.gl -.-,L Thi iitcerflffithi {he " ‘ " dwdsemfikd THE INTERACTION 0F SPIN WAVES WITH DRIFTED CONDUCTION ELEGTRONS presented by Bruce C. Bur-key has been accepted towards fulfillment of the requirements for Ph.Do degree in P1123108 Z r I Majg professor Date ML 0-169 ABSTRACT 'THE INTERACTION OF SPIN WAVES WITH DRIFTED CONDUCTION ELECTRONS by Bruce C. Burkey The elementary excitations of a ferromagnetic or ferrimagnetic material are magnetoelastic waves if the material has appropriate magnetoelastic properties. A magnetoelastic wave may be predominantly spin wave or it may be an admixture of both a spin wave and an elastic wave, depending on the magnetic, elastic and magneto- elastic characteristics of the material and the wave vector of the excitation. If, in addition, the material has appropriate deformation potential or piezoelectric properties, and is a semiconductor, the magnetoelastic excitations may interact with the drifted charged par— ticles, the drifting being caused by an externally applied electric field. The purpose of the present work is to investigate theoretically the interaction between the predominantly spin wave magnetoelastic waves and drifted charged particles. The model material considered is an n—type ferro- magnetic semiconductor with idealized magnetoelastic, piezoelectric and deformation potential properties. Bruce C. Burkey A continuum approach is taken. The equations of motion for the magnetization, lattice and conduction electrons are derived, linearized and solved simultaneously with the continuity equation and Maxwell's equations for perturbations from the steady state conditions. The dispersion relation is obtained and the magnetoelastic branches of the spectrum are analyzed for various values of the conduction electron density, drift velocity and mean free time. Two types of instability in the magnetoelastic waves are predicted. The first occurs when the conduc- tion electron drift velocity becomes greater then the magnetoelastic wave phase velocity. This is analogous to ultrasonic amplification. The second occurs when the conduction electron drift velocity becomes large enough to doppler shift the frequency of the plasma wave into coincidence with the frequency of the oppositely directed magnetoelastic wave. .1-hsmh_£m,g. gun may?“ o"J '. wk. g!!_ ;,., DRIFT!” CONDUCTION ELECTRONS By g5 Bruce C?” rkey A THESIS Submitted to Michigan State University in partial fulfillment of the requirements for the degree of DOCTOR OF PHILOSOPHY Department of Physics 1967 ACKNOWLEDGEMENT I wish to express my sincere appreciation to ’ Professor Michael J. Harrison for his continued encourage- ment and guidance throughout this work. To my wife, Marilyn, for her continued moral support and for typing the manuscript, I am most grateful. For financial support, I thank the National Science Foundation. ii v. e h 8 “‘V ' ‘ »¥" l - -. I ‘l‘ ."i‘fv' ts IV {"1 .l l‘; n 1 ,. -x I. II. III. VI. TABLE OF CONTENTS Page Introduction...IOCIOOIOOOOOCIOII... ...... O llllll .0. 1 Equations of Motion ........ ... ..... . ............... 8 Equation of Motion of the Magnetization.. ...... 8 Elastic Equation of Motion ..... .... ............ 15 The Kinetic Equation. ................ ..... 22 Linearization of the Equations of Motion ....... 26 Supplementary Equations ..... ....... ............ 29 The Dispersion Relation ............................ 31 Analysis of the Dispersion Relation ................ 38 Zero Magnetoelastic Coupling ................... 40 Zero Electromagnetic Coupling .......... . ....... 48 A. Magnetoelastic Wave Amplification ......... 53 B. Resonant Interaction Between 122 and I1L" 63 Discussion of Results .............................. 71 Summary. .................................. 71 Material Considerations ........................ 73 Bibliography ....................................... 75 iii . Ar: I. 3"“ >4. . "Iii 34-13?" trigger ‘ P1,)(g351 , whisk; measures that past-v“ of In 6) depending explicitly an electron _. p .‘oa‘ld‘fiifit VSIQGIEY... I‘o‘i cos. ocean so... so. cacti“. .EW 3:... Plot of log X1 and log X? for kkcr°"" 62 . 5. Plot of the relative increment of the magnetoelastic wave due to a resonant interaction with the doppler shifted plasma waveoo....I.......OQOOOOOOIOOOODIOOI 69 iv ‘tdh‘ g} .-., '_-. . I a .gIchltho finib anglb é-uvfi-lsocial-nei-stsn-tcc-s V17t00l'07331'1b .WH fr nor #2 arts _ . _.A ; r I (a. ----- I Introduction Many of the properties of ferromagnetic, ferri- magnetic and antiferromagnetic single crystals can be understood in terms of a model in which the low lying excited states of the system correspond to the waves of magnetization called Spin waves. When quantized, these spin waves are called magnons, and they represent the elementary excitations of the magnetic system. At sufficiently low temperatures, the Spin waves very nearly represent eigenstates of the magnetic system and propagate in the form of weakly attenuated waves. At any given temperature the average excitation ampli- tude of any spin wave mode is an equilibrium thermo— dynamic property of the magnetic system which manifests itself by contributing to the deviation of the magneti- zation from its saturation value. However, the spin wave amplitude can be influenced by another factor, namely, an appropriate charged particle stream. As a consequence of an interaction with an appropriate charged particle stream, a spin wave mode may become unstable; it may either grow Spatially in the direction of propagation or grow in time. The purpose of the present study is to investigate a specific mechanism for the production of a coherent spin wave instability. Such an instability may prove useful as a tool in the investigation of both the spin wave energy spectrum and spin wave relaxation processes. In addition, a Spin wave instability might have device applications. Theoretical studies of spin wave instabilities have been published by several investigators.l-4 In each of these theories the predicted instability arises from a coherent interaction with a compensated charged particle stream. The interaction is a result of the coupling of the magnetic forces associated with the charged particle stream and the spin wave. Akhiezer, Baryakhtar and Peletminskii studied1 spin wave instabilities in both ferromagnets and anti- ferromagnets. They assume an infinite, single domain magnetic medium with collinear internal dc magnetic field, given saturation magnetization, and dc electron stream. The equation of motion of the magnetization, Newton's law for the electron motion, Maxwell's equations and the continuity equation are linearized for perturbations of the fields from their dc conditions. Harmonic space and time dependence is assumed and a dispersion relation obtained. The analysis of the Edmund Iqrrl’ ”was; amt . . In" "thhdsfud on! ‘uIIf 3." ' 96“»: (Hill: "1-: 3‘ Tc“; :rtzm‘T spin wave branch of this dispersion relation discloses that under certain resonance conditions, spin wave instabilities can occur. The effect