SPIN WAVE RESISTIVITY IN FERROMAGNETIC Tb

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SPIN WAVE RESISTIVITY IN FERROMAGNETIC

Tb

N. Andersen, H. Smith

To cite this version:

JOURNAL DE PHYSIQUE

Colloque C6, supplément au n° 8, Tome 39, août 1978, page C6-824

SPIN WAVE RESISTIVITY IN FERROMAGNETIC Tb

N.H. Andersen and H. Smith

