitude, -15 dBm in this case. The amplification imbalance allows us to change the phase sensitivity of the circuit and globally bias the state of the spins. The consequences of the bias are shown in Fig.2(b) and (c). In (b), we observe that the modulation harmonics present in the spectra depend on the spin state, with the biased signal containing a central carriercorresponding to ωrefthat is absent for the unbiased solutions. ωbias, whose frequency is the same as the reference signal’s, drives the oscillators as they all acquire the same phase (i.e. align their spin direction). We complement this picture in (c) using the time traces of the RF pulses colored with their respective instantaneous phase and associated graph. It is clear that despite antiferromagnetic coupling, we are able to induce a change in ordering in both possible directions using ωbias alone. Combining time trace and spectrum analyses, we have different tools to study the effectiveness of the biasing as well as develop differentiation and operation protocols that allow us to program more complex problems. Although this is very promising in terms of exploration of higher complex- ity schemes, unexpected states also appear for intermediate amplitudes of h. While it is expected that there will be a sudden breaking of up and down spin symmetry at h= 2, a gradual spin flipping occurs. Mixed spin states appear at intermediate values of ωbiasamplitude ( ≈-20 dBm), as seen in Fig.3(a), resembling a chain with magnetic domains. Although the system is indeed synchronized with ωbias, as shown in fig.3(b), these 3+1 states (three spins up and one down, or vice versa) are not a minimum energy solution of eq.1 and suggest an unintended increase of the degrees of freedom of the system. In Fig.3(c) we show the phase diagram for a ring-shaped 4 spin system under an external Zeeman field where one of the spin’s amplitude |S1|can be less than one, i.e. one spin is shorter. The shortening of S1results in a phase transition of the 4-spin system and appearance of the 3+1 states. In fact, from the time traces shown in Fig.3, we can directly observe that the spins have different electrical amplitudes and the odd one out has the smallest one. Since electrical amplitude and spin amplitude are proportional to each other, the emergence of the 3+1 states can give us information about the operation of our circuit and its limitations.02π πPulse phase-20 dBm Pulse amplitude (V) 50 300 100 150 200 250 Time (ns) AFM FM 0.60.81.0 0.0 0.5 1.0 1.5 2.03+1 Zeeman fieldSpin amplitde |s1|(a) (b) (c)3.11 3.15 ωrefPSD (dBm/Hz) Frequency (GHz)-100-500 h>0h<0h>0h<0Fig. 3. Mixed artificial spin states for ωbias with -20 dBm. (a) Colored time traces of the RF spins. The appearance of 3+1 states is evidence of additional degrees of freedom in the Hamiltonian of the system. (b) Phase diagram of a ring-shaped 4 spin Ising machine with variable spin amplitude. The 3+1 states are a consequence of phase-dependent amplification of the spins. (c) PSD for both solutions with 3+1 states, we observe the same synchronization as in fig.2(b), but the modulation peaks are characteristic to these solutions. Probing into the origin of this amplitude mismatch, we propose that the emergence of the 3+1 states depends on the non-linearity of the saturation of the LNA. Since we are injecting additional power to the system with the Zeeman signal, the LNA saturates and thus gain compression occurs beyond its linear range. Even if they are very close in the power transfer curve [12], its non-linearity results in spins of different signs being amplified differently with the minority spin shortening. An amplifier with a higher linear regime would mitigate this state degeneration. If compression gain is unavoidable, a stronger coupling between spins and a smoother saturation curve for the LNA would suppress 3+1 states. Dig- ital feedback using a field programmable gate array (FPGA) has been employed successfully in previous time-multiplexed Ising machines to improve amplitude stability [13] and can also be used instead to the delay line to modify pairwise or all- to-all coupling. These findings can be implemented in future circuit designs to improve the quality and complexity of the solutions with larger amounts of spins. It is worthwhile to mention that spectral analysis can help us understand the synchronization dynamics of the system as well as use for soution differentiation. As we see in fig.2(b) and fig.3(b), the peaks at ωrefshow that ωbias drives the Time (ns)350 1000.00.5 -0.50.00.5 -0.5 150 200 250 300Pulse amplitude (V) h>0h<0-15 dBm-23 dBm -23 dBmPSD (dBm/Hz) 3.10 3.14 ωref Frequency (GHz)-100-500 -100-500 -15 dBm (a) (b) 02π πPulse phaseFig. 4. Five spin states at two different ωbias amplitudes. (a) Time traces at -15 dBm and -20 dBm. We can observe that although the bias manages to stabilize the phase, the solution is not phase binarized. (b) Power spectral densities (PDS) of the solutions at the same amplitudes. We see that each solution is synchronized to ωbias has a characteristic spectrum. oscillators. Additionally, both biased solutions have their own characteristic spectrum. Thus, including this information in the digital feedback can allow us to design differentiation metrics and stopping conditions for relaxation and annealing protocols in bigger and more complex systems. Finally, we tried to recreate a stable globally biased 3+2 solution with five spins as it would appear in a spin ring. Evidently, a ring with an odd number of spins will not have a stable unbiased solution as the phase difference on each circulation period will never be zero. It was clear that for a large enough bias, we do see all spins parallel to each other, as shown for -15 dBm in fig.4(a). The phase transition mentioned before and shown in fig.3 could be an indication that a 3+2 state could be viable and could allow us to construct a magnetic system analog with two clearly defined domains. An amplitude of -23 dBm (fig.4(a)), is unable to produce such solution because the spins do not synchronize with the external field (we can observe that the highest peak in fig.4(b) is not atωref). Instead, the resulting state’s phase is not binarized and produces spins do not comply with the definition of eq.1. We believe that frustration is responsible for this phase slip and stable solutions are achievable with digital coupling and individual bias, both implementable with the aforementioned FPGA. We have shown the broad features of a globally biased artificial spin state space composed of RF pulses in a YIG waveguide. Despite antiferromagnetic coupling between near- est neighbors in a 4 spin ring, we are able to induce fer- romagnetic ordering by injecting an external signal of the same frequency to emulate the role of the Zeeman term in the Ising Hamiltonian. Intermediate values of this Zeeman signal introduce degeneracy in the amplification of the pulses and, consequently, 3+1 spin states. These effects be mitigatedby alternative amplification schemes whose implementation would guide future work in enabling all-to-all spin coupling for tackling non-trivial optimization tasks. The present work improves upon the emerging technology of commercially feasible IM hardware accelerators. The SWIM concept has a high potential for further scaling in terms of spin capacity, physical size and low-power low-footprint circuits for applied combinatorial optimization. REFERENCES [1] N. Mohseni, P. L. McMahon, and T. Byrnes, “Ising machines as hard- ware solvers of combinatorial optimization problems,” Nature Reviews Physics , vol. 4, no. 6, pp. 363–379, 2022. [2] A. Litvinenko, R. Khymyn, V . H. Gonz ´alez, A. A. Awad, V . Ty- berkevych, A. Slavin, and J. ˚Akerman, “A spinwave ising machine,” arXiv preprint arXiv:2209.04291 , 2022. [3] D. I. Albertsson, M. Zahedinejad, A. Houshang, R. Khymyn, J.˚Akerman, and A. Rusu, “Ultrafast ising machines using spin torque nano-oscillators,” Applied Physics Letters , vol. 118, no. 11, p. 112404, 2021. [4] A. Houshang, M. Zahedinejad, S. Muralidhar, R. Khymyn, M. Rajabali, H. Fulara, A. A. Awad, J. ˚Akerman, J. Checi ´nski, and M. Dvornik, “Phase-binarized spin hall nano-oscillator arrays: Towards spin hall ising machines,” Physical Review Applied , vol. 17, no. 1, p. 014003, 2022. [5] T. Honjo, T. Sonobe, K. Inaba, T. Inagaki, T. Ikuta, Y . Yamada, T. Kazama, K. Enbutsu, T. Umeki, R. Kasahara, K. ichi Kawarabayashi, and H. Takesue, “100,000-spin coherent ising machine,” Science Ad- vances , vol. 7, no. 40, p. eabh0952, 2021. [6] K. Tatsumura, M. Yamasaki, and H. Goto, “Scaling out ising machines using a multi-chip architecture for simulated bifurcation,” Nature Elec- tronics , vol. 4, no. 3, pp. 208–217, 2021. [7] A. Litvinenko, R. Khymyn, V . Tyberkevych, V . Tikhonov, A. Slavin, and S. Nikitov, “Tunable magnetoacoustic oscillator with low phase noise,” Physical Review Applied , vol. 15, no. 3, p. 034057, 2021. [8] A. Litvinenko, S. Grishin, Y . P. Sharaevskii, V . Tikhonov, and S. Niki- tov, “A chaotic magnetoacoustic oscillator with delay and bistability,” Technical Physics Letters , vol. 44, no. 3, pp. 263–266, 2018. [9] A. Lucas, “Ising formulations of many np problems,” Frontiers in physics , vol. 2, p. 5, 2014. [10] F. Barahona, “On the computational complexity of ising spin glass models,” Journal of Physics A: Mathematical and General , vol. 15, no. 10, p. 3241, oct 1982. [Online]. Available: https: //dx.doi.org/10.1088/0305-4470/15/10/028 [11] Y . Inui, M. D. S. H. Gunathilaka, S. Kako, T. Aonishi, and Y . Yamamoto, “Control of amplitude homogeneity in coherent ising machines with artificial zeeman terms,” Communications Physics , vol. 5, no. 1, p. 154, Jun 2022. [Online]. Available: https://doi.org/10.1038/s42005-022-00927-x [12] MiniCircuits. Coaxial low noise amplifier zx60-83ln12+. [Online]. Available: https://www.minicircuits.com/pdfs/ZX60-83LN12+.pdf [13] H. Takesue, K. Inaba, T. Inagaki, T. Ikuta, Y . Yamada, T. Honjo, T. Kazama, K. Enbutsu, T. Umeki, and R. Kasahara, “Simulating ising spins in external magnetic fields with a network of degenerate optical parametric oscillators,” Phys. Rev. Appl. , vol. 13, p. 054059, May 2020. [Online]. Available: https://link.aps.org/doi/10.1103/PhysRevApplied. 13.054059

2023-08-15

A spinwave Ising machine (SWIM) is a newly proposed type of time-multiplexed hardware solver for combinatorial optimization that employs feedback coupling and phase sensitive amplification to map an Ising Hamiltonian into phase-binarized propagating spin-wave RF pulses in an Yttrium-Iron-Garnet (YIG) film. In this work, we increase the mathematical complexity of the SWIM by adding a global Zeeman term to a 4-spin MAX-CUT Hamiltonian using a continuous external electrical signal with the same frequency as the spin pulses and phase locked with with one of the two possible states. We are able to induce ferromagnetic ordering in both directions of the spin states despite antiferromagnetic pairwise coupling. Embedding a planar antiferromagnetic spin system in a magnetic field has been proven to increase the complexity of the graph associated to its Hamiltonian and thus this straightforward implementation helps explore higher degrees of complexity in this evolving solver.

Global biasing using a Hardware-based artificial Zeeman term in Spinwave Ising Machines

2308.07718v1

Coupling function from bath density of states S. Nemati1, C. Henkel1andJ. Anders1;2 1University of Potsdam, Institut f ur Physik und Astronomie, 14476 Potsdam, Germany. 2Department of Physics and Astronomy, University of Exeter, Stocker Road, Exeter EX4 4QL, UK. PACS 03.65.Yz { Decoherence; open systems; quantum statistical methods PACS 63.20.Dj { Phonon states and bands, normal modes, and phonon dispersion PACS 67.57.Lm { Spin dynamics Abstract { Modelling of an open quantum system requires knowledge of parameters that specify how it couples to its environment. However, beyond relaxation rates, realistic parameters for specic environments and materials are rarely known. Here we present a method of inferring the coupling between a generic system and its bosonic (e.g., phononic) environment from the experimentally mea- surable density of states (DOS). With it we conrm that the DOS of the well-known Debye model for three-dimensional solids is physically equivalent to choosing an Ohmic bath. We further match a real phonon DOS to a series of Lorentzian coupling functions, allowing us to determine coupling parameters for gold, yttrium iron garnet (YIG) and iron as examples. The results illustrate how to obtain material-specic dynamical properties, such as memory kernels. The proposed method opens the door to more accurate modelling of relaxation dynamics, for example for phonon-dominated spin damping in magnetic materials. Introduction. { Quantum technologies face many chal- lenges, often arising due to the unavoidable coupling of any system to its environment. The prediction of their dynam- ics requires open quantum system methods that include such coupling eects, for example the Caldeira-Leggett model [1] and the spin-boson model [2]. These methods are success- fully employed in many physical contexts, e.g., quantum optics [3{5], condensed matter [6{11], quantum computa- tion [12{14], nuclear physics [15] and quantum chemistry [16]. For instance, modelling circuit quantum electrodynam- ics with the spin-boson model shows that the heat trans- port of a superconducting qubit within a hybrid environment changes signicantly, depending on the qubit-resonator and resonator-reservoir couplings [6]. In the mathematical treatment of an open quantum sys- tem, a coupling function C!is typically introduced that de- scribes how strongly the system interacts with bath degrees of freedom (DoF). Its functional form determines the temporal memory of the bath and whether the noise is coloured or not [1, 2, 17], critically aecting the system dynamics [8, 18, 19]. A large body of theoretical results exist for various toy mod- els that make specic assumptions on the coupling function C![1,2,20]. However, a major drawback is a somewhat lack- ing connection to system- or material-specic characteristics to which these methods could be applied: for a given DoF, in a given material, which coupling function C!should one choose to model its dynamics? An alternative approach is taken in the condensed matter literature, where open quantum systems are usually char- acterized by the density of states (DOS) of their environ-ment [21]. Measurement of, for example, the phonon DOS is well-established using dierent inelastic scattering tech- niques [22, 23]. Modes in the environment typically cou- ple to the system with a wave vector-dependent strength gk[2, 24, 25], which in many cases can be captured by a frequency-dependent g!. In this paper, we present a useful relation that trans- lates the coupling function C!of an open quantum system into an experimentally measurable DOS D!, and vice versa. While a similar relation has previously been reported for one- dimensional quantum spin impurities [26,27], the relation ob- tained here is valid for a generic system coupled to a bosonic bath, capturing dimensionality and anisotropy. It paves the way to parametrizing realistic coupling functions for a range of applications, for example, for spins in a magnetic material that experience damping through the coupling to the crystal lattice [17, 28] or for nitrogen vacancy centers, a solid-state analogue of trapped atoms, whose coherence lifetime in op- tical transitions is also limited by interaction with phonons [29, 30]. The link is explicitly established for a generic quan- tum system that couples locally to a bosonic environment. Extensions to other environments, such as fermionic environ- ments, will be possible using similar arguments. The paper is organised as follows: we rst introduce the two approaches involving D!andC!, respectively. Setting up the dynamics of the environment, we evaluate its memory kernel and establish the link between D!andC!, allowing for generalg!. In the second part of the paper, we choose a atgfor simplicity, and illustrate the application of the rela- tion with a few examples. We demonstrate that the widely p-1arXiv:2112.04001v2 [quant-ph] 15 Dec 2022S. Nemati et al. Fig. 1: Schematic picture of two equivalent approaches to mod- elling the open quantum systems. (a) Wave vector approach: Each bath frequency !includes several wave vectors fkgwhere each bath wave vector kcouples to the system with strength gk. (b) Frequency approach: Every bath frequency !couples to the system with a strength given by C!. used Debye approximation is equivalent to the well-known Ohmic coupling function. While this approximation suces at low frequencies, experimental DOS show peaks at higher frequencies, leading to non-trivial dissipation regimes. We parametrize two measured phonon DOS, those of gold and iron (see Supplementary Material (SM)), and one theoret- ically computed phonon DOS of yttrium iron garnet (YIG) and extract key parameters for the corresponding coupling functionsC!. These give direct insight into the impact of memory for any phonon-damped dynamics in these materi- als. Two approaches. { The Hamiltonian of a quantum sys- tem in contact with a bath is ^Htot=^HS+^HB+^HSB; (1) where the bath Hamiltonian ^HBand the system Hamiltonian ^HSmay contain the internal interactions among their own components. The system-bath interaction is assumed to be of product form, ^HSB=^S
^B; (2) where ^Sis a (Hermitian) system operator and ^Bis a bath operator, each with dscomponents. The form of the bath Hamiltonian ^HBand of the bath operator ^Bdepends on the context. We consider here a bosonic bath, i.e. an innite set of harmonic oscillators. In the literature, one can broadly distinguish two representations of the bath, working either in wave vector (WV) or frequency (F) space, as illustrated in Fig. 1. The wave vector approach is common in condensed matter physics [2, 21] where the bath Hamiltonian is expressed as asum over all possible modes k ^HWV B =X kh!k ^by k^bk+1 2 : (3) Here!=!kgives the dispersion relation of a normal mode with wave vector kand^bk(^by k) are bosonic annihi- lation (creation) operators of a mode excitation with com- mutation relations [^bk;^by k0] =kk0. Usually one consid- ers a three-dimensional ( 3D) structure with wave vectors k= (kx;ky;kz). For example, in a cubic 3D lattice with number of lattice sites N, lattice constant aand volume V=Na3, each component of kruns through the range 3p N1 2;:::; 0;:::;3p N1 2 2 3p Na. For largeNandV, and for any function f(!k)that only depends on the frequency !k, one can approximate sums over the wave vectors as 1 VX kf(!k)=Zd3k (2)3f(!k) =:Z d!D!f(!):(4) This equation denes D!as the DOS per unit volume of bath modes at frequency ![21]. For bosonic baths, we choose the standard interaction [2] where the bath operator ^Bis linear in the bosonic mode operators (single phonon processes), ^BWV=1p VX kk^bk+h.c.; (5) wherek=k hg2 k=(2!k)
1=2withkads-dimensional unit polarisation vector [1] and gkthe wave vector-dependent coupling, see Fig. 1. Eq. (2) may be generalized to the sit- uation that several system components ^Smare located at dierent positions Rm, and sum over interaction terms, i.e. ^HSB=P m^Sm
^B(Rm). The eld operators would then beR-dependent, i.e. ^BWV(R) =1p VP kk^bkeik
R+h.c.. For simplicity, we will concentrate in the following on just one system site and drop summation over magain. Another approach to setting up the bath Hamiltonian ^HB and the interaction ^HSBis based on a frequency expansion often employed in the open quantum systems literature [1,2]. In contrast to Eq. (3), here ^HBis written directly as a sum or integral over frequencies, ^HF B=1 2Z1 0d! ^P2 !+!2^X2 ! ; (6) where ^P!and ^X!are3D [in general, d-dimensional ( dD)] momentum and position operators, respectively, for the bath oscillator with frequency !. Their components obey [^X!;j;^P!0;l] = ihjl(!!0). In this approach, the bath operator in Eq. (2) is often chosen as [17] ^BF=Z1 0d!C!^X!; (7) where the coupling function C!(in general a dsdtensor) is weighting the system-bath coupling at frequency !. The p-2Coupling Function From Bath Density Of States system operators couple isotropically to the bath if C!CT != 1dsC2 !. The scalar coupling function C!is related to the bath spectral density J!, which alternatively quanties the eect of the environment on the system as J!/C2 !=![1,2]. The bath dynamics can be categorised [2] based on the low- ! exponent of the spectral density, J!/!s, into three dierent classes, called Ohmic ( s= 1), sub-Ohmic ( s<1), and super- Ohmic (s>1). The dierence between wave vector approach and fre- quency approach is that at a xed frequency !, there is in Eq. (7) just one bath operator ^X!that couples to the sys- tem, while according to Eq. (5), the interaction is distributed over several wave vector modes kwith weighting factors k, their number being set by the DOS D!(see Fig. 1). We now want to address the question of the connection between the DOS D!and the coupling function C!. To achieve this, we consider one relevant quantity in both ap- proaches and equate the corresponding formulas. In the fol- lowing, we choose the memory kernel Kwhich encodes the response of the bath to the system operator ^S. Note that the choice of ^Bin Eq. (5) restricts the discussion to the lin- ear response of the bath, as is reasonable for a bath that is thermodynamically large [1, 2]. Memory kernel in both approaches. { To nd an ex- plicit relation in the wave vector approach for the dynamics of the bath operator ^BWVin Eq. (5), the starting point is the equation of motion for ^bk, d^bk dt=i!k^bk+i hp Vy k
^S; (8) whose retarded solution contains two terms ^bk(t) = ^bk(0) ei!kt(9) +i hp Vy k
Zt 0dt0^S(t0) ei!k(tt0): Therefore, the time evolution of the bath operator can be written as ^BWV(t) = ^BWV induced (t) + ^BWV response (t). The rst term represents the internally evolving bath which is given by ^BWV induced (t) =1p VP k^bk(0)ei!tk+h.c., while ^BWV response (t)contains information about the system's past trajectory, ^BWV response (t) =Z1 0dt0KWV(tt0)^S(t0); (10) whereKWV(tt0)is the memory kernel (a tensor), KWV(tt0) =(tt0) VX kg2 kky ksin!k(tt0) !k:(11) Here, thekhave been expressed by the unit polarisation vectorsk[see after Eq. (5)] and (tt0)is the Heaviside function, which ensures that the bath responds only to the past state of the system, i.e. t0<t. For large volume V, the summation over kin Eq. (11) can be transformed into a frequency integration as in Eq. (4). Theprojection on polarization vectors, averaged over an isofre- quency surface , is taken into account by a ( dsds) posi- tive Hermitian matrix g2 !M!= ( )1R d g2 kky k, where the matrixM!is normalized to unit trace and g2 !is a scalar. With these notations, the memory tensor in the wave vector approach is KWV(tt0) = (tt0)Z1 0d!g2 !M!D!sin!(tt0) !:(12) Turning now to the frequency approach, the dynamics of the bath operator ^X!in Eq. (7) follows a driven oscillator equation d2^X! dt2+!2^X!=CT !^S: (13) Its exact solution is ^X!(t) = ^X!(0) cos!t+^P!(0) sin!t +Z1 1dt0G!(tt0)CT !^S(t0); (14) whereG!(tt0) = (tt0) sin!(tt0)=!is the retarded Green's function. Inserting this solution in Eq. (7) leads again to induced and response evolution parts given, respectively, by^BF induced (t) =R1 0d! ^X!(0) cos!t+^P!(0) sin!t and ^BF response (t) =Z1 0d!Z1 0dt0G!(tt0)C!CT !^S(t0):(15) Comparing with Eq. (10) one can identify the memory kernel tensor in the frequency approach as KF(tt0) =Z1 0d!C!CT !G!(tt0): (16) Coupling function C!versus DOS D!. { Since Eqs. (12) and (16) describe the same memory eects, we may set them equal, leading to C!CT !=g2 !M!D!: (17) This relation links the system-bath couplings in the two ap- proaches, i.e. the DOS D!is proportional to the Hermitian "square" of the coupling function C!. This is the rst result of the paper. The result (17) may be applied to any quantum system that interacts linearly with a bosonic bath. For instance, magnetic materials in which spins ^Srelax in contact with a phonon reservoir have been studied extensively [13, 17, 31]. The noise-aected occupation of fermionic modes in a double quantum dot [32], and the behaviour of an impurity in a Bose- Einstein condensate environment [33] are other examples. Note that in Eq. (17) the dimension of the system is either smaller or equal to the dimension of the bath, i.e. dsd. A rectangular ( dsd) coupling matrix C!may model a graphene-on-substrate structure, where the electronic system (ds= 2) is in contact with a 3D phononic bath [34]. An example for equal dimensions is a 3D spin vector that couples to a3D phononic environment [17]. p-3S. Nemati et al. Specic examples. { In this second part of the paper, we wish to use Eq. (17) to obtain coupling function estimates from experimentally measurable quantities. To do so we have to drop generality and make a number of simplifying assump- tions. First, we assume isotropic coupling to an isotropic bath, and setC!= 1dsC!with scalar C!, andM!= 1ds=ds. Second, for simplicity, we assume a frequency-independent g so that the frequency-dependent impact of the coupling is captured by D!alone. This is a common approximation for quantum optics systems [35,36], while examples of condensed matter systems where this approximation holds over a range of frequencies are limited [2]. Whenever a non-trivial g!is known for a specic context, such as a power law behaviour /!p, this can be included in Eq. (17) separately from the DOS'!-dependence. To establish such g!requires micro- scopic models for specic physical situations, which make several approximation steps. For example, a derivation of the electron-phonon interactions in quantum dots [25] was given in [24]. These assumptions reduce Eq. (17) to the scalar equation C2 !=g2 dsD!; (18) with system dimension ds. We will base the following dis- cussion of examples for the coupling functions C!on this simpler scalar form. Debye approximation. { In condensed matter physics, the Debye model is used to describe the phonon contribu- tion to a crystal's thermodynamic properties. It assumes an acoustic dispersion, i.e. !=cjkjwith an averaged sound speedc, resulting in 3D in [21] DDeb !=3!2 22c3(!D!): (19) Here!Dis the Debye frequency, i.e. the maximum bath frequency, which in practice is taken to be near the edge of the Brillouin zone. For example, for gold, see Fig. 2 (a), the Debye model ts the DOS data reasonably well in frequency regionIup to1:4 THz . For the Debye DOS, our relation Eq. (18) implies the cou- pling function (setting ds=d= 3) CDeb !=g!p 22c3(!D!): (20) The scaling of CDeb ! implies that the spectral density J(!)/ C2 !=!is Ohmic, i.e. J(!)/!. Hence, the 3D Debye model with constant coupling gin the wave vector approach captures the same relaxation dynamics as an Ohmic bath in the frequency approach. Beyond 3D cubic lattices, D!will depend on the dimen- sionality and lattice symmetry. What happens if the lattice is eectively two- or one-dimensional? To answer this, let us imagine adD isotropic lattice with volume V=Nad. The volume element of such a lattice in k-space corresponds to 0.0 2.0 4.0 6.0 8.0 /2[THz] 05101520D[arb.u.] (a) I II III Expt. Debye two-peak -2.0 0.0 2.0 4.0 tt/prime[ps] -0.40.00.40.81.2K(tt/prime)[arb.u.] (b) Debye two-peakFig. 2: (a) Debye DOS (pink solid line, Eq. (19)) and two-peak Lorentzian DOS (blue solid line, Eq. (22)) tted to a measured phonon DOS for gold (red dots) reported as in Ref. [37]. The Debye frequency for gold is !D=2= 3:54 THz given in Ref. [21]. Fit specied peak frequencies !0;j, widths jand peak ratios Aj=A1are given in Table 1. The grey dashed lines separate three frequency regimes discussed in the main text. (b) Memory kernels K(tt0)corresponding to Debye DOS and two-peak Lorentzian DOS. ddk= dkd1dkwhere d= 2;2;4is thedD solid angle ford= 1;2;3, respectively. Analogously to the 3D lattice, using the acoustic dispersion with an averaged sound speed c, one nds the dD Debye DOS D(d) != d!d1 (2c)d(!D!): (21) Via Eq. (18) we obtain the power-law C!/!(d1)=2for the corresponding coupling functions which implies spectral densitiesJ(!)/!d2. Thus, isotropic baths in 2D or 1D behave in a distinctly sub-Ohmic way. Inferring coupling functions from DOS data. { Here we wish to go beyond the conceptually useful Debye model, and fully specify the functional form of C!, given experi- mentally accessible DOS data that characterise the phononic environment. A generic feature of real materials is a structured DOS, which shows several peaks [37, 38]. Sums of Lorentzian or Gaussian functions are two convenient candidates to approx- imate such peaky shaped densities [39]. Here, we t experi- mentally measured DOS for gold [37] (and iron [38] in SM) p-4Coupling Function From Bath Density Of States and theoretically computed DOS for YIG [40] to a function consisting of multiple Lorentzians, DLor !=6A1 g2X j=1Ajj A1!2 (!2 0;j!2)2+ 2 j!2: (22) The ts, see Figs. 2 (a), 3 and gure in SM, reveal the mate- rial specic peak frequencies !0;j, peak widths jand peak ratiosAj=A1, see Table 1 and tables in SM, while the rst peak amplitude A1remains undetermined. Fixing A1would require information additional to the DOS, such as the sys- tem's relaxation rate due to interaction with the phonon bath. Note that phonon DOS are generally slightly temperature de- pendent [38]. Hence the t parameters in Eq. (22) will be (usually weak) functions of temperature, a dependence that only matters when a large range of temperatures is consid- ered. Table 1: Fit parameters of two-peak Lorentzian matched to the experimentally measured DOS for gold reported in Ref. [37] (see Fig. 2 (a)). peak frequency width ratio j!0;j=2[THz] j=2[THz]Aj=A1 1 2.11 1.3 1 2 4.05 0.56 0.15 The peak widths in Eq. (22) determine a characteristic memory time 1=j. However, beyond this single timescale number, the functional dependence of the memory is fully determined by the kernel Eq. (12), which for multi-peak Lorentzians is proportional to KLor(tt0)/X jAjej(tt0) 2sin(!1;j(tt0)) !1;j(tt0); (23) with!1;j=q !2 0;j2 j=4. The degree of memory intro- duced by this kernel into a system's dynamics could be quan- tied in terms of several non-Markovianity measures, see e.g. [41{44]. For gold, Fig. 2 (a) shows the phonon DOS measured by Mu~ noz et al. [37], together with our two-peak Lorentzian t. The t gives good agreement in all frequency regimes, with a slightly slower decay in region III than the measured DOS. For a system coupled to phonons in gold, the peak widths (see Table 1) immediately imply a characteristic memory time in the picosecond range. The relevant kernel is shown (blue) in Fig. 2 (b) for the two-peak tted DOS of gold shown in (a). Using the Debye model instead would give a qualitatively dierent behaviour: the pink curve shows a distinctly slower long-time tail, due to the sharp cuto at the Debye frequency. Note also that without any cuto, the kernel would be K(t t0)/@t0(tt0), implying no memory [17]. In contrast, the Lorentzian t (blue) provides a quantitatively accurate memory kernel. Our approach may provide a more realistic picture of the magnetization dynamics based on actual material data. YIG 0 5 10 15 20 25 /2[THz] 0510152025D[arb.u.] Theo. 18-Lor FitFig. 3: Illustration of eighteen-peak Lorentzian DOS, Eq. (22), (orange curve) tted to the theoretically predicted phonon DOS D!for YIG (cyan curve) reported in Ref. [40]. The grey dashed line shows a single-peak Lorentzian t. The tted peak frequen- cies!0;j, widths jand amplitude ratios Aj=A1can be found in Table 2 in the SM. [45,46] is a typical magnetic insulator in which the relaxation of a spin DoF ^Sis dominated by the coupling to phonons [47], similar to magnetic alloys like Co-Fe [48], while in metal- lic materials, the coupling to electrons is more relevant [49]. Fig. 3 illustrates a theoretically computed DOS for YIG [40] with a t that contains eighteen Lorentzians. (Parameters are displayed in Table 2 in the SM.) In this t, a few nega- tive amplitudes Ajin Eq. (22) are needed to reproduce the gap near 16 THz , however, the total D!remains positive. Using additional information of the typical Gilbert damping parameter for this material [50], also the peak amplitude A1 can be determined (see the SM). More generally, via Eq. (18) the parameters of the multi- peak DOS (22) immediately specify the functional form of the couplingC!of a system to a phononic bath in real materials. This second result of the paper will be useful for modelling the Brownian motion of spins [17,51] and in applications such as quantum information processing with solid-state spin systems [52]. Conclusion. { We have derived the general relation (17) that translates the function C!, determining the coupling of a generic system to a bosonic bath at various frequencies, into the density of states D!of the latter. This was achieved by evaluating the memory kernel of dynamical bath variables in two equivalent approaches. Several applications of the rela- tion were then discussed. We demonstrated how for systems damped by phonons in 3D with a frequency-independent g, Debye's quadratic DOS captures the same physics as a linear coupling function C!which corresponds to an Ohmic spec- tral density. Secondly, we have established how to infer C! from the measured DOS of a material, such that it re ects the specic properties of the material. Given that real mate- rials have densities of states with multiple peaks, the typical p-5S. Nemati et al. picture which emerges from our general relation (17) is that the coupling function is non-Ohmic and memory eects in the system dynamics become important. The corresponding time scales (in the ps range, e.g., for gold in Fig. 2 (b)) can be conveniently determined by tting multiple Lorentzians to the bath DOS. Future work could address how to extend relation (17) to systems interacting with multiple independent baths. This should be suitable for non-equilibrium settings involving dif- ferent temperatures [53], as used in heat transport [54]. The impact of memory may also change the behaviour of systems like superconducting qubits or two-level systems that are in contact with two baths [55, 56]. Acknowledgments. { We thank Jorge A. Mu~ noz, Lisa M. Mauger and Brent Fultz for sharing their experimental data. 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[57]Mauger L. M. ,Phonon thermodynamics of iron and ce- mentite (California Institute of Technology; PhD thesis) 2015. [58]Hauser C., Richter T., Homonnay N., Eisenschmidt C., Qaid M., Deniz H., Hesse D., Sawicki M., Ebbing- haus S. G. andSchmidt G. ,Sci. Rep. ,6(2016) 1. p-7S. Nemati et al. 0 3 6 9 12 /2[THz] 0.00.030.060.09D[arb.u.] (a)Expt. Debye one-peak 0 3 6 9 12 /2[THz] 0.00.030.060.09D[arb.u.] (b)Expt. three-peak 0 3 6 9 12 /2[THz] 0.00.030.060.09D[arb.u.] (c)Expt. Theory five-peak Fig. 4: Various ts for the phonon DOS in iron, compared to experimental measurements (red dots in (a), (b) and (c)) reported in Ref. [38]. (a) The pink and green curves arise from Debye model(Eq. (19)) and the single-peak Lorentzian ( = 1 in Eq. (22)), respectively. (b) A three-peak Lorentzian (Eq. (22) ts the experimental data well. (c) The black curve is a ve-peak Lorentzian Eq. (22) tted to the theoretical phonon DOS (cyan curve) reported in Ref. [57]. Experimental data are shifted vertically for clarity. The t parameters for the Lorentzian peaks are given in Table 2. Supplementary Material. { Fig. 4 shows the phonon DOS for iron, a well-known magnetic conductor. In this g- ure, we use experimental data reported in Ref. [38] and mi- croscopic theoretical results calculated in Ref. [57]. Fig. 4 (a) shows ts with the Debye DOS [Eq. (19) of the main paper] and a Lorentzian DOS [Eq. (22) with a single peak]. The De- bye cut-o frequency is taken from the Debye temperature (420 K ) according to h!D=kBTD. Both models reproduce the quadratic scaling at low frequencies, but the Lorentzian also captures the rst two peaks. It does not properly capture the data for large frequencies, however. In Fig. 4 (b), a three-peak Lorentzian [Eq. (22)] ts the measured DOS spectrum remarkably well. It slightly devi- ates at higher frequencies because the Lorentzians decay less rapidly. Therefore, one can conclude that all three models are reliable ts to the measured data at low frequencies. How- ever, if details are required in a broader frequency range (i.e. at shorter time scales), the three-peak Lorentzian DOS will be the better choice. Fig. 4(c) illustrates the additional structure in the DOS that becomes visible when tting to the theoretical DOS of Ref. [57]. Here, we tted a ve-peak Lorentzian [see Table 2] to reproduce two additional shoulders. Compared to the ex- perimental data, these are probably hidden by the nite res- olution of the measurement. Taking experimental data with such a broadening at face value, one would therefore produce a tted coupling function with a somewhat shorter memory time compared to the real material. Fit parameters are given in Table 2. In Table 3, we give the tting parameters for the DOS of the material Yttrium Iron garnet (YIG). This is a magnetic material which has phonon-damped spin dynamics. Here, an absolute magnitude for the peak amplitudes can be extracted using the known Gilbert damping parameter '5104 [50] and the electron gyromagnetic ratio e'28109Hz=T [58]. We consider for the system operator ^Sin Eq. (2), as in Ref. [17], a spin vector smultiplied by the gyromagnetic ratio e(withjsj= h=2). For the eighteen-peak t, in Fig. 3 of the main paper, we determine the absolute peak height to beA1'71:14 (rad THz T)2=meV . The other t param- eters are given in Table 3. For the single-peak t, whichTable 2: Parameters of Lorentzian ts matched to experimentally measured Ref. [38] and theoretically calculated Ref. [57] DOS for iron. single peak [Fig. 4(a)] peak frequency width ratio j!0;j=2[ THz] j=2[ THz]Aj=A1 1 6.27 3.71 1.00 three peaks [Fig. 4(b)] 1 5.23 2.04 1.00 2 6.77 1.74 0.50 3 8.45 0.71 0.62 ve peaks [Fig. 4(c)] 1 4.67 1.87 1.00 2 5.46 0.74 0.34 3 6.63 1.41 1.20 4 8.03 0.78 0.27 5 8.49 0.44 0.68 is easier to use in simulations, the absolute peak height is A1'1194:19 (rad THz T)2=meV and the other t param- eters are given in Table 3. Table 3: Fit parameters of single-peak and eighteen-peak Lorentzians { shown in Fig. 3 of the paper { matched to the the- oretical DOS for YIG reported in Ref. [40]. single peak peak frequency width ratio j!0;j=2[ THz] j=2[ THz]Aj=A1 1 5.91 12.4 1.00 eighteen peaks 1 2.56 0.99 1.00 2 3.66 1.35 16.20 3 4.89 1.22 10.10 4 6.45 0.55 1.47 5 7.16 0.99 7.75 6 8.10 1.20 10.60 7 9.20 1.18 11.50 8 10.20 0.54 1.70 9 10.80 1.82 11.30 10 12.60 1.67 33.20 11 13.70 0.83 9.07 12 13.80 3.80 {86.60 13 14.40 1.30 19.60 14 16.10 1.07 {13.70 15 16.40 1.83 40.10 16 18.70 1.46 6.08 17 20.10 0.94 4.27 18 20.90 0.45 2.20 p-8

2021-12-07

Modelling of an open quantum system requires knowledge of parameters that specify how it couples to its environment. However, beyond relaxation rates, realistic parameters for specific environments and materials are rarely known. Here we present a method of inferring the coupling between a generic system and its bosonic (e.g., phononic) environment from the experimentally measurable density of states (DOS). With it we confirm that the DOS of the well-known Debye model for three-dimensional solids is physically equivalent to choosing an Ohmic bath. We further match a real phonon DOS to a series of Lorentzian coupling functions, allowing us to determine coupling parameters for gold, yttrium iron garnet (YIG) and iron as examples. The results illustrate how to obtain material-specific dynamical properties, such as memory kernels. The proposed method opens the door to more accurate modelling of relaxation dynamics, for example for phonon-dominated spin damping in magnetic materials.

Coupling function from bath density of states

2112.04001v2

Chiral spin-wave velocities induced by all-garnet interfacial Dzyaloshinskii-Moriya interaction in ultrathin yttrium iron garnet lms Hanchen Wang,1,Jilei Chen,1, 2,Tao Liu,3,Jianyu Zhang,1Korbinian Baumgaertl,2 Chenyang Guo,4Yuehui Li,5, 6Chuanpu Liu,1, 3Ping Che,2Sa Tu,1Song Liu,7Peng Gao,5, 6, 8 Xiufeng Han,4Dapeng Yu,7, 5Mingzhong Wu,3Dirk Grundler,2, 9and Haiming Yu1,y 1Fert Beijing Institute, BDBC, School of Microelectronics, Beihang University, Beijing 100191, China 2Laboratory of Nanoscale Magnetic Materials and Magnonics, Institute of Materials (IMX), School of Engineering, Ecole Polytechnique F ed erale de Lausanne (EPFL), 1015 Lausanne, Switzerland 3Department of Physics, Colorado State University, Fort Collins, Colorado 80523, USA 4Beijing National Laboratory for Condensed Matter Physics, Institute of Physics, University of Chinese Academy of Sciences, Chinese Academy of Sciences, Beijing 100190, China 5Electron Microscopy Laboratory, School of Physics, Peking University, Beijing 100871, China 6International Center for Quantum Materials, School of Physics, Peking University, Beijing 100871, China 7Shenzhen Institute for Quantum Science and Engineering (SIQSE), and Department of Physics, Southern University of Science and Technology (SUSTech), Shenzhen 518055, China 8Collaborative Innovation Center of Quantum Matter, Beijing 100871, China 9Institute of Microengineering (IMT), School of Engineering, Ecole Polytechnique F ed erale de Lausanne (EPFL), 1015 Lausanne, Switzerland (Dated: December 24, 2019) Spin waves can probe the Dzyaloshinskii-Moriya interaction (DMI) which gives rise to topological spin textures, such as skyrmions. However, the DMI has not yet been reported in yttrium iron garnet (YIG) with arguably the lowest damping for spin waves. In this work, we experimentally evidence the interfacial DMI in a 7 nm-thick YIG lm by measuring the nonreciprocal spin-wave propagation in terms of frequency, amplitude and most importantly group velocities using all electrical spin-wave spectroscopy. The velocities of propagating spin waves show chirality among three vectors, i.e. the lm normal direction, applied eld and spin-wave wavevector. By measuring the asymmetric group velocities, we extract a DMI constant of 16 J/m2which we independently conrm by Brillouin light scattering. Thickness-dependent measurements reveal that the DMI originates from the oxide interface between the YIG and garnet substrate. The interfacial DMI discovered in the ultrathin YIG lms is of key importance for functional chiral magnonics as ultra-low spin-wave damping can be achieved. Spin waves (or magnons) [1{3] are collective magnetic excitations that can propagate in metals [4] and also in insulators [5]. Intensive research eorts have been made to investigate spin waves in yttrium iron garnet Y3Fe5O12(YIG) [6, 7] which exhibits the lowest damping that is promising for low-power consumption magnonic devices [8{10]. Most previous works were conducted on bulk or thick YIG lms where the Damon-Eshbach (DE) spin-wave chirality [11{13] is well known for magneto- static surface spin waves as illustrated in Fig. 1(a). How- ever, the DE spin-wave chirality was negligible [14] in the thin YIG lms which were recently achieved with high quality for on-chip magnonic devices [15, 16]. We report a dierent type of spin-wave chirality scaled up in ultrathin YIG lms, which we attribute to the interfacial Dzyaloshinskii-Moriya interaction (DMI) [17, 18]. Very recently, domain wall motion in Tm 3Fe5O12(TmIG) on gadolinium gallium garnet (GGG) suggested interfacial DMI consistent with the Rashba eect at oxide-oxide in- terfaces [19{21]. However, independent evidence for the DMI was not provided. Spin waves provide a prevail- ing methodology [22{28] for probing the DMI in metallicmultilayers. DMI is the fundamental mechanism to form chiral spin textures, such as skyrmions [29{31]. Most previous studies on the interfacial DMI focus on mea- suring frequency shifts fbetween counter-propagating spin waves (with wavevectors + kandk) using Brillouin light scattering (BLS) [25, 27] or all-electrical spin-wave spectroscopy (AESWS) [26, 28]. It has been theoretically predicted that the interfacial DMI can generate not only a frequency shift, but also asymmetric spin-wave group velocities [24]. So far, there has been no experimental ob- servation of nonreciprocal spin-wave characteristics and group velocities induced by interfacial DMI in bare ul- trathin YIG lms on GGG. In this letter, we report chiral propagation of spin waves in a 7 nm-thick YIG lm on a (111) GGG sub- strate [32]. The spin waves propagating in the chirally- favored direction are found to be substantially faster than in its counter direction as illustrated in Fig. 1(b). The asymmetry in group velocities vgis characterized by AESWS [4, 16, 26, 33] to be approximately 80 m/s. The chiral spin-wave velocities in YIG lms can be ac- counted for by an interfacial DMI and a DMI constant ofarXiv:1910.02599v2 [cond-mat.mtrl-sci] 21 Dec 20192 H+k (a) H-k YIG YIG H YIG GGG(b) symmetry breaking+k -k(c) n nH vgDMI FIG. 1. (a) Damon-Eshbach spin-wave chirality in thick YIG lms. (b) An illustrative diagram of the chiral propagation of spin waves in an ultrathin YIG lm. The group velocities of spin waves propagating in + kandkdirections are dierent. The inset shows a right-handed chirality among three vectors, i.e. the lm normal direction n, applied eld Hand DMI- induced drift group velocity vDMI g. (c) A high-angle annular dark-eld image is taken at the YIG/GGG interface of the 7 nm-thick YIG sample. The scale bar is 5 nm. 16J/m2estimated from the experiments. By integrat- ing dierent antennas, we vary the spin-wave wavevectors and obtain an asymmetric spin-wave dispersion that can be tted well using the DMI constant extracted from the chiral spin-wave velocities. Five thin-lm YIG samples with thicknesses of 7 nm, 10 nm, 20 nm, 40 nm and 80 nm are investigated. The DMI-induced f[23, 24] and vg increase when the lm thickness decreases, which demon- strates that the DMI in YIG is of interfacial type. We evidence the DMI in the ultrathin YIG lms indepen- dently by BLS revealing nonreciprocal spin-wave disper- sion relations. The DMI constants extracted from the BLS and the AESWS measurements performed on the 10 nm-thick YIG sample agree well. Our discovery makes chiral magnonics [34{36] a realistic vision as bare YIG oers ultra-low spin-wave damping and thereby enables the plethora of suggested devices that functionalize for instance unidirectional power ow, magnon Hall eect and nontrivial refraction of spin waves. The YIG lms were grown on GGG substrates by mag- netron sputtering [32]. The damping parameter for the 7 nm-thick YIG lm is extracted from the ferromagnetic resonance measurements to be = 6.42:1104[37]. The DE spin-wave chirality [11{13] [Fig. 1(a)] is negli- gible when the thickness is 7 nm [14]. However, a new type of spin-wave chirality might arise in the presence of DMI where spin waves propagating in opposite directions not only show amplitude nonreciprocity and frequency shifts, but also chiral spin-wave velocities. They are at- tributed to a DMI-induced drift group velocity whose di- rection follows a right-handed rule [inset of Fig. 1(b)].Figure 1(c) shows a high-angle annular dark-eld im- age near the YIG/GGG interface. To measure the spin wave group velocities, two nano-stripelines (NSLs) are integrated on the 7 nm-thick YIG lm to excite and de- tect spin waves [37, 39]. The spin-wave propagation dis- tances= 2m. The spin-wave transmission spectra S12(S21) suggest that spin waves are excited by NSL2 (NSL1) and detected by NSL1 (NSL2), which is dened as +k(k) spin-wave propagating directions [37]. The measured spectra with an external eld swept from -50 mT to 50 mT are shown in Supplementary Material Fig. S3 [37]. Chiral propagation of spin waves is clearly ob- served with respect to the wavevector kand applied eld H. This chirality is manifested in the angle-dependent measurement shown in Fig. 2. Figure 2(a) shows angle- dependent spectra for S12(+k), where clear asymmetry is observed. The transmission spectra show contrast os- cillations that indicate the phase variation of propagating spin waves [4, 16, 33]. In Fig. 2(b), we show a single spec- trum at 90, where a peak-to-peak frequency span
f+ is marked indicating a phase change of 2 . According to v+() g =d! dk=2
f+() 2=s=
f+()s; (1) the group velocity v+ gfor +kspin waves is estimated as v+ g312 m/s, and v gforkspin waves is estimated asv g236 m/s. Interestingly, for -90, i.e. a reversed applied eld H, the situation is nearly mirrored (Fig. 2). This demonstrates that spin waves propagating in two opposite directions (+ kandk) exhibit dierent group velocities which can be reversed by changing the sign of the applied eld and thereby chiral spin-wave velocities are observed. Figure 3 shows vg=v+ gv gextracted from the eld- dependent measurements [37] based on Eq. 1 as a func- tion of the eld applied in 90. Near zero eld, the group velocities are reciprocal. However, at positive elds S12 is faster than S21and at negative elds S21is faster than S12, i.e. chiral spin-wave velocities are observed [37]. The vgshows a distinctive step-like eld dependence. The- oretical studies [24] have predicted that interfacial DMI can introduce a drift group velocity vDMI g. We indeed observevDMI g experimentally and extract vDMI g = 40:8 m/s by tting the experimental results in Fig. 3 with an empirical equation vg= 2vDMI gtanhH H0 ; (2) whereH0is a tting eld value above which the velocity dierence saturates. The sign of vDMI gis determined by a chiral relation among three unit vectors, i.e. the lm nor- mal vector ^n, applied eld ^H, and spin-wave wavevector ^k[inset of Fig. 1(b)]. According to previous theoretical studies [24, 36], one can write vDMI g=h ^n^H
^ki2 MSD; (3)3 (a) -180 -90 0 903.2f (GHz)3.43.6S12 θ (deg)0 4 IMG (arb. units)-4180 2 -2 (c) -180 -90 0 903.2f (GHz)3.43.6S21 θ (deg)0 4 IMG (arb. units)-4180 2 -2Δf-Δf+(b) (d)k H θ n FIG. 2. Spin-wave transmission spectra S12(a) andS21(c) measured as a function of the eld angle . The eld is xed at 44 mT.is dened as the angle between Handk. Single spectra at 90(dashed lines in (a) and (c)) are shown for S12 (b) andS21(d). The peak-to-peak frequencies
f+0:15 GHz and
f0:11 GHz are extracted for the estimation of spin-wave group velocities v+ gandv g. The 260 nm-wide striplines are used in the experiments [37]. whereDthe DMI constant, MSsaturation magnetiza- tion. At a positive eld, for example, the drift group velocity is towards the + kdirection and therefore the spin-wave group velocity in the + kdirection is faster than in the kdirection. Consequently, vgshows apositive sign and is twice the vDMI g. Based on Eq. 3 and considering an MS= 141 kA/m [32], we can deduce a DMI constant D= 16J/m2. The origin of a nite H0is however unclear. It is not observed in the micro- magnetic simulations [40] (green dashed line in Fig. 3) showing a sharp step-like eld dependence [37]. Consid- ering that the magnetization loop measured by the vi- brating sample magnetometer [37] is not fully closed at low eld, we speculate H0to be a nite eld required to completely eliminate small domain structures induced by surface roughness. To study the kdependence, we integrated coplanar waveguides (CPWs) on the 7 nm-thick YIG lm [37]. Two distinct modes are observed and attributed to the k1= 3:1 rad/m andk2= 9:1 rad/m, identied by Fourier transformation [4, 16, 26, 33, 37]. We summarize -50 -25 0 25-100 δvg (m/s)100 field (mT)500 -5050 FIG. 3. The asymmetric group velocity vg=v+ gv gas a function of the applied eld. The eld is swept from -50 mT to 50 mT. Black circles are data points calculated using the values ofv+ gandv gextracted from experiments [37]. The red line is a t using Eq. 2. The green dashed line is the micromagnetic simulation results [37]. data from NSL and CPW samples in Fig. 4(a) and ob- serve clear asymmetry. The spin-wave dispersion relation f(k) [Fig. 4(a)] is calculated based on f= 0 2 H+2A 0MSk2 H+MS+2A 0MSk2 +M2 S 4 1e2kt
1 2 +h ^n^H
^ki D MSk; (4) where the gyromagnetic ratio, A= 0:371011J/m the exchange stiness constant [41], t= 7 nm the lm thickness. The rst term is the non-chiral contribution from the dipole-exchange spin waves [42] in a DE con- guration [11, 41, 43]. The second term originates from the interfacial DMI. The chirality is determined by the vector relation ( ^n^H)
^kand its amplitude is decided by the DMI strength D. By taking the DMI constant D= 16J/m2extracted from vg, the calculated dis-persion relations agree reasonably well with the experi- mental data points [Fig. 4(a)]. We rotate Hin the lm plane with respect to kand measure vgas a function of [Fig. 4(b)]. The sinusoidal angular dependence is ob- served consistent with the vector relation ( ^n^H)
^k. In addition to the 7 nm-thick YIG lm, we also measured spin-wave propagation in YIG lms with other thick- nesses [37]. The thickness dependence of vgis shown in Fig. 4(c), where the asymmetry in group velocities4 enlarges with decreasing thickness. The observed thick- ness dependence is in opposite to that of the surface anisotropy-induced nonreciprocity [37, 44, 45]. This ob- servation indicates that the observed DMI is an interfa- cial eect. The DMI constant Dcan then be expressed asD=Di t[26], where Dithe interfacial DMI parameter andthe characteristic length of the interface, which de- pends on the details of the interface, such as roughness. With a rough estimation of being the lattice constant of YIGa= 12:4A [46] and a linear tting of Fig. 4(c), we derive an interfacial DMI parameter Di= 90J/m2. The DMI-induced frequency shifts f[25{27] are also ob- served [37]. (a) -10 0 5f (GHz)2.2 k (rad/μm) (c) 0 0.05 0.10δvg (m/s)100 1/t (nm-1)0.1502.4 -5 101.82.0 50+20 mT -20 mT(b) -180 0 90δvg (m/s)50 θ (deg) (d) -90 -45 0β1.0 θ (deg)90-1.0100 -90 180-100 0.50 -50 450 -0.5 FIG. 4. (a) The asymmetric spin-wave dispersion in the pres- ence of an interfacial DMI. The red dots (black squares) are experimental data extracted with an applied eld of +20 mT (-20 mT). The red line (black line) is the calculated disper- sion using a DMI constant of 16 J/m2for +20 mT (-20 mT). (b) Angle-dependent asymmetric group velocities vg.is de- ned in Fig. 2(a). The eld is set at 44 mT. Black open circles are data points extracted from the experiments and the red line is a sinusoidal tting. (c) vgfor samples with dierent thicknesses of 7 nm, 10 nm, 20 nm, 40 nm and 80 nm with an applied eld of 20 mT. The red line is a linear t to the experimental data. (d) Spin-wave amplitude nonreciprocity  measured as a function of on the 10 nm-thick sample. The eld is xed at 10 mT. The 730 nm-wide striplines are used in the experiments [37]. The transmission spectra shown in Fig. 2 exhibit a clear amplitude nonreciprocity. The nonreciprocity pa- rameter=S12S21 S12+S21[47] is extracted as a function of (Fig. 4(d)). At = 90,increases when lms go thicker [37] due to the DE nonreciprocity [14]. The re- maining sizable amplitude nonreciprocity for 7 nm-thick YIG lm may also be resulted from the asymmetric decay of propagating spin waves [48], but most likely due to the excitation characteristics given by the antenna [12, 49]. Interestingly, an unexpected increase of nonrecirpocity occurs around 30[Fig. 4(d)] [37], which may result fromthe interfacial DMI. The spin-wave dispersions at 30be- come rather at and therefore only a small group veloc- ityv0 gremains [37, 42, 43, 50]. If the DMI-induced drift group velocity vDMI gv0 g, the nonreciprocity is enhanced due to the interfacial DMI [37, 51]. We conrmed the interfacial DMI in ultrathin YIG lms independently by BLS. The BLS spectra measured on the 10-nm thick YIG showed clear frequency shifts f between the Stokes and Anti-Stokes peaks [37], indicating an asymmetric dispersion f(k) induced by the DMI. We extractffrom the BLS data and plot them as a func- tion of the spin-wave wavevector kin Fig. 5, where f increases with an increasing k. This is consistent with the linear relationship given by f=2 MSDk[23{27]. Fit- ting thekdependent frequency shifts we extract a DMI constant of 10.3 0:8J/m2for the 10 nm-thick sample. This value is in good agreement with 9.9 1:9J/m2ex- tracted from the chiral spin-wave velocities measured by the AESWS [Fig. 4(c)]. The BLS spectra taken on a 7 nm-thick YIG lm exhibit a small signal-to-noise ra- tio [37]. Still we observe a clear frequency shift of up to about 0.15 GHz at k= 13:3 rad/m that yields a DMI constant of 14.2 4:2J/m2. 0 5 10 150δf (GHz)0.15 k (rad/μm)200.10 0.05 FIG. 5. Frequency shift f(black squares) measured on the 10 nm-thick YIG sample in an applied eld of 80 mT with BLS in re ection geometry at four dierent incident angles probing spin waves with wavevectors k= 4:6 rad/m, 9.1 rad/m, 13.3 rad/ m and 17.1 rad/ m [37]. The red line is a linear t to the experimental data. We now discuss the origin of the interfacial DMI in ultrathin YIG lms. In general, the interfacial DMI stems from the inversion symmetry breaking in the lm normal direction ^nand the spin-orbit coupling. In these samples, the upper surface is either exposed to air or cov- ered by antennas. The observed eect does not change with dierent adhesion layers [37], which indicates that the DMI does not originate from the top but from the bottom surface of YIG consistent with TmIG/GGG samples reported recently [19{21]. The DMI may be enhanced by a heavy metal layer [52, 53] on YIG, but consequently the damping will be severely aected [54]. The energy dispersive X-ray spectroscopy and geometric5 phase analysis [37] are conducted and no signicant Gd diusion [55] is found in the 7 nm-thick YIG lm grown by sputtering. Oxide interfaces are demonstrated experimentally to exhibit Rashba splitting, for example in LaAlO 3/SrTiO 3[56]. The Rashba-induced DMI is predicted at oxide-oxide interfaces [57] and is further demonstrated by simulations to form chiral magnetic textures, such as skyrmions [58]. It has been manifested by rst-principles calculations that Rashba-induced DMI exists in graphene-ferromagnet interface [59], even without strong spin-orbit coupling. For semiconductor heterostructures consisting of materials with dierent band gaps it is reported that the band osets in conduc- tance (and valence bands) are relevant for the Rashba eect [60]. The insulators YIG and GGG exhibit dier- ent band gaps [61] and corresponding band osets are likely. To fully understand the origin of the interfacial DMI at the YIG/GGG interface, more studies such as rst-principals calculations [59, 62] and X-ray magnetic dichroism [63, 64] are required, which are beyond the scope of this work. It would also be instructive to study the substrate dependence of the spin-wave nonre- ciprocity. However, it proves to be highly challenging to fabricate low-damping YIG on conventional substrates, such as Si [65] and GaAs [66]. The nonreciprocity eect induced by the dipolar interaction between top and bottom layers should be negligible since the parallel magnetic conguration can be established with a small eld [37, 67] To summarize, we have observed chiral spin-wave velocities in ultrathin YIG lms induced by DMI attributed to the YIG/GGG interface. The drift group velocity is about 40 m/s yielding a DMI constant of 16J/m2. The chirality is ruled by the vector relation (^n^H)
^k, veried by angle-dependent measurements. The DMI-induced chiral propagation of spin waves in the magnetic insulator YIG oers great prospects for chiral and spin-texture based magnonics [68{71]. The authors thank R. Duine, M. Kuepferling and A. Slavin for their helpful discussions and H.-Z. Wang for her help on the illustration. Financial support by NSF China under Grant Nos. 11674020 and U1801661, 111 talent program B16001, the National Key Research and De- velopment Program of China No. 2016YFA0300802 and 2017YFA0206200, ANR-12-ASTR-0023 Trinidad and by SNF via project 163016 and sinergia grant 171003 Nanoskyrmionics is gratefully acknowledged. T.L. and M.W. were supported by the U.S. National Science Foun- dation (EFMA-1641989) and the U.S. Department of Energy, Oce of Science, Basic Energy Sciences (de- sc0018994). 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2019-10-07

Spin waves can probe the Dzyaloshinskii-Moriya interaction (DMI) which gives rise to topological spin textures, such as skyrmions. However, the DMI has not yet been reported in yttrium iron garnet (YIG) with arguably the lowest damping for spin waves. In this work, we experimentally evidence the interfacial DMI in a 7~nm-thick YIG film by measuring the nonreciprocal spin wave propagation in terms of frequency, amplitude and most importantly group velocities using all electrical spin-wave spectroscopy. The velocities of propagating spin waves show chirality among three vectors, i.e. the film normal direction, applied field and spin-wave wavevector. By measuring the asymmetric group velocities, we extract a DMI constant of 16~$\mu$J/m$^{2}$ which we independently confirm by Brillouin light scattering. Thickness-dependent measurements reveal that the DMI originates from the oxide interface between the YIG and garnet substrate. The interfacial DMI discovered in the ultrathin YIG films is of key importance for functional chiral magnonics as ultra-low spin-wave damping can be achieved.