of finite conduc- tivity is neglected in their treatment. Baryakhtar and Makhmudox generalized2 the analysis of reference (1) to the case wherein the dc electron stream is not necessarily collinear with the saturation magnetization and the internal dc magnetic field. The generalization is only for the case of a ferromagnet. Makhmudox and Baryakhtar studied3 Spin wave insta- bilities initiated by drifted conduction electrons in ferromagnetic and antiferromagnetic semiconductors. Their analysis is the same as that of reference 1 except that the Boltzmann equation is substituted for Newton's law, and the condition 60P viJ s1, (1) where ViJ are the deformation potential constants, I! ll 813 are the lattice strain components and e is the magnitude of the electronic charge. The deformation potential interaction between conduction electrons and lattice strain is manifested in many phenomena. With respect to the present investigation, the most important of these phenomena is the acoustoelectric effect.7 The piezoelectric effect can occur in crystals which lack a center of inversion symmetry. A uniform strain can then cause a separation of the + and — charges, which results in an internal electric field. Conversely, an internal electric field produces a uniform strain. The piezoelectric effect is described [V by the piezoelectric equations of state _ D —1 Tkl “ cijkl Sij 441i) hnkl Dn (2) —1 S,M Em — (6 )mn Dn - hmij S13 (3) D -1\S,M where Tij’ Sij’ Di’ E1, Cijkl’ hijk and ( 6 ‘mn are respectively, the stress components, strain components, electric displacement components, electric field com— ponents, elastic constants, piezoelectric constants and the components of the inverse lattice dielectric tensor. The piezoelectric interaction between conduction electrons and lattice strain is manifested in many phenomena. With respect to the present study the most inportant of these phenomena is ultrasonic amplification8_9 which occurs in certain piezoelectric semiconductors. The interaction between the lattice strain and the magnetization occurs through the magnetoelastic interaction. This interaction is thought to have its origin in the spin-orbit coupling, as does the magneto- crystalline anisotropy. Following Kittello the form of the interaction energy density is obtained by a Taylor series expansion of the magnetocrystalline anisotropy energy density, and is Hme = biJkl «k 0(1 81;! (4) where bijkl are the magnetoelastic constants, 0(1 are the components of a unit vector parallel at every point to the instantaneous magnetization and S13 are the strain components. Magnetoelasticity is manifested in many phenomena. With respect to the present study the most important of these phenomena is the spin wave- phonon interaction11 which causes the normal modes of the magnetoelastic system to be admixtures of spin waves and lattice waves.12 The purpose of the present study is the investiga- tion of this alternative phonon-dependent mechanism for coupling spin waves and a charged particle stream. The ultimate objective is to demonstrate that instabilities may occur in the spin wave portion of the excitation spectrum of the system. The approach to the problem is straightforward. The equations of motion of the lattice and magnetization are assumed along with Maxwell's equations, Newton's law for the electronic motion, and the continuity equation. The equations are linearized with respect to perturbations from the dc conditions. A dispersion relation is formed by assuming harmonic space and time dependence for the variables. The dispersion relation is then analyzed for changes in the spin wave frequency due to the interaction with the lattice vibrations and drifted conduction electrons. The material envisaged for the model is an n-type ferromagnetic semiconductor with idealized magnetoelastic, deformation potential, piezoelectric, and elastic properties. The idealization is necessary to make the problem tractable. To avoid the necessity of considering boundary conditions, the medium is assumed to be infinite in extent. II Equations of Motion Equation of Motion of the Magnetization. From a macroscopic point of View a ferromagnetic may be considered as a continous medium characterized by a magnetic moment density or magnetization M(£) which in the general case is a vector function of position 3. Following KittellB, the differential equation of motion of the magnetization is determined from the equation dfl_ 3 rh —dt-" E’J/fi¥mag d r . (5) Equation (5) is the Heisenberg equation of motion for a quantum mechanical operator M. In equation (5) 2¥mag is the part of the total energy density which involves the magnetization. To determine 2imag’ the physical model under con- sideration must be more explicitly described. With respect to the magnetic properties, the medium is a ferromagnetic semiconductor with appropriate magnetoelastic properties. The ferromagnetism is assumed to be a consequence of the exchange interaction between electrons which are localized 8 . ‘ _ , VJ~ 'C'. "_ 0 v - w. lemmas? t. wtlv “: mt": thy-«owns “mi ."".l“f’ ‘I‘G"!‘.(‘,' fi‘".b"fil 9!." {'3' “ 3 28 17"" "IILRCéJ . t‘ams;£:_ , in the lattice and which, therefore, do not contribute to the conduction current density. The semiconducting property is meant to imply that the medium contains a compensated low density "free" electron gas, hereafter referred to as the conduction electrons. In the present model the conduction electrons do not interact directly with the localized ferromagnetic electrons through, for example, the s-d exchange interaction. The conduction electrons do, however, interact indirectly with the magnetic electrons. This indirect interaction is electro- magnetic in nature and is responsible for eddy current damping of spin waves. For this reason, the necessity of including the electromagnetic interaction should be emphasized. In addition to its interaction with the conduction electrons, the magnetization is assumed be be strongly coupled to the lattice strain by virtue of the magneto— elastic interaction. Above saturation, i.e., for single domain ferromagnets, the state of strain of the medium can be influenced by the magnetization in a number of ways. Three effects are distinguished:lbr linear magneto— striction (anisotropic strain), which arises from the strain dependence of the exchange energy density; volume magnetostriction, which arises from the strain dependence of the dipolar or demagnetizing energy in a finite sample. 10 The effects due to volume magnetostriction are assumed to be small and are therefore neglected in the present study. The effects caused by the form effect are also neglected in this study because the ensuing theory is developed for an infinite medium where the form effect vanishes. In what follows, the term "magnetoelastic interaction" refers only to linear magnetostriction (anisotropic strain). In addition to the electromagnetic and magnetoelastic interactions with the magnetization, any disturbance of the uniformity of the magnetization results in an increase in the exchange energy density. The energy density, ;§/mag’ associated with the magnetization becomes ?