Magnon polarons in the spin seebeck effect
Citation for published version (APA): Kikkawa, T., Shen, K., Flebus, B., Duine, R. A., Uchida, K. I., Qiu, Z., Bauer, G. E. W., & Saitoh, E. (2016). Magnon polarons in the spin seebeck effect. Physical Review Letters, 117(20), [207203]. https://doi.org/10.1103/PhysRevLett.117.207203
DOI: 10.1103/PhysRevLett.117.207203
Document status and date: Published: 10/11/2016
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Download date: 28. Sep. 2021 week ending PRL 117, 207203 (2016) PHYSICAL REVIEW LETTERS 11 NOVEMBER 2016
Magnon Polarons in the Spin Seebeck Effect
† Takashi Kikkawa,1,2,* Ka Shen,3 Benedetta Flebus,4 Rembert A. Duine,4,5 Ken-ichi Uchida,1,6,7, Zhiyong Qiu,2,8 Gerrit E. W. Bauer,1,2,3,7 and Eiji Saitoh1,2,7,8,9 1Institute for Materials Research, Tohoku University, Sendai 980-8577, Japan 2WPI Advanced Institute for Materials Research, Tohoku University, Sendai 980-8577, Japan 3Kavli Institute of NanoScience, Delft University of Technology, Lorentzweg 1, 2628 CJ Delft, The Netherlands 4Institute for Theoretical Physics and Center for Extreme Matter and Emergent Phenomena, Utrecht University, Leuvenlaan 4, 3584 CE Utrecht, The Netherlands 5Department of Applied Physics, Eindhoven University of Technology, P.O. Box 513, 5600 MB Eindhoven, The Netherlands 6PRESTO, Japan Science and Technology Agency, Saitama 332-0012, Japan 7Center for Spintronics Research Network, Tohoku University, Sendai 980-8577, Japan 8Spin Quantum Rectification Project, ERATO, Japan Science and Technology Agency, Sendai 980-8577, Japan 9Advanced Science Research Center, Japan Atomic Energy Agency, Tokai 319-1195, Japan (Received 8 July 2016; published 10 November 2016)
Sharp structures in the magnetic field-dependent spin Seebeck effect (SSE) voltages of Pt=Y3Fe5O12 at low temperatures are attributed to the magnon-phonon interaction. Experimental results are well reproduced by a Boltzmann theory that includes magnetoelastic coupling. The SSE anomalies coincide with magnetic fields tuned to the threshold of magnon-polaron formation. The effect gives insight into the relative quality of the lattice and magnetization dynamics.
DOI: 10.1103/PhysRevLett.117.207203
The spin Seebeck effect (SSE) [1–19] refers to the s (a) magnetization generation of a spin current (J ) as a result of a temperature LW V ∇T gradient ( ) in magnetic materials. It is well established LV for magnetic insulators with metallic contacts, at which a EISHE L magnon flow is converted into a conduction-electron spin T magnon TA phonon current by the interfacial exchange interaction [20] and Js detected as a transverse electric voltage via the inverse x ∇T spin Hall effect (ISHE) [21–27] [see Fig. 1(a)]. The SSE y provides a sensitive probe for spin correlations in magnetic z H LA phonon materials [8,9,12–15]. magnon polaron The ferrimagnetic insulator yttrium-iron-garnet Y3Fe5O12 (b) 0.0355 (c) (YIG) is ideal for SSE measurements [19], exhibiting a long 1.0 – magnon-propagation length [28 30], high Curie temperature magnon 0.0350 k1 LA phonon ∼560 0.56 0.58 (THz) ( K) [31], and high resistivity owing to a large band gap (THz) π 0.5 π /2 (∼2.9 eV) [32]. The magnon and phonon dispersion rela- /2 (d) TA phonon 0.2750 tions in YIG arewell known [33–38]. The magnon dispersion µ0H = 1.0 T in the relevant regime reads 0.2730 k2 0510 4.44 4.46 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiq ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi k (108 m-1) k (108 m-1) 2 2 2 ωk ¼ D k þγμ0H D k þγμ0H þγμ0Mssin θ; ð1Þ ex ex FIG. 1. (a) The longitudinal SSE in the Pt=YIG=GGG E sample, where ISHE denotes the electric field induced by the ISHE. The close-up of the upper (lower) right shows a where ω, k, θ, γ, and μ0Ms are the angular frequency, wave vector k with length k, angle θ with the external magnetic schematic illustration of a propagating magnon and TA (LA) phonon. (b) Magnon [Eq. (1) with μ0Ms ¼ 0.2439 T, field H (of magnitude H), gyromagnetic ratio, and saturation μ H ¼ 1 0 θ ¼ π=2 ω ¼ c k – 0 . T, and ], TA-phonon ( ⊥ ), and magnetization, respectively [33 36]. The exchange stiffness LA-phonon (ω ¼ cjjk) dispersion relations for the parameters D coefficient ex as well as the transverse-acoustic (TA) and in Table I. (c),(d) Magnon polarons at the (anti)crossings longitudinal-acoustic (LA) sound velocities for YIG are between the magnon and TA-phonon branches at (c) lower summarized in Table I and the dispersion relations are and (d) higher wave numbers, where k∥xˆ (θ ¼ π=2 and ϕ ¼ 0) plotted in Fig. 1(b). and H∥zˆ.