Chiral spin-wave velocities induced by all-garnet interfacial Dzyaloshinskii-Moriya interaction in ultrathin yttrium iron garnet films

1910.02599v2

1 Independent tuning of electronic properties and induced ferromagnetism in topological insulators with heterostructure approach Zilong Jiang†, Cui-Zu Chang‡, Chi Tang†, Peng Wei‡, Jagadeesh S. Moodera‡, §, and Jing Shi†* †Department of Physics and Astronomy, University of California, Riverside, CA 92521 ‡Francis Bitter Magnetic Lab, Massachusetts Institute of Technology, Cambridge, MA 02139 §Department of Physics , Massachusetts Institute of Technology, Cambridge, MA 02139 ABSTRACT : The quantum anomalous Hall effect (QAHE) has been recently demonstrated in Cr- and V-doped three -dimensional topological insulators (TIs) at temperatures below 100 mK. In those materials, the spins of unfilled d-electrons in the transition metal dopants are exchange coupled to develop a long - range ferro magnetic order , which is essential for realizing QAHE . However, t he addition of random dopants does not only introduce excess charge carriers that require readjusting the Bi/Sb ratio , but also unavoidably introduce s paramagnetic spins that can adversely affect the chiral edge transport in QAHE . In this work, we show a heterostructure approach to independent ly tune the electronic and magnetic properties of the topological surface states in (Bi xSb1-x)2Te3 without resorting to random doping of transition metal elements . In heterostructures consisting of a thin (Bi xSb1-x)2Te3 TI film and yttrium iron garnet (YIG), a high Curie temperature (~ 550 K) magnetic insulator , we find that the TI surface in contact with YIG becomes ferromagnetic via proximity coupling which is revealed by the anomalous Hall effect (AHE) . The Curie temperature of the magnetized TI surface ranges fro m 20 to 150 K but is uncorrelated with the Bi fraction x in (Bi xSb1-x)2Te3. In contrast , as x is varied , the AHE resistivity scales with the longitudinal resistivity . In this approach, we decouple the electronic properties from the induced 2 ferromagnetism in TI . The independent optimiz ation provides a pathway for realizing QAHE at higher temperatures , which is important for novel spintronic device applications . TOC GRAPHIC KEYWORDS : Topological insulator, magnetic proximity effect, ferrimagnetic insulator, anomal ous Hall effect, heterostructures, quantum anomalous Hall effect. Quantum anomalous Hall effect ( QAHE ) requires both a spontaneous ferromagnetic order and a topological non trivial inverted band structure .1-3 To introduce the ferromag netic order, random do ping of transi tion metal elements, e.g. Cr or V, has been employed .4-7 Although t he Curie temperature ( TC) of the magnetic TI can be as high as ~ 30 K,4,7 QAHE only occurs at temperatures two orders of magnitude below TC.7-10 While the mechanism of this large discrepancy remains elusive , in order to observe QAHE at higher temperatu res, it is essential that the exchange interaction in magnetic TI is drastically increased ; in the meantime, the magnetic disorder needs to be greatly reduced . For the random doping approach , it is rather difficult to accomplish these two objectives . An alternative way to address both issues simultaneously is to couple a non -magnetic TI to a high TC magnetic insulator to induce strong exchange interaction via the proximity effect .11-13 The magnetic proximity effect is a well -known phenomenon that has been intensely investigated .14-20 By proximity coupling, the surfa ce layer of TI acquires a magnetic order without being exposed to any random magnetic impurities .16-18 This heterostructure approach was previously adopted in ref. [ 16] using EuS, a ferromagnetic insulator with a band gap of 1.6 3 eV and a TC of ~ 16 K , grown on a 20 quintuple -layer (QL) thick Bi 2Se3. Both m agnetoresistance and a small AHE signal at low temperatures were ascribed to an induced magnetization in Bi 2Se3. Alternatively, YIG has a much larger band gap (~ 2.8 5 eV) and a much higher TC (~ 550K) , and therefore is a better magnetic insulator for heterostructures . In Bi2Se3/YIG, the magnetoresistance observed at low temperatures indicates an interaction effect between the two materials .17,18 We observed suppressed weak anti -localization in 20 QL -Bi2Se3 on YIG.17 Based on our recent successes in Pd/YIG ,20 Bi2Se3/YIG ,17 and graphene/ YIG,15 here we demonstrate that high-quality heterostructures of 5 QL thick (Bi xSb1-x)2Te3 grown on atomically flat YIG films exhibits induced ferromagnetism at the TI surface . By var ying the ratio of Bi to Sb, we can effectively tune the electronic properties such as the carrier density and resistance of the TI layer without affecting the magnetic properties of YIG or the induced magnetic layer in TI . The a tomically flat YIG films (~20 nm) were first epitaxi ally grown on (111) - gadolinium gallium garnet ( GGG ) substrates via pulsed laser deposition as described previously .17,20 Room temperature ferromagnetic resonance (FMR) , vibrating sample magnetometry (VSM ), and atomic force microscopy (AFM) measurements have been performed on all samples (see Supporting Information ). YIG films show clear in-plane magnetic anisotropy (Fig. 1b) and the bulk magnetization value (4Ms ~ 2000 Oe). Then they were transferred to an ultra -high vacuum molecular beam epitaxy (MBE) system for TI film growth . After high temperature annealing (degas ing), 5 QL-thick TI films were grown on YIG and then capped with a 10 nm thick epitaxial Te layer . The sharp and streaky reflection high energy electron diffraction (RHEED) pattern (Fig. 1d) of the 5 QL TI indicates a flat surface and well-defined single crystal structure. High single crystal quality was also confirmed by x -ray diffraction (XRD) on a 20 QL TI on (111) -orient ed YIG/GGG , as shown in Fig. 1e. All peaks can be identified with (00n) diffraction 4 of (BixSb1-x)2Te3, while the YIG/GGG shows the (444) diffraction peak and the ( 001) peak of the Te capping layer is also present . No other phase is observed in the TI/YIG film according to the XRD data. The zoom -in low-angle XRD s can near the (003) -peak shows multiple Kiessig fringes on both sides, further revealing excellent layered structures of TI films on (111) -oriented YIG and good TI/YIG interface correlation . For transport studies, Hall bars of 900 µm ×100 µm were fabricated by photolithography and etching the TI layers by inductively coupled plasma ( ICP). For selected samples, a 50 nm thick Al 2O3 layer was grown as a top gate dielectric by atomic layer deposition (ALD), and a 80 nm thick Ti/Au layer was deposited by electron beam evaporation to form a top -gated device (Fig. 1a). In this work, we have prepared multiple (Bi xSb1-x)2Te3 samples w ith six different Bi fractions , i.e. x=0, 0.16, 0.24, 0.26, 0.36, and 1. By varying Bi content , the carrier concentration s of TI samples are systematically controlled ,21,22 so that the position of the Fermi level is tuned from the bulk valence band (e.g. x=0), through the band gap (intermediate x’s), and to the bulk conduction band (e.g. x=1). The Fermi level position tuning allows us to control the relative contributions to electrical transport from the bulk and surface states. For the surface state-dominated samples (e.g. x=0.24), we can continuously fine tune the Fermi level across the Dirac point by electrostatic gating . Figure 2a displays th e temperature dependence of five TI/YIG samples . For the Bi 2Te3 (x=1) and Sb 2Te3 (x=0) samples at the extreme doping level s, the resistivity (Rxx) is lower than that of the other three samples over the entire temperature range. Moreover, these two samples show metallic behaviors, i.e. dRxx/dT > 0 over the most temperature range , while the other three samples show a stronger insulating tendency , i.e. dRxx/dT < 0, due to depletion of bulk carriers at lower temperatures . Among these three insulating samples, the x=0.26 sample has the highest Rxx 5 at 2 K, reaching 8.4 k. Fig. 2b summarize s both the 2 K resistivity and the carrier density vs. the Bi fraction x. As x increases from 0 to ~ 0.16, Rxx increases and the carriers are holes from the Hall measurements . The hole carrier density decreases as x approaches 0. 16. The increasing Rxx, decreasing 2D carrier density , and the insulating behavior of x= 0.16 sample all indicate that the Fermi level shifts up from the bulk valence band into the bulk band gap .21 As x increases further, the Rxx continually increases , and the 2D hole density passes the minimum and then carriers switch to electrons. These facts suggest that the Fermi level passes the Dirac point of the topological surfaces states between x~ 0.16 and 0. 26. As x increases further, the Fermi level shifts up more and finally enter s the bulk conduction band as x approaches 1. Fig. 2c illustrates a schematic band diagram when x is varied .23 The actual band structure and the precise Fermi level position for each x require detailed first-principles calculations. Nevertheless , the relative position of the Fermi level wi th respect to the Dirac point for different x values can be qualitatively determined from our experimental data . To probe the proximity induced ferromagnetism in the TI surface cont acting YIG, we focus on the non linear Hall signal by removing the dominant linear ordinary Hall background signal .15,16 The inset of Fig. 3a shows a comparison between Bi 2Te3/YIG and Bi 2Te3/Si. Both have strong linear Hall signals with negative slopes as expected for an n-type Bi 2Te3. The carrier density of TI layer is 4.4x1013/cm2 for Bi 2Te3/YIG and 5. 4x1013/cm2 for Bi2Te3/Si, which is not very sensitive to substrate. However, as the linear background is rem oved, the remaining Hall signal from the two samples shows a distinct difference. While Bi2Te3/Si does not have any definitive non linear signal left, Bi2Te3/YIG has a clear non linear component with a saturation feature . In general, non-linearity in Hall voltages can arise from co-exist ing two types of carriers. In fact, su ch non -linearity is often observed when the Fermi level is in the vicinity of the Dirac 6 point where both electrons and holes are present .24 These two samples are clearly in the single carrier type regime; t herefore, we exc lude the two -carrier possibility. The shape of the non linear Hall signal in Bi2Te3/YIG resembles that of the YIG hysteresis loop in perpendicular magnetic fields ( Fig. 1b). We thus assign this nonlinear signal a contribution from AHE . Further evidence will be discussed shortly . Although it is not straightforward to determine the exact physical origin of the observed AHE, it is known that AHE must stem from ferromagnetism in conduct ors.25 Since the underlying YIG is found to remain insulating (resistance >40 GΩ) when measured after the TI growth , etching, and device fabrication are completed , we exclude that the YIG surface itself becomes conducting and contributes to the AHE signal. Moreover , since YIG is grown at ~700 º C while TI is grown at ~ 250 º C later, we do not expect a ny significant diffusion of Fe atoms to dope the TI to turn it to ferromagnetic. Hence, we conclude that the bottom metallic surface of the TI film becomes ferromagnetic via the proximity coupling just as what has been observed in other systems .15,16,19,20 Note that in the Bi 2Te3 sample (x=1) the saturation value of the AHE magnitude RAHE is only ~ 0.015 much smaller than Rxx. In fact , all samples show clear AHE signals at 2 K. More importantly, as the carrier type switches as x goes from 0.16 to 0.26, the sign of the AHE resistivity remains the same. Since the ordinary Hall effect arises from the Lorentz force associated with an external magnetic field such as the stray magnetic field from do main boundaries, if the nonlinear Hall signal observe d here is due to the ordinary Hall effect, its sign would change as the carrier type switches. The absence of the sign change further confirms the AHE nature of the nonlinear Hall signal , i.e. it is a consequence of the induced ferromagnetic surface of TI .25 Moreover, RAHE follows the same trend as that of Rxx (Fig. 3b), i.e. the more insulating samples show ing larger RAHE. In x= 0.26 sample, RAHE jump s to near ly 2 Ω, over two 7 order s of magnitude larger than in Bi2Te3 (x=1). After passing the crossover point, RAHE decrease s to 1.5 Ω at x=0.16 and finally drop s to 0.12 Ω at x=0 (Sb 2Te3) which is only 10% of the maximum value. Quantitatively , the correlation between Rxx and RAHE can be better seen in Fig. 3c where a power -law with an exponent ~ 2 best fit s the data. Since 𝜌𝑥𝑦≪𝜌𝑥𝑥, and 𝜎𝑥𝑦=−𝜌𝑥𝑦 𝜌𝑥𝑥2+𝜌𝑥𝑦2, this power -law suggest s that the AHE conductivity is nearly independent of xx. Since x is the controlling parameter, the quadratic relation between xx and xy suggests that xy is constant (shown in the inset of Fig. 3c) , which rules out the skew scattering mechanism . The induced ferromagnetism arises from the hybridization between the boundary layers of the two materials in the TI/YIG heterostructures ; therefore , the resulting exchange coupling is expected to be weaker than that in the interior of YIG .16 Additionally , less than ideal interface s can further weaken the exchange coupling strength . To quantify the exchange interaction of the proximity effect , we measure th e AHE signal as the temperature is increased until it vanishes . We define it as the ferromagnetic ordering temperature or TC for the induced magnetic layer in TI. The TC of all TI/YIG samples is above 20 K and can be as high as ~ 150 K, as shown in Fig. 3d. There seems to be no correlation between TC and x or the carrier concentration (see Fig. S5). Instead, t his sample -to-sample variation may be attributed to variations in the state of the TI-YIG interface. Although i t is not possible to pinpoint the most important factor (e.g. oxidation state and surface termination) responsible for the exchange strength , there is no fundamental reason that the TC should be limited to 15 0 K. Future improvement of interface quality is expected to result in a higher TC of the magneti zed layer. For the insulating samples whose Fermi level is located in the bulk band gap of the TI , we can further fine tune the position with a gate to access the surface states at different energies . Fig. 8 4a shows the gate voltage dependence of Rxx for the x=0.24 sample which has a top gate above a 50 nm thick Al 2O3 insulator. At zero gate voltage , i.e. Vg= 0 V, the temperature dependence shows a bulk insulating behavior and the ordinary Hall data indicate s the p-type conduction . Therefore, t he Fermi level is located just below the Dirac point in the band gap . As the gate voltage is swept from negative to positive , the resistivity reaches a peak (Rxx ~ 22 k) at Vg ~ 25 V and the carrier type switches from the p- to n-type as indicated by the ordinary Hall background . As Vg is swept , the Fermi level moves upwards and passes the Dirac point which coincides with the maximum in resistivity . The AHE data for two representative gate voltages, i.e. 0 and 40 V, are displayed in Fig. 4b. Although t he ordinary Hall slope has opposite signs (not shown) , the sign of the AHE remains the same on both sides of the Dirac point , which is consistent with the doping dependence discussed earlier. Interestingly , in samples with widely different doping levels, not only is the relative Fermi level position with respect to the Dirac point, but also the band structure is different .21-23 Here in one sample, the electr ostatic gating only shifts the Fermi level in a fixed band structure. Therefore, the fact that the AHE sign remains unchanged across the Dirac point is robust. Unlike in Cr - or V-doped TI, the ferromagnetism is only induced at the bottom surface of the TI layer in TI/YIG heterostructures. Nevertheless, we have demonstrated an alternative rout e of introducing stronger exchange interaction to TI surface states by proximity coupling . Stronger excha nge should lead to a large r topological gap, therefore a higher te mperature at which QAHE occurs . In the meantime, by independently optimizing the electronic properties of the TI , we can reduce disorder, e specially magnetic disorder, and simultaneously tune the Fermi level position in TI without affecting the induced ferromagnetism . 9 SUPPORTING INFORMATION AVAILABLE Material growth, device fabrication and additional figures are given in the Supporting Information document. This material is available free of charge via the Internet at http://pubs.acs.org . AUTHOR INFORMATION Corresponding Authors: *email: jing.shi@ucr.edu Note: The authors declare no competing financial interest. ACKNOWLEDGMENTS We would like to thank V. Aji for fruitful discussions, and W. Beyermann, C.N. Lau, D. Humphrey, R. Zheng, M. Aldosary & K. Myhro for the technical assistance. Work by Z.L.J. was supported by the UC Lab Fees under Award # 12 -LR-237789. Work by C.T. was supported by NSF ECCS under Award # 1202559. Work by J.S. was supported by the U.S. Department of Energy (DOE), Office of S cience, Basic Energy Sciences (BES) under Award # DE -FG02 - 07ER46351. C. Z. C, P. W. and J. S. M. would like to thank support from the STC Center for Integrated Quantum Materials under NSF Grant No. DMR -1231319, NSF DMR Grants No. 1207469 and ONR Grant No. N00014 -13-1-0301. 10 REFERENCES (1) Yu, R. ; Zhang, W.; Zhang, H. J.; Zhang, S. C.; Dai, X.; Fang, Z. Science 2010 , 329, 61- 64. (2) Nomura, K.; Nagaosa, N. Phys. Rev. Lett. 2011 , 106, 166802 -4. (3) Liu, C. X.; Qi, X. L.; Dai, X.; Fang, Z.; Zhang, S. C. Phys. Rev. Lett. 2008 , 101, 146802 . (4) Chang, C. -Z.; Zhang, J.; Liu, M.; Zhang, Z.; Feng, X.; Li, K.; Wang, L. -L.; Chen, X.; Dai, X.; Fang, Z.; Qi, X. -L.; Zhang, S. -C.; Wang, Y.; He, K.; Ma, X. -C.; Xue, Q. -K. Adv. Mater. 2013 , 25, 1065 -1070 . (5) Kou, X.; Lang, M.; Fan, Y.; Jiang, Y.; Nie, T.; Zhang, J.; Jiang, W.; Wang, Y.; Yao, Y.; He, L.; Wang, K. L. ACS Nano 2013 , 7, 9205−9212 . (6) Kou, X.; He, L.; Lang, M.; Fan, Y.; Wong, K.; Jiang, Y.; Nie, T.; Jiang, W.; Upadhyaya, P.; Xing, Z.; Wang, Y.; X iu, F.; Schwartz, R. N.; Wang, K. L. Nano Lett. 2013 , 13, 4587−4593. (7) Chang, C. -Z.; Zhao, W.; Kim, D. Y.; Zhang, H. J.; Assaf, B. A.; Heiman, D.; Zhang, S. C.; Liu, C. X.; Chan, M. H. W.; Moodera, J. S. Nat. Mat. 2015 , 14, 473 -477. (8) Chang, C. Z.; Zhang, J.; Feng, X.; Shen, J.; Zhang, Z.; Guo, M.; Li, K.; Ou, Y.; Wei, P.; Wang, L.; Ji, Z.; Feng, Y.; Ji, S.; Chen, X.; Jia, J.; Dai, X.; Fang, Z.; Zhang, S.; He, K.; Wang, Y.; Lu, L.; Ma, X.; Xue, Q. -K. Science 2013 , 340, 167−170. (9) Kou, X. ; Guo, S. -T.; Fan, Y.; Pan, L.; Lang, M.; Jiang, Y.; Shao, Q.; Nie, T.; Murata, K.; Tang, J.; Wang, Y.; He, L.; Lee, T. -K.; Lee, W.; Wang, K. L. Phys. Rev. Lett. 2014 , 113, 137201 . (10) Checkelsky, J. G. ; Yoshimi, R.; Tsukazaki, A.; Takahashi, K. S.; Kozuka , Y.; Falson, J.; Kawasaki, M.; Tokura, Y. Nat. 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(18) Lang, M. ; Montazeri, M.; Onbasli, M.; Kou, X.; Fan, Y.; Upadhyaya, P.; Yao, K.; Liu, F.; Jiang, Y.; Jiang, W.; Wong, K.; Yu, G.; Tang, J.; Nie, T.; He, L.; Schwartz, R.; Wang, Y.; Ross, C.; Wang, K. L. Nano Lett. 2014 , 14, 3459 -3465 . (19) Huang, S.Y. ; Fan, X.; Qu, D.; Chen, Y. P.; Wang, W.; Wu, J.; Chen, T.; Xiao, J.; Chien, C. L. Phys. Rev. Lett. 2012 , 109, 107204 . (20) Lin, T.; Tang, C. ; Shi, J. Appl. Phys. Lett. 2013 , 103, 132407 . (21) Zhang, J.; Chang, C. Z.; Zhang, Z.; Wen, J.; Feng, X.; Li, K.; Liu, M.; He, K.; Wang, L.; Chen, X.; Xue, Q.; Ma, X.; Wang, Y. Nat.Commun. 2011 , 2, 574 . (22) Kong, D.; Chen, Y.; Cha, J. J.; Zhang, Q.; Analytis, J. G.; Lai, K.; Liu, Z.; Hong, S. S.; Koski, K. J.; Mo, S.-K.; Hussain, Z.; Fisher, I. R.; Shen, Z. -X.; Cui, Y. Nat. Nanotechnol. 2011 , 6, 705−709. 12 (23) Zhang, H.; Liu, C. -X.; Qi, X. -L.; Dai, X.; Fang, Z.; Zhang, S. -C. Nat. Phys. 2009 , 5, 438−442 . (24) Bansal, N. ; Kim, Y.; Brahlek, M.; Edrey, E.; Oh, S. Phys. Rev. Lett. 2012 , 109, 116804 . (25) Nagaosa , N.; Sinova, J.; Onoda, S.; MacDonald, A. H.; Ong, N. P. Rev. Mod. Phys. 2010 , 82, 1539 -1592 . 13 Figure 1. Device schematics and properties of (Bi xSb1-x)2Te3/YIG thin film s. a, Schematic picture of a top -gated 5QL -(Bi xSb1-x)2Te3/YIG device for magneto -transport measurements with a side view. b, 300 K magnetic hysteresis loops measured by VSM with in -plane and perpendicular magnetic fields. The out -of-plane curve indicates a sat uration field ~2700 Oe which slightly varies in different YIG samples. c, Tetradymite -type crystal structure of (Bi xSb1- x)2Te3 consisting of quintuple layers. Bi atoms are partially replaced by Sb atoms. d, RHEED pattern of MBE -grown 5 QL -(Bi xSb1-x)2Te3 on YIG/GGG showing highly ordered flat crystalline surface. e, X-ray diffraction result of a typical 20 QL-(Bi xSb1-x)2Te3 grown on YIG/GGG. The inset shows a zoom -in view of the (003) peak and clear Kiessig fringes. a e 10 20 30 40 50 60 70103104105106107108 5 6 7 8 9 10103104105Intensity (a.u) ()003GGG (444) YIG (444)Te (001) 002100180015006 0012009 003 () Intensity (a.u) c b d 14 Figure 2 . Evolu tion of the longitudinal resistivity and carrier density with Bi fraction in five 5QL (Bi xSb1-x)2Te3/YIG films . a, Temperature dependent longitudinal resistance for five 5QL (Bi xSb1-x)2Te3/YIG samples with x varying from 0 to 1. b, Longitudinal resistance and carrier density vs. Bi fraction. c, Schematic electronic band structure of (Bi xSb1-x)2Te3 indicating the shift of the Fermi energy as x is varied. b 0.0 0.2 0.4 0.6 0.8 1.00246810 Rxx (k) x (Bi fraction)p n -60-3003060n2D (x1012/cm2) a c 15 Figure 3 . Proximity induced ferromagnetism in (Bi xSb1-x)2Te3/YIG films . a, A comparison of nonlinear Hall resistivity after the linear Hall background is removed in Bi 2Te3/YIG and Bi2Te3/Si. The inset shows the total Hall data for Bi2Te3/YIG and Bi 2Te3/Si. b, AHE resistance and longitudinal resistance vs. Bi fraction (bottom axis) and carrier density (top axis) . c, Log-log plot for AHE resistance vs. longitudinal resistance for five samples. The slope of the red line is 1.99±0.2. The inset shows σ xy, the AHE conductivity, as the Bi fraction is varied. d, AHE resistance of (Bi 0.16Sb0.84)2Te3/YIG sample measured from 2 to 200 K. The inset is the saturated AHE resistance as a function of the temperature showing a TC of ~150 K. b a c d 0.0 0.2 0.4 0.6 0.8 1.00.00.51.01.52.0n2D (x1012/cm2) RAHE () x (Bi fraction)p n50 6 -5 -45 04812 Rxx (k) -6 -3 0 3 6-0.02-0.010.000.010.02 -6 -3 0 3 6-10-50510 Rxy () H (kOe)Bi2Te3/YIG Bi2Te3/Si Bi2Te3/SiRAHE () H (kOe)Bi2Te3/YIG 1 100.010.11 0.0 0.2 0.4 0.6 0.8 1.010-410-310-2 xy (e2/h) Bi fraction (x) RAHE () Rxx (k)RAHE~Rxx1.99±0.2 x=1x=0x=0.36x=0.16x=0.2616 Figure 4. AHE of (Bi 0.24Sb0.76)2Te3/YIG film s tuned with gate voltage. a, The gate voltage dependence of longitudinal resistance for x=0.24 sample. The blue squares and green circles represent 0 and 40 V top gate voltages, respectively. The inset is the optical image of the top - gated device. b, AHE resistance at different gate voltages with different carrier types: 0 V (hole side) and 40 V (electron side). Solid lines are guides for the eyes. The inset shows schematic picture of the relative Fermi level position at 0 and 40 V top gate voltages, respectively. b a

2015-08-19

The quantum anomalous Hall effect (QAHE) has been recently demonstrated in Cr- and V-doped three-dimensional topological insulators (TIs) at temperatures below 100 mK. In those materials, the spins of unfilled d-electrons in the transition metal dopants are exchange coupled to develop a long-range ferromagnetic order, which is essential for realizing QAHE. However, the addition of random dopants does not only introduce excess charge carriers that require readjusting the Bi/Sb ratio, but also unavoidably introduces paramagnetic spins that can adversely affect the chiral edge transport in QAHE. In this work, we show a heterostructure approach to independently tune the electronic and magnetic properties of the topological surface states in (BixSb1-x)2Te3 without resorting to random doping of transition metal elements. In heterostructures consisting of a thin (BixSb1-x)2Te3 TI film and yttrium iron garnet (YIG), a high Curie temperature (~ 550 K) magnetic insulator, we find that the TI surface in contact with YIG becomes ferromagnetic via proximity coupling which is revealed by the anomalous Hall effect (AHE). The Curie temperature of the magnetized TI surface ranges from 20 to 150 K but is uncorrelated with the Bi fraction x in (BixSb1-x)2Te3. In contrast, as x is varied, the AHE resistivity scales with the longitudinal resistivity. In this approach, we decouple the electronic properties from the induced ferromagnetism in TI. The independent optimization provides a pathway for realizing QAHE at higher temperatures, which is important for novel spintronic device applications.

Independent tuning of electronic properties and induced ferromagnetism in topological insulators with heterostructure approach

1508.04719v1

Controlling magnon-photon coupling in a planar geometry Dinesh Wagle, Anish Rai, Mojtaba T. Kaffash and M. Benjamin Jungfleisch Department of Physics and Astronomy, University of Delaware, Newark, DE 19716, USA E-mail: mbj@udel.edu January 2024 Abstract. The tunability of magnons enables their interaction with various other quantum excitations, including photons, paving the route for novel hybrid quantum systems. Here, we study magnon-photon coupling using a high-quality factor split-ring resonator and single-crystal yttrium iron garnet (YIG) spheres at room temperature. We investigate the dependence of the coupling strength on the size of the sphere and find that the coupling is stronger for spheres with a larger diameter as predicted by theory. Furthermore, we demonstrate strong magnon-photon coupling by varying the position of the YIG sphere within the resonator. Our experimental results reveal the expected correlation between the coupling strength and the rf magnetic field. These findings demonstrate the control of coherent magnon-photon coupling through the theoretically predicted square-root dependence on the spin density in the ferromagnetic medium and the magnetic dipolar interaction in a planar resonator.arXiv:2402.03071v1 [cond-mat.mtrl-sci] 5 Feb 2024Controlling magnon-photon coupling in a planar geometry 2 1. Introduction Systems with strong light-matter interaction play an important role in quantum information, communications, and sensing applications such as the quantum internet [1], quantum memory [2], quantum transduction [3], and hybrid quantum devices [4]. Magnetic materials are ideal candidates for achieving control of strong light-matter interaction because they can have spin densities many orders of magnitude higher than that of dilute spin ensembles [5]. Moreover, magnetic media can easily be controlled by external stimuli such as magnetic fields. In particular, the quanta of collective spin excitations in magnetic materials (i.e., magnons) can interact with microwave photons through light-matter interaction, leading to magnon polaritons [6] which host a wealth of interesting physics [7, 8, 9, 10, 11, 12, 13, 14, 15, 16]. Magnon-photon coupling relies on the dipolar (Zeeman) interaction between the spins and the magnetic component of the electromagnetic wave of the photon modes. Phenomenologically, coherent magnon- photon coupling can be described using the coupled harmonic oscillator model given by the equation: f±=1 2" (f0+fr)±r f0−fr2 +g π2# , (1) where f±,f0(independent of the applied field), and fr(dependent on the applied field) are the hybridized modes, uncoupled resonance frequency of the photon mode, and the ferromagnetic resonance (FMR) mode, respectively. The coupling strength g/2πgoverns the extent of coupling, which is larger for systems with a larger number of spins ( N): g/2π∝√ N[17]. The efficiency of energy exchange between the subsystems of a magnonic hybrid system depends on the strength of coherent magnon-photon coupling. Controlled magnon-photon coupling has previously been demonstrated experimentally in 3D SRR modemagnon mode𝑔𝜅p𝜅m(a)(b) Figure 1. (a) Schematic illustration of coupling process between SRR mode (photon) and FMR mode (magnon). κpandκmare the microwave photon and magnon dissipation rates, respectively, and g(= 2 πgeff) is the magnon-photon coupling strength. (b) Representation of the experimental setup. The resonator consists of a split-ring in the vicinity of the microwave feedline. A YIG sphere is placed on the top of SRR. The experiments are conducted by recording the microwave absorption as a function of the magnetic field.Controlling magnon-photon coupling in a planar geometry 3 rectangular cavities of different sizes using yttrium iron garnet (YIG) spheres with varying diameters [9, 10] and in a 3D cylindrical cavity by varying the angle between the external magnetic field and the microwave field [18]. Similar physics has been demonstrated in planar-geometry resonator-based systems, which exhibit higher coupling strengths than those of the 3D cavity resonator-based systems [19, 20, 21, 22, 23], by varying the position of a YIG thin film [24] . Furthermore, it was shown that the coupling can be controlled in a split-ring resonator with a nonuniform microwave magnetic field by changing the track width of the split-ring [25]. Recently, magnon-photon coupling was shown to be controllable in a NiFe/Pt-superconducting cavity hybrid system by changing the dc current applied to the Py/Pt bilayer [26] and in a YIG-superconducting cavity hybrid system by varying the temperature [27]. Moreover, the ultrastrong coupling regime was reached in superconductor/ferromagnet multilayered heterostructures [28]. The control of such coupling by spin torque was recently theoretically demonstrated [29, 30]. Furthermore, it was shown that a distinct magnon-photon dissipative coupling regime can be achieved, in which a level attraction can be observed [31, 32]. More recently, the suppression of coherent magnon-photon coupling due to non-linear spin wave interactions at high microwave power was shown [33]. Additionally, the coupling of magnon excitation with a superconducting qubit [34] and the magnon-photon coupling on a coplanar superconducting resonator [35, 36, 37] were demonstrated. Magnons are bosons, which can couple with other excitations such as photons [38, 39], phonons [40, 41], and magnons [42, 43] or simultaneously couple to both photons and phonons forming a tripartite hybrid system [44, 45, 46]. Magnon modes can be controlled using an external magnetic field. On the other hand, photon modes can be manipulated by engineering the resonator design using different geometries and materials. In particular, the photon properties can be altered using periodic structures and Bragg gratings, for example in a photonic crystal [47, 48, 49]. Ultra-strong magnon- photon coupling has been demonstrated in such photonic crystals with a point defect consisting of a ferromagnetic material [50, 51]. Furthermore, photonic crystals have been used to enable strong photon-phonon interaction, e.g., [52, 53, 54]. Very recently, hybrid coherent control of magnons in a ferromagnetic phononic resonator has been achieved by laser pulse excitation of a 1D galfenol nanograting [55]. Despite the time and effort invested, efficient control of magnon-photon coupling remains challenging. At the same time, the ability to efficiently control the coupling is essential for realizing on-chip hybrid magnonic devices [56]. Here, we demonstrate strong magnon-photon coupling in an on-chip planar split- ring resonator (SRR)/YIG sphere system. Our results indicate that coherent magnon- photon coupling can efficiently be controlled by (i) increasing the YIG sphere diameter (corresponding to a higher number of spins) upto a critical size for a given resonator, (ii) choosing the appropriate SRR dimensions and resonance frequency, and/or (iii) increasing the spatial mode overlap between magnon mode in the YIG sphere and the microwave magnetic field.Controlling magnon-photon coupling in a planar geometry 4 This article is structured in the following fashion. In Section 2, we introduce the experimental configuration and setup, in which two SRRs of different resonating frequencies are used to study the position and volume dependence of YIG spheres on the coherent magnon-photon coupling. In Section 3, we discuss our findings, and in Section 4, we summarize our work. 2. Experimental configuration and setup A schematic illustration of the experimental setup is shown in Fig. 1. The magnonic hybrid system comprises an SRR loaded with epitaxial YIG spheres of varying diameters between 0.2 mm and 1 mm. The resonator mode with a dissipation rate κp/2πcouples with the YIG magnon mode with a coupling constant geff(=g/2π), while the YIG sample dissipates its energy at a rate κm/2πas schematically illustrated in Fig. 1(a). The experiment requires a high-quality resonator to confine electromagnetic waves by reflecting them back and forth between the boundaries. The SRR is located in the vicinity of a microwave (MW) feedline, as shown in Fig. 1(b). Ansys HFSS finite element simulations were used to optimize the SRR dimensions for the desired frequency. To perform the simulation, we set the SRR geometries, assigned the material properties to each SRR component, and defined the wave ports and boundary conditions. Based on the simulation results, we fabricated two SRRs with different resonance frequencies and, hence, different inner dimensions by etching one side of Rogers RO3010 laminate with a dielectric constant of 10.20 ±0.30 and copper thickness of 17.5 µm that is coated on each side of the substrate. The first SRR has the following dimensions: outer and inner widths are a= 4.5 mm and b1= 1.5 mm, respectively, while the gap between the SRR and the feedline is gp= 0.2 mm, and the feedline’s width is w= 0.4 mm, see Fig. 2(a). This SRR resonates at ∼5.05 GHz [see Fig. 2(c)] and has a quality factor, Q1= 99.1±0.8. The experimentally observed and Ansys HFSS simulated SRR resonances are shown in Figs. 2(b) and (c), respectively, with the corresponding fittings which are in good agreement with one another: the values for simulated and fabricated SRR lie within 4% of one another. Increasing the inner width of the SRR to b2= 2.5 mm while keeping all other parameters the same decreases the resonating frequency to ∼3.7 GHz; see Fig. 2(d). This SRR has a slightly lower quality factor, Q2= 74.8±0.5. Figures 2(e) and (f) show the experimental and Ansys HFSS simulated SRR resonances for the larger inner ring. The corresponding fitting of simulated and fabricated SRR are in good agreement; the values lie within 2.5% of one another. Microwave spectroscopy measurements were performed in the frequency domain using a vector network analyzer (VNA). The microwave signal was applied to one end of the feedline, and the transmission coefficient (S 21) was measured at the receiving port as a function of MW frequency and applied biasing field. The nominal microwave power was -15 dBm. The biasing field was applied in the plane of the resonator and perpendicular to the feedline, as shown in Fig. 1(b). Ferrimagnetic insulator YIG (Y 3Fe5O12) single crystal spheres of different diametersControlling magnon-photon coupling in a planar geometry 5 FeedlineSRRwgpgpab1(a) FeedlineSRRwab2(d)0-5-100-5-10S21(dB)S21(dB)4.55.05.54.55.5 0-5-100-5-10 3.23.64.0 3.54.0f(GHz)f(GHz)f(GHz)f(GHz)SimulationExperiment SimulationExperiment(b) (e)(c) (f) aa gpgp Figure 2. (a), (d) Top view of two split-ring resonators used in the experiment with the dimensions defined in the text. (b), (e) Resonator modes of the two SRRs with the corresponding Lorentzian fit (red line). The resonating frequency of the SRR decreases from∼5.05 GHz to ∼3.7 GHz by changing the inner width of the SRR from b 1= 1.5 mm to b 2= 2.5 mm. (c), (f) Ansys HFSS simulations of two resonators with the corresponding Lorentzian fits (red lines). The Ansys HFSS simulation gives resonating modes of (5 .261±0.002) GHz and (3 .786±0.004) GHz for the two designs, respectively. were used as a magnon source. The spherical geometry of the sample rules out an inhomogeneous demagnetization field [57]. We studied spheres of different diameters, i.e., the number of spins, in a planar resonator to test the coupling strength dependence on the number of spins. Furthermore, we investigated the dependence of coupling strength on the rf-magnetic field distribution, hrf, by changing the position of the YIG sphere with respect to the SRR.Controlling magnon-photon coupling in a planar geometry 6 3. Results A typical room temperature avoided level crossing spectrum for a YIG sphere of radius 0.75 mm and SRR with ∼5.05 GHz resonance frequency is shown in Fig. 3. Figure 3(a) shows a false color-coded microwave absorption spectrum of the magnon-photon coupling where the color represents the transmission parameter (S 21). We observe an avoided crossing in the field/frequency region where we expect a crossing between the uncoupled magnon mode (magnetic-field dependent mode) and the microwave photon mode (horizontal, field-independent mode). This avoided crossing corresponds to the hybridization between the two modes, creating two new dynamic hybrid modes, indicating the formation of a magnon polariton. The red dotted line in Fig. 3(a) is a fit to Eq. (1) considering the ferromagnetic resonance condition for a spherical sample fr=γ′H, where γ′is the reduced gyromagnetic ratio, and His the effective magnetic field. The parameters extracted are γ′= 2.907±0.002 GHz/kOe, which is close to the values reported for YIG spheres [58] and geff= 96.5±1.5 MHz, which is larger than the value measured in a 3D-cavity for the same size of the sphere at low temperature [10]. The cooperativity C relates the coupling rate ( geff) to the losses ( κp/2πandκm/2π): C = g2 eff/(κp/2π)(κm/2π). For our system, the coupling strength geff(= 96.5 MHz) is larger than the losses κp/2π(= 51 MHz) and κm/2π(= 20 MHz). Hence, C >1, which means that the coupling lies in the strong coupling regime. We show exemplary frequency-dependent line plots of the transmission parameter (S21) in Fig. 3(b) at different magnetic field magnitudes. When the field is swept from higher to lower values, the FMR mode approaches the SRR mode in frequency, while its intensity (S 21) increases. At ∼173.6 mT, the frequency gap is minimum and the two modes switch their intensities. The transduction between the magnon and photon modes is most efficient in this hybridized state. As the field is lowered further, the separation between the modes increases, with the lower-frequency mode – the FMR mode – having a lower intensity than the higher-frequency mode – the SRR mode. To investigate the dependence of the coupling strength on the number of spins involved in the coupling, we performed the same type of experiment for different YIG sphere diameters. For this purpose, the YIG spheres were placed in the resonator’s center. Figure 4(a) shows the variation of the coherent coupling strength with the square root of the effective volume of the YIG sphere. The effective volume is defined as the active material participating in the coupling. The effective volume was determined by calculating the microwave magnetic field ( hrf) distribution using Ansys HFSS; see Fig. 4(b). The inset shown in Fig. 4(a) is the volume integral of the hrffield for the 0.75 mm diameter sphere. As is evident from the figure, we can see that approximately one- fourth of the entire volume (the lower portion of the sphere) experiences a stronger hrf. Based on these modeling results, we define this one-fourth of the entire volume as the effective volume. Figure 4(a) shows that the variation of coherent coupling strength on the square root of the effective volume is linear, following the predicted√ Ndependence since the effective volume of the sphere Veff∝N;N- number of spins. Hence, theControlling magnon-photon coupling in a planar geometry 7 4.65.05.4min.max.|S21|(dB) |S21|(dB)(a) (b)FMRSRR FMRSRRHybridmodes 200160180µ0H(mT) f(GHz)187.2 mT180.4 mT174.5 mT173.6 mT170.4 mT167.2 mT162.4 mT157.6 mTf(GHz)5.255.004.75 Figure 3. (a) A typical avoided level crossing spectrum where magnon mode and SRR mode strongly couple. The spectrum was obtained using a YIG sphere (of radius 0.75 mm) placed in the center of the ring. The false color represents the S21transmission parameter. The dotted red curve is the fit based on Eq. (1). (b) Transmission parameter S 21as a function of frequency at different biasing magnetic field magnitudes. larger the YIG sample, the larger the effective volume, and the stronger the coupling, providing a control method of the coupling strength. Furthermore, we investigated the position dependence of the magnon-photon coupling along the x-axis of the SRR (as is shown in Fig. 5). This position-dependent experiment was conducted using a YIG sphere with a diameter of 0.75 mm and an SRR with a resonance frequency of ∼3.7 GHz. Figure 5(a) compares the experimentally observed geff-dependence (red data points, left axis) with the rf magnetic field ( hrf) distribution obtained by Ansys HFSS simulation for an SRR with resonance frequencyControlling magnon-photon coupling in a planar geometry 8 Figure 4. (a) Coherent coupling strength as a function of the square root of the effective volume of the YIG sphere (V a) in an SRR (of frequency ∼5.05 GHz). The inset shows the hrfdistribution over the volume of the sphere with a diameter of 0.75 mm and (b) rf magnetic field distribution at 5.05 GHz obtained by Ansys HFSS simulations. at∼3.7 GHz (blue data point, right axis). The simulated position-dependent rf magnetic field distribution along the x-axis was extracted from the two-dimensional field distribution shown in Fig. 5(b). Our results reveal that the coherent coupling strength geffvaries non-monotonically [see Fig. 5(a)]. Furthermore, we find a direct correlation between the coupling strength and the magnitude of the microwave magnetic field. This indicates a stronger mode overlap between the spins in the YIG sphere and the magnetic component of the electromagnetic wave created by the SRR photon when the rf magnetic field hrfis stronger. The maximum coupling strength (57 MHz) is observed close to the inner wall of the SRR-ring where hrfis maximum [see Fig. 5(a)]. As we move further away from this position, the coupling strength decreases. In other words, the weaker the magnetic field generated by the SRR mode, the smaller will be the coupling strength, enabling an active control of the coupling strength by varying the spatial location of the YIG sphere on the SRR. We also note that the observed correlation of the magnon-photon coupling strength with the created rf magnetic field could be utilized as a magnetic field sensor in a properly calibrated system. When comparing the rf magnetic field distributions for the two resonator designs [Figs. 4 (b) and 5(b)], it becomes clear that the modeled microwave magnetic field strength hrfin the center of the ring is greater for the 5 .05 SRR than for the 3 .7 GHz SRR. This means a stronger coupling of a particular YIG sphere with a given diameter can be achieved with the 5 .05 GHz SRR compared to the 3 .7 GHz SRR [Fig. 5(b)]. We extracted a coupling strength of g5.05≈96 MHz [see experimental results in Fig. 3(a)] for a 0.75 diameter sphere coupled to the 5 .05 GHz, while a coupling strength of g3.7≈20 MHz was found for the 3 .7 GHz resonator [see Fig. 5(a) for x=4 mm, which is approximately in the center of the SRR]. In the following, we determine the expected ratio between the coupling strengths of the two resonators for a given YIG sphere diameter (0.75 mm). The coupling strength is given by geff∝p ωVm/Va, where ωis theControlling magnon-photon coupling in a planar geometry 9 Figure 5. (a) Coherent coupling strength (red) and normalized hrfintensity (blue) as a function of position ( x) of YIG sphere (of diameter 0.75 mm) in an SRR (of frequency ∼3.7 GHz). The shaded areas represent the positions of the feedline and the ring boundaries. (b) Rf magnetic field distribution obtained by Ansys HFSS simulation for the 3.7 GHz SRR. The color scales in Fig. 5(b) and Fig. 4(b) are normalized to the same value, so they are directly comparable. resonator frequency, VaandVmare the mode volume of the resonator and YIG sphere volume, respectively [9]. Here, the resonator mode volume can be approximated by the volume integral of the hrffield distribution over the volume of the inner space of the resonator; the dependence on Vmcancels out when we calculate the ratio of the coupling strengths since the same YIG sphere diameter was used. With this approximation, one can express the ratio of gefffor two resonators as g5.05/g3.7= 3.86, which is close to the ratio extracted from the experiment ( g5.05/g3.7≈96/20 = 4 .8). We note that Shi et al. [25] observed a different behavior: when the YIG sample is placed in the highly non-uniform microwave magnetic field region near the edge of the SRR, a larger coupling was observed for the lower-frequency resonator. To confirm if we find the same behavior, we tested the coupling strength when the YIG sphere was placed at the edge of the resonators facing the feedline (not in the center of the ring as discussed above) and found an agreement with Shi et al.’s observation: the coupling strength for the 5.05 GHz SRR was 15 MHz when the sample is placed close to the SRR edge, while a larger coupling strength of 37 MHz was found for the 3.7 GHz SRR at the same position [see also position dependence in Fig. 5(a)]. Finally, we note that this behavior can be expected based on the correlation between the coupling strength and the microwave magnetic field discussed above: hrfhas a non-uniform position dependence that is maximum near the edge of the SRR [see Fig. 5(a)]. 4. Outlook In conclusion, we demonstrated the coherent coupling of magnons with the microwave magnetic field in an SRR/YIG sphere hybrid system at room temperature. Our results show that the coherent coupling can be controlled by varying the YIG sphere diameter and, hence, the number of spins participating in the coupling modifying the modeControlling magnon-photon coupling in a planar geometry 10 overlap through dipolar (Zeeman) interaction between the spins in the YIG sphere with the magnetic component of the electromagnetic wave of the SRR photon mode. By comparing two distinct SRRs with different resonances, we furthermore confirmed the theoretically expected dependence of the coupling strength on both the resonating frequency and the effective mode volume of the resonator. 5. Acknowledgment We acknowledge support by the U.S. Department of Energy, Office of Basic Energy Sciences, Division of Materials Sciences and Engineering under Award DE-SC0020308. 6. 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2024-02-05

The tunability of magnons enables their interaction with various other quantum excitations, including photons, paving the route for novel hybrid quantum systems. Here, we study magnon-photon coupling using a high-quality factor split-ring resonator and single-crystal yttrium iron garnet (YIG) spheres at room temperature. We investigate the dependence of the coupling strength on the size of the sphere and find that the coupling is stronger for spheres with a larger diameter as predicted by theory. Furthermore, we demonstrate strong magnon-photon coupling by varying the position of the YIG sphere within the resonator. Our experimental results reveal the expected correlation between the coupling strength and the rf magnetic field. These findings demonstrate the control of coherent magnon-photon coupling through the theoretically predicted square-root dependence on the spin density in the ferromagnetic medium and the magnetic dipolar interaction in a planar resonator.