/mag = 7yz + )%;e + ;y;x (6) where }{Z is the Zeeman contribution due to the electro- magnetic interaction with the magnetization, Az/me is the magnetoelastic contribution, and is the exchange ex contribution. For general crystal symmetry the exchange energy density is QM QM Z/ex = _%— Alk iii? jifi (7) 'ur “ " .- a- ' miflssafis' seem: at 11 where Alk are the exchange energy constants. To simplify the calculation, isotropic symmetry is assumed for the exchange energy density, and therefore, equation (7) 13,15 becomes 2iex = A“? (VMX)2 + (VMy)2 + (VMZ)2 (8) where A is the exchange energy constant and Mo is the saturation magnetization. A magnetic field §_at a given point produces a torque on the magnetization density at that point even when E is partially produced by the surrounding magneti— zation. Thus the field E contains an externally applied component which is constant in space and time and a self— ‘consistent ac component. self—consistent because it satifies Maxwell's equations. In a finite medium the internal dc magnetic field is altered by the demagnetizing field; however, in the following calculation the demagne— tizing field is neglected. The interaction energy density between M and g is taken as a Zeeman energy and is given by 77/Z=—M'§- (9) Following Kittel,13 the form of the magnetoelastic energy density is obtained by a Taylor series expansion of the magnetocrystalline anisotropy energy density, and 12 is defined by Wme = ”4'2 Z 0 iJkl biJkl Si M M ' (10) J k 1 Here Ml are the magnetization components, 513 are the strain components and b are the magnetoelastic ijkl coupling constants. The strain components are defined an AR __1-__1 __J 813 - 2 AXJ *' 8x1) (11) where R is the displacement of a point in the medium from its equilibrium position. If magnetoelastic isotropy is assumed, the only nonzero magnetoelastic coupling 16 constants are b1111 = b2222 = b3333 = b1331 = b3131 = b1313 2 b3113 = b3223 = b2323 = b3232 = (12) b2332 = b1212 = b2112 = b1221 = b2112 ‘ However, we shall now introduce an essential anisotropy in the magnetoelastic properties by assuming that the only nonzero magnetoelastic coupling constants are b = (13) b1313 = b1331 b3113 = b3131 As shown later, this choice of magnetoelastic coupling constants effectively couples a z—directed spin wave to 13 only one z-directed transverse lattice vibration. Also, the strain independent part of the magnetocrystalline anisotropy energy density is purposely neglected. Following Herring and Kittel13 the equation of motion of the magnetization is obtained from the Hamil- tonian density through equation (5). The commutator which must be evaluated is the following: ictflffir' {Amy was)“ was)“ (“~sz (14) _ AB R + 2bM02 (3; + 3x? )MxMz —— (MXHX+ MyHy +MZHZ) , The components of the magnetization satisfy the commuta— tion relation mmxmr) = — val/my) (Sm—r) (15) where l/ is the magnetomechanical ratio. Both H and R are assumed to commute with M After evaluating equation (14) and substituting into equation (5), the equation of motion of the magnetization becomes M _ g5?— = - 2AVM02(EXV2E>-YEXE+32 2W): + ‘37 . ., . . . 5: x : ...:"fi ""Pf‘ . 5!: .-P'.‘*' W: ‘33?) .6321;ng '31 15'. - , u ’ ,. i.eliirj? .Eneg , .A-fi;__- 1,.a._iihs..g Q ’o o 14 where linearization of the magnetoelastic term has already been accomplished by dropping products of the strain with either Mx or My and by approximating Mz by Mo. In equation (16) 22 y-direction. It should be noted at this point that is a unit vector in the Herring and Kittel simply postulate the magnetic field term, —YM)(_H, in the equation of motion of the magne- tization. 13 Elastic Equation of Motion Let ZKL be the part of the Hamiltonian density which involves the lattice strain. The equation of motion of the components of the lattice displacement vector 3 is where [/9 is the mass density of the medium. Equation (12), which is the Hamilton equation of motion for the momentum density /K>)R1/’)t , has its origin in classical field theory. The energy density 7¥L is discussed in the following paragraphs. In addition to the magnetoelastic energy density discussed in the preceeding section and the elastic energy density, the Hamiltonian density, 2%L’ must contain a contribution arising from the interaction between the lattice strain and the conduction electrons. As stated in the introduction, this interaction is assumed to be a consequence of either the deformation potential or the piezoelectric effect. Therefore, WL2fiel+Wme+ fidefpot+ fiéiezo 15 16 where flyél, Jyme’ (ydef pot and (ypiezo refer to the elastic, magnetoelastic, deformation potential and piezoelectric energy densities respectively, and where fiGw is given by equation (10). The general expression for the elastic energy density is -113 7921 ‘ 2 cijkl Sig Skl (18) D ijkl stant electric displacement. where c are the elastic constants measured at con- In a phenomenological theory of the deformation potential, a lattice strain changes the energy of a charge carrier, which may be either an electron or a hole, by5 86 = q V381) Sij (19) where q is the charge, V§3> the deformation potential tensor and Si is the lattice strain. For a medium .1 with N conduction electrons per unit volume, the inter— action energy density becomes ;Qgef pot = (—e) N viJ siJ . (20) 17 Holes are not included in the ensuing theory. If the charge density becomes nonuniform, a net body force on the lattice may occur as a result of the deformation potential. This force can be easily obtained by substi— tuting the expression for 42éef pot into equation (17). The piezoelectric contribution to the Hamiltonian density is obtained from the sum of the elastic and electric energy densities. That sum is given by U=—21——TS+1ED (21) where Tij’ 815’ E1 and D1 are respectively, the stress components, the strain components, the electric field components and the electric displacement field components. 6 Following Mason , the appropriate variables to describe the Hamiltonian density are the strain components 813’ and the electric displacement field components, D1‘ The dependent variables T13 and E1 are then related to the independent variables 3 and Di' In the notation 13 of Mason, these relations are _ D _ .L Tkl " cijkl 813 2477 hnkl Dn (22) — D E = ( 6 1) D — h S (23) m mn n mij ij 18 are the piezoelectric constants, cD are the 13 D iJkl elastic constants, and (6‘1)mn are the inverses of the where h m lattice dielectric constants. After substituting equa— tions (22) and (23) into equation (21), equation (21) becomes _ 1 D 1 -1 D U ’ T 313 cklij Skl + 37 D1( 5 )in Dn 1 _ 751 D1 hm 33k . (24) The piezoelectric contribution to the Hamiltonian density is then identified as fly — __ .l. D h s (25) piezo “ Hi i iJk jk ' Using equations (10), (18), (20) and (25) the ex- pression for 7¥L becomes D -2 — _1_ ”L ‘ 2 Cijkl 313 Skl + Mo bijkl Sij Mk I"11 (25) 1 ‘ Til Di hijk Sjk - e N V13 813 After equation (26) is substituted into equation (17) and all the operations carried out, the following elastic equation of motion is obtained 19 2 2 3 R1 D o) Rk ‘2 3 P3? = Cw 3?