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Interaction of spin waves with the conduction electron current and the ferromagnetic resonance linewidth in metals
Volume 74A, number 6
PHYSICS LETrERS
10 December 1979
INTERACTION OF SPIN WAVES WITH THE CONDUCTION ELECTRON CURRENT AND THE FERROMAGNETIC RESONANCE LINEWIDTH IN METALS 0. OLKHOV and S. SEMIN Institute of Chemical Physics, USSR Academy of Sciences, Moscow, USSR Received 20 April 1979
It is shown that the interaction of spin waves with the conduction electron current does not lead to any new mechanism of linewidth broadening different from the well-known exchange—conductivity mechanism.
In spite of numerous investigations not all physical origins of linewidth broadening mechanisms in ferromagnetic metals are established yet. Such a situation stimulates studies of other possible mechanisms and requires a more careful consideration of the mechanisms reported earlier (see, for example, refs.[1—6]). One of the mechanisms discussed earlier in detail considers the interaction of ferromagnetic spins with the conduction electron current [7—9] The theoretical expressions seem to indicate a substantial contribution of this mechanism to the linewidth but this disagrees with the experimental results [9,10]. Earlier the authors of ref. [11] asserted the equivalence of this and the exchange—conductivity mechamsm [12]. This was deduced from the approximate calculation of the decrement of the spin wave damping (by substituting a lorentzian function instead of the 6-function in the energy conservation law). The equivalence of these mechanisms will be shown here by means of an exact solution of the equation of motion of the magnetization vector, the Maxwell equations and the kinetic equation for the conduction electrons. It will be shown that the apparent difference between the mechanisms is due to an incorrect use of perturbation theory under conditions at which it is invalid and also to an incorrect use of the expression connecting the FMR linewidth with the decrement of the spin wave damping. spin waves with the current has .
been considered earlier in terms ofa hamiltonian formalism using the second-order approximation of quan-
turn statistical perturbation theory [7—9]. But it is the long wave spin vibrations with k 6_i (k is the wave vector, 6 the skin depth) that are excited in the FMR experiments. At not too low temperatures (for a normal skin effect) the electron free path is 1 ~6. Therefore, under these conditions spin waves with A ~ 1 are excited (A is the wave length). But in this case perturbation theory calculations are inapplicable. This situation is the same as in the well-known sound absorption problem: for high frequencies the damping of sound waves is calculated in terms of perturbation theory [131 but for low frequencies the damping is obtained directly from the kinetic equation for phonons [14]. It will be shown that for our problem the spin wave damping is best calculated using the equations of motion, i.e. the equation of motion of the magnetization M, the kinetic equation for the conduction electrons and the Maxwell equations. Note that, as shown in ref. [15] the equation of motion approach and correct perturbation theory calculations of quasiparticle damping are equivalent over wide ranges of frequency h~ ~ k~ T and wave vectors k ~ kt (where k* is the wave vector of the particles involved in the coffisions). We now use the Maxwell equations and the equation of motion ofthe magnetization vectorMin the form ,
,
aM/at ‘y{M X [H+ (21/14) V2MJ } where ‘y is the gyromagnetic ratio, M
0 is the saturation 435
Volume 74A,
number 6
PHYSICS LETTERS
10 December 1979
magnetization, I is the exchange constant. We then use the linearised equations for the deviation m M
of the spin wave damping due to the interaction with the current
M0. Substituting m in the form of a plane wave into the above equation for M andcalculations into the Maxwell equations we obtain by standard [16] the spin
1’k
—
2 =
~“k
~8ire
,
wave spectrum: 60k
ck2
= ~
2 ~“~k 2 = 2
leokfk
—
.
k2
—
Here EF isbethe Fermiinenergy. Fork suit may written the form (see also ref. [11]) 0 <
ik~ 2Q
4~g~~M0k
41rgI2~M k2
—
—
0k 0k~cos ik~
—1
X 1, k >~.‘1 (3a) X 4k1/3ir, k 0 <
(1)
Here k~= 4lrwka(w, k)/c2, o((.~.,k) is the conductivity, 0k the angle between k and the internal static magnetic field H
0, fk
(Aklcok) (1
+ 2cos
==
~‘
=
l61r27MO—~—~fk .