Controlling magnon-photon coupling in a planar geometry

2402.03071v1

Magnon spin transport driven by the magnon chemical potential in a magnetic insulator L.J. Cornelissen,1K.J.H. Peters,2G.E.W. Bauer,3, 4R.A. Duine,2, 5and B.J. van Wees1 1Physics of Nanodevices, Zernike Institute for Advanced Materials, University of Groningen, Nijenborgh 4, 9747 AG Groningen, The Netherlands 2Institute for Theoretical Physics and Center for Extreme Matter and Emergent Phenomena, Utrecht University, Leuvenlaan 4, 3584 CE Utrecht, The Netherlands 3Institute for Materials Research and WPI-AIMR, Tohoku University, Sendai, Japan 4Kavli Institute of NanoScience, Delft University of Technology, Delft, The Netherlands 5Department of Applied Physics, Eindhoven University of Technology, PO Box 513, 5600 MB Eindhoven, The Netherlands We develop a linear-response transport theory of diusive spin and heat transport by magnons in magnetic insulators with metallic contacts. The magnons are described by a position dependent temperature and chemical potential that are governed by diusion equations with characteristic relaxation lengths. Proceeding from a linearized Boltzmann equation, we derive expressions for length scales and transport coecients. For yttrium iron garnet (YIG) at room temperature we nd that long-range transport is dominated by the magnon chemical potential. We compare the model's results with recent experiments on YIG with Pt contacts [L.J. Cornelissen, et al. , Nat. Phys. 11, 1022 (2015)] and extract a magnon spin conductivity of m= 5105S/m. Our results for the spin Seebeck coecient in YIG agree with published experiments. We conclude that the magnon chemical potential is an essential ingredient for energy and spin transport in magnetic insulators. I. INTRODUCTION The physics of diusive magnon transport in magnetic insulators, rst investigated by Sanders and Walton [1], has been a major topic in spin caloritronics since the discovery of the spin Seebeck eect (SSE) in YIG jPt bi- layers [2{4]. This transverse voltage generated in plat- inum contacts to insulating ferromagnets under a temper- ature gradient can be explained by thermal spin pumping caused by a temperature dierence between magnons in the ferromagnet and electrons in the platinum [4{7]. The magnons and phonons in the bulk ferromagnet are con- sidered as two weakly interacting subsystems, each with their own temperature [1]. Homan et al. explained the spin Seebeck eect in terms of the stochastic Landau- Lifshitz-Gilbert equation with a noise term that follows the phonon temperature [8]. Recently, diusive magnon spin transport over large distances has been observed in YIG that was driven ei- ther electrically [9, 10], thermally [9] or optically [11]. Notably, our observation of electrically driven magnon spin transport was recently conrmed in a Pt jYIGjPt tri- layer geometry[12, 13]. Here we argue that previous the- ories cannot explain these observations, and therefore do not capture the complete physics of magnon transport in magnetic insulators. We present arguments in favor of a non-equilibrium magnon chemical potential and work out the consequences for the interpretation of experiments. Magnons are the elementary excitations of the mag- netic order parameter. Their quantum mechanical cre- ation and annihilation operators fulll the boson com- mutation relations as long as their number is suciently small. Just like photons and phonons, magnons at ther- mal equilibrium are distributed over energy levels ac- cording to Planck's quantum statistics for a given tem-peratureT. This is a Bose-Einstein distribution with zero chemical potential, because the energy and there- fore magnon number is not conserved. Nevertheless, it is well established that a magnon chemical potential can parametrize a long-living non-equilibrium magnon state. For instance, parametric excitation of a ferromagnet by microwaves generates high energy magnons that thermal- ize much faster by magnon-conserving exchange interac- tions than that their number decays [14]. The resulting distribution is very dierent from a zero-chemical po- tential quantum or classical distribution function, but is close to an equilibrium distribution with a certain tem- perature and nonzero chemical potential. The breakdown of even such a description is then indicative of the cre- ation of a Bose (or, in the case of pumping at energies much smaller than the thermal one, Rayleigh-Jeans [15]) condensate. This new state of matter has indeed been observed [16]. Here we argue that a magnon chemical potential governs spin and heat transport not only un- der strong parametric pumping, but also in the linear response to weak electric or thermal actuation [17]. The elementary magnetic electron-hole excitations of normal metals or spin accumulation has been a very fruit- ful concept in spintronics [18]. Since electron thermaliza- tion is faster than spin- ip decay, a spin polarized non- equilibrium state can be described in terms of two Fermi- Dirac distribution functions with dierent chemical po- tentials and temperatures for the majority and minority spins. We may distinguish the spin (particle) accumula- tion as the dierence between chemical potentials from thespin heat accumulation as the dierence between the spin temperatures [19]. Both are vectors that are gen- erated by spin injection and governed by diusion equa- tions with characteristic decay times and lengths. The spin heat accumulation decays faster than the spin par-arXiv:1604.03706v1 [cond-mat.mes-hall] 13 Apr 20162 ticle accumulation, since both are dissipated by spin- ip scattering, while the latter is inert to energy exchanging electron-electron interactions. Here we proceed from the premise that non-equilibrium states of the magnetic order can be described by a Bose-Einstein distribution function for magnons that is parametrized by both temperature and chemical potential, where the latter implies magnon number conservation. We therefore dene a magnon heat accumulation Tmas the dierence between the tempera- ture of the magnons and that of the lattice. The chemical potentialmthen represents the magnon spin accumu- lation , noting that this denition diers from that by Zhang and Zhang [20], who dene a magnon spin accu- mulation in terms of the magnon density. The crucial parameters are then the relaxation times governing the equilibration of Tmandm. When the magnon heat accumulation decays faster than the magnon particle ac- cumulation, previous theories for magnonic heat and spin transport should be doubted [1, 5{7, 21]. The relaxation times are governed by the collision integrals that include inelastic (one, two and three magnon scatterings involv- ing phonons) and elastic two and four-magnon scattering processes. At room temperature, two-magnon scattering due to disorder is likely to be negligibly small compared to phonon scattering. Four-magnon scattering only re- distributes the magnon energies, but does not lead to momentum or energy loss of the magnon system. Pro- cesses that do not conserve the number of magnons are caused by either dipole-dipole or spin-orbit interaction with the lattice and should be less important than the magnon-conserving ones for high quality magnetic ma- terials such as YIG. At room temperature, the magnon spin accumulation is then essential to describe diusive spin transport in ferromagnets. Here we revisit the linear response transport theory for magnon spin and heat transport, deriving the spin and heat currents in the bulk of the magnetic insulator as well as across the interface with a normal metal con- tact. The magnon transport is assumed to be diusive. Formally we are then limited to the regime in which the thermal magnon wavelength  and the magnon mean free path`(the path length over which magnon momen- tum is conserved) are smaller than the system size L. The wavelength of magnons in YIG in a simple parabolic band model and is a few nanometers at room tempera- ture. Boona et al. [22] nd that `at room temperature is of the same order. As in electron transport in mag- netic multilayers, scattering at rough interfaces is likely to render a diusive picture valid even when the formal conditions for diusive bulk transport are not met. Un- der the assumptions that magnons thermalize eciently and that the mean-free path is dominated by magnon- conserving scattering by phonons or structural and mag- netic disorder, we nd that the magnon chemical poten- tial is required to harmonize theory and experiments on magnon spin transport [9]. This paper is organized as follows: We start with a brief review of diusive charge, spin and heat transportin metals in Sec. II A. In Sec. II B, we derive the linear response expressions for magnon spin and heat currents, starting from the Boltzmann equation for the magnon distribution function. We proceed with boundary con- ditions at the PtjYIG interface in Sec. II C. In Sec. II D we provide estimates for relaxation lengths and trans- port coecients for YIG. The transport equations are analytically solved for a one-dimensional model (longitu- dinal conguration) in Sec. III A. In Sec. III B we imple- ment a numerical nite-element model of the experimen- tal geometry and we compare results with experiments in Sec. III C. We apply our model also to the (longitu- dinal) spin Seebeck eect in Sec. III D. A summary and conclusions are given in Sec. IV. Generation Absorption Platinum YIG PlatinumMx yz jsjsjcinjcoutjm FIG. 1. Schematic of the 1D geometry [13, 20]. A charge currentjin cis sent through the left platinum strip along + y. This generates a spin current js=jxz=jin ctowards the YIGjPt interface and a spin accumulation, injecting magnons into the YIG with spin polarization parallel to the magne- tization M. The magnons diuse towards the right YIG jPt interface, where they excite a spin accumulation and spin cur- rent into the contact. Due to the inverse spin Hall eect, this generates a charge current jout calong theydirection. Note that if Mis aligned alongz, magnons are absorbed at the injector and created at the detector. II. THEORY We rst review the diusion theory for electrical magnon spin injection and detection as published by one of us in [17, 23]. By introducing the magnon chemical potential this approach can disentangle spin and heat transport in contrast to earlier treatments based on the magnon density [20] or magnon temperature [1, 5{7] only. We initially focus on the one-dimensional geometry in Fig. 1 with two normal metal (Pt) contacts to the mag- netic insulator YIG. We express the spin currents in the bulk of the normal metal contacts and magnetic spacer, and the interface. While Ref. [17] focussed on the chemi- cal potential, here we include the magnon temperature as well. At low temperatures the phonon specic heat has been reported to be an order of magnitude larger than the magnon one [22]. The room-temperature phonon mean3 free path (that provides an upper bound for the phonon collision time) of a few nm [22] corresponds to a sub- picosecond transport relaxation time for sound velocities of 103104m/s. From the outset, we therefore take the phonon heat capacity to be so large and the phonon mean free path and collision times so short that the phonon distribution is not signicantly aected by the magnons. The phonon temperature Tpis assumed to be either a xed constant or, in the spin Seebeck case, to have a constant gradient. For simplicity, we also disregard the nite thermal (Kapitza) interface heat resistance of thephonons [24]. A. Spin and heat transport in normal metals There is much evidence that spin transport in metals is well described by a spin diusion approximation. Spin- ip diusion lengths of the order of nanometers reported in platinum betray the existence of large interface con- tributions [25], but the parameterized theory describes transport well [26]. The charge ( jc;), spin (j) and heat (jQ;) current densities in the normal metals, where the spin polarization is dened in the coordinate system of Fig. 1, are given by (see e.g. [27]) jc;=e@eeS@TeSH 2 @ ; 2e ~j=e 2@SH @ eSHSSN @ Te; jQ;=e@TeeP@eSH 2PSN @ : (1) Here,e,Te, anddenote the electrochemical poten- tial, electron temperature, and spin accumulation, re- spectively. The subscripts ;; 2fx;y;zgare Carte- sian components in the coordinate system in Fig. 1, indicating current direction and spin polarization.  is the Levi-Civita tensor and the summation conven- tion is assumed throughout. The charge, spin, and heat current densities are measured in units of A/m2, J/m2 and W/m2;respectively, while both the electrochemical potential and the spin accumulation are in volts. The charge and spin Hall conductivities are eandSH, both in units of S/m. Thermoelectric eects in metals are gov- erned by the Seebeck coecient Sand Peltier coecient P=STe. Similarly, we allow for a spin Nernst eect via the coecient SSNand the reciprocal spin Etting- shausen eect governed by PSN=SSNTe. We assume, however, that spin-orbit coupling is weak enough so that we can ignore spin swapping terms, i.e., terms of the formj@and their Onsager reciprocal [28]. The spin heat accumulation in the normal metal and therefore spin polarization of the heat current are disregarded for simplicity [19]. ~andeare Planck's constant and the electron charge. The continuity equation @te+r
je= 0 expresses conservation of the electric charge density e. The electron spin and heatQeaccumulations relax to the lattice at rates sand QT;respectively: @ts+1 ~@j=2se ; (2) @tQe+r
jQ=QTCe(TeTp); (3) where the non-equilibrium spin density s= 2e,Ce is the electron heat capacity per unit volume, and the density of states at the Fermi level. Inserting Eq. (1)leads to the length scales `s=p e=(4e2s) and `ep=p e=(QTCe) governing the decay of the electron spin and heat accumulations, respectively. At room tem- perature, these are typically `Pt s= 1:5 nm,`Pt ep= 4:5 nm for platinum [21, 29], and `Au s= 35 nm,`Au ep= 80 nm for gold [21, 30]. B. Spin and heat transport in magnetic insulators Magnonics traditionally focusses on the low energy, long wavelength regime of coherent wave dynamics. In contrast, the basic and yet not well tested assumption underlying the present theory is diusive magnon trans- port, which we believe to be appropriate for elevated tem- peratures in which short-wavelength magnons dominate. Diusion should be prevalent when the system size is larger than the magnon mean free path and magnon ther- mal wavelength (called magnon coherence length in [5]). Magnons carry angular momentum parallel to the mag- netization ( z-axis). Oscillating transverse components of the angular momentum can be safely neglected for system sizes larger than the magnetic exchange length, which is on the order of ten nanometer in YIG at low external magnetic elds [8]. Not much is known about the scattering mean-free path, but extrapolating the results from Ref. [22] to room temperature leads to an estimate of a few nm. Dipo- lar interactions aect mainly the long wavelength coher- ent magnons that do not contribute signicantly at room temperature. Thermal magnons interact by strong and number-conserving exchange interactions. In the Ap- pendix the magnon-magnon scattering rate is estimated4 as (T=Tc)3kBT=~[31, 32] or a scattering time of 0 :1 ps for YIG with Curie temperature Tc500 K at room temper- atureT= 300 K, where TTmTp. According to the Landau-Lifshitz-Gilbert phenomenology [33] the magnon decay rate is GkBT=~[32], with Gilbert damping con- stantG1041 for YIG. Hence, the ratio between the scattering rates for magnon non-conserving to con- serving processes is G(Tc=T)31 at room tempera- ture. These numbers justify the second crucial premise of the present formalism, viz. very ecient, local equi- libration of the magnon system. Since a spin accumu- lation in general injects angular momentum and heat at dierent rates, we need at least two parameters for the magnon distribution f, i.e. an eective temperature Tm and a non-zero chemical potential (or magnon spin accu- mulation)min the Bose-Einstein distribution function nB f(x;) =nB(x;) = em(x) kBTm(x)11 ; (4) wherekBis Boltzmann's constant. Both magnon accu- mulationsTmTpandmvanish on in principle dierent length scales during diusion. Assuming an isotropic (cu- bic) medium, the magnon spin current ( jm, in J/m2) and heat current densities ( jQ;m, in W/m2) in linear response read 0 @2e ~jm jQ;m1 A=0 @mL=T ~L=2e m1 A0 @rm rTm1 A; (5) wheremis measured in volts, mis the magnon spin conductivity (in units of S/m), Lis the (bulk) spin See- beck coecient in units of A/m, and mis the magnonic heat conductivity in units of Wm1K1. Magnon- phonon drag contributions jm;jQ;m/rTpare assumed to be absorbed in the transport coecients since Tm Tp. The spin and heat continuity equations for magnon transport read 0 @@m @t+1 ~r
jm @Qm @t+r
jQ;m1 A=0 @T QQT1 A0 @m@m @m Cm(TmTp)1 A; (6) in whichmis the non-equilibrium magnon spin den- sity andQmthe magnonic heat accumulation. Cmis the magnon heat capacity per unit volume. The rates  and QTdescribe relaxation of magnon spin and tem- perature, respectively. The cross terms (decay or gener- ation of spins by cooling or heating of the magnons and vice versa) are governed by the coecients Tand Q. Eqs. (5) and (6) lead to the diusion equations 0 @e kB eT=kB 11 A0 @r2m r2Tm1 A= 0 @e=`2 mkB= `TT2
e= kB`Q2 m
1=`2 mp1 A0 @m TmTp1 A;(7) Te TmTp μsμmNM FI lslmleplmp ∆μ∆Tme jxzMFIG. 2. (Color online) Length scales at normal metaljferromagnetic insulator (NM jFI) interfaces in Fig. 1. Assuming a constant gradient of the phonon temperature Tp and disregarding Joule heating, the electron temperature Te and magnon temperature Tmrelax on length scales `epand `mp. A signicant phonon heat (Kapitza) resistance would cause a step in Tpat the interface. The spin Hall eect in the normal metal drives a spin current jxztowards the in- terface, which will be partially transmitted to the magnon system (causing a non-zero magnon chemical potential in the FI) and partially re ected back into the NM (causing a non- zero electron spin accumulation in the NM). The electron spin accumulation s=zand the magnon chemical potential m relax on length scales `sand`m, respectively. with four length scales and two dimensionless ra- tios.`m=r m=(2e) @m @m1 and`mp = p m=(QTCm) are the relaxation lengths of, respec- tively, magnon chemical potential and temperature with equilibrium values m= 0 andTm=Tp(see Fig. 2). The length scales `T=p kBm=(2e2TCm) and`Q=r em=(~kBQ) @m @m1 arise from the non-diagonal cross terms. The dimensionless ratio =eL=(kBmTp) is a measure for the relative ability of chemical-potential and temperature gradients to drive spin currents. Simi- larly,T=~kBL=(2em) characterizes the magnon heat current driven by chemical potential gradients relative to that driven by temperature gradients. C. Interfacial spin and heat currents The electron and magnon diusion equations are linked by interface boundary conditions. Spin currents and ac- cumulations are parallel to magnetization direction of the5 ferromagnet along the z-direction. We assume that the exchange coupling dominates the coupling between elec- trons and magnons across the interface. A perturbative treatment of the exchange coupling at the interface leads to the spin current [34, 35] jint s=~g"# 2e2sZ dD() (ez)  nBem kBTm nBez kBTe ;(8) whereg"#is the real part of the spin mixing conduc- tance in S/m2,s= S=a3the equilibrium spin densityof the magnetic insulator and S is the total spin in a unit cell with volume a3. The density of states of magnonsD() =p 
= 42J3=2 s for a dispersion ~!k=Jsk2+
. The spin wave gap
is governed by the magnetic anisotropy and the applied magnetic eld. In soft ferromagnets such as YIG
1 K, which we dis- regard in the following since we focus on eects at room temperature (see e.g. Ref. [8]). The heat current is given by inserting =~into the integrand of Eq. (8). Linearizing the above equation we nd the spin and heat currents across the interface [17] 0 @jint s jint Q1 A=3~g"# 4e2s30 @e(3=2)5 2kB(5=2) 5 2ekBT ~(5=2)35 4k2 BT ~(7=2)1 A0 @zm TeTm1 A: (9)  =p 4Js=(kBT) is the magnon thermal (de Broglie) wavelength (the factor 4 is included for convenience). These expressions agree with those derived from a stochastic model [5] after correcting numerical factors of the order of unity. In YIG at room temperature 1 nm. The term proportional to zcorresponds to the spin transfer (absorption of spin current by the uc- tuating magnet), while that proportional to mis the spin pumping contribution (emission of spin current by the magnet). The prefactor 1= s3
can be under- stood by noting that s3is the eective number of spins in the magnetic insulator that has to be agitated and appears in the denominator of Eq. (9) as a mass term. In the macrospin approximation this term would be re- placed by the total number of spins in the magnet. From Eq. (9) we identify the eective spin mixing conductance gsthat governs the transfer of spin across the interface by the chemical potential dierence
= zm. In units of S/m2 gs=33 2
2sg"# 3: (10) Using the material parameters for YIG from Tab. II and the expression for the thermal De Broglie wavelength given above, we nd gs= 0:06g"#at room temperature [21, 36].gsscales with temperature like (T=Tc)3=2, but it should be kept in mind that the theory is not valid in the limitsT!TcandT!0:It is nevertheless consis- tent with the recently reported strong suppression of gs at low temperatures [10]. D. Parameters and length scales In this section we present expressions for the transport parameters derived from the linearized Boltzmann equa-tion for the magnon distribution function and present numerical estimates based on experimental data. 1. Boltzmann transport theory Magnon transport as formulated in the previous sec- tion is governed by the transport coecients m,L,m, four length scales `m,`mp,`Tand`Q, and two dimen- sionless numbers andT. In the Appendix we derive these parameters using the linearized Boltzmann equa- tion in the relaxation time approximation. We consider four interaction events: i) elastic magnon scattering by bulk impurities or interface disorder, ii) magnon dissipa- tion by magnon-phonon interactions that annihilate or create spin waves and/or inelastic scattering of magnons by magnetic disorder, iii) magnon-phonon interactions that conserve the number of magnons, and iv) magnon- magnon scattering by magnon-conserving exchange scat- tering processes, see also Sec. II B The magnon energy and momentum dependent scat- tering times for these process are el,mr,mp, andmm: At elevated temperatures they should be computed at magnon energy kBTand momentum ~=. Magnon- magnon interactions that conserve momentum do not di- rectly aect transport currents, so the total relaxation rate is 1== 1=el+ 1=mr+ 1=mp. The transport coecients and length scales de- rived in the appendix are summarized in Tab. I. The Einstein relation m= 2eDm@m=~@mcon- nects the magnon diusion constant Dmdened by jm=Dmrmwith the magnon conductivity, where @m=@m=eLi1=2(e
=kBT)=(4Js) and Lin(z) is the poly-logarithmic function of order n. We observe that the magnon spin diusion length `mp is smaller than the magnon decay length `msince the6 Symbol Expression Magnon thermal De- Broglie wavelengthp 4Js=(kBT) Magnon spin conduc- tivitym 4(3=2)2e2Js=(~23) Magnon heat conduc- tivitym35 2(7=2)Jsk2 BT=(~23) Bulk spin Seebeck co- ecientL 10(5=2)eJskBT=(~23) Magnon thermal ve- locityvth 2pJskBT=~ Magnon spin diu- sion length`m vthq 2 3mr Magnon-phonon re- laxation length`mpvthq 2 3(1=mr+ 1=mp)1 Magnon spin-heat re- laxation length`T `m=p Magnon heat-spin re- laxation length`Q `m=pT 5 2(5=2)=(3=2) T2 7(5=7)=(7=2) TABLE I. Transport coecients and length scales [17] as de- rived in Appendix A. latter is proportional to mr, whereas`mpis limited by both magnon conserving and non-conserving scattering processes. Furthermore, 1 =mrcan be estimated by the Landau-Lifshitz-Gilbert equation as GkBT=~[32], where the Gilbert constant Gat thermal energies is not necessarily the same as for ferromagnetic resonance. 2. Clean systems In the limit of a clean system 1 =el!0. At suciently low temperatures the magnon-conserving magnon-phonon scattering rate 1 =mpT3:5[37] (see also the Appendix) loses against 1 =mrGkBT=~since Gis approximately temperature independent. Then all lengths=G10m for YIG at room temperature and withG= 104from FMR [8]. The agreement with the observed signal decay [9] is likely to be coincidental, however, since the spin waves at thermal energies have a much shorter lifetime than the Kittel mode for which Gis measured. mestimated using the FMR Gilbert damping is larger than the experimental value by several orders of magnitude, which is a strong indication that the clean limit is not appropriate for realistic devices at room temperature.3. Estimates for YIG at room temperature The phonon and magnon inelastic mean free paths de- rived from the experimental heat conductivity appear to be almost identical at low temperatures up to 20 K [22] but could not be measured at higher temperatures. Both are likely to be limited by the same scattering mech- anism, i.e. the magnon-phonon interaction. We as- sume here that the magnon-phonon scattering of ther- mal magnons at room temperature is dominated by the exchange interaction (which always conserves magnons) rather than the magnetic anisotropy (which may not conserve magnons) [38]. Then, mpand extrapo- lating the low temperature results to room temperature leads to an `mpof the order of a nm, in agreement with an analysis of spin Seebeck [6] and Peltier [21] experi- ments. The associated time scale mp10:1 ps is of the same order as mmestimated in Sec. II B. On the other hand, mr1 ns from G104and there- fore`mvthpmpmr0:11m. The observed magnon spin transport signal decays over a somewhat longer length scale ( 10m). Considering that the esti- matedmris an upper limit, our crude model apparently overestimates the scattering. An important conclusion is, nonetheless, that `m`mp, which implies that the magnon chemical potential carries much farther than the magnon temperature. Withmp0:11 ps we can also estimate the magnon spin conductivity e2Js=~23105106 S/m, in reasonable agreement with the value extracted from our experiments (see next section). III. HETEROSTRUCTURES Here we apply the model, introduced and parameter- ized in the previous section, to concrete contact geome- tries and compare the results with experiments. We start with an analytical treatment of the one-dimensional ge- ometry, followed by numerical results for the transverse conguration of top metal contacts on a YIG lm with nite thickness. Throughout, we assume |motivated by the estimates presented in the previous section| that the magnon-phonon relaxation is so ecient that the magnon temperature closely follows the phonon temperature, i.e. Tm=Tp(only in section III C 3 we study the implica- tions of the opposite case, i.e. Tm=Tpandm= 0). This allows us to focus on the spin diusion equation for the chemical potential m. This approximation should hold at room temperature, while the opposite regime `mp`mmight be relevant at low temperatures or high magnon densities: when the magnon chemical potential is pinned to the band edge, transport can be described in terms of the eective magnon temperature. The interme- diate regime `mp`min which both magnon chemical potential and eective temperature have to be taken into account, is left for future study.7 Symbol Value Unit YIG lattice constant a 12 :376 A Spin quantum number per YIG unit cellS 10 - Spin wave stiness con- stant in YIGJs 8:4581040Jm2 YIG magnon spin diu- sion length`m 9:4 m YIG spin conductivity m 5105S/m Real part of the spin mixing conductanceg"#1:61014S/m2 Platinum conductivity e 2:0106S/m Platinum spin relaxation length`s 1:5 nm Platinum spin Hall angle  0:11 - TABLE II. Selected parameters for spin and heat transport in bilayers with magnetic insulators and metals. a, S and Jsare adopted from [39], `sandfrom [21, 29], and eis extracted from electrical measurements on our devices [9]. Note that our values foreand`sare consistent with Elliot-Yafet scattering as the dominant spin relaxation mechanism in platinum [40]. The mixing conductance, magnon spin diusion length, and the magnon spin conductivity are estimated in the main text. A. One-dimensional model We consider rst the one-dimensional geometry shown in Fig. 1. We focus on strictly linear response and there- fore disregard Joule heating in the metal contacts as well as thermoelectric voltages by the spin Nernst and Et- tingshausen eects. The spin and charge currents in the metal are then governed by 0 @jc 2e ~js1 A=0 @eSH SHe1 A0 @@ye 1 2@xz1 A; (11) where the charge transport is in the y-direction, spin transport in the x-direction, and the electron spin ac- cumulation is pointing in the z-direction. The spin and magnon diusion equations reduce to @2s @x2=z `2s; (12) @2m @x2=m `2m: (13) The interface spin currents Eq. (8) provide the boundary conditions at the interface to the ferromagnet, while all currents at the vacuum interface vanish. Eqs. (9) and (10) lead to the interface spin current density jint s= gs int zint m
, wheregsis dened in Eq. (10).1. Current transfer eciency The non-local resistance Rnlis the voltage over the detector divided by current in the injector, also referred to as non-local spin Hall magnetoresistance (see below). The magnon spin injection and detection can also be ex- pressed in terms of the current transfer eciency , i.e. the absolute value of the ratio between the currents in the detector and injector strip [20] when the detector circuit is shorted. =Rnl=R0for identical Pt contacts with resistance R0. In Fig. 3 we plot the calculated as a function of distance dbetween the contacts for a Pt thick- nesst= 10 nm and parameters from Table II. decays algebraically/1=dwhend`m;which implies diu- sion without relaxation, and exponentially for d`m. The calculated order of magnitude already agrees with experiments [9]. The 0s in Ref. [20] are three orders of magnitude larger than ours due to their much weaker relaxation. lmlm FIG. 3. The current transfer eciency (non-local resistance normalized by that of the metal contacts) as a function of distance between the contacts in a Pt jYIGjPt structure cal- culated in the 1D model. Parameters are taken from Tab. II and the Pt thickness t= 10 nm. The dashed lines are plots of the functions C1=d(red dashed line) and C2exp (d=`m) (blue dashed line) to show the dierent modes of signal de- cay in dierent regimes: diusive 1 =ddecay ford < `mand exponential decay for d>`m. The constants C1andC2were chosen to show overlap with for illustrative purposes, but have no physical meaning. The origin of the small is caused by the ineciency of the spin-Hall mediated spin-charge conversion. The ratio between the spin accumulations in injector and detector s=det s=inj sis much larger than and discussed in Sec. III C 2.8 2. Spin Hall magnetoresistance The eective spin mixing conductance gsgoverns the amount of spin transferred across the interface between the normal metal and the magnetic insulator. While gs cannot be extracted from measurements directly, it is re- lated to the spin mixing conductance g"#via Eq. (10). In order to determine g"#we measured the spin Hall mag- netoresistance (SMR) [41, 42] in devices of Ref. [9]. The SMR is dened as the relative resistivity change in the Pt contact between in-plane magnetization parallel and normal to the current,
=. The expression for the magnitude of the SMR reads [43]
 =2`s t2`sg"#tanh2t 2`s e+ 2`sg"#cotht `s; (14) wheret= 13:5 nm is the platinum thickness. Figure 4 shows the experimental SMR as a function of platinum strip width. As expected
== (2:60:09)104 does not depend on the strip width. Using Eq. (14) and the values for `s,andeas indicated in Tab. II, we ndg"#= (1:60:06)1014S/m2;which agrees with previous reports [29, 42, 44]. In Chen et al. 's zero-temperature theory [43] the spin current generated by the spin Hall eect in Pt is perfectly re ected when spin accumulation and magnetization are collinear. As discussed above, at nite temperature a fraction of the spin current is injected into the ferromag- net in the form of magnons. This implies that the SMR should be a monotonously decreasing function of temper- ature. This has been found for high temperatures [45], but the decrease of the SMR at low temperatures [46] hints at a temperature dependence of other parameters such as the spin Hall angle. The current transfer eciency can be interpreted as a non-local version of the SMR [10] The SMR is caused by the contrast in spin current absorption of the YIG jPt interface when the spin accumulation vector is normal or parallel to the magnetization M. In the non-local geometry, we measure the voltage in contact 2 that has been induced by a charge current (in the same direction) in contact 1. Since gs<g"#the relationj
=jmust hold even in the absence of losses in the ferromagnet and detector. This indeed agrees with our data. 3. Interface transparency The analytical expression for in the one-dimensional geometry is lengthy and omitted here, but it can be sim- plied for special cases. In the the limit of a large bulk magnon spin resistance, the interface resistance can be disregarded. The decay of the spin current is then domi- nated by the bulk spin resistance and relaxation of both FIG. 4. Experimental spin Hall magnetoresistance (SMR) as a function of platinum strip width. The black squares (left axis) show absolute resistance changes
RSMRdevided by the device length (18 m) in units of /m . The red dots (right axis) show the relative resistivity changes
=. materials. When m=`m;e=`sgs =2`mem t 2m+ `m `s2 2esinh1d `m; (15) where the Pt thickness is chosen t`sand=SH=e is the spin Hall angle. When d`mwe are in the purely diusive regime with algebraic decay /1=d. Exponen- tial decay with characteristic length `mtakes over when d&`m. In our experiments (see Tab. II) meand `m`s, so =2`2 sm `mtesinh1d `m: (16) On the other hand, when m=`m;e=`sgsthe in- terfaces dominate and =2g2 s`2 s`m temsinh1d `m; (17) with identical scaling with respect to d, but a dier- ent prefactor. According to the parameters in Tab. II m=`me=`sgs, so spin injection is limited by the interfaces due to the small spin conductance between YIG and platinum. B. Two-dimensional geometry Experiments are carried out for Pt jYIGjPt with a lat- eral (transverse) geometry in which the platinum injec- tor and detector are deposited on a YIG lm. The two- dimensional model sketched in Fig. 5 captures this cong- uration but cannot be treated analytically. We therefore developed a nite-element implementation of our spin9 diusion theory by the COMSOL Multiphysics (version 4.3a) software package, extending the description of spin transport in metallic systems [47] to magnetic insulators. The nite-element simulations of the spin Seebeck [6] and spin Peltier [21] eects in Pt jYIG focussed on heat trans- port and were based on a magnon temperature diusion model. Here we nd that neglecting the magnon chem- ical potential underestimates spin transport by orders of magnitude, because the magnon temperature equili- brates at a length scale `mpof a few nanometers and the magnon heat capacity and heat conductivity are small [22]. The magnon chemical potential and the associated non-equilibrium magnons, on the other hand, diuse on the much longer length scale `m. In order to model the experiments in two dimen- sions, we assume translational invariance in the third direction, which is justied by the large aspect ratio of relatively small contact distances compared with their length. With equal magnon and phonon temperatures everywhere, the magnon transport in two dimensions is governed by 2e ~jm=mrm; r2m=m `2m; (18) wherer=x@x+z@z. The particle spin current js= (jxx;jzx) in the metal is described by 2e ~js=e 2rx; r2x=x `2s; (19) wherexis thex-component of the electron spin accumu- lation. The spin-charge coupling via the spin Hall eect is implemented by the boundary conditions in Sec. III B 2, while the inverse spin Hall eect is accounted for in the calculation of the detector voltage, see Sec. III B 5). The estimates at the end of the previous section justify disre- garding temperature eects. 1. Geometry In order to accurately model the experiments, we de- ne two detectors (left and right) and a central injector, introducing the distances dleftanddrightas in Fig. 5. We generate a short (A) and a long distance (B) geometry. The injector and detectors are slightly dierent as sum- marized in table III. The YIG lm thicknesses are 200 nm for (A) and 210 nm for (B). The YIG lm is chosen to be long compared to the spin diusion length ( wYIG= 150 m) in order to prevent nite-size artifacts.Pt width Pt thickness Distances w(nm) t(nm) d(m) Geometry A 140 13 :5 0 :25 Geometry B 300 7 2 42:5 TABLE III. Properties of geometry sets A and B. 2. Boundary conditions Sending a charge current density jcin the +y-direction through the platinum injector strip generates a spin ac- cumulation sat the YIGjplatinum interface by the spin Hall eect (shown in Fig. 5). This is captured by Eqs. (1) that predict a spin accumulation at the Pt side of the in- terface of [21] sxjinterface= 2jc`s etanht 2`s ; (20) which is used for the interface boundary condition of the magnon diusion equation. Here, we assume that the contact with the YIG does not signicantly aect the spin accumulation [43], which is allowed for the collinear conguration since gs< e=`s. The spin orientation of spoints alongx, parallel to the YIG magnetization. A charge current I= 100A generates spin accumulations in the injector contact of A s= 9:6V andB s= 7:7V for geometries A and B, respectively. The uncovered YIG surface is subject to a zero cur- rent boundary condition ( r
n)s=0, where nis the surface normal. 3. The YIGjPt interface The interface spin conductance gsis modelled by a thin interface layer, leading to a spin current jint s= int s@x=@z, with spin conductivity int s=gstint. When the interface thickness tintis small compared to the plat- inum thickness tPtwe can accurately model the Pt jYIG interface without having to change the COMSOL code. Varying the auxiliary interface layer thickness between 0:5<tint<2:5 nm, the spin currents vary by only 0 :1%. In the following we adopt tint= 1:0 nm. Finally, with Eq. (10) gs= 0:06g"#andg"#from Sec. III A 2 we get gs= 9:61012S/m2. 4. Magnon chemical potential prole A representative computed magnon chemical potential map is shown in Fig. 6(a), while dierent proles along the three indicated cuts are plotted in Fig. 6(b)-(d). The magnon chemical potential along xand atz=1 nm (i.e. 1 nm below the surface of the YIG) in Fig. 6(b)10 Detector YIGInjector Interface layer/uni03BCsDetectordrightdleft tintt tYIGw w w xz y MInterface layerjzxleftjzxright Interface layer FIG. 5. Schematic of the 2D geometry. The relevant dimensions are indicated in the gure. The spin accumulation arising from the charge current through the injector, s, is used as a boundary condition on the YIG jPt interface. The interface layer is used to account for the eect of nite spin mixing conductance between YIG and platinum. /uni03BCm (/uni03BCV) 059 1234678 −15 −10 −5 0 5 10 1502468 x (µm)µm (µV)Linecut along x −200 −100 00510 z (nm)µm (µV)Injector linecut −200 −100 00123 z (nm)µm (µV)Left detector linecuta b c dx (nm)0 -200 400 -400 200z (nm) -200015213 FIG. 6. (a) Two-dimensional magnon chemical potential dis- tribution for geometry (A) with dleft= 200 nm and dright= 300 nm. The lines numbered 1,2,3 indicate the locations of the proles plotted in gures (b),(c),(d), respectively. is characterized by the spin injection by the center elec- trode. Globally, mdecays exponentially with distance from the injector on the scale of `m. We also observe that the left and right detector contacts at x=200 nm andx= 300 nm, respectively, act as sinks that visibly suppress but do not quench the magnon accu- mulation. The nite mixing conductance and therefore magnon absorption are also evident from the proles alongzin Figs. 6(c) and 6(d): The magnon chemical po- tential changes abruptly across the YIG jPt interface by the relatively large interface resistance g1 s. The magnon chemical potential is much smaller than the magnon gap(1 K). We are therefore far from the threshold for current-driven instabilities such as magnon condensation and/or self-oscillations of the magnetization [32]. 5. Detector contact and non-local resistance The spin current density in the detectors is governed by the spin accumulation according to hjzxi=e 2AZ A@x @zdA0; (21) which is an average over the detector area A=wt. The observable non-local resistance Rnl(normalized to device length) in units of /m Rnl=hjzxi eI: (22) is compared with experiments in the next section. C. Comparison with experiments 1. Two-dimensional model Fig. 7 compares the simulations as described in the previous section with our experiments [9]. Fig. 7(a) is a linear plot for closely spaced Pt contacts while Fig. 7(b) shows the results for all contact distances on a loga- rithmic scale. The magnon spin conductivity mand the magnon spin diusion length `mare adjustable pa- rameters; all others are listed in Table. II. We adopted m= 5105S/m and`m= 9:4m as the best t values that agree with the estimates in Ref. [9] and Sec. II D. At large contact separations in geometry (B), the sig- nal is more sensitive to the bulk parameters `mandm than the interface gs. When contacts are close to each other, the interfaces become more important and the re- sults depend sensitively on gsandmas compared to `m. For very close contacts ( d<500 nm) the total spin resistance of YIG is dominated by the interface and our11 b a FIG. 7. (a) Computed non-local rst harmonic signal as a function of distance on a linear scale. The red open circles show the results for sample (A), while black open squares represent sample (B). The blue triangles are the experimental results [9]. The red dashed line is a 1 =dt of the numerical results for (A). (b) Same as (a) but on a logarithmic scale. model calculations slightly underestimate the experimen- tal signal and, in contrast to experiments, deviate from thed1t that might indicate an underestimated gs: However, a larger gswould lead to deviations at interme- diate distances (1 <d< 5m). 2. Spin transfer eciency and equivalent circuit model The spin transfer eciency s=det s=inj s, i.e. the ratio between the spin accumulation in the injector and that in the detector, can be readily derived from the ex- periments by Eq. (20). From the voltage generated in the detector by the inverse spin Hall eect VISHE [48] det s=2t L1 +e2t=`s 1et=`s
2VISHE; (23) wherelis the length of the metal contact. The spin transfer eciency therefore reads s=t `s2Rnl Rdet et=`s+ 1
e2t=`s+ 1
et=`s1
3; (24) whereRnl=VISHE=Iis the observed non-local resistance andRdetthe detector resistance. Figure 8a shows the ex- perimental data converted to the spin transfer eciency as a function of distance dthat is tted to a 1D magnon spin diusion model that does not include the interfaces [9]. When d!0 and interfaces are disregarded, sdi- verges. This artifact can be repaired by the equivalent spin-resistor circuit in Fig. 8(b) according to which s=Rs Pt Rs YIG+ 2Rs int+ 2RPts; (25) whereRs Pt=`s=(Ainttanh(t=`s)) is the spin resistance of the platinum strip [48], Rs int= 1=(gsAint) is interface RintsRintsRYIGs µsinjµsdet RPtsa b10−310−210−110010110210−610−510−410−310−210−1100 Distance (µm)µsdet / µsinj Experimental data Fit from Ref. 9 Circuit model (no relaxation) RPtsFIG. 8. (Color online) (a) Experimental and simulated spin transfer eciency s=det s=inj s. The blue solid line is a t by the 1D spin diusion model [9]. Since here interfaces are disregarded det s!inj sfor vanishing contact distances. The red dashed line are obtained from the equivalent circuit model in (b) with spin resistances Rs Xdened in the text. This model includes gsbut is valid for d<`monly since spin relaxation is disregared. The interfaces lead to a saturation ofsat short distances.12 spin resistance and Rs YIG=d=(mAYIG) is the magnonic spin resistance of YIG. AYIG=ltYIGis the cross-section of the YIG channel and Aint=wlis the area of the PtjYIG interfaces. The parameters in Tab. II lead to the red dashed line in Fig. 8(a), which agrees well with the experimental data for d < `m. No free parameters were used in this model, since we adopted m= 5105S/m as extracted from our 2D model in the previous section. The model predicts that the spin transfer eciency should saturate for d.100 nm for gs= 9:61012S/m2. A predicted onset of saturation at 200 nm is not con- rmed by the experiments, which as pointed out already in the previous section, could imply a larger gs. Experi- ments on samples with even closer contacts are dicult but desirable. Based on the available data we predict that the eciency saturates at s= 4103. The charge transfer eciency (dened in Sec. III A 1) would be max- imized at5105, which is still below the SMR
== 2:6104, as predicted in Sec. III A 2. 3. Magnon temperature model We can analyze the experiments also in terms of magnon temperature diusion [1] as applied to the spin Seebeck [5, 6] and spin Peltier [21] eects. Commu- nication between the platinum injector and detector is possible via phonon and magnon heat transport: The spin accumulation at the injector can heat or cool the magnon/phonon system by the spin Peltier eect. The diusive heat current generates a voltage at the detector by the spin Seebeck eect. However, pure phononic heat transport does not stroke with the exponential scaling, but decays only logarithmically (see below). The magnon temperature model (which describes the magnons in terms of their temperature only) can give an exponen- tial scaling, but in order to agree with experiments, the magnon-phonon relaxation length must be large such thatTm6=Tpover large distances. This is at odds with the analysis by Schreier et al. and Flipse et al. . How- ever, we can test this model by, for the sake of argument, increasing this length scale by four orders of magnitude to:`mp= 9:4m and completely disregard the magnon chemical potential. The spin Peltier heat current Qinj SPE is then [21] Qinj SPE=LsTinj s 2Aint; (26) whereLsis the interface spin Seebeck coecient, Ls= 2g"# ~kB=(eMs3) [5, 6, 21], and Ms=BS=a3is the saturation magnetization of YIG. The equivalent circuit is based on the spin Peltier heat current and the spin thermal resistances of the YIG jPt interfaces and the YIG channel. This allows us to nd Tme, the temperature dierence between magnons and electrons at the detector interface, which is the driving force for the SSE in this model. The equivalent thermal resistance circuit is shown in Fig. 9(b). Relaxation is disregarded, so the model is RintthRintthRYIGth QSPEinja b Tm-e10−310−210−110010110210−710−610−510−410−310−210−1 Distance (µm)µsdet / µsinj Experimental data κm=1e−2 W/(mK) κm=1e−1 W/(mK) κm=1 W/(mK)FIG. 9. (a) Results of the thermal model for m= 102 W/(mK) (red curve), m= 101W/(mK) (green curve) and m= 1 W/(mK) (black curve). Plotted on the y-axis is the spin transfer eciency resulting from the thermal model, th=det s=inj s. The blue squares represent the experimen- tal data. (b) The equivalent thermal resistance model. The denitions of the thermal resistances used in the model are given in the main text. At the thermal grounds in the circuit, the temperature dierence between magnons and electrons (Tme) is zero. only valid for d < ` mp. The interface magnetic heat resistance is given by Rth int= 1=(I sAint), withI sequal to [5, 6, 21] I s=h e2kBT ~BkBg"# Ms3; (27) and where Bis the Bohr magneton. The YIG heat resistance Rth YIG =d=(mAYIG) and from the thermal circuit model we nd that Tme= Qinj SPE Rth int
2= Rth int+Rth YIG
, which generates a spin ac- cumulation in the detector by the spin Seebeck eect det s=Tmeg"# ~kB Ms34 e`s tanht 2`s1 +e2t=`s 1et=`s
2: (28) The thus obtained spin transfer eciency this plotted in Fig. 9(a) as a function of the magnon spin conduc- tivitym. Form0:11 W/(mK) reasonable agree- ment with the experimental data can be achieved. While Schreier et al. argued that mshould be in the range 102103W/(mK)), mfrom Tab. I is also of the order of 1 W/(mK) at room temperature. Hence, the13 magnon temperature model can describe the non-local experiments, provided that the magnon-phonon relax- ation length `mpis large. However, from the expression for`mpthat we gave in Tab. I we nd that `mp10m corresponds to mpmr1 ns andm104W/(mK), which is at least three orders of magnitude larger than even the total YIG heat conductivity, and is clearly unre- alistic. Thus, requiring `mp10m while maintaining m1 W/(mK) is inconsistent. Also, an `mpof the order of nanometers as reported by Schreier et al. and Flipse et al. is dicult to reconcile with the observed length scale of the order of 10 m. Up to now we disregarded phononic heat transport. As argued, the interaction of phonons with magnons in the spin channel is weak, but the energy transfer can be ecient. The spin Peltier eect at the contact generates a magnon heat current that decays on the length scale `mp, heating up the phonons that subsequently diuse to the detector, where they cause a spin Seebeck eect. The magnon system is in equilibrium except at distances from injector and detector on the scale `mpthat we argued to be short. In this scenario there is no non-local magnon transport in the bulk at all, but injector and detector communicate by pure phonon heat transport. However, this mechanism does not explain the exponential decay of the non-local signal: the diusive heat current emitted by a line source, taking into account that the GGG substrate has a heat conductivity close to that of YIG [6], decays only logarithmically as a function of distance. D. Longitudinal spin Seebeck eect The spin Seebeck eect is usually measured in the lon- gitudinal conguration, i.e. samples with a YIG lm grown on gadolinium gallium garnet (GGG) and a Pt top contact, for which our one-dimensional model [17] ap- plies. A recent study extracted the length scale of the lon- gitudinal spin Seebeck eect from experiments on sam- ples with various YIG lm thicknesses [49]. A length of the order of 1 m was found. Similar results were ob- tained by Kikkawa et al. [50]. We assume a constant gradient ( TLTR)=d< 0;where TL;TRare the temperatures at the interfaces of YIG to GGG,platinum, respectively, with Tmeverywhere equi- librized toTp;and disregard the Kapitza heat resistance, cf. Fig. 10(a). At the YIG jGGG interface the spin cur- rent vanishes. Figs. 10 illustrate the magnon chemical potential prole on the YIG thickness das well as the transparency of the Pt jYIG interface for four limiting cases, i.e. for opaque ( gs< m=`m) and transparent (gs> m=`m) interfaces and a thick ( d > `m) and a thin (d<`m) YIG lm, in which analytic results can be derived. We dene a spin Seebeck coecient as the normalized inverse spin Hall voltage VISHE=tyin the platinum lm of lengthtydivided by the temperature gradient
T=d; d > lm YIG PtChemical potential μmd < lm 0 GGGYIG Pt GGG YIG PtChemical potential μm 0GGG YIG PtGGG0 0Opaque Transparenta b c dTR TLFIG. 10. Magnon chemical potential munder the spin See- beck eect for a linear temperature gradient in YIG, in the limit of: (a) an opaque interface and thick YIG, (b) an opaque interface and thin YIG, (c) a transparent interface and thick YIG and (d) a transparent interface and thin YIG. In all four cases,mchanges sign somewhere in the YIG. For higher in- terface transparency (larger gs), the zero point shifts closer to the PtjYIG interface. with
T=TLTRand average temperature T0: SSE=dVISHE ty
T: (29) Assuming that the Pt spin diusion length `sis much 0 2 4 6 8 100.000.050.100.150.200.250.300.35 d/Slash1lmΣSSE/LParen1ΜV/Slash1K/RParen1 FIG. 11. Normalized spin Seebeck coecient as a function of the thickness of the magnetic insulator in the direction of the temperature gradient. Parameters taken are from Tab. II, together with a Pt thickness of t= 10 nm and temperature of 300 K. The value for the bulk spin Seebeck coecient Lis taken from the expression in Tab. I with = 0:1 ps. shorter than its lm thickness twe nd the analytic ex-14 pression SSE=gs`s`mLh coshd `m1i teT0h gs`mcoshd `m+m 1 +2gs`s e sinhd `mi: (30) In Fig. 11 SSEis plotted as a function of the relative thicknessd=`mof the magnetic insulator in the transport direction, Pt thickness of t= 10 nm and T0= 300 K. We adoptLfrom Table I and a relaxation time mp0:1 ps and the parameters from Tab. 11. The normalized spin Seebeck coecient saturates as a function of don the scale of the magnon spin diusion length `m. While ex- periments at T0250 K report somewhat smaller length scales than our `m;our saturation SSE0:11V/K is of the same order as the experiments [51]. In the limit of an opaque interface, SSEsaturates to SSE(d`m) =gs`s`mL tT0em=gs`s e`m tkB e; (31) in terms of the dimensionless ratio from Eq. (7). For a transparent interface with `m`sandme, the result is governed by bulk parameters only: SSE(d!1 ) =`sL tT0e: (32) This model for the spin Seebeck eect is oversimpli- ed by assuming a vanishing magnon-phonon relaxation length and disregarding interface heat resistances. The gradient in the phonon temperature can give rise to a spin Seebeck voltage [52] even when bulk magnon spin transport is frozen out by a large magnetic eld. Never- theless, it is remarkable that it gives a reasonable quali- tative description for the spin Seebeck eect with input parameters adapted for electrically-driven magnon trans- port. We conclude that also in the description of the spin Seebeck eect the magnon chemical potential can play a crucial role. IV. CONCLUSIONS We presented a diusion theory for magnon spin and heat transport in magnetic insulators actuated by metal- lic contacts. In contrast to previous models, we focus on the magnon chemical potential. This is an essential ingredient because under ambient conditions `m> `mp, i.e., the magnon chemical potential relaxes over much larger length scales than the magnon temperature. We compare theoretical results for electrical magnon injec- tion and detection with non-local transport experiments on YIGjPt structures [9], for both a 1D analytical and a 2D nite-element model. In the 1D model we study the relevance of interface- vs. bulk-limited transport and nd that, for the mate- rials and conditions considered, the interface spin resis- tance dominates. For the limiting cases of transparentand opaque interfaces the spin transfer eciency de- cays algebraically /1=das a function of injector-detector distancedwhend<`m, and exponentially with a char- acteristic length `mford>`m. A 2D nite element model for the actual sample cong- urations can be tted well to the experiments for dier- ent contact distances, leading to a magnon conductivity m= 5105S/m and diusion length `m= 9:4m. The experiments measure rst and second order har- monic signals that are attributed to electrical magnon spin injection/detection and thermal generation of magnons by Joule heating with spin Seebeck eect de- tection, respectively. Here, we focus on the linear re- sponse that we argue to be dominated by the diusion of a magnon accumulation governed by the chemical po- tential, rather than the magnon temperature. However, we applied our theory also to the standard longitudinal (local) spin Seebeck geometry. We nd the same length scale`mand a (normalized) spin Seebeck coecient of SSE0:11V/K ford`m;which is of the same order of magnitude as the observations [49]. ACKNOWLEDGMENTS We would like to acknowledge H. M. de Roosz and J.G. Holstein for technical assistance, and Yaroslav Tserkovnyak, Arne Brataas, Scott Bender, Jiang Xiao, and Benedetta Flebus for discussions. This work is part of the research program of the Foundation for Funda- mental Research on Matter (FOM) and supported by NanoLab NL, EU FP7 ICT Grant No. 612759 In- Spin, Grant-in-Aid for Scientic Research (Grant Nos. 25247056, 25220910, 26103006) and the Zernike Institute for Advanced Materials. RD is member of the D-ITP con- sortium, a program of the Netherlands Organization for Scientic Research (NWO) that is funded by the Dutch Ministry of Education, Culture and Science (OCW). Appendix A: Boltzmann transport theory Here we derive our magnon transport theory from the linearized Boltzmann equation in the relaxation time ap- proximation, thereby introducing and estimating the dif- ferent collision times. 1. Boltzmann equation Eqs. (5,6,7) are based on the Boltzmann equation for the magnon distribution function f(x;k;t): @f @t+@f @x
@!k @k= in[f]out[f]; (A1) where in= in el+ in mr+ in mp+ in mmand out= out el+ out mr+ out mp+ out mmare the total rates of scatter- ing into and out of a magnon state with wave vector15 k, respectively. The subscripts refer to elastic magnon scattering at defects, magnon relaxation by magnon- phonon interaction that do not conserve magnon number, magnon-conserving inelastic and elastic magnon-phonon interactions, and magnon number and energy-conserving magnon-magnon interactions. We discuss them in the following for an isotropic magnetic insulator and in the limit of small magnon and phonon numbers. The elastic magnon scattering is given by Fermi's Golden rule as out el=2 ~X k0Vel kk02(~!k~!k0)f(k;t); (A2) whereVel kk0is the matrix element for scattering by de- fects and rough boundaries [23, 37] of a magnon with momentum ~kto one with ~k0at the same energy. in elis obtained from this expression by interchanging kandk0. In the presence of the in-scattering term (vertex correc- tion) in elthe Boltzmann equation is an integrodierential rather than a simple dierential equation. Gilbert damping parameterizes the magnon dissipa- tion into the phonon bath. According to the linearized Landau-Lifshitz-Gilbert equation [32] out mr= 2G!kf(k;t): (A3) Since the phonons assumed at thermal equilibrium with temperature Tp, in mris obtained by substituting f(k;t)!nB(~!k=kBTp) in out mr. Magnon-conserving magnon-phonon interactions with matrix elements Vmp kk0qgenerate the out-scattering rate out mp=2 ~X k0;qVmp kk0q2 (~!k~!k0q) f(k;t)((1 +f(k0;t)) 1 +nBq kBTp ;(A4) whereq=~cjqjis the acoustic phonon dispersion with sound velocity cand momentum q:The \in" scattering rate in mp=2 ~X k0;qVmp kk0q2 (~!k~!k0q) f(k0;t)((1 +f(k;t))nBq kBTp : (A5) Finally, the four-magnon interactions (two magnons in, two magnons out) generate out mm=2 ~X k0;k00;k000Vmm k+k0;kk0;k00k0002 (~!k+~!k0~!k00~!k000)(k+k0k00k000) f(k;t)f(k0;t)[1 +f(k00;t)][1 +f(k000;t)];(A6) while in mmfollows by exchanging k k00, and k0and k000. Disregarding umklapp scattering, the magnon- magnon interactions conserve linear and angular momen- tum.Vmmtherefore depends only on the center-of-massmomentum and the relative magnon momenta before and after the collision, which implies that mmdoes not af- fect transport directly (analogous to the role of electron- electron interactions in electric conduction). The collision rates govern the energy and momentum-dependent collision times a(k;~!) (with a2fel;mr;mp;mmg). These are dened from the \out" rates via 1 a(k;~!)=out a f(k;t); (A7) replacingf!nB(~!k=kBTp) and ~!kwith ~!where phonons are involved. Here we are interested mainly in thermal magnons for which the relevant collision times are evaluated at energy ~!=kBTand momentum k= 1. Then 1=mrGkBT=~. Elastic magnon scattering can be parameterized by a mean-free-path `el=el(k;~!)@!k=@k, and therefore 1 =el(k;~!) = 2`1 elp Js!=~orel=`el=vm, wherevm= 2p Js!=~is the magnon group velocity. Estimates for `elrange from 1m [23] under the assumption that `mis due to Gilbert damping and disorder only, to 500 m [37]. Therefore el10105ps. Since we deduce in the main text that at room temperature mpis one to two orders of magnitude smaller than this el, we completely disregard elastic two-magnon scattering in the comparison with ex- periments. We adopt the relaxation time approximation in which the scattering terms read [f] =1 el fnB~!km kBTm +1 mr fnB~!k kBTp +1 mp fnB~!km kBTp +1 mm fnB~!km kBTm : (A8) The distribution functions here are chosen such that the elastic scattering processes stop when fapproaches the Bose-Einstein distribution with local chemical potential m6= 0;in contrast to the inelastic scattering that cause relaxation to thermal equilibrium with the lattice and m= 0:Similarly, the temperatures Tpvs.Tmare chosen to express that the scattering exchanges energy with the phonons or keeps it in the magnon system, respectively. The Boltzmann equation may be linearized in terms of the small perturbations, i.e. the gradients of temperature and chemical potential. The local momentum space shift fof the magnon distribution function f(x;k) =@nB ~!k kBTp @~!k@!k @k
 rxm+~!krxTm Tp ; (A9)16 where 1== 1=mr+ 1=mp. The magnon spin and heat currents Eq. (5) are obtained by substituting finto jm=~Zdk (2)3f(k)@!k @k; (A10) jQ;m=Zdk (2)3f(k)~!k@!k @k: (A11) The magnon spin and heat diusion Eqs. (6) are ob- tained by a momentum integral of the Boltzmann equa- tion (A8) after multiplying by ~and~!k, respectively. The local distribution function in the collision terms con- sists of the sum of the \drift" term fand the Bose- Einstein distribution with local temperature and chemi- cal potential f(k;t) =f+nB((~!km(x))=kBTm(x))) (A12) We reiterate that the relatively ecient magnon conserv- ingmlimits the energy, but not (directly) the spin dif- fusion. 2. Magnon-magnon scattering rate The four-magnon scattering rate is believed to e- ciently thermalize the local magnon distribution to the Bose-Einstein form [31, 32]. At room temperature the leading-order correction to the exchange interaction in the presence of magnetization textures reads Hxc=Js 2sZ dxs(x)
r2s(x); (A13) where s(x) (s=jsj=S=a3) is the spin density. By the Holstein-Primako transformation the spin lowering operator reads ^ s=sxisy=q 2s^ y^ ^ 'p 2s^ ^ y^ ^ =2p 2sin terms of the bosonic creation ( ^ y) and annihilation ( ^ ) operators. Hxccan be approximated as a four-particle point-like interaction term HmmgZ dx^ y^ y^ ^ ; (A14) wheregkBT=s is the exchange interaction strength at thermal energies. Using Fermi's Golden Rule for this interaction yields collision terms as Eq. (A6) with Vmm g: 1 mm(k;~!)g2 ~X k0;k00;k000(~!k+~!k0~!k00~!k000) (k+k0k00k000)nB~!k0 kBTp  1 +nB~!k00 kBTp 1 +nB~!k000 kBTp : (A15)The momentum integrals can be estimated for thermal magnons with k= 1and~!=kBTand 1 mmg2 6kBT ~T Tc3kBT ~; (A16) with Curie temperature kBTcJss2=3. With param- eters for YIG Jss2=3=kB200 K, which is the cor- rect order of magnitude. The T4scaling of the four- magnon interaction rate results from the combined ef- fects of the magnon density of states (magnon scattering phase space) and energy-dependence of the exchange in- teractions. While the magnon-magnon scattering is ecient at thermal energies, it becomes slow at low energies close to the band edge due to phase space restrictions and leads to deviations from the Bose-Einstein distribution functions that may be disregarded at room temperature. 3. Magnon-conserving magnon-phonon interactions At thermal energies and large wave numbers the magnon-conserving magnon-phonon scattering [37] is dominated by the dependence of the exchange interac- tion on lattice distortions rather than magnetocrystalline elds. Since we estimate orders of magnitude, we disre- gard phonon polarization and the tensor character of the magnetoelastic interaction and start from the Hamilto- nian Hmp=B sZ dxs(x)
r2s(x)0 @X 2fx;y;zg@R @x1 A; (A17) whereBis a magnetoelastic constant. The scalar lat- tice displacement eld Rcan be expressed in the phonon creation and annihilation operators ^yand^as R=s ~2 2h ^+^yi ; (A18) whereis the phonon energy and the mass density. By the Holstein-Primako transformation introduced in the previous section we nd to leading order HmpBZ dx r^ y
 r^ ~2 0 @X 2fx;y;zg@^ @x1 A+h:c: (A19) This Hamiltonian is the scattering potential in the matrix elements of Eq. (A5) Vmp kk0q2 B2~2q2 q(k
k0)2(kk0q) (A20) which by substitution and in the limit  p, where p=~c=kBTpis the phonon thermal de Broglie wave- length, leads to 1 mpB2 ~~ kBT21 45p; (A21)17 In the opposite limit  p 1 mpB2 ~~ kBT21 72p: (A22) At room temperature  pand fora3= 1024kg both expressions lead to mp= 10(Js=B)2ns [38]. We could not nd estimates of Bfor YIG in the literature. In iron, exchange interactions change by a factor of twoupon small lattice distortion
aa[53]. While the au- thors of this latter work nd that this does not strongly aect the Curie temperature, it leads to fast magnon- phonon scattering as we show now. Namely, B a@Js=@
aj
a=0aJs=
a, so thatmp= 10(
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2016-04-13

We develop a linear-response transport theory of diffusive spin and heat transport by magnons in magnetic insulators with metallic contacts. The magnons are described by a position dependent temperature and chemical potential that are governed by diffusion equations with characteristic relaxation lengths. Proceeding from a linearized Boltzmann equation, we derive expressions for length scales and transport coefficients. For yttrium iron garnet (YIG) at room temperature we find that long-range transport is dominated by the magnon chemical potential. We compare the model's results with recent experiments on YIG with Pt contacts [L.J. Cornelissen, et al., Nat. Phys. 11, 1022 (2015)] and extract a magnon spin conductivity of $\sigma_{m}=5\times10^{5}$ S/m. Our results for the spin Seebeck coefficient in YIG agree with published experiments. We conclude that the magnon chemical potential is an essential ingredient for energy and spin transport in magnetic insulators.

Magnon spin transport driven by the magnon chemical potential in a magnetic insulator

1604.03706v1

Measurement of the intrinsic damping constant in individual nanodisks of YIG and YIG jPt C. Hahn,1V.V. Naletov,1, 2, 3G. de Loubens,1O. Klein,1O. d'Allivy Kelly,2A. Anane,2R. Bernard,2E. Jacquet,2 P. Bortolotti,2V. Cros,2J.L. Prieto,4and M. Mu~ noz5 1)Service de Physique de l' Etat Condens e (CNRS URA 2464), CEA Saclay, 91191 Gif-sur-Yvette, France 2)Unit e Mixte de Physique CNRS/Thales and Universit e Paris Sud 11, 1 av. Fresnel, 91767 Palaiseau, France 3)Institute of Physics, Kazan Federal University, Kazan 420008, Russian Federation 4)Instituto de Sistemas Optoelectr onicos y Microtecnolog a (UPM), Madrid 28040, Spain 5)Instituto de Microelectr onica de Madrid (CNM, CSIC), Madrid 28760, Spain (Dated: 4 June 2022) We report on an experimental study on the spin-waves relaxation rate in two series of nanodisks of diameter =300, 500 and 700 nm, patterned out of two systems: a 20 nm thick yttrium iron garnet (YIG) lm grown by pulsed laser deposition either bare or covered by 13 nm of Pt. Using a magnetic resonance force microscope, we measure precisely the ferromagnetic resonance linewidth of each individual YIG and YIG jPt nanodisks. We nd that the linewidth in the nanostructure is sensibly smaller than the one measured in the extended lm. Analysis of the frequency dependence of the spectral linewidth indicates that the improvement is principally due to the suppression of the inhomogeneous part of the broadening due to geometrical connement, suggesting that only the homogeneous broadening contributes to the linewidth of the nanostructure. For the bare YIG nano-disks, the broadening is associated to a damping constant = 4
104. A 3 fold increase of the linewidth is observed for the series with Pt cap layer, attributed to the spin pumping eect. The measured enhancement allows to extract the spin mixing conductance found to be G"#= 1:55
1014 1m2for our YIG(20nm)jPt interface, thus opening large opportunities for the design of YIG based nanostructures with optimized magnetic losses. Yttrium iron garnet (Y 3Fe5O12), commonly referred as YIG, is the champion material for magneto-optical appli- cations as it holds the highest gure of merit in terms of low propagation loss. It is widely used in high-end mi- crowave and optical-communication devices such as l- ters, tunable oscillators, or non-reciprocal devices. It is also the material of choice for magnonics1, which aims at using spin-waves (SW) (or their quanta magnons) to carry and process information. The development of this emerging eld is presently limited by the damping con- stant of SW. Recently it was proposed that spin-current transfer generated by spin Hall eect from an adjacent layer can partially or even fully compensate the intrinsic losses of the traveling SW beyond the natural decay time. The achievement of damping compensation by pure spin current in 1.3 m thick YIG covered by Pt was reported by Kajiwara et al.2, although attempts to reproduce the results have so far failed3,4. Given that the spin-orbit torque is purely interfacial in such hybrid system, it is crucial to work with nanometer-thick lms of epitaxial YIG. Primarly, because the interface spin-current trans- fer scales inversely with the YIG thickness5. Secondly, TABLE I. Magnetic parameters of the 20 nm thick YIG lm. 4M s(G) ex(nm) (rad.s1.G1) 2:1
1034
10415 1 :79
107because it permits nano-patterning of the YIG and thus the engineering of the spin-wave (SW) spectra through spatial connement6{8. To the best of our knowledge, however, there is no report yet9on the measurement of the dynamical properties on submicron size nanostruc- ture patterned out of YIG ultrathin lms. Beneting from our recent progress in the growth of very high quality YIG lms by pulsed laser deposition (PLD)4, here we will demonstrate that these ultra-thin YIG lms (thicknesses ranging from 20 to 4 nm) can be reliably nano-patterned into sub-micron size nano-disks. In the following, we will perform a comparative study of the linewidth on these nanostructures. We will ana- lyze the dierent contributions to the damping by sepa- rating the homogeneous from the inhomogeneous broad- ening and quantify the spin-pumping contribution when an adjacent metallic layer is added. We will also eval- uate the consequences of the damages produced by the lithographic process on the dynamical response of these devices. It will be shown that the dierent alterations produced by chemical solvent, heat treatment, amor- phization and redeposition of foreign elements during the lithography, edge roughness etc... do not produce any increase to the linewidth, oering great hope for incor- poration of these YIG lms in magnonics. The 20 nm thick YIG has been grown by PLD on a (111) Gd 3Ga5O12substrate following the preparation de- scribed in Ref.4. A 13 nm thick Pt layer was then de- posited by sputtering on half of the YIG surface. OnearXiv:1402.3630v1 [cond-mat.mtrl-sci] 15 Feb 20142 FIG. 1. Optical image of the YIG disks, with (red) and with- out (blue) Pt on top, placed below a microwave antenna. The direction of the microwave magnetic eld is indicated by a blue arrow. The external bias magnetic eld is oriented per- pendicularly to the surface. TABLE II. Comparative table of the measured and predicted (analytical and SpinFlow simulations)10resonance values for the SW modes. These predictions are uncorrected by the additional stray eld of the MRFM probe. The last row dis- play the SWs eigen-values obtained by adjusting both the radius of the disks (respectively = 700;520 and 380 nm) and the probe sample separation (respectively h= 3:0;2:0 and 1.5m). The SWs are labeled by their azimuthal and radial number ( `;n)10. (nm) modefexp(GHz) simu. analyt. t 700 (00) 8.74 8.54 8.69 8.76 700 (01) 9.14 9.04 9.10 9.16 700 (02) 9.65 9.66 9.68 9.72 500 (00) 9.10 8.73 8.90 9.05 500 (01) 9.72 9.51 9.63 9.69 500 (02) 10.52 10.51 10.72 10.65 300 (00) 9.58 9.19 9.47 9.51 300 (01) 10.57 10.66 11.25 10.63 slab of 15 mm was cut to perform standard magnetic hysteresis and ferromagnetic resonance (FMR) measure- ments. The measured magnetic parameters of the bare YIG lm are summarized in Table I. On the remaining piece, a series of YIG nanodisks has been subsquently patterned using standard electron lithography and dry ion etching. After the YIG lithography, an insulating layer of 50 nm SiO 2was deposited on the whole surface and a 150 nm thick and 5 m wide microwave Au-antenna was deposited on top. Fig. 1 shows an optical image of the sample and the antenna pattern. The series of de- creasing diameters bare YIG nanodisks is placed on the left. The spacing between the disks is 3 m. The series of disks on the right side mirrors the rst one and has a 13 nm Pt layer on top. Here we concentrate on the disks enclosed in the rectangular area, with nominal diameter =700, 500 and 300 nm. To measure the FMR-spectra of the nanodisks buriedunder the microwave antenna, we use a ferro-magnetic resonance force microscope (f-MRFM)11. It is based on measuring the de ection of a MFM-cantilever with a magnetic Fe particle of about 800 nm diameter axed to the tip. The tip magnetic dipole moment senses the stray eld produced by the perpendicular component M z of the magnetization of the magnetic nanodisks, which is modulated by the exciting microwave power at the me- chanical frequency of the cantilever. FIG. 2. Mechanical-FMR spectra at H0=4.99 kOe of the 700, 500 and 300 nm diameter YIG disks arranged in rows by decreasing lateral size. The spectra of the pure YIG disks are shown in the left column (blue), while the ones covered with 13 nm of Pt are shown in the right column (red). In Fig. 2 we show f-MRFM spectra recorded for dier- ent diameters (by row) on both the YIG and YIG jPt nan- odisks (by column). The spectra correspond to the spin- wave eigenmodes of the disks biased by a perpendicular magnetic eld H0= 4:99 kOe. The largest peak at low- est frequency stems from the lowest energy FMR-mode, the so-called uniform mode. The smaller peaks at higher energy correspond to higher order modes. The splitting corresponds to the quantization of the SW wavenumber /n= (nbeing an integer) in the radial direction11. One can thus infer from the peak separation the lateral size of the disk. Using the literature12,13value for the YIG exchange length  ex= 15 nm, a t of the peak separation leads to an eective connement of respec- tively 700, 520, 380 nm for our 3 disks, assuming total pinning at the disk edge (see Table II). This is in very good agreement with nominal sizes targeted by the pat- terning. Dierences with the nominal value are due both to imperfection of the lithographic process and the dipo- lar pinning condition14, which scales as the aspect ratio. Conning a spin wave in a smaller volume leads also to an overall increase of the exchange and self-dipolar energy, and thus a shift of the fundamental mode with increasing energy. We have checked that the position of the peak is compatible with the magnetic parameters shown is Ta- ble I assuming that the center of the probe is approached from 3.0 to 1.5 m when moved from the largest to the3 FIG. 3. (a) Low-power spectrum of the reference YIG lm recorded at f0= 8:2 GHz. Lineshape of the uniform mode measured at H0= 4:99 kOe on the 700 nm disk bare (b, blue circles) and with (c, red circles) Pt. (d) Dependence of the linewidth on the resonance frequency. smallest disk. From the comparative measurements presented in Fig. 2, we see that the peak position is not altered by the addition of the Pt layer ontop of the YIG disk whereas its linewidth is clearly increased. To emphasize this result, Fig. 3 shows the linear f-MRFM spectra of the 700 nm bare YIG (b, blue dots) and of the YIG jPt (c, red dots) disks. For comparison the spectrum measured on the extended YIG thin lm is shown in (a, black). It was recorded with a transmission line Au-antenna of 500 m width and an in-plane external eld oriented perpendic- ularly to the exciting microwave eld, similarly to the conguration used in Ref.3. A rst striking result of Fig. 3 is obtained by compar- ing the spectra measured in extended lm (a) and the nanostructure (b) for the bare YIG: the nanopatterning improves the linewidth7. One can observe that, while the lineshape of the resonance has a Lorentzian shape in the nanostruture (continuous line), the peak shape is asymetric in the YIG lm. We attribute it to inhomoge- neous broadening. This is conrmed by performing ex- periments on dierent slabs of the reference lm, which actually yield dierent lineshapes (not shown). The usual method of separating the homogeneous contribution from the inhomogeneous one is to study the frequency depen- dence of the half-linewidth
f=2. The slope gives , the Gilbert damping constant, while the zero frequency inter- cept gives the inhomogeneous contribution. In Fig. 3(d), we have thus plotted the full-linewidth
fof the ex- tended reference lm at dierent frequencies using thesame color as in Fig. 3(a). The linewidth varies almost linearly with the frequency. We extract from the slope the damping 4
104, while the intercept at zero fre- quency indicates the amount of inhomogeneity of the res- onance:
f=2.5 MHz (or
H=1 Oe). On the same Fig. 3(d), we show using red dots the frequency dependence of the linewidth of the YIG jPt nanostructure. The t of the slope yields YIGjPt= 13
104. The striking feature is that a linear t now in- tercepts with the origin of coordinates. It means that the linewidth measured in the nanostructures directly yields the homogeneous contribution. For the bare YIG nan- odisk it was only possible to reliably extract the linear linewidth at one point at 8.2 GHz (blue dot). Inter- estingly enough the slope of the straight line from this point to the origin is exactly that of the line tted to the extended lm data. We speculate by analogy to the Pt/YIG case that the linewidth in the YIG nanostructure is purely homogeneous in nature, while the intrinsic part of the damping has been unaected by the lithographic process. In Fig. 3, we also see that the linewidth of the 700 nm YIGjPt disk is 22 MHz (c) i.e.about three times wider than that of the 700 nm YIG disk, which is 7 MHz (b). The in uence of an adjacent Pt layer is shown to increase the damping threefold through spin-pumping eect15. The characteristic parameter for the eciency of spin transfer across the interface and the accompanying in- crease of damping in YIG is the spin mixing conductance G"#. One can directly evaluate the increase of damp- ing induced by the presence of the Pt layer. As we nd YIG= 4
104andYIGjPt= 13
104, we deduce a large spin-pumping contribution sp= 9
104that adds to the intrinsic damping : YIGjPt=YIG+sp. From this, one can calculate the spin mixing conductance according to16: G"#=sp4MSt gBG0; (1) wheret= 20 nm is the YIG thickness, gthe elec- tron Land e factor, Bthe Bohr magneton, and G0= 2e2=hthe quantum of conductance. This corresponds to:G"#= 1:55
1014 1m2for our YIGjPt interface. In summary, we have conducted a study of spin wave spectra of individual submicron YIG and YIG jPt disks. We nd that the litography process does not broaden the linewidth and on the contrary, the linewitdh decreases compared to the extended lm. The in uence of an adja- cent Pt layer on the YIG through the spin pumping eect is investigated and quantied to increase the damping 3 fold. As a non zero spin mixing conductance is deter- mined, these experiments pave the way for observation of inverse spin Hall eects in a YIG jPt nanodisk access- ing its individual spin wave modes. Most importantly, thanks to the small volume and purely intrinsic damp- ing of the YIG sample, we will address in future studies the in uence of direct spin Hall eect on the linewidth4 of these YIG nano-disks as it was demonstrated in all- metallic NiFe/Pt dots17. ACKNOWLEDGMENTS This research was supported by the French Grants Trinidad (ASTRID 2012 program) and by the RTRA Triangle de la Physique grant Spinoscopy. We also ac- knowledge usefull contributions from C. Deranlot, A.H. Molpeceres and R. Lebourgeois. 1A. A. Serga, A. V. Chumak, and B. Hillebrands, Journal of Physics D: Applied Physics 43, 264002 (2010) 2Y. Kajiwara, K. Harii, S. Takahashi, J. Ohe, K. Uchida, M. Mizuguchi, H. Umezawa, H. Kawai, K. Ando, K. Takanashi, S. Maekawa, and E. Saitoh, Nature (London) 464, 262 (2010) 3C. Hahn, G. de Loubens, O. Klein, M. Viret, V. V. Naletov, and J. Ben Youssef, Phys. Rev. B 87, 174417 (2013) 4O. d'Allivy Kelly, A. Anane, R. Bernard, J. Ben Youssef, C. Hahn, A. H. Molpeceres, C. Carretero, E. Jacquet, C. Der- anlot, P. Bortolotti, R. Lebourgeois, J.-C. Mage, G. de Loubens, O. Klein, V. Cros, and A. Fert, Applied Physics Letters 103, 082408 (2013) 5J. Xiao and G. E. W. Bauer, Phys. Rev. Lett. 108, 217204 (2012) 6J. Jorzick, S. O. Demokritov, B. Hillebrands, M. Bailleul, C. Fer- mon, K. Y. Guslienko, A. N. Slavin, D. V. Berkov, and N. L. Gorn, Phys. Rev. Lett. 88, 47204 (2002)7G. de Loubens, V. V. Naletov, O. Klein, J. B. Youssef, F. Boust, and N. Vukadinovic, Phys. Rev. Lett. 98, 127601 (2007) 8F. Guo, L. M. Belova, and R. D. McMichael, Phys. Rev. Lett. 110, 017601 (2013) 9P. Pirro, T. Brcher, A. V. Chumak, B. Lgel, C. Dubs, O. Surzhenko, P. Grnert, B. Leven, and B. Hillebrands, Applied Physics Letters 104, 012402 (2014), http://scitation.aip.org/ content/aip/journal/apl/104/1/10.1063/1.4861343 10V. V. Naletov, G. de Loubens, G. Albuquerque, S. Borlenghi, V. Cros, G. Faini, J. Grollier, H. Hurdequint, N. Locatelli, B. Pigeau, A. N. Slavin, V. S. Tiberkevich, C. Ulysse, T. Valet, and O. Klein, Phys. Rev. B 84, 224423 (Dec 2011), http: //link.aps.org/doi/10.1103/PhysRevB.84.224423 11O. Klein, G. de Loubens, V. V. Naletov, F. Boust, T. Guillet, H. Hurdequint, A. Leksikov, A. N. Slavin, V. S. Tiberkevich, and N. Vukadinovic, Phys. Rev. B 78, 144410 (2008) 12M. Sparks, Ferromagnetic relaxation theory (McGraw-Hill, New York, 1964) 13R. W. Damon and H. Van De Vaart, Journal of Applied Physics 36, 3453 (1965), http://scitation.aip.org/content/ aip/journal/jap/36/11/10.1063/1.1703018 14K. Y. Guslienko, S. O. Demokritov, B. Hillebrands, and A. N. Slavin, Phys. Rev. B 66, 132402 (Oct 2002), http://link.aps. org/doi/10.1103/PhysRevB.66.132402 15Y. Tserkovnyak, A. Brataas, and G. E. W. Bauer, Phys. Rev. B 66, 224403 (2002) 16B. Heinrich, C. Burrowes, E. Montoya, B. Kardasz, E. Girt, Y.-Y. Song, Y. Sun, and M. Wu, Phys. Rev. Lett. 107, 066604 (2011) 17V. Demidov, S. Urazhdin, H. Ulrichs, V. Tiberkevich, A. Slavin, D. Baither, G. Schmitz, and S. O. Demokritov, Nature Mater. (London) 11, 1028 (2012)

2014-02-15

We report on an experimental study on the spin-waves relaxation rate in two series of nanodisks of diameter $\phi=$300, 500 and 700~nm, patterned out of two systems: a 20~nm thick yttrium iron garnet (YIG) film grown by pulsed laser deposition either bare or covered by 13~nm of Pt. Using a magnetic resonance force microscope, we measure precisely the ferromagnetic resonance linewidth of each individual YIG and YIG{\textbar}Pt nanodisks. We find that the linewidth in the nanostructure is sensibly smaller than the one measured in the extended film. Analysis of the frequency dependence of the spectral linewidth indicates that the improvement is principally due to the suppression of the inhomogeneous part of the broadening due to geometrical confinement, suggesting that only the homogeneous broadening contributes to the linewidth of the nanostructure. For the bare YIG nano-disks, the broadening is associated to a damping constant $\alpha = 4 \cdot 10^{-4}$. A 3 fold increase of the linewidth is observed for the series with Pt cap layer, attributed to the spin pumping effect. The measured enhancement allows to extract the spin mixing conductance found to be $G_{\uparrow \downarrow}= 1.55 \cdot 10^{14}~ \Omega^{-1}\text{m}^{-2}$ for our YIG(20nm){\textbar}Pt interface, thus opening large opportunities for the design of YIG based nanostructures with optimized magnetic losses.

