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Magnetostatic Spin Waves
Kirk T. McDonald Joseph Henry Laboratories, Princeton University, Princeton, NJ 08544 (September 15, 2002)
1 Problem
Magnetostatics can be defined as the regime in which the magnetic fields B and H have no time dependence, and “of course” the electric fields D and E have no time dependence either. In this case, the divergence of the fourth Maxwell equation,
∇ × H =
4 π c
Jfree +
c
∂D
∂t
(in Gaussian units) implies that, ∇ · Jfree = 0, (2)
i.e., that the free currents flow in closed loops. Likewise, the time derivative of the fourth Maxwell equation implies that Jfree has no time dependence in magnetostatics. Often, magnetostatics is taken to be the situation in which ∇ · Jfree = 0 and D, E and Jfree have no time dependence, without explicit assumption that B and H also have no time dependence. Discuss the possibility of waves of B and H, consistent with the latter definition of magnetostatics [1]. Consider two specific examples of “magnetostatic” waves in which Jfree = 0:
- Ferromagnetic spin waves in a medium subject to zero external field, but which has a uniform static magnetization that is large compared to that of the wave. That is, M = M 0 ˆz + mei(k·r−ωt), where m M 0. Here, the quantum-mechanical exchange interaction is the dominant self interaction of the wave, which leads to an effective magnetic field in the sample given by Beff = α∇^2 m, where α is a constant of the medium.
- Waves in a ferrite cylinder in a uniform external magnetic field parallel to its axis, supposing the spatial variation of the wave is slight, so the exchange interaction may be ignored. Again, the time-depedendent part of the magnetization is assumed small compared to the static part. Show that the waves consist of transverse, magnetostatic fields that rotate with a “resonant” angular velocity about the axis.
In practice, the spin waves are usually excited by an external rf field, which is to be neglected here.
2 Solution
2.1 General Remarks
In both definitions of magnetostatics the electric field E has no time dependence, ∂E/∂t = 0, so the magnetic field B obeys ∂^2 B/∂t^2 = 0, as follows on taking the time derivative of
Faraday’s law,
∇ × E = −
c
∂B
∂t
(in Gaussian units). In principle, this is consistent with a magnetic field that varies linearly with time, B(r, t) = B 0 (r) + B 1 (r)t. However, this leads to arbitrarily large magnetic fields at early and late times, and is excluded on physical grounds. Hence, any magnetic field B that coexists with only static electric fields is also static. There remains the possibility of a “magnetostatic wave” in a magnetic medium that involves the magnetic field Hwave and magnetization density Mwave that are related by,
0 = Bwave = Hwave + 4πMwave. (4)
If there are no free currents in the medium, and any electric field is static, then the fourth Maxwell equation is simply, ∇ × H = 0, (5)
which defines a subset of magnetostatic phenomena.