— * “obsikl—as‘wl’ 1 3D1 3N _ W hlij 3x3 — eViJ ...de . (27) The elastic equations specified by equation (27) and the analysis of the diapersion relation are considerably sim- plified if the appropriate choice is made for the magneto- elastic, elastic, piezoelectric and deformation potential constants. This choice and the resulting elastic equa- tions are discussed in the remainder of this section. The magnetoelastic constants, bijkl’ for this cal— culation were discussed in the development of the equation of motion of the magnetization and are specified by equation (13). The elastic constants for an elastically isotropic medium are assumed. This assumption implies that the only nonzero elastic constants are c * cD — cD — cD 11 ‘ 1111 ‘ 2222 ‘ 3333 C = CD :3 CD = CD = 12 1122 2211 1133 CD = CD = CD 3311 3322 2233 2O 11 12 C = —— = C = 44 2 2323 cD = cD = cD = 2323 3223 3232 CD = CD : CD = CD — 1331 1313 3113 3131 D _ D _ D _ D 01212 c1221 " c2112 ’ c2112 ‘ (28) For a more convenient notation, define 0( and K? so that 0 il 11 201+”) C 12 3K? - (29) An examination of equation (16) reveals that a spin wave propagating in the z-direction is coupled to only an x—polarized transverse lattice vibration propagating in the z—direction. Therefore, to create an interaction between the spin wave and the drifted conduction electrons, the x-polarized transverse lattice vibration propagating in the z-direction must couple with a conduction electron density variation in the z—direction or an electric displacement field gradient in the z—direction, as can be seen by inspection of equation (27). This model 21 coupling is accomplished by choosing the only nonzero deformation potential and piezoelectric constants to be 13 31 and = = o h h313 h331 (3 ) respectively. For the model described above, equation (27) reduces to 2 ARK _ 2 3 /° 3.2 - “W + (“Wm B) ._ i 3% __ evil + 2bM-2 —3—-(MM ) 47- 2 dz 92 x Z 2 312 y = d 2R + (0(+ 2fl)—3—(V .R) (31) J 2 )7 by t 92R 2 2 9 : + O<+ '— )0 9t? dVRZ ( 2fl)92(v R) __ .2. QDZ _. evi3E 2bm'2-4a—(M M ) The Kinetic Equation The conduction electrons are acted upon by several forces. The Newton's law equation of motion for a con- duction electron is 3!. e "‘Tt‘MX-vm =-e§—-c—1x2 HIV: mv + e Vij $78iJ -— No §7N .. 77 . (32) The left hand side is the total time derivative following an electron. The velocity 1 is treated as a field which is a function of spatial coordinates and time. The terms on the right hand side represent respectively, the force due to the electric field, the Lorentz force, the force due to the deformation potential, the diffusion force, and the momentum relaxation force. The origin of the first two forces and the last is obvious. The form of both the deformation force and the diffusion force is discussed in the following paragraphs. Due to the existence of the deformation potential, the lattice strain may affect the energy of an electron. If the strain is nonuniform, the electron may experience a force which tends to move it to a position of lower potential energy. If, for a conduction electron, the 22 23 energy arising from the deformation potential is given by equation (19), then the equivalent potential is q6 = ViJ 313 . (33) The resulting force on the conduction electron in a non- uniform strain field becomes E =(-e) (—v¢) = eViJVSij. (34) This method of treating the deformation potential force on the conduction electrons has been used in the treat- 17 ment of both ultrasonic amplification and the acousto- electric effect.7’18 To discuss the nature of the diffusion force in equation (32), the particle current density J is assumed to obey a diffusion law % = — DVN BE where D is the diffusion constant and §7N is the electron concentration gradient. If the electron density is N, the effective force acting on the electrons to produce the diffusion flow is defined by 2h 24 1 eD VN = nggd (36) where the same scattering mechanism determines the electrical conductivity as regulates the diffusive flow of electrons under a concentration gradient. From equation (36) the diffusion force is Ed = " jffir' l7N . (37) The Einstein relation D = kBT (mobility) (38) is assumed valid where kB is the Boltzmann constant, T is the temperature of the electron distribution and the mobility is defined as the drift velocity per unit force and in this case is assumed to be P/m. The diffusion force may be rewritten in terms of the temperature II I 1., N VN (39) where the conduction electron density and its gradients satisfy the condition VN << NO (40) 'M electron gass‘ J ;F_velocity distribution, then kBT is a measure“ ' .,: the mean square component of the thermal velocity along some direction. Therefore kBT = mvg (41) ' 2 where VT velocity, and so equation (39) becomes is the mean square component of the thermal £1. . Ed-—No VN. (2) Linearization of the Equations of Motion Equations (16), (31) and (32) are linearized by substituting E ;> £0 + E E0 = 33 E0 2220+2 20= 23B. 1=>10+1 xo= 23vo E €> £210 + E E0 = 33 Ho E=>Mo+fl Mo= 9.3M. 2220+2 20= 23D, N =>NO +N M$R (43) where E, B, X, M, M, 2, and N on the right hand side are perturbations from the dc conditions specified by EC, go, 20’ Mo, Mo, 20 and NO respectively, where 23 is a unit vector in the z-direction. In the linearization process products of perturbed quantities are neglected. In this approximation equations (16), (31) and (32) become 26 27 .5; z—YHOMy + )’MO(Hy + M2 V My) (44) 0 3M $3, : yHoMx - Y 0(Hx + STA—2 V2 Mx) R R + 2bY( (93: + 33:) (“5) 3‘3 /0 913:: wve Rx + (0(4- 2fl):§—x(v 3) D _ 1%. 99: -—e v Ag + 213M;1 —M’E‘ (L16) 32R 2 3 P431? = «(V Ry + (X‘F Qfl )SV-(V'E) (47) SR. \ 0/ n- to II h 3Dz 9N — 3M _ "ET—r 3X -—-eV—§;—(+2bMOl "—3—: (48) Supplementary Equations The fields M, M, E, 2, g, X and N of equations (h4)-(49) must also satisfy Maxwell's equations. Also, the continuity equation for the conduction electrons must be obeyed. Therefore, the following equations must be satisfied 333' VXE = " 713—6—‘6 (51) 3.D VXH : _ 4Ze(NOl+Nl/‘o) +% '5‘;- (52) (NOX+NX0)+ N = o (53) 3t where the ac current density has been written 1 = (-e) (Nox+Nzo) . (54) In addition, the piezoelectric equation of state6 13 (55) mn n mij 29 lattice dielectric tensor. For the model “w;tion equation (55) becomes ' R R .E. = —é—_12 -s3h(%—;‘+ 33:). (55) if the tensor ( 6'1): is assumed to be diagonal with n 1 equal diagonal elements -€— , and the piezoelectric constants are given by equation (30). III The Dispersion Relation Equations (44), (45), (46), (47), (48), (49), (51), (52), (53) and (56) constitute a set of linear and homogeneous partial differential equations for the field variables which describe the postulated magnetic-elastic- electric system. To avoid the necessity of considering the boundary conditions, only oscillations of wavelength A satisfying the condition A<< L will be investigated, where "L" is a characteristic dimension of the medium. Furthermore, since the magnetization and elastic equa- tions of motion are not valid for oscillations of wave- length the order of the interatomic spacing "a", the condition }fi))a must be imposed upon the oscillations under investigation. Under these conditions the normal modes of all the time dependent field quantities may be taken to be plane waves, regardless of the shape of the medium. The set of differential equations , (44), (45), (46), (47), (48), (49), (51), (52), (53) and (561 are reduced to a set of linear and homogeneous algebraic equations for the Fourier components of the fields by assuming 31 of the form exp (ikz-iwt) for M, .