Measurement of the intrinsic damping constant in individual nanodisks of YIG and YIG{\textbar}Pt

1402.3630v1

IEEE MAGNETICS LETTERS, Volume 6 (2015) Page 1 of 4 (actually, 4 pages maximum; IEEE will place article number here) 1949-307X ©2015 IEEE. Personal use is permitted, bu t republication/redistribution requires IEEE permiss ion. See http://www.ieee.org/publications standards/publ ications/rights/index.html for more information. (In serted by IEEE) Magnetodynamics ___________________________________ ________________________ All-angle collimation for spin waves J. W. Kłos, P. Gruszecki, A. E. Serebryannikov, M. Krawczyk Faculty of Physics, Adam Mickiewicz University in P ozna ń, Umultowska 85, 61-614 Pozna ń, Poland Received xxx, revised xxx, accepted xxx, published xxx, current version xxx. (Dates will be inserted b y IEEE; “published” is the date the accepted preprint is posted on IEEE Xplore ®; “current version” is the date the typeset version is posted on Xplore ®). Abstract— We studied the effect of collimation for monochrom atic beams of spin waves, resulting from the refraction at the interface separating two magnetic half-planes. The collimation was observed in broad range of the angles of incidence for homogenous Co and Py half-planes, due to significant intrinsic anisotropy of spin wave propagation in these materi als. The effect exists for the sample saturated by in- plane magnetic field tangential to the interface. T he collimation for all possible angles of incidence was found in the system where the incident spin wave is refracted on the interface between homogeneous and periodically patterned layers of YIG. The refractio n was investigated by the analysis of isofrequency dispersion contours of both pairs materials, i.e., uniform YIG/patterned YIG and Co/Py, which are calculated with the aid of the plane wave method. B esides, the refraction in Co/Py system was studied using micromagnetic simulations. Index Terms— Magnetodynamics, Spin Waves, Collimation, Antidots Lattices I. INTRODUCTION The propagation of spin waves (SWs) in ferromagneti c films, with the saturated magnetic field applied in the plane of the film, is highly anisotropic. The dispersion rel ation (frequency vs. wave vector) for SWs varies with cha nges of direction of the wave vector with respect to the ex ternal magnetic field [Stancil 2009]. Such intrinsic aniso tropy is typical for magnonic systems [Gieniusz 2013]. The a nisotropy of propagation can be also introduced by periodic p atterning of the ferromagnetic film [Tacchi 2015]. This leads to the breaking of the symmetry of the dispersion relation even for the configuration which has initially isotropic dis persion, i.e. for a homogeneous film with out-of-plane oriented magne tic field that is known as forward configuration. The refraction effects on the interfaces between an isotropic media are complex and sometimes counterintuitive at specific conditions [Foteinopoulou 2003, Gralak 2000]. Such effects were intensively explored in photonic systems where the most common way to introduce the anisotropy is patternin g of the structure in the form of a photonic crystal [Joanno poulos 2008]. Here we adopt the ideas from photonics that are related to beam refraction [Luo 2002, Chigrin 2004] and collimation [Wu 2012] to magnonic systems, where th e intrinsic anisotropy of solid film (absent in photo nic systems) can also be exploited. The goal of this paper is to study possibility of collimation for SWs propagating in t hin ferromagnetic films, which is, so far, not complete ly utilized. In magnonics, the control of SWs propagation direction has been discussed in connection with the caustic effect [Ve erakumar Corresponding author: J. W. Kłos(klos@amu.edu.pl). 2006, Schneider 2010] where the two preferred directions of propagation exist. This is related to hyperbolic di spersion of in-plane magnetized thin films. We are interested i n collimation, when only one direction of propagation is supported. To achieve this effect we need a system characterized by dispersion relation which has flat isofrequency dispersion contour perpendicular to th e required direction of the wave propagation. This can be gain ed in periodic system with rectangular lattice with stro ng and week coupling in two orthogonal directions [Kumar 2015]. In our study, we obtain suitable dispersion by applying ma gnetic field in the plane of the system. It makes the propagatio n of spin waves in orthogonal directions distinguishable (due to introduction of anisotropic dipolar interactions). Moreover, we show that rectangular lattice is not necessary and collimation can be obtained for geometries of higher symmetry. We consider two cases when collimation can appear: i) due to peculiar dispersion relation of homogeneous ferr omagnetic film, and ii) due to tailored magnetic properties o f the ferromagnetic film by its regular patterning. In th e patterned medium the dispersion relation of SWs can be shaped by adjusting structural parameters. We show, that the periodicity of the patterned medium enables all-angle collimati on, i.e., beams incident on the interface between the homogen eous and the periodic medium may formally propagate alon g the normal direction in the exit medium even of the inc idence algle is close to 90° limit. We perform the analysis of t he refraction with the aid of isofrequency dispersion contours (I FCs) [Gralak 2003, Jiang 2013]. For the flat interface, the tang ential component of the wave vector and, hence, the tangen tial component of the phase velocity must be conserved. Using this condition, the direction of the group velocity of the refracted wave can be found. It is determined by th e direction Page 2 of 4 IEEE MAGNETICS LETTERS, Volume 6 (2015) ————————————————————————————————————– normal to the IFC at the point on this contour wher e the tangential component of the phase velocity is the s ame as in the medium from which the wave is incoming. In turn , the directions of incidence and refracted beams are giv en by the directions of the corresponding group velocities. T herefore, the direction of the beam refraction can be deduced by the geometrical analysis of the IFCs shape for two homo geneous media at the opposite sides of the flat interface. Similar analysis can be performed for the wave incident on the interface between homogeneous and periodic media. However, in this case the wave vector can be shifte d by any reciprocal lattice vector’s tangential to the inter face due to discrete translational symmetry along the interface . We’ve used this approach to adjust the geometry of the sy stem and find the IFCs suitable for obtaining the all-angle collimation in yttrium iron garnet (YIG) based antidots lattice. In this letter, we will discuss the geometry of pla nar magnonic systems with a single internal interface. Then, we consider the SW dispersion and micromagnetic simula tions (MSs) results for a system comprising only homogene ous films. Finally, we present a detailed analysis of d ispersion for a system containing a square-lattice magnonic cryst al. The paper will be finished with short conclusion. II. THE MODEL We consider two planar systems that consist of two half- planes. The first system is composed of two uniform half- planes of Co and Py films (see Fig.1b). For this sy stem the collimation is limited to some range of incidence a ngels of SWs propagating from the side of Co. In turn, the a ll-angle collimation is showed for the structure made of the YIG with one half-plane in the form of a homogeneous film an d the second half-plane in the form of a square lattice o f cylindrical antidots (see Fig.2a and Fig.3c). The interface wit h solid YIG is along one of the principal directions of the squ are lattice, i.e., is parallel to the x-axis. The magnetic field H0 is applied along the interface for both systems. To describe the SW dynamics, we used Landau-Lifshit z equation (LLE) with exchange and dipolar interactio ns included. The two-dimensional dispersion relations (for both homogeneous and patterned media) were calculated us ing plane wave method (PWM) for the saturation state of the static magnetization [Rychly 2015]. The simulations of refraction and reflection of the Gaussian beams wer e performed with the aid of MSs using Mumax3 [Vanstee nkiste 2014] package solving the full LLE based on finite- difference time-domain method. Detailed algorithm of MSs and especially description of Gaussian beams of SWs gen eration in MSs is presented in [Gruszecki 2015]. III. THE RESULTS AND DISCUSSION The collimation effect can be observed for a magne tic planar system without periodic patterning. Fig.1 pr esents the simulated refraction of the Gaussian beam of SWs on the interface between Co and Py. Both materials, and es pecially Co, are anisotropic in terms of SW propagation. The other factor which increases the anisotropy of Co (in ref erence to Py) is the fact that the selected frequency, 20 GHz , is much closer to the FMR frequency of Co than to that of P y, due to higher magnetization saturation. As a result we obs erve the highly anisotropic IFC for Co (drawn in red in Fig. 1a), exhibiting hyperbolic sections. The IFC for Py has stadium-like shape. To observe the collimation of SWs incident f rom Co after refraction at the interface with Py, we use t he nearly flat sections of the stadium-like IFC (drawn in blue in Fig. 1a). However, the observed collimation is restricted her e to some range of incidence angles around the direction norm al to the interface, i.e., for the SWs with small tangential component of the phase velocity. This condition means that we ar e using the hyperbolic section of the IFC in Co in order to obt ain collimation in Py. The limited collimation incidenc e angle is close to the parabolic point [Chigrin 2004] of IFCs , which delimits the areas of the hyperbolic dispersion. Fig. 1. (color online) Refraction of the SWs with f requency 20 GHz on the interface between Co and Py films (a) from disp ersion results obtained by using PWM and (b) from MSs where the am plitude of the out-of-plane component of magnetization was visuali zed. The thickness of the films is 12 nm; the external magne tic field (0.15 T) is applied along the interface. Directions of the inci dent and reflected beams are marked by red arrows, direction of the re fracted beam is marked by black arrow, for Co and Py respectively. The directions of phase velocity are marked by orange and gray arrows , for Co and Py, respectively. θi and φi denote the angle of incidence and the angle spanned between wave-vector ki and normal to the interface. Note the difference in divergence of the outgoing beams in C o and Py. Fig. 1b demonstrates refraction and reflection of t he SW's Gaussian beam at the interface between the Co and P y thin films that are simulated with the aid of Mumax3 pac kage. Both films were 12 nm thick, 8 µm wide and 8 µm lon g (discretized using 8 nm × 8 nm × 12 nm rectangular unit cells), adjusting each other at one straight horizo ntal interface. The system was assumed to be uniformly a nd stably magnetized, and simulated under the presence of a small (0.15 T), static, in-plane, directed parallel to the interface, magnetic field. In the Co film, far from the edges, the Gaussian beam of SWs was continuously excited, with the width of 0.5 µm and at the frequency f=20 GHz. The wave vector of the incident beam, ki, creates an angle φi=45° with the normal to the interface. To stay in the linear regime, the amplitude of the excitation was much smaller than t he saturation magnetization of Co. In calculations, ty pical magnetic parameters were assumed: saturation IEEE MAGNETICS LETTERS, Volume 6 (2015) Page 3 of 4 ————————————————————————————————————– magnetizations MS,Co =1.45 ×10 6 A/m, MS,Py =0.7 ×10 6 A/m and exchange constants Aex,Co =3 ×10 -11 J/m, Aex,Py =1.1 ×10 -11 J/m for Co and Py, respectively. Simulations of the SWs propagation were carried out at a finite value of t he damping parameter, α=0.0005, for the Co and Py films. In Co the phase velocity of SWs (marked by orange arrow, being perpendicular to the wave-fronts and p arallel to the wave vector ki) is almost perpendicular to the group velocity (marked by red arrow and denoting the dire ction of an energy propagation), see Fig.1b. This effect is a result of the hyperbolic-like IFC (red line in Fig.1a) occurr ing in Co with the chosen parameters. Similar analysis of MSs’ res ults and IFC of Py shows that the angle between group and ph ase velocities is much smaller, but still significant. As seen in Fig.1b, the group velocity in Py is almost perpendi cular to the interface in a range of the angles φi up to ~60°. This can be concluded from Fig.1a and was confirmed by multiple MSs for various angles of incidence. The difference in divergence of the beams propagati ng in Co and Py is worth to notice. It results from diffe rent values of phase velocities for SWs of the same frequency in C o and Py. On both sides of Co/Py interface we have then d ifferent ratios of wavelength to width of spin wave beam, i. e. in the case of Co the ratio is greater and due to that div ergence of beam is stronger [Saleh 2007]. Nevertheless, in met allic ferromagnets, especially in Co, the damping is high . This significantly limits the proposed collimation scena rio for practical realization. Collimation of SWs can be obtained and the mentione d restriction related to damping can be mitigated wit h the use of magnonic crystals based on thin YIG film [Yu 2014]. The two dimensional plot of the dispersion relation of the planar square, antidots lattice based on YIG film in Fig.2 a is presented in Fig.2b. For the magnetic field applied in-plane, the dispersion is strongly anisotropic for lower ma gnonic bands [Kłos 2012], where dipolar interactions preva il over the exchange ones. The magnetic field applied in the x-direction supports the propagation of SWs (non-zero slope of the dispersio n relation) along the y-direction in low frequency range. This is mostly caused by the profile of demagnetizing field loweri ng the effective magnetic field in the rows between antido ts [Tacchi 2015] which are aligned in the y-direction. It is also clearly visible in Fig.2b, where the narrower frequency gap s and wider bands resulting in larger group velocities ar e observed along this direction. The top of the first band and the bottom of the second band have a peculiar shape, i.e., the di spersion is almost independent on the kx component of the wave vector. This means that group velocity, being the gradient of the dispersion relation, is directed almost exactly alo ng the y- direction, independently on the direction of the co rresponding phase velocity. The Bloch waves of different k-vectors will then propagate mostly in the y-direction, enabling collimation. To investigate this effect, we’ve considered the re fraction of the plane waves entering the patterned YIG film from the homogeneous region of this material. We have chosen arbitrarily chosen on of the frequencies of 9.1 GHz , laying within the second band in the center of the region where the frequency of SWs is almost independent on kx and varies almost linearly with ky (see white dashed line in Fig.2b). The top of the first band could also be considered to g ain this effect but this band is much narrower and character ized by lower group velocities and much narrower range of l inear decadence of the frequency on ky component of the wave vector. This basically limits its application to co llimation of monochromatic beams only. Fig.3a presents the IFCs for the homogeneous film o f YIG (red line) and for the patterned one (black lines, extracted from the magnonic band structure shown in Fig.2b). The IFC for the homogeneous layer of YIG is approximately c ircular because of the weak intrinsic anisotropy. It is cau sed by small magnetization saturation of YIG and a comparatively high value of the considered frequency, which is signifi cantly above the FMR frequency. Therefore the phase velocity and group velocity are almost collinear for the SWs at this and higher frequencies. Fig. 2. (color online) (a) The structure of the mag nonic antidots lattice in the form of YIG film of thickness 12 nm patterne d by cylindrical antidots (diameter 84.6 nm) arranged in square latt ice (lattice constant 150 nm). The values of magnetic parameters for YIG are: MS=0.194x10 6 A/m, Aex,Co =0.40 ×10 -11 J/m. (b) Dispersion relation for the considered structure with external field applie d along one principal direction of antidots lattice (i.e. x-direction), calculated using PWM. The figure presents four lowest (and the part of fi fth) magnonic bands over the whole first Brillouin zone. The white dash ed line in (b) marks the IFC of the SW at 9.1 GHz. The IFCs at 9.1GHz for the antidots lattice in Fig. 3a are approximately straight and stretch between the boun daries of the first Brillouin zone. Due to periodicity of the dispersion relation in dependence on the wave vector, the IFCs are extended infinitely in the reciprocal space, withou t any gaps. Also the group velocity (marked by black arrows) al most coincides with one of two opposite defections (y or –y) and has a nearly constant value at any point of the IFC . This peculiar shape of the IFC is responsible for all-an gle collimation. Indeed, there is no restriction regard ing the tangential component of the phase velocity that wou ld result from the dispersion of the patterned YIG and make o btaining of this effect impossible. Therefore, the SWs incom ing form the homogeneous YIG propagate in the direction norm al to the interface, independently on their incidence ang le. It is illustrated in Fig.3b, where IFCs are shown togethe r with Page 4 of 4 IEEE MAGNETICS LETTERS, Volume 6 (2015) ————————————————————————————————————– directions of the phase and group velocities in the media on the both sides of the virtual interface, for three selected incidence angles. One more illustration of the coll imation effect is schematically presented in Fig.3c, where the directions of incident and refracted waves are impo sed on the sketch of the considered structure. The studies bas ed on geometrical analysis of the shape of IFCs can be supplemented by the MSs results for refraction of G aussian beams, as it was done above for the interface of tw o homogenous materials, i.e. Co and Py. Fig. 3. (color online) (a) The IFCs for the antidot s lattice presented in Fig. 2a and solid thin film of YIG at the frequency 9.1 GHz are shown in the wave vector plane with black and red solid l ines, respectively. Arrows show directions of the group velocity at dif ferent values of kx. (b), (c) Schematics of all-angle collimation effect arising due to the flat IFCs for antidots lattice; beams coming from the so lid film side are collimated in the patterned YIG film with antidots lattice into one beam propagating along the normal to the virtual interfa ce (θref =0), independently on the incidence angle θin . The angle φin corresponds to the angle spanned between exemplary wave vector (or phase velocity) of incident SWs and normal to the interfa ce. IV. CONCLUSION We have demonstrated the effect of all-angle collim ation for SWs in YIG based square-lattice magnonic crystal. I t is shown that for the SWs incidents from the homogeneous reg ion of YIG at an arbitrary angle, the outgoing wave propag ates in the properly patterned area of the same material in the normal direction. This enables collimation without any res triction, except for that related to the coupling efficiency at the virtual interface. We supplemented this investigation by th e analysis of SW refraction on the interface between two homog eneous materials: Co and Py. It was found in this structur e, that the collimation effect can exist in a limited range of the incidence angle variation, being caused by intrinsic anisotro py of magnetic films. ACKNOWLEDGMENT The funding from Polish National Science Centre (DEC-2-12/07/E/ST3/00538) and EU’s Horizon2020 prog ramme under the Marie Skłodowska-Curie grant No644348 is acknowledge d. REFERENCES Chigrin D N (2004), “Radiation pattern of a classic al dipole in a photonic crystal: Photon focusing,” Phys. Rev. E 70, 056611, doi: 10. 1103/PhysRevE. 70.056611. Foteinopoulou S, Soukoulis C M (2003), “Electromagne tic wave propagation in two-dimensional photonic crystals: A study of anomal ous refractive effects,” Phys. Rev. A 79, 033829, doi: 10.1103/PhysRevB.72.1 65112. Gieniusz R, Ulrichs H, Bessonov V D, et al. (2013), ” Single antidot as a passive way to create caustic spin-wave beams in yttrium iro n garnet films,” Appl. Phys. Lett. 102, 102409, doi:10.1063/1.4795293. Gralak B, Enoch S, and Tayeb G (2006), Superprism ef fects and EGB antenna applications“ chap.10 in ”Metamaterials - Physics an d Engineering Explorations,” Eds. Engheta N, Ziolkowski W R, John Wiley & Sons, pp. 261-284. Gruszecki P, Dadoenkova Yu S, Dadoenkova N N, et. a l (2015),“Influence of magnetic surface anisotropy on spin wave reflection from the edge of ferromagnetic film,” Phys. Rev B 92,054427, doi:10. 1103/PhysRevB.92. 054427. Gralak B, Enoch S, and Tayeb G (2000), “Anomalous r efractive properties of photonic crystals,” J. Opt. Soc. Am. A 17, 1012. Jiang B, Zhang Y, Wang Y, et. al (2012), “Equi-freq uency contour of photonic crystals with the extended Dirichlet-to-Neumann wav e vectoreigenvalue equation method,” J. Phys. D: Appl. Phys. 45, 06530 4, doi:10.1088/0022- 3727/45/6/065304. Joannopoulos J D, Johnson S G, Winn J N, et. al (20 08 ), “Photonic Crystals:Molding the Flow of Light,” Princeton University Press. Kumar D, Adeyeye A O (2015), “Broadband and total au tocollimation of spin waves using planar magnonic crystals,” J. Appl. Phy s. 117, 143901, doi: 10.1063/1.4917053 . Kłos J W, Sokolovskyy M L, Mamica S, et. al (2012), ”The impact of the lattice symmetry and the inclusion shape on the spectrum of 2 D magnonic crystals,” J. Appl. Phys. 111, 123910, doi:10.1063/ 1.4729559. Luo C, Johnson S G, Joannopoulos J D, and Pendry J B (2002), “All-angle negative refraction without negative effective inde x,” Phys. Rev. B 65,201104, doi: 10.1103/PhysRevB.65.201104. Rychły J, Gruszecki P, Mruczkiewicz M, et. al (2015), ”Magnonic crystals — prospective structures for shaping spin waves in na noscale,” Low Temp. Phys. 41, 959. Saleh B E A, Teich M C (2007), “Beam optics” chap.3 in ”Fundamentals of Photonics, 2nd Edition,” Eds. Saleh B E A, Teich M C , John Wiley & Sons, pp. 75-85. Schneider T, Serga A A, Chumak A V, et. al, “Nondiff ractive Subwavelength Wave Beams in a Medium with Externally Controlled An isotropy,” (2010) Phys. Rev. Lett. 104, 197203, doi: 10.1103/PhysRevL ett.104.197203 . Stancil D, Prabhakar A (2009), “ Spin waves – theory and applications ,” Springer. Tacchi S, Gruszecki P, Madami M, el. al (2015), “Univ ersal dependence of the spin wave band structure on the geometrical charact eristics of two- dimensional magnonic crystals,” Sci. Rep. 5, 10367, doi:10.1038/srep10367. Vansteenkiste A, Leliaert J, Dvornik N, et. al (201 4), ”The design and verification of MuMax3,” AIP Adv. 4, 107133, doi: 10.1063/1.4899 186. Veerakumar V and Camley R E (2006), “Focusing of Spi n Waves in YIG Thin Films,” IEEE Trans. Mag. 42, 3318, doi: 10.1109/TMAG. 2006.879624 Wu Z H, Xie K, Yang H J, et. al (2012), “All-angle self-collimation in two- dimensional rhombic-lattice photonic crystals,” J. O pt. A 14, 15002, doi:10.1088/2040-8978/14/1/015002. Yu H, Kelly O A, Cros V, et. al (2014) “Magnetic thi n-film insulator with ultra-low spin wave damping for coherent nanomagnonics,” Sci. Rep. 4, 6848, doi: 10.1038/srep06848.

2017-03-16

We studied the effect of collimation for monochromatic beams of spin waves, resulting from the refraction at the interface separating two magnetic half-planes. The collimation was observed in broad range of the angles of ncidence for homogenous Co and Py half-planes, due to significant intrinsic anisotropy of spin wave propagation in these materials. The effect exists for the sample saturated by in plane magnetic field tangential to the interface. The collimation for all possible angles of incidence was found in the system where the incident spin wave is refracted on the interface between homogeneous and periodically patterned layers of YIG. The refraction was investigated by the analysis of isofrequency dispersion contours of both pairs materials, i.e., uniform YIG/patterned YIG and Co/Py, which are calculated with the aid of the plane wave method. Besides, the refraction in Co/Py system was studied using micromagnetic simulations.

All-Angle Collimation for Spin Waves

1703.05548v1

In uence of interface condition on spin-Seebeck eects Z. Qiu1, D. Hou1, K. Uchida2;3, and E. Saitoh1;2;4;5 1WPI Advanced Institute for Materials Research, Tohoku University, Sendai 980-8577, Japan 2Institute for Materials Research, Tohoku University, Sendai 980-8577, Japan 3PRESTO, Japan Science and Technology Agency, Saitama 332-0012, Japan 4CREST, Japan Science and Technology Agency, Tokyo 102-0076, Japan 5Advanced Science Research Center, Japan Atomic Energy Agency, Tokai 319-1195, Japan E-mail: qiuzy@imr.tohoku.ac.jp Abstract. The longitudinal spin-Seebeck eect (LSSE) has been investigated for Pt/yttrium iron garnet (YIG) bilayer systems. The magnitude of the voltage induced by the LSSE is found to be sensitive to the Pt/YIG interface condition. We observed large LSSE voltage in a Pt/YIG system with a better crystalline interface, while the voltage decays steeply when an amorphous layer is introduced at the interface articially. PACS numbers: 85.75.d, 72.25.b, 72.15.Jf, 73.50.Lw Keywords : Interface structure, spin-Seebeck eect, thermoelectric conversion, spin currentarXiv:1410.1655v1 [cond-mat.mtrl-sci] 7 Oct 2014Author guidelines for IOP journals in LATEX 2" 2 1. Introduction The spin-Seebeck eect (SSE) enables the generation of a non-equilibrium spin current by a temperature gradient [1{21]. When a conductor is attached to a magnetic material, the SSE induces a spin-current injection into the conductor. The SSE is of crucial importance in spintronics [22, 23] and spin caloritronics [24, 25], since it enables simple and versatile generation of spin currents from heat. The spin current induced by the SSE is converted into electric voltage via the inverse spin-Hall eect (ISHE) owing to the spin orbit interaction in the conductor [20, 21, 26{35]. The combination of SSE and ISHE provides a new way of thermoelectric conversion. Although this phenomenon has been observed in many magnetic materials [4{13], the thermoelectric conversion eciency of the SSE device is still small and its improvement is essential for realizing future spin-current-driven thermoelectric generators [10]. The thermoelectric voltage in a SSE device is aected by several factors, such as the spin Hall angle of the conductor, the quality and magnetic propeties of the magnetic material, and the conductor/magnet interface condition. The conductor/magnet interface condition in SSE devices should govern the injection eciency of a thermal spin current across the interface, which is described in terms of the spin-mixing conductance [19,21]. In our previous work, we found that the spin-mixing conductance of a Pt/yttrium iron garnet (YIG) spin-pumping system is sensitive to the atomic-scale interface condition, and a well-controlled and highly-crystalline interface is a prerequisite for a large spin-mixing conductance [28]. However, few works have involved the in uence of the interface condition on the SSE [36], and it is yet to be investigated systematically. In this work, we systematically study the in uence of the interface condition on the SSE in Pt/YIG systems, which have been widely used for investigating the SSE [8{18]. We controlled the Pt/YIG interface condition in atomic scale, and measured the ISHE voltage induced by the SSE with changing the interface conditions. These measurements enable to form a perceptual understanding of the in uence of the interface condition on the SSE and of the spin-injection mechanism at the conductor/magnet interface. 2. Experimental In this study, we employ a longitudinal conguration to investigate the SSE in the Pt/YIG system [8,17]. The longitudinal SSE (LSSE) device consists of a ferromagnetic or ferrimagnetic insulator covered with a paramagnetic conductor lm (Fig. 1). When a temperature gradient rTis applied to the ferromagnetic layer perpendicular to the paramagnet/ferromagnet interface, a spin current is injected into the paramagnetic layer via the SSE. Then an ISHE-induced electric eld EISHEis generated in the paramagnetic layer according to the relation EISHE/Js; (1) where Js, and denote the spatial direction of the thermally generated spin current and the spin-polarization vector of electrons in the paramagnetic layer, respectively.Author guidelines for IOP journals in LATEX 2" 3 V H TPt YIG xyzΔ Figure 1. A schematic illustration of LSSE. rTandHdenote the temperature gradient and the magnetic eld, respectively. The preparation process of our Pt/YIG samples is as follows. The YIG lms were grown on (111) gadolinium gallium garnet (GGG) substrates by using a liquid phase epitaxy (LPE) method in PbO-B 2O3 ux at the temperature of 1210 K. The thickness of the YIG lms is about 4.5 m. The surfaces of the YIG lms were annealed under oxygen pressure of 5 105Torr at 1073 K for 2 hours to improve the crystal structure, or were bombarded by ion beams to create amorphous layers [28]. After the annealing or ion beam bombardment process, 10 nm-thick Pt lms were deposited on those YIG lms with the same condition by using a pulse laser deposition system at room temperature in the same vacuum chamber. The crystalline characterization of the Pt/YIG samples was carried out by means of an X-ray diraction (XRD) method and a high-resolution transmission electron microscopy (TEM). The lattice constant was calculated from the XRD and electron diraction patterns. The magnetization ( M)-magnetic eld ( H) curve of the YIG lm was measured by using a vibrating sample magnetometer. The LSSE measurements were performed by using an experimental system similar to that described in Ref. [17]. The Pt/YIG sample with the size of 2 6 mm2was sandwiched between two AlN heat baths, of which the temperatures were stabilized to 300 K+
Tand 300 K, respectively. The temperature gradient was generated by a Peltier thermoelectric module. The electric voltage dierence Vbetween the ends of the Pt layer was measured by using a micro-probing system, where a distance between the two probes is about 5 mm. 3. Results and discussion First of all, we evaluate the quality of the YIG lms grown by the LPE method. Figure 2(a) shows the TEM image of the YIG/GGG interface; the image conrms that the YIG lm is epitaxially grown on the GGG substrate, and no defect is observed owing to the small lattice-constant mismatch between YIG and GGG [37,38]. The XRD patterns of the YIG lm in Fig. 2(b) show a good single crystal characteristic, where only the (444)Author guidelines for IOP journals in LATEX 2" 4 
ON Figure 2. (a) A cross-sectional high-resolution TEM image of the YIG/GGG interface. (b) The comparison of the XRD patterns for the YIG lm and the GGG substrate. The inset shows the XRD pattern for the YIG lm in a large 2 region. (c) The comparison of the rocking curves between the YIG lm and the GGG substrate. (d) TheM-Hcurve for the YIG lm. peak is observed. The lattice constants of the YIG lm and GGG substrate determined by the XRD results are 12.376 A and 12.384 A, respectively, both of which are in good agreement with the values found in previous literature [37, 38]. The YIG lms have almost the same crystal quality as the GGG substrate because of the same full width at half maximum of the rocking curves (Fig. 2(c)). The M-Hcurve of the YIG lm in Fig. 2(d) shows that the coercive force of the YIG lm is very small (smaller than 5 Oe) and the saturation magnetization is 4 Ms=1710 G (close to the bulk value [39]). These sample characterizations clearly conrm that our YIG lms are of very high quality. Figure 3(a) shows the cross-sectional structure of the prepared Pt/YIG interface measured by a high-resolution TEM. The crystal perfection of the YIG surface was found to be well kept even after forming the Pt lm; the YIG layer has a perfect single- crystalline structure with a [111] direction perpendicular to the Pt/YIG interface. The Pt layer has a typical polycrystalline structure without a preferred orientation. The Pt/YIG interface used in this study has the same high quality as that shown in Ref. [28]. By using such Pt/YIG devices, we measured the LSSE voltage VLSSE. In Fig. 3(b), we show VLSSE in the Pt/YIG sample with the highly-crystalline interface as a function of the temperature dierence
Tat the magnetic eld H=100 Oe, measured when the magnetic eld was applied along the + ydirection. As shown in Fig. 3(b), the sign of VLSSE for nite values of
Tin the Pt/YIG device is reversed by reversing therTdirection. This result indicates that the observed voltage signals are attributed to the LSSE, on the basis that the direction of the thermally generated spin current at the Pt/YIG interface is reversed by reversing
T. Figure 3(c) shows the voltage signal Vas a function of Hfor various values of
T. The sign of Vis reversedAuthor guidelines for IOP journals in LATEX 2" 5 by reversing Hat nite values of
T, consistent with the feature of the ISHE-induced electric eld (see Eq. (1)). The shape of the V-Hcurves is in good agreement with that of theM-Hcurve for the YIG lm (compare Figs. 2(b) and 3(c)), conrming that the observed voltage Vis associated with the magnetization reversal of the YIG layer. Now we investigate the in uence of the Pt/YIG interface condition on the LSSE voltage. To compare the magnitude of the LSSE voltage, we normalize VLSSE with the temperature dierence
T. TheVLSSE=
Tfor the Pt/YIG sample with the highly- crystalline interface is about 3.6 V=K. This value is 45% larger than that for the sample without the annealing process (Fig. 4(d)), in which an amorphous layer with the thickness of <1 nm was observed at the Pt/YIG interface (Fig. 4(a)). This enhancement ofVLSSE=
Tis attributed to the improvement of the spin mixing conductance, which determines the transfer eciency of spin currents across the Pt/YIG interface. [28] To further investigate the relation between the interface condition and the LSSE voltage, we measure the dependence of VLSSE on the thickness of the amorphous interlayer in the Pt/YIG samples. We found that the thickness of the amorphous layer at the Pt/YIG interface can be changed by ion-beam bombardment; the thickness of the amorphous layer increases with increasing the acceleration voltage (P ion) of the ion beam (Fig. 4 (a), (b) and (c)). Although it is dicult to quantitatively evaluate the properties of the amorphous layers, we believe that its macroscopic magnetization is extremely depressed because of the broken crystal structure, while localized spins may be alive because of the presence of iron atoms. TheHdependence of VLSSE=
Tfor various amorphous-layer thicknesses are plotted in Fig. 4(d). The magnitude of VLSSE=
Tin the Pt/YIG samples decreases exponentially with increasing the thickness dof the amorphous layer, indicating that the amorphous layer blocks the spin injection from YIG to Pt. By tting the d dependence of VLSSE=
Twith an exponential function Aexp (d=) withAbeing an adjustable parameter, the decay length is estimated to be 2.3 nm. The estimated  value for the amorphous layer is greater than those for other non-magnetic insulator oxide barriers estimated from the spin-pumping measurements using Pt/oxide/YIG systems [40], implying that the remaining localized spins in the amorphous layer may reduce a potential barrier for spin currents [41]. Nevertheless, our experimental results show that clean Pt/YIG interfaces are necessary for realizing ecient spin injection. 4. Conclusion In this work, the in uence of the interface crystal structure on the SSE has been investigated by using Pt/YIG bilayer LSSE devices. We found that the ISHE voltage induced by the LSSE is very sensitive to the interface condition of the YIG lm. The large LSSE voltage was found to be achieved by forming a highly-crystalline Pt/YIG interface in an atomic scale. Our systematic experiments show that the magnitude of the LSSE voltage decreases exponentially with increasing the thickness of the articial amorphous layer formed at the top surface of the YIG lm.Author guidelines for IOP journals in LATEX 2" 6 VLSSE Figure 3. (a) A cross-sectional high-resolution TEM image of the Pt/YIG interface, of which the YIG was annealed before the Pt layer was formed. (b) The
Tdependence ofVLSSE for the Pt/YIG LSSE device, shown in (a), measured when a temperature gradientrTis applied along the + z(left) andz(right) directions. (c) Hdependence ofVfor the Pt/YIG LSSE device, shown in (a), for various values of
T. Acknowledgements This work was supported by CREST \Creation of Nanosystems with Novel Functions through Process Integration", PRESTO \Phase Interfaces for Highly Ecient Energy Utilization", Strategic International Cooperative Program ASPIMATT from JST, Japan, Grant-in-Aid for Young Scientists (B) (26790038), Young Scientists (A) (25707029), Challenging Exploratory Research (26600067), Scientic Research (A) (24244051), Scientic Research on Innovative Areas \Nano Spin Conversion Science" (26103005) from MEXT, Japan, the Tanikawa Fund Promotion of Thermal Technology, the Casio Science Promotion Foundation, and the Iwatani Naoji Foundation.Author guidelines for IOP journals in LATEX 2" 7 d~7 nm d~14 nm d~1 nm 5 nm 0 5 10 15 1234 Exp. A exp( d / λ) d (nm) VLSSE /∆T (µV/K) -100 -50 0 50 100 -4 -2 024 Anealled No treatment Pion =1.5 kV Pion =2.0 kV Pion =3.0 kV Pion =4.0 kV H (Oe) V/∆T (µV/K) (a) (d) (e)(b) (c) Figure 4. (a)-(c) The cross-sectional high-resolution TEM images of the Pt/YIG samples, of which the YIG lms were prepared without any anealling process (a), with the ion beam bombardment at 2 kV-acceleration voltage (b), and with the ion beam bombardment at 4 kV-acceleration voltage (c). The ion bombardment was performed before the Pt layers were formed. (d) The Hdependence of V=
Tin the Pt/YIG devices with various interfaces. (e) The dependence of VLSSE=
Ton the amorphous- layer thickness d. The solid curve is the tting result using an exponential function Aexp (d=), whereAandare adjustable parameters. References [1] J C Slonczewski. Current-driven excitation of magnetic multilayers. Journal of Magnetism and Magnetic Materials , 159(12):L1 { L7, 1996. [2] H Adachi, K Uchida, E Saitoh, J Ohe, S Takahashi, and S Maekawa. Gigantic enhancement of spin Seebeck eect by phonon drag. Applied Physics Letters , 97(25):252506, 2010. [3] K Uchida, H Adachi, T An, T Ota, M Toda, B Hillebrands, S Maekawa, and E Saitoh. Long-range spin Seebeck eect and acoustic spin pumping. Nature Materials , 10(10):737{741, 2011. [4] K Uchida, S Takahashi, K Harii, J Ieda, W Koshibae, K Ando, S Maekawa, and E Saitoh. Observation of the spin Seebeck eect. Nature , 455(7214):778{781, 2008. [5] C M Jaworski, J Yang, S Mack, D D Awschalom, J P Heremans, and R C Myers. Observation of the spin-Seebeck eect in a ferromagnetic semiconductor. 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Nanoscale Spin Seebeck Rectier: Controlling Thermal Spin Transport across Insulating Magnetic Junctions with Localized Spin. Physical Review B , 89(21):214407, June 2014.

2014-10-07

The longitudinal spin-Seebeck effect (LSSE) has been investigated for Pt/yttrium iron garnet (YIG) bilayer systems. The magnitude of the voltage induced by the LSSE is found to be sensitive to the Pt/YIG interface condition. We observed large LSSE voltage in a Pt/YIG system with a better crystalline interface, while the voltage decays steeply when an amorphous layer is introduced at the interface artificially.

Influence of interface condition on spin-Seebeck effects

1410.1655v1

Spin Seebeck effect and ballistic transport of quasi-acoustic magnons in room-temperature yttrium iron garnet films Timo Noack1, Halyna Yu. Musiienko-Shmarova1, Thomas Langner1, Frank Heussner1, Viktor Lauer1, Bj¨ orn Heinz1, Dmytro A. Bozhko1, Vitaliy I. Vasyuchka1, Anna Pomyalov2, Victor S. L’vov2, Burkard Hillebrands1 and Alexander A. Serga1 1Fachbereich Physik and Forschungszentrum OPTIMAS, Technische Univer sit¨ at Kaiserslautern, 67663 Kaiserslautern, Germany 2Department of Chemical Physics, Weizmann Institute of Science, Rehovot 76100, Israel E-mail: tnoack@rhrk.uni-kl.de Abstract. We studied the transient behavior of the spin current generated by the longitudinal spin Seebeck effect (LSSE) in a set of platinum-coated yt trium iron garnet (YIG) films of different thicknesses. The LSSE was induced by means of pulsed microwave heating of the Pt layer and the spin currents w ere measured electrically using the inverse spin Hall effect in the same layer. We demonstrate that the time evolution of the LSSE is determined by the evolution of the thermal gradient triggering the flux of thermal magnons in t he vicinity of the YIG/Pt interface. These magnons move ballistically within th e YIG film with a constant group velocity, while their number decays exponentiall y within an effective propagation length. The ballistic flight of the magnon s with energies above 20 K is a result of their almost linear dispersion law, similar to t hat of acoustic phonons. By fitting the time-dependent LSSE signal for different film thicknesses varying by almost an order of magnitude, we found that th e effective propagation length is practically independent of the YIG film thick ness. We consider this fact as strong support of a ballistic transport scenario – t he ballistic propagation of quasi-acoustic magnons in room temperature YIG. Keywords : Spin Seebeck effect, magnons, spin diffusion, yttrium iron g arnet, ballistic transport PACS numbers: 75.30.Ds., 75.40.Gb, 75.47.Lx, 75.50.Gg, 75.70 .-i, 75.76.+j arXiv:1802.09593v1 [cond-mat.mes-hall] 26 Feb 2018Spin Seebeck effect and ballistic transport of quasi-acoust ic magnons in room-temperature YIG films 2 1. Introduction A permanently growing interest in the field of spin-caloritronics, which combines thermoelectrics with spintronics and nanomagnetism, underlines the importance of spin currents [1] as an alternative to charge currents for the utilization in logic devices [2, 3, 4]. This is due to zero Joule heating and the wide spectrum of methods to generate and manipulate spin currents. The spin current may arise in the form of charge currents with opposite flow directions for spin up and spin down carriers, or it can consist of magnons – the quanta of collective spin excitations [3, 5]. Among other methods, the magnon current can be created by a thermal gradient induced in a ferromag- net exposed to a magnetic field [6, 7]. This effect is re- ferred to as the longitudinal spin Seebeck effect (LSSE) [8, 9]. Despite extensive studies, the exact microscopic mechanism responsible for the magnon-mediated spin Seebeck effect (SSE) is not yet completely clarified. In particular, some fundamental transport properties of magnetic materials, related to the LSSE, require fur- ther experimental and theoretical clarification. Among them is the timescale of the formation of the LSSE. The temporal dynamics of the LSSE is tightly connected with fundamental properties of a magnon gas such as the magnon mean free path. This physical quantity is crucial for the understanding of the transport proper- ties of magnetic materials and the general peculiarities of magnon-phonon interaction [10, 11, 12, 13, 14, 15], as well as for the engineering of efficient LSSE-based spin-caloritronic devices [16, 17]. Typically, the spin Seebeck effect is indirectly de- tected by an electric voltage in a thin film of a heavy normal metal (e.g. Pt, Pd, W) deposited on the surface of a magnetic material. This voltage appears as a re- sult of the conversion of a spin current into an electric one by the inverse spin Hall effect (ISHE) due to spin- dependent electron scattering. The temporal charac- teristics of ISHE voltages are determined both by the electron dynamics [18] in the metal and by the magnon dynamics in the magnetic media. It has been experimentally demonstrated [19] by using coherently excited magnons, that in low-damping magnetic materials, such as epitaxial YIG films, the temporal profile of the ISHE voltage is dominated by the magnon dynamics in the magnetic insulator, rather than by the very fast electron dynamics in the nor- mal metal. Similarly, it has been shown [20] that the LSSE dynamics is strongly influenced by the transport of thermal magnons inside the magnetic material and, thus, depends on the temporal development of the tem- perature gradient in the magnetic material close to the YIG/Pt interface. Recently, a number of studies on the time depen- dency of the LSSE in YIG/Pt bilayers have been pre- Figure 1. a) A schematic view of the experimental setup used for the investigation of the temporal evolution of the LSSE voltage. b) Sketch of the sample holder with the mounted YIG/Pt sample. c) Structure of the YIG/Pt sample, showing the relative orientation of the bias magnetic field H, the thermal gradient ∇T(t, x) created by the microwave heating of the platinum cover, the magnon carried spin current Js(t, x), and the electric field EISHE(t) induced in the Pt layer due to the inverse spin Hall effect [9]. sented [20, 21, 22, 23, 24, 25] and a rise time of the LSSE-induced ISHE voltage ( VLSSE) of a few hundred nanoseconds was reported. Furthermore, a magnon propagation length of about L∼500 nm was deter- mined from the time-dependent measurements [20, 22] in a 6.7 µm thick YIG film, using a diffusion model of thermally driven magnons. However, these values of the propagation length are not unanimous and vary in the literature [20, 23, 25, 26, 27, 28] between several hundred nanometers and a few micrometers. In this article, we study the transient evolution of the LSSE voltage VLSSE in YIG films of different thick- nesses. Using a modified magnon transport model [20] with the time decay defined by a ballistic flight model, we find that the magnon propagation length is prac- tically independent from the YIG film thickness and represents a material property of YIG. 2. Experimental setup A schematic representation of the experimental setup is shown in Fig. 1. In this experiment, the microwave- induced heating of the Pt layer is used to create the thermal gradient in YIG/Pt bilayers [22]. The Pt film is heated by eddy currents, induced in the metal by a microwave field. This field is created by a 600 µm wide microstrip transmission line placed below the samples. In all measurements, the microstrip is driven by a microwave generator that produces microwaves at aSpin Seebeck effect and ballistic transport of quasi-acoust ic magnons in room-temperature YIG films 3 fixed frequency of 6.875 GHz. The 10 µs-long microwave heating pulses are ap- plied with a repetition rate of 1 kHz to provide the system with sufficient time for cooling down after ev- ery heating cycle. The applied microwave power is set to 30 dBm. To avoid possible reflections of microwave energy, a matched 50 Ohm load is connected at the end of the microstrip line. The samples are placed on top of the microwave antenna with the platinum layer fac- ing downwards [see Fig. 1(b)]. A few micrometer thick insulating layer was used to prevent galvanic contact between the platinum and the microstrip line. All samples used in the experiments have the same structure, shown in Fig. 1(c): The YIG films of dif- ferent thicknesses were grown in the (111) crystallo- graphic plane by liquid phase epitaxy on a 500 µm thick gadolinium gallium garnet (GGG) substrate. A 10 nm- thick Pt layer was deposited on top of the YIG films using sputter deposition. The thermal gradient generates a spin current across the YIG/Pt interface, shown in Fig. 1(c) by a violet arrow. Due to the spin-dependent scattering of spin polarized conducting electrons in the Pt layer, this spin current is converted into a charge current in-plane to the metal layer. This effect is known as the in- verse spin Hall effect (ISHE) and leads to an electric potential perpendicular to the external magnetic field. Therefore, the resulting DC-voltage is proportional to the number of magnons transferring its angular mo- mentum to electrons at the YIG/Pt interface. The ISHE voltage is given by: VISHE∝ΘSHE(JS×σ)l, where Θ SHEis the spin Hall angle, which defines the ef- ficiency of the ISHE in platinum, JSis the spin current, σis the spin polarization and l= 3 mm is the distance between the electric contacts, as shown in Fig. 1(b). Due to the small amplitude of the ISHE voltage of several microvolts, the electric signal from the YIG/Pt sample is amplified by using a DC voltage amplifier. Before the amplification, the signal is sent through a low-pass filter (DC-400 MHz) to avoid a possible disturbance of the sensitive receiving circuit by strong microwave pulses. Finally, the pulsed DC signal is displayed on the oscilloscope together with a reference microwave pulse, which has been reflected from the sample and afterwards directed to a HF diode detector by a microwave circulator (cf. Fig. 1(a)). Figure 1(b) shows the geometry of the sample holder. The magnetic field His oriented in the sample plane along the microstrip direction. The magnetic field strength is H= 250 Oe for all measurements. This value was chosen to avoid both resonant and parametric excitations of spin waves in the YIG film by the microwave magnetic field. The absence of such excitation processes is evidenced by the fact that the shapes of the observed LSSE pulses were identicalfor 250 Oe and for over 2500 Oe, where the spin-wave spectrum is shifted up so high that the frequency of the input microwave pulses lies well below the bottom of the spectrum. 3. Time-resolved investigation of the YIG thickness dependent temporal behavior of the longitudinal spin Seebeck effect The main goal of this work is the investigation of the time dependent behavior of the LSSE voltage for differ- ent YIG thicknesses. The experiment has been carried out for eleven different samples with YIG thicknesses between 150 nm and 53 µm. For each sample, the evo- lution of the LSSE-induced ISHE voltage VLSSE result- ing from a 10 µs-long heating pulse is analyzed. In order to eliminate nonmagnetic thermoelectric contri- butions to VLSSE, the LSSE-induced ISHE voltage was measured twice for opposite orientations of the mag- netic field H. Next, the difference V+H LSSE -V−H LSSE is calculated. Since the LSSE changes its sign with the magnetic field orientation, this voltage difference gives twice the value of the LSSE voltage with accompanying static effects be removed. The normalized time profiles of LSSE voltages for the selected YIG thicknesses are shown in Fig. 2. The time evolution of the voltages clearly depends on the YIG-layer thickness. For the thinnest samples with a YIG thickness of 150 nm (not shown) and 300 nm, al- most rectangular profiles are observed (cf. Fig. 2). For these thicknesses, the voltage is rising almost instan- taneously and reaches its saturation level already after a few nanoseconds. With increasing YIG thickness, the voltage profiles deviate from the rectangular shape. The slower rising time and later saturation of the sig- Figure 2. Temporal evolution of VLSSE for different YIG thick- nesses d. Four dependencies VLSSE (t, d) for d= 0.3, 1.26, 6.7 and 30µm are shown by solid lines. The LSSE voltage build-up be- comes visibly slower with increasing film thickness. The dashed lines represent the simulation of the LSSE dynamics using Eq. 1.Spin Seebeck effect and ballistic transport of quasi-acoust ic magnons in room-temperature YIG films 4 nals is clearly correlated with the YIG film thickness in the range from 150 nm to 23 µm. In contrast, for the three thickest YIG films (23 µm, 30 µm and 53 µm) we observe practically identical LSSE signal profiles. These results can be understood by considering the motion of the magnons in the sample. The LSSE is a magnon transport process driven by the temperature gradient, leading to a spatial distribution of the ther- mally excited magnons. A flow of magnons arises to compensate this spatial inhomogeneity. Since the heat- ing of the Pt-layer is homogeneous in the contact area, the thermal gradient is created perpendicular to the interface. Thus, the diffusion process is also oriented perpendicular to the interface and therefore limited by the thickness of the YIG layer. The magnons, excited farther away from the YIG/Pt interface, have to prop- agate a different distance than the magnons excited in its vicinity. Hence, they contribute to the LSSE volt- age with smaller amount at a later point in time. This effect is particularly pronounced for thick YIG sam- ples. Additional influence on the transient behavior of the LSSE voltage can be attributed to the thickness- dependence of the evolution of the thermal gradients in the samples. The contributions of these factors are discussed below. 4. Extraction of the effective magnon propagation length To obtain more quantitative predictions regarding the VLSSE time dependence and the magnon propagation lengths, we performed simulations of the heat dy- namics for the sample structure shown in Fig. 1(c). This was done by using the numerical simulation soft- ware COMSOL MultiphysicsR/circlecopyrtand by solving the one- dimensional heat transport problem analytically. The used thermal parameters of the materials are taken from Ref. [25]. The results of both calculations are nearly identical and consistent with the previous cal- culations in Ref. [20]: The rather slow temperature changes in the YIG film, developing on the millisec- ond time scale, are accompanied by a fast nano- and microsecond dynamics of the temperature gradient ∇T, which strongly depends on the distance from the YIG/Pt interface x(see Fig. 3). In Refs. [20, 25], the spin Seebeck voltage was considered as a combination of interface and YIG bulk effects. Taking into account that the rise time of the thermal gradient at the interface is much faster than the observed rise time of the LSSE and as a first attempt to understand the data, we use here only the bulk effect for the fitting of the measured data: VLSSE(t)∝/integraldisplayd 0∇Tx(x, t) exp/parenleftbigg−x L/parenrightbigg dx . (1)Figure 3. Spatial distribution of the temperature gradient ∇Tx(x, t) analytically calculated at different moments of time tfor the sample structure shown in Fig. 1(c). Here Lis the effective magnon propagation length, tis the time and dis the thickness of the YIG film. The length Lin Eq. 1 should be determined by some interactions in the YIG films. Without going into details of the magnon-magnon, magnon-phonon and other interactions, one expects that Lshould be independent of the film thickness, as long as L≪d. This expectation is a general physical requirement. For instance, in real gases, the mean free path of the atoms or molecules does not depend on the size of the con- tainer as long as it is much smaller than the container size. This requirement will serve us as a criterion in comparing different dynamical models for the LSSE signals evolution. Using the simulated temporal evolution of ∇T, we calculated the bulk term for different magnon propa- gation lengths L. Finally we used the method of least squares to determine the best value of Lfor every pro- file. The measured dynamics of the normalized spin Seebeck voltage is plotted in Fig. 2 together with the simulated bulk terms dynamics. By using this fitting procedure, we determined the effective magnon propa- gation length Lfor every YIG film thickness with an ac- curacy of 20 nm. The obtained values of Lare shown in Fig. 4 by empty blue diamonds. These values clearly in- crease with the thickness of the investigated YIG films. The observed behavior obviously contradicts the previously mentioned physical requirement. Most likely it is caused by the fact that such an impor- tant factor as the magnon propagation dynamics is not accounted in Eq. 1. In fact, it is supposed that the magnons which are driven somewhere in the sam- ple by a local temperature gradient are immediately contributing to the ISHE. In order to improve the model we need to involve the propagation time of these magnons to the YIG/Pt interface τmand to take intoSpin Seebeck effect and ballistic transport of quasi-acoust ic magnons in room-temperature YIG films 5 Figure 4. The magnon propagation length L, defined using three different models: The simple “no-delay” model [20, 22] Eq. (1) (empty blue diamonds), the magnon delay model Eq. (2) with diffusive magnon delay (empty black triangles) and the ballistic propagation model (filled red circles). The horizontal dashed line marks the saturation level L= 435 nm. The numbers near the vertical dotted lines represent the thickness of the measured YIG films in micrometers. account their movement through a spatially varying temperature gradient. Following Hioki et al. [25], the magnon delay can be accounted by a simple modifica- tion of Eq. 1: VLSSE(t)∝/integraldisplayd 0∇Tx(x, t−τm) exp/parenleftbigg−x L/parenrightbigg dx . (2) The exact form of τmwas defined in Ref. [25] in the framework of a diffusion propagation model as τm= x2/(lmvm), where lmis the mean free path of the ther- mal magnons and vmis their group velocity. This ve- locity can be calculated using experimental [29, 30] and theoretical [31, 32, 33, 34, 35] data of the magnon spec- tra in YIG. It is known, that starting from about 1 THz (≈20 K) and up to the end of the first Brillouin zone at about 6.5 THz ( ≈300 K), the lowest magnon branch, which is mostly populated at room temperature, has an almost linear dispersion relation ω(k). Therefore, for the thermal magnons in this branch, the magnon ve- locity is constant in a wide frequency range. Its value along the [111] crystallographic direction, normal to the surfaces of all of our films, is about vm= 104m/s. The question about the value of the magnon mean free path lmis much more complicated. In Ref. [15, 25], a value of lm≈1 nm was assumed at room tempera- ture. Taking these values of lmand vm, we fitted our experimental data by Eq. 2. The resulting behavior of L, shown by empty black triangles in Fig. 4, is qual- itatively the same as in the “no delay” case of Eq. 1. This result is not very surprising: The 1 nm value for lmwas obtained by Boona and Heremans [15] by com- parison of phonon and magnon contributions to the specific heat and to the thermal conductivity of a bulkYIG sample and by a consequent interpolation of the low temperature (2 K–20 K) magnon-related data to the room temperature range. However, the relevance of such an interpolation is not completely clear due to the strong differences in the relaxation and spec- tral characteristics of low- and high-energy magnons [36]. Boona and Heremans emphasized that their “es- timate is conservative, especially at room temperature, where SSE experiments are typically conducted”. We also have to notice that the diffusive delay in Eq. 2 accounts exclusively for the diffusive spreading of the magnon package. This makes the applicability of the simplified diffusion model in Eq. 2 rather questionable. On the other hand, the linear dispersion relation of the lowest magnon mode allows us to suggest an al- ternative “ballistic” model of the magnon propagation. In the ballistic approach, the magnon delay is defined asτm=x/vm. At room temperature the magnons pop- ulate the lowest magnon mode and at high frequencies their velocity obeys the relation: ω(k)≈vmk. Indeed, for the “acoustic” magnons with ω(k) =vmk, the con- servation laws for the dominant four-magnon scattering ω(k1) +ω(k2) =ω(k3) +ω(k4), k1+k2=k3+k4, (3) are satisfied only if k1∝ba∇dblk2∝ba∇dblk3∝ba∇dblk4, i.e. when all magnons propagate in the same direction, with the same velocity vm=k1/k1. The same condition k1∝ba∇dblk2∝ba∇dblk3is correct for the three-magnon processes ω(k1) +ω(k2) =ω(k3), k1+k2=k3, (4) as well as for the processes with any other number of magnons (see, e.g., Chapter 1 in the book [37]). This means that the package of magnons propa- gates ballistically with the velocity vmand all types of interaction processes within the “acoustic” magnon mode with the linear dispersion law do not change the direction and the value of the propagation veloc- ityvm, leading only to an evolution of the package shape in the k-space during the ballistic flight. In ad- dition, the dominating four-magnon processes (Eq. 3) preserve the total number of magnons in the pack- age, while the three-magnon processes, that change their number, are much less probable for the exchange magnons. In such a situation, the two-particle scat- tering on crystal defects alongside with the Cherenkov radiation [10] and the four-magnon interaction between the “acoustic” magnons and the “optical” magnons with ω(k)≈const .can be seen as the dominant mecha- nism, restricting the propagation length of the thermal magnons. In view of the aforesaid, we used the ballistic model to fit the experimental VLSSE waveforms and to determine the propagation length Lfor the different YIG thicknesses. The result is presented in Fig. 4 bySpin Seebeck effect and ballistic transport of quasi-acoust ic magnons in room-temperature YIG films 6 the filled red circles. Due to the rather weak depen- dence of the obtained values on the YIG-film thickness, the length Lwas determined with an accuracy of 1 nm. As it is clearly seen in the figure, as the YIG thickness grows by a fraction of eight (from 150 nm to 1.2 µm), the propagation length increases only by about 3.5% (from 410 nm to 428 nm). The magnon propagation length Lis therefore almost independent of the YIG film thickness and can be considered as an inherent property of YIG. Under such an assumption the prop- agation length should become independent of the film thickness for L≪d, as observed: For the films with a thickness d≥4.1µm,Lsaturates near 435 nm (see Fig. 4). This value agrees well with both our previ- ous estimations [20, 22] and the results of other groups [26, 38]. The slight increase in Lin the smaller thick- ness range can be related to, e.g., the relative decrease in the density of crystal defects at large distances from the YIG/GGG interface in thicker epitaxial YIG films. 5. Summary In this article, we studied the influence of the magnetic insulator thickness in YIG/Pt bilayers on the tempo- ral dynamics of the longitudinal spin Seebeck effect (LSSE). A microwave-induced heating technique has been used to generate a thermal gradient across the bi- layer interface. The experiment demonstrates a strong dependence of the time evolution of the LSSE signal on the magnetic layer thickness. An increase of the YIG thickness from 150 nm to 53 µm leads to a 7-fold increase in the rise time of the detected LSSE voltage. The experimental data have been precisely fitted us- ing a model which assumes ballistic motion of thermal magnons in a temperature gradient. The average magnon propagation length of about 425 nm was found to be almost independent of the YIG film thickness. This fact strongly supports the sug- gested simple ballistic model of the magnon propaga- tion in room-temperature YIG films. Acknowledgments Financial support by Deutsche Forschungsgemein- schaft (DFG) within Priority Program 1538 “Spin Caloric Transport” (project SE 1771/4-2) and DFG project INST 248/178-1 as well as technical support from the Nano Structuring Center, TU Kaiserslautern are gratefully acknowledged. References [1] Maekawa S, Valenzuela S O, Saitoh E and Kimura T(eds) 2017 Spin Current, 2nd ed. (Series on Semiconductor Science and Technology ) (Oxford: Oxford University Press) p 544[2] Sinova J and ˇZuti´ c I 2012 New moves of the spintronics tango Nat. Mater. 11368 [3] Chumak A V, Vasyuchka V I, Serga A A and Hillebrands B 2015 Magnon spintronics Nat. Phys. 11453 [4] Sato K and Saitoh E (eds) 2015 Spintronics for Next Generation Innovative Devices (Wiley Series in Materials for Electronic & Optoelectronic Applications ) (Chichester: John Wiley & Sons) p 280 [5] Kajiwara Y, Harii K, Takahashi S, Ohe J, Uchida K, Mizuguchi M, Umezawa H, Kawai H, Ando K, Takanashi K, Maekawa S and Saitoh E 2010 Transmis- sion of electrical signals by spin-wave interconversion in a magnetic insulator Nature 464262 [6] Uchida K, Xiao J, Adachi H, Ohe J, Takahashi S, Ieda J, Ota T, Kajiwara Y, Umezawa H, Kawai H, Bauer G E W, Maekawa S and Saitoh E 2010 Spin Seebeck insulator Nat. Mater. 9894 [7] Bauer G E W, Saitoh E and van Wees B J 2012 Spin caloritronics Nat. Mater. 11391 [8] Uchida K, Adachi H, Ota T, Nakayama H, Maekawa S and Saitoh E 2010 Observation of longitudinal spin-Seebeck effect in magnetic insulators Appl. Phys. Lett. 97172505 [9] Schreier M, Bauer G E W, Vasyuchka V I, Flipse J, Uchida K, Lotze L, Lauer V, Chumak A V, Serga A A, Daimon S, Kikkawa T, Saitoh E, van Wees B J, Hillebrands B, Gross R and Goennenwein S T B 2015 Sign of inverse spin Hall voltages generated by ferromagnetic resonance and temperature gradients in yttrium iron garnet |platinum bilayers, J. Phys. D: Appl. Phys. 48025001. [10] R¨ uckriegel A, Kopietz P, Bozhko D A, Serga A A and Hillebrands B 2014 Magnetoelastic modes and lifetime of magnons in thin yttrium iron garnet films Phys. Rev. B89184413 [11] Bozhko D A, Clausen P, Melkov G A, L’vov V S, Pomyalov A, Vasyuchka V I, Chumak A V, Hillebrands B and Serga A A 2017 Bottleneck accumulation of hybrid magnetoelastic bosons Phys. Rev. Lett. 118237201 [12] Sandweg C W, Kajiwara Y, Chumak A V, Serga A A, Vasyuchka V I, Jungfleisch M B, Saitoh E and Hillebrands B 2011 Spin pumping by parametrically excited exchange magnons Phys. Rev. Lett. 106216601 [13] Uchida K, Adachi H, An T, Ota T, Toda M, Hillebrands B, Maekawa S and Saitoh E 2011 Long-range spin Seebeck effect and acoustic spin pumping Nat. Mater. 10737 [14] Kikkawa T, Shen K, Flebus B, Duine R A, Uchida K, Qiu Z, Bauer G E W and Saitoh E 2016 Magnon polarons in the spin Seebeck effect Phys. Rev. Lett. 117207203 [15] Boona S R and Heremans J P 2014 Magnon thermal mean free path in yttrium iron garnet Phys. Rev. B 9064421 [16] Langner T, Kirihara A, Serga A A, Hillebrands B and Vasyuchka V I 2017 Damping of parametrically excited magnons in the presence of the longitudinal spin Seebeck effect Phys. Rev. B 95134441 [17] Jungfleisch M B, An T, Ando K, Kajiwara Y, Uchida K, Vasyuchka V I, Chumak A V, Serga A A, Saitoh E and Hillebrands B 2013 Heat-induced damping modification in yttrium iron garnet/platinum hetero-structures Appl. Phys. Lett. 102, 062417 [18] Stern N P, Steuerman D W, Mack S, Gossard A C and Awschalom D D 2008 Time-resolved dynamics of the spin Hall effect Nat. Phys. 4843 [19] Jungfleisch M B, Chumak A V, Vasyuchka V I, Serga A A, Obry B, Schultheiss H, Beck P A, Karenowska A D, Saitoh E and Hillebrands B 2011 Temporal evolution of inverse spin Hall effect voltage in a magnetic insulator- nonmagnetic metal structure Appl. Phys. Lett. 99, 182512 [20] Agrawal M, Vasyuchka V I, Serga A A, Kirihara A, Pirro P, Langner T, Jungfleisch M B, Chumak A V,Spin Seebeck effect and ballistic transport of quasi-acoust ic magnons in room-temperature YIG films 7 Papaioannou E Th and Hillebrands B 2014 Role of bulk-magnon transport in the temporal evolution of the longitudinal spin-Seebeck effect Phys. Rev. B 89224414 [21] Roschewsky N, Schreier M, Kamra A, Schade F, Ganzhorn K, Meyer S, Huebl H, Gepr¨ ags S, Gross R and Goennenwein S T B 2014 Time resolved spin Seebeck effect experiments Appl. Phys. Lett. 104202410 [22] Agrawal M, Serga A A, Lauer V, Papaioannou E Th, Hillebrands B and Vasyuchka V I 2014 Microwave- induced spin currents in ferromagnetic-insulator |normal- metal bilayer system Appl. Phys. Lett. 105092404 [23] Ritzmann U, Hinzke D and Nowak U. 2014 Propagation of thermally induced magnonic spin currents Phys. Rev. B 89024409 [24] Schreier M, Kramer F, Huebl H, Gepr¨ ags S, Gross R, Goennenwein S T B, Noack T, Langner T, Serga A A, Hillebrands B and Vasyuchka V I 2016 Spin Seebeck effect at microwave frequencies Phys. Rev. B 93224430 [25] Hioki T, Iguchi R, Qiu Z, Hou D, Uchida K and Saitoh E 2017 Time-resolved study of field-induced suppression of longitudinal spin Seebeck effect Appl. Phys. Express 10 73002 [26] Kehlberger A, Ritzmann U, Hinzke D, Guo E-J, Cramer J, Jakob G, Onbasli M C, Kim D H, Ross C A, Jungfleisch M B, Hillebrands B, Nowak U and Kl¨ aui M 2015 Length scale of the spin Seebeck effect Phys. Rev. Lett. 11596602 [27] Cornelissen L J, Liu J, Duine R A, Youssef J Ben and Van Wees B J 2015 Long-distance transport of magnon spin information in a magnetic insulator at room temperature Nat. Phys. 111022 [28] Chavez-Angel E, Zarate R A, Fuentes S, Guo E J, Kl¨ aui M and Jakob G 2017 Reconstruction of an effective magnon mean free path distribution from spin Seebeck measurements in thin films New J. Phys. 19013011 [29] Plant J S 1977 Spinwave dispersion curves for yttrium iron garnet J. Phys. C: Solid State Phys. 10(1977) 4805 [30] Princep A J, Ewings R A, Ward S, T´ oth S, Dubs C, Prabhakaran D and Boothroyd A T 2017 The full magnon spectrum of yttrium iron garnet npj Quantum Mater. 263 [31] Brinkman W F and Elliott R J 1966 Space group theory for spin waves J. Appl. Phys. 371457 [32] Kolokolov I V, L’vov V S and Cherepanov V B 1983 Spin-wave spectra and thermodynamics of yttrium iron garnet—a twenty-sublattice ferrimagnet Sov. Phys.– JETP 57605 [33] Cherepanov V, Kolokolov I and L’vov V 1993 The saga of YIG: spectra, thermodynamics, interaction and relaxation of magnons in a complex magnet Phys. Rep. 22981 [34] Barker J and Bauer G E W 2016 Thermal spin dynamics of yttrium iron garnet Phys. Rev. Lett. 117217201 [35] Xie L-S, Jin G-X, He L, Bauer G E W, Barker J and Xia K 2017 First-principles study of exchange interactions of yttrium iron garnet Phys. Rev. B 9514423 [36] Kolokolov I V, L’vov V S and Cherepanov V B 1984 Magnon interaction and relaxation in yttrium iron garnet, a twenty-sublattice ferromagnet Sov. Phys.– JETP 591131 [37] L’vov V S 1994 Wave Turbulence Under Parametric Excita- tion: Application to Magnets (Springer Series in Nonlin- ear Dynamics) (Springer-Verlag Berlin Heidelberg) p 330 [38] Althammer M 2018 Pure spin currents in magneti- cally ordered insulator/normal metal heterostructures arXiv:1802.08479v1

2018-02-26

We studied the transient behavior of the spin current generated by the longitudinal spin Seebeck effect (LSSE) in a set of platinum-coated yttrium iron garnet (YIG) films of different thicknesses. The LSSE was induced by means of pulsed microwave heating of the Pt layer and the spin currents were measured electrically using the inverse spin Hall effect in the same layer. We demonstrate that the time evolution of the LSSE is determined by the evolution of the thermal gradient triggering the flux of thermal magnons in the vicinity of the YIG/Pt interface. These magnons move ballistically within the YIG film with a constant group velocity, while their number decays exponentially within an effective propagation length. The ballistic flight of the magnons with energies above 20K is a result of their almost linear dispersion law, similar to that of acoustic phonons. By fitting the time-dependent LSSE signal for different film thicknesses varying by almost an order of magnitude, we found that the effective propagation length is practically independent of the YIG film thickness. We consider this fact as strong support of a ballistic transport scenario - the ballistic propagation of quasi-acoustic magnons in room temperature YIG.