(3b)
The main result is that the expression (3a) for k i~ exactly coincides with the relation which was
obtained in refs. [7—9] in the framework of secondorder which was considered by theperturbation authors as a theory “new” and result in comparison with the exchange—conductivity mechanism. But, as noted
2 0k’
2
“k
Ok) + (Bk/eOk) sin
Ak gp~(H 0+ 21k /M0) + Bk Bk
=
2irgp~MAsin 2
0k’
-
where eok = A~ B~denotes the spin wave energy in nonconducting crystals [16] ~B is the Bohr magneton, g is the spectroscopic splitting factor. Note that expression (1) for the spectrum is the basis of the calculation of the linewidth due to exchange—conductivity [9,12]. We obtain u(w, k) from the kinetic equation for the electrons (using the r-approximation), —
aN~
aN~
aN~ N~ N~ —
—
0
at
+ v~, a~ + eE
ap
=
{exp[(E~
—
r
—
where the electric field E Npçj
above, in the FMR experiments it is the spin waves mal skin the case calculations of with k ~ effect). 1_i thatThis are means excitedthat (in the of the northe FMR linewidth performed in refs. [7,8] in terms of perturbation theory were incorrect because the expression for T’k for k ~ 1_i differs from the correct expression for I’ for k ~ (see eqs. (3a, b)). And this was one of the reasons leading to the illusion of the presence of some new mechanism of linewidth broadening different from the exchange—conductivity mechanism. Besides, another and more serious reason for the
p)IkBT]
erroneous result was the incorrect use of the wellknown relation for the wave vector excited in FMR
‘
[9], 2 k~(4~M k 0/Mi),
exp(k~t ikr) and —
(4)
for the calculation of the linewidth ~.Haccording to
+
is the equilibrium electron distribution. A simple calculation yields
the formula [9] MI (aw/aH)’rk~.
a(w, k) =
Relation (4) results from the equations of motion considered above [9]. Therefore substitution of the correct expression for r~into eqs. (4), (5) should lead only to an identity (as may be seen by using expression (3b) for I’k). The correct interpretation of relation (4) is that it allows obtaining qualitatively the FMR linewidth for a known wave vector of the excited waves. This k is
GO,
wi ~ I. ki
~
1, (2)
3ir/4k1, wr ‘~ ki, ki ~‘ 1 2n-r/m is the static conductivity, n is the Here a0 =density, e electron m the electron mass, I the electron free path. The free electron model is used for the calculations. From eqs. (1) and (2) we obtain for the decrement =
436
.
(5)
Volume 74A, number 6
PHYSICS LETTERS
to be defined from the dispersion relation w = Re wk, where o is the frequency of the rf field and tik is the spin wave spectrum which includes damping due to the considered relaxation mechanism. In our case we obtain for k UShg eq. (1) for ok = $r, H -# 47rMo: k2 m 4nMo(ow/Z)1’2. By substituting this k into relation (4) we obtain the well-known Ament-Rado expression for the linewidth due to exchange-conductivity (or the interaction of the spin waves with the current): AH=4l&+. The above consideration seems to show that the socalled interaction of spin waves with the conduction electron current does not yield a new mechanism of linewidth broadening because the correct theory shows it to coincide exactly with the exchange-conductivity mechanism.
10 December 1979
References [ 11 B. Heinrich, D. Fraitova and V. Kambersky, Phys. Stat. Sol. 23 (1967) 501. [ 21 V. Kambersky, Can. J. Phys. 48 (1970) 2906. 131 V. Karenman and R Range, Phys. Rev. 136 (1972) 2769. [4] 0. Olkhov, Fiz. Tverd. Tela (USSR) 20 (1978) 605. [ 5) 0. Olkhov and S. Semin, Fia. Met. Metaloved. (USSR) 46 (1978) 909. [6] V. Lutovinov, M. Reiaer and M. Savchenko, Proc. 20th Conf. on Low temperature physics (Moscow, 1979). [71 E. Abrahams, Phys. Rev. 98 (1955),387. [ 81 T. Kasuya, Prog. Theor. Phys. (Japan) 12 (1954) 803. [91 E. Turov, Ferromagnitnyi Rezonans, ed. S.V. Vonsovskij (Moscow, 1962) Ch. 5. [lo] Z. Frait and II. MacFaden, Phys. Rev. 1394 (1965) 1173. [ 1 l] V. Kambersky, B. Heimich and D. Fraitova, Phys. Lett. 23 (1966) 26. [12] W. Ament and G. Rado, Phys. Rev. 97 (1955) 1558. [ 131 L. Landau and G. Rumer, Phys. Z. Sowjetunion 11 (1937) 18. [14] A. Akhiezer, Zh. Eksp. Teor. Fii 8 (1938) 1318. [ 151 V. Lutovinov, 0. Olkhov and S. Semin, Zh. Eksp. Teor. Fiz 72 (1977) 2275. [ 161 A. Akhieaer, V. Barlakhtar and S. Peletminscy, The spin waves (Nauka, 1967; Pergamon, 1971).