2.2 Ferromagnetic Spin Waves
Consider a ferromagnetic material that consists of a single macroscopic domain with magne- tization density M = M 0 ˆz + m(r, t), where M 0 is constant and m M 0. We suppose there are no external electromagnetic fields. Associated with the magnetization density M are magnetic fields B and H whose values depend on the geometry of the sample. We suppose that the weak time-dependent magnetic fields due to m lead to even weaker time-dependent electric fields, such that the situation is essentially magnetostatic. The consistency of this assumption will be confirmed at the end of the analysis. The ferromagnetism is due to electron spins, whose dominant interaction is the quantum mechanical exchange interaction, in the absence of external fields. For a weak perturbation m of the magnetization, the exchange interaction preserves the magnitude of the magnetization, so its time evolution has the form of a precession [2],
dM dt
= Ω × M. (6)
As this is the same form as the precession of a magnetic moment in an external magnetic field [3], the precession vector Ω is often written as a gyromagnetic factor Γ = e/ 2 mec ≈ 107 Hz/gauss times an effective magnetic field Beff (or Heff ). Here, e > 0 and me are the charge and mass of the electron, and c is the speed of light. For a weak perturbation in an isotropic medium [2], Beff = α∇^2 m, (7)
where α is a constant of the medium. Then, the equation of motion of the magnetization m is,
dm dt
= αΓ∇^2 m × M. (8)
between 1 for a disk and 0 for a cylinder. The perturbation m exists only inside the sample, but the corresponding perturbations b and h exist outside the sample as well. Inserting eqs. (14) and (15) in the equation of motion (13), we keep only the first-order terms to find, − i ω m = Γ ˆz × (M 0 h − Hz m), (16)
whose components are,
mx = i
ω
(M 0 hy − Hz my),
my = −i
ω
(M 0 , hx − Hz mx), (17) mz = 0. (18)
We solve for m in terms of h as,
mx = αhx − iβhy , my = iβhx + αhy , (19)
where,
α =
Γ^2 Hz M 0 Γ^2 H z^2 − ω^2
, β =
ΓM 0 ω Γ^2 H z^2 − ω^2
For later use, we note that in cylindrical coordinates, (r, θ, z), eq. (19) becomes,
mr = αhr − iβhθ, mθ = iβhr + αhθ. (21)
As we are working in the magnetostatic limit (12), we also have that,
∇ · b = ∇ · (h + 4πm) = 0, ∇ × h = 0. (22)
Hence, the perturbation h can be derived from a scalar potential,
h = −∇φ, (23)
and so, ∇^2 φ = 4π∇ · m. (24)
Outside the sample the potential obeys Laplace’s equation,
∇^2 φ = 0 (outside), (25)
while inside the sample we find, using eq. (19),
(1 + 4πα)
∂^2 φ ∂x^2
∂^2 φ ∂y^2
∂^2 φ ∂z^2
= 0 (inside). (26)
The case of an oblate or prolate spheroid with axis along the external field has been solved with great virtuosity by Walker [6], following the realization that higher-order modes deserved discussion [7]. Here, we content ourselves with the much simpler case of a long cylinder whose axis is along the external field, for which the lowest-order spatial mode was first discussed by Kittel [8]. We consider only the case of waves with no spatial dependence along the axis of the cylinder. With these restrictions, both eqs. (25) and (26) reduce to Laplace’s equation in two dimensions. We can now work in a cylindrical coordinate system (r, θ, z), where appropriate 2-D solutions to Laplace’s equation have the form,
φ(r < a, θ) =
∑ (^) rn an^
(An einθ^ + Bn e−inθ^ ), (27)
φ(r > a, θ) =
∑ (^) an rn^
(An einθ^ + Bn e−inθ^ ), (28)
which is finite at r = 0 and ∞, has period 2π in θ, and is continuous at the boundary r = a. The boundary conditions at r = a in the magnetostatic limit (22) are that br and hθ are continuous. The latter condition is already satisfied, since hθ = −(1/r) ∂φ/∂θ. We note that, br = hr + 4πmr = (1 + 4πα)hr − 4 πiβhθ , (29)
recalling eq. (21). Using eqs. (27) and (28), we find that continuity of br at r = a requires,
∑ (^) n a
[
(1 + 2πα + 2πβ)An einθ^ + (1 + 2πα − 2 πβ)Bn e−inθ^
]
Nontrivial solutions are possible only if 2π(α ± β) = −1, in either of which case there is an infinite set of modes that are degenerate in frequency. Using eq. (20), we find the “resonance” frequency to be, ω = ±Γ(H 0 + 2πM 0 ), (31)
noting that for a cylinder the demagnetization factor is Nz = 0, so that Hz = H 0 , as is readily deduced by elementary arguments. Since we consider frequency ω to be positive, we see that the two solutions (31) correspond to two signs of H 0 , and are essentially identical. For spheroidal samples, the modes are enumerated with two integer indices, and are not all degenerate in frequency, as discussed in [6]. We close our discussion by showing that the electric field of the wave is much smaller than the magnetic field. The scalar potential for mode n is,
φn(r < a) =
rn an^
ei(nθ−ωt), φn(r > a) =
an rn^
ei(nθ−ωt). (32)
We see that for n > 0 the potential rotates with angular velocity Ωn = ω/n about the z-axis. The potential is maximal at r = a, so consistency with special relativity requires that,
v(r = a) =
aω n
c, (33)
which appears to have been (barely) satisfied in typical experiments [8]. We also see that for high mode number the spatial variation of the wave becomes rapid, and the neglect of the exchange interaction is no longer justified.