Spin Seebeck effect and ballistic transport of quasi-acoustic magnons in room-temperature yttrium iron garnet films

1802.09593v1

Magnetoresistance of heavy and light metal/ferromagnet bilayers Can Onur Avci, Kevin Garello, Johannes Mendil, Abhijit Ghosh, Nicolas Blasakis, Mihai Gabureac, Morgan Trassin, Manfred Fiebig, and Pietro Gambardella Department of Materials, ETH Z urich, H onggerbergring 64, CH-8093 Z urich, Switzerland (Dated: 14 July 2018) We studied the magnetoresistance of normal metal (NM)/ferromagnet (FM) bilayers in the linear and nonlinear (current-dependent) regimes and compared it with the amplitude of the spin-orbit torques and thermally induced electric elds. Our exper- iments reveal that the magnetoresistance of the heavy NM/Co bilayers (NM = Ta, W, Pt) is phenomenologically similar to the spin Hall magnetoresistance (SMR) of YIG/Pt, but has a much larger anisotropy, of the order of 0 :5 %, which increases with the atomic number of the NM. This SMR-like behavior is absent in light NM/Co bi- layers (NM = Ti, Cu), which present the standard AMR expected of polycrystalline FM layers. In the Ta, W, Pt/Co bilayers we nd an additional magnetoresistance, directly proportional to the current and to the transverse component of the mag- netization. This so-called unidirectional SMR, of the order of 0.005 %, is largest in W and correlates with the amplitude of the antidamping spin-orbit torque. The unidirectional SMR is below the accuracy of our measurements in YIG/Pt. 1arXiv:1510.06285v1 [cond-mat.mes-hall] 21 Oct 2015The interconversion of charge and spin currents is a central theme in spintronics. In normal metal (NM)/ferromagnet (FM) bilayers, the conversion of a charge current into a spin current driven by the spin Hall (SHE)1and Rashba-Edelstein eects2leads to strong spin- orbit torques (SOT),3{11which are widely studied for their role in triggering magnetization switching7,12,13, magnetic oscillations14, and related applications.15,16Additionally, it has been shown that the spin currents induced by a charge current can have a signicant back- action on the longitudinal charge transport, leading to changes of the resistance of NM/FM bilayers that depend on the relative orientation of the magnetization in the FM and spin- orbit coupling (SOC) induced spin accumulation in the NM.17{23 A direct unequivocal demonstration of such a back-action eect is the spin Hall magne- toresistance (SMR) reported for FM insulator /NM bilayers, namely YIG/Pt and YIG/Ta,17{21 in which complications due to the anisotropic magnetoresistance (AMR) of metallic FM are either absent or restricted to proximity eects in the NM.24For a charge current directed alongx, the SMR is proportional to m2 y, the square of the in-plane component of the mag- netization transverse to the current, and is typically of the order of 0.01-0.1 % of the total resistance. In the simplest spin diusion model, the SMR is associated to the re ection (absorption) of a spin current at the NM/FM interface when the spins are collinear (or- thogonal) to the FM magnetization, leading to an increase (decrease) of the conductivity due to the inverse SHE in the NM layer.17SMR-like behavior has been observed also in all metal NM/FM systems such as Pt/Co/Pt, Pt/NiFe/Pt, Pt/Co, Ta/Co, and W/CoFeB layers.22,25{28In this case, however, the SMR cannot be easily singled out due to the AMR of the FM and magnetoresistive contributions induced by spin scattering at the NM/FM interface independent of the SHE.25 Recently, an additional magnetoresistance has been reported in Pt/Co and Ta/Co bilay- ers, which depends in magnitude and sign on the product ( j^z)
m, where jis the current density and mthe unit vector of the magnetization in the FM.22This expression describes a resistance that depends linearly on the applied current and my(Fig. 1a), and is therefore a nonlinear eect as opposed to the SMR and AMR, which are both current-independent and proportional to m2 yandm2 x, respectively, as imposed by the Onsager relations in the linear transport regime.29This so-called unidirectional spin Hall magnetoresistance (USMR) is associated to the modulation of the NM/FM interface resistance due to the SHE-induced spin accumulation, similar to the mechanism leading to the current-in-plane giant magne- 2toresistance in FM/NM/FM multilayers, but orders of magnitude smaller.22The USMR depends on the thickness of the NM and is about 0.002-0.003 % of the total resistance in Ta/Co and Pt/Co for j= 107A/cm2. An analogous eect has been reported in param- agnetic/ferromagnetic GaMnAs bilayers, where the USMR is signicantly larger (0.2 % for j= 106A/cm2) due to the much smaller conductivity of semiconductors relative to metals.23 These studies reveal that nonlinear phenomena must be taken into account to achieve a full description of the charge-spin conversion in NM/FM systems. New insight may be gained by comparing such eects to the magnetoresistance and SOT, particularly on the nature of the interface resistance, spin accumulation, and material parameters governing them. Further, the USMR oers a way of detecting the magnetization direction of a single FM layer using a two terminal geometry that is otherwise not accessible by conventional magnetoresistance eects. Understanding the role of dierent NM and FM and searching for systems with larger USMR is a prerequisite to achieve these goals. Here, we study the magnetoresistance of NM/FM bilayers where the NM has both weak (Ti, Cu) and large (W, Ta, Pt) SOC, as well as low (Cu, Pt) and high (Ti, W, Ta) resistivity. We nd that both the SMR-like magnetoresistance and nonlinear USMR are larger in the strong SOC materials, reaching 0.5 % and 0.004 % of the total resistance, respectively. The USMR of W/Co is about a factor two (three) larger with respect to Ta/Co (Pt/Co) of equal thickness, in agreement with the larger eective spin Hall angle of W ( SH= 0:330:05) estimated from the amplitude of the antidamping SOT in this system. The USMR is found to correlate with the magnitude of the antidamping SOT in the NM/FM layers. Additionally, to separate the USMR from thermomagnetic voltage contributions, we evaluate the electric eld due to the anisotropic Nernst (ANE) and spin Seebeck eect (SSE), and show that this correlates with the resistivity of the NM layer. These data are compared to measurements of a YIG/Pt bilayer. Our samples are NM(6 nm)/Co(2.5 nm)/Al(1.6 nm) layers with NM = Ti, Cu, W, Ta, Pt grown by dc magnetron sputtering on oxidized Si wafers. A 1 nm-thick Ta buer layer was deposited before the Cu/Co bilayer in order to improve the wetting of the substrate by Cu. The Al capping layer was oxidized by exposure to a radio-frequency O plasma. All samples present isotropic in-plane (easy-plane) magnetization as expected for a polycrystalline Co lm. Additionally, a Y 3Fe5O12(111) (YIG) (90 nm)/Pt(3 nm) bilayer was grown on a Gd3Ga5O12(111) oriented substrate by a combination of in situ DC sputtering for the 3metal and pulsed laser deposition for the epitaxial garnet lm growth. The crystalline quality and topography of the YIG lm was veried using x-ray diraction and atomic force microscopy, respectively. The as-grown layers were then patterned by optical lithography and ion milling in the form of Hall bars of nominal width w= 410m and length l= 5w. The Hall bars were mounted on a motorized stage allowing for in-plane ( ') and out-of-plane ( ) rotation (see Fig. 1b), and placed in an electromagnet producing elds of up to 1.7 T. The experiments were performed at room temperature using an ac current of amplitudej= 107A/cm2and frequency !=2= 10 Hz. The rst and second harmonic resistances, R!andR2!, corresponding to the conventional (current-independent) resistance and nonlinear (current-dependent) resistance, respectively, and the Hall resistances RH !and RH 2!were measured by Fourier analysis of the voltages VandVHshown in Fig. 1b (see Ref. 22 for more details). Figure 2 shows the angular dependence of the resistance of (a) Cu/Co and (b) W/Co bilayers measured by sweeping the external eld in the xy,zx, andzyplanes dened in Fig. 1b. The magnetoresistance of the two samples (top panels) is representative of the strong dierence between bilayers composed of light and heavy NM: Cu/Co displays the typical AMR of polycrystalline FM layers characterized by Rx>RyRz, whereRidenotes the resistance measured for msaturated parallel to i=x;y;z , whereas W/Co displays SMR- like behavior, with RxRz> Ry. Thezymagnetoresistance is ( RzRy)=Rz= 0:4 % in W/Co, similar to that reported for other metal systems22,25{27and a factor 15 larger than the SMR of our reference YIG/Pt sample. Figure 3 resumes the behavior of the dierent NM/FM bilayers. We nd that the zy(zx) magnetoresistance increases (decreases) with increasing atomic number of the NM, conrming that the unconventional angular dependence of NM/FM bilayers is related to SOC. The largest zymagnetoresistance is observed for the NM with the largest spin Hall angles, namely W and Pt, consistently with the SMR model.17According to this model, the variations of the SMR between the same NM and dierent FM (as between Pt/Co and Pt/YIG) and between dierent NM and the same FM (as between W/Co and Pt/Co) may be attributed to changes of the real part of the spin mixing conductance,17,30which sensitively depends on the material choice and interface properties.31,32However, these arguments alone are not sucient to conclude that thezyanisotropy is entirely due to the SMR in all-metal systems, as the anisotropic interface scattering proposed by Kobs et al.25and other interface contributions33may in uence the 4magnetoresistance. The second harmonic signals, reported in the lower panel of Fig. 2b, reveal another striking dierence between light and heavy NM systems, namely the presence of a nonlinear resistance in W/Co, which is absent in Cu/Co. We observe that R2!has a large variation (13:4 m ) in the xyandzyplanes and negligibly small variation in the zxplane.R2!is found to be proportional to myonce incomplete saturation of the magnetization in the zy plane is taken into account (due to the competition between the demagnetizing eld of Co and the external eld). This signal is compatible with both the USMR and a thermomagnetic contribution due to the anomalous Nernst eect (ANE) for a temperature gradient rTk z.22In a recent work we have shown that asymmetric heat dissipation towards the air and substrate side gives rise to such an out of plane temperature gradient in NM/FM bilayers, which is more pronounced when the conductivity of the top layer is larger than that of the bottom layer,34as is the case in W/Co. The ANE signal, however, can be accurately quantied by Hall resistance measurements and separated from the USMR.22 By saturating the magnetization along xwe have quantied the transverse ANE resistance as 0.97 m . Since the ANE is due to an electric eld ErT/ rTm, we calculate its longitudinal contribution by using the ratio between the longitudinal and transverse resistance (
R!=
RH !l=w), which gives 4.09 m , about 30% of the total R2!shown in Fig. 2b. We thus deduce an USMR value in the W/Co bilayer of RUSMR 2! = 9:3 m and the USMR ratio
RUSMR=R= 0:004 %, where
RUSMR=RUSMR 2!(+my)RUSMR 2!(my). The same procedure was used to estimate the ANE and USMR of all the bilayers studied in this work. The values of
RUSMR=RandErTobtained for dierent NM are compared in Fig.4a and b. We nd that the USMR is about a factor two (three) larger in W/Co with respect to Ta/Co (Pt/Co), and has opposite sign in Ta/Co and W/Co relative to Pt/Co. Although the amplitude of
RUSMR=Rdepends on the ratio between the thickness of the NM, tNM, and the spin diusion length, NM,22the similar NMof Ta, W, and Pt indicates that the USMR is strongly enhanced in W, which we associate to the larger spin Hall angle of -phase W relative to Ta and Pt.35Contrary to the USMR, we nd that the ANE scales with the resistivity of the layers independently of the SOC in the NM. The ANE-induced electric eld is of the order of 1 V/m for Ti, Ta, and W, all of them highly resistive metals withexceeding 100 
cm, whereas negligible ANE signals are detected in NM with low 5resistivity, where the current is shunted towards the NM side of bilayer. This indicates that the dominant ANE contribution comes from "bulk" Co and is largest when the current ows through the top FM layer. As noted in the introduction, the spin currents induced by charge ow are responsible for the USMR as well as SOT. In the following, we compare the magnitude of these eects in dierent layers. The antidamping and eld-like SOT, TADandTFL, were measured using har- monic Hall voltage analysis,8,9carried out simultaneously with the resistance measurements. The details of such measurements in samples with in-plane magnetization are outlined in Ref. 34. Figure 4a evidences a clear correlation between TADand
RUSMR=Rin W/Co and Pt/Co, and, to a lesser degree, in Ta/Co. Both quantities have negligible amplitude in the light NM, as expected due to the small SOC of these systems. Assuming that TADis driven by the SHE of the NM and a transparent interface, the torque amplitude can be expressed as an eective spin Hall angle14SH=2e hMstFM j[1sech(tNM NM)]TAD, where0Ms= 1:5 T is the saturation magnetization of Co, tNM= 6 nm,W= 1:6 nm,Pt= 1:1 nm, and Ta= 1:5 nm.22,27We thus obtain SH(W)= 0:330:05,SH(Pt)= 0:100:02, and SH(Ta)= 0:090:02, comparable to previous reports.8,12,35,36 The correspondence between the USMR and SHof W shows that materials with large spin Hall angles are required to enhance this eect. The fact that
RUSMR=Rof Pt is smaller than Ta whereas SH(Pt)>SH(Ta), on the other hand, is attributed to the dilution of the USMR in highly conducting NM layers when tNM> NM,22as is the case here for tNM= 6 nm, although other eects may also play a role, such as spin memory loss at the NM/FM interface.37Additionally, we observe that TFLis much smaller than TADin the heavy NM, as expected when the thickness of the FM exceeds 1 nm,9,34and has no apparent relationship to the USMR. The latter observation suggests that the USMR depends mainly on the real part of the spin mixing conductance, which is proportional to TAD, similar to the SMR.17 Finally, we show that the USMR is absent when the FM is an insulator such as YIG, as expected by analogy to the current-in-plane giant magnetoresistance. We use YIG/Pt as a model FM insulator/NM system with well-characterized SMR and thermomagnetic properties,17{20and show that no signicant USMR signal can be detected in this system. Figures 5a and b show the angular dependence of the longitudinal ( R!) and transverse ( RH !) resistance of a YIG(90 nm)/Pt(3 nm) bilayer in the xyplane. In both channels we measure a 6signal consistent with the SMR, namely R!(1m2 y) = cos2'andRH !mxmy= sin 2', with ratio
R!=
RH != 4:3l=was determined from optical microscopy. From the longitudinal measurement we calculate the resistivity of Pt, Pt= 56:5 cm, and the SMR ratioRx;zRy Rx;z = 2:7
104, both within the range of literature values reported for samples with comparable Pt and YIG thickness.19,38 The second harmonic resistances, R2!andRH 2!, are shown in Fig. 5c and d. The an- gular variation of R2!in thexyplane is about a factor ten smaller relative to R!and is proportional to my= sin'. This signal has the symmetry expected of the USMR as well as the spin Seebeck eect (SSE) due to an out of plane thermal gradient.20,39Similar to the ANE in FM metals, the SSE voltage appears in the longitudinal channel when mkyand in the transverse channel when mkx. Accordingly, we observe that RH 2!is proportional to mx= cos'in Fig. 5d and estimate the electric eld due to the SSE as ErT= 0:68 V/m. We can thus calculate the thermal contribution to R2!by rescaling the transverse resis- tanceRH 2!by the factor
R!=
RH !, as done in the case of the NM/FM metal layers. The comparison between R2!andRH 2!in Fig. 5 shows that
R2!=
RH 2!
R!=
RH !to within 10 % accuracy. Therefore, we conclude that most of the R2!signal is of thermal origin and not related to the USMR. The small discrepancy between the longitudinal and rescaled transverse nonlinear signals can be explained by several factors, for example by considering that the current spreading in the Hall branches can decrease Joule heating in the Hall cross with respect to the central region of the Hall bar, thus reducing the thermal voltage in the transverse measurement with respect to the longitudinal one. Alternatively, a small proximity-induced magnetization in Pt could couple to the spin accumulation due to the SHE and give rise to the USMR. Overall, our data show that the USMR in YIG/Pt, if it exists, is much smaller compared to NM/FM metal systems. In summary, we have measured the angular dependence of the magnetoresistance in light and heavy metal/FM layers in the linear and nonlinear response regimes. The resistance of Pt/Co, W/Co, and Ta/Co bilayers depends strongly on the magnetization orientation in the plane perpendicular to the current direction, akin to the spin Hall magnetoresistance (SMR) in YIG/Pt, but with magnetoresistance ratios 15 times as large, of the order of (RzRy)=Rz= 0:5 %. This ratio increases with the atomic number of the NM, whereas the light NM/Co bilayers (NM = Ti, Cu) present the usual AMR expected of polycrys- talline FM layers, characterized by RzRy. Thermomagnetic eects typied by the ANE 7correlate with the resistivity of the NM rather than SOC. In the Ta, W, Pt/Co bilayers we nd an additional nonlinear magnetoresistance, which depends linearly on the current and on they-component of the magnetization. This so-called USMR, of the order of 0.005 %, is enhanced by a factor 2-3 in W/Co relative to Pt/Co and Ta/Co and correlates with the amplitude of the AD spin-orbit torque, whereas it shows no apparent relationship to the FL spin-orbit torque. The USMR is below the accuracy of our measurements in YIG/Pt. These results suggest that NM with large spin Hall angles and NM/FM interfaces with large and real spin mixing conductance are required to enhance the USMR. ACKNOWLEDGMENTS This research was supported by the Swiss National Science Foundation (Grant No. 200021-153404) and the European Commission under the Seventh Framework Program (spOt project, Grant No. 318144). REFERENCES 1M. Dyakonov and V. Perel, Phys. Lett. A 35, 459 (1971). 2V. M. Edelstein, Sol. St. Comm. 73, 233 (1990). 3A. Manchon and S. Zhang, Phys. Rev. B 78, 212405 (2008). 4P. M. Haney, H.-W. Lee, K.-J. Lee, A. Manchon, and M. Stiles, Phys. Rev. B 87, 174411 (2013). 5K. Ando, S. Takahashi, K. Harii, K. Sasage, J. Ieda, S. Maekawa, and E. Saitoh, Phys. Rev. Lett. 101, 036601 (2008). 6I. M. Miron, G. Gaudin, S. Auret, B. Rodmacq, A. Schuhl, S. Pizzini, J. Vogel, and P. Gambardella, Nature Mater. 9, 230 (2010). 7I. M. Miron, K. Garello, G. Gaudin, P.-J. 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Castel, B. van Wees, and J. B. Youssef, Phys. Rev. B 87, 184421 (2013). 39M. Schreier, N. Roschewsky, E. Dobler, S. Meyer, H. Huebl, R. Gross, and S. T. Goen- nenwein, Appl. Phys. Lett. 103, 242404 (2013). 10FIG. 1. (a) Illustration of the SHE-induced spin accumulation at the NM/FM interface. Parallel (antiparallel) alignment of the magnetization with respect to the spin accumulation gives rise to a decrease (increase) of the longitudinal resistance or USMR. (b) Schematics of the measurement geometry. 11569.8571.3572.8Rω (Ω)0 90180270360-14014R2ω (mΩ)θ ,ϕ (degrees)W(6nm)/Co(2.5nm)8 3.3883.4683.54(b)Rω (Ω)(a)0 90180270360-101 xy scan zx scan zy scanR2ω (mΩ)θ ,ϕ (degrees)Cu(6nm)/Co(2.5nm)FIG. 2. Angular dependence of the resistance ( R!, top panels) and nonlinear resistance ( R2!, bottom panels) of (a) Cu/Co and (b) W/Co bilayers with dimensions w= 10,l= 50m in an external eld of 1.7 T. Ti/CoCu/CoTa/CoW/CoPt/CoYIG/Pt0.00.20.40.6ΔR/R (%) (Rx-Ry)/Rz (Rx-Rz)/Rz (Rz-Ry)/Rz FIG. 3. Anisotropy of the magnetoresistance in the xy,zx, andzyplanes derived from the angle- dependent curves shown in Fig. 2a. 12Ti/CoCu/CoTa/CoW/CoPt/Co024( b) |ΔRUSMR/R| |TAD| |TFL||ΔRUSMR/R| (x10-5)0 24 SOT (mT)T i/CoCu/CoTa/CoW/CoPt/Co3080130180ρ (µΩcm)0 .00.40.81.2| E∇T| (V/m)(a)FIG. 4. (a) USMR and spin-orbit torques measured in dierent NM/Co bilayers. (b) Resistivity of the bilayers and electric eld eld induced by the ANE. The current density is j= 107A/cm2 in all cases. 13809.9810.0810.1( d)(a)( b)Rω (Ω)( c)YIG(90nm)/Pt(3nm)- 0.020.000.02RHω (Ω)0 90180270360-10010R2ω (mΩ)ϕ (degrees)090180270360-202RH2 ω (mΩ)ϕ (degrees)FIG. 5. Angular dependence of the longitudinal resistance (a) and transverse resistance (b) of YIG(90 nm)/Pt(3 nm) with dimensions w= 10,l= 50m measured in an external eld of 0.2 T. Nonlinear longitudinal resistance (c) and transverse resistance (d). The curve in (c) is an average over twenty consecutive measurements to improve the signal-to-noise ratio. 14

2015-10-21

We studied the magnetoresistance of normal metal (NM)/ferromagnet (FM) bilayers in the linear and nonlinear (current-dependent) regimes and compared it with the amplitude of the spin-orbit torques and thermally induced electric fields. Our experiments reveal that the magnetoresistance of the heavy NM/Co bilayers (NM = Ta, W, Pt) is phenomenologically similar to the spin Hall magnetoresistance (SMR) of YIG/Pt, but has a much larger anisotropy, of the order of 0.5%, which increases with the atomic number of the NM. This SMR-like behavior is absent in light NM/Co bilayers (NM = Ti, Cu), which present the standard AMR expected of polycrystalline FM layers. In the Ta, W, Pt/Co bilayers we find an additional magnetoresistance, directly proportional to the current and to the transverse component of the magnetization. This so-called unidirectional SMR, of the order of 0.005%, is largest in W and correlates with the amplitude of the antidamping spin-orbit torque. The unidirectional SMR is below the accuracy of our measurements in YIG/Pt.

Magnetoresistance of heavy and light metal/ferromagnet bilayers

1510.06285v1

arXiv:2206.12769v1 [quant-ph] 26 Jun 2022Tripartite high-dimensional magnon-photon entanglement inPT-symmetry broken phases of a non-Hermitian hybrid system Jin-Xuan Han1, Jin-Lei Wu1,∗Yan Wang1, Yan Xia2, Yong-Yuan Jiang1, and Jie Song1,3,4,5† 1School of Physics, Harbin Institute of Technology, Harbin 1 50001, China 2Department of Physics, Fuzhou University, Fuzhou 350002, C hina 3Key Laboratory of Micro-Nano Optoelectronic Information S ystem, Ministry of Industry and Information Technology, Harbin 15 0001, China 4Key Laboratory of Micro-Optics and Photonic Technology of H eilongjiang Province, Harbin Institute of Technology, Harbin 150001, China 5Collaborative Innovation Center of Extreme Optics, Shanxi University, Taiyuan, Shanxi 030006, People’s Republic of China Hybrid systems that combine spin ensembles and superconduc ting circuits provide a promising platform for implementing quantum information processing . We propose a non-Hermitian magnon- circuit-QED hybrid model consisting of two cavities and an y ttrium iron garnet (YIG) sphere placed in one of the cavities. Abundant exceptional points ( EPs), parity-time ( PT)-symmetry phases and PT-symmetry broken phases are investigated in the parameter s pace. Tripartite high- dimensional entangled states can be generated steadily amo ng modes of the magnon and photons inPT-symmetry broken phases, corresponding to which the stable quantum coherence exists. Re- sults show that the tripartite high-dimensional entangled state is robust against the dissipation of hybrid system, independent of a certain initial state, and i nsensitive to the fluctuation of magnon- photon coupling. Further, we propose to simulate the hybrid model with an equivalent LCRcircuit. This work may provide prospects for realizing multipartite high-dimensional entangled states in the magnon-circuit-QED hybrid system. I. INTRODUCTION Hybridizing two or more quantum systems can com- bine complementary advantages of different systems and improve the multi-task processing ability, which is the key to realizing quantum information processing [ 1–8]. Strikingly, more macroscopic objects, such as supercon- ducting circuits possessing advantages of flexibility, scal- ability and tunability [ 9–11], arestronglycoupled to elec- tromagnetic fields, making them easy to entangle to- gether even with shorter coherence times [ 12,13]. How- ever, microscopicsystems (such asspin ensembles), natu- rally decoupled well from their environmentand reaching relativelylong coherence times [ 14,15], can be integrated into a circuit by means of techniques of trapping and doping. Consequently, a hybrid system can combine the rapid operationsof superconducting circuits and the long coherence time of spins. Among possible materials of spin ensembles, a single- crystal yttrium iron garnet (YIG) sphere has shown up recently as a promising candidate for hybrid systems, benefiting from excellent characteristics of low magne- tization damping, long life, easy adjustment as well as strong coupling between magnon and photon [ 16–19]. Coherent and dissipative couplings have been identified experimentally in coupled magnon-photon hybrid sys- tems [17,20–30]. In the earlier studies, the coherent coupling between modes of photon and magnon has been proved by the anticrossing or the level repulsion of two ∗jinleiwu@126.com †jsong@hit.edu.cncoupled modes at/near their common resonance frequen- cies [31–35]. In contrast to the anticrossing level shown in coherently coupled magnon-photon systems, dissipa- tivecoupledsystemsexhibit thelevelattractionatexcep- tional points (EPs) [ 25–27,36,37], which opens a new avenue for exploring non-Hermitian quantum physics and parity-time ( PT) symmetry in dissipative coupled magnon-photon systems. So far, many applications and effects have been ex- plored based on dissipative coupled magnon-photon sys- tems, for example nonreciprocal microwave engineer- ing [38], generation of the steady entangled state [ 39], quantum sensing [ 40], distant magnetic moments [ 41,42] and anti- PTsymmetry [ 43]. Recently, Yuan etal[39] re- ported that a high-fidelity Bell state of magnon and pho- ton can be generated in the PT-symmetry broken phase. Also, it has been proposed that tripartite entanglement among the deformation mode, magnetostatic mode and microwave cavity mode may be realized in a cavity magnomechanical system via magnetostrictive interac- tion and magnetic dipole interaction. Comparing with bipartite and tripartite binary entangled states, multi- partite high-dimensional entangled states, which can en- hance the violations of local realism [ 45] and the security of quantum cryptography [ 46–49], have attracted much interest owing to the larger channel capacity of quan- tum communicationand the higherefficiency ofquantum information processing. To this end, high-dimensional entanglement has been not only generated theoretically in various physical platforms [ 50–55], but also investi- gated experimentally in photonic systems [ 56,57], cold atoms [58,59] and trapped ions [ 60]. In this paper, we propose a non-Hermitian magnon-2 circuit-QED hybrid system consisting of two cavities and an YIG sphere inside one of cavities. We derive ana- lytically an effective Hamiltonian of the hybrid magnon- photonsystemanditsenergyspectra,andthenabundant EPs,PT-symmetry phases and PT-symmetry broken phases are investigated in the parameter space. In PT- symmetry broken phases, through the Zeeman effect be- tween photon and magnon modes and the electric dipole interaction between modes of photons, a tripartite high- dimensional entangled state can be generated steadily among the modes of magnon and photon, corresponding to which the stable quantum coherence exists. Our work may facilitate potential applications of magnon-circuit- QED hybrid systems in quantum information processing, because of the following advantages and interests. First, by varying systematic parameters, there are abundant EPs,PT-symmetry phases and PT-symmetry broken phases in comparison with Refs. [ 39,61–65]. Second, the steady quantum coherence among the modes of magnon and photon exists in PT-symmetry broken phases, with respect to which tripartite high-dimensional entangled states can be generated. However, both of quantum co- herence and entanglement states appear with intense os- cillations in PT-symmetry phases, which is contrary to the universal viewpoint that the state is unstable as the PTsymmetry is broken. This anomaly can be further understood by the competition of the evolution of non- Hermitian system and particle number conservation of the hybrid system. Finally, the fidelity of tripartite high- dimensional entangled state and the quantum coherence are robust to the dissipation of hybrid system, indepen- dent of a certain initial state, and insensitive to the fluc- tuation of magnon-photon coupling. By comparing with a previous study in the magnon-cavity QED hybrid sys- tem [44], the present scheme for generating tripartite en- tanglement is originated from the Zeeman effect between the modes of photon and magnon via non-Hermitian coupling and the electric dipole interaction between the modes ofphotonsin the magnon-circuit-QEDhybridsys- tem. The entanglement resulted from the evolution of non-Hermitiansystemisnotonlytripartitebutalsohigh- dimensional. Therefore, the present work may provide prospects for realizing multipartite high-dimensional en- tangled states in the magnon-circuit-QED hybrid system andfurtherapplicationsinquantuminformationprocess- ing. II. THEORETICAL MODEL AND IMPLEMENTATION IN NANOCIRCUIT A. Model and Hamiltonian As illustrated in Fig. 1(a), we considerfirst an abstract model of a three-mode coupled magnon-photon hybrid system including two microwave photon modes aandb in cavitiesAandB, respectively, and one magnon mode cin the YIG sphere, where the photon mode ais coupledcavity B cavity A (c) cavity A cavity Bexternal magnetic field ݔ ݕ YIG sphere(b) P1 P2 YIG ܴെ ܴ ଵ ܮെ ܮ ଵܥܴଵ ܮଵCouplingܴെ ܴ ଵ ܮെ ܮ ଵ ܥCܴ ܮ ܥMicrowave photon modeMicrowave photon mode cMagnon mode (a) ࣘ capacitor FIG. 1. (a) Schematic diagram for couplings of the mi- crowave photon mode ato the microwave photon mode bwith strength rand the magnon mode cwith the direct strength gor feedback action strength geiφ. (b) Schematic layout of the proposed hybrid system. The design of planer cross-line microwave circuit cavity is composited by a Michelson-type microwave interferometer with two short-terminated verti cal arms and two horizontal arms, in which port 1 and port 2 are connected to a vector network analyzer to enable microwave- transmission measurements [ 66,67]. The planar cavities A andBplaced in the x−yplane are coupled to each other by the inter-cavity capacitor. The YIG sphere magnetized to saturation, where the Kittel mode of magnon is excited by a an external magnetic field B0along the zdirection, is mounted at the center of planar microstrip-cross junction i n the cavity A. (c) Equivalent circuit of the coupled magnon- photon system. Circuit elements are used to model the cavity A(B) and the YIG sphere. to the photon (magnon) mode b(c) via the electric (mag- netic) dipole interaction with net coupling strength r(g). In particular, there is a coherent coupling between the microwavephotonmode aandthemagnonmode cdueto the Amp´ ere effect and the Faraday effect, corresponding to the coupling strength g. Meanwhile, due to the effect of Lenz’s law, the microwave current in the cavity Aalso creates a back action on the the YIG sphere to impede the magnetization of the magnon mode, which leads to a dissipative magnon-photon coupling, corresponding to thecouplingstrength geiφ. Thephasedifference φresults from the competition between the coherent and dissipa- tive couplings [ 26]. Such a model of the magnon-photon system can be described by a non-Hermitian interaction3 Hamiltonian ˆH=ωaˆa†ˆa+ωbˆb†ˆb+ωcˆc†ˆc+g(ˆaˆc†+eiφˆa†ˆc) +r(eiθˆaˆb†+e−iθˆa†ˆb), (1) where ˆa(ˆb) and ˆcare annihilation operators of the uni- form precession modes for the photon mode a(b) and the magnon mode c.θis a phase shift coming from the crosstalk effects between the fields produced inside the cavitiesAandB. Whenthedrivingtorqueonthemagne- tization ofmagnonfrom Amp´ ere’slawis more(less) than the retarding torque on the magnetization of magnon from Lenz’s law, φ= 0 (φ=π) corresponds to a purely coherent (dissipative) coupling. B. Implementation in nanocircuit According to existing circuit-QED technologies, one can construct a magnon-circuit-QED hybrid system to implement the model proposed above via arranging pla- nar microwave cavities AandBand the YIG sphere, as shown in Fig. 1(b). The electric dipole interaction be- tween the cavities AandBcan be realized by using an inter-cavity capacitor. The magnons embodied by a col- lective motion of a large number of spins in a ferrimagnet can be provided by an YIG sphere. At the same time, an external magnetic field B0along thezdirection mag- netizes the YIG sphere to saturation and then induces the Kittel mode of magnons in the YIG sphere [ 68]. The Zeeman effect is achieved by placing an YIG sphere in- side the cavity Aso that it overlaps with the microwave magnetic field of the cavity A. This setup can be con- tinuously tuned for the YIG sphere position to change the local field and the coupling effect, which can further realize dissipative coupling or coherent coupling between the YIG sphere and the cavity A. Inordertodescribequantitativelynon-Herimitiancou- pling in the hybrid system, both the dissipative cou- pling and the coherent coupling can be realized by an LCR-circuit model, corresponding to the existence of the resistance-dominated coupling and the inductance- dominated coupling, respectively [ 27]. Equivalently, the hybrid system can be quantized into a series of circuit components of the capacitance, inductance and resis- tance in Fig. 1(c). The dissipative and coherent cou- plings between the YIG sphere and the cavity Acan be modeled by a mutual resistance R1and a mutual induc- tanceL1, respectively [ 27]. Also, the cavity-cavity cou- pling can be modeled by an inter-cavity capacitor. As long as the inter-cavity capacitance Cis much smaller than the capacitance Ca(Cb) of the cavity A(B), pho- tons can hop between the cavities AandB[69] with the inter-cavity coupling strength r≈2Z0Cωaωb,Z0 being the characteristic impedance of the transmission line [70]. Based on the standard process quantization of anLCR-circuit [71], the frequencies of the cavities Aand Band the YIG sphere are expressed as ωa= 1/√LaCa,ωb= 1/√LbCbandωc= 1/√LmCm, respectively, and the damping rates are γa=Ra/2Laωa,γb=Rb/2Lbωb andγm=Rm/2Lmωc, respectively. The Hamiltonian of the hybrid system shown in Fig.1(b) reads in an ideal situation as ˆH=ˆHYIG+ˆHA,B+ˆHA−c+ˆHA−B.(2) ˆHYIGis the free Hamiltonian of the YIG sphere, and ˆHA,Bis the free Hamiltonian of the cavities AandB. ˆHA−c(ˆHA−B) is the interaction Hamiltonian of the cav- ityAand the YIG sphere (the cavity B), respectively. Concretely, ˆHYIG=−/summationdisplay ig∗µBB0·ˆSi−J/summationdisplay i,jˆSi·ˆSj ˆHA,B=/summationdisplay n=a,b1 2/integraldisplay (ǫ0E2 n+1 µ0B2 n)dxdydz, ˆHA−c=−/summationdisplay iˆSi·H, ˆHA−B=C/integraldisplay ˙BadSa·/integraldisplay ˙BbdSb. (3) ForˆHYIG,Jis the exchangeconstant, g∗theg-factor,µB the Bohr magneton, B0the external magnetic field along thezaxis in order for the YIG sphere to be magnetized, andˆSi≡(ˆSxi,ˆSyi,ˆSzi) the Heisenberg spin operator for thei-th site. For ˆHA,B,Ea(b)andBa(b)are respective componentsofelectric field and magnetic field in the cav- ityA(B), andǫ0andµ0are vacuum permittivity and susceptibility, respectively. For ˆHA−c,His the corre- sponding magnetic field acting on the spin. For ˆHA−B, Cis the inter-cavity capacitance, and/integraltext˙BndSnis the voltage profile in the cavity n(n=a,b) [72,73]. The Heisenberg operators can be expressed as bosonic operatorsˆciandˆc† ibyusingtheHolstein-Primakofftrans- formation [ 74] ˆSz i=S−ˆci†ˆci, ˆS+ i= ˆci/radicalBig 2S−ˆci†ˆci, ˆS− i= ˆci†/radicalBig 2S−ˆci†ˆci (4) whereSis the total spin on each site and ˆS±≡ ˆSx i±iˆSy i. The bosonic operators are related to the spin-wave operators by Fourier transformation ˆci= 1/√ N/summationtext qce−iqc·riˆcqc(ˆc† i= 1/√ N/summationtext qceiqc·riˆc† qc), ˆcqc(ˆc† qc)representingtheannihilation(creation)operator in the spin-wave mode with wave vector qc[75,76]. Sub- stituting these operators into ˆHYIG, the Hamiltonian of magnetostatic modes in the YIG sphere under the static limit [76] can be written as ˆHYIG=/summationtext qcωqcˆc† qcˆcqc. Then the magnetic fields of cavities AandBcan be quan- tized as Bn=i/summationtext qn/radicalbig ωqn/4Vnqn×[u(qn)ˆnqneiqn·r− u∗(qn)ˆn† qne−iqn·r] (n=a,b), where u(qn) is the com- plex amplitude of field in the cavity, ˆ nqnand ˆn† qnthe4 annihilation and creation operators of the cavity at fre- quencyωqnwith wave vector qn, andVnthe volume of the cavity. By substituting magnetic fields into ˆHA,B, we obtain the quantized Hamiltonian of microwaves ˆHA,B=/summationtext qa,qbωqaˆa† qaˆaqa+ωqbˆb† qbˆbqb[77]. However, there are not only a coherent coupling on ac- count of the Amp´ ere effect and the Faraday effect, but also a dissipative coupling on account of Lenz effect be- tween the YIG sphere and cavity A[26]. In this case, the oscillating field Hacting on the local spin includes a direct action of the microwave h1and a reaction field of the precessing magnetization h2=h1δeiΦ[26–29], where δand Φ are the relative amplitude and phase of the two fields, respectively. In the presence of both coherent and dissipative couplings, according to the standard quanti- zation process and the rotating-waveapproximation, one can recast the Hamiltonian ( 2) into ˆHYIG=/summationdisplay qcωqcˆc† qcˆcqc, ˆHA,B=/summationdisplay qa,qbωaˆa† qaˆaqa+ωbˆb† qbˆbqb, ˆHA−c=/summationdisplay qa,qcgqa,qc(ˆaqaˆc† qc+eiφˆa† qaˆcqc), ˆHA−B=/summationdisplay qa,qbrqa,qb(eiθqa,qbˆaqaˆb† qb+e−iθqa,qbˆa† qaˆbqb),(5) whereφ= 2arctan[ −δsinΦ/(1 +δcosΦ)] is a tun- able phase [ 26–29,39],θqa,qba phase coming from the two rf-fields in the cavities AandBjoined by microstrips with the inter-cavity capacitor, gqa,qcthe magnon-photon coupling strength between the magneto- static mode cqcand the microwave cavity mode aqa, and rqa,qbthe photon-photon coupling strength between mi- crowave cavity modes aqaandbqb. The magnon-photon coupling strength via Zeeman effect and the photon- photon coupling strength via electric dipole interaction can be expressed as [ 69,73,78] gqa,qc=g∗µB√ 2S 2/integraldisplay Vdr Ba(r)·sqc(r), rqa,qc=1 2Z0C/radicalBig ωqa(ωqb+∆ab)φAφB,(6) whereBa(r) is the strength of microwave magnetic field in the cavity Aat the position rof the spin, sqc(r) the orthonormal mode function describing the spatial profile of the amplitude and phase for the magnetostatic mode cqc,Z0the characteristic impedance of the transmission line, ∆abthe frequency detuning between the cavities A andBarising from the external magnetic field B0, and theφa(φb) the classical mode function of cavity A(B). Themagnon-photoncouplingisdissipativeonlywhenthe YIG sphereismounted atthecenterofplanermicrostrip- crossjunction in the cavity A. Otherwise, it is a coherent coupling [ 27]. It is supposed that the microwavemagnetic field in the cavityAis uniform throughout the YIG sphere, so themagnetic dipole coupling vanishes except for the uniform magnetostatic mode, i.e., the Kittel mode of magnon at frequency ωc[78]. The photon then only interacts strongly with the magnon around the Gamma point (i.e, the Kittel mode) so as to match the resonant/near- resonant frequency ωa(ωb) in the cavity A(B) andωc in the YIG sphere [ 39,79,80]. Thus the sum in Eq. ( 5) can be removed to obtain Eq. ( 1). It should be empha- sized that the frequency ωbcan be tuned by the external magnetic field, which can further affect the cavity-cavity detuning. C. Energy spectrum The Hamiltonian ( 1) is non-Hermitian, but there are still real eigenvalues [ 81], because Hermicity is not a nec- essary condition for a physical quantum theory [ 81]. Im- portantly, under the condition of φ=nπ(n= 0,1,2...), the Hamiltonian ( 1) isPTsymmetric, [ ˆH,PT] = 0, wherePandTare parity inversion and time rever- sal operators, respectively. When the Hamiltonian ( 1) holds real eigenvalues, the phase of the system can be regarded as the PT-symmetry phase. However, the PT-symmetry broken phase is characterized by complex eigenvalues [ 82]. When the YIG sphere is mounted at the center of microstrip-cross junction in the cavity A, the dissipative magnon-photon coupling is absolutely dominant owing to the Lenz effect inducing a feedback microwave current on the YIG sphere and in turn impeding the excitation of the magnon, which results in the phase φ=π[26,27]. In this situation, we can obtain analytically three complex eigenvalues ωn(n= 1,2,3) of Hamiltonian ( 1) ω1=A 3−21/3(3B−A2) 3E+E 21/3, ω2=A 3−(1+i√ 3)(3B −A2) 3×22/3×E1/3−(1−i√ 3)E1/3 6×21/3, ω3=A 3−(1−i√ 3)(3B −A2) 3×22/3×E1/3−(1+i√ 3)E1/3 6×21/3,(7) whereA=ωa+ωb+ωc,B=g2 1eiφ+r2−ωcωa−ωcωb− ωaωb,C=−ωcωaωb+ωcr2+g2 1eiφωb,D=−A2B2+ 4B3−4A3C+ 18ABC+ 27C2andE= 2A3−9AB − 27C+ 3√ 3√ D. We plot the real and imaginary parts of the three eigenvalues ωn, respectively, represented by solid and dotted lines in the y-zplane of Fig. 2. Specif- ically, the real and imaginary parts of energy spectrum versus the magnon-photon coupling strength g/2πand the frequency detuning of the magnon-photon coupling ∆ = (ωa−ωc)/2gare also plotted in Figs. 2(a)-(c) and Figs.2(d)-(f), respectively. We take the photon-photon couplingstrength r/2π= 50MHzand θ= 1.1πforexam- ple and select three decreasing magnon-photon coupling strength ranges g/2π∈[0,70] MHz in (a) and (d), g/2π ∈[0,25] MHz in (b) and (e), and g/2π∈[0,10] MHz in (c) and (f), respectively. It is noted that the value5 FIG. 2. Energy spectrum versus the magnon-photon coupling s trengthg/2πand the frequency detuning of the magnon-photon coupling ∆ = ( ωa−ωc)/2gby setting three decreasing magnon-photon coupling streng th ranges g/2π∈[0,70] MHz in (a) and (d),g/2π∈[0,25] MHz in (b) and (e), and g/2π∈[0,10] MHz in (c) and (f), respectively. (a)-(c) Real parts of th e eigenvalues. (d)-(f) Imaginary parts of the eigenvalues. PT-symmetry broken phases in the x−zplane: (a) and (d) ∆ ∈[−1.03,1.03] with g/2π= 70 MHz; (b) and (e) ∆ ∈[−1.38,0)∪(0,1.38] with g/2π= 25 MHz; (c) and (f) ∆ ∈[−2.24,−0.88)∪(0.88,2.24] with g/2π= 10 MHz. PT-symmetry phases in the x−zplane: (a) and (d) ∆ ∈[−3,−1.03)∪(1.03,3] withg/2π= 70 MHz; (b) and (e) ∆∈[−4,−1.38)∪(1.38,4] withg/2π= 25 MHz; (c) and (f) ∆ ∈[−4,−2.24)∪(−0.88,0.88)∪(2.24,4] withg/2π= 10 MHz. Parameters: r/2π= 50 MHz, φ=πandθ= 1.1π. of coupling phase θhas no impact on the energy spec- trum of the hybrid system. Obviously, the real parts of energy spectrum show two surfaces of eigenvalues merge into one hybrid surface in Figs. 2(a)-(c). In other words, Fig.2demonstrates a typical spectrum of level attrac- tion (φ=π). As shown in Figs. 2(d)-(f), when the eigenvalues are real (imaginary), PT-symmetry ( PT- symmetry broken) phases are appeared. The phase tran- sitions at EPs are shown by black solid points. Depend- ing on ranges of magnon-photon coupling strength, the regions of PT-symmetry, PT-symmetry broken phases and the number of EPs can be changed. For convenience, we take the x−zplane as an ex- ample to identify the regions of PT-symmetry and PT- symmetry broken phases. In Figs. 2(a) and (d) with the magnon-photon coupling strength g/2π= 70 MHz, when ∆∈[−3,−1.03)∪(1.03,3] (real eigenvalues) and ∆ ∈ [−1.03,1.03] (imaginary eigenvalues), there are two PT- symmetry phases and one PT-symmetry broken phase, respectively, with the separation of two EPs at ∆ = ±1.03. In Figs. 2(b) and (e) with the magnon-photon coupling strength g/2π= 25 MHz, three EPs (∆ = −1.38,0,1.38)divide the PT-symmetry broken area into two adjacent parts in the region of ∆ ∈(−1.38,1.38). Two parts of PT-symmetry phases exist in the region of ∆∈[−4,−1.38]∪[1.38,4]. In Figs. 2(c) and (f) with the magnon-photon coupling strength g/2π= 10 MHz,by continuously decreasing the magnon-photon coupling strength, it is distinct that two parts of PT-symmetry broken phases are separated further than Figs. 2(b) and (e). And three parts of PT-symmetry phases are sepa- rated by four EPs at ∆ = ±0.88 and±2.24, respectively. III. TRIPARTITE HIGH-DIMENSIONAL ENTANGLED STATES AND ROBUSTNESS Inthissection, weusethemodeltogeneratethetripar- tite qubit entangled state and high-dimensional entan- gled states. Then we discuss the robustness of the entan- gled states and quantum coherence in the PT-symmetry broken phases. A. Generation of tripartite qubit entangled state We focus on the generation of tripartite qubit entan- gled state among the modes of magnon and photon in PT-symmetry broken phases. Firstly, the evolution of the hybrid system can be evaluated by using the master equation [ 83] ∂ˆρ ∂t=−i[ˆH1,ˆρ]−i{ˆH2,ˆρ}+2i/an}bracketle{tˆH2/an}bracketri}htˆρ,(8)6 0.00 0.05 0.10 0.15 0.200.00.20.40.60.81.0 FIG.3. Populationevolutionofthetargettripartiteentan gled state|φ1/angbracketright= (|˜10/angbracketrightA1,c+i|˜01/angbracketrightA1,c)/√ 2 andits threecomponent states, based on the initial state |001/angbracketrighta,b,cinPT-symmetry broken phases. Parameters: ωc/2π= 6 GHz, ωa/2π= ωb/2π= 5.95 GHz, g/2π= 6 MHz, r/2π= 50 MHz and θ= 1.1π. where ˆρis the density matrix of the system, ˆH1(ˆH2) is a Hermitian (anti-Hermitian) operator by recasting the effective Hamiltonian as ˆH1≡(ˆH+ˆH†)/2 (ˆH2≡ (ˆH−ˆH†)/2),/an}bracketle{tˆH2/an}bracketri}ht= tr(ˆρˆH2), and the brackets [ ·] and {·}represent the commutator and anti-commutator, re- spectively. Specially, the resulting equation is nonlinear in the quantum state ˆ ρby adding the third term so as to preserve tr(ˆ ρ) = 1. By solving the evolution of density matrix ρgoverned by Eq. (6), an initial pure state |φ0/an}bracketri}ht=|001/an}bracketri}hta,b,cwith a mean particle number N=/an}bracketle{tˆa†ˆa+ˆb†ˆb+ˆc†ˆc/an}bracketri}ht= 1 is taken as an example. Then we introduce nonlocal modes ˆA1,2=ˆa±ˆbe−iθ √ 2. (9) And assume that ωa=ωb=ωand the large detuning condition |ω−ωc−r| ≫ |g|. After transforming into the interaction picture of the nonlocal modes, the interaction Hamiltonian reads as ˆHeff=geff(ˆA† 1ˆc+ˆA1ˆc†eiφ), (10) whereˆA1= (ˆa+ˆbe−iθ)/√ 2 andgeff=g/√ 2. Ac- cordinlgly, the hybrid system evolves in the finite sub- space{|˜01/an}bracketri}htA1,c,|˜10/an}bracketri}htA1,c}, where |˜0/an}bracketri}htA1and|˜1/an}bracketri}htA1are Fock states of the mode A1, represented by |00/an}bracketri}hta,band (|10/an}bracketri}hta,b+eiθ|01/an}bracketri}hta,b)/√ 2, respectively, on the basis of {|0/an}bracketri}hta,|1/an}bracketri}hta,|0/an}bracketri}htb,|1/an}bracketri}htb}. By solving the eigenequation ˆH|φm/an}bracketri}ht=ωm|φm/an}bracketri}ht, eign- states of the hybrid system are obtained |φm/an}bracketri}ht= cosθm|˜10/an}bracketri}htA1,c+eiψmsinθm|˜01/an}bracketri}htA1,c,(11) whereθmandψmare related with eiψmtanθm= (ωk− ω)/g(k= 1,2,3...). If the initial state is ˆ ρ0= |˜01/an}bracketri}htA1,c/an}bracketle{t˜01|, the time-dependent density matrix can be0.00 0.05 0.10 0.15 0.200.00.20.40.60.81.0Entanglement(a) 0.00 0.05 0.10 0.15 0.200.00.20.40.60.81.0Entanglement(b) FIG. 4. Time dependence of the collective coherence of the system and the fidelity of |φ1/angbracketright= (|˜10/angbracketrightA1,c+i|˜01/angbracketrightA1,c)/√ 2, based on the initial state |001/angbracketrighta,b,c(a) in the PT-symmetry broken phase and (b) in the PT-symmetry phase. Pa- rameters: (a) ωc/2π= 6 GHz, ωa/2π=ωb/2π= 5.95 GHz, g/2π= 6 MHz, r/2π= 50 MHz and κa/2π= κb/2π=γm/2π= 0.1g; (b)ωc/2π= 6 GHz, ωa/2π= ωb/2π= 6 GHz, g/2π= 6 MHz, r/2π= 50 MHz and θ= 1.1π. further written as [ 83] ˆρ=/summationtext k,jpk,je−iωkjt|φk/an}bracketri}ht/an}bracketle{tφj|/summationtext k,jpk,je−iωkjttr(|φk/an}bracketri}ht/an}bracketle{tφj|),(12) whereωk,j=ωk−ω∗ jandpk,jare expansion coef- ficients. As t→ ∞, the steady density matrix is ˆρ(∞) =|φ1/an}bracketri}ht/an}bracketle{tφ1|. On the one hand, the steady state is|φ1/an}bracketri}ht= (|˜10/an}bracketri}htA1,c+i|˜01/an}bracketri}htA1,c)/√ 2, which is a Bell state in the general form on the basis of {|˜01/an}bracketri}htA1,c,|˜10/an}bracketri}htA1,c}. On the other hand, it is a tripartite entangled state with |φ1/an}bracketri}ht=|100/an}bracketri}hta,b,c/2 +eiθ|010/an}bracketri}hta,b,c/2 +i|001/an}bracketri}hta,b,c/√ 2 on the basis of {|0/an}bracketri}hta,|1/an}bracketri}hta,|0/an}bracketri}htb,|1/an}bracketri}htb,|0/an}bracketri}htc,|1/an}bracketri}htc}. In Fig.3, we plot numerically population evolution of the system, including the ideal entangled state and other evolution states in PT-symmetry broken phases. By sat- isfying the large detuning condition, we set ωc/2π= 6 GHz,ωa/2π=ωb/2π= 5.95 GHz,g/2π= 6 MHz andr/2π= 50 MHz that is accessible in [ 21,22,27, 28,69,70,84]. The fidelity is formulated as F(t) = tr/radicalbig /an}bracketle{tφ1|ˆρ(t)|φ1/an}bracketri}ht, where|φ1/an}bracketri}htand ˆρ(t) are the target state and the time-dependent density matrix of the system by7 solving Eq. ( 8). It is identified that the population of |010/an}bracketri}hta,b,cand|100/an}bracketri}hta,b,chastheexactlyidenticalevolution with population 0 .25 while the population of the initial state|001/an}bracketri}hta,b,cbecomes 0.5 at the time T= 0.2µs. Ev- idently, the population of the target state |φ1/an}bracketri}htevolves from about 0.7 to 1, signifying the successful creation of a steady tripartite entangled state in PT-symmetry broken phases. In fact, in a tripartite system, quantum coherence may exist due to the collective participation of several subsys- tems, or can be attributed to coherence located within thesubsystems. Therefore,themagnon-photonentangle- ment can be quantified through the collective coherence, which are given by expression [ 85] C=/radicalbigg S(ˆρ+ ˆρπ 2)−S(ˆρ)+S(ˆρπ) 2,(13) in whichSis the von Neumann entropy, ˆ ρthe density matrix of the hybrid system, and the closest product state ˆρπ≡ˆρmin= ˆρa⊗ˆρb⊗ˆρc. Figures 4(a) and (b) show the time evolution of fidelity (thin dotted line) of |φ1/an}bracketri}htand the collective coherence (thin solid line) with- out losses in the PT-symmetry broken phase and the PT-symmetry phase, respectively. Apparently, in the PT-symmetry broken phase with complex eigenvalues ωn,FandCapproach one and 0.806, respectively, when the system without losses evolves to the tripartite en- tangled state |φ1/an}bracketri}ht. Nevertheless, the system enters into thePT-symmetry phase as the eigenvalues ωnare real. The hybrid system has no gain modes, which results in theunstabledynamicsofcollectivecoherenceandfidelity. Thus, thesteadytripartiteentangledstateandthecollec- tive coherencecanbe steadyin the PT-symmetrybroken phase but oscillate in the PT-symmetry phase. Actually, this anomaly can be further understood by thecompetitionoftheevolutionofnon-Hermitiansystem and the particle number conservation in the hybrid sys- tem. When the hybrid system lies in the PT-symmetry broken phase, the evolution of the system guarantees not only the process of gain and loss but also the particle conserving process. The coexistence of two processes will render the initial state to evolve in the steady target en- tangled state. As for the PT-symmetry phase, evolution of the hybrid system does not involve gain and loss, and hence can not render the system to a steady state. The oscillation phenomenon is analogous to the unitary evo- lution oftraditional Hermitian systems with a real beam- splitter type interaction through the particle conserving process [ 86]. B. Generation of tripartite high-dimensional entangled states In the following, we consider the hybrid system evolv- ing in the Hilbert subspace of N >1, when the system is under the PT-symmetry broken phase.0.00 0.05 0.10 0.15 0.200.00.20.40.60.81.0(a) 0.00 0.05 0.10 0.15 0.200.00.20.40.60.81.0(b) FIG. 5. Population evolution of the target tripartite high- dimensional entangled states and their component states fo r (a)|φ2/angbracketright=|˜20/angbracketrightA1,c/2−i|˜11/angbracketrightA1,c/√ 2− |˜02/angbracketrightA1,c/2 based on the initial state |002/angbracketrighta,b,cand (b) |φ3/angbracketright=i√ 2|˜03/angbracketrightA1,c/4 +√ 6|˜12/angbracketrightA1,c/4−i√ 6|˜21/angbracketrightA1,c/4−√ 2|˜30/angbracketrightA1,c/4 based on the initial state |003/angbracketrighta,b,c. Parameters: ωc/2π= 6 GHz, ωa/2π= ωb/2π= 5.95 GHz, g/2π= 6 MHz, r/2π= 50 MHz and θ= 1.1π. Firstly, we take an example of an initial pure state |ψ2/an}bracketri}ht=|˜02/an}bracketri}htA1,cwith a mean particle number N= 2. By solving the master equation ( 6), a steady tripartite three-dimensional entangled state can be generated by satisfying the large detuning condition, which is repre- sented as |φ2/an}bracketri}ht=1 2|˜20/an}bracketri}htA1,c−i√ 2|˜11/an}bracketri}htA1,c−1 2|˜02/an}bracketri}htA1,c,(14) where|˜2/an}bracketri}htA1can be represented by ( |20/an}bracketri}hta,b+ eiθ√ 2|11/an}bracketri}hta,b+e2iθ|02/an}bracketri}hta,b)/2, on the basis of {|0/an}bracketri}hta,|1/an}bracketri}hta,|2/an}bracketri}hta,|0/an}bracketri}htb,|1/an}bracketri}htb,|2/an}bracketri}htb}. In Fig. 5(a), with the same parameters as in Fig. 3, we numerically plot the time evolution of populations for the target entangled state |φ2/an}bracketri}ht, the initial state |ψ2/an}bracketri}ht, and other evolution states. The populations of |101/an}bracketri}hta,b,cand |011/an}bracketri}hta,b,c(|020/an}bracketri}hta,b,cand|200/an}bracketri}hta,b,c) have the identical trend reaching 0.25 (0.06). The population of initial state|002/an}bracketri}hta,b,cevolve from 1 to 0.25, and the population of|110/an}bracketri}hta,b,cis close to 0.16 finally. Obviously, the population of |φ2/an}bracketri}htreaches nearly 1 and remains steady at the end of evolution, which indicates the successful creation of the tripartite high-dimensional entangled state.8 0.00 0.05 0.10 0.15 0.200.00.20.40.60.81.0Entanglement(a) 0.00 0.05 0.10 0.15 0.200.00.20.40.60.81.0Entanglement(b) FIG. 6. Time dependence of the collective coherence and fidelity of tripartite high-dimensional entangled states ( a) |φ2/angbracketright=|˜20/angbracketrightA1,c/2−i|˜11/angbracketrightA1,c/√ 2− |˜02/angbracketrightA1,c/2 based on the initial state |002/angbracketrighta,b,cand (b) |φ3/angbracketright=i√ 2|˜03/angbracketrightA1,c/4 +√ 6|˜12/angbracketrightA1,c/4−i√ 6|˜21/angbracketrightA1,c/4−√ 2|˜30/angbracketrightA1,c/4 based on the initial state |003/angbracketrighta,b,cin thePT-symmetry broken phase. Parameters: ωc/2π= 6 GHz, ωa/2π=ωb/2π= 5.95 GHz, g/2π= 6 MHz, r/2π= 50 MHz, θ= 1.1πand κa=κb=γm= 0.1g. Next, by setting initially a mean particle number N= 3 and choosing an initial state as |ψ3/an}bracketri}ht=|˜03/an}bracketri}htA1,c, we obtain a tripartite four-dimensional entangled state |φ3/an}bracketri}ht=√ 2 4(√ 3|˜12/an}bracketri}htA1,c−i√ 3|˜21/an}bracketri}htA1,c−|˜30/an}bracketri}htA1,c+i|˜03/an}bracketri}htA1,c), (15) where|˜3/an}bracketri}htA1can be represented by ( e2iθ√ 6|12/an}bracketri}hta,b+ eiθ√ 6|21/an}bracketri}hta,b+√ 2|30/an}bracketri}hta,b+e3iθ√ 2|03/an}bracketri}hta,b)/4, on the ba- sis of{|0/an}bracketri}hta,|1/an}bracketri}hta,|2/an}bracketri}hta,|3/an}bracketri}hta,|0/an}bracketri}htb,|1/an}bracketri}htb,|2/an}bracketri}htb,|3/an}bracketri}htb}. Also, the population of the initial state, the target state and other evolution states are exhibited in Fig. 5(b) with the same parametersasFig. 5(a). The fidelity of |φ3/an}bracketri}htreachesunity and remains stable at the end of evolution. Thus, the result reveals that the tripartite high-dimensional entan- gled state can be generated in this scheme. Figures 6(a) and (b) show the time evolution of the collective coher- ence (thin solid line) and fidelity (thin dotted line) with |φ2/an}bracketri}htand|φ3/an}bracketri}htin thePT-symmetry broken phase, respec- tively. As expected, the fidelity of |φ2/an}bracketri}htand|φ3/an}bracketri}htcan be both achieved by unity in the PT-symmetry broken phase, when not considering losses. The final collective coherence of |φ2/an}bracketri}htand|φ3/an}bracketri}htare 0.902 and 0.937, respec- FIG. 7. Collective coherence and the fidelity of the tar- get tripartite high-dimensional entangled state |φ3/angbracketrightas func- tions of the frequency detuning of the magnon-photon cou- pling ∆. Parameters: ωc/2π= 6 GHz, ωa/2π=ωb/2π= 5.95 GHz, g/2π= 6 MHz, r/2π= 50 MHz, θ= 1.1πand Ttotal= 0.2µs. tively, and keep steady. The tripartite entangled state has the larger collective coherence when the initial state is of a larger mean particle number. In order to represent more intuitively the relation among the frequency detuning of the magnon-photon coupling, the collective coherence, and the fidelity of tri- partite high-dimensional entangled state, a full phase di- agramof the system is shown in Fig. 7. InPT-symmetry broken phases, the collective coherence remains the max- imum and steady value, independent of the magnitude of detuning. Meanwhile, the fidelity becomes unity un- der the condition of ∆ ≈ ±2.5. We find that the col- lective coherence and the fidelity are of oscillations and cannot reach the maximal value in the PT-symmetry phases. Therefore, the EPs at ∆ = ±1.9 and±3.1 play an important role of critical point whether the target tri- partite high-dimensional entanglement can be generated steadily. In Fig.8, we numerically work out the time evolution of the collective coherence with the total number of par- ticles from 1 to 6. Apparently, the collective coherence has the lager value with the increase of N. Neverthe- less, when N= 2,3,4,5,6, the collective coherence in- creases slowly and lies in the position between 0.937 and 0.972. By magnifying the time range T∈[0.12,0.17]µs, tripratite high-dimension entangled states can be ob- tainedwiththecollectivecoherenceof0.902,0.937,0.955, 0.965 and 0.972, respectively, when N= 2,3,4,5,6. As for the maximal tripartite high-dimensional entangled states|Ψn/an}bracketri}ht=1√N+1(/summationtextN k=0|k,k,k/an}bracketri}hta,b,c), their collective coherence can reach 0.941, 0.970, 0.983, 0.989 and 0.993, respectively, corresponding to N= 2,3,4,5,6, calcu- lated by Eq. ( 13). In fact, tripartite high-dimensional entangled states proposed here are not the standard ones |Ψn/an}bracketri}ht, and hold collective coherence slightly less than the maximal ones for the integer N∈[2,6]. However, tri- partite high-dimensional entangled states |φn/an}bracketri}hthave also enough capacity to complete tasks of quantum informa-9 0.00 0.05 0.10 0.15 0.200.00.20.40.60.81.0 0.120 0.145 0.1700.880.941.00 FIG. 8. Collective coherence as the function of evolution ti me with the total number of particles from N= 1 toN= 6 based on the initial state |00N/angbracketrighta,b,c. Parameters: ωc/2π= 6 GHz,ωa/2π=ωb/2π= 5.95 GHz, g/2π= 6 MHz, r/2π= 50 MHz and θ= 1.1π. tion. Besides, the tripartite high-dimensional entangled state|φn/an}bracketri}htcan still collapse possibly to entanglement states, such as |˜1/an}bracketri}htA1,|˜2/an}bracketri}htA1and|˜3/an}bracketri}htA1, when quantum measurements are introduced. C. Robustness of entanglement In this subsection, we discuss the robustness of the tri- partite high-dimensionalentangledstate againstenviron- ment noises. Three dominant influence aspects are con- sidered: (i) losses of cavity A, cavityBand the magnon with decayrates κa,κbandγm, respectively; (ii) different initialstates; (iii)unexpectedmagnon-photoncoupling g. When the hybrid system with PT-symmetry broken phases is in an ideal environment, the result shows that the collective coherence and the fidelity of tripartite entangled states |φ1/an}bracketri}ht,|φ2/an}bracketri}htand|φ3/an}bracketri}htremain stable in Figs.4(a),6(a) and6(b). In order to give a quantita- tive illustration of the effects of losses of cavity modes and the magnon mode, the dynamics of the lossy sys- tem is governed by adding Lindblad operators of cavity modes and the magnon mode into the master equation ∂ˆρ ∂t=−i[ˆH1,ˆρ]−i{ˆH2,ˆρ}+2i/an}bracketle{tˆH2/an}bracketri}htˆρ+γmD[ˆc] +κaD[ˆa]+κbD[ˆb], (16) whereD[ˆA] =ˆAˆρˆA†−ˆA†ˆAˆρ/2−ˆρˆA†ˆA/2 withˆA= ˆa,ˆbor ˆcis a Lindblad operator [ 87] describing the loss effect of the cavityA, cavityBor the magnon, respectively. For convenience, we assume that κa=κb=γm= 0.1gthat is accessible in experiment [ 21,22]. Through solving the master equation with losses, we can obtain time evolu- tion of the fidelity and collective coherence of |φ1/an}bracketri}htwith thick lines in Fig. 4(a),|φ2/an}bracketri}htand|φ3/an}bracketri}htwith thick lines in Figs.6(a) and (b), respectively. The fidelity of |φ1/an}bracketri}ht,|φ2/an}bracketri}ht and|φ3/an}bracketri}htare0.926,0.856and0.793,respectively. Andthe0.0 0.5 1.0 1.5 2.0 2.5 3.00.00.20.40.60.81.0 FIG. 9. Infidelity of the steady tripartite high-dimensiona l entangled state versus the decay ratio with different magnon - photon coupling strengths. We choose κa=κb=γm=γ andgeff/2π= 212 MHz, 42.4 MHz, 21.2 MHz and 4.2 MHz, respectively. Parameters: ωc/2π= 6 GHz, ωa/2π=ωb/2π= 5.95 GHz, g/2π= 6 MHz, r/2π= 50 MHz, θ= 1.1πand Ttotal= 0.2µs. collective coherenceof |φ1/an}bracketri}ht,|φ2/an}bracketri}htand|φ3/an}bracketri}htare 0.678, 0.794 and 0.848, respectively. Tripartite high-dimensional en- tangled state |φ3/an}bracketri}htis most affected by losses due to the largest particle number N. In Fig.9, we plot the infidelity (1 −F) of tripar- tite high-dimensional entangled state |φ3/an}bracketri}htat the end of evolution time Ttotal= 0.2µs as a function of losses of the cavities and the magnon with different coupling strengthsgeffin thePT-symmetry broken phase. As the decay rate γincreases, the infidelity of tripartite entan- gled state shows a trend of increase. When geff/2π= {42.4,21.2,4.2}MHz, the three lines of infidelities are of insignificant differences, indicating that the magnon- photoncouplingstrengthhaslittleeffectonthefidelityas 4.2MHz/lessorequalslantgeff/2π/lessorequalslant42.4MHz. However, underthe con- dition ofgeff/2π= 212 MHz, the infidelity reaches 0.37 at the point γ/geff= 0, because of the magnon-photon coupling strength mismatching the large detuning con- dition|ω−ωc−r| ≫ |g|. The fidelity of the tripartite high-dimensional entangled state |φ3/an}bracketri}htcan be over 90% withγ/geff<0.1, when the large detuning condition is well satisfied. In the protocol above, the tripartite high-dimensional entangled state |φ3/an}bracketri}htis achieved through setting the ini- tial state as |ψ3/an}bracketri}ht=|003/an}bracketri}hta,b,c. To test dependence on the initial state, we plot the time evolution of the fi- delity and collective coherence for |φ3/an}bracketri}htwith different ini- tial states satisfying the condition of the mean particle numberN= 3 in Figs. 10(a) and (b), respectively. Inter- estingly, both of high fidelity and steady collective coher- ence can be attained by choosing different initial states to meet the condition of N= 3. In particular, different initial states for the collective coherence have almost the10 0.00 0.05 0.10 0.15 0.200.00.20.40.60.81.0(a) 0.00 0.05 0.10 0.15 0.200.20.40.60.81.0(b) FIG. 10. Time dependence of (a) fidelity of the tripartite high-dimensional entangled state |φ3/angbracketrightand (b) collective co- herence with different initial states by satisfying the cond i- tion of the mean particle number equal to 3. Parameters: ωc/2π= 6 GHz, ωa/2π=ωb/2π= 5.95 GHz, g/2π= 6 MHz, r/2π= 50 MHz and θ= 1.1π. identical evolution trend. Thus, the dynamic evolution of the system is independent of a certain initial state, which can be satisfied with the condition of ∂N/∂t= 0 proved by the commutation relation [ N,ˆH] = 0. Accord- ing to various of initial states in Fig. 10(a), the shortest evolution time for achieving unity fidelity of |φ3/an}bracketri}htis dif- ferent, determined by the population of the initial state atT= 0. Therefore, we can choose the initial state which is of the largest population in the tripartite en- tangled state (e.g., |101/an}bracketri}hta,b,c,|002/an}bracketri}hta,b,cand|011/an}bracketri}hta,b,cfor |φ2/an}bracketri}ht;|111/an}bracketri}hta,b,c,|102/an}bracketri}hta,b,cand|012/an}bracketri}hta,b,cfor|φ3/an}bracketri}ht) so that the target entangled state with the saturation fidelity can be generated more efficiently. Due to influences of external temperature and experi- mental techniques, the initial state of the hybrid system may not be prepared perfectly as a pure state. Thus, it is convenient to introduce a parameter pthat characters the purity of initial state so as to identify the robust- ness of generating the desired high-dimensional entan- glement. The initial state is assumed with a mixed state represented as ρ0=p|˜03/an}bracketri}htA1,c/an}bracketle{t˜03|+ (1−p)|˜30/an}bracketri}htA1,c/an}bracketle{t˜30|. As shown in Fig. 11(a), we plot the fidelity of |φ3/an}bracketri}htver- sus the evolution time Tand the purity pof initial state with respect to |˜03/an}bracketri}htA1,c. We can learn that the fidelity is not dependent of a certain initial state but the evo- FIG. 11. (a) The fidelity of the steady high-dimensional en- tangled state |φ3/angbracketrightversus the evolution time and the purity pof the initial state with respect to |˜03/angbracketrightA1,c. The red solid line represents 0.99 fidelity contour line of the steady high - dimensional entangled state |φ3/angbracketright. (b) Time evolution of the fidelity for the steady high-dimensional entangled state |φ3/angbracketright with the initial density: ρ′ 0=/summationtext3 n=0pn|˜0n/angbracketrightA1,c/angbracketleft˜0n|. Parame- ters:ωc/2π= 6 GHz, ωa/2π=ωb/2π= 5.95 GHz, g/2π= 6 MHz,r/2π= 50 MHz and θ= 1.1π. lution time. The non-Hermitian property of the hy- brid system in the PT-symmetry broken phase results in three eigenmodes, whose imaginary parts are posi- tive, negative and zero, respectively. The eigenmode with a positive (negative) imaginary part is correspond- ing to a gain (loss) mode. The particle number of the system in the gain mode will increase until all parti- cles are in this state. That is, the hybrid system be- haves as a attractor, while in the PT-symmetry broken phase each mixed state is asymptotically purified to a ground state [ 83]. Specially, we plot a 0.99 fidelity con- tour line of the steady high-dimensional entangled state |φ3/an}bracketri}htin Fig.11(a). When the initial state is pure with p= 1, the shortest evolution time is 0.06 µs to reach the fidelity F= 0.99. By comparison with the initial density matrix ˆ ρ0= 0.2|˜03/an}bracketri}htA1,c/an}bracketle{t˜03|+0.8|˜30/an}bracketri}htA1,c/an}bracketle{t˜30|, the steady high-dimensional entangled state with F= 0.99 requiresT= 0.08µs at least. The red line indicates that a larger proportion of |˜03/an}bracketri}htA1,cin the initial state demands a shorter evolution time to realize a satura-11 0.0 0.2 0.4 0.6 0.8 1.00.920.940.960.981.00Entanglement FIG. 12. Collective coherence and the fidelity of the steady tripartite high-dimensional entangledstateas functions ofdis- orderδof the magnon-photon coupling strength g. Parame- ters:ωc/2π= 6 GHz, ωa/2π=ωb/2π= 5.95 GHz, g/2π= 6 MHz,r/2π= 50 MHz, θ= 1.1πandTtotal= 0.2µs. tion fidelity of the tripartite high-dimensional entangled state. On the other hand, we further consider an ini- tial state as ρ′ 0=/summationtext3 n=0pn|˜0n/an}bracketri}htA1,c/an}bracketle{t˜0n|where the total particle number is less than 3. In Fig. 11(b), the steady entangled state is always of a high fidelity at the end of evolution, showing independence of creating the tripar- tite high-dimensional entangled state on a certain initial state with the total particle number being either equal to or less than 3. Furthermore, in Fig. 11(b), we can increase the proportion of the state |˜03/an}bracketri}htA1,cin the initial state so as to shorten the evolution time to attain the saturation fidelity of |φ3/an}bracketri}ht, when the initial state is mixed with|˜01/an}bracketri}htA1,c,|˜02/an}bracketri}htA1,cand|˜03/an}bracketri}htA1,c. In addition, the existence of some circuit imperfections in the magnon-circuit-QED hybrid system, such as the mutual inductance, the self-inductance of circuit, and the unstable external magnetic field, may cause a varia- tion in ideal couplings. In order to study the robustness of the protocol against the variation of cavity-magnon coupling strength, we add a random disorder into cou- pling strength g′=g(1+ rand[ −δ,δ]) where rand[ −δ,δ] is to pick up a random number in the range of [ −δ,δ]. The relation among the fidelity of steady entangled state |φ3/an}bracketri}ht, collective coherence and the disorder δ∈[0,1] is exhibited in Fig. 12. It is the disorder that is ran-domly sampled 51 times, and then the fidelity and col- lective coherence are taken as an average of the 51 re- sults. Learning from the red-diamond line unchanged with varying δ, the collective coherence is independent of the disorder of cavity-magnon coupling strength, which reflects the robustness due to the non-Hermitian prop- erty ofPT-symmetry broken phases. Under the con- dition ofδ∈[0,0.5], the fidelity always remains above 99.5%. Nevertheless, the fidelity oscillates obviously be- tween 94.5% and 99.2% withδ∈[0.65,1], on account of fluctuations of parameter gaffecting approximate con- ditions to produce an effective Hamiltonian ( 8) for the steady tripartite high-dimensional entangled state. IV. CONCLUSION To summary, we have proposed a non-Hermitian model of the magnon-circuit-QED hybrid system. There are the steady quantum coherence and tripartite high- dimensional entangled states among the modes of magnon and photon in PT-symmetry broken phases in the proposed system. The tripartite high-dimensional entangled state and the quantum coherence are robust to the dissipation of hybrid system and the fluctuation of magnon-photon coupling. Besides, the tripartite high- dimensional entangled state is independent of a certain initial state. We take into account the experimental con- siderations, including the implementation of the model, the design of equivalent circuit diagram and the real- ization of non-Hermitian coupling between the modes of magnon and photon in the circuit. This work provides a new approach to generate tripartite high-dimensional entangled states and is expected to be helpful for real- izing tripartite and even multipartite high-dimensional entangled states in the non-Hermitian system with the hybridization of the magnon and the circuit-QED sys- tem. ACKNOWLEDGEMENTS This work was supported by National Natural Science Foundation of China (NSFC) (Grant No.11675046), Pro- gram for Innovation Research of Science in Harbin In- stitute of Technology (Grant No. A201412), and Post- doctoral Scientific Research Developmental Fund of Hei- longjiang Province (Grant No. LBH-Q15060). [1] G.Khitrova, H.M. Gibbs, F. Jahnke, M. Kira, andS.W. Koch,Rev. Mod. Phys. 71, 1591 (1999) . [2] J. M. Raimond, M. Brune, and S. Haroche, Rev. Mod. Phys. 73, 565 (2001) . [3] Z.-L. Xiang, S. Ashhab, J. Q. You, and F. Nori, Rev. Mod. Phys. 85, 623 (2013) .[4] M. Aspelmeyer, T. J. Kippenberg, and F. Marquardt, Rev. Mod. Phys. 86, 1391 (2014) . [5] X. Gu, A. F. Kockum, A. Miranowicz, Y. xi Liu, and F. Nori, Phys. Rep. 718-719 , 1 (2017) . [6] J. Q. You and F. Nori, Nature474, 589 (2011) . [7] J. Q. You and F. Nori, Phys. Today 58, 42 (2005) .12 [8] I. Buluta, S. Ashhab, and F. Nori, Rep. Prog. Phys. 74, 104401 (2011) . [9] A. Blais, R.-S. Huang, A. Wallraff, S. M. 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2022-06-26

Hybrid systems that combine spin ensembles and superconducting circuits provide a promising platform for implementing quantum information processing. We propose a non-Hermitian magnoncircuit-QED hybrid model consisting of two cavities and an yttrium iron garnet (YIG) sphere placed in one of the cavities. Abundant exceptional points (EPs), parity-time (PT )-symmetry phases and PT -symmetry broken phases are investigated in the parameter space. Tripartite highdimensional entangled states can be generated steadily among modes of the magnon and photons in PT -symmetry broken phases, corresponding to which the stable quantum coherence exists. Results show that the tripartite high-dimensional entangled state is robust against the dissipation of hybrid system, independent of a certain initial state, and insensitive to the fluctuation of magnonphoton coupling. Further, we propose to simulate the hybrid model with an equivalent LCR circuit. This work may provide prospects for realizing multipartite high-dimensional entangled states in the magnon-circuit-QED hybrid system.