437
PHYSICS LETrERS
10 December 1979
INTERACTION OF SPIN WAVES WITH THE CONDUCTION ELECTRON CURRENT AND THE FERROMAGNETIC RESONANCE LINEWIDTH IN METALS 0. OLKHOV and S. SEMIN Institute of Chemical Physics, USSR Academy of Sciences, Moscow, USSR Received 20 April 1979
It is shown that the interaction of spin waves with the conduction electron current does not lead to any new mechanism of linewidth broadening different from the well-known exchange—conductivity mechanism.
In spite of numerous investigations not all physical origins of linewidth broadening mechanisms in ferromagnetic metals are established yet. Such a situation stimulates studies of other possible mechanisms and requires a more careful consideration of the mechanisms reported earlier (see, for example, refs.[1—6]). One of the mechanisms discussed earlier in detail considers the interaction of ferromagnetic spins with the conduction electron current [7—9] The theoretical expressions seem to indicate a substantial contribution of this mechanism to the linewidth but this disagrees with the experimental results [9,10]. Earlier the authors of ref. [11] asserted the equivalence of this and the exchange—conductivity mechamsm [12]. This was deduced from the approximate calculation of the decrement of the spin wave damping (by substituting a lorentzian function instead of the 6-function in the energy conservation law). The equivalence of these mechanisms will be shown here by means of an exact solution of the equation of motion of the magnetization vector, the Maxwell equations and the kinetic equation for the conduction electrons. It will be shown that the apparent difference between the mechanisms is due to an incorrect use of perturbation theory under conditions at which it is invalid and also to an incorrect use of the expression connecting the FMR linewidth with the decrement of the spin wave damping. spin waves with the current has .
been considered earlier in terms ofa hamiltonian formalism using the second-order approximation of quan-
turn statistical perturbation theory [7—9]. But it is the long wave spin vibrations with k 6_i (k is the wave vector, 6 the skin depth) that are excited in the FMR experiments. At not too low temperatures (for a normal skin effect) the electron free path is 1 ~6. Therefore, under these conditions spin waves with A ~ 1 are excited (A is the wave length). But in this case perturbation theory calculations are inapplicable. This situation is the same as in the well-known sound absorption problem: for high frequencies the damping of sound waves is calculated in terms of perturbation theory [131 but for low frequencies the damping is obtained directly from the kinetic equation for phonons [14]. It will be shown that for our problem the spin wave damping is best calculated using the equations of motion, i.e. the equation of motion of the magnetization M, the kinetic equation for the conduction electrons and the Maxwell equations. Note that, as shown in ref. [15] the equation of motion approach and correct perturbation theory calculations of quasiparticle damping are equivalent over wide ranges of frequency h~ ~ k~ T and wave vectors k ~ kt (where k* is the wave vector of the particles involved in the coffisions). We now use the Maxwell equations and the equation of motion ofthe magnetization vectorMin the form ,
,
aM/at ‘y{M X [H+ (21/14) V2MJ } where ‘y is the gyromagnetic ratio, M
0 is the saturation 435
Volume 74A,
number 6
PHYSICS LETTERS
10 December 1979
magnetization, I is the exchange constant. We then use the linearised equations for the deviation m M
of the spin wave damping due to the interaction with the current
M0. Substituting m in the form of a plane wave into the above equation for M andcalculations into the Maxwell equations we obtain by standard [16] the spin
1’k
—
2 =
~“k
~8ire
,
wave spectrum: 60k
ck2
= ~
2 ~“~k 2 = 2
leokfk
—
.
k2
—
Here EF isbethe Fermiinenergy. Fork suit may written the form (see also ref. [11]) 0 <
ik~ 2Q
4~g~~M0k
41rgI2~M k2
—
—
0k 0k~cos ik~
—1
X 1, k >~.‘1 (3a) X 4k1/3ir, k 0 <
(1)
Here k~= 4lrwka(w, k)/c2, o((.~.,k) is the conductivity, 0k the angle between k and the internal static magnetic field H
0, fk
(Aklcok) (1
+ 2cos
==
~‘
=
l61r27MO—~—~fk .