‘L’ M, _B_, 2, M and N. The resultant set of algebraic equations is -1 -- 1wa = —YH°My 4404011y — 2mmo 1:214}, — 1am = YH M -— w H + 2AYM'1k2M y o x o x o x + 21 mex _ 2 11m - —/°A?Rx —o< k Rx— T77 Dz — ikeVN + 2ikbMolMx 211 - 1:212 flu y - — o< y .705sz = —o(k2Rz —(O(+2/6)k2Rz (57) (58) (59) (50) (61) m(-iw+ ikvo )vy = — _ _ e m(-ia)+ikvo)vz _ ?Dz + iehka 2 mv mv z —— erznx — 1k T N ‘T No t p. N U ll I [.0- 6 N II Mo 6 I w 1:! lo 4 (64) (65) (56) (57) (68) (69) — 1ka = — L—LZTCN—Oe vy —— 170153, (70) __ 4/ICNoe Vz __ [fire—63b __ 1:.) DZ = o (71) Bx = Hx + 4fiMx (72) 1'5y = Hy + 4TMy (73) A nontrivial solution of these equations exists if the determinant of the coefficients is equal to zero. Setting the determinant equal to zero gives an equation of the form F(k,(x)) = o . (74) Equation (74) is the dispersion relation for the excita— tions of the system. If the following definitions are made w = YHO + 2AYM;l k2 (75) S 02 _ 411Noe P E m D _ 2Y’b2k2 o ._ 1" Mo 4% = 4FVMO 2 _ ac Vt - 7 K2 = (h2 + V2k2) .. 2 4”th _ K2 w2v2k2 __.___E_t_______ 2 2 2 - NY w’+ (78) (79) (80) (81) (82) (83) I i 2 _ ’ 2 95 = W+Tiw§w wwp ,(84) 1 (k2,; _ Q2” wz+ 1— + o) ) + wlw2 e 1‘ B p the dispersion relation becomes (0.) was -c.)M¢+ )(w—ws + wM¢-><“2- 13"? -I”) II c (85) + DO[—2ws+wM(¢++¢_)] The dispersion relation is a 12th degree polynomial equation in aJand a 14th degree polynomial equation in k, with complex coefficients. Therefore, an analytical solution is impossible. The alternative methods of analysis are l) a machine calculation or 2) approximating the frequency of the branch of interest. The second of these alternatives is the approach to be taken. The dispersion relation may be analyzed either by solving for (0 as a function of k or for k as a function of Cd , that is w = mm 01" k = K“(¢J) 37 where x denotes the various modes. The approach taken in the following analysis is to focus attention on the time dependence of the field quantities and analyze the dispersion relation by solving for complex frequency go as a function of real wave vector k. The most general expression for the frequency is therefore w(k) = wr(k) +iA)i(k) where the mode index has been omitted and where 60F and CHE are real functions of k. As a consequence of the choice of time dependence of the field variables, i.e. exp(irk —-i6)t) , the mode either is transient and decays if 6Qi< O or is unstable if 601 > o. IV Analysis of the Dispersion Relation The dispersion relation may be most easily investi— gated under two conditions, namely, 1) D0 = O and 2) “III: e o. The condition D0 = 0 implies that the magnetoelastic coupling constant b is zero. In this case the magnetization does not interact with the lattice strain. Therefore, any change in the spin wave frequency “E is of electromagnetic origin and arises from the ac magnetic field which is a consequence of Maxwell's equations. The electromagnetic effects on the spin wave frequency arise from the torque terms YMOHy and - YMOHx in the magnetization equations of motion, equations (44) and (45). The condition wh¢i = O effectively eliminates the electromagnetic change in the spin wave frequency. In the case of spin waves propagating in the direction of the external constant magnetic field, the condition. QQII = O is equivalent to the magnetostatic approximation. In the magnetostatic approximation, Ampere's and Faraday's laws are replaced by VXM : O and V- M : 0. Therefore, under the condition €JM¢+ :-O in equation (85), the change in 38 39 the spin wave frequency, (1%, is entirely a consequence of the interaction of the magnetization with the lattice strain, and through the lattice strain, with the drifted conduction electrons. The electromagnetic change in the Spin wave frequency always occurs, although for certain wave vector and conduction electron densities it is in- significant. In such regions the dispersion relation may be analyzed for LCM ¢; :70. The next section in this chapter is devoted to the analysis of the dispersion relation under the condition of zero magnetoelastic coupling. The last section in this chapter is devoted to the analysis of the disper- sion relation under the conditions which permit neglect- ing the electromagnetic coupling. Zero Magnetoelastic Coupling For Do 0, the dispersion relation, equation (85), factors into the three equations w+ ag—wM¢+=o . (86) w -— ws + mm (L = o (87) 422 — kevg _ 5) = o (88) Equation (88) describes the interaction between an x-polarized transverse phonon propagating in the z— direction and the drifted conduction electron system moving in the positive z-direction. As shown below, equa- tion (88) describes the interaction between a z-directed transverse phonon and doppler shifted, damped plasma waves. This interpretation becomes apparent when equation (88) is rewritten in the form 40 41 where ‘Ilr and (2L are defined by 2 22 l i = t Q + kv -— +kv _-—-- 0 I): p T 472 O 2? (9) The quantities 'I1r and IQIL are now seen to be solutions to equation (89) when K = O, and represent plasma waves traveling parallel and anti parallel to the positive z-axis respectively. If the following conditions are valid 27‘kvo << 1, (91) then the plasma oscillations are highly damped. The frequency of the transverse phonon propagating in the positive z—direction is approximated by K a)k v &)-kv ‘ p i 2 2 2kvt[(kvt - kvo)(kvt -kvo+ IT) — w —k v] 2 2 2 2 t 42 Equation (92) is obtained by dividing both sides of equation (89) by the factor (7.) +kvt)(w- limo—DI) and then approximating the right hand side of the re- sulting equation by replacing a) by kvt. This approxi— mation technique is valid if the coupling parameter K is small and if the plasma frequencies .Ilr and 'IlL are not close to kvt. The latter condition is fulfilled if the plasma modes are highly damped since .1) and 1" (1L will contain a large imaginary part. In terms of the parameter A \Mflbh.is defined by III <1 0 I H A < d. and is a measure of the ratio of the conduction electron drift velocity to the phonon phase velocity, the real and imaginary parts of the frequency become k2V2 2 Rev: 2 1 + 2 ‘— A 2 Re to: kv 1 _ _K_ “)2. wp (93) t 2 knvé 2 lcv; 2 A‘k‘v€ (1+ 2 _ A 2) + 4 v12 (013 mp 60p / 43 kv t Im z.) = In, 35— P (94) 2 kevg 2 kevg 2 A2k2vE (l + 2 --13 2) + 4 2 mp cop cap 7‘ If Vo/Vt > 1, then A )0 and Imw )0. Therefore, when the conduction electron.drift velocity exceeds the phonon phase velocity, the phonon becomes unstable. This instability corresponds to ultrasonic amplification. It was first seen by Hutson, et a1.,8 and the theory was developed later by White.9 Equations (86) and (87) describe the excitations of a ferromagnetic semiconductor. Both equations are 4th degree polynomials in CO, and therefore could be solved exactly by an algebraic procedure. In the present study only the Spin wave branches of the dispersion relations are of interest. Since the analysis of the spin wave branch of equation (86) is similar to the analy- sis of equation (87), only equation (87) will be inves— tigated. Under the conditions 44 equation (87) describes two modes which are 1) a spin wave propagating in the positive z-direction and 2) a circularly polarized electromagnetic wave propagating in the z-direction.2O To determine the effect of the drifted conduction electrons on the spin wave frequency, equation (87) must be solved for a));£ 0 under the second of the conditions given by equation (95). To make this analysis, equation (87) is rewritten in the form (96) (02 [(w— kvo) + -%— ——C¢)B] — 02(CJ—kvo) (Cd— 6%) = -C%“ -———__———___———_2-2——-._———————2___jf mgw—kvo) (k5c —w2)(w-kvo+7,—LJB) or better, in this form (97) 2 2 (w—kv )(w—c.) — ) [(CJ- (4)8)(02—1520— — wEMM] .402 0 s (“II/I :0, Under the conditions of equation (95) the right hand term of equation (97) vanishes and the left hand term has the solution (4) zw . s Equation (97) is of the form D(k,CJ) - G(k,w) ll 0 Both D(k,CO) and G(k,CJ) may be expanded for frequencies near (J8. This procedure yields the equation 3D 0 = D(k,ws) +(w“‘t)a_qu .... s w w 9G 8) -— G(k, E.,) +( _wS)—3?Jw +~- ' (9 3 But D(k,Cds) = 0 under the second of the conditions given by equation (95), and if all but the first terms of the expansions in equation (98) may be neglected, the Spin wave frequency becomes (4)-“) 23M (99) or Under the additional condition 2 2 r (ws-kvo—wB) << 1 equation (100) becomes _ 0056 2 «2 kvO kvo Cd - (108 + COM( 2k2)(wp( )(1 _ E)(l — ws __ /\ 2 CJSCJM kVo Elwp R202 (1 _ (’0 ) A" O —l: I I 8 the imaginary part of the frequency is _ A 2 (4)3 COM Im O) — €'(Cdp_:F;F7_ZX (101) (102) 47 For [3 ).O the spin wave becomes unstable. This result 3 was recently predicted by Baryakhtar. If the conduction electron drift velocity is zero, A: —l and equation (102) describes eddy current damping of spin waves. Also, under appropriate conditions, the effect of the drifted conduction electrons on both the real and the imaginary part of the spin wave frequency can be made quite small. The conditions already specified are 2 ems 22 << 1 k c wp7<1 ?(ws— lure—mp?“ 1 Thus, by inspection of equations (101) and (102) it is apparent that if Cdg==IaiM and couple p A ("—ZE;§——)( &%)( ) <<< 1 then the electromagnetically caused change in the spin wave frequency is very small. Zero Electromagnetic Coupling The electromagnetic effects on the Spin wave fre- quency Cd; can be eliminated by simply letting QM ¢t = 0 in equation (85). In the previous section the electro- magnetic effects were discussed, and shown to be small under certain conditions. Therefore, under these con- ditions the electromagnetic effects may be justifiably omitted and the dispersion relation reduces to 2 2 2 2 2 (w -—cos)(w —kvt —yz) = - 2DOCJS. (103) Equation (103) describes the interaction of spin waves with the lattice strain and through the lattice strain, with the drifted conduction electrons. Equation (103) may be rewritten [(we—wgmi kevf) — 21:0 cos] (w—QLMw—flr) 2 2 2 2 __ 2 2 — K wpk vt(w —ws) (104) 48 49 where jQr and 'I)L are given by equation (90). Equation (104) is a 6th degree polynomial equation in co , and therefore an approximation technique is necessary. In the limit of zero electromechanical coupling constant K,the dispersion relation factors into the equations 2 2) (we—wifi a)? —k v -—2Do as = o (105) and (w—fler—QL) = 0. (106) Equation (106) is the dispersion relation for the plasma oscillations and equation (105) describes the magneto— elastic excitations. Schlomannl2 has analyzed the magneto- elastic spectrum of a medium with isotropic magnetic, magnetoelastic and elastic properties. It should be noted, however, that equation (105) refers to a medium with the idealized magnetoelastic characteristics given by equation (13). The magnetoelastic frequencies, which are solutions of equation (105), are w=:f21 and w=:n, 50 where The frequencies Ill and 512 are shown plotted in figure 1 for the parameters as = 1010 + .1k2 sec‘1 2 Vt : 109/ cm/Sec /9 = 5.2 gm/cm3 M0 = 140 oersted Y' = 1.76 107 oersted.1 sec-l b2 = 1014/3.87 ergg/cm6 which approximately correspond to those of yttrium iron garnet. Only the region about k = kcr has been plotted since the greatest change from the pure spin wave and pure elastic wave frequencies occurs in that region. Here kcr is the crossover wave vector and is defined as the smaller of the two solutions of the equation 608%”) = kervt . (108) For the parameters listed above. kcr = 109/20m_1. 50 where .1. .Q = l—(w2+kvt)i—:— [(w:—£v§)2+8pows] 2.(107) The frequencies 111 and 112 are shown plotted in figure 1 for the parameters as = 1010 + .1142 sec"1 2 vt = 109/ cm/sec /o = 5.2 gm/cm3 M0 = 140 oersted Y’ = 1.76 107 oersted_1 sec-1 b2 = 1014/3.87 ergg/cm6 which approximately correspond to those of yttrium iron garnet. Only the region about k = kcr has been plotted since the greatest change from the pure spin wave and pure elastic wave frequencies occurs in that region. Here kcr is the crossover wave vector and is defined as the smaller of the two solutions of the equation (4)3(kcr) = kcrvt . (108) For the parameters listed above, kcr = 109/20m_1. 1. 1. 1| 1 O .8 k/k %_L____L_____1__I__I___1 .7 .8 .9 1 1.1 1.2 1.3 CI' Figure l. The magnetoelastic spectrum near the crossover wave vector. 52 In terms of the magnetoelastic frequencies equation (104) may be written (w—nlu w+91>(w—n2)( w+fl2)(w-flr)( w— h.) = Kamikevzwe— 00:) . (109) The spirit of the procedure for solving this dispersion relation is to consider the electromechanical coupling constant as a small parameter and to determine its effect on the unperturbed plasma and magnetoelastic frequencies. As K2 is "turned on", the magnetoelastic modes begin to interact with the plasma modes as a con- sequence of either the piezoelectric effect or the de- formation potential. The effect of this interaction on the magnetoelastic and plasma frequencies is assumed to be small. If after calculating the corrected frequencies, they are found to differ considerably from the unper— turbed frequencies, then the approximation technique may be invalid. Two cases of interest are discussed in the following two sections, A and B. The first is the instability in either I11 or.{22 which is analogous to ultrasonic amplification. The second is a resonant interaction 53 between .{22 and .rlL under the condition that 511.18 doppler shifted into coincidence with .