Tripartite high-dimensional magnon-photon entanglement in PT -symmetry broken phases of a non-Hermitian hybrid system

2206.12769v1

Thermal spin current generation in the multifunctional ferrimagnet Ga 0:6Fe1:4O3 Alberto Anad on,1Elodie Martin,1Suvidyakumar Homkar,2Benjamin Meunier,2Maxime Verges,1Heloise Damas,1Junior Alegre,1Christophe Lefevre,2Francois Roulland,2Carsten Dubs,3Morris Lindner,3Ludovic Pasquier,1Olivier Copie,1Karine Dumesnil,1Rafael Ramos,4Daniele Preziosi,2 S ebastien Petit-Watelot,1Nathalie Viart,2and Juan-Carlos Rojas-S anchez1 1Institut Jean Lamour, Universit e de Lorraine CNRS UMR 7198, Nancy, France 2Universit e de Strasbourg, CNRS, IPCMS, UMR 7504, F-67000 Strasbourg, France 3INNOVENT e.V. Technologieentwicklung, Jena, Germany 4Centro de Investigaci on en Qu mica Biol oxica e Materiais Moleculares (CIQUS), Departamento de Qu mica-F sica, Universidade de Santiago de Compostela, 15782, Spain. (Dated: July 5, 2022) 1arXiv:2206.13426v2 [cond-mat.mtrl-sci] 3 Jul 2022Abstract In recent years, multifunctional materials have attracted increasing interest for magnetic mem- ories and energy harvesting applications. Magnetic insulating materials are of special interest for this purpose, since they allow the design of more ecient devices due to the lower Joule heat losses. In this context, Ga 0:6Fe1:4O3(GFO) is a good candidate for spintronics applications, since it can exhibit multiferroicity and presents a spin Hall magnetoresistance similar to the one observed in a yttrium iron garnet (YIG)/Pt bilayer. Here, we explore GFO utilizing thermo-spin measurements in an on-chip approach. By carefully considering the geometry of our thermo-spin devices we are able to quantify the spin Seebeck eect and the spin current generation in a GFO/Pt bilayer, obtaining a value comparable to that of YIG/Pt. This further conrms the promises of an e- cient spin current generation with the possibility of an electric-eld manipulation of the magnetic properties of the system in an insulating ferrimagnetic material. I. INTRODUCTION The search for multifunctional materials is nowadays a hot topic in spintronics[1, 2]. Currently, functional devices are typically made of a bilayer composed of a non-magnetic material with large spin-orbit coupling (NM) and a ferromagnetic material (FM). These types of devices allow functionalities such as the manipulation of the FM magnetization by the spin Hall eect (SHE)[3{5] in the NM or energy harvesting by employing its in- verse counterpart, the inverse spin Hall eect[6]. Insulating magnetic materials (FMI) are preferred for this purpose to pave the way towards low dissipation spintronics devices[3]. Additional functionalities like the possibility of the electric eld control of the magnetic properties of such systems could be given to these heterostructures by the introduction of multifunctional magnetic materials, opening the possibility of designing more ecient and versatile devices[7, 8]. In spin Seebeck experiments[9{12] a thermal gradient is typically applied in the out- of-plane direction of the magnetic thin lm, generating a spin current owing alongside this direction. In insulating ferromagnetic materials, this spin current is carried by spin collective excitations, also called magnons, and can be injected into an adjacent layer such alberto.anadon@univ-lorraine.fr, juan-carlos.rojas-sanchez@univ-lorraine.fr 2as Pt in the case of this study, and is then converted into a charge current via the inverse spin Hall eect (ISHE)[13{15]. This conversion occurs through the spin-orbit interaction of conduction electrons, which can be strong in heavy metals like Pt and is given by[16]: EISHE =2e ~
SHJs; (1) where EISHE is the electric eld produced by the ISHE, eand~are the electron charge and reduced Plank constant, is the resistivity of the Pt layer, SHis the spin Hall angle, Jsis the spin current injected into Pt and its spin polarization. Until now, yttrium iron garnet (YIG) has been the cornerstone material in thermo-spin phenomena due to its insulating nature and its unique magnetic properties such as low damp- ing and coercive eld. Here, we have studied the thermo-spin current generation in bilayers composed of Pt and the multifunctional magnetoelectric oxide Ga 0:6Fe1:4O3(GFO)[17, 18]. Engineering of thermo-spin devices with properly chosen dimensions allowed an accurate de- termination of the thermo-spin voltages necessary to calculate the spin Seebeck coecient, and therefore granting a comparison with other systems. Indeed, the spin Seebeck eect (SSE) as well the spin Hall magnetoresistance [19{21] in GFO system are largely compara- ble with YIG-based ones. Furthemore, to corroborate our experimental nding we resorted to nite element simulations of the thermal prole to obtain an accurate heat ux in both GFO and YIG layers. II. METHODS A. Growth and structural characterization GFO lms were prepared by pulsed laser deposition (PLD) on SrTiO 3(111) substrates (Furuuchi Chemical Corporation, Japan, with root mean square roughness lower than 0.15 nm) maintained at 900 °C. The KrF excimer laser ( = 248 nm) was operated with a uence of 4 J/cm2[21] and a repetition rate of 2 Hz. The growth was done from a stoichiometric GFO target in an atmosphere of 0.1 mBar of O 2. The YIG lm was grown by liquid phase epitaxy on 3-inch (111)-oriented gadolinium gallium garnet (GGG) substrate from PbO- B2O3-based high-temperature solution (HTL) at about 800 °C using a standard dipping technique [22, 23]. During the deposition time of 90 seconds, the substrate was rotated in 3FIG. 1. Structural characterization. a) X-ray diraction 2scan showing the Ga 0:6Fe1:4O3 (004) peak, the Pt (111) peak and the associated Laue oscillations, indicating high crystalline coherence in the lm and b) crystal structure of Ga 0:6Fe1:4O3. In the crystallographic structure of GFO there are four dierent cationic sites that can be occupied by the Ga3+and Fe3+cations, named Ga1, Fe1, Ga2, and Fe2. Ga1 is a tetrahedral site, and the Fe1, Ga2, and Fe2 are non- equivalent octahedral sites. the solution at 33 rpm and then pulled out. Subsequently, the solution residues were spun o from the sample surface above the HTL and the sample was pulled out of the hot heating zone. Platinum layers for the GFO-based samples were deposited by PLD in situ in order to avoid any surface contamination. The deposition was carried out at room temperature in order to avoid any metal/oxide interdiusion, under vacuum (base pressure of 2 x 108 mbar) and with a deposition rate of 0.06 nm/s, as in [21]. Structural characterization of the samples was done by X-ray diraction 2scans using a Rigaku Smart Lab diractometer equipped with a rotating anode (9 kW) and monochro- mated copper radiation (1.54056 A). The2scan shown in gure 1 indicates that the system grows following the (SrTiO 3) STO(111)//GFO(001)/Pt(111) structure, i.e., with the GFO lm oriented along the [001] direction. The presence of Laue oscillations for the Pt (111) re ection is an indication of a smooth Pt/GFO interface. The thermo-spin devices were made using conventional UV lithography. We have chosen the following stacking to perform our main experiments: STO//GFO(64 nm)/Pt(5 nm) and GGG//YIG(140 nm)/Pt(5 nm). The Pt thin lm is rst patterned by ion milling. Then, an 4FIG. 2. Thermo-spin measurements. a) Longitudinal Spin Seebeck eect conguration. b) Transversal voltage as a function of the in-plane eld in a thermo-spin measurement with a heater current of 100 mA. insulating SiO 2layer with a thickness of 75 nm is grown by RF sputtering using a Si target and Ar+and O2plasma. In a third and last step the Au heater (Ti(10 nm)/Au(150 nm)) is evaporated using a conventional evaporator. Both the heater and the sample are patterned with four pads to measure their resistance using four probes for more precise estimation. The dimensions of the active part of the heater and the sample are 330 10m2and 27010 m2respectively. B. Thermo-spin measurements and estimation of heat transport parameters Thermo-spin measurements are carried out in these devices using an electromagnet to apply an external in-plane magnetic eld (H) as shown in gure 2 (a). A DC current is passed through the heater and after 5 minutes of stabilization, the resistances of the sample and the heater are monitored using I-V measurements to avoid spurious contributions from thermal voltages. The thermo-spin voltage is monitored using a Keithley 2182a nano voltmeter. Numerical simulations based on nite element method have been performed by COMSOL multiphysics, coupling the Electric Currents and Heat Transfer modules, in order to quantify thermal gradients in GFO on STO substrate and in YIG on GGG substrate (see supporting information). The cross-plane thermal conductivity of the GFO thin lm was measured using the 3!method. In this method a thin metal resistor simultaneously serves as a heater and a thermometer, it has been previously employed to determine the thermal conductivity of bulk and thin lm materials, details of the measurements can be found elsewhere [24{26]. For these measurements a Pt resistor (100 nm thick, 10 um width, 1 mm length with 10 nm of Cr for adhesion) is deposited on the bare GFO/STO heterostructure and the 3 !voltage 5response to an AC current with frequency is measured. The thermal parameters of the other layers are obtained from literature[27{33] and detailed in the supporting information. III. RESULTS A. Thermo-spin voltage in GFO/Pt and YIG/Pt We have performed thermo-spin measurements in a GFO/Pt and YIG/Pt systems by observing the thermally induced voltage upon sweeping H as shown in gure 2(a) for GFO, obtaining a typical hysteresis loop-like curve that follows the magnetization of the GFO thin lm[21] (gure 2(b)) for a heater current of 100 mA. We have performed these measurements for H applied at dierent directions in YIG and GFO and observed an isotropic behaviour of the SSE voltage. By monitoring the magnitude of the thermo-spin voltage at saturation for dierent heater powers we can observe that the voltage dierence between the saturation at positive and negative elds scales linearly with the power applied to the heater, as expected by the origin of the spin current generated in GFO. This is shown in the insets of gure 3 for both GFO and YIG systems. We also observe here that the order of magnitude of the SSE voltage is similar in both systems, although in the GFO the oset voltage of the loop and the coercive FIG. 3. Spin Seebeck eect in GFO and YIG. A comparable spin Seebeck voltage is obtained in the GFO/Pt and in the YIG/Pt system varying the heater power. The insets show the linear behaviour of the thermo-spin voltage with the heater power. 6eld are signicantly larger than in YIG. While the elucidation of the origin of such oset voltage is out of the scope of this paper, we argue that it might be related to a persistent pyroelectric current due to the polar nature of the GFO[34]. B. Quantitative comparison of the spin Seebeck eect in YIG and GFO The determination of the spin Seebeck coecient is normally subjected to large uncer- tainty due to the low reproducibility of the experimental conditions. Typically, in literature, the SSE coecient is dened as the ratio between the SSE voltage and the thermal gradient in the FM thin lm normalized by the sample resistance[35]. SrT SSE=VISHE rT
RPt
l; (2) where VISHEis the thermo-spin voltage dierence between positive and negative saturation elds divided by two, rT is the thermal gradient through the FMI, R Ptis the 4-probe resistance in the Pt using I-V measurements and lis the distance between the voltage contacts. Two steps can be taken to overcome the issue of low reproducibility in SSE measure- ments: rst, the heat ux in the system can be considered instead of an estimation of the thermal gradient[36], and second, it is possible to maximize the reproducibility of the ther- mal conditions by using on-chip devices for a consistent thermal contact and dimensions of the system[37{39]. Following both of these steps, we propose to dene the SSE coecient considering the heat ux instead of the thermal gradient in the FMI (Sq SSE) as follows: Sq SSE=VISHE q
RPt; (3) whereqis the heat ux through the FMI. To reliably compare the thermo-spin voltage in the GFO/Pt and YIG/Pt systems we compute the Sq SSEfrom equation 3 and obtain a comparable value for both systems as depicted in table I. We obtain a Sq SSE= 2.90.3fV
m2 W
for GFO, whereas for YIG the value is slighty larger, 3.6 0.4. This suggests that the eciency of the spin current generation in GFO is similar to that of YIG, opening new possibilities in spin caloritronics using insulating magnetic materials with predicted tunable magnetic properties by electric elds. 7FIG. 4. Thermal simulation prole for the STO//GFO/Pt thermo-spin device. Cal- culated local temperature within the device that results from heating: a) (X0Z) cut plane and b) (0YZ) cut plane. The insets show the same distribution near the device (top left) and at an expanded range through the whole system (top right). c) z-component of heat ux in the out-of plane direction at the center of the device as calculated by the numerical method. The inset ex- tends the range in depth including the substrate, showing that the heat ux close to the bottom of the substrate vanishes. d) Temperature in the out-of-plane direction at the center of the device as calculated by the numerical method. Using the thermal conductivity for both FMI we can also estimate the thermal gradient in the FMI layers considering Fourier's law and a one-dimensional thermal ux ( q=rT) to compare with other studies in literature as shown in table I. We observe that the temperature dierence between the upper and lower boundaries of the FMI in both systems are similar 8in magnitude under these considerations. To assess possible deviations from this simplied model due to the geometry of the device and possible thermal losses, we have performed a nite element simulation using COMSOL. In this simulation, we introduce the geometry of the system and use a combination of the electric currents and heat transfer modules and we obtain a temperature dierence in the FMI layer slightly smaller but comparable to the one calculated by Fourier's law for both systems. Figures 4(a) and (b) show respectively the local temperature distribution in the device in both X0Z and 0YZ cut planes. They show that the temperature gradient direction is mostly out of plane within the device, as expected. The heat ux through the GFO layer is almost constant as shown in gure 4(c) indicating that the thermal losses are not relevant and the direction of the thermal gradient is almost completely out of plane in the GFO. Figure 4(d) shows the simulated thermal prole in the whole device. Pt( cm) Sq SSE(fV
m2 W
)FMI (W/m
K)rTFourier FMI (K)rTCOMSOL FMI (K) SrT SSE(pV K
) GFO/Pt 27.5 2.90.3 41 0.251 0.191 90 10 YIG/Pt 25.3 3.60.4 8.5 0.252 0.190 160 20 TABLE I. Thermo-spin and thermal transport parameters for YIG and GFO at room temperature . Electrical resistivity, heat ux spin Seebeck coecient, thermal conductivity of the FMI, estimated temperature dierence between the upper and lower boundaries of the FMI layer for a heater current of 100 mA using Fourier's law and COMSOL simulations and thermal gradient spin Seebeck coecient. The thickness of the FMI are 64 and 140 nm for the GFO and YIG lms respectively. In the case of GFO, we have estimated the thermal conductivity using the 3 !method to beGFO= 41 W/mK, signicantly smaller than that of YIG[27]. The rest of the values considered have been extracted from literature [27{33]. Following these estimations, the value of the SSE considering the thermal gradient in the FMI can be recovered to compare with other studies. The calculated values of SrT SSEare 9010 and 16020pV K
respectively. The rest of the parameters to obtain them are shown in table I. We obtain a value of SrT SSE within the same order of magnitude and a similar value of Sq SSEfor both materials, showing that for both methods of estimating the spin current generation point towards the interest in using GFO as a promising functional material in insulating spintronics. 9IV. CONCLUSIONS In summary, we have explored the thermo-spin properties of the multi-functional material Ga0:6Fe1:4O3in the form of thin lm for energy harvesting and thermal management appli- cations. This material provides additional functionality in terms of ferroelectricity while maintaining the electrically insulating behaviour compared to other materials such as yt- trium iron garnet. By using an on-chip approach to increase reproducibility and carefully considering the heat ux and thermal gradients in both systems, we nd that the spin cur- rent generation by thermal excitation is comparable to the one of yttrium iron garnet. This observation supports the promises of an ecient spin current generation with the possibility of an electric-eld manipulation of the magnetic properties of the system in a new insulating ferrimagnetic material. Our results show that this material can be exploited in spintronics and spin caloritronics applications. ACKNOWLEDGMENTS This work was funded by the French National Research Agency (ANR) through the ANR-18-CECE24-0008-01 `ANR MISSION' and the No. ANR-19-CE24-0016-01 `Toptronic ANR'. S.H. acknowledges the Interdisciplinary Thematic Institute QMat, as part of the ITI 2021 2028 program of the University of Strasbourg, CNRS and Inserm, supported by IdEx Unistra (ANR 10 IDEX 0002), SFRI STRAT'US project (ANR 20 SFRI 0012), ANR-11-LABX-0058 NIE, and ANR-17-EURE-0024 under the framework of the French Investments for the Future Program. M. L. acknowledges the funding by the German Bun- desministerium f ur Wirtschaft und Energie (BMWi) - 49MF180119. R.R. acknowledges support from the European Commission through the project 734187-SPICOLOST (H2020- MSCA-RISE-2016), the European Union's Horizon 2020 research and innovation programme through the MSCA grant agreement SPEC-894006, Grant RYC 2019-026915-I funded by the MCIN/AEI/10.13039/501100011033 and by \ESF investing in your future", the Xunta de Galicia (ED431B 2021/013, Centro Singular de Investigaci on de Galicia Accreditation 2019- 2022, ED431G 2019/03) and the European Union (European Regional Development Fund - ERDF). 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Basso, Scientic Reports 7, 46752 (2017), arXiv:1701.03285. [37] S. M. Wu, F. Y. Fradin, J. Homan, A. Homann, and A. Bhattacharya, Journal of Applied Physics 117, 17C509 (2015). [38] Y. Luo, C. Liu, H. Saglam, Y. Li, W. Zhang, S. S.-L. Zhang, J. E. Pearson, B. Fisher, T. Zhou, A. Bhattacharya, and A. Homann, Phys. Rev. B 103, L020401 (2021). [39] C. Liu, Y. Luo, D. Hong, S. S.-L. Zhang, H. Saglam, Y. Li, Y. Lin, B. Fisher, J. E. Pearson, J. S. Jiang, H. Zhou, J. Wen, A. Homann, and A. Bhattacharya, Science Advances 7, eabg1669 (2021), https://www.science.org/doi/pdf/10.1126/sciadv.abg1669. [40] C. Guillemard, S. Petit-Watelot, S. Andrieu, and J.-C. Rojas-S anchez, Applied Physics Let- ters113, 262404 (2018), https://doi.org/10.1063/1.5079236. [41] S. M. Wu, F. Y. Fradin, J. Homan, A. Homann, and A. Bhattacharya, Journal of Applied Physics 117, 17C509 (2015), 1501.07599. [42] K. Uchida, M. Ishida, T. Kikkawa, a. Kirihara, T. Murakami, and E. Saitoh, Journal of physics. Condensed matter : an Institute of Physics journal 26, 343202 (2014). 13SUPPORTING INFORMATION A. Methods t (nm)(W/m
K) Cp (J/Kg
K)(Kg/m3) GFO 64 41 (this paper) 680* 5 530 YIG 140 8.5[27] 600[28] 5 170 STO 100000 12[29, 30] 437[30, 31] 4 891 GGG 100000 7.5[32] 380[33] 7 080 SiO2 75 1.3 680 2 270 Pt 5 69.1 130 21 450 Au 150 310 130 19 300 Air 225 0.0262 1 003 1.225 Table S1. Parameters for the thermal simulations using COMSOL.* Specic heat capacity (Cp) of GFO is not available in literature to our knowledge, so the Cp of SiO 2was used. Thermal gradients were calculated in GFO on STO substrate and in YIG on GGG sub- strate by nite element method (FEM) simulations in COMSOL Multiphysics software by coupling Electric Currents and Heat transfer modules. An electric current density j c= 6
10 A/m2 ows into the Au wire while temperature is xed at 293.15 K on the bottom of the substrate. We used thermal insulation condition on all the external boundaries (except the one whose temperature is xed at room temperature) and thin layer condition for the Pt layer. We considered a Pt(5 nm) / SiO2(75 nm) / Au(150 nm) wire (27 x 1 um ²) on either GGG (100 m) / YIG (140 nm) or STO (100 m) / GFO (64 nm). All the parameters of interest for each material are indicated in table S1. B. Finite element simulation of the thermal prole in YIG A nite element simulation was also carried out using the parameters of the YIG/Pt sample as shown in table S1. The resulting thermal and heat ux proles are similar to the ones in GFO and can be observed in gure S1. 14Fig. S1. Thermal simulation prole for the GGG//YIG/Pt thermo-spin device. Cal- culated local temperature within the device that results from heating with a) (X0Z) cut plane and b) (0YZ) cut plane. The insets show the same distribution near the device (top left) and at an expanded range through the whole system (top right). c) z-component of heat ux in the out-of plane direction at the center of the device as calculated by the numerical method. The inset presents the same information in the stack including the substrate. d) Temperature in the out-of-plane direction at the center of the device as calculated by the numerical method. 15

2022-06-27

In recent years, multifunctional materials have attracted increasing interest for magnetic memories and energy harvesting applications. Magnetic insulating materials are of special interest for this purpose, since they allow the design of more efficient devices due to the lower Joule heat losses. In this context, Ga$_{0.6}$Fe$_{1.4}$O$_3$ (GFO) is a good candidate for spintronics applications, since it can exhibit multiferroicity and presents a spin Hall magnetoresistance similar to the one observed in a yttrium iron garnet (YIG)/Pt bilayer. Here, we explore GFO utilizing thermo-spin measurements in an on-chip approach. By carefully considering the geometry of our thermo-spin devices we are able to quantify the spin Seebeck effect and the spin current generation in a GFO/Pt bilayer, obtaining a value comparable to that of YIG/Pt. This further confirms the promises of an efficient spin current generation with the possibility of an electric-field manipulation of the magnetic properties of the system in an insulating ferrimagnetic material.

Thermal spin current generation in the multifunctional ferrimagnet Ga$_{0.6}$Fe$_{1.4}$O$_{3}$

2206.13426v2

Compact tunable YIG-based RF resonators José Diogo Costa,a)Bruno Figeys, Xiao Sun, Nele Van Hoovels, Harrie A. C. Tilmans, Florin Ciubotaru, and Christoph Adelmannb) Imec, 3001 Leuven, Belgium Wereportonthedesign, fabrication, andcharacterizationofcompacttunableyttrium iron garnet (YIG) based RF resonators based on µm-sized spin-wave cavities. Induc- tive antennas with both ladder and meander configurations were used as transducers between spin waves and RF signals. The excitation of ferromagnetic resonance and standing spin waves in the YIG cavities led to sharp resonances with quality factors up to 350. The observed spectra were in excellent agreement with a model based on the spin-wave dispersion relations in YIG, showing a high magnetic field tunability of about 29 MHz/mT. a)Author to whom correspondence should be addressed. E-mail: diogo.costa@imec.be b)christoph.adelmann@imec.be 1arXiv:2101.02909v4 [physics.app-ph] 27 Apr 2021Yttrium iron garnet (YIG, Y 3Fe5O12) radiofrequency (RF) resonators based on YIG spheres are a well-established technology and are employed in a broad range of applications, including RF filters and oscillators.1–5These technologies benefit from the low intrinsic magnetic loss of YIG, leading to quality ( Q) factors of up to several thousand. Moreover, the resonance frequency of filters and oscillators can be tuned over a wide frequency by means of an applied magnetic field. Yet, YIG spheres have mm-scale diameters and require even larger transducers for input and output RF signals in addition to the system to apply magnetic fields. More recently, the development of thin film deposition techniques, in particular liquid phase epitaxy (LPE),6has led to an increased interest in planar YIG resonators.7,8This has resulted in considerable reduction of size and form factor,9–12culminating in Qfactors reaching several thousand for groove-based cavities.13–15However, the area of such devices was still in the square mm range, with YIG thicknesses of several µm, and usually operating in a flip-chip configuration. In the last decade, the field of magnonics has seen tremendous progress and spin-wave devices have been miniaturized to µm and sub- µm dimensions.16–21Nonetheless, while YIG micro-andnanostructureshavebeenstudiedintensivelyinrecentyears,22–25theminiaturiza- tion of YIG-based filters and resonators to sub-mm dimensions has received little attention so far.26,27Here, we report on the fabrication and characterization of YIG thin film RF resonators that are based on spin-wave cavities with µm lateral dimensions. This approach allows for a compact device design, with Qfactors of up to 350 and large magnetic field tunability. A model of the spin-wave dispersion relations of the structures is in excellent agreement with the observed resonator spectra. These results show such devices are promis- ingtoreducethefootprintofYIG-basedfilters, whilemaintaininghigh Qfactorsandtunable electrical output. The devices were based on 800 nm thick (111) YIG films (Innovent Technologieentwick- lung, Jena, Germany) deposited by LPE on (111) gadolinium gallium garnet (GGG) sub- strates. The films showed low Gilbert damping of 1104.28A combination of e-beam lithography, wet etching, and lift-off was used for device fabrication. The spin-wave cavities were microfabricated by wet etching (H 3PO4, 130C) using a SiO 2hardmask. After pla- narization by spin-on carbon, the 100 nm thick Au antenna transducers were defined by a lift-off process. 2CONFIDENTIALconfidential3 (b)(a) YIG magnetic cavity 10 µmAntenna YIG Cavity Hext(c) h w G S G Hext Hrf Hrf Meander antennaYIG cavity Ladder antennaFIG. 1. (a) Schematic of a YIG cavity resonator. Examples of standing spin-wave modes inside the YIG cavity are represented by dashed lines. (b) Inductive antennas used in this study: meander and ladder antennas. (c) Schematic of experimental setup (top), including the probing scheme and the external electromagnet, as well as optical micrograph (bottom) of a meander antenna resonator. The red box indicates the position of the YIG cavity with width wand height h. Figure 1(a) depicts a schematic of a YIG cavity resonator along with the direction of the applied external magnetic field. Spin-wave cavity modes (dashed lines) are excited inside the cavity by inductive antennas, which generate oscillating Oersted fields from RF currents. Two distinct types of antennas were used in this study: meander and ladder structures [Fig. 1(b)]. Both meander and ladder antennas consisted each of Nidentical wires [ N= 4 in Fig. 1(b)] with a width of 1 µm and a pitch of 2 µm. Figure 1(c) shows an optical micrograph of processed device including a meander antenna. The dashed red line indicates theregionwheretheYIGcavity( 832µm2)islocated. Thedifferencebetweenthetwotypes ofantennasliesinthedirectionoftheRFcurrent: whileinladderantennas, currentsinwires flow in the same direction with the same phase, currents in adjacent meander antenna wires have a phase shift of ,i.e.they flow in opposite directions. Hence, local magnetic fields are always in the same direction for ladder antennas but alternating for meander antennas. The RF response of the devices was assessed by measuring the RF reflection ( S11- parameter) with a Keysight E8363B network analyzer (input RF power 17dBm) at fre- 3CONFIDENTIALconfidential0.2 0.5 1.0 2.0 5.0 -0.2j0.2j -0.5j0.5j -1.0j1.0j -2.0j2.0j -5.0j5.0j L: h= 64 µm L: h= 128 µm L: h= 256 µm (c)FIGURE 2 5.5 6.0 6.5 7.0 7.5-10.0-9.5-9.0-8.5-8.0 5.5 6.0 6.5 7.0 7.5-10-50 f (GHz)(a)h= 64 µm h= 128 µm h= 256 µm |S11| (dB) Ladder antenna Meander antenna h= 32 µm |S11| (dB)(b)M: h = 32 µm FIG. 2. RF characteristics of YIG cavity resonators. Measured magnitude of the S11-parameter vs. frequency for resonators with (a) ladder antennas (L) and (b) a meander antenna (M), for indicated cavity heights h(w= 16 µm). (c) Smith chart representation of the RF measurements matched to a 50 impedance ( f= 10MHz to 15 GHz). For ladder antennas, external bias field 0Hext= 145 mT; for the meander antenna, 0Hext= 163mT. quencies between 10 MHz and 15 GHz. The antennas were connected between signal and ground paths of 50 coplanar waveguides and contacted by GSG probes [see Fig. 1(c)]. During the measurements, an external magnetic field was applied along the antenna wires [see Fig. 1(c)] using an electromagnet, calibrated by a Hall effect Gauss meter. Figures 2(a) and (b) show as examples the return loss of three ladder-antenna resonators ( N= 8) with different cavity heights ( w= 16 µm) as well as of a meander-antenna resonator ( N= 8, w= 16 µm,h= 32 µm), respectively, after de-embedding the contact pad parasitics. Sharp resonances are observed at frequencies between 6.1 and 6.2 GHz, next to a series of weaker resonances. A Smith chart representation of the device impedances is shown in Fig. 2(c). Theintrinsic Qfactoroftheladder-antennadevicesvariedbetweenabout200and350, which was obtained for the largest device. The RF absorption by the YIG cavity increased with h due to the increasing magnetic volume. By contrast, the Qfactor for the meander-antenna resonator was lower, about 120. 4CONFIDENTIALconfidential 0.000.020.040.060.08 0.000.020.040.060.08 5.0 5.5 6.0 6.5 7.00.00.51.01.52.0 5.0 5.5 6.0 6.5 7.00.00.51.01.52.0 DE mode BV mode Cavity modes Meander Ladder DE mode BV mode Cavity modesFIGURE 3B 8f (GHz) f (GHz)k (rad/µm) k (rad/µm)(b) (a) (c) (d) (e) (f) Re[S11(Hext)–S11(0)] HRFRe[S11(Hext)–S11(0)] Fourier transform DE mode BV mode H(9,1) FMR FMR(7,1) (5,1) (3,1) (1,1) (3,1) (1,1) (3,1)(5,1)(1,1)FIG. 3. RF characteristics of YIG cavity resonators ( 1632µm2cavity size) with (a) ladder and (c) meander antennas. (b) schematic representation of the studied devices (top), the exciting Oersted field (bottom), as well as BV and DE spin-wave configurations. Dispersion relations for the resonators with (d) ladder and (f) meander antennas, with DE (solid lines, m= 1), BV (dashed lines,m= 1), and excited cavity modes (stars) with mode numbers (n;m).0Hext= 145and 163 mT for ladder and meander antenna resonators, respectively. (e) Excitation spectrum due to the Oersted field for ladder (blue line) and meander antennas (red line). We now discuss in more detail the different resonance spectra for the devices including both ladder and meander antennas. Resonator spectra are shown in Figs. 3(a) and 3(c) for ladder ( 0Hext= 145mT) and meander ( 0Hext=163 mT) antennas, respectively. The cavity area was wh= 1632µm2in both cases and all applied magnetic fields were sufficient to saturate the magnetization. In addition to the main resonance, the spectra showed several additional weaker resonances that are well separated in the case of the ladder antenna. This behavior can be understood by considering the spin-wave dispersion relations in a planar rectangular YIG cavity with dimensions `1`2, which are given by21,29 5fn;m=1 2q (!0+!Mexk2 tot)(!0+!Mexk2 tot+!MF); (1) with!0= 0Hext,!M= 0Ms, and the abbreviations F=P+ sin2 1P(1 + cos2(kM)) +!MP(1P) sin2(kM) !0+!Mexk2 tot ;(2) P= 11edktot dktot: (3) Here,ktot=p k2 n+k2 mwithkn=n=` 1andkm=m=` 2the quantized wavenumbers in the two cavity directions. nandmare the mode numbers. denotes the angle between the magnetization and the normal to the waveguide, Mis the angle between the magnetization and the direction of kn, andk= arctan(km=kn). In our devices, the symmetry between the two confinement directions is broken by the direction of the magnetic field and two types of spin-wave modes need to be distinguished: (i) confined backward volume (BV) modes with the direction of knparallel toHext,`1=h, `2=w,==2,M= 0; and (ii) confined Damon-Eshbach (DE) modes with the direction ofknperpendicular to Hext,`1=w,`2=h,==2,M==2[see Fig. 3(b)]. Frequencies and wavenumbers of resulting discrete modes for different modes with (n;m = 1)(saturation magnetization Ms= 130kA/m, exchange constant A= 3:5pJ/m,= 1104) are represented in Figs. 3(d) and 3(f) for the two devices and 0Hext= 145mT and 163 mT, respectively. In addition, Figs. 3(d) and 3(f) also show the continuous dispersion relations of BV and DE spin waves (blue/red dashed and solid lines, respectively), from which the cavity modes are derived. A comparison with the experimental spectra in Figs. 3(a) and (c) shows excellent agree- ment with the frequencies of both BV and DE spin-wave cavity modes with m= 1. Further insight into the excited resonances, their relative amplitudes, and the dependence on the antenna design can be gained by considering the excitation efficiency of a spin-wave (cavity) mode by the Oersted field from an inductive antenna, which is given by21,30 n/Z VHRF(x)
m(x)d3x; (4) withHRF(x)the exciting Oersted field, m(x)the dynamic magnetization of the spin-wave mode, and Vthe cavity volume. In wavevector space, the excitation efficiency is thus given by a Fourier transform of the magnetic excitation field. To a first approximation, the 6magnetic field underneath a wire antenna with width dcan be written as HRFIRF=2d in the wire region and 0 outside, as illustrated in Fig. 3(b) for both ladder and meander antennas. Here, IRFis the RF current. The resulting spatial Fourier spectra of the excitation fields transverse to the wires are shown in Fig. 3(e) for both ladder and meander antennas ( N= 8). This direction cor- responds to M==2and thus to the excitation of DE-like spin-wave cavity modes (see above). The spectra show that ladder antennas efficiently excite DE cavity modes with small mode numbers (small wavenumbers k, large wavelengths) but cannot excite modes with higher k. A comparable result is found for the Fourier transform along the wires of the ladder antenna (not shown), which describes the coupling to BV cavity modes. As a result, the main resonance in the experimental spectrum in Fig. 3(a) can be attributed to a superposition of the DE and BV ferromagnetic resonances (FMR)—which are nondegener- ate due to the finite dimensions of the rectangular YIG cavity—and the first BV spin-wave cavity mode (n= 1;m= 1). Additional resonances at lower frequency correspond to higher order BV spin-wave modes with increasing n, whereas only one higher-order DE mode was clearly observed at higher frequencies. We note that due to symmetry reasons, only odd BV and DE cavity modes can be excited. BV modes with m= 3follow a dispersion relation that is approximately 200 MHz above the m= 1curve. Therefore, the broadening of the peak at6:2GHz may be attributed to the superposition with the (n= 1;m= 3)mode. Modes with m> 1andn>1cannot be distinguished as they are superposed to the main resonance as well as the m= 1modes, and their intensity decreases rapidly with increasing n(as form= 1) andm. Bycontrast,meanderantennaspreferentiallyexciteDEcavitymodeswithlargerwavenum- bers around a maximum determined by the wire pitch. Moreover, due to the opposite direc- tions of the magnetic fields underneath adjacent wires, the meander antenna cannot excite BV modes since the average magnetic field transverse to the wires is zero. Therefore, the spectrum consists of DE modes with increasing mode number nuntil the dispersion relation becomes flat at high wavenumbers. The main resonance in the spectrum in Fig. 3(c) thus consists of a superposition of a large number of BV cavity modes large nand thus nearly continuous k. In addition, modes with large (odd) mcan also be expected to contribute to the main resonance. As a result, the Qfactor of the main resonance is lower for meander antennas than for ladder antennas. Note that the Qfactor of resonators with meander 7CONFIDENTIALconfidential2 4 6 8 10 12-1.0-0.50.050 100 150 200 250 30046810 Ladder MeanderTunability: Ladder - 29.3 MHz/mTMeander - 28.0 MHz/mTFIGURE 4 9f (GHz) µ0Hext(mT) (b)(a) f (GHz)73 mT198 mT 232 mT257 mT|S11| (dB)FIG. 4. Tunability of YIG cavity resonators. (a) Resonance frequency vs.external magnetic bias field for resonators ( 1632µm2cavity) with ladder and meander antennas, as indicated. The solid lines represent best linear fits to the data. (b) Magnitude of the experimental S11-parameter vs. frequency for the ladder antenna resonator for different magnetic bias fields. antennas can be optimized by reducing the wire pitch, which reduces the excitation of DE cavity modes with low wavenumbers. One of the key advantages of YIG resonators is their tunability by a magnetic field. This is illustrated in Fig. 4(a), which shows the dependence of the measured main resonance frequency on the applied magnetic field for ladder and meander antennas. In both cases, the dependence on the studied magnetic field range was linear in the studied range with slopes of 29.3 MHz/mT (ladder) and 28.0 MHz/mT (meander). The tunability was very similar to that of devices based on bulk YIG,5,31demonstrating the device miniaturization does not affect the tunability. The slight dependence on the antenna design can be attributed to the different magnetic field dependence of the relevant spin-wave cavity modes. Figure 4(b) 8CONFIDENTIALconfidential5.5 6.0 6.5 7.00.00.10.20.30.4 FIGURE 5 5.5 6.0 6.5 7.00.000.030.060.090.12 f (GHz)Re[S11(Hext) –S11(0)] Re[S11(Hext) –S11(0)] (b)(a) w= 16 µmhN = 8h = 256 µm h = 128 µm h = 64 µm h = 32 µm h = 16 µm h = 8 µmw = 64 µmN = w/2 µm wh = 40 µmw = 32 µm w = 16 µm w = 8 µmFIG. 5. RF characteristics of YIG cavity resonators with ladder antennas for different cavity dimensions. (a) Different cavity widths wfor constant antenna wire density ( N=w=2µm) and cavity height ( h= 40 µm). (b) Different cavity heights for constant cavity width ( w= 16 µm, N= 8). In all cases, 0Hext= 145mT. shows the corresponding measured return loss for the ladder-antenna resonator, indicating that the device impedance varies only weakly with external bias field. These results indicate that ladder antennas lead to a better-defined device response with a sequence of well-marked and sharp resonances. In the following, we focus on the signal optimization of ladder structures. Figure 5(a) shows the frequency response of resonators with identical antenna wire width (1 µm) and pitch (2 µm), identical cavity height h= 40 µm, but different cavity width w. Maintaining a constant wire density for increasing wwas achieved by setting the number of wires in each resonator to N=w=2µm. In this case, increasingwleads to two competing effects: (i) an increase of the magnetic volume and transducer size; and (ii) the redistribution of the total current in an increasing number of 9parallel wires, which lowers the Oersted field underneath each individual wire. Whereas (i) increases the RF absorption by the cavity, (ii) reduces the external excitation. Both effects tend to compete with each other and, as a result, an optimum width of w= 16 µm (N= 8) was observed, as shown in Fig. 5(a). As expected, the separation between adjacent DE cavity modes decreased for larger wand several peaks became superimposed for the largest cavity. Narrower cavities also showed reduced resonator Qfactors, possibly due to edge effects or processing imperfections. The effect of varying the cavity height his illustrated in Fig. 5(b). In this case, longer antennas overlap with a larger magnetic volume without reducing the driving Oersted field, leading to increased RF absorption by the longer YIG cavity. Thus, the strongest RF absorptionwasobtainedforthelargestcavitywith h= 256 µm, inkeepingwiththeresultsin Fig. 2(a). The main resonance frequency increased slightly with h, which may be attributed tosmallchangesintheinternalmagneticfieldresultingintheobservedshiftofthedispersion relation as a function of the cavity length.17 In conclusion, we have studied the characteristics of RF resonators based on YIG cavities with areas down to 128 µm2(104mm2). Both ladder- and meander-type antennas were used as transducers between the RF and spin-wave domains. The resonators showed Q factors up to 350, depending on the YIG cavity dimensions, and a magnetic field tunability of about 29 MHz/mT. The reduced Q factor with respect to bulk or mm-size YIG resonators can be attributed to the combination of larger intrinsic damping of thin film YIG as well as edge effects at the cavity boundaries. The observed frequency dependence of the resonators was in excellent agreement with the spin-wave dispersion relations in YIG, indicating that the different transducers excited distinctly different cavity modes. Concretely, the model indicated that ladder antennas mainly coupled to ferromagnetic resonance, whereas meander antennas excited standing spin-wave modes with large mode numbers n. A figure of merit of resonators for RF filters can be defined as FoM =k2 eQuwithQu the unloaded Qfactor of the resonance and k2 ethe effective coupling coefficient, which can be deduced from the admittance Y11of the resonator. Acoustic resonators of similar size used in RF filers possess FoMs of 50–200.32By contrast, our resonators have FoMs5–10, which is still an order of magnitude smaller, mainly due to the intrinsically smaller energy transfer from inductive antennas into the spin-wave system with respect to (piezoelectric) transducers used in acoustic filters. Future improvements of k2 emay come from the use of 10magnetoelectric transducers, which still remain to be brought to maturity.21However, to cover a large amount of RF bands on one chip, many different filters typically need to be co-packaged into one system. By contrast, one to several tunable spin-wave resonators inte- grated in single filter can replace a large filter bank, consisting of many acoustic resonators, and therefore compensate for the lower FoM. The results thus show that µm-sized YIG res- onators may find applications in future miniaturized magnetically tunable RF filters. The small device size and form factor of the resonators are also of interest for future integrated systems in a package combining different RF components. All data needed to support the conclusions are present in the paper. Additional data may be requested from the authors. This work has received funding from the imec.xpand fund. The authors would like to thank Patrick Vandenameele and Peter Vanbekbergen for their support of the project as well as Xavier Rottenberg, Kristof Vaesen, and Barend van Liempd for many valuable discussions. J.D.C. acknowledges financial support from the European Union MSCA-IF Neuromag under grant agreement No. 793346. FC’s and CA’s contributions have been funded in part by the European Union’s Horizon 2020 research and innovation program within the FET-OPEN project CHIRON under grant agreement No. 801055. REFERENCES 1C. Kittel, Phys. Rev. 73, 155 (1948). 2J.F. Dillon, Phys. Rev. 112, 59 (1958). 3P.S. Carter, IRE Trans. Microw. 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Phys. 43, 264002 (2010). 17S.O. Demokritov and A.N. Slavin, Magnonics: From Fundamentals to Applications (Springer, Berlin, Heidelberg, 2013). 18A.V. Chumak, V.I. Vasyuchka, A.A. Serga, and B. Hillebrands, Nature Phys. 11, 453 (2015). 19D. Sander, S.O. Valenzuela, D. Makarov, C.H. Marrows, E.E. Fullerton, P. Fischer, J. McCord, P. Vavassori, S Mangin, P. Pirro, B. Hillebrands, A.D. Kent, T. Jungwirth, O. Gutfleisch, C.G. Kim, and A. Berger, J. Phys. D: Appl. Phys. 50, 363001 (2017). 20G. Talmelli, T. Devolder, N. Träger, J. Förster, S. Wintz, M. Weigand, H. Stoll, M. Heyns, G. Schütz, I.P. Radu, J. Gräfe, F. Ciubotaru, and C. Adelmann, Sci. Adv. 6, eabb4042 (2020). 21A. Mahmoud, F. Ciubotaru, F. Vanderveken, A.V. Chumak, S. Hamdioui, C. Adelmann, and S. Cotofana, J. Appl. Phys. 128, 161101 (2020). 22C. Hahn, V.V. Naletov, G. de Loubens, O. Klein, O. d’Allivy Kelly, A. Anane, R. Bernard, E. Jacquet, P. Bortolotti, V. Cros, J.L. Prieto, and M. Muñoz, Appl. Phys. Lett. 104, 152410 (2014). 23N. Zhu, H. Chang, A. Franson, T. Liu, X. Zhang, E. Johnston-Halperin, M. Wu, and H.X. Tang, Appl. Phys. Lett. 110, 252401 (2017). 24B. Heinz, T. Brächer, M. Schneider, Q. Wang, B. Lägel, A.M. Friedel, D. Breitbach, S. Steinert, T. Meyer, M. Kewenig, C. Dubs, P. Pirro, and A.V. Chumak, Nano Lett. 20, 4220 (2020). 25Q. Wang, M. Kewenig, M. Schneider, R. Verba, F. Kohl, B. Heinz, M. Geilen, M. Mohseni, B. Lägel, F. Ciubotaru, C. Adelmann, C. Dubs, S.D. Cotofana, O.V. Dobrovolskiy, T. Brächer, P. Pirro, and A.V. Chumak, Nature Electron. 3, 765 (2020). 26G. Yang, J. Wu, J. Lou, M. Liu, and N.X. Sun, IEEE Trans. Magn. 49, 5063 (2013). 27S. Dai, S.A. Bhave, and R. Wang, IEEE Trans. Ultrason. Ferroelectr. Freq. Control 67, 2454 (2020). 28C. Dubs, O. Surzhenko, R. Linke, A. Danilewsky, U. Brückner, and J. Dellith, J. Phys. D: 12Appl. Phys. 50, 204005 (2017). 29B.A. Kalinikos and A.N. Slavin, J. Phys. C: Sol. State Phys. 19, 7013 (1986). 30V.E. Demidov and S.O. Demokritov, IEEE Trans. Magn. 51, 0800215 (2015). 31K.-W. Chang and W.S. Ishak, 1986 IEEE MTT-S Intern. Microw. Symp. Dig., 473 (1986). 32Y. Liu, Y. Cai, Y. Zhang, A. Tovstopyat, S. Liu, and C. Sun, Micromachines 11, 630 (2020). 13

2021-01-08

We report on the design, fabrication, and characterization of compact tunable yttrium iron garnet (YIG) based RF resonators based on $\mu$m-sized spin-wave cavities. Inductive antennas with both ladder and meander configurations were used as transducers between spin waves and RF signals. The excitation of ferromagnetic resonance and standing spin waves in the YIG cavities led to sharp resonances with quality factors up to 350. The observed spectra were in excellent agreement with a model based on the spin-wave dispersion relations in YIG, showing a high magnetic field tunability of about 29 MHz/mT.