(3b)
The main result is that the expression (3a) for k i~ exactly coincides with the relation which was
obtained in refs. [7—9] in the framework of secondorder which was considered by theperturbation authors as a theory “new” and result in comparison with the exchange—conductivity mechanism. But, as noted
2 0k’
2
“k
Ok) + (Bk/eOk) sin
Ak gp~(H 0+ 21k /M0) + Bk Bk
=
2irgp~MAsin 2
0k’
-
where eok = A~ B~denotes the spin wave energy in nonconducting crystals [16] ~B is the Bohr magneton, g is the spectroscopic splitting factor. Note that expression (1) for the spectrum is the basis of the calculation of the linewidth due to exchange—conductivity [9,12]. We obtain u(w, k) from the kinetic equation for the electrons (using the r-approximation), —
aN~
aN~
aN~ N~ N~ —
—
0
at
+ v~, a~ + eE
ap
=
{exp[(E~
—
r
—
where the electric field E Npçj
above, in the FMR experiments it is the spin waves mal skin the case calculations of with k ~ effect). 1_i thatThis are means excitedthat (in the of the northe FMR linewidth performed in refs. [7,8] in terms of perturbation theory were incorrect because the expression for T’k for k ~ 1_i differs from the correct expression for I’ for k ~ (see eqs. (3a, b)). And this was one of the reasons leading to the illusion of the presence of some new mechanism of linewidth broadening different from the exchange—conductivity mechanism. Besides, another and more serious reason for the
p)IkBT]
erroneous result was the incorrect use of the wellknown relation for the wave vector excited in FMR
‘
[9], 2 k~(4~M k 0/Mi),
exp(k~t ikr) and —
(4)
for the calculation of the linewidth ~.Haccording to
+
is the equilibrium electron distribution. A simple calculation yields
the formula [9] MI (aw/aH)’rk~.
a(w, k) =
Relation (4) results from the equations of motion considered above [9]. Therefore substitution of the correct expression for r~into eqs. (4), (5) should lead only to an identity (as may be seen by using expression (3b) for I’k). The correct interpretation of relation (4) is that it allows obtaining qualitatively the FMR linewidth for a known wave vector of the excited waves. This k is
GO,
wi ~ I. ki
~
1, (2)
3ir/4k1, wr ‘~ ki, ki ~‘ 1 2n-r/m is the static conductivity, n is the Here a0 =density, e electron m the electron mass, I the electron free path. The free electron model is used for the calculations. From eqs. (1) and (2) we obtain for the decrement =
436
.
(5)
Volume 74A, number 6
PHYSICS LETTERS
to be defined from the dispersion relation w = Re wk, where o is the frequency of the rf field and tik is the spin wave spectrum which includes damping due to the considered relaxation mechanism. In our case we obtain for k UShg eq. (1) for ok = $r, H -# 47rMo: k2 m 4nMo(ow/Z)1’2. By substituting this k into relation (4) we obtain the well-known Ament-Rado expression for the linewidth due to exchange-conductivity (or the interaction of the spin waves with the current): AH=4l&+. The above consideration seems to show that the socalled interaction of spin waves with the conduction electron current does not yield a new mechanism of linewidth broadening because the correct theory shows it to coincide exactly with the exchange-conductivity mechanism.
10 December 1979
References [ 11 B. Heinrich, D. Fraitova and V. Kambersky, Phys. Stat. Sol. 23 (1967) 501. [ 21 V. Kambersky, Can. J. Phys. 48 (1970) 2906. 131 V. Karenman and R Range, Phys. Rev. 136 (1972) 2769. [4] 0. Olkhov, Fiz. Tverd. Tela (USSR) 20 (1978) 605. [ 5) 0. Olkhov and S. Semin, Fia. Met. Metaloved. (USSR) 46 (1978) 909. [6] V. Lutovinov, M. Reiaer and M. Savchenko, Proc. 20th Conf. on Low temperature physics (Moscow, 1979). [71 E. Abrahams, Phys. Rev. 98 (1955),387. [ 81 T. Kasuya, Prog. Theor. Phys. (Japan) 12 (1954) 803. [91 E. Turov, Ferromagnitnyi Rezonans, ed. S.V. Vonsovskij (Moscow, 1962) Ch. 5. [lo] Z. Frait and II. MacFaden, Phys. Rev. 1394 (1965) 1173. [ 1 l] V. Kambersky, B. Heimich and D. Fraitova, Phys. Lett. 23 (1966) 26. [12] W. Ament and G. Rado, Phys. Rev. 97 (1955) 1558. [ 131 L. Landau and G. Rumer, Phys. Z. Sowjetunion 11 (1937) 18. [14] A. Akhiezer, Zh. Eksp. Teor. Fii 8 (1938) 1318. [ 151 V. Lutovinov, 0. Olkhov and S. Semin, Zh. Eksp. Teor. Fiz 72 (1977) 2275. [ 161 A. Akhieaer, V. Barlakhtar and S. Peletminscy, The spin waves (Nauka, 1967; Pergamon, 1971).
437