References
[1] The regime in which electric fields have time dependence, but magnetic fields do not, is explored in K.T. McDonald, An Electrostatic Wave (July 28, 2002), http://physics.princeton.edu/~mcdonald/examples/bernstein.pdf
[2] See, for example, sec. 69 of E.M. Lifshitz and L.P. Pitaevskii, Statistical Physics, Part 2 (Butterworth Heinemann, 1996).
[3] For an example of this phenomenon, see K.T. McDonald, Wave Amplification in a Magnetic Medium (May 1, 1979), http://physics.princeton.edu/~mcdonald/examples/magnetic_waves.pdf
[4] F. Bloch, Zur Theorie des Ferromagnetismus, Z. Phys. 61 , 206 (1930), http://physics.princeton.edu/~mcdonald/examples/QM/bloch_zp_61_206_30.pdf
[5] The magnetic field H (and B = H− 4 πM) inside a spheroid with uniform magnetization M = M 0 ˆz along its axis can be deduced from chap. 5, probs. 80 and 82 of W.R. Smythe, Static and Dynamic Electricity, 3rd^ ed. (McGraw-Hill, 1968). If we denote the ratio of the axial length of the spheroid to its diameter by c, then c = 0 is a disk, 0 < c < 1 is an oblate spheriod, c = 1 is a sphere, 1 < c < ∞ is a prolate spheroid, and c = ∞ is a cylinder. For an oblate spheroid of aspect ratio c, the “radial” coordinate is ς = c/
1 − c^2 , and the magnetic field due to the uniform magnetization is/
H = − 4 πM
1 − ς
[
(1 + ς^2 ) cot−^1 ς − ς
]}
For example, a disk with c = 0 has ς = 0 also, and H = − 4 πM, B = 0. For a sphere, c = 1 and ς → ∞, in which limit cot−^1 ς → 1 /ς − 1 / 3 ς^3 , so that H = − 4 πM/ 3 and B = 8πM/3. For a prolate spheroid of aspect ratio c, the “radial” coordinate is η = c/
c^2 − 1, and,
H = − 4 πM
1 − η
[
(1 − η^2 ) coth−^1 η + η
]}
For a cylinder with c → ∞, we have η = 1, coth−^1 η = 0, and H = 0, B = 4πM. The fields for a sphere can also be obtained from the limit c → 1, η → ∞ and coth−^1 η → 1 /η + 1/ 3 η^3. The expressions in braces in eqs. (44) and (45) correspond to the demagnetization factor Nz introduced in the main text.
[6] L.R. Walker, Magnetostatic Modes in Ferromagnetic Resonance, Phys. Rev. 105 , 390 (1957), http://physics.princeton.edu/~mcdonald/examples/EM/walker_pr_105_390_57.pdf Ferromagnetic Resonance: Line Structures, J. Appl. Phys. 29 , 318 (1958), http://physics.princeton.edu/~mcdonald/examples/EM/walker_jap_29_318_58.pdf
[7] J.E. Mercereau and R.P. Feynman, Physical Conditions for Ferromagnetic Resonance, Phys. Rev. 104 , 63 (1956), http://physics.princeton.edu/~mcdonald/examples/EM/mercereau_pr_104_63_56.pdf
[8] C. Kittel, On the Theory of Ferromagnetic Resonance Absorption, Phys. Rev. 73 , 155 (1948), http://physics.princeton.edu/~mcdonald/examples/EM/kittel_pr_73_155_48.pdf
[9] R. Becker, Electromagnetic Fields and Interactions (Dover Publications, 1982), sec. 87, vol. 1.