Compact tunable YIG-based RF resonators

2101.02909v4

1 Circular displacement current induced anomal ous magneto -optical effect s in high index Mie resonators Shuang Xia1,2, Daria Ignatyeva3,4,5, Qing Liu6, Hanbin Wang7, Weihao Yang1,2, Jun Qin*1.2, Yiqin Chen6, Huigao Duan *6, Yi Luo7, Ondřej Novák8, Martin Veis *8, Longjiang Deng1,2, Vladimir I. Belotelov*3,4,5 and Lei Bi*1,2 1National Engineering Research Center of Electromagnetic Radiation Control Materials, University of Electronic Science and Technology of China, Chengdu 610054, China 2State Key Laboratory of Electronic Thin -Films and Integrated Devices, University of Electronic Science and Technology of China, Chengdu, 610054, China 3Lomonosov Moscow State University, Leninskie Gory, Moscow 119991, Russia 4Vernadsky Crime an Federal University, 295007 , Simferopol, Russia 5Russian Quantum Center, 143025 , Moscow, Russia 6College of Mechanical and Vehicle Engineering National Engineering Research Center for High Efficiency Grinding, Hunan University Changsha, 410082, China 7Microsystem and Terahertz Research Center, China Academy of Engineering Physics, Chengdu 610200, China 8Charles University of Prague, Faculty of Mathematics and Physics, Ke Karlovu 3, 12116 Prague 2, Czech Republic *Email: qinjun@uestc.edu.cn , duanhg @hnu.edu.cn , v.belotelov@rqc.ru , veis@karlov.mff.cuni.cz, bilei@uestc.edu.cn 2 ABSTRACT : Dielectric Mie nanoresonators showing strong light -matter interaction at the nanoscale may enable new functionality in photonic devices. Recently , strong magneto -optical effects have been observed in magneto -optical nanophotonic devices due to the electromagnetic field localization . However , most reports so far have been focused on the enhance ment of conventional magneto -optical effects . Here , we report the observation of circular displacement current induced anomalous magneto -optical effects in high-index -contrast Si/Ce:YIG /YIG/SiO 2 Mie resonators . In particular, giant modulation of light intensity in transverse magnetic configuration up to 6.4 % under s-polarized incidence appears , which is non -existent in planar magneto -optical thin films. Apart from that, w e observe a large rotation of transmitted light polarization in the longitudinal magnetic configuration under near normal incidence conditions , which is two orders of magnitude higher than for a planar magneto -optical thin film . These phenomena are essentially originated from the unique circular displacement current when exciting the magnetic resonance modes in the Mie resonators , which changes the incident electric field direction locally. Our work indicates an uncharted territory of light polarization control based on the complex modal profiles in all -dielectric magneto -optical Mie resonators and metasurfac es, which opens the door for versatile control of light propagation by magnetization for a variety of applications in vectoral magnetic field and bio sensing, free space non -reciprocal photonic devices, magneto -optical imaging and optomagneti c memories . KEY WORDS : all-dielectric metasurfaces, Mie resonances, magnetic oxides, Magneto -optical Kerr effects Dielectric Mie nanoresonators showing strong light -matter interaction at the subwavelength scale have attracted great research interest recently1-4. Compared to plasmonic devices in which metal 3 nanostructures are used to confine electromagnetic fields at the nanoscale , all-dielectric Mie resonators show several important difference s, including lower optical absorption loss, much large r field penetration into the dielectric nanostructures, and the existence of unique magnetic resonance modes5, 6. These features make dielectric Mie resonators and metasurfaces a fertile playground to discover novel photonic phenomena. Observations such as magnetic mirror7, 8, directional scattering9, 10, enhanced optical nonlinear effects11, 12 and photoluminescence13 have been demonstrated, which are promising for future nanophotonic device applications . Recently, enhancement of the magneto -optical effects ha s been reported in dielectric photonic nanostructures . Several theoretical work s predict strong Faraday rotation enhancement in all - dielectric magneto -optical nanostructures14, 15, as well as large Magneto -optical Kerr Effect (MOKE) enhancement in all -dielectric gratings.16 Experimentally, all -dielectric magneto -optical metasurfaces featuring quasi -waveguide modes have been fabricated17, 18. Transverse magneto - optical intensity effect is observed upon the excitation of waveguide modes19, 20. Meanwhile, silicon Mie resonators are reported to enhance the F araday rotation and transverse magneto -optical intensity effect in a 5 nm thick metallic nickel film21, 22. Electric dipole resonance induced circular birefringence and dichroism are observed in a perpendicular magnetic anisotropic Pt/Co film23. Howe ver, a high index contrast , all-dielectric magnetic metasurface exhibiting strong Mie resonance modes have not been considered in this respect . How the Mie resonance modes influence the magneto -optical effects remain s largely unexplored. Here, we observe the effect of the circular displacement current in the high index contrast Mie resonators on the magneto -optical activity. The effect is disclosed on the structure of Si nanodisks on a magneto -optical bilayer with Y3Fe5O12(YIG) and Ce1Y2Fe5O12 (Ce:YIG ) films deposited on a 4 quartz substrate (Fig. 1a ). In the near infrared range Ce:YIG has a prominent magneto -optical activity much larger than that for YIG, therefore in this case most of the magneto -optical response comes from Ce:YIG . Due to the large index contrast, the metasurface exhibits strong Mie resonance modes including the magnetic dipole (MD) mode, electric dipole (ED) mode, electric quadru pole (EQ) mode , magnetic quadrupole (MQ) mode , as well as waveguide (WG) modes. The highly confined electromagnetic field in Ce:YIG leads to strong light -matter interaction and circular displacement currents which induce anomalous magneto -optical effects in transversal and longitudinal magnetic configurations. It should be emphasized that these effects are absent or negligible in planar Ce:YIG thin films . These results indicate promising potential of controlling light propagation by utilizing the complex Mie resonance modes in all-dielectric magneto -optical Mie resonators and metasurfaces . Our work may inspire the development of novel magneto - nanophotonic devices such as vectoral magnetic field sensing, free space non -reciprocal photonic devices, magneto -optical imaging and optomagnetic memories. 5 RESULTS AND DISCUSSION Figure 1. Device structure and mode analysis of the a ll-dielectric magneto -optical metasurface (a) Schematic diagram of the metasurface magnetized in Voigt configurations for s-polarized TMOKE and LMOKE -T measurements . The device consist s of periodic silicon nanodisk s on Ce:YIG/YIG bilayer thin films fabricated on a quartz substrate. ( b) Top-view SEM image of a fabricated sampl e. The inset shows a zoom -in image of several Si nanodisks. (c) Measured and simulated transmission spectra of the device under normal incidence and linear polarization. (d)(e) Simulated transmission spectra of the array with R=230 nm cylinder radius as a function of the wavelength and incident angle s for s-polarized and p -polarized incidence respectively using rigorous coupled wave analysis (RCWA) method . The white dash lines indicate the dispersion of the WG mode s in the metasurface calculated using the planar waveguide theory. (f) Analytical multipole analysis of scattering spectra for an individual silicon nanodisk (height 170 nm, radius 230 nm) embedded into a low -index medium (n=1.46) . The device consist s of periodic Si (n=3.17) nanodisk Mie resonators on Ce:YIG (n=2.30) /YIG (n=2.30) bilayers fabricated on double side polished SiO 2 (n=1.46) substrate s (Fig. 1a) . The YIG and Ce:YIG thin films were fabricated by pulsed laser deposition (PLD) and annealing (see Methods ). The thickness es of the YIG and Ce:YIG layers were 50 nm and 200 nm, respectively. The X-ray diffraction and Faraday rotation characterizations are shown in Supplementary Fig. S1. The Si nanodisk s were fabricated by plasma enhanced chemical vapor deposition (PECVD) of amorphous silicon thin films followed by electron beam lithography ( EBL ) and reactive ion etching (RIE) (see details in Supplementary Fig. S2 and Method s). The period, radius and thickness of the Si pillars are P=750 nm, R=230 nm and H=170 nm , respectively (Fig. 1b). In order to study the magneto -optical effects in transversal (TMOKE) and longitudinal (LMOKE -T) configurations , we magnetized the device along y - and x- directions , respectively (Fig. 1a). 6 We firstly simulated the transmission spectrum of the sample using finite element method (COMSOL MUITIPHYSICS ) (see details in Supplementary Fig. S3). According to the directions of the applied magnetic field, the permittivity tensor of Ce:YIG and YIG takes the form :24, 25 xx z y MO z yy x y x zzaM aM aM aM aM aM = − − − (1) where iiand iaM (i stands for x, y or z) represent the diagonal and off diagonal permittivity tensor elements , respectively . Fig. 1c shows the transmi ttance spectrum of the magneto -optical Mie resonators under normal incidence with linear polarized light along x-direction . Three resonance peaks at wavelengths of 1100 nm, 1310 nm and 1420 nm are clearly seen . The experimental transmi ttance spectr um (red dotted line in Figure 1c) is in a very good agreement with the simulation results (see measurement set -up and details in Methods ). The small difference of transmission intensity and resonant wavelengths between the experiment and simulation may be attribute d to the sample imperfections during the fabrication process , including slight size deviations, non -vertical sidewall s etc. Simulated transmission spectra as a function of wavelengths and incident angle s for s-polarized and p -polarized incident light (Fig. 1d, e, respectively ) demonstrate different dispersive behavior of the resona nces. To identify their character w e calculated the dispersion relation of the waveguide modes according to the device geometry and using the planar waveguide theory18, 19 26 (see white dashed lines in Fig. 1d, e). These modes correspond to different diffraction orders of the Si nanodisk grating as labeled in the figures (see Methods ). Several modes with narrow linewidth agree well with the calculated waveguide modes. However, other modes cannot be explained by waveguide modes, for example the modes (transmittance dips) at high incident angles label ed by black dashed lines for s -polarized incidence, whose wavelengths seem to be independent o n the incident angle , and the full width at half maximum ( FWHM) is quite broad . These modes can be 7 attributed to different Mie resonance modes as indicated by m ultipole analysis results for normal incidence as shown in Fig. 1f (see Methods ). The scattering cross -section is computed on the basis of a multipole decomposition by considering an individual silicon nanodisk embedded into a low- index medium (n=1.46)9. The multipole analysis agrees with experimental observations despite of a blue shift of the Mie resonance wavelengths , which is due to the more complicated surrounding medium environment including multilayer YIG/CeYIG films, also the coupling effect between different nanodisks. For s -polarized incidence , the Mie resonance wavelengths are almost angle independent, as shown by black dashed lined in Fig. 1 d27. As the incident angle incr eases, ED and MD resonances show a tendency to couple with each other, and the MQ resonance becomes increasingly apparent28. Characteristic circular electric field and displacement current distribution are observed both in Si and Ce:YIG for MD and MQ resonances, as shown in Supple mentary Fig. S5. For the p-polarized incidence, EQ and ED resonance wavelengths are almost independent on the incident angles for small incident angles . The mode at 1310 nm for the non-zero angles of incidence splits into two separate modes27. This dispersion relationsh ip is consistent with the TM (1,0) WG modes , where (1,0) indicate the grating diffraction order29-32. Therefore, in the case of small incident angles , the Mie resonance and these WG modes couple with each other , which also explains why the resonance peak s of the mode s in p-polarization is broader than that in s - polarization for small incident angles33. As the angle of incidence increases, the contribution of the WG mode s dominates and the width of the resonance peak gradually narrow s. The near -field distributions of these modes with s - and p - polarization under different incident angles are shown in detail s in Supplementary ( Figs. S5 and S6). 8 Figure 2. Reflection spectr um and giant TMOKE for s-polarized incident light (a) Schematic diagram of s -polarized TMOKE characterization set -up. All measurements are obtained for s -polarized light and =45° incidence angle. (b) Measured s -polarized TMOKE and reflection spectra of the metasurface compared with bare Ce:YIG /YIG thin film s. The error bars are the standard deviation of 5 consecutive measurements. (c) Measured TMOKE hysteresis for the metasurface and the bare MO film s at 1230 nm wavelength. (d) Simulated reflection spectra a nd TMOKE responses of the metasurface for s -polarized light and =45° incidence angle. Next, we characterize the transverse magneto -optical Kerr effect ( TMOKE ) spectr a for s- polarized light incident at =45°. The incidence plane is parallel to the (1,0) direction of the Si nanodisk grating . The schematic diagram of s -polarized TMOKE measurement is shown in Fig. 5a. The value of TMOKE is defined as the relative change in reflectivity due to the magnetization switch along the incidence plane normal direction34, 35: ()() ()()R M R Mδ=2R M +R M+ − − +− (2) According to Fig. 1 d, the reflect ion spectrum in Fig. 2b (black curve) show s high reflectivity at 1100 nm, 12 50 nm and 13 70 nm corresponding to the EQ, M Q and MD modes , respectively. And a relatively weak resonance peak at 1420 nm wavelength is attributed to the ED mode . The electric and magnetic near -field distribution corresponding to these resonances are detailed in 9 Supplemen tary Fig. S5. In the planar magneto -optical thin films, it is well known that there is no TMOKE under s - polarized incident light36, 37, as confirmed by the bl ue line in Fig. 2b measured on bare Ce:YIG /YIG thin film s, which is zero within the measurement error . However, a large s -polarized TMOKE is observed in the magneto -optical Mie resonators , as shown by the red line in Fig. 2b. Importantly, we observe high reflectivity up to 73 % together with strong s -polarized TMOKE up to =2.7 % at the M Q mode wavelength of 1250 nm . Stronger s-polarized TMOKE up to =6.4 % at 1275 nm wavelength and =-6.4 % at 1170 nm wavelength are also observed around the MQ resonances , which are attributed to low reflectance (~10 %, smaller value of the denominator in equation (3) ) induce d enhancement of the TMOKE , namely the optical contribution38. A TMOKE peak of =- 1.4 % also appears at 1375 nm wavelength with 61 % reflectivity, corresponding to the MD mode. Note a sign change of the TMOKE is observed for both MQ and MD modes, which is caused by a sign change of the numerator in Eq. (3). Meanwhile , the s -polarized TMOKE is also non-zero, but weaker at EQ and ED resonances. These measurement results agree very well with simulation using COMSOL as shown in Fig. 2d (see details in Methods ). To confirm our observation, we also measured the TMOKE hysteresis of the structure and film using a custom -built magneto -optical characterization set up (See Supplementary Fig. S8), as shown in Fig. 2c. A clear TMOKE hysteresis is observed for the metasurface sample with up to 3 % intensity variation at 1230 nm , which is in drastic contrast with the thin film sample which showed no hysteresis at all . The hysteresis resembles the hysteresis of the CeYIG /YIG films with t he in - plane magnetization. It should be noted that these s -polarized TMOKE values are even higher than recently reported conventional p -polarized TMOKE in all -dielectric magneto -optical gratings18, 19, 36. We also measured and simulated the TMOKE spectrum under conventional p-polarized incidence for the thin film and metasurface as discussed i n Supplementary Fig. S7. Giant p -polarized TMOKE up to ~20 % is also observed , which are attributed to waveguide mode induced TMOKE enhancement, as also discussed in previous publications .19 10 Figure 3. Transmission spectra and giant LMOKE -T under p -polarzed incidence. (a) Schematic diagram of LMOKE -T characterization set -up. All measurements are obtained for p -polarized light and under =3° incidence angle. (b) Measured LMOKE -T (red line with dots) and transmission spectra (black line) of the fabricated metasurfaces compared with bare Ce :YIG thin films ( blue line). (c) Measured LMOKE -T hysteresis loops for the metasurface with R=230 nm and the bare MO film at 13 20 nm wavelength. (d) Simulated transmission and LMOKE -T spectra of the metasurface. Next , we study the LMOKE effect in transmission mode with sample configuration shown in Fig. 3a. The complex LMOKE -T angles L can be expressed as38: arctan( )ps L pptit = + = (3) where and are the LMOKE -T angle and ellipticity, respectively. And tps and tpp represent Fresnel transmission coefficient s for s-polarized and p-polarized light respectively for p-polarized incidence . For symmetry considerations, a non -trivial LMOKE -T can only be observed when [k×N]≠036, where k is the incident wave vector and N is the sample surfac e normal vector . Therefore , we measure d the LMOKE -T and corresponding transmi ssion spectr a of the sample under =3°, a small off-normal incident angle , as shown in Fig . 3a. Fig. 3b shows the transmission and LMOKE -T 11 spectra. Two waveguide modes are observed at 12 50 nm and 1320 nm respectively as indicated by Fig. 1e . As discussed in Fig. 1 e, the MD mode hybridizes with the WG mode at a small incident angle of 3 °. For such a small angle, the LMOKE -T of the bare Ce:YIG /YIG thin film is almost negligible (<10-3 deg). This value is smaller than our measurement noise, therefore it is only numerically simulated and shown by the blue line in Fig. 3b. Interestingly, the LMOKE -T shows a giant enhancement when exciting the resonanc e modes. A large LMOKE -T angle up to =0.086 deg is observed at 1320 nm wavelength, which is about two orders higher compared to bare Ce:YIG /YIG films with same thickness es of =9×10−4 deg. Enhancement of LMOKE -T is also observed at 1250 nm and 1418 nm wavelength but with a lower amplitude, corresponding to the other waveguide mode and hybridized ED-WG mode respectively . This result can be better observed by comparing the LMOKE -T hysteresis l oops between the metasurface and the film as shown in Fig. 3c. A clear hysteresis resembling the in -plane magnetization hysteresis of the Ce:YIG thin film is observed. The LMOKE -T angle of 0.086 deg is even comparable to the Faraday rotation angle of a bare film at the same wavelength as shown in Supplementary Fig. S1b. The simulated LMOKE - T spectrum is displayed in Fig. 3d, which shows similar characteristic s with experiment resul ts, despite of sharper peaks and larger rotation angle values. This is because in the case of simulation, the transmittance at resonance s is almost zero, leading to a much large r optical contribution . The difference of transmission intensity between experiment and simulation may be caused by sample imperfections originated from the fabrication process. 12 Figure 4. Mechanism of the anomalous MO effects in the MO Mie resonators . Electric field distribution for (a) MD resonance (b) ED resonance for θ=3°, p-polarized incidence (LMOKE -T case) and (c) MQ resonance (d) ED resonance for θ=45 °, s-polarized incidence ( TMOKE case ) respectively. The pink arrows indicate the electric field vectors in Z -X plane. The vectors of different lengths at the bottom left corner of each figure indicate the relative magnitudes of the volume integral of E x, E y and E z components in the magneto -optical layers . Schematic diagram of the relationship between electric field vectors, magnetic field direction and propagation direction are shown for (e) MD/MQ and (f) ED modes respectively. To understand the mechanism of the observed magneto -optical effect s, we consi der the electric and magnetic near -field distribution for different resonance modes as shown in Fig. 4 . Fig. 4a and 4b show the electric field profile under p-polarized incidence at θ=3° for the MD -WG and ED -WG mode wavelengths respectively (the LMOKE -T case) . Fig. 4c and 4d show the modal profile under s-polarized incidence at =45° for the MQ and ED resonance wavelengths respectively ( the TMOKE case ). As shown in figure 4a and 4c, for the MD/MQ mode s, the resonance s show characteristic circular electric field distribution . Thus the Ez field is significantly enhanced , inducing circular displacement currents in Ce:YIG . Whereas f or the ED resonances shown in Fig. 4b and 4d , 13 the electric dipole resonance induces mostly Ex or Ey fields in Ce:YIG due to the oscillated dipole resonance behavior. Considering the magneto -optical effect, only the electric field perpendicular to the applied magnetic field shows the magneto -optical effect s, as indicated by the form of the permittivity tensors in Eq. (1). Therefore, Ez make s a main contribution for the near normal incident LMOKE -T and s-polarized TMOKE. We can quantitatively compare the E x, Ey, Ez field intensity by performing an area integral of |E| inside the Ce:YIG layer. The integrated Ez intensity is 1.34 times higher than Ex at the MD-WG mode wavelength , whereas the intensity of Ex is 4.6 times larger than Ez at the ED-WG resonance mode for θ=3°. For θ=45° incident angl e of the s-polarized TMOKE configurations, the intensity of Ez is comparable with Ey at the MQ mode wavelength, whereas the intensity of E y is 1.87 times larger than E z at the ED resonance . Fig. 4e and 4f show the relationship between the incident wavevector , the electric field vector and the applied magnetic field direction s. For the resonance s dominated by MD /MQ, the p-polarized /s-polarized incident light generates E z component of the displacement current , leading to enhancement of Kerr effect when the magnetic field is along the x (LMOKE -T configuration ) or y directions ( TMOKE configuration ). On the contrary, the electric field distribution for ED resonance modes is different. The electric fields remain mostly along x- or y-directions. Nevertheless, a small amount of E z field is also observed for these modes, therefore we do see TMOKE/LMOKE -T enhancement at around ED resonances but with a much lower amplitude compared to MD/MQ modes. These observations again highlight the important role of circular displacement currents to the observed anomalous magneto - optical effects. The observation of several anomalous magneto -optical effects in high index contrast, all - dielectric magneto -optical metasurfaces indicate unprecedented opportunity of using the complex 14 modal profiles in high -index -contrast all -dielectric Mie resonators to control light propagation by magnetization and vice versa . Study on other dielectric resonance modes can be envisioned for future works , such as anapole modes39, Fano resonance modes40, supercavity mode s41 as well as coupled Mie resona nce modes42. Ultra -high quality factor modes such as the bound state s in the continuum modes43 can a lso be explored. Note the observ ed magneto -optical effects are rooted in localized Mie resonance mod es, which is very different from several previous proposals of enhancing magneto -optical effect s by propagating waveguide modes29, 30, 44. This new mechanism offers a possibility to construct advanced magneto -optical materials by locally des ign the structure at the subwavelength sc ale, leading to a variety of possibilities to control the wave front by specifically designed magneto -optical Mie resonators and metasurface s. On the other hand, the complex field profile also indicates rich physics of manipulating spin using femtosecond optical pulses, i.e., ultrafast optomagnetic effects in high index contrast Mie resonators45. The low optical absorption, strong field localization , broad angular and frequency width and designable modal profiles may enable more efficient all optical magnetization switch for future spintronic devices. These possibilities make high index contrast, all -dielectric magneto -optical metasurfaces promising for a variety of applications including vectoral magnetic field sensing, free space non -reciprocal photonic devices, magneto -optical ima ging and optomagneti c memories . CONCLUSIONS In summary , we observe anomalous magneto -optical effects including giant s-polarized TMOKE and LMOKE -T in high index contrast magneto -optical Mie resonators , which are not achievable in bulk or planar magneto -optical materials . The se magneto -optical effects are originated from the unique circular displacement currents associated with MD or MQ modes in high index 15 contrast all -dielectric Mie resonators . A giant s-polarized TMOKE up to 6.4 % and n early two orders of magnitude enhancement of the LMOKE -T under near normal incidence conditions are observed experimentally . Our results indicate the possibility of utilizing the complex Mie resonance modes to realize novel magneto -optical effects , which will allow unprecedented opportunity to control light propagation with magnetization and vice versa . 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All -Dielectric Active Terahertz Photonics Driven by Bound States in the Continuum. Adv Mater 31, e1901921 (2019). 44. Bsawmaii, L., Gamet, E., Royer, F., Neveu, S. & Jamon, D. Longitudinal magneto -optical effect enhancement with high transmission through a 1D all -dielectric resonant guided mode grating. Opt Express 28, 8436 -8444 (2020). 45. Wu, Y .H., Kang, L., Bao, H.G. & Werner, D.H. Exploiting Topological Properties of Mie - Resonance -Based Hybrid Me tasurfaces for Ultrafast Switching of Light Polarization. Acs Photonics 7, 2362 -2373 (2020). 46. Tian, J. et al. Active control of anapole states by structuring the phase -change alloy Ge2Sb2Te5. Nat Commun 10, 396 (2019). 47. Pagani, Y ., Van Labeke, D., Gu izal, B., Vial, A. & Baida, F. Diffraction hysteresis loop modeling in magneto -optical gratings. Opt Commun 209, 237 -244 (2002). METHODS Numerical Simulation Numerical simulations were carried out using finite element method via COMSOL MUITIPHYSICS. Periodic boundary conditions were applied in both x and y directions for the unit cell. Perfect Match Layer (PML) were set along the z direction at the upper and lo wer boundaries of the structure. The refractive index of the quartz substrate was set as 1.46 in our model. As for the permittivity tensor of Ce:YIG films, we adopted the detailed parameters from reference 2 4. The mesh sizes varied slightly depending on th e refractive index of the 18 material to ensure the accuracy of the results. The transmittance and reflectivity of the metasurfaces w ere obtained by extracting the S parameters of the scattering matrix. Then the TMOKE and LMOKE -T angle can be calculated using Eq. (3) and Eq. (4), respectively . The RCWA method was also applied for angle dependent transmission spectrum simulations. And the calculated results agree well with numerical simulation by finite element analysis using COMSOL (seen in Supplementary Fig. S4). Calculation of Waveguide Mode Dispersion To further understand the WG modes, we use the equivalent medium method to treat the structure of ‘Si nanodisk s + magneto -optical film’ as an equivalent slab waveguide core surrounded by air and SiO 2 claddings. According to the planar waveguide theory18, 19 26, the dispersion equation of the modes can be defined as: 2 2 2 2 2 2 2( sin ) +( ) ( )xym m nPP += (4) where λ is wavelength of incident light from the free space, θ is the angle of incidence, mx/y is the diffraction orders along x and y directions, P is the period of the Si nanodisk s, 𝑛𝛽 is the effective refractive index of the guided mode, 2π𝑛𝛽/λ is the wave vector of the guided mode. Firstly, we assume that the resonance at 1310 nm wavelength under normal incidence is a waveguide mode, then we can obtain the 𝑛𝛽 by equation 22()xy n m mP=+ . With this fixed 𝑛𝛽 , the trend of waveguide mode as a function of angles and wavelengths can be calculated via 22sin ( )y xm mnPP= − − , as shown in Fig. 1e and 1f. Multipole Decomposition By extracting the value of electric field E(r) through simulation calculation, we defined the induced polarization current density [ 1] 0r iωεε = − −J(r) (r) E(r) , then the electric (),a l m and magnetic ( , )b l m spherical multiple coefficients can be calculated as46: -1ˆ [ ( ) ( )] (cos )( )! ()( , ) exp( ) ' ( )ˆ ˆ (cos ) (cos ) ( 1)( )!sinm l l l l 3 mm l 0 llψkrψ'' kr P θlm ikηa l m im d ψkr d im2πE Pθθ Pθ l l l mkr dθ θ + +− − =− − ++ r J(r) r J(r) J(r) φφ (5) ( )! ()ˆ ˆ ( , ) exp( ) ( ) (cos ) (cos )sin ( 1)( )!l 1 2 mm l l l 0lm ikη im db l m im j kr P θθ Pθ2πE θ dθ l l l m+− − = − + ++ J(r) J(r) φ φ (6) 19 where 0E is the electric field amplitude of the incident plane wave; η is the impedance of free space; ()lj kr is the spherical Bessel function of the first kind while (cos )m lPθ is the associated Legendre polynomials. Then the total contribution to the scattering cross section scatC of the Si disk can now be written as ( )( ( , ) ( , ) )22 l scat l 1 m l 2πC 2l 1 a l m b l mk = =− = + + (7) For the sake of simplicity, we investigate the scattering response of a single silicon disk in homogeneous medium with the refractive index of 1.46 . Sample Fabrication The magnetic oxides were deposited on 10 mm × 10 mm double -polished quartz substrate by pulsed laser deposition (TSST PLD, Netherlands) equipped with 248 nm KrF excimer laser. A layer of 50 nm YIG was first dep osited on SiO 2 at room temperature with the oxygen pressure of 5 mTorr. Then a rapid thermal annealing process in oxygen atmosphere of 2 Torr at 900 °C for 480 s was performed to ensure crystallization. Subsequently, a 200 nm thick Ce:YIG layer was deposited at 750 °C with the oxygen pressure of 10 mTorr using 2.03 J/cm2 power density. After deposition, the film was cooled down in the main chamber at the rate of 5 °C/min. The XRD pattern and magneto -optical response of the oxide film grown on quartz can be seen in Fig. S1 . The -Si nanodisk arrays were fabricated by electron beam lithography (EBL, Raith). First, a layer of uniform amorphous silicon was deposited by plasma enhanced chemical vapor deposition (PECVD). Then the pillar -shaped pa tterns with an area of 200 m × 200 m was prepared by electron -beam lithography using HSQ resist (XR -1541006) followed by deep reactive ion etching (DRIE). The etching gas and specific flow ratio applied in the experiment were CH 3F3: SF6: O 2 = 30: 30: 5. Optical and Magneto -optical Measurement The transmittance spectra and LMOKE -T effects of the metasurfaces were measured by a custom -built characterization set -up as shown in Fig. S8 . The incident light was provided by a supercontinuum laser (NKT Photonics) connected with a spectrometer. Then the beam was incident on the sample surface through an aperture and a Glan -Taylor calcite polarizer to ensure linear polarization in the x direct ion. A pair 20 of lenses of the same focal length were equipped on both sides of the sample to achieve confocal effect. The light spot can be focus ed in to the size of 10 m to meet the test requirements . LMOKE -T spectra were obtained with the applied magnetic field along x -direction. The magnetic field is generated by an electromagnet with the maximum magnetic field of 1T. The reflection and TMOKE spectra were characterized on a spectroscopic ellipsometer (J. A. Woollam RC2). The reflect ance of the silicon wafer with a 25 nm silicon dioxide layer was firstly measured , then we used this as the baseline to obtain the reflect ion spectra of the structure at the 45˚ incident angle . The applied magnetic field of 3 kOe is along the in -plane y direction, which was provided by a neodymium iron boron permanent magnet . By changing the position of the permanent magnet, the reflecti on spectr a of the structure under the positive and negative magnetic field were measured , then the spectra of TMOKE were calculated according to equation 3. As for the TMOKE hysteresis , it was also measured by the custom -built characterization set -up. The magnetic field generated by an electromagnet is applied in y -direction and the reflectivity of the sample changed with the applied magnetic field was measured . Then the hysteresis was obtained via using the equation ()() ()R M R 0δ=R0−47, where R(0) is the reflectivity through the non -magnetized sample . DATA A V AILABILITY All the data generated and analyzed during this study are included in the article and its Supplementary Information. Source data are provided with this paper. ACKNOWLEDGEMENTS The authors are grateful for support by the National Natural Science Foundation of China (NSFC) (Grant Nos. 51972044 and 52021001), Ministry of Science and Technology of the People’s Republic of China (MOST) (Grant Nos. 2016YFA0300802 and 2018YFE0109200), Sich uan Provincial Science and Technology Department (Grant Nos. 2019YFH0154 and 2020ZYD015), the Open -Foundation of Key Laboratory of Laser Device Technology, China North Industries Group Corporation Limited (Grant No. 200900), and the Fundamental Research Fu nds for 21 the Central Universities (Grant No. ZYGX2020J005) . D.O.I. and V .I.B. acknowledges financial support by the Ministry of Science and Higher Education of the Russian Federation, Megagrant project N 075 -15-2019 -1934 . AUTHOR CONTRIBUTIONS S.X., J.Q., L.D. and L.B. designed the experiments . Q.L., Y.C. and H .D. fabricated the magnetic all-dielectric metasurface samples . S.X. and W .Y . deposited the magneto -optical films and measured the transmittance , reflectance and magneto -optical spectra . H. W. and Y . L. helped with the material and device structure characterizations. D.O.I. and V .I.B. calculate d the angle -resolved transmittance spectra using RCWA and the dispersion of the WG modes. O.N. and M .V. helped the theoretical analysis and the set-up of the home -made magneto -optical spectrum characterization stage . All the authors contributed to the preparation of the manuscript . COMPETING INTERESTS The authors declare no competing interests. ADDITIONAL INFORMATION Supplementary Information Correspondence and requests for materials should be addressed to J.Q., V .I.B., Y .C., M.V . or L. B.

2021-08-02

Dielectric Mie nanoresonators showing strong light-matter interaction at the nanoscale may enable new functionality in photonic devices. Recently, strong magneto-optical effects have been observed in magneto-optical nanophotonic devices due to the electromagnetic field localization. However, most reports so far have been focused on the enhancement of conventional magneto-optical effects. Here, we report the observation of circular displacement current induced anomalous magneto-optical effects in high-index-contrast Si/Ce:YIG/YIG/SiO2 Mie resonators. In particular, giant modulation of light intensity in transverse magnetic configuration up to 6.4 % under s-polarized incidence appears, which is non-existent in planar magneto-optical thin films. Apart from that, we observe a large rotation of transmitted light polarization in the longitudinal magnetic configuration under near normal incidence conditions, which is two orders of magnitude higher than for a planar magneto-optical thin film. These phenomena are essentially originated from the unique circular displacement current when exciting the magnetic resonance modes in the Mie resonators, which changes the incident electric field direction locally. Our work indicates an uncharted territory of light polarization control based on the complex modal profiles in all-dielectric magneto-optical Mie resonators and metasurfaces, which opens the door for versatile control of light propagation by magnetization for a variety of applications in vectoral magnetic field and biosensing, free space non-reciprocal photonic devices, magneto-optical imaging and optomagnetic memories.

Circular displacement current induced anomalous magneto-optical effects in high index Mie resonators

2108.00615v1

Magnon based logic in a multi-terminal YIG/Pt nanostructure Kathrin Ganzhorn,1, 2,Stefan Klingler,1, 2Tobias Wimmer,1, 2Stephan Gepr ags,1 Rudolf Gross,1, 2, 3Hans Huebl,1, 2, 3,yand Sebastian T. B. Goennenwein1, 2, 3,z 1Walther-Meiner-Institut, Bayerische Akademie der Wissenschaften, 85748 Garching, Germany 2Physik-Department, Technische Universit at M unchen, 85748 Garching, Germany 3Nanosystem Initiative Munich, 80799 M unchen, Germany Boolean logic is the foundation of modern digital information processing. Recently, there has been a growing interest in phenomena based on pure spin currents, which allow to move from charge to spin based logic gates. We study a proof-of-principle logic device based on the ferrimagnetic insulator Yttrium Iron Garnet (YIG), with Pt strips acting as injectors and detectors for non- equilibrium magnons. We experimentally observe incoherent superposition of magnons generated by dierent injectors. This allows to implement a fully functional majority gate, enabling multiple logic operations (AND and OR) in one and the same device. Clocking frequencies of the order of several GHz and straightforward down-scaling make our device promising for applications. In the eld of spintronics, the generation and de- tection of spin (polarized) currents have been studied extensively1{8, with the aim to use the spin degree of freedom to improve electronic devices. In particular, magnonic devices based not on the ow of electrical charges but on the ow of quantized excitations of the spin system (magnons) have been considered for the transmission and processing of information. In this con- text, dierent propositions for the implementation of magnonic logic elements and their integration into the present electronic circuits have been made9{13, using e.g. the phase or the amplitude of magnon modes to en- code a logical bit. One important step towards magnonic logic is the implementation of a majority gate11, with three inputs and one output. The output of the majority gate assumes the same logical state ("0 or "1") as the ma- jority of the input signals. Additionally, one of the input channels can be used as a control channel to switch be- tween "OR" and "AND" operations (see Tab. I), allowing for multiple types of logic operations in a single struc- ture. Recently, a design for an all magnonic majority gate based on single magnon mode operations has been proposed14,15, using the phase of coherent spin waves (magnons) to encode the logical "1" and "0". However, in order to realize this magnonic majority gate, phase sensitive generation and detection of the spin waves is required. Furthermore, the selected magnon modes and therefore the functionality of the logic gate depend cru- cially on the waveguide geometry, making down-scaling challenging. In this letter, we present a proof-of-principle device implementing a multi-terminal magnonic majority gate based on the incoherent superposition of magnons, re- quiring neither microwave excitation nor phase sensi- tive detection of the spin waves. For the implementa- tion, we exploit the recently discovered magnon mediated magnetoresistance (MMR)16,17, which was rst measured in two parallel Pt strips separated by a distance dof a few 100 nm, deposited onto an Yttrium Iron Garnet (Y3Fe5O12, YIG) thin lm. By driving a charge current through the rst strip, i.e. the injector, an electron spinaccumulation proportional to the driving current is gen- erated in the Pt at the interface via the spin Hall eect. This spin accumulation is then converted into a magnon accumulation in the YIG beneath the injector18,19, which diuses across the ferrimagnetic insulator. We assume that the generated non-equilibrium magnons have the same polarization direction as the initial spin accumu- lation in the injector strip. If the second Pt strip, i.e. the detector, is within the magnon diusion length, the non-equilibrium magnons in YIG are converted back into an electron spin accumulation in the detector Pt strip. This spin accumulation induces a charge current via the inverse spin Hall eect (ISHE), which is detected as non- local voltage in open circuit conditions. Here, we extend the basic two Pt strip structure to a device with three magnon injectors and one detector as shown schemat- ically in Fig. 1 (a). We observe incoherent superposi- tion of magnons generated by dierent injector strips, i.e. the phase of the excited magnons is not relevant. We show that the detected non-local ISHE voltage is sensi- tive to the number and polarization of all non-equilibrium magnons accumulating beneath the detector. Based on these ndings we are able to experimentally implement a four-strip magnon majority gate, which is integrated into an electronic circuit. The YIG/Pt four-strip device we studied was fabri- cated starting from a commercially available 2 m thick YIG lm grown onto 111 oriented Gd 3Ga5O12via liquid phase epitaxy. After Piranha cleaning and annealing of the YIG to improve the interface quality (see Ref. 16 for details), a 10 nm thick Pt lm was deposited on top of the YIG using electron beam evaporation. For the magnon injection and detection, we patterned four Pt strips with a width ofw= 500 nm and a center-to-center separation ofd= 1m using electron beam lithography followed by Ar ion etching. An optical micrograph of the nal device is shown in Fig. 1 (b). The two center strips (strip 1 and 2 in Fig. 1 (b)) have a length of l= 162 m and the outer strips (strip C on the left and strip 3 on the right) are 148 m long. The wiring scheme for the measure- ments is also sketched in Fig. 1 (b). All experiments arearXiv:1604.07262v1 [cond-mat.mes-hall] 25 Apr 20162 a) -1.0-0.5 0.0 0.5 1.0-5000500 -1.0-0.5 0.0 0.5 1.0-5000500V2(nV) Gleichung y=a+b*x Gewichtung KeineGewichtung FehlerderSummeder Quadrate6.69324E-18 1.07849E-17 PearsonR -0.99999 -0.99999 Kor.R-Quadrat 0.99998 0.99997 Wert Standardfehler V\-(out)SchnittpunktmitderY-Ac hs e-4.0586E-10 2.16347E-10 Steigung -3.76726E7 45806.36661 SchnittpunktmitderY-Ac hs e-2.8806E-10 2.74625E-10 Steigung -3.71509E7 58145.36296V2(nV) -1.0 -0.5 0.0 1.0 0.5500 0 -500V2 (nV) b) -1.0-0.5 0.0 0.5 1.0-5000500V2(nV)500 0 -500c)V2 (nV) -1.0-0.5 0.0 0.5 1.0-5000500V2(nV)500 0 -500V2 (nV)d)+I1A1= +I3A3 +I1A1= -I3A3 IiAi (10-14 A·m2)V2(0, I3A3)V2(I1A1, 0) YIGPty xz control C injector 1 injector 3detector 2 I3IcV2 I1 e) 10µmwd + -+ -+ - + - FIG. 1. (a) Schematic of the YIG/Pt nanostructures con- sisting of four Pt strips with width wand center-to-center separation d. We label the strips as C (control), 1 (injector), 2 (detector) and 3 (injector) from left to right. (b) Optical micrograph of the YIG/Pt device: the bright strips are the Pt strips, the dark parts the YIG. Current sources are attached to the injector strips 1 and 3 as well as to the control strip. The ensuing non-local voltage drop V2is recorded at strip 2. (c) The non-local voltage V2(I1A1;0) while current-biasing only along strip 1, and V2(0; I3A3) while driving strip 3 are measured as a function of the applied current and are repre- sented by blue open squares and orange open circles, respec- tively. The blue and orange lines represent linear ts to data. Note that we quote I
Aas the relevant bias, since the spin Hall current across the entire interface area Acontributes to the MMR. (d) V2(I1A1; I3A3) measured while current-biasing strips 1 and 3 with the same current magnitude and polarity is represented by the purple open squares. The purple line is a linear t. (e) The green open squares represent V2measured when biasing strips 1 and 3 with currents of opposite polarity. In this conguration, the non-local voltage goes to 0 within experimental error. performed at T= 275 K with an external magnetic eld 0H= 1 T applied in the thin lm plane perpendicular to the strips. In the rst part of the experiments we demonstrate the incoherent superposition of magnons in the simple three- strip device with strips 1, 2, and 3 (black wiring in Fig. 1 (b)): for the generation of the magnon spin accumula- tion beneath strip 1 and 3 we drive the charge currents I1andI3through the respective strips. As the number of magnons generated beneath one Pt strip is proportional to the electron spin accumulation at the interface, which in turn scales with the charge current I owing in the strip times the area Aof the YIG/Pt interface18, the dif- ferent lengths of injector strips 1 and 3 need to be takeninto account for quantitative analysis. In order to de- tect the spin accumulation beneath strip 2, we therefore measure the open-circuit voltage V2(I1A1;I3A3) using a nanovoltmeter. To increase the measurement precision, we use the current switching method described in Ref. 16. Firstly, we focus on measuring the voltage response V2(I1A1;I3A3) while applying a current bias only to strip 1 (I16= 0;I3= 0) or to strip 3 ( I1= 0;I36= 0), in order to compare the injection eciency of both strips. The results are shown in Fig. 1 (c). As expected, V2scales linearly with I1andI3, respectively, since the non-local voltage is proportional to the number of non-equilibrium magnons accumulating beneath strip 218,19. Since at zero bias current no magnons are injected, V2(0;0) = 0. For positive driving currents along the + y-direction in Fig.1 (a), the spin accumulation in the Pt is polarized along the +x-direction and thus we assume the magnons generated beneath the injector to be polarized along the same di- rection. These magnons diuse to the detector (strip 2) and via the inverse spin Hall eect induce a negative volt- ageV2, which is consistent with previous measurements using the same conguration16. By inverting the driving current, the spin viz. magnon polarizations are inverted and consequently a positive non-local voltage is recorded. Taking these properties together, our detection method is therefore sensitive to the number of non-equilibrium magnons reaching the detector as well as their magnetic polarization. Note also that the magnons diuse isotrop- ically, i.e. to the left as well as to the right in Fig. 1 (a). Therefore, it does not matter for the sign of the detected voltage whether the injector strip is on the right or left side of the detector strip. Within the experimental uncertainty of 5 nV of our setup (limited by the nanovoltmeter noise and/or ther- mal stability of the setup), V2(I1A1;0) =V2(0;I3A3) for I1A1=I3A3, showing that the magnon generation e- ciency of both strips is identical. To quantify this in more detail, we t V2(I1A1;0) =1
I1A1andV2(0;I3A3) = 3
I3A3to the data. We nd 1=3:77107V=Am2 and3=3:72107V=Am2, showing that 1;3are iden- tical within less that 2%. Next, we investigate the non-local voltage V2(I1A1;I3A3) while simultaneously biasing strips 1 and 3. For identical current polarity ( I1A1=I3A3) equal numbers of magnons with identical polarization are generated beneath both strips. The results are shown in Fig. 1 (d) as purple open squares. The purple line represents a linear t to the data with a slope of 1+3=7:53107V=Am2. Assuming an incoherent su- perposition of the magnons created beneath strip 1 and strip 3 we expect 1+3=1+3, which is in good agree- ment with the experimental data. This result is further corroborated by the V2values obtained for opposite cur- rent directions in strip 1 and strip 3, i.e. I1A1=I3A3, shown in Fig. 1 (e). A linear t to the data reveals a slope of 0:05107V=Am2and therefore 31=31. Additional measurements were conducted using dierent driving current amplitudes and polarities for strips 1 and3 3 (not shown here). From these data, we consistently ndV2(I1A1;I3A3) =1
I1A1+3
I3A3. Taken together, we observe incoherent superposition of non-equilibrium magnons, injected independently by the dierent injector Pt strips. Control C Injector 1 Injector 3 Detector 2 1 1 1 1 1 0 1 1 1 1 0 1 1 0 0 0 0 1 1 1 0 0 1 0 0 1 0 0 0 0 0 0 Ic I1 I3 V2 + + + + + + + + + + + + + + + + + TABLE I. Truth table of a majority gate14. The bottom half of the table shows the sign of the bias currents for the input channels as well as the sign of the resulting non-local voltage V2in a MMR based majority gate. Positive and negative bias currents in the injectors are dened as logical "1" and "0" respectively. Since a positive bias current in the injector yields a negative non-local voltage V216, we dene V2<0 as "1" and V2>0 as "0". We now show that a magnon-based majority gate can be implemented in the four-strip nanostructure, with three input and one output channel as shown in Fig. 1 (a). A majority gate returns true if more than half the inputs are true, otherwise it returns false, as shown in Tab. I. The third input can be used as a so-called con- trol channel (wire "C" in Fig. 1 (a)), which allows for switching between "AND" and "OR" operations. We dene positive and negative currents in the injectors as the logical "1" and "0" respectively. Since for positive bias currents a negative non-local voltage is detected, for the output signal V2<0 is dened as "1" and V2>0 as "0". The bias currents for input 1 and 3 are cho- sen such thatjI1A1j=jI3A3j= 0:811014Am2and V2(I1A1;I3A3) is subsequently measured for xed val- ues ofIc=150A. The resulting voltage is shown as black open squares in Fig. 2 for dierent input combina- tions (IcI1I3). For convenience, we do not quote the real charge current values, but rather the corresponding bit values (0 or 1), with the control bit marked in red. For example, if all three injectors are biased with a posi- tive current, corresponding to the input (111), a negative voltage is detected at the output, corresponding to a log-ical "1". The experimental data successfully reproduces the truth table (Tab. I): by changing the control bit from "1" to "0", one can switch from an "OR" to an "AND" operation (left and right side of the graph respectively). The majority function is furthermore visible in the sign of the output signal (green and orange area in Fig. 2), as the output mirrors the majority of the input signals. -0.50.5V2 (µV)AND OR “0” “1” (000) (010) (011)(110)(001)(100) (111)(101) 0.0 Ic= 100µA Ic= 150µA0 2 4 6 8-0.50.00.5V_2(nV)150µA FIG. 2. Test measurement of the four-strip majority gate. The detected non-local voltage V2is depicted for dierent input signals ( IcI1I3), where the bit of the control channel is marked in red. The magnitudes of the injector currents I1andI3are kept constant, and V2is measured for control current magnitudes Ic= 150 and 100 A (black and blue symbols). The experimental data faithfully reproduce all the properties of a majority gate. Since the control channel is further away from the de- tector, due to the exponential decay of the magnon ac- cumulation with distance16,17, the number of magnons from the control channel reaching the detector is smaller by about a factor 5. However, the actual amplitude of the control signal Icis not crucial for the function of the ma- jority gate and can be further reduced as shown in Fig. 2 for 100 A (blue triangles): the majority gate function is not perturbed by changing the amplitude of the con- trol bias current. We can therefore choose any amplitude forIcas long as the induced ISHE voltage is detectable. This allows for a high exibility in the device operation. In this context, another advantage of this magnon based logic gate is the possibility to reprogram the device by simply interchanging the injector, detector and control channels. To guarantee the functionality of the majority gate, the input bias currents only need to be adapted to the geometry of the device, in particular to the distance between the injector and detector strips. Note that logic operations are already feasible in a three strip device with two injectors and one detector. In this case the threshold chosen for the detector deter- mines whether the gate performs "AND" or "OR" opera- tions. However, the four strip device allows for the same4 functions, without requiring a redenition of the output threshold. Apart from exibility, an important aspect for the ap- plication of spintronic logic is the clocking frequency. The relevant time scale for a MMR based majority gate is dominated by two processes: the generation of a spin/magnon accumulation beneath the injector and the diusion of the incoherent magnons. It has been shown experimentally that the spin Hall eect induced spin accumulation persists up to frequencies of at least a few GHz20, corresponding to spin accumulation build up times well below a nanosecond. Concerning the prop- agation of the spin waves, the average group velocity of magnons in YIG is of the order of 1 m/ns12. In the device shown in Fig. 1, the magnons therefore have a lower limit of about 2 ns for their travel time from the control to the detector strip, resulting in a maximum switching frequency f= 1=Tof about 500 MHz. This can be further improved by changing the design of the four-strip device. For example reducing the width of the Pt strips to 50nm and their center-to-center separation to 100nm already leads to a factor 10 increase of the switching frequency, to several GHz. This value is com- parable to clocking frequencies in current devices and to those expected in other spintronic based logic gates21. Down-scaling not only leads to faster switching, but also decreases the device footprint, and increases the energy eciency of the logic gate: for smaller distances between strips, the output signal increases exponentially17, suchthat the required bias currents are lower. Since the log- ical bit is encoded in the polarization of the generated magnons and not in the frequency, phase or amplitude of the spin waves, down-scaling does not perturb the func- tionality of the logic gate. Note that in the short distance limit, assuming ballistic magnon transport and neglect- ing interface losses, the non-local voltage is limited by the bias voltage times 2 H, whereHis the spin Hall angle in the normal metal. This corresponds to the conversion ef- ciency from charge to spin and back to a charge current in the Pt layer. In summary, we have measured the non-local voltage (the magnon mediated magnetoresistance) in a YIG/Pt device with multiple magnon injectors and observe in- coherent superposition of the non-equilibrium magnon populations. The measurements show that the detected non-local voltage is sensitive to the number of magnons reaching the detector and their polarization. Based on the incoherent superposition of spin waves, we imple- mented a fully functional four-strip majority gate. The logical bit in this device is encoded in the polarization of the magnons, which are injected using a dc charge cur- rent. The output can be read out as a dc non-local volt- age, enabling a simple integration into an electronic cir- cuit. Clocking frequencies of the order of several GHz and straightforward down-scaling make the device promising for applications. This work is nancially supported by the Deutsche Forschungsgemeinschaft through the Priority Programm Spin Caloric Transport (GO 944/4). kathrin.ganzhorn@wmi.badw.de yhuebl@wmi.badw.de zgoennenwein@wmi.badw.de 1E. Saitoh, M. Ueda, H. Miyajima, and G. Tatara, Appl. Phys. Lett. 88, 182509 (2006), http://dx.doi.org/10.1063/1.2199473. 2Y. Tserkovnyak, A. Brataas, and G. E. W. Bauer, Phys. Rev. Lett. 88, 117601 (2002). 3O. Mosendz, J. E. Pearson, F. Y. Fradin, G. E. W. Bauer, S. D. Bader, and A. Homann, Phys. Rev. Lett. 104, 046601 (2010). 4F. D. Czeschka, L. Dreher, M. S. Brandt, M. Weiler, M. Al- thammer, I.-M. Imort, G. Reiss, A. Thomas, W. Schoch, W. Limmer, H. Huebl, R. Gross, and S. T. B. Goennen- wein, Phys. Rev. Lett. 107, 046601 (2011). 5K. Uchida, H. Adachi, T. Ota, H. Nakayama, S. Maekawa, and E. Saitoh, Appl. Phys. Lett. 97, 172505 (2010). 6M. Weiler, M. Althammer, F. D. Czeschka, H. Huebl, M. S. Wagner, M. Opel, I.-M. Imort, G. Reiss, A. Thomas, R. Gross, and S. T. B. Goennenwein, Phys. Rev. Lett. 108, 106602 (2012). 7M. Althammer, S. Meyer, H. Nakayama, M. Schreier, S. Altmannshofer, M. Weiler, H. Huebl, S. Gepr ags, M. Opel, R. Gross, D. Meier, C. Klewe, T. Kuschel, J.- M. Schmalhorst, G. Reiss, L. Shen, A. Gupta, Y.-T. Chen, G. E. W. Bauer, E. Saitoh, and S. T. B. Goennenwein, Phys. Rev. B 87, 224401 (2013).8H. Nakayama, M. Althammer, Y.-T. Chen, K. Uchida, Y. Kajiwara, D. Kikuchi, T. Ohtani, S. Gepr ags, M. Opel, S. Takahashi, R. Gross, G. E. W. Bauer, S. T. B. Goen- nenwein, and E. Saitoh, Phys. Rev. Lett. 110, 206601 (2013). 9M. P. Kostylev, A. A. Serga, T. Schneider, B. Leven, and B. Hillebrands, Appl. Phys. Lett. 87, 153501 (2005). 10T. Schneider, A. A. Serga, B. Leven, B. Hillebrands, R. L. Stamps, and M. P. Kostylev, Appl. Phys. Lett. 92, 22505 (2008). 11A. Khitun, M. Bao, and K. L. Wang, IEEE Trans. Magn. 44, 2141 (2008). 12A. V. Chumak, A. A. Serga, and B. Hillebrands, Nature Communications 5, 4700 (2014). 13K. Wagner, K. K akay, K. Schultheiss, A. Henschke, T. Se- bastian, and H. Schultheiss, Nature Nanotechnology ad- vance online publication , (2016). 14S. Klingler, P. Pirro, T. Br acher, B. Leven, B. Hillebrands, and A. V. Chumak, Appl. Phys. Lett. 105, 152410 (2014). 15S. Klingler, P. Pirro, T. Br acher, B. Leven, B. Hillebrands, and A. V. Chumak, Appl. Phys. Lett. 106, 212406 (2015). 16S. T. B. Goennenwein, R. Schlitz, M. Pernpeintner, K. Ganzhorn, M. Althammer, R. Gross, and H. Huebl, Appl. Phys. Lett. 107, 172405 (2015). 17L. J. Cornelissen, J. Liu, R. A. Duine, J. B. Youssef, and B. J. van Wees, Nature Physics 11, 1022 (2015). 18S. S.-L. Zhang and S. Zhang, Phys. Rev. B 86, 2144245 (2012). 19S. S.-L. Zhang and S. Zhang, Phys. Rev. Lett. 109, 096603 (2012). 20J. Lotze, H. Huebl, R. Gross, and S. T. B. Goennenwein, Phys. Rev. B 90, 174419 (2014). 21R. L. Stamps, S. Breitkreutz, J. Akerman, A. V. Chu-mak, Y. Otani, G. E. W. Bauer, J.-U. Thiele, M. Bowen, S. A. Majetich, M. Kl aui, I. L. Prejbeanu, B. Dieny, N. M. Dempsey, and B. Hillebrands, J. Phys. D: Appl. Phys. 47, 333001 (2014).

2016-04-25

Boolean logic is the foundation of modern digital information processing. Recently, there has been a growing interest in phenomena based on pure spin currents, which allow to move from charge to spin based logic gates. We study a proof-of-principle logic device based on the ferrimagnetic insulator Yttrium Iron Garnet (YIG), with Pt strips acting as injectors and detectors for nonequilibrium magnons. We experimentally observe incoherent superposition of magnons generated by different injectors. This allows to implement a fully functional majority gate, enabling multiple logic operations (AND and OR) in one and the same device. Clocking frequencies of the order of several GHz and straightforward down-scaling make our device promising for applications.

Magnon based logic in a multi-terminal YIG/Pt nanostructure

1604.07262v1

Magnon-phonon relaxation in yttrium iron garnet from rst principles Yi Liu,1Li-Shan Xie,1Zhe Yuan,1and Ke Xia1, 2 1The Center for Advanced Quantum Studies and Department of Physics, Beijing Normal University, 100875 Beijing, China 2Synergetic Innovation Center for Quantum Eects and Applications (SICQEA), Hunan Normal University, Changsha 410081, China (Dated: November 5, 2018) We combine the theoretical method of calculating spin wave excitation with the nite-temperature modeling and calculate the magnon-phonon relaxation time in the technologically important ma- terial Yttrium iron garnet (YIG) from rst principles. The nite lifetime of magnon excitation is found to arise from the uctuation of the exchange interaction of magnetic atoms in YIG. At room temperature, the magnon spectra have signicant broadening that is used to extract the magnon- phonon relaxation time quantitatively. The latter is a phenomenological parameter of great im- portance in YIG-based spintronics research. We nd that the magnon-phonon relaxation time for the optical magnon is a constant while that for the acoustic magnon is proportional to 1 =k2in the long-wavelength regime. Yttrium iron garnet (Y 3Fe5O12, YIG), a ferrimagnetic insulator, has been extensively applied in the spintronics experiments, such as the investigations of the spin Seebeck eect,1spin Hall magnetoresistance2and cavity magnon polariton.3In particular, YIG is an ideal magnetic material for transport study of a pure spin current because electronic transport can be completely eliminated. For example, a spin wave is allowed to propagate in YIG over a long distance4,5due to its ultralow Gilbert damping6of YIG. Unlike the ferromagnetic metals, where the Gilbert damping is dominated by the conduction electrons near the Fermi level,7,8spin-lattice interaction is the main mechanism of dissipating angular momentum and magnetic energy during the magnetization dynamics of YIG. Fundamental research on the spin-lattice interaction can be traced back to the early works in 1950s, when Abrahams and Kittel developed a phenomenological theory of the magnetoelastic eects in magnetic metals.9{11They introduced a phenomenological relaxation time to characterized the spin-lattice interaction. Later Sanders and Walton performed an experimental measurement investigating the magnon-phonon relaxation time mpin magnetic insulators.12Recently,mpis found to be crucially important to describe many YIG-based spin transport phenomena.13{15Nevertheless, a quantitative estimation of the phenomenological relaxation time in a complex magnetic material like YIG is essentially nontrivial; mpcan be expressed in terms of the magnetoelastic coupling constants, which are, however, still short of reliable evaluation experimentally or theoretically. The present work aims at a quantitative evaluation of the magnon-phonon relaxation time in YIG using a physically transparent and numerically reliable method. More specically, we combine the rst-principles method of calculating magnon spectrum16and a nite-temperature modeling by displacing atoms from their equilibrium positions,17,18so as to obtain the uctuation of the exchange interaction induced by lattice vibration. Such a computational scheme allows us to determine the magnon excitation mediated by phonons without imposing any phenomenological parameters. As a consequence, the uctuation of the exchange interaction results in a broadening of the magnon spectrum. Then we are able to extract the magnon-phonon relaxation time based on this broadening. The electronic structure of the bulk YIG is calculated self-consistently based upon the generalized gradient ap- proximation of the density functional theory, which has been implemented in the Vienna ab initio simulation package (VASP).21,22We chooseUJ= 2:7 eV in all the calculations to account for the strongly correlated electronic states. The experimental lattice constant 12.375 A is used for the body-centered cubic (bcc) unit cell, which contains 80 atoms in total. At the ground state, a Fe atom on one of the twelve dsites in the unit cell has a magnetic moment aligned oppositely with the moment of a Fe atom on one of the eight asites, whose magnitudes are both approximately 5 B. The other atoms are not magnetized. By ipping one magnetic moment of the Fe atoms, we are able to obtain the total energy of the corresponding metastable state. Comparing the energy dierence between the metastable states and the ground state, we can calculate the exchange constants between a pair of Fe atoms both on the dorasites (JddandJaa), or on two dierent sites ( Jad). The computational details to determine the exchange constants can be found in Ref.16. Having obtained the exchange constants, we determine the magnon spectrum using the \frozen magnon method".23,24As an example, the magnon spectra of the perfectly crystalline YIG are shown by the black and grey curves in Fig. 1(a) that are in good agreement with the experimental measurement (blue dots).19,20 Note that these (black and grey) magnon spectra in Fig. 1(a) are obtained without any phonons corresponding to the case at zero temperature. To examine the in uence of phonons, we apply the recently developed computational scheme to account for the temperature-induced atomic vibration,17,18i.e., at nite temperature, atoms in the bulk YIG displaced away from their equilibrium positions in a crystalline lattice. The magnitude of the displacements increases with increasing temperature. In the practical calculation, the displacement uiof thei-th atom with its massarXiv:1710.02647v1 [physics.comp-ph] 7 Oct 20172 Mifollows a random Gaussian distribution and we have the statistical mean square of the displacements determined by the Debye model, hjuij2i=9~2 MikBD T2 2 DZD T 0x ex1dx+1 4! : (1) In this paper, we focus on the properties at room temperature, T= 300Kand the Debye temperature  = 403 K is chosen from experiment.25It is worth noting that we do not take the temperature-induced magnon-magnon interaction into account in our calculation, which has been investigated with atomistic spin dynamics simulation.26 Strictly speaking, the exchange interaction of each pair of magnetic Fe atoms in the disordered YIG depends on the specic distance between them. So there may be dierent numerical values of Jad,Jdd, orJaain a unit cell N110Γ001H05101520ω (THz)Γ111P05101520ω (THz)(a) (b) FIG. 1: (a) Calculated magnon spectra of YIG. At zero temperature, the dispersion of the acoustic magnon mode and the lowest-energy optical mode are shown as the black curves, while the other modes are plotted as grey curves. The experimental data (blue dots) are shown for comparison.19,20The brown curves correspond to the calculated acoustic and optical magnon modes using 40 random phonon congurations at room temperature. (b) The same as (a) but the calculated acoustic and optical magnon spectra are replotted a spectrum with errorbars (shadows) based on the 40 brown curves in (a).3 of YIG with lattice disorder. When applying the method in Ref.16 to determine the J's, we do not distinguish the distance-dependent exchange interaction, but instead average over pairs of Fe atoms in one disordered conguration to obtain an eective value of Jad,Jdd, and Jaa. Using the eective exchange constants, we calculate the corresponding magnon spectrum for this disordered conguration. In practice, we consider 40 dierent disordered congurations and the resulting acoustic and the lowest-energy optical magnon dispersions are plotted by the brown curves in Fig. 1(a). The other optical branches of magnons obtained at room temperature are now not shown for simplicity. The brown curves basically superimpose on the zero-temperature spectra curves (the black ones) and show signicant spread in energy. The spread of the magnon dispersion can be interpreted in terms of a spectral function A(k;!), which denes the probability density of the magnon mode having the wavevector kand the frequency !. Following a standard expression using the retarded Green's function,27the spectral function has the form of a Lorentzian function: A(k;!)/X i i;k (!~!i;k)2+ 2 i;k; (2) where ~!i;kis the eective (median) frequency of the i-th branch magnon and i;kis the width of this mode. In the case of our 40 dierent congurations, !n i;kwithn= 1;:::;40, we determine ~ !i;kat each kpoint and the width, respectively, by ~!i;k= median(!n i;k); (3) and i;k= median(j!n i;k~!i;kj): (4) A careful numerical test shows that ~ !i;kand i;kfor the acoustic and the lowest-energy optical magnons are both well converged with 40 congurations. We then plot ~ !i;kand i;kof the calculated room-temperature spectra in Fig. 1(b) as the brown curves with error bars. The eective magnon frequency ~ !i;kagrees globally with the zero temperature spectra despite of a slight blue shift, which is discussed later. Such global agreement indicates that the exchange interaction between the Fe atoms in YIG does not change much due to the temperature-induced ionic vibration, and justies our computational framework of applying the frozen lattice disorder. The magnon frequency at room temperature in Fig. 1 is slightly higher than that at zero temperature. The blue shift is larger for the optical mode and at the edge of the Brillouin zone. To understand the phonon-induced blue shift of the magnon frequency, we systematically calculate the exchange interaction as a function of the lattice constant of YIG, as shown in Fig. 2(a). All these J's decrease as the lattice constant increases and Jadis much larger in magnitude than the others. Using the calculated dominant Jad, we can determine the Gr uneisen constant at the equilibrium lattice constant ( a0), m=@lnJad=@lnVjV=a3 0= 3:07. It agrees well with experimental values 3.13 and 3.26 (Ref.28,29). The calculated magnon spectra, as plotted in Fig. 2(b), show a monotonic decrease in frequency as the lattice constant increases (along the direction of the arrow). Note that the dependence of J's on the lattice constant is not perfectly linear. A detailed inspection shows that the derivative @J=@a decreases as aincreases. Therefore, compressing the lattice leads to a larger rise in the exchange energy than the energy reduction by expanding the lattice by the same amount. This is also re ected by the magnon spectra in Fig. 2(b), where the magnon frequency dierences are not equal but becomes larger at small lattice constants. On the other hand, the lattice vibration can be described by a harmonic potential in the lowest order approximation, where the atom displacement subject to a Gaussian distribution is symmetric with respect to its equilibrium position. As a consequence, the displacements that decrease interatomic distance lead to the rise in the magnon frequency, which is larger than the energy reduction caused by the increase of interatomic distance. This explains the slight blue shift of the magnon spectra at room temperature. Though phonons do not dramatically in uence the magnon dispersion, they give rise to a signicant broadening of the spectra indicating a nite lifetime of the magnon due to the magnon-phonon interaction. We extract the broadening of the spectra
!for the acoustic and the lowest-energy optical magnon modes out of the 40 room- temperature congurations. Figure 3(a) shows the calculated
!for both acoustic and optical magnons, both of which increase monotonically with jkjand exhibit very little anisotropy. We further replot
!as a function of !in the inset of Fig. 3(a), where all the data points fall into a single continuous curve indicating that the magnon-phonon relaxation time only depends on the magnon energy. This may not be surprising since the broadening of magnon excitation results from the phonon-induced uctuation of the exchange interaction, which in turn only depends on the energy.30In addition, as the frequency increases, the density of states of phonons increases monotonically in this frequency range resulting in an increasing magnon-phonon scattering rate. This is the reason why
!increases monotonically with the frequency. In Fig. 3(b),
!is replotted in the logarithmic scale to see the asymptotic behavior4 N110Γ001H05101520ω (THz)Γ111Pa340.60.8J (meV)-0.3 -0.2 -0.1 0 0.1 0.2 0.3a-a0 (Å)0.050.06JadJddJaa(a) (b) FIG. 2: (a) Calculated exchange interaction in YIG as a function of the lattice constant. The dashed lines illustrate the linear dependence. (b) Calculated spectra of the acoustic and the lowest-energy optical magnons of YIG as a function of the lattice constant. The black grey lines are the same as in Fig. 1. The spectra of dierent colors corresponds to articially changed lattice constants in (a). Thermal lattice disorder is not included in these calculations.5 0.01 0.1 1k (π/a)10-410-310-210-1100∆ω (THz)001110111110k (106cm-1) 10-410-310-210-1100101OpticalAcoustic∆ω (meV)k2k001 2k (π/a)0123∆ω (THz)001110111 04812160 5 10 15 20ω (THz)012∆ω (THz)OpticalAcoustic∆ω (meV)OpticalAcoustic(a) (b) FIG. 3: (a) Calculated broadening of the magnon spectrum of YIG,
!, at room temperature as a function of k. Inset:
! replotted as a function of magnon frequency !. (b)
!(k) replotted in a log-log scale. The black lines indicate a constant
! and a quadratic dependence on kfor the optical branch and the acoustic branch, respectively.6 in the long wavelength limit, where we can nd a quadratic dependence on kof
!for the acoustic branch and a constant
!= 1:48 meV for the optical branch, as illustrated by the black solid lines. The magnon-phonon relaxation time mpcan be estimated from the broadening
!of the magnon spectrum using the uncertainty principle, i.e.
!mp~. For the acoustic magnon at small k, we havemp/k2up to about k==2a. Note that this relation qualitatively agrees with the phonon-induced absorption rate of sound waves in solids,311/!2= (ck)2, wherecis the velocity of the sound wave. The latter was used to estimate the magnon relaxation rate.13If we choose a specic wave vector, for instance, from Ref.32, k= 5:67105cm1, the magnon- phonon relaxation time can be estimated as mp= 0:2 ns. It is worth mentioning that the quadratic dependence of the relaxation time mpon the wavevector kmay not be extrapolated to a much smaller kin the dipolar magnon regime, where the magnon frequency is dominated by magnetic dipole-dipole interaction that is not included in our calculation. For the lowest-energy optical magnon, the minimum broadening at small kis a constant,
!= 1:48 meV, corresponding to the magnon-phonon relaxation time mp= 4:41013s. This is rather small because the long- wavelength optical magnons have the relatively high frequency corresponding to a large density of state of phonons in the same frequency range. In this case, the magnon-phonon scattering rate becomes quite large. It is interesting to note that a higher order dependence, k4, of the magnon-phonon scattering rate, would be obtained if one employs the oversimplied Heisenberg model of a simple cubic ferromagnet to describe phonon- mediated exchange interaction in YIG, i.e. Hmp=1 2X i;jJ(jrirjj)Si
Sj: (5) Here phonons contribute to the change of exchange coupling through atomic displacements ui=riRifrom the equilibrium positions Ri, J(jrirjj) =J(jRiRjj) +rJ
(juiujj) +:::: (6) By expressing the displacements in terms of phonon eigen modes18and the spins in terms of magnons, one can determine the rate for a magnon of kscattered to k0by a phonon q=k0k, which has a k4dependence.33 However, the oversimplied model does not include the optical phonons that are populated at room temperature, or the multiple magnetic Fe atoms in a unit cell that may introduce the internal degrees of freedom for the relaxation. Instead, our calculation take the material-specic electronic and magnetic structures of YIG into account as well as the temperature-induced ionic vibrations and is therefore more realistic. To formulate the lower order k2dependence of the magnon-phonon scattering rate that we obtained in our calculations calls for further work. The mechanism of magnon relaxation at nite temperature under the current study is the phonon-induced uctu- ation of the exchange interaction, which is much larger in energy than the spin-orbit interaction. The latter is not included in our calculation due to the fact that both the calculated magnetic moments and the exchange interaction are hardly in uenced by the spin-orbit interaction. In contrary to magnetostriction resulting from the magnetoelastic interaction, whose origin is magnetic anisotropy, as reviewed in Ref.10, our results suggest that the dominant eect of phonon upon magnon relaxation arises from the modied exchange interaction by lattice vibration. We would also like to emphasize that the Gilbert damping of YIG arises partly from the magnon-phonon relaxation mechanism and partly from the magnon-magnon relaxation.26The latter is beyond the scope of the present work. In conclusion, we have investigated the magnon-phonon relaxation in YIG at nite temperature by calculating its magnon spectrum with frozen thermal lattice disorder from rst principles. The ionic vibrations in the Debye model is employed to model the phonons in YIG at room temperature. The uctuation of the exchange interaction between magnetic Fe atoms in YIG is found to be the main mechanism of magnon-phonon interaction. The magnon frequencies are slightly blue shifted by the phonons associated with a signicant broadening in the magnon spectra. The latter is used to extract the magnon-phonon relaxation time mp. At smallk,mpfor the acoustic magnon is proportional to k2while that for the lowest-energy optical magnon is nearly a constant. The authors would like to thank Simon Streib for sharing their model results and technical suggestions, and thank Ka Shen for helpful discussions. This work was partly supported by the National Natural Science Foundation of China (Grants No. 61604013) and the Fundamental Research Funds for the Central Universities (Grants No. 2016NT10). 1K Uchida, J Xiao, H Adachi, J Ohe, S Takahashi, J Ieda, T Ota, Y Kajiwara, H Umezawa, H Kawai, G E W Bauer, S Maekawa, and E Saitoh, \Spin seebeck insulator," Nature Materials 9, 1{4 (2010). 2H. Nakayama, M. Althammer, Y.-T. Chen, K. Uchida, Y. Kajiwara, D. Kikuchi, T. Ohtani, S. Gepr ags, M. Opel, S. Taka- hashi, R. Gross, G. E. W. Bauer, S. T. B. Goennenwein, and E. Saitoh, \Spin hall magnetoresistance induced by a nonequilibrium proximity eect," Phys. Rev. Lett. 110, 206601 (2013).7 3Lihui Bai, M. Harder, Y. P. Chen, X. Fan, J. Q. Xiao, and C.-M. Hu, \Spin pumping in electrodynamically coupled magnon-photon systems," Phys. Rev. Lett. 114, 227201 (2015). 4L. J. Cornelissen, J. Liu, R. A. Duine, J. Ben Youssef, and B. J. van Wees, \Long-distance transport of magnon spin information in a magnetic insulator at room temperature," Nature Physics 11, 1022{1026 (2015). 5Brandon L. Giles, Zihao Yang, John S. Jamison, and Roberto C. Myers, \Long-range pure magnon spin diusion observed in a nonlocal spin-seebeck geometry," Phys. Rev. B 92, 224415 (2015). 6Yiyan Sun, Young-Yeal Song, Houchen Chang, Michael Kabatek, Michael Jantz, William Schneider, Mingzhong Wu, Helmut Schultheiss, and Axel Homann, \Growth and ferromagnetic resonance properties of nanometer-thick yttrium iron garnet lms," Appl. Phys. Lett. 101, 152405 (2012). 7V Kambersk y, \On ferromagnetic resonance damping in metals," Czech. J. Phys. 26, 1366{1383 (1976). 8K. Gilmore, Y. U. Idzerda, and M. D. Stiles, \Identication of the dominant precession-damping mechanism in fe, co, and ni by rst-principles calculations," Phys. Rev. Lett. 99, 027204 (2007). 9Elihu Abrahams and C. Kittel, \Spin-lattice relaxation in ferromagnets," Phys. Rev. 88, 1200{1200 (1952). 10C. Kittel and Elihu Abrahams, \Relaxation process in ferromagnetism," Rev. Mod. Phys. 25, 233{238 (1953). 11C. Kittel, \Interaction of spin waves and ultrasonic waves in ferromagnetic crystals," Phys. Rev. 110, 836{841 (1958). 12D. J. Sanders and D. Walton, \Eect of magnon-phonon thermal relaxation on heat transport by magnons," Phys. Rev. B 15, 1489{1494 (1977). 13C. Vittoria, S. D. Yoon, and A. Widom, \Relaxation mechanism for ordered magnetic materials," Phys. Rev. B 81, 014412 (2010). 14Andreas R uckriegel, Peter Kopietz, Dmytro A. Bozhko, Alexander A. Serga, and Burkard Hillebrands, \Magnetoelastic modes and lifetime of magnons in thin yttrium iron garnet lms," Phys. Rev. B 89, 184413 (2014). 15L. J. Cornelissen, K. J. H. Peters, G. E. W. Bauer, R. A. Duine, and B. J. van Wees, \Magnon spin transport driven by the magnon chemical potential in a magnetic insulator," Phys. Rev. B 94, 014412 (2016). 16Li-Shan Xie, Guang-Xi Jin, Lixin He, Gerrit E. W. Bauer, Joseph Barker, and Ke Xia, \First-principles study of exchange interactions of yttrium iron garnet," Phys. Rev. B 95, 014423 (2017). 17Yi Liu, Anton A. Starikov, Zhe Yuan, and Paul J. Kelly, \First-principles calculations of magnetization relaxation in pure fe, co, and ni with frozen thermal lattice disorder," Phys. Rev. B 84, 014412 (2011). 18Yi Liu, Zhe Yuan, R. J. H. Wesselink, Anton A. Starikov, Mark van Schilfgaarde, and Paul J. Kelly, \Direct method for calculating temperature-dependent transport properties," Phys. Rev. B 91, 220405 (2015). 19J S Plant, \Spinwave dispersion curves for yttrium iron garnet," Journal of Physics C: Solid State Physics 10, 4805 (1977). 20J S Plant, \'pseudo-acoustic' magnon dispersion in yttrium iron garnet," Journal of Physics C: Solid State Physics 16, 7037 (1983). 21G. Kresse and J. Hafner, \Ab-initio molecular-dynamics for liquid-metals," Phys. Rev. B 47, 558{561 (1993). 22G. Kresse and J. Furthm uller, \Ecient iterative schemes for ab initio total-energy calculations using a plane-wave basis set," Phys. Rev. B 54, 11169{11186 (1996). 23S. V. Halilov, A. Y. Perlov, P. M. Oppeneer, and H. Eschrig, \Magnon spectrum and related nite-temperature magnetic properties: A rst-principle approach," Europhys. Lett. 39, 91 (1997). 24S. V. Halilov, H. Eschrig, A. Y. Perlov, and P. M. Oppeneer, \Adiabatic spin dynamics from spin-density-functional theory: Application to fe, co, and ni," Phys. Rev. B 58, 293 (1998). 25K. B. Modi, M. C. Chhantbar, P. U. Sharma, and H. H. Joshi, \Elastic constants determination for fe3+ substituted yig through infra-red spectroscopy and heterogeneous metal mixture rule," Journal of Materials Science 40, 1247{1249 (2005). 26Joseph Barker and Gerrit E. W. Bauer, \Thermal spin dynamics of yttrium iron garnet," Phys. Rev. Lett. 117, 217201 (2016). 27Sebastian Doniach and Ernst H. Sondheimer, Green's Functions For Solid State Physicists (Imperial College Press, London, 1998). 28D Bloch, \The 10/3 law for the volume dependence of superexchange," J. Phys. Chem. 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2017-10-07

We combine the theoretical method of calculating spin wave excitation with the finite-temperature modeling and calculate the magnon-phonon relaxation time in the technologically important material Yttrium iron garnet (YIG) from first principles. The finite lifetime of magnon excitation is found to arise from the fluctuation of the exchange interaction of magnetic atoms in YIG. At room temperature, the magnon spectra have significant broadening that is used to extract the magnon-phonon relaxation time quantitatively. The latter is a phenomenological parameter of great importance in YIG-based spintronics research. We find that the magnon-phonon relaxation time for the optical magnon is a constant while that for the acoustic magnon is proportional to $1/k^2$ in the long-wavelength regime.