{22 A. Magnetoelastic Wave Amplification For this case the following assumption is made: (6.): +k2v2)4?2<< 1, Plasma oscillations are highly damped under this condi— tion. The imaginary part of both 'IQr and flL is large. I‘ either .f21 or .(22 which are real. If, in addition, the splitting of the magnetoelastic frequencies at the Therefore, I) and _flL cannot be approximately equal to magnetoelastic crossover wave vector is sufficiently large, then..f)1 and .f22 are not degenerate. Under these conditions, the effect of the piezoelectric inter— action on the magnetoelastic frequencies may be determined analogously to the effect of the piezoelectric inter— action on the elastic frequency which was discussed on page 34. As was done in the case of elastic waves, equation (109) is written K2 m2 21:2 Q2 km}? S) ((0—01) =__._.___—p_————S—————— (110) (w + Dlxwe-ngww-npnw -I)L> OI' K2:covak2(w2- we) «0- ()2) = ————————————5———— (111) (nl—0L) K20 2V 2K2 ( [12—02) (0 -n2> = ——————p—:———2—S—— (113) 202M122 {n ><02—- .01. MHz—0L) where I)j_and.{)2 are plotted in figure 1 and given analytically in equation (107). After substituting the appropriate expressions for_{21,and JQ'L and defining the parameters Al’ fll’ (gland A2, fig 62 by the relations 55 kvo £3 kvo '—‘_ = 1 + , -—— : l + A I11 1 572 2 ,\ ’ 2 -4 4 1‘ .021 2 2 2 2 2 2 52 = 602 + k VT , 62 = (JP +1; VT (11“) 1 2 2 .0l .0, equations (112) and (113) become 2 Q2 2 2 2 2 (£0.12) = __Liw_ W1 (115) l 2 2 2 2 2 2fl1(Q1-D2) (A1—51)+/°’1 Al [E (116) A2 The right hand side of equations (115) and (116) give the corrections to the magnetoelastic frequencies.f21and _{22 resulting from the interaction with the drifting conduction electrons. The imaginary part of the frequency of the upper magnetoelastic mode is obtained from equation (115) and is 2 222 2 2 W_‘~>_t_ 53.1.2.1 Jig—m, 2 3 2 2 2 22 2 2 D1(D1‘fl2) (131‘ 51) .../@101 The parameter ‘Al is a measure of the ratio of the con- duction electron drift velocity to the phase velocity of the magnetoelastic wave of frequency .(21. If kv°>'f2f then Al) O and from equation (117), Imw> O. This is Just the condition for an instability in the magneto— elastic wave characterized by the frequency 11 . A similar condition applies to the second branch of the magnetoelastic spectrum with frequency .f22. In figure 2 is plotted the quantity Pl( A1) defined by P1(A1) = fll A1 (118) 2_ 2 + 2 2 < Al 61) 6’1 Al Pl(‘A]) represents that part of In1a) in equation (117) Figure 2. Plot of P1X [1, which measures that part of Im co depending explicitly on electron drift velocity. 58 which depends explicitly on the drift velocity. In plotting figure 2, the following conditions have been assumed: fll >> 1 2 2 31 >> 6’1 —1 The maximum of Pl( A1) is (2 6:) and occurs for A1 = (SQ/fl . Figure 2 indicates that the Im C.) l 1 changes sign at A1 = 0, and therefore a magnetoelastic mode characterized by frequency.121 changes from a transient to an unstable mode. Using A1 = 612/61 in equation (117), the maximum value of the imaginary part of the frequency is 2 2 2 2_ (.32 2V 2 (Imwznax=fl11(T(k—:2—)(——£2%—i2) (1 + k 2t ).1 (119) I21 (I21 422) up The parameter which determines the magnitude of the in- stability is not ImGU) but rather the relative increment of the amplitude of the wave. The relative increment is defined by 59 relative increment = —lfll¥l . (120) Re 4) The maximum relative increment for the mode characterized by 12] becomes 2 2 maximum relative _ K2 k VT _1 increment for 11 "_jf ,A{ (1 + 2 ) (121) l 1 g) p where )(lis defined 2 2 k2vi ( I11 - cg ) ____________§_. . (122) 2 2 1 .(21 (91— 122) In equation (120) the Rea) has been replaced by _I21, rather than using the real part of the frequency given by equation (115). A similar analysis can be made for the mode characterized by .112 with frequency given by equation (116). The maximum relative increment is given by equation (121) where ;(l is replaced by )(2 which is defined X 2 2 2 2 kvt (122—08) __——— ' (123) 2 fl: (hi—:12) ll 60 For the maximum relative increments of the modes characterized by 111 and _{)2 become and respectively. The functions X1 and .X; are shown plotted versus k/kcr in figures 3 and 4. Figure 3 is for k é kcr and figure 4 is for k > kcr' Both figures 3 and 4 are plotted for the values of the parameters listed on page 50. From figures 3 and 4 it is apparent that )(1 falls off very rapidly for decreasing k, reaching, at -6 k/k = ,1 , a value about 10 times the value at or kcr' Similarly, )(2 falls off very rapidly for increas- ing k above kcr’ reaching, at k/kcr : 10, a value about 10-3 times the value at k . Since these two regions or of k for 111 and 112 respectively are the spin wave like portions of the magnetoelastic spectrum, the relative increment of the spin wave amplitude becomes very small for modes away from the crossover wave vector region. til?) No ..vax pom mx moH use fix 3 so 38 m. m. a. m. .m enemas No Ml 62 no . xAx to.“ «X woa new HX woa .Ho no: 5: onswfim m w s w m e m m H JIJIJIJIJIJ|IJ|JIIJI nox\x fix MOH 63 Akhiezer1 suggests a lower limit of 10-2 or 10—3 for the increment of the spin wave amplitude if this effect is to be observed. If K2/4 3.1, then this limit is surpassed for k only slightly away from kcr' From figures 3 and 4 it is apparent that )(l approaches unity for k > kc r and )(2 approaches unity for k ( kcr' In these k regions I11 and _f22 respec- tively describe transverse phonons. The instability then corresponds to ultrasonic amplification. B. Resonant Interaction Betweenfl2 and '[2L‘ For sufficiently large drift velocities, the plasma wave propagating in the negative z—direction with frequency .Illlgiven by equation (90) may be doppler shifted so that the real part of “121. comes into coincidence with 122. This resonant condition results in an interaction between the doppler shifted plasma wave and the magnetoelastic wave. Since in this study we are primarily interested in spin wave phenomena, we shall analyze the consequences of this resonant inter— action only for _f22 and only for k > kcr where _1122 is predominantly spin wave. As in the calculation in part A the electromagnetic effects are neglected by setting (JM¢+ :0. To consider the plasma oscillation as sufficiently long lived to be well defined, the 64 condition must apply to the conduction electron plasma. We now assume that the drift velocity V0 is large enough for IQIL and 112 to have equal real parts, and divide both sides of equation (109) by the comoination 2 2 (co —_Ql)( c0 -nr)(w +n2). In the same spirit as the calculation in part A the right hand side of the resulting equation is then approx- imated by replacing CO by either 1) 2 or '{ZIK This procedure yields a quadratic equation in CJ , which is 2 Q2 2 2 2 K (.0 ) (w— 112—)(60 ”1.) = _______&___,(124) 2 2 2 [22m2 “91491: n?) If the quantity Q(k) is defined 2222 2 2 vak((Z—CJ) Q = ———R—-—-————t 2 S (125) 65 and the parameter (3 introduced to measure the resonance condition, then equation (124) has solutions 1 l 2 (J = -§(_Q +0) 