0031-9007=16=117(20)=207203(6) 207203-1 © 2016 American Physical Society week ending PRL 117, 207203 (2016) PHYSICAL REVIEW LETTERS 11 NOVEMBER 2016
(a) H < HTA H = HTA H > HTA (c) (e) H < HLA H = HLA H > HLA (g) 6.0 at H magnon 6.0 at H TA TA LA point LA phonon point 4.0 phonon touch 4.0 (µV) touch (µV) V 2.0 TA V 2.0
at 0.1 T frequency phonon frequency magnon 2.6 T 9.3 T 0 00 0 1.0 2.0 000 0 1.0 2.0 wave vector wave vector wave vector ΔT (K) wave vector wave vector wave vector ΔT (K) (b) (d) (f) (h) T = 50 K 5.2 T = 50 K T = 50 K 5.2 T = 50 K 5.0 5.0 ΔT = 1.73 K 5.0 5.0
4.8 4.8 Δ ΔT = 1.73 K 0 T = 1.73 K 0 (µV) (µV) (µV) (µV) 5.2 V V V
4.6 V 4.6 0.2 4.8 0.2 Δ T = 0 K -4.8 ΔT = 0 K -5.0 ΔT = 1.73 K 0 -5.2 0 -5.0 -10 0 10 -0.2 -0.2 -5 0 5 2.0 2.5 3.0 -10-5 0 501 9.0 9.5 10.0 µ0H (T) µ0H (T) µ0H (T) µ0H (T)
FIG. 2. (a) Magnon and TA-phonon dispersion relations for YIG when H
In this Letter, we report the observation of a resonant reproducible. A magnified view of the V-H curve is shown enhancement of the SSE. The experimental results are well in Fig. 2(d), where the anomaly is marked by a blue reproduced by a theory for the thermally induced magnon triangle. Since the structures scale with ΔT [see Figs. 2(c) flow in which the magnetoelastic interaction is taken into and 2(d)], they must stem from the SSE. H account. We interpret the experiments as evidence for a The peak appears for the field TA at which according to strong magnon-phonon coupling at the crossings between the parameters in Table I the magnon dispersion curve the magnon and phonon dispersion curves, i.e., the formation touches the TA-phonon dispersion curve. By increasing H, of hybridized excitations called magnon polarons [40,41]. the magnon dispersion shifts toward high frequencies due The sample is a 5-nm-thick Pt film sputtered on the (111) to the Zeeman interaction (∝ γμ0H), while the phonon surface of a 4-μm-thick single-crystalline YIG film grown dispersion does not move. At μ0H ¼ 0, the magnon branch on a single-crystalline Gd3Ga5O12 (GGG) (111) substrate intersects the TA-phonon curve twice [see Fig. 2(a)]. With by liquid phase epitaxy [42]. The sample was then cut into a increasing H, the TA-phonon branch becomes tangential rectangular shape with LV ¼ 4.0 mm (length), LW ¼ to the magnon dispersion at μ0H ¼ 2.6 T and detaches at 2.0 mm (width), and LT ¼ 0.5 mm (thickness). SSE mea- higher fields [see Fig. 2(a)]. If the anomaly is indeed linked surements were carried out in a longitudinal configuration to the “touch” condition, there should be another peak [1,19] [see Fig. 1(a)], where the temperature gradient ∇T is associated with the LA-phonon branch. Based on the applied normal to the interfaces by sandwiching the sample parameters in Table I, we evaluated the magnon−LA- μ H ∼ 9 3 between two sapphire plates, on top of the Pt layer (at the phonon touch condition at 0 LA . T. We then bottom of the GGG substrate) stabilized to TH (TL) with a upgraded the equipment with a stronger magnet and temperature difference ΔT ¼ TH − TL ð>0Þ. ΔT was subsequently investigated the high-field dependence of measured with two calibrated Cernox thermometers. the SSE. A uniform magnetic field H ¼ Hzˆ was applied by a Figure 2(f) shows the dependence VðHÞ of the Pt=YIG superconducting solenoid magnet. We measured the dc sample at T ¼ 50 K, measured between μ0H ¼ 10.5 T. μ H ∼ 9 3 electric voltage difference V between the ends of the Pt Indeed, another peak appeared at 0 LA . T precisely layer with a highly resolved field scan, i.e., at intervals of at the estimated field value at which the LA-phonon branch 15 mT and waiting for ∼30 sec after each step. Figure 2(b) shows the measured VðHÞ of the Pt/YIG T ¼ 50 TABLE I. Parameters for the magnon and phonon dispersion sample at K. A clear signal appears by applying relations of YIG [34–39]. the temperature difference ΔT and its sign is reversed when reversing the magnetization. The magnitude of V at Symbol Value Unit μ0H ¼ 0.1 T is proportional to ΔT [see Fig. 2(c)]. These −6 2 Exchange stiffness Dex 7.7 × 10 m =s results confirm that V is generated by the SSE [19]. 3 TA-phonon sound velocity c⊥ 3.9 × 10 m=s Owing to the high resolution of H, we were able to 3 LA-phonon sound velocity cjj 7.2 × 10 m=s resolve a fine peak structure at μ0H ∼ 2.6 T that is fully
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(a) H < H H = H H < H < H H = H (b) TA TA TA LA LA 0.5 touches the magnon dispersion [see Fig. 2(e)], sharing the /2π magnon point at 0.1 T point touch ) 0.4 at HTA characteristic features of the SSE; i.e., it appears only when touch 0.3 at HLA LA V/K 0.2 ΔT ≠ 0 ΔT TA (µ and exhibits a linear- dependence [see Figs. 2(g) S 0.1 μ H>9 3 V H and 2(h)]. For 0 . T the - curves remain smooth. 0 kkkk000 0204060 We carried out systematic measurements of the temper- Tavg (K) (c) (d) (e) H H 0.40 ature dependence of the SSE enhancement at TA and LA. Tavg = 49.61 K 0.34 0.20 0.32 Figure 3(c) shows the normalized SSE voltage S ≡ 0 0.32 -0.20 0.30 ðV=ΔTÞðLT=LVÞ H as a function of for various average -0.40 0.30 T ½≡ðT þ T Þ=2 0.20 sample temperature avg H L . The amplitude 0.20 34.38 K 0.18 0.10 of the SSE signal monotonically decreases with decreasing 0 0.18 -0.10 0.16 T in the present temperature range [8,9] [see Fig. 3(b)]. -0.20 Importantly, the two peaks in S at H and H exhibit 0.10 16.24 K 0.07 TA LA 0.08 T 0 0.06 different dependences [see Figs. 3(c), 3(d), and 3(e)]. The 0.07 H 0.05 peak shape at TA becomes more prominent with decreas- -0.10 0.06 0.10 T T 13.22 K ing and it is the most outstanding at the lowest . On the 0.06 0.07 S H ∼10 0 other hand, the peak at LA is suppressed below K 0.05 0.06 (µV/K) and it is almost indistinguishable at the lowest T. This S -0.10 0.04 0.05 different T dependence can be attributed to the different 10.48 K 0.05 0.05 H ¼ H 0.04 energy scale of the branch crossing point for TA and 0 H ¼ H – 0.04 LA. The frequency of the magnon LA-phonon -0.05 0.03 0 53 ¼ 26 ≡T 0.04 0.04 intersection point is . THz K( MLA), and it 6.30 K 0.02 0.03 is more than 3 times larger than that of the magnon–TA- 0 0.03 phonon intersection point (0.16 THz). Therefore, for -0.02 0.02 -0.04 0.02 T
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(a) (b)Experiment (c) Calculation We assume diffuse transport that at low temperatures 49.61 K 0.4 50 K T = 13 K 0.31 0.2 is limited by elastic magnon and phonon impurity with MEC |v mag/v ph| = 10 0.29 0 H ¼ 22.53 22.53 scattering [45]. We employ the Hamiltonian imp with MEC P P P 49.61 K 0.4 50 K † ph † mag 10 |vmag/vph| = 1 0.32 0 c v 0 c 0 þ 0 a v 0 a 0 HTA 0.2 μ k;k μk k;k μk k;k k k;k k , where, assum- without MEC 0.30 0 ph ph mag mag Enhancement ing s-wave scattering, vk;k0 ¼ v and vk;k0 ¼ v denote 99.510 99.510 ) 13.22 K ) 0.4 13 K -1 0.06 -1 K K 0.2 -1 the phonon and magnon impurity scattering potentials, -1 s s 0 -1 0.04 -1 V/K) m m respectively. We compute the spin current driven by a 22.53 22.53 22 22 HLA (µ 13.22 K 0.4 13 K
S 0.07 10 temperature gradient [6,16] and thereby the SSE in the 10 ( 5 ( 0.2 xx xx ζ relaxation-time approximation of the linearized Boltzmann 0.05 ζ 0 99.510 99.510 equation. The linear-response steady-state spin current 3.46 K 0.4 3.5 K JsðrÞ¼−ζ · ∇T is governed by the SSE tensor ζ: 0.02 0.2 0 0 Z 22.53 22.53 3 3.46 K 0.4 3.5 K d k X ×10 ζ ¼ Ws τ ð∂ Ω Þð∂ Ω Þ∂ fð0Þj : 0.02 0.2 αβ 3 ik ik kα ik kβ ik T ik T¼TðrÞ 0 0 ð2πÞ 0 i 051099.510 99.510 µ H (T) µ H (T) µ0H (T) 0 0 ð2Þ FIG. 4. (a) Calculated SSC ζxx at T ¼ 13 K as a function of H, s 2 Here, Wik ¼jh0jakjψ ikij is the intensity of the ith with (red solid curve and blue circles) and without (red dashed magnon polaron and τik is the relaxation time curve) magnetoelastic coupling (MEC). The red solid curve and towards the