Magnon-phonon relaxation in yttrium iron garnet from first principles

1710.02647v1

arXiv:1706.07559v1 [cond-mat.mtrl-sci] 23 Jun 2017Detection of induced paramagnetic moments in Pt on Y 3Fe5O12 via x-ray magnetic circular dichroism Takashi Kikkawa,1,2,∗Motohiro Suzuki,3Jun Okabayashi,4Ken-ichi Uchida,1,5,6,7Daisuke Kikuchi,1,2,†Zhiyong Qiu,2and Eiji Saitoh1,2,7,8 1Institute for Materials Research, Tohoku University, Send ai 980-8577, Japan 2WPI Advanced Institute for Materials Research, Tohoku Univ ersity, Sendai 980-8577, Japan 3Japan Synchrotron Radiation Research Institute (JASRI), S ayo, Hyogo 679-5198, Japan 4Research Center for Spectrochemistry, The University of To kyo, Bunkyo, Tokyo 113-0033, Japan 5National Institute for Materials Science, Tsukuba 305-004 7, Japan 6PRESTO, Japan Science and Technology Agency, Saitama 332-0 012, Japan 7Center for Spintronics Research Network, Tohoku Universit y, Sendai 980-8577, Japan 8Advanced Science Research Center, Japan Atomic Energy Agen cy, Tokai 319-1195, Japan (Dated: August 28, 2018) Magnetic moments in an ultra-thin Pt film on a ferrimagnetic i nsulator Y 3Fe5O12(YIG) have been investigated at high magnetic fields and low temperatur es by means of X-ray magnetic circular dichroism (XMCD). We observed an XMCD signal due to the magne tic moments in a Pt film at the PtL3- andL2-edges. By means of the element-specific magnetometry, we fo und that the XMCD signal at the Pt L3-edge gradually increases with increasing the magnetic fiel d even when the field is much greater than the saturation field of YIG. Importantly , the observed XMCD intensity was found to be much greater than the intensity expected from the Pauli paramagnetism of Pt when the Pt film is attached to YIG. These results imply the emergence o f induced paramagnetic moments in Pt on YIG and explain the characteristics of the unconvent ional Hall effect in Pt/YIG systems. PACS numbers: 75.70.-i, 75.47.-m, 85.75.-d I. INTRODUCTION Paramagnetic-metal/ferromagnetic-insulator het- erostructures provide a unique platform to explore spin-current phenomena such as spin pumping [1, 2], spin-transfer torque [1–8], and spin Seebeck effect [9–13]. In this structure, itinerant electrons in the paramagnet and magnons in the ferromagnet interact with each other via the interfacial spin-exchange interaction [14]. One of the most widely-used heterostructures for study- ing spin-current phenomena is a Pt/Y 3Fe5O12(YIG) junction system. Pt is a paramagnetic metal exhibiting high spin-charge conversion efficiency due to its strong spin-orbit interaction [15–19]. YIG is a ferrimagnetic insulator with a high Curie temperature ( ∼560 K) [20, 21] and extremely high resistivity [22]. Therefore, this structure enables pure detection of spin-current phenomena free from conduction-electrons’ contribution in YIG. In Pt/YIG junction systems, an unconventional mag- neticfield Hdependence ofHalleffectshasbeenobserved [23–28]. Although Pt films on non-magnetic substrates such as SiO 2and Y 3Al5O12(YAG) show a conventional H-linear response due to the normal Hall effect, the mea- sured Hall effect in Pt/YIG systems exhibits two anoma- lous behaviors: (i) in a low- Hrange (H/lessorsimilar2 kOe), the Hdependence of the Hall voltage looks reflecting the magnetization process of YIG and (ii) in a high- Hrange much greater than the saturation field of YIG ( ∼2 kOe), the magnitude of the Hall voltage nonlinearly increaseswith increasing H. The behavior (ii) becomes prominent at low temperatures especially below 100 K. Since both the behaviors (i) and (ii) become more outstanding with decreasing the Pt thickness, these unconventional Hall effects should be related to interface effects at Pt/YIG. To explain the unconventional Hall effects, several sce- narioshavebeen proposed. Asapossibleoriginofthe be- havior(i), Chen et al.[29] and Zhang et al.[30] indepen- dently suggested a role of nonequilibrium spin currents combined with interfacial spin-mixing effects. On the other hand, Huang et al.[23, 31, 32] and Guo et al.[33] pointed out a roleofstatic Pt ferromagnetisminduced by magnetic proximity effects at the Pt/YIG interface. The latter scenario is based on the fact that Pt is near the Stoner ferromagnetic instability [34–36], and thereby it might be magnetized due to static proximity effects near the interface, as reported in various Pt/ferromagnetic- metal junction systems [37–40]. A possible mechanism for the behavior (ii) was discussed by Shimizu et al.[24] and Linet al.[41] in terms of independent paramagnetic moments in Pt, which cause skew scatterings for itiner- ant electrons in Pt [41–47]. However, there is no direct evidence for the existence of such paramagneticmoments in Pt on YIG. To study magnetic properties of Pt films on YIG, X- ray magnetic circular dichroism (XMCD) [48–52] mea- surements have been conducted at Pt L3- andL2-edges. Gepr¨ ags et al.[53, 54] tried to detect magnetic prox- imity effects at Pt/YIG interfaces by means of XMCD measurements, but did not observe any XMCD signals in Pt(1.6−10 nm)/YIG-film samples at room tempera-2 ture within the margin of experimental error ( <0.003± 0.001µB). In contrast, Lu et al.[32] reported a finite XMCD signal in a Pt(1 .5 nm)/YIG-film sample at 300 K and 20 K, at which total magnetic moment values per Pt atom were estimated to be 0 .054µBand 0.076µB, respectively. These results indicate that magnetic prop- erties of Pt on YIG are sensitive to the qualities of the Pt film and Pt/YIG interface. However, since all the previ- ous studies focused only on the behavior (i), the XMCD measurements were performed only in the low field range (H <6 kOe) [32, 53, 54]. In the present study, we mea- sured XMCD in a Pt film on a YIG substrate at high magnetic fields and low temperatures. We observed a clearXMCD signal at the Pt L3- andL2-edgesand found that the signal suggests induced paramagnetic moments in Pt, providing an important clue to understand the un- conventional Hall effect in the Pt/YIG systems. II. SAMPLE PREPARATION, CHARACTERIZATION, AND EXPERIMENTAL SETUP ToperformXMCD measurements, wepreparedajunc- tion system comprising a Pt film and a YIG substrate. The nominal thickness of the Pt film is 0 .5 nm, thin- ner than that used for the previous XMCD studies [32, 53, 54], because the unconventional Hall effect is enhanced with decreasing the Pt thickness [32]. We used a single-crystalline YIG substrate grown by a flux method, commercially available from Ferrisphere Inc., USA, and cut it into a rectangular shape with the size of 8×8×1 mm3. As shown in Fig. 1(a), the magnitude of the magnetization Mfor the YIG at the temperature T= 5.5 K increases with increasing the magnetic field Hfrom zero and saturates at around H= 2 kOe, where the saturation Mvalue was observed to be ∼5µB, con- sistent with a literature value [13, 20]. Before putting a Pt film, the 8 ×8 mm2surface [(111) surface] of the YIG slabwasmechanicallypolished with sandpapersandalu- mina slurry; the resultant surface roughness of the YIG slab isRa∼0.2 nm, as shown in an atomic force mi- croscope (AFM) image in Fig. 1(b). The YIG slab was cleaned with acetone and methanol in an ultrasonic bath and, then, cleaned with so-called Piranha etch solution (a mixture of H 2SO4and H 2O2at a ratio of 1:1) [26, 55]. The Piranha etch solution may remove organic matter attached to the YIG surface [55] and hence does not af- fect the surface roughness and morphology of the YIG, which were confirmed by our AFM measurements. We deposited a Pt film on the YIG surface by rf magnetron sputtering with an rf power density of 0 .88 W/cm2in an Ar atmosphere of 4 .8 mTorr at room temperature, which results in a deposition rate of 0 .028 nm/sec. The Pt/YIG interface and the morphologyof the Pt film were evaluated by means of transmission electron microscopy-40 -20 0 20 40-505 -4 -2 0 2 4-505T = 5.5 K M (μ B/Y3Fe 5O12 ) H (kOe) Low H(a) M-H curve for YIG (b) AFM image of YIG surface (nm)0 0.2 0.4 0.6 0.8 1.00.20.40.60.82 0 (μm) (μm) HM YIGPt (c) Cross-sectional TEM image of Pt/YIG sample 50 nm 5 nmYIGPt FIG. 1: (a) M-Hcurve for the YIG slab when the magnetic fieldHwas applied along the [111] direction. The inset to (a) shows the magnified view of the M-Hcurve in the low- H range (|H|<5 kOe). The M-Hcurve was measured with a vibrating sample magnetometer. (b) An AFM image of the polished (111) surface of the YIG slab before the Piranha treatment, where the surface roughness is Ra∼0.2 nm. Sim- ilar AFM image and roughness value were confirmed also for the YIG surface with the Piranha treatment. (c) A cross- sectional TEM image of the Pt/YIG sample. The inset shows a blowup of the Pt/YIG interface. (TEM). As shown in Fig. 1(c), we confirmed a flat YIG surface and also a clear Pt/YIG interface. Due to its island-growth nature [39], the Pt film was found to con- sist of discontinuous clusters with a typical thickness of ∼2 nm, which may reduce an interface-to-volume ratio compared to ideal flat films and may make proximity- induced Pt magnetic moment densities be cluster-size dependent. Nevertheless, the observed clear contact be- tween the Pt and YIG allows us to evaluate the char- acteristic of the magnetic proximity effect in the present systemat highmagneticfields andlowtemperatures. We also prepared a control sample comprising a (nominally) 0.5-nm-thick Pt film on a non-magnetic YAG (111) sub- strate with the size of 8 ×8×0.5 mm3. The Pt films on the YIG and YAG substrates were deposited at the same time in our sputtering chamber. Figure 2 shows the experimental setup for the XMCD measurements. The XMCD experiments were performed at the beam line BL39XU of SPring-8 synchrotron radi- ation facility using the fluorescence detection mode [38]. X-ray absorption spectra (XAS) were recorded at the Pt L3-edge (2p3/2→5dvalence, 11570 eV) and Pt L2-edge (2p1/2→5dvalence, 13282 eV) with circularly-polarized X-rays while reversing their helicity at 1 Hz. Here, by measuring the XAS for the two circular polarizations al-3 Circularly-polarized X-raynSilicon drift detector H10 X-ray fluorescence Pt filmY3Fe 5O12 (YIG) (111) slab or Y3Al 5O12 (YAG) (111) slab FIG. 2: A schematic illustration of the experimental setup f or the XMCD measurements. A magnetic field, H, parallel or antiparallel to the incoming X-ray beam, was applied to the Pt/YIG or Pt/YAG sample at an angle of 10◦to the normal direction nof the sample surface. most at the same time, we can exclude time-dependent artifacts. The circularly-polarizedX-rays with a high de- gree of polarization ( ≧95 %) were generated by using a diamond X-ray phase retarder. The Pt Lα(for the Pt L3-edge) and Lβ(for the Pt L2-edge) fluorescences were measured with a silicon drift detector. In order to per- form XMCD measurements in the configuration similar to the set-up for the Hall measurements, an out-of-plane magnetic field, H, was applied to the samples using a split-type superconducting magnet, where the angle be- tweenHand the direction normal to the sample surface nwas set to be 10◦. The direction of the angular mo- mentum vector of the incident X-ray beam was parallel or antiparallel to the Hdirection. The XMCD signal was obtained by taking a difference of the XAS recorded with the opposite helicities at H= 50 kOe. The mea- surements of element-specific magnetization curves were performed at the constant energy of E= 11568 eV (the XMCD peak position for the Pt L3-edge) with sweeping theHvalue between ±50 kOe. The temperature was fixed atT= 5.5 K in all the measurements. III. RESULTS AND DISCUSSION Figure 3(a) shows the XAS for the Pt L3- andL2- edges of the Pt/YIG and Pt/YAG samples, where the XAS edge jump is normalized to 1 (2 .22−1) for the L3- edge (L2-edge) [35, 56]. The whiteline intensity, the ra- tio of the absorption maximum at the L3-edge to the edge jump, is estimated to be 1.35 for the Pt/YIG sam- ple. This value is consistent with that for the Pt(1 .6− 10 nm)/YIG systems reported in Refs. 53 and 54, indi- cating a mainly metallic state of our Pt films (note that the whiteline intensity was reported to be 1 .25−1.30, 1.50, and 2 .20 for metallic Pt foil, PtO 1.36, and PtO 1.6, respectively [54]). The quality of our Pt films was also confirmed by the clear oscillation in the extended X-ray absorption fine structure (EXAFS), marked with trian- gles in Fig. 3(a), which is identical with metallic Pt [54].11560 11600012 13260 13300 -0.0100.01 -0.0500.050.10 XAS (arb. units) Pt L3-edge Pt L2-edge 11560 11600-0.0100.01 Photon energy (eV)XMCD T = 5.5 K H = 50 kOeXMCD integral Pt/YIG Pt/YAG Pt/YIG Pt/YAGH > 0 H < 0㧖(a) (b) Pt/YIG FIG. 3: (a) The normalized XAS for the Pt L3- andL2- edges of the Pt/YIG and Pt/YAG samples at T= 5.5 K and H= 50 kOe. The XAS edge jump, definedas the difference in theXASintensitybetween11540 eV(13255 eV)and11610 eV (13325 eV), is normalized to 1 (2 .22−1) for the L3-edge (L2- edge) according to Refs. 35 and 56. The XAS offset value for theL3-edge (L2-edge) of the Pt/YIG sample is set to be 0 (1) [39, 52–54, 56]. Similarly, the XAS offset value for theL3-edge (L2-edge) of the Pt/YAG sample is set to be 0 (1) and then, for clarity, is shifted along the vertical axi s by a constant value of 0.75 (0.75). The triangles indicate the oscillation structures in the EXAFS region. The peak structure marked with an asterisk in the XAS at 11607 eV for thePt/YAGsampleappearsduetoelasticdiffractionfromthe YAGsubstrate[54]. (b)TheXMCDspectraandtheirintegral for the Pt L3- andL2-edges of the Pt/YIG sample. The blue (red) arrow indicates the negative (positive) XMCD signal a t the PtL3-edge (L2-edge). The inset to (b) shows the XMCD spectra for the Pt L3-edge of the Pt/YIG sample measured atH=±50 kOe, where the blue arrows indicate the XMCD signal. In (b), the XMCD spectra for the Pt L3-edge of the Pt/YAG sample are also plotted. Figure 3(b) shows the XMCD spectra for the Pt/YIG sample at T= 5.5 K and H= 50 kOe. We observed small but finite XMCD signals with a negative sign at theL3-edge (11568 eV) and with a positive sign at the L2-edge (13280 eV). The sign of the XMCD signal was found to be reversed by reversing the Hdirection [see the inset to Fig. 3(b)]. These results confirm that the4 observed XMCD signals are of magnetic origin; the Pt film on YIG is slightly magnetized under such a high field and the Mdirection of the Pt responds to the H direction. Importantly, the XMCD intensity relative to the XAS edge jump at H= 50 kOe for the Pt/YIG sample (∼5×10−3) is several times greater than that for the Pt/YAG sample ( ∼1×10−3) [Fig. 3(b)] and a Pt foil ( ∼ 1×10−3) [56]. The result indicates that, due to the YIG contact, the Pt film acquires magnetic moments greater than the Pauli paramagnetic moments [52, 56, 57]. By XMCD sum-rule analysis [58, 59], the averaged Pt mag- netic moment per Pt atom in the whole volume of the Pt film on YIG at H= 50 kOe was estimated to be mtot=morb+meff spin= 0.0212±0.0015µB, wheremorb= 0.0017±0.0006µBandmeff spin= 0.0195±0.0014µBare the orbital and effective spin magnetic moments per Pt atom, respectively (see Appendix). To clarify the Hdependence of the magnetic moments in the Pt film on YIG, we measured the element-specific magnetization (ESM) curve at the Pt L3-edge for the same Pt/YIG sample at T= 5.5 K (see Fig. 4). Im- portantly, the magnitude of the XMCD signal was ob- served to increase gradually and almost linearly with H. This behavior cannot be explained by the Pt ferromag- netism induced by the magnetic proximity effect, since the XMCD signal due to the proximity-induced Pt fer- romagnetism, if it exists, reflects the M-Hcurve of the YIG substrate surface and the Mof the YIG saturates at around H= 2 kOe [see Fig. 1(a)] [60]. Significantly, the slope of the XMCD signal with respect to Hfor the Pt/YIG sample was found to be 5 .7 times greater than that for the Pt foil where only the Pauli paramagnetism of Pt appears (see Fig. 4). The result shows that the large XMCD slope observed in the Pt/YIG sample can- not be explained also by the simple Pauli paramagnetic moments. The above ESM results suggest that the Pt film on YIG acquires paramagnetic moments greater than the Pauli paramagnetic moments under such high magnetic fields. In the following, we discuss a possible origin of the induced moments. In general, induced Pt magnetic moments in a Pt/ferromagnet interface are attributed to direct interaction between 5 d-electrons in the Pt and spin-polarized electrons in the ferromagnet[50], and thus theM-H(ESM) curve for the induced magnetic moment in the Pt reflects the magnetization process of the mag- net which is coupled to the Pt [39]. This indicates that the largeparamagneticmoments observedin the Pt/YIG sample may originate from magnetic coupling between Pt and interfacial magnetic moments at the Pt/YIG in- terface whose magnetization process exhibits the para- magnetic behavior. The followings are possible candi- dates which may explain the appearance of such inter- facial magnetic moments: (1) interdiffusion of ions and alloying at the Pt/YIG interface; (2) local amorphous-40 -20 0 20 40-0.0100.01XMCD (×
-1) H (kOe)Pt foilPt/YIG Pt/YAGPt L3-edge, T = 5.5 K 0.005 -0.005 0 -4 -2 0 2 4XMCD HLow H FIG. 4: The ESM curveatthe Pt L3-edge ofthePt/YIG sam- ple atT= 5.5 K and the fixed photon energy of 11568 eV, where the blue dots and blue solid line represent the exper- imental data and the linear-fitting result, respectively. T he XMCD data at the Pt L3-edge of the Pt/YAG sample (gray circle), estimated from Fig. 3(b), and of a Pt foil at T= 10 K (gray dashed line), taken from Ref. 56, are also plotted. The inset shows the magnified view of the ESM curve of the Pt/YIG sample in the low- Hrange (|H|<5 kOe). The XMCD values of the vertical axes are multiplied by −1 for clarity, since the sign of the XMCD signal at the Pt L3-edge is negative [see Fig. 3(b)] structures of the YIG surface, which were proposed to exhibit magnetic properties different from the YIG bulk [62, 63]; (3) Y, Fe, and O vacancies of the YIG surface [61, 63, 64]; and (4) unintentional formations of free Fe ions on the YIG surface [41]. Future detailed studies on magnetic properties of the Pt/YIG interface are desir- able for full understanding of the origin of the observed paramagneticbehaviorin Pt. Furthermore, the relevance between the Pt paramagnetic moments and the cluster- like-growthfilmstructure[Fig. 1(c)] andpossiblecluster- sizedependenceofthePtparamagneticmomentdensities are issues to be addressed by experimental and theoreti- cal approaches. Finally, we comment on the relevance between the in- duced paramagnetic moments and the unconventional HalleffectunderhighmagneticfieldsobservedinPt/YIG systems. We found that the gradual increasing behavior of the induced Pt paramagnetic moments with Hshown in Fig. 4 is similar to that of the Hall effect in the Pt/YIG systems at high magnetic fields and low tem- peratures [25, 26, 28], suggesting that the paramagnetic moments are relevant to the Hall effect. It has been re- ported that, if paramagnetic moments exist in a con- ductor, they induce anomalous Hall effects due to the skew-scattering mechanism [42–47]. This suggests that the unconventional Hall effect under high magnetic fields in the Pt/YIG systems can be attributed to the induced paramagnetic moments. This scenario is supported also by the recent reports by Miao et al.[25, 28]. They re-5 ported that the Hall effect in the Pt/YIG systems un- der high magnetic fields is well consistent with that in a diluted-Fe-doped Pt film and a pure Pt film on a diluted- Fe-doped SiO 2film [25, 28], where Fe impurities exhibit a paramagnetic behavior in Pt [65–71], suggesting a role of the induced paramagnetic moments in the mechanism ofthe unconventionalHalleffects in the Pt/YIG systems. IV. CONCLUSION In this study, we estimated the magnetic moments in an ultra-thin Pt film on a ferrimagnetic insulator Y3Fe5O12(YIG) at high magnetic fields (up to 50 kOe) and at low temperatures (5 .5 K) by means of X-ray magnetic circular dichroism (XMCD). We observed an XMCD signal due to magnetic moments in the Pt film at the PtL3- andL2-edges. The measurements of element- specific magnetization curves at the Pt L3-edge reveal unconventional paramagnetic moments of which the in- tensity is greaterthan that expected from the Paulipara- magnetism ofPt. Our experimentalresults reported here provide an important clue in unravelingthe nature of the unconventional Hall effects observed in Pt/YIG systems at high magnetic fields and low temperatures. ACKNOWLEDGMENTS The synchrotron radiation experiments were per- formed at the beam line BL39XU of SPring-8 syn- chrotronradiation facility with the approval ofthe Japan Synchrotron Radiation Research Institute (JASRI) (Pro- posal Nos. 2013B1910, 2014A1204, 2015A1178, and 2015A1457). The authors thank K. S. Takahashi, S. Shimizu, Y. Shiomi, T. Niizeki, T. Ohtani, T. Seki, S. Ito, D. Meier, T. Kuschel, and S. T. B. Goennenwein for valuable discussions and N. Kawamura for experi- mental assistance. This work was supported by ER- ATO “Spin Quantum Rectification Project” (No. JP- MJER1402) and PRESTO “Phase Interfaces for Highly Efficient Energy Utilization” (No. JPMJPR12C1) from JST, Japan, Grant-in-Aid for Scientific Research on In- novative Area “Nano Spin Conversion Science” (No. JP26103005), Grant-in-Aid for Scientific Research (A) (No. JP15H02012) from JSPS KAKENHI, Japan, NEC Corporation, and The Noguchi Institute. T.K. is sup- ported by JSPS through a research fellowship for young scientists (No. JP15J08026). APPENDIX: XMCD SUM RULE ANALYSIS We estimate the averageorbital, effective spin, and to- tal magnetic moments, morb,meff spin, andmtot, per Pt atom in the whole volume of the Pt film on YIG fromthe integrated XAS and XMCD spectra by using the fol- lowing sum rules [58, 59]: (a) (c)(b) (d) -20 0 20 4000.20.40.6 L2-edge EʵE0 (eV)XAS (arb. units) Pt/YIG Au foil Difference -20 0 20 4000.51.01.5 L3-edge EʵE0 (eV)XAS (arb. units) Pt/YIG Au foil Difference -20 0 20 4000.51.01.5 L3-edge EʵE0 (eV) XAS (arb. units) Pt foil Au foil Difference -20 0 20 4000.20.40.6 L2-edge EʵE0 (eV) XAS (arb. units) Pt foil Au foil Difference FIG. 5: [(a) and (b)] The normalized XAS for the (a) Pt L3-edge and (b) Pt L2-edge of the Pt/YIG sample. [(c) and (d)] The normalized XAS for the (c) Pt L3-edge and (d) Pt L2-edge of the Pt foil. The horizontal axes are shifted with respect to the energy at the inflection point E0for the Pt spectra. In (a) and (b) [(c) and (d)], the XAS for the Au L3- andL2-edges of the Au foil and difference in the XAS between the Pt/YIG sample (Pt foil) and Au foil are also plotted, where the energy scale of the Au XAS is expanded by afactorof1.07totakeintoaccountthedifferenceinthelatt ice constant between Pt and Au [72] and is shifted in energy on the basis of the oscillation structures in the EXAFS region o f the Pt spectra. morb=−2 3∆IL3+∆IL2 IL3+IL2nhµB, (1) meff spin=−∆IL3−2∆IL2 IL3+IL2nhµB, (2) whereIL3(IL2) is the XAS integral summed over the PtL3-edge (L2-edge) after subtracting the contribution coming from electron transitions to the continuum, ∆ IL3 (∆IL2) is the integral of the XMCD spectra for the Pt L3-edge (L2-edge),nhis the number of holes in the Pt 5dband, and µBis the Bohr magneton. We first calcu- late the value of the XAS integral r=IL3+IL2for the Pt/YIG sample. To remove the X-ray absorption due to the electron transitions to the continuum, we subtract the XAS of an Au foil from those of the Pt film accord- ing to the method proposed in Refs. 52, 56, and 72. To do this, the energy scale of the XAS of the Au foil was expanded by a factor of 1.07 to take into account the difference in the lattice constant between Pt and Au and to align in energy with that of the Pt film. The XAS6 of the Au foil is normalized to the edge jump (see Fig. 5). The area differences between the Pt-film and Au- foil spectra for both the L3- andL2-edges, i.e., the blue dashed curves in Figs. 5(a) and 5(b), were integrated between −20 eV≦E−E0≦20 eV, where E0is the energy at the inflection point of the Pt XAS. As a result, the XAS integral for the Pt/YIG sample was estimated to ber= 10.2 eV. The absolute value of nhfor the Pt film on YIG was calculated following the methodology described in Ref. 56. First, we determined the scal- ing factor C−1= ˜nh/rPt−foil= 0.112 holes/eV, where ˜nh=nPt h−nAu h= 0.98 is the difference between the 5dholes of the Pt metal nPt h(= 1.73) and the Au metal nAu h(= 0.75).rPt−foil= 8.72eVis theXAS-integralvalue estimated from the XAS of the Pt foil [Figs. 5(c) and 5(d)]. Second, we calculated the nhvalue of the Pt film asnh=nAu h+C−1r= 1.89. Using the estimated r= IL3+IL2,nh, and the integrals of the XMCD spectra for the PtL3- andL2-edges(∆ IL3and ∆IL2) [see Fig. 3(b)], the orbital and effective spin magnetic moments were re- spectively determined to be morb= 0.0017±0.0006µB andmeff spin= 0.0195±0.0014µBfrom Eqs. (1) and (2). This result leads to the total magnetic moment of mtot=morb+meff spin= 0.0212±0.0015µB. 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2017-06-23

Magnetic moments in an ultra-thin Pt film on a ferrimagnetic insulator Y$_3$Fe$_5$O$_{12}$ (YIG) have been investigated at high magnetic fields and low temperatures by means of X-ray magnetic circular dichroism (XMCD). We observed an XMCD signal due to the magnetic moments in a Pt film at the Pt $L_{3}$- and $L_{2}$-edges. By means of the element-specific magnetometry, we found that the XMCD signal at the Pt $L_{3}$-edge gradually increases with increasing the magnetic field even when the field is much greater than the saturation field of YIG. Importantly, the observed XMCD intensity was found to be much greater than the intensity expected from the Pauli paramagnetism of Pt when the Pt film is attached to YIG. These results imply the emergence of induced paramagnetic moments in Pt on YIG and explain the characteristics of the unconventional Hall effect in Pt/YIG systems.

Detection of induced paramagnetic moments in Pt on Y$_3$Fe$_5$O$_{12}$ via x-ray magnetic circular dichroism

1706.07559v1

Hybrid nanodiamond-YIG systems for efficient quantum information processing and nanoscale sensing P. Andrich,1C. F. de las Casas,1X. Liu,1H. L. Bretscher,1J. R. Berman,1F. J. Heremans,1, 2P. F. Nealey,1, 2and D. D. Awschalom1, 2 1Institute for Molecular Engineering, University of Chicago, Chicago, Illinois 60637, USA 2Materials Science Division, Argonne National Laboratory, Argonne, Illinois 60439, USA The nitrogen-vacancy (NV) center in diamond has been extensively studied in recent years for its remarkable quantum coherence properties that make it an ideal candidate for room temperature quantum computing and quantum sensing schemes. However, these schemes rely on spin-spin dipolar interactions, which require the NV centers to be within a few nanometers from each other while still separately addressable, or to be in close proximity of the diamond surface, where their coherence properties significantly degrade. Here we demonstrate a method for overcoming these limitations using a hybrid yttrium iron garnet (YIG)-nanodiamond quantum system constructed with the help of directed assembly and transfer printing techniques. We show that YIG spin-waves can amplify the oscillating field of a microwave source by more than two orders of magnitude and efficiently mediate its coherent interactions with an NV center ensemble. These results demonstrate that spin- waves in ferromagnets can be used as quantum buses for enhanced, long-range qubit interactions, paving the way to ultra-efficient manipulation and coupling of solid state defects in hybrid quantum networks and sensing devices. I. INTRODUCTION Remarkable advancements have recently been made in the use of paramagnetic defects in semiconductors for quantum information and quantum sensing applications[1–6], laying the foundation for the next generation of computing machines and nanosensors. In particular, point defects in diamond and silicon carbide have attracted considerable attention because of their optical addressability and long quantum coherence persisting at and above room temperature. However, chal- lenges remain in their adoption as qubits in functional quan- tum information processing (QIP) devices and quantum sen- sors. Many QIP applications[7, 8] rely on dipolar interactions between the qubits, requiring them to be within a few nanome- ters from each other while still allowing for their individual manipulation with optical and microwave fields. Similarly, sensing applications are based on the detection of the dipo- lar fields generated by target spins external to the diamond lattice, calling for the qubits to be positioned within a few nanometers from the surface, where their coherence proper- ties are strongly diminished[9, 10]. Hybrid ferromagnet-solid state qubit quantum systems have recently been proposed as an architecture with great poten- tial to tackle some of these limitations by introducing quan- tum buses between distant qubits[11], or by enhancing the sensitivity capabilities of quantum sensors[12]. In this work we use spin-waves (SWs) excited in a yttrium iron garnet (YIG) thin film to mediate long-range coherent interactions between a microwave source and nitrogen-vacancy (NV) cen- ters in nanodiamonds (NDs), demonstrating a fundamental step towards obtaining interconnected quantum networks and quantum sensing devices based on paramagnetic defects in the solid state. We show, in particular, that surface Damon- Eshbach spin-wave (DESW) modes excited in the YIG film result in the amplification of a microwave field created by a microstrip line (MSL) antenna by more than two orders of magnitude. We use this amplification to obtain SW mediated resonant driving of NV centers located in an area hundredsof microns wide and to demonstrate highly efficient coherent control using a power that is over three orders of magnitude lower than what is necessary using the microwave field gen- erated directly by the antenna. To accurately determine the origin and extent of the observed phenomena we rely on a novel transfer printing technique that enables the precise con- trol of the NDs position. The strong microwave field amplifi- cation and the coherent nature of the SW-NV center interac- tions demonstrate the possibility of relaxing the distance re- quirements imposed by direct dipolar coupling and enhancing the NV centers sensitivity by using ferromagnetic thin films as quantum buses for inter-qubits and qubit-target spin con- nections. II. EXPERIMENTAL APPARATUS To investigate the SW properties and their interactions with NV centers we use the setup shown in Fig. 1a. A pair of Ti/Au MSLs 5µm wide and 200nm thick, separated by 100µm is lithographically patterned on the surface of a 3:08µm thick YIG layer epitaxially grown on a gadolinium gallium garnet substrate (GGG). The MSLs are tilted with respect to the sam- ple edges to avoid SW reflections. YIG is chosen as a sub- strate because of its small damping parameter for spin wave propagation in the GHz frequency range[13], which makes it ideal for studying long-range interactions. Positioned in contact with the YIG layer is an array of commercial NDs (Adamas Technology, 500NV centers per particle) embed- ded on the surface of a 300µm thick strip of polydimethyl- siloxane (PDMS), which was fabricated through chemical pat- tern directed assembly[14] of NDs on a silicon substrate fol- lowed by transfer printing[15] with PDMS as described in Supplementary Section 2. This portable and reusable system allows us to control the position of the NDs with respect to the MSLs and to easily locate and address single nanoparti- cles, which is critical for interpreting our measurements. Ad- ditionally, the flexibility of the PDMS guarantees the presence of close contact between the NDs and the YIG substrate. InarXiv:1701.07401v1 [quant-ph] 25 Jan 20172 0 50 100 150 200 2501.52.02.53.03.5 Field (G)Frequency (GHz) Microstrip linesYIG PDMSNanodiamonds Bextθ 0 50 100 150 200 2501.52.02.53.03.5 Field (G)Frequency (GHz) Relative transmission (a.u.) 00.10.20.30.40.50.60.70.80.91.0a b c d µmµm 4 8 12 16481216 Photoluminescence (a.u.) 012345678 FIG. 1. Setup and spin-wave dispersion relation of the system. a, Sample schematics. The sample is a 3:08µm thick single-crystal YIG film with a300µm thick PDMS strip laid on top. The PDMS layer contains an array of NDs that are in contact with the YIG substrate. Two 5µm wide microstrip lines (MSL) are patterned 100µm apart on the YIG to apply microwave fields. The microwave magnetic fields (circles) and the direction of the propagating spin-waves are indicated. An external magnetic field (Bext) is applied at an angle with respect to the MSLs. The arrows in the ND represent the NV center spins, with each ND containing hundreds of NV centers. b, Spatial photoluminescence scan of ND arrays. c, Simulated spin wave spectrum for our sample in the case of magnetic field parallel to the MSLs. The dashed lines enclose the range of frequencies where the NV centers ground state spin resonances lay. d, Microwave transmission spectrum between the MSLs as a function of the externally applied magnetic field for = 0. The data for 0gauss was subtracted from the data at higher fields to eliminate features that are not field dependent. Fig. 1b we show a photoluminescence (PL) spatial scan of a typical section of the ND array as collected through a custom- built confocal microscopy apparatus (Supplementary Section 2). III. SPIN-WA VE SPECTRUM As we are interested in the effects of the resonant interac- tions between SWs and NV centers we first calculate and ex- perimentally measure the SW spectrum of our system to ascer- tain where it overlaps with the NV center’s spin resonances. While the direction of SW propagation is always orthogonal to the SML, SW modes with different dispersion relations and magnetization profiles across the ferromagnetic layer can be excited depending on the relative orientation of the externally applied magnetic field B and the direction of propagation of the SWs[16]. Here we primarily focus on DESWs (unless oth- erwise stated), which are excited when the external magnetic field is in the plane of the YIG film and parallel to the MSL (= 0). We select these modes as their energies lie closest to the NV center spin ground state transitions at the magneticfields used in this work (B = 0 to250gauss) and their sur- face nature could provide the strongest interaction with exter- nal spins. We calculate the theoretical spectrum of the DESW following the approach detailed in Supplementary Section 3 and report the result of this calculation in Fig. 1c. In the same figure (dashed lines) we identify the frequency range for the NV center spin ground state resonances, which is enclosed by the resonance spectrum for a defect aligned with the external magnetic field. Even though the nanoparticles have random crystal orientations, their resonances inherently fall within this range[17]. We also experimentally measure the SW modes dispersion using microwave transmission measurements between the two MSLs[18] (Supplementary Section 4). The zero-field mea- surement is used as a reference for the ones at higher fields to eliminate the features in the spectrum that are not magnetic field dependent. The data in Fig. 1d shows good agreement with the calculated spectrum. In order to interpret the results described in what follows it is important to note that DESW excitations with higher frequencies (at a fixed external field) are associated with larger wave vectors k(Supplementary Sec-3 0 50 100 150 200 2502.22.42.62.83.03.2 Field (G)Frequency (GHz) ODMR Contrast (%) -1.8-1.6-1.4-1.2-1.0-0.8-0.6-0.4-0.2 0 50 100 150 200 2502.22.42.62.83.03.2 Field (G)Frequency (GHz) ODMR Contrast (%) -3.5-3.0-2.5-2.0-1.5-1.0-0.50 0 50 100 150 200 2502.22.42.62.83.03.2 Field (G)Frequency (GHz) ODMR Contrast (%) -1.0-0.8-0.6-0.4-0.20 0 50 100 150 200 2502.22.42.62.83.03.2 Field (G)Frequency (GHz) ODMR Contrast (%) -6-5-4-3-2-10Bext Bext Bext4 mW 40 µWa b c 32 mW 4 mW FIG. 2. ODMR spectra of NV centers in NP-P for different orientations of the external magnetic field and microwave power. a, Spectra collected using 4mW and 40µW in the case of = 0.b,c, Spectra collected for =using 32mW and for =/2 using 4mW respectively. tion 3), and display a more pronounced surface confinement as both the magnetization oscillations within the ferromagnetic layer and the field generated outside decay exponentially with characteristic length 1/kin the direction orthogonal to the YIG surface[16]. IV . OPTICALLY DETECTED RESONANT INTERACTION To study the extent of the DESW-NV centers interaction, we perform optically detected magnetic resonance (ODMR) measurements using one of the MSLs to drive microwave fields. Fig. 2a shows the magnetic field configuration used for these measurements and the ODMR spectra obtained ona nanoparticle labeled NP-P, located 40µm away from the MSL. On the left we report the results obtained using 4mW of microwave power, which is the lowest power that still re- solves the NV center‘s resonances at low fields. When we increase the external magnetic field we observe a broad fea- ture of increasing frequency that intersects the NV center reso- nances following the SW resonances’ expected behavior. Re- cent studies have shown the presence of less prominent off- resonant features in the NV centers‘ ODMR spectrum[19, 20] ascribed to the shortening of the NV center longitudinal (T 1) spin coherence time caused by the broad spectrum magnetic field noise introduced by the excited spin waves. Because of the extensive quenching effect on the spin coherence of the4 0 0.1 0.2 0.3 0.4 0.5-50510 Microwave pulse length ( µs)Relative photoluminescence (a.u.)32 µW10 µW3 µW0 0.2 0.4 0.6 0.8 1-0.500.51.0 Microwave pulse length ( µs)Relative photoluminescence (a.u.)1 µW, Spin-Wave Driving4000 µW, Antenna Driving 0 1 2 3 4 500.20.40.60.81.0 Free evolution time ( µs)Relative photoluminescence (a.u.)CPMG3 Hahna b c d20 40 60 8050100150200250300350400450 Distance from antenna ( µm)Microwave amplification0 0.5 1.0-0.6-0.30 MW time ( µs)Rabi Contrast (%)235 µm 1 µW FIG. 3. Time resolved measurements and spatial dependence of the microwave field amplification. a, Rabi oscillations measured on NP-Q (20µm away from the MSL) at the same microwave driving frequency ( 2:862GHz) but different external magnetic field and microwave power. The top curve is measured in the antenna driving regime at 15gauss and using ( 4mW) of microwave power while the bottom curve is collected in the pure SW driving regime at 145gauss and using 1µW of microwave power. b, Microwave field amplification obtained when using SW driving (high fields) over antenna driving (low fields) as a function of the NDs distance from the antenna. The error bars reflect imprecisions in the determination of the Rabi frequencies. In the inset we show the Rabi signal obtained at high fields when the particle is 235mm away from the antenna using 1µW of microwave power. c, Rabi oscillations measured on NP-Q at a fixed external magnetic field ( 120gauss) and different microwave powers. d, Hahn-echo and CPMG3 pulsed measurements that show robust multi-pulse control of the NV centers. Both sets of data are renormalized and fit to exp[-(t/T2)], where= 1and2for the Hahn and CPMG3 case respectively. From these fits we obtain T2;Hahn =1:54µs and T 2;CPMG 3=2:78µs. NV centers, it is not possible to isolate the effect of resonant interactions between the SWs and the NV centers at this mi- crowave power level. On the right side of Fig. 2a, we present the ODMR spec- trum obtained using 40µW of microwave power. Under this condition, the off-resonant effects are not visible but some of the NV center ODMR features are still prominent in the field-frequency subspace where they cross the spin wave res- onances. We stress that at magnetic fields below 50gauss, no ODMR is visible due to the microwave magnetic field gener- ated by the antenna being insufficiently strong to directly drive the NV centers. Only at higher fields does the microwave an- tenna create spin waves that strongly interact with the NV cen- ters at its resonant frequency, creating optical contrast. This result clearly indicates the presence of a purely SW drivenexcitation. We note that only the lower branches of the NV centers ground state spin transitions are visible. We attribute this phenomenon in part to the fact that the upper branches are not intersected by the SW resonances, as can be inferred from Fig. 1b. Additionally, the transmission of the MSLs also decreases at higher frequencies resulting in a lower excitation efficiency. While we cannot resolve the effect of separate SW excitations, this is due to the coarse magnetic field steps used here ( 10gauss). When finer steps are taken, the discrete nature of the SW spectrum becomes clearly visible (Supplementary Section 6). Moreover, we note in Fig. 2a that the interaction appears to be stronger for the SWs associated with larger wave vectors, as can be deduced from the decrease in ODMR con- trast at higher fields, where the NV centers resonances cross lower kmodes. This is consistent with a stronger surface con-5 finement of the magnetization oscillations for higher values of k. We investigate the effect of the magnetic field orientation on the SW-NV resonant interaction for the cases =and =/2. In these conditions the excited spin waves have dif- ferent dispersion relations and magnetization profiles, which allows us to analyze the dependence of the SW-NV interac- tions on these properties. In Fig. 2b we show the ODMR data collected for the =case using 32mW of microwave power, which is the minimum power needed to resolve the NV cen- ters resonances at all fields. The strong reduction in the PL quenching can be explained in light of the non-reciprocal na- ture of the DESW modes[21], which implies that a =ro- tation of the magnetic field results in a decrease of the SWs excitation efficiency[22, 23] and in a drastic change in the SWs amplitude profile, which is confined to the opposite sur- face of the ferromagnetic layer[24]. In the case of ==2, pure backward volume magnetostatic spin waves (BVMSW) are excited[16, 24]. These modes have lower resonant fre- quencies (Supplementary Section 5) than the DESW and are characterized by a sinusoidal magnetization oscillation pro- file across the thickness of the ferromagnetic layer. In Fig. 2c we present the ODMR spectrum collected using 4mW of mi- crowave power. The extent to which the PL is affected is re- markably smaller than for the = 0 case. While the excitation efficiency for DESW and BVMSW is different, this alone can- not explain the two orders of magnitude increase in power re- quired to observe ODMR contrast in the latter case[25]. The difference in the frequencies of the two sets of modes also does not justify the absence of a region of strong PL quench- ing in the magnetic field range we studied, particularly con- sidering the broadband nature of the off-resonant effects. To- gether, the measurements presented in Fig. 2 demonstrate the ferromagnetic nature of the enhanced microwave-NV cen- ter interactions and that the surface confinement of DESW greatly contributes to this enhancement. V . HYBRID COHERENT DRIVING OF NV CENTERS We establish the viability of hybrid YIG-ND systems for quantum information and sensing applications by demonstrat- ing coherent Rabi driving of NV centers using DESWs. By measuring Rabi oscillations on multiple NDs at magnetic fields below and above the threshold for SW-NV interaction, we determine that the SWs can be the source of coherent con- trol and quantify the enhancement that the SW driving pro- vides over using the microwave field generated by the antenna. In contrast to previous demonstrations of off-resonant interac- tions between YIG and NDs[19, 20], these results show that SWs can serve as long range buses for coherent microwave signals without quenching the quantum coherence of the NV centers. In order to isolate the effect of the SW mediated driving from other parameters that influence the SW-NV in- teraction, such as the frequency dependence of microwave power transmission of the MSL and the NV center orientation, we focus on a nanoparticle (NP-Q) that contains NV centers aligned nearly perpendicularly to the external magnetic field. For these NV centers, the lower branch of the ground statespin transition can assume the same energy at low and high fields (Supplementary Section 7), which allows us to compare measurements collected from the same subset of the NV cen- ter ensemble using the same microwave frequency. In Fig. 3a we report the data collected using a driving field resonant with the analyzed NV transition both at 15and 145gauss, using 4mW and 1µW of power respectively. These measurements clearly show that it is possible to coherently drive the NV cen- ters exclusively using interactions with the propagating SW. Moreover, this data shows a SW mediated amplification of the driving microwave magnetic field by a factor of 100 (Sup- plementary Section 8). This amplification factor is a func- tion of the nanoparticles distance from the antenna because, while the antennas field sharply decays as the inverse of the distance from it, the magnetic field generated by the propa- gating spin waves is only limited by the YIGs large spin wave propagation length. This effect is illustrated in Fig. 3b where we show the behavior of the microwave field amplification for NP-Q as a function of its distance from the antenna. When we translate the ND from 20to80µm away from the MSL, the microwave magnetic field amplification increases roughly linearly to>350, suggesting that the SW decay length in the YIG substrate is significantly larger than 80µm. To illustrate this point and to showcase the long-range nature of the SW- NV centers interaction we show the Rabi measurement (inset of Fig. 3b) collected 235µm away from the antenna using 1µW of microwave power. We note that at distances >80µm, it was not possible to observe direct driving induced by the an- tenna’s electromagnetic field because of the limited available microwave power making it impossible to determine the am- plification factor. The effect of the microwave power on the SW mediated co- herent driving is portrayed in Fig. 3c, where we show a series of measurements collected on NP-Q. The Rabi frequency in- creases linearly with the square root of the input power in the range used in this work, even in the pure SW driving regime (Supplementary Section 9), suggesting that we can neglect the impact of nonlinear effects. This allows us to associate the effect described in Fig. 3b only to the difference in the spatial profiles of the antenna and SW fields, independently of the microwave power used for each measurement. In the data in Fig. 3c, we also notice a reduction in the temporal ex- tent of the Rabi oscillations with increasing microwave pow- ers. This phenomenon is likely the result of an increase in the non-resonant magnetic noise (see effect of higher microwave power on the ODMR data) that adversely affects the coher- ences of the NV centers, and can be counteracted by using higher quality NDs with controlled geometry and NV center density, which possess longer coherence times[26]. We further demonstrate the robustness of the spin wave me- diated coherent control using advanced multi-pulse dynam- ical decoupling protocols that are the basis for sensing and quantum computing applications. In Fig. 3d we show the result obtained in the pure SW driving regime for an addi- tional nanoparticle (NP-R) 70µm away from the MSL us- ing5µW of microwave power. The ability to extend the coherence time using multi-pulse sequences demonstrates full control of the NV centers through the pure SW driving. We6 NanodiamondSpin-waveTarget spin YIGa b NV center driving Cavity pumping Polarization Read-out Time FIG. 4. Conceptual representation of a hybrid sensing scheme based on remote spin-wave driving of NV centers. a, Schematic of the proposed design for the sensing device. b, Pulse sequence for the detection of electronic spins through indirect coherent driving of the NV centers. note that, at the microwave power levels required to perform these measurements, the spin coherence times are not signifi- cantly altered by the presence of the ferromagnet, as it is clear from measuring T2 on a ND first while in contact with YIG and then with a non-ferromagnetic GGG substrate (Supple- mentary Section 10). VI. ENHANCED SENSING WITH MICROWA VE MAGNETIC FIELD AMPLIFICATION The results we present suggest that hybrid YIG-ND systems can be useful for enhancing the NV center sensing capabili- ties by relaxing the limits imposed by the NV-target dipolar interactions. To illustrate this concept, we propose a device (Fig. 4a) in which target spins (e.g. free radicals in a biolog- ical molecule) are placed on top of a magnonic cavity that is connected by a SW waveguide to a distant ND. Following the pulse sequence shown in Fig. 4b, a microwave field resonant with the target spins Zeeman transition but not with the SW cavity mode, which is resonant with a NV centers transition, induces Rabi oscillations of the target spins. When the period- icity of these oscillations matches the frequency of the cavitymode, the latter is progressively pumped by the microwave field generated by the precessing target spins, until the rate of SWs leaking from the cavity matches the rate of pumping. The NV centers are then initialized using a laser pulse, be- fore interacting with the spin waves leaking out of the cavity for a variable time . Finally, the state of the NV centers is read out optically with a second laser pulse. This measure- ment scheme is equivalent to performing SW mediated Rabi driving on the NV centers and contains information on the species of the target spins through the resonant nature of their driving, and on their concentration through the Rabi oscilla- tions frequency for a fixed pumping time. We note that similar systems can be developed to obtain long distance interactions between NV centers[11]. VII. CONCLUSION We demonstrate the use of a hybrid YIG-ND system to ob- tain long-range, purely SW mediated coherent control of di- amond NV centers. We show that propagating surface SWs in a YIG thin film can locally amplify an antenna driven mi- crowave field by more than two orders of magnitude, and this amplification persists up to hundreds of micrometers away from the microwave source and it is only limited by the SW propagation length. Additionally, we demonstrate the via- bility of using strong SW-NV center coherent interaction to implement advanced dynamical decoupling schemes. Further enhancement of this interaction can be achieved by using an antenna designed[27] to create higher wave-vector SW modes with even more surface confinement. The use of engineered NDs with controllable NV center density, position, and orien- tation, as well as much improved coherence times[26], could greatly improve the use of the YIG-ND platform for quantum information and sensing applications while remaining com- patible with the flexible PDMS membrane used here. The re- cent advent of spin-wave waveguides[28, 29] and cavities[30] composed of microfabricated YIG films also suggests that separate elements within a nanoparticle array could be ad- dressed individually with a local YIG based microwave source for applications where a global microwave antenna is not ideal. Finally, the demonstration of low power control of solid state paramagnetic defects illustrates the potential of hybrid ferromagnet-qubit platforms as promising pathways for en- ergy efficient classical and quantum spintronics devices. The authors thank B.B. Zhou and M. Fukami for useful dis- cussions. This work was supported by the Army Research Of- fice through the MURI program W911NF-14-1-0016 and U.S. Air Force Office of Scientific Research FA8650-090-D-5037. P.F.N., F.J.H. and D.D.A. were supported by the US Depart- ment of Energy, Office of Science, Basic Energy Sciences, Materials Sciences and Engineering Division. [1] H. Bernien et al . Heralded entanglement between solid-state qubits separated by three metres. Nature ,497, 86-90 (2013). [2] M. Veldhorst et al. A two-qubit logic gate in silicon. Nature , 526, 410-414, (2015).[3] P. V . Klimov, A. L. Falk, D. J. Christle, V . V . Dobrovitski and D. D. 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2017-01-25

The nitrogen-vacancy (NV) center in diamond has been extensively studied in recent years for its remarkable quantum coherence properties that make it an ideal candidate for room temperature quantum computing and quantum sensing schemes. However, these schemes rely on spin-spin dipolar interactions, which require the NV centers to be within a few nanometers from each other while still separately addressable, or to be in close proximity of the diamond surface, where their coherence properties significantly degrade. Here we demonstrate a method for overcoming these limitations using a hybrid yttrium iron garnet (YIG)-nanodiamond quantum system constructed with the help of directed assembly and transfer printing techniques. We show that YIG spin-waves can amplify the oscillating field of a microwave source by more than two orders of magnitude and efficiently mediate its coherent interactions with an NV center ensemble. These results demonstrate that spin-waves in ferromagnets can be used as quantum buses for enhanced, long-range qubit interactions, paving the way to ultra-efficient manipulation and coupling of solid state defects in hybrid quantum networks and sensing devices.

Hybrid nanodiamond-YIG systems for efficient quantum information processing and nanoscale sensing

1701.07401v1

1 SCiEntifiC REPORTS | 7: 11511 | DOI:10.1038/s41598-017-11835-4www.nature.com/scientificreportsStrong coupling of magnons in a YIG sphere to photons in a planar superconducting resonator in the quantum limit R. G. E. Morris , A. F. van Loo , S. Kosen & A. D. Karenowska We report measurements made at millikelvin temperatures of a superconducting coplanar waveguide resonator (CPWR) coupled to a sphere of yttrium-iron garnet. Systems hybridising collective spin excitations with microwave photons have recently attracted interest for their potential quantum information applications. In this experiment the non-uniform microwave field of the CPWR allows coupling to be achieved to many different magnon modes in the sphere. Calculations of the relative coupling strength of different mode families in the sphere to the CPWR are used to successfully identify the magnon modes and their frequencies. The measurements are extended to the quantum limit by reducing the drive power until, on average, less than one photon is present in the CPWR. Investigating the time-dependent response of the system to square pulses, oscillations in the output signal at the mode splitting frequency are observed. These results demonstrate the feasibility of future experiments combining magnonic elements with planar superconducting quantum devices. Magnons are elementary excitations of magnetically ordered materials. For more than half a century, their study has been recognized as a fertile area of research; both for its rich fundamental physics, and its potential techno-logical applications. Experimentally, the magnon modes that can be accessed in a given magnetic sample depend on its geometry and magnetic environment 1, 2. While magnons can be studied in a wide range of magnetic materials, one in particular stands out for its remarkable properties. Yttrium-iron garnet (YIG) is a ferrimagnetic garnet with extremely low magnon damp - ing3. This low loss, in combination with very low electrical conductivity, makes it ideal for the production of high-Q microwave resonators and waveguides. Magnon systems based on YIG have been used for some decades as the basis for commercial radio-frequency devices and components4–6. Much more recently, the use of such systems in the development of devices for quantum computation has begun to attract significant attention7. This has led to a number of investigations into architectures for coupling magnons to photons in ways that can easily be extended into the quantum regime, in the new field of quantum magnonics. To date, work towards quantum magnonics includes room-temperature 8–11 investigations into the coupling of standing magnon modes to photons in 3D electromagnetic cavities, as well as several experiments performed at low temperatures approaching the quantum regime 12–16. In addition, the coupling of a superconducting qubit to magnons via a 3D cavity17 has been demonstrated. However, there has been little work done with planar superconducting structures18, perhaps due to the difficulty of keeping films superconducting in the magnetic fields required for magnon excitation. Quantum devices built using these 2D systems offer the ability to use qubit-specific control lines to precisely control qubit states 19, typically have high qubit-resonator coupling, and are well-suited to integration with other electronic components20. As well as opening up the possibility for novel hybrid devices incorporating magnonic systems, coupling such structures to YIG would offer new tools to study the quantum physics of magnons. In this paper we investigate the coupling of magnons in a YIG sphere to photons in a coplanar waveguide res- onator (CPWR). Our experimental geometry allows us access to a variety of magnon modes with different char - acteristic frequencies, and to identify some of these modes based on numerical calculations. We also investigate the time-dependent behaviour of the system and extend our measurements into the quantum regime by driving Clarendon Laboratory, Department of Physics, University of Oxford, Oxford, UK. Correspondence and requests for materials should be addressed to R.G.E.M. (email: richard.morris@physics.ox.ac.uk )Received: 27 April 2017 Accepted: 31 August 2017 Published: xx xx xxxxOPEN www.nature.com/scientificreports/2 SCiEntifiC REPORTS | 7: 11511 | DOI:10.1038/s41598-017-11835-4the CPWR such that it contains, on average, less than one excitation. This work validates our understanding of the behaviour and interactions of magnon systems at millikelvin temperatures, and is a necessary step in the development of experimental systems coupling magnons to 2D circuit quantum electrodynamic (cQED)systems. Methods Our experiments are performed using a half-wavelength CPWR fabricated from a 150 nm film of superconduct-ing niobium. The CPWR has a frequency of 4.572 GHz and a quality factor of 1220 at zero applied magnetic field. A monocrystalline YIG sphere of diameter 0.25 mm, obtained from Ferrisphere Inc. 21, is glued onto the centre of the CPWR where the amplitude of its magnetic field is maximal. The sample is mounted inside an oxygen-free copper box which is thermally anchored to the mixing chamber plate of a dilution refrigerator with a base tem-perature of approximately 10 mK. This low temperature is necessary to ensure that there is a negligible thermal population of excitations in the frequency range we investigate. The position of the sample is such that it sits at the centre of a superconducting magnet providing a field parallel to the length of the CPWR and in the plane of the niobium film. A schematic of the experimental setup and the room temperature microwave apparatus is shown in Fig. 1a. We are also able to measure the phase of the transmitted signal by comparing the output signal from the sample with a downconverted reference signal from the microwave source (not shown in Fig. 1). Experiments are performed by measuring the complex transmission of the CPWR as a function of fre - quency, magnetic field and input signal power. Approximately 70 dB of attenuation between the output of the room-temperature microwave source and the input port of the CPWR ensures that the electrical noise tempera-ture of the input signals is comparable to the thermodynamic temperature of the sample. From the output of the CPWR, signals are amplified by approximately 40 dB at 4 K before reaching the room temperature part of the measurement instrumentation where, after mixing to an intermediate frequency and undergoing further ampli - fication and filtering, they are digitized at 2.5GHz using a fast data acquisition card. Measurements were typically averaged between ten thousand and one hundred thousand times. The frequencies of the magnon modes in the YIG sphere are functions of the magnetic bias field, and can therefore be shifted by adjusting the current in the superconducting magnet. Since the plane of the CPWR is aligned parallel to the applied magnetic field, it remains superconducting up to relatively high fields, making it possible to bring a variety of magnon modes into resonance with it. By calculating the shape of the resonance microwave magnetic field of the CPWR and the magnetisation within the YIG sphere for a given magnon mode, we can find their relative orientations and thus estimate the relative strength of the coupling for different magnon modes. All the data displayed in this study, as well as details of the calculations performed, can be obtained from the corresponding author on request. Results Theoretical calculations. The microwave field of the CPWR in the region of the sphere was computed using HFSS22 (see Fig. 2a). In contrast with the experimental geometries used in some of the previous investi- gations on this topic8, 11–13, 17, the magnetic field of the CPWR is strongly inhomogeneous in the region around the centre conductor, making it highly non-uniform in the region of the YIG sphere. This allows us to address a significant number of different magnon modes in the sphere. The magnon modes in ferromagnetic spheres are traditionally described using three indices (n, m, r)1, where n and m refer to the order of Legendre polynomials used in determining the resonance condition, and r distin- guishes between modes when there are multiple solutions for the same values of n and m. The mode frequencies and the associated distribution of magnetisation transverse to the magnetic field can be calculated from the theory presented in ref. 23. None of the modes relevant to this experiment have multiple solutions, so in what fol- lows modes will simply be identified by their n and m indices. The relative coupling strength of different magnon Figure 1. Illustrations of the experimental apparatus used. (a) The sample is mounted inside a dilution refrigerator where a superconducting magnet provides a magnetic field, allowing tuning of the magnon mode frequencies. Coaxial transmission lines connect the input to a microwave source, and the output to downconversion instrumentation. Approximately 70 dB of attenuation is present in the input line (60 dB from fixed attenuators and 10 dB from the coaxial cables). (b) Schematic showing the scale of the YIG sphere, with diameter 250 μm, compared to the CPWR, with centre conductor width 10 μm.www.nature.com/scientificreports/3 SCiEntifiC REPORTS | 7: 11511 | DOI:10.1038/s41598-017-11835-4modes can be estimated from the dot product of the magnetisation with the CPWR magnetic field over the sphere, as illustrated for the (2, 2) magnon mode in Fig. 2b. The results of relative coupling calculations for magnon modes in three families up to n = 10 are summarised in Fig. 2c. The (n , n) modes consistently couple most strongly to the CPWR. The (1, 1) mode, also known as the Kittel or ferromagnetic resonance mode, has the strongest coupling of all. This mode corresponds to in-phase precession of all spins in the sphere. Experimental measurements. To observe the coupling between the YIG sphere and the CPWR experi- mentally, we measure the complex transmission (S21) of microwave radiation through the CPWR as a function of frequency and magnetic field. The results of this experiment are summarised in Fig. 3. Each data point in the 2D plots is averaged 4000 times. The results are consistent over a range of temperatures from 10 mK to 5 K, and are reproducible in measurements made over a period of several months. As the field increases above 120 mT, photons in the CPWR couple to different magnon modes in the YIG sphere. This is observed as a series of avoided crossings, with the Kittel mode, labelled (1, 1) in Fig. 3a, being the most pronounced. Apart from the Kittel mode, at least ten additional splittings are visible. As discussed above, we identify the strongly coupled modes by considering the magnetisation distribution and fit their frequencies using the ( n, n) mode family, which converges at high n, as seen in the data. We also observe some weakly coupled modes of similar slope attributable to other mode families in the sphere, such as those plotted in Fig. 2c. Discussion There is good agreement between the calculated frequencies of the (n, n) modes and our experimental data up to the (4, 4) mode, for a saturation magnetisation of 170 kA/m. For higher modes, the observed frequencies begin to diverge from the predicted frequencies, which we attribute to the effect of magnetocrystalline anisotropy. This Figure 2. Results of the theoretical analysis relating to the magnon modes in the YIG sphere. (a) The microwave magnetic field distribution above the centre of the CPWR. A circle of equivalent size to the YIG sphere used in the experiment is superimposed. As can be seen, the field inside the sphere is highly non-uniform. (b) The overlap of the CPWR magnetic field with the (2, 2) magnon mode in the sphere. The arrows indicate the pattern of magnetisation in the sphere for this mode, and the colour scale reflects the dot product of the CPWR field and sphere magnetisation at each point. (c) The relative coupling of four families of sphere modes with the CPWR field is estimated for values of n up to 10. Lines are guides to the eye. The (n, n − 1) and (n, n − 2) families are scaled as indicated to make them more visible. The (n, n) modes consistently couple most strongly, which remains the case for small displacements of the sphere from the centre of the CPWR.www.nature.com/scientificreports/4 SCiEntifiC REPORTS | 7: 11511 | DOI:10.1038/s41598-017-11835-4effect is likely to be especially significant at low temperatures as the first-order anisotropy constant of YIG is approximately three times larger near zero Kelvin than at room temperature24. If the bias magnetic field is along either the hard or easy axis of the crystal, the effect of anisotropy is to shift the frequencies of all modes in the spectrum by a constant factor25. However, in the case of a general orientation, the effects become much more difficult to calculate and analytic expressions for the magnon mode frequencies exist for only the first few low-order modes 26. In our experiment, it was not practical to determine and align the orien- tation of the crystal axes of the sphere, making it impossible to calculate the exact frequencies taking anisotropy into account. The fitted lines in Fig. 3a use only a constant offset to account for anisotropy, which works well for the first few modes. The challenge of determining the effect of the anisotropy on higher-order mode frequencies also makes it difficult to identify the weakly coupled modes observed in the data with particular mode families. The value of 170 kA/m used for the saturation magnetisation is significantly below the literature value of 197 kA/m at 4.2 K 27. Our value is derived from the spacing of the (1, 1) and (2, 2) modes, where analytic expres- sions for the effects of anisotropy do exist. We were therefore able to calculate that anisotropy effects are not suffi-cient to explain this spacing, necessitating the use of a lower value of saturation magnetisation in our calculations. Figure 3b shows a high-resolution measurement of the avoided crossing between the CPWR resonance and the Kittel mode. This crossing is fitted in Fig. 3c using an equation derived from the input-output formalism 12: ωκ ωω κ= −− −+κ ωω−−γS i() () (1)g i21ext re xt2 ()intm2 mm 2 In this fit we account for the effect of anisotropy on the frequency of the Kittel mode ωm since an analytic expres- sion exists. Both amplitude and phase measurements were fitted simultaneously (phase information is not plotted in Fig. 3), resulting in a linewidth of γ m/2π= 2.97 MHz for the Kittel mode, and internal and external linewidths of κint/2π = 1.39 MHz and κext/2π= 1.34 MHz respectively for the CPWR. The average residual of the fit is below 4%, and deviations are only significant around the weaker features above and below the FMR field, which are not Figure 3. Results of microwave spectroscopy measurements of the sample. (a) Experimental transmission through the CPWR as a function of frequency and applied magnetic field at an input power of approximately −80 dBm. More than ten anticrossings are visible, along with a number of weaker features. The (n, n) modes are plotted using the equations in ref. 23 with a saturation magnetisation of 170 kA/m and a constant offset to account for the effects of magnetocrystalline anisotropy. (b) A high-resolution measurement of the (1, 1) anticrossing. The splitting is 16.34 MHz and two of the weaker features are also visible at higher and lower magnetic field. (c) A fit to the crossing in panel (b) using Eq. 1. The two weaker crossings are not modelled.www.nature.com/scientificreports/5 SCiEntifiC REPORTS | 7: 11511 | DOI:10.1038/s41598-017-11835-4accounted for by Eq. 1. The coupling strength gm/2π= 8.17 MHz is larger than either of the individual linewidths, indicating that the system is in the strong coupling regime. At these magnetic fields, the CPWR frequency is shifted down slightly from its zero-field value to ωr = 4.568 GHz. The trend in measured mode coupling strengths mirrors that predicted in Fig. 2c. By reducing the input power, the average number of photons in the CPWR can be reduced to less than one; the so-called quantum limit. Even at these low powers, high levels of averaging (7.5 × 105 averages per data point) allow us to detect strong coupling between CPWR photons and magnon modes (see Fig. 4a), demonstrating the viability of the system in the context of potential quantum information processing applications. We can also excite the YIG-CPWR system with a short square pulse having a carrier frequency between the two split levels (e.g. 4.570 GHz at 149 mT). After such a pulse, we see oscillations in the output signal with fre - quency equal to the splitting. These oscillations decay with an exponential envelope that is due to a combination of the intrinsic damping of the CPWR and the damping of magnons in the YIG sphere. The temporally narrow square pulse has a sufficiently wide frequency spectrum to simultaneously excite both levels, so that these oscilla-tions are interpreted as being due to beating between the frequencies of the two levels. This interpretation is rein-forced by the reduction in amplitude of the oscillations as the pulse length increases, reducing its spectral width. In summary, we have demonstrated strong coupling of a magnonic resonator, consisting of a YIG sphere, to a coplanar waveguide resonator typical of those used in cQED experiments. The coupling strength of 8.17 MHz for the Kittel mode is in the strong coupling regime. We have also shown strong coupling to a series of other non-uniform modes of oscillation in the YIG sphere, which were identified as the (n, n) modes. The range and tunability of excitations available in magnonic systems presents interesting possibilities for building hybrid quantum systems. Our work differs from previous work in the chosen geometries of the mag-nonic and photonic resonators. While the work by Huebl et al . 18 did previously investigate the coupling of a CPWR to a magnonic system, that work focused on the FMR mode in a slab of gallium-doped YIG. Here, we show that the highly inhomogeneous field close to the CPWR allows for coupling to a variety of modes with different coupling strengths. The low magnon damping in our experiment also allows us the fully resolve the anticrossing, in contrast to the previous work. Planar superconducting structures differ in various important ways from three-dimensional cavities for use in quantum information applications, as described in the introduction. By using superconducting qubits designed to work in sufficiently high magnetic fields 28, it is possible to integrate magnonic elements with planar quantum information devices. Our experiment therefore demonstrates the possibility of building a different class of hybrid quantum devices than those previously investigated 12, 17. References 1. Walker, L. R. Magnetostatic modes in ferromagnetic resonance. Phys. Rev. 105, 390–399, https://doi.org/10.1103/PhysRev.105.390 (1957). 2. Damon, R. W . & Eshbach, J. R. Magnetostatic modes of a ferromagnetic slab. Journal of Applied Physics 31, S104–S105, https://doi. org/10.1063/1.1984622 (1960). 3. Cherepanov, V ., Kolokolov, I. & L ’vov, V . The saga of YIG: Spectra, thermodynamics, interaction and relaxation of magnons in a complex magnet. Physics Reports 229, 81–144, https://doi.org/10.1016/0370-1573(93)90107-o (1993). 4. Chumak, A. V ., Vasyuchka, V . I., Serga, A. A. & Hillebrands, B. Magnon spintronics. 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Realization of microwave quantum circuits using hybrid superconducting-semiconducting nanowire josephson elements. Phys. Rev. Lett. 115, 127002, https://doi.org/10.1103/PhysRevLett.115.127002 (2015). Acknowledgements This work was supported by Engineering and Physical Sciences Research Council grant EP/K032690/1. S.K. would also like to thank the Indonesia Endowment Fund for Education for its support. Author Contributions A.K. conceived the experiment; R.M., A.v.L., and S.K. carried out the measurements; R.M. and S.K. performed the calculations; R.M. prepared the manuscript; all the authors reviewed the manuscript. Additional Information Competing Interests : The authors declare that they have no competing interests. Publisher's note: Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations. 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2016-10-31

We report measurements of a superconducting coplanar waveguide resonator (CPWR) coupled to a sphere of yttrium-iron garnet. The non-uniform CPWR field allows us to excite various magnon modes in the sphere. Mode frequencies and relative coupling strengths are consistent with theory. Strong coupling is observed to several modes even with, on average, less than one excitation present in the CPWR. The time response to square pulses shows oscillations at the mode splitting frequency. These results indicate the feasibility of combining magnonic and planar superconducting quantum devices.

Strong coupling of magnons in a YIG sphere to photons in a planar superconducting resonator in the quantum limit

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