1—2‘ [(flg—HL) +“QJ 2 L .O .02 i , 2 1 =——Q+A-n7—)i——FA+AA ) l+—————— 2 2(122 2 2’92 fl§<—A+ —1—) where A is defined by kv d = ]_ + g +‘A 2 (127) 2 and 6 is defined by 2 2 2 2 Q) + k v g: __P_._—2 T_———212 . (128) .Q 421) Since 4 Q << 1 2 i 2 J72('- A “+Er7rifii) 66 for k >/ k r’ the square root in equation (126) may be c expanded in a binomial series. Retaining only the first two terms in that series yields frequencies i w=_{)2+—g~————2——— (129) (A +——§—2> 4711 2 and (—A— A1 ) w=n2<1+A—2’jfl>—-9— 2’”? -(130) 2 flg<42+ 12 2) 47‘ _(2 Equation (129) gives the frequency of the magnetoelastic wave and equation (130) gives the frequency of the plasma wave. This association is easily determined by setting the electromechanical coupling constant equal to zero. The imaginary part of the frequency of the magneto— elastic wave is 2 2 2 2 A 2 2 2—1 Imwlf k Vt)( mfg?) 02 ((1 + a? flE‘A) -(131) 2 2 2 2 2V2 é— .Q2 Ill-{)2 1 + k T _ 1 2 ’\2 2 Q 4/ w 67 This quantity is positive for k equal to and slightly greater than kcr’ therefore implying an instability. The relative increment of the amplitude of the magneto- elastic wave is 2 2 2 2 (1 +4 7‘ S) A )‘1 Inlé) _ §__ ’\ 2 .02 - 2 (up! X2 ————————-——-—k2v2 % (132) 1 + g — ___2L___ w 47‘? cf 9 p where >(2 is defined by equation (123). For a given k, equation (131) has a gaussian shape centered about A: O. For the parameters listed on page 50 and 11 _ wp = 2.24 10 sec 1 _ 7 VT _ 10 cm/sec kk = .4 / CI‘ 1 10 _ ‘02 = 1.02 10 sec 1 X2 = .01 T :: 10-11 sec the relative increment of the amplitude of the 68 magnetoelastic wave becomes 2 1m6.) : 1; 2.23 2 -1 —2 2 ——2.21 (.o1)(1 +.011A ) (133) and is plotted versus vO/y in figure 5. where v is the ph ph phase velocity of the magnetoelastic wave of frequency 10 _ .{22 = 1.02 10 sec 1. Figure 5 indicates that Ich/Cf22 has a maximum at vo/v h = 49.5. This ratio of the P drift velocity to the phase velocity is determined from the resonance condition which is Dab-4 V (4)2 + k2Vi. 1 ‘79— : 1 + —-—2———-T -- 7—? ' (1314') ph £2 4 T He The maximum increment of the magnetoelastic wave amplitude occurs for A :0, and is 2 A (Immmax = 52- X2 Cap! . (135) .(22 1 +k2V2I2 _ l .2— 42 2 (.J 4 ( (A) .m>m3 mEmmHQ poumfinm pmammop on» Sea; coapompmucfi pcmcommp m 0» map m>w3 ofipmwflmoumcmwe on» mo pamsmpocfi m>HpmHms mnp mo poam .m ossmfim om om ON 00 Om 0: CM ON OH ..IIIIIIIJIIll—lll—Il—IIIHI'In—llllj SQ?\0> o elastic wave of frequency V Discussion of Results Summary Two different spin wave instabilities were discussed in the previous chapter. Both involve an interaction be— tween drifted conduction electrons and a Spin wave which is mediated by lattice strain. Both types of instability occur because the spin wave contains one or more components of elastic strain and can therefore create a longitudinal electric field, given appropriate piezoelectric or defor— mation potential properties. The first type of insta— bility was discussed in Chapter IV in the section entitled "Magnetoelastic Wave Amplification." This instability results from a wave-stream interaction. The electrically active magnetoelastic wave causes periodic perturbations in the charge density of the electron stream. The electric field associated with this periodic disturbance then generates a magnetoelastic wave. The second type of in- stability was discussed in Chapter IV in the section entitled "Resonant Interaction Between .122 and IlL." This instability occurs when the electronic drift velocity is large enough to doppler shift the backward plasma wave 71 72 into resonance with the magnetoelastic wave propagating in the direction of the electronic drift velocity. Under the resonance condition anomalous behavior of the system occurs and energy is transferred from the electrons to the magnetoelastic wave. The magnitude of the instability depends strongly on the magnitude of the elastic strain component in the mag- netoelastic wave since the longitudinal electric field associated with the wave depends on the elastic strain, for a given electromechanical coupling constant. There- fore, the magnitude of the instability of a wave on the spin wave like portion of the magnetoelastic spectrum, and characterized by wave vector k, becomes smaller for wave vectors farther from the crossover wave vector. In addition, for a given wave vector, the magnitude of the instability becomes larger as the magnetoelastic coupling constant becomes larger. Finally, the magnitude of the instability is directly proportional to the square of the electromechanical cou- pling constant. This constant determines the magnitude of the longitudinal electric field created by the strain component in the magnetoelastic wave. Therefore, the larg- er the electromechanical coupling constant, the more elec— trically active the magnetoelastic wave will be, and the stronger the coupling to the conduction electrons will be. Material Considerations The observation of the two lattice strain dependent instabilities requires a material with a rather compli— cated set of properties. The material must be either ferromagnetic or ferrimagnetic, and also possess a strong magnetoelastic interaction. In addition, it must possess a strong deformation potential, or it must be piezo- electric. Finally, it must be a semiconductor in the sense that it contains conduction electrons. All these properties are strongly symmetry dependent in real materials. In the present study, the assumptions regarding the magnetic, elastic, magnetoelastic, piezo- electric and deformation potential constants were made for the sake of simplifying the analysis of the dispersion relation. However, it is anticipated that the essential features of this model will apply to a real material. The relative increment of the amplitude of the mag- netoelastic wave is proportional to the square of the electromechanical coupling constant. If piezoelectricity provides the electromechanical coupling, then the electro- mechanical coupling constant is K 73 74 and if the deformation potential provides the coupling, then the electromechanical coupling constant is 2 2 K2 = (V k 2 .. 2 4 n and is therefore wave vector dependent and so not really a constant. The piezoelectric constant h is about 2x10)4 dyne/statcoulomb. The deformation potential multiplied by the magnitude of the electronic charge is usually about 10 electron volts.19 Therefore, the wave vector for which the deformation potential coupling is equal to the piezoelectric coupling is found from k = 31 V and is 6 k = 10 cm‘l. Hence, to observe an instability in a wave with wave 1 vector less than 106cm- , the suitable material would probably be piezoelectric. One such piezoelectric ferromagnet which may be suitable for the investigation of the instabilities is Ga Fe 0 .21 2-x x 3 10. ll. 12. 13. 14. 15. 16. VI Bibliography A. I. Akhiezer, V. G. Baryakhtar and S. V. 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Mason, Physical Acoustics Vol. III b, Academic Press, New York, N. Y., Chpt. 4, 1965. 75 |}T *9 “'1111(1))gyrgfflngzym)ES