Magnons and their interactions with phonons and photons C. Kittel To cite this version: C. Kittel. Magnons and their interactions with phonons and photons. J. Phys. Radium, 1959, 20 (2-3), pp.145-147. 10.1051/jphysrad:01959002002-3014500. jpa-00236005 HAL Id: jpa-00236005 https://hal.archives-ouvertes.fr/jpa-00236005 Submitted on 1 Jan 1959 HAL is a multi-disciplinary open access L’archive ouverte pluridisciplinaire HAL, est archive for the deposit and dissemination of sci- destinée au dépôt et à la diffusion de documents entific research documents, whether they are pub- scientifiques de niveau recherche, publiés ou non, lished or not. The documents may come from émanant des établissements d’enseignement et de teaching and research institutions in France or recherche français ou étrangers, des laboratoires abroad, or from public or private research centers. publics ou privés. LE JOURNAL DE PHYSIQUE ET LE RADIUM TOME 20, FÉVRIER 1959, 145 MAGNONS AND THEIR INTERACTIONS WITH PHONONS AND PHOTONS By C. KITTEL, Department of Physics, University of California, Berkeley 4, California (U. S. A.). Résumé. 2014 Cette étude est une mise au point de la théorie des ondes de spin et des effets phy- siques liés à ces ondes. On y montre que la condition aux limites à la surface mx = my = 0 permet l’excitation des ondes de spin par un champ uniforme de fréquence radio. Waring et Jarrett, ainsi que Seavey et Tannenwald, ont observé ces excitations. Abstract. 2014 A review is given of the theory of spin waves and of the principal physical effects associated with spin,waves. It is shown that the boundary condition mx = my = 0 at the surface permits the excitation of spin waves by a uniform rf field. These excitations have been observed by Waring and Jarrett and by Seavey and Tannenwald. A magnon is a quantized spin wave, just as a domain patterns and, further from the obser- photon is a quantized electromagnetie wave and a vations of Rado [15] and others on exchange"effects phonon is a quantized elastic_wave. Magnons in ferromagnetic resonance in metals. Granted exist only in’an idealization, just as"photons and that the energy of a non-uniform distribution of phonons_exist only in the idealization that terms magnetization directions is higher than for a in the ’complete hamiltonian representinglinter- uniform distribution, we are led directly by a well- actions or collisions,amongthe particles are neglect- known argument to the Landau and Lifshitz [6] ed. Two, phonons do not exist independently in expression the actual anharmonic potentials in real crystals, nor do two magnons exist entirely independently, but for both particles it can be shown that in the for the exchange energy density in a cubic crystal ; limit of small excitation amplitudes the interaction here oc is- the unit vector in the direction of the corrections are small. For magnons this has been loèal magnetization. Granted Eq" (1), we are then shown [1] by Bethe, Hulthén, and Dyson; Bethe led by purely macroscopic reasoning to a theory calculated explicitly the exact energy eigenvalues of magnon fields. for two magnons on a line of atoms, and the inter- The classical exchange hamiltonian density equi- action terms are found to be very small. The valent to Eq. (1) in the limit of small amplitudes bearirig of the electron gas in metals on the exis- 1 -- -- 1 nB tence of phonons and magnons [2] probably does not upset the wave aspect of the phonon and fields. This is the fact that magnon explained by = Mx, and = is the conjugate the is much than where Ç My/yfg plasma frequency very greater momentum [7] ; here y = ge/2mc. The the and the density Debye frequency exchange frequency ; Zeeman lagrangian leading to this result for 7c it is known from neutron and experimentally, [3] gives the correct spin resonance_frequency for a work that elastic waves have a in x-ray reality uniformly magnetized ferromagnet in a magnetic metals up to the top of the frequency spectrum, field. The equations of motion in hamiltonian mea- approximately 1013 cps. Neutron diffraction form are surements [4] at Harwell give evidence of the existence of spin waves in iron. It is perhaps more convincing to begin the treat- ment of magnons from a macroscopic viewpoint, just as the idea of a phonon originated with Debye in the quantization of a macroscopic elastic wave in an essentially homogeneous medium. We assert Thus, using Eq. (2), first that work must be done to distort or twist the local direction of magnetization from one point to another in a ferromagnetic specimen, just as work must be done to set up a mechanical deformation or If we look for solutions of the form ei(cùg--k.r) , we an The most strain in elastic medium. convincing find from Eqs. (5) and (6) that macroscopic evidence for