equilibrium (Planck) distribution function the blue circles are computed for ratios of the scattering potentials jvmag=vphj¼10 jvmag=vphj¼1 fð0ÞðrÞ¼ð ðℏΩ =½k TðrÞ Þ − 1Þ−1 of and , respectively. The blue ik exp ik B . The relaxation dashed curve is a blowup of the difference between the red solid τ i τ−1 ¼ time Pik of the th magnon polaron reads ik and dashed curves. (b) Experimental S and (c) theoretical ζxx after 2 ð2π=ℏÞ jhψ 0 jH jψ ij δðℏΩ − ℏΩ 0 Þ jk0 jk imp ik ik jk . The subtraction of the zero MEC results. strong-coupling (weak scattering) approach is valid when τ−1 ≪ ΔΩ ΔΩ ∼30 K clearly increases with increasing H [Fig. 3(c)]. We ik1;2 , where is the energy gap at the anticross- tentatively attribute this discrepancy to an additional spin ing points k1;2. We disregard the Gilbert damping that is very small in YIG. current caused by the paramagnetic GGG substrate that, From the experiments we infer the scattering parameters when transmitted through the YIG layer, causes an addi- 2 −5 −2 tional voltage. Wu et al. [7] found a paramagnetic SSE jvmagj ¼ 10 s [28] and jvmag=vphj¼10; i.e., the signal in a Pt=GGG sample proportional to the induced magnons are more strongly scattered than the phonons. magnetization (∼a Brillouin function for spin 7=2) [7]. The computed longitudinal spin Seebeck coefficient (SSC) Indeed, the increase of S in the present Pt=YIG=GGG ζxx [Eq. (2)] is plotted in Fig. 4(a). Switching on the sample is of the same order as the paramagnetic SSE in a magnetoelastic coupling increases the SSC especially at the Pt=GGG sample [8]. “touching” magnetic fields H and H . At these points TA LA In conclusion, we observed two anomalous peak struc- the group velocity of the magnon is identical to the sound tures in the magnetic field dependence of the SSE in YIG velocity. Nevertheless, spin transport can be strongly that appear at the onset of magnon-polaron formation. The jvmag=vphj modified when the ratio differs from unity. experimental results are well reproduced by a calculation in The SSC can be enhanced or suppressed compared to its which magnons and phonons are allowed to hybridize. Our purely magnonic value. A high acoustic quality as implied results show that the SSE can probe not only magnon jvmag=vphj¼10 by is beneficial for spin transport and dynamics but also phonon dynamics. The magnitude and enhances the SSC by hybridization, as illustrated by shape of the anomalies contain unique information about Fig. 4(a). When magnon and phonon scattering potentials the sample disorder, depending sensitively on the relative jvmag=vphj¼1 would be the same, i.e., , the anomalies scattering strengths of the magnons and phonons. vanish identically [see the blue circles in Fig. 4(a)]. The difference between the calculations with and without MEC The authors thank S. Daimon, J. Lustikova, L. J. agrees very well with the peak features on top of the smooth Cornelissen, and B. J. van Wees for valuable discussions. background as observed in the experiments, see Figs. 4(b) This work was supported by PRESTO “Phase Interfaces for and 4(c). We can rationalize the result by the presence of a Highly Efficient Energy Utilization” from JST, Japan, magnetic disorder that scatters magnons but not phonons. Grant-in-Aid for Scientific Research on Innovative Area Finally, we address the SSE background signal. The “Nano Spin Conversion Science” (Grants No. 26103005 overall decrease of the calculated ζxx is not related to and No. 26103006), Grant-in-Aid for Scientific Research the phonons, but reflects the field-induced freeze-out of the (A) (Grants No. 15H02012 and No. 25247056) and (S) magnons (that is suppressed in thin magnetic films [8]). In (Grant No. 25220910) from MEXT, Japan, NEC the experiments, on the other hand, the global S below Corporation, The Noguchi Institute, the Dutch FOM
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