our assertion comes from the observation of Bloch walls of finite thickness in Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphysrad:01959002002-3014500 146 which is the dispersion relation for spin waves. The strongest effects of the magnon-phonon cou- Our discussion has been entirely parallel to the pling occur when both the wavelengths and fre- theory of phonon fields in a homogeneous medium. quencies of the magnons and phonons are equal. We have not assumed any particular atomic model. We have found a number of interesting properties The dispersion relation (ô = (2y AI Ms) k2 is of the motion : identical with that of de Broglie waves for a non- (a) It appears that thin ferrite crystal sections relativistic particle of mass might make effective magnetostrictive oscillators at microwave frequencies if the spin system is driven at the spin resonance frequency. and we may take over to magnons many of the (b) The absorption of ultrasonic waves is very results we know concerning the motion of electron large when the resonance conditions are satisfied. wave packets. The group velocity of a spin packet This is probably the explanation of the effect disco- is vered by Friedberg that the thermal conductivity of ferrites at helium temperatures depends on the magnetic field. (c) The rotatory dispersion of ultrasonic waves is large. Even quite far from the spin resonance fre- quency the rotation of the plane of polarization of degenerate transverse elastic waves is large. The rotation of the plane of polarization is the basis for application as a non-reciprocal mechanical circuit element. A whole class of new devices, such as m - k N 106 For 1011 sec-1, we have cm" 1 and acoustic gyrators, becomes possible in vjg - 106 cm/sec. Their dispersive properties principle. make it difficult to observe the motion of a directly Excitation caf magnons by a uniform rf field [9]. of waves. The Sixtus-Tonks and packet pure spin - The excitation of magnetostatic modes by a non- related experiments deal with the propagation of uniform rf field has been discussed White and Bloch walls. by Solt, Walker and others. We are not concerned here with the White-Solt effect, but with the exci- Interaction of magnons and long wavelength tation of exchange modes by a uniform rf field. phonons. - It is interesting to consider the inter- The latter effect has probably been observed by action of waves and ultrasonic waves in ferro- spin Jarrett and in a of This is treated Waring [10] crystal NiMno3, magnetic crystals [7]. problem We have an the follow- in form proposed explanation along most simply by setting up hamiltonian the lines : It is to excite and of motion of the two of ing possible exchange equations types fields, in a a the first-order inter- magnetostatic magnons ferromagnet by including magnetoelastic uniform rf field provided that spins on the surface action as deduced from magnetostrictive measu- of the interactions rements. The hamiltonian ae be specimen experience anisotropy density may different from those acting on spins in the interior. written as the sum of Zeeman, exchange, magneto- The concept of surface anisotropy was first intro- elastic interaction and phonon terms : duced by Néel. Although the proportion of surface atoms in an actual be less than 1 : 104 of the small deflection of the specimen may Assuming (a) amplitudes total number of a modest surface aniso- from the z-axis and elastic iso- atoms, yet magnetization (b) can have a far-reaching effect on the tran- we have tropy tropy, sition probabilities for the excitation of spin waves in the microwave region. The essential point is simple : the surface anisotropy acts to pin down the surface spins. For a line of length L with the origin on one end, the modes will tend to have the form sin p1CzIL, where p is an integer. The modes of odd p will have a non-vanishing interaction with Here wo = Y Ho; (a), = Y M. ; b2 is a magneto- a uniform rf field, because the instantaneous trans- elastic constant ; p is the density ; oc and g are verse magnetic moment does not sum to zero over elastic constants ; 7ri is a conjugate phonon momen- the line. tum density ; and A surface anisotropy of the order of 10- 2 of the isotropic exchange energy suffices to pin down the surface spins, according to the detailed calcu- lation. The separation of successive magnon exci- is a strains component. tations in a crystal of thickness 3 X 10"4 cm will 147 be of the order of 2p oersteds, where p is the mode Mr. Nagamiya. - How could you pin the spins on index. Observation of the Jarrett-Waring effect the surface ? Are you able to control the condi- should be a very good direct method for the deter- tion of the pinning ? mination of the interaction constant A exchange - Surface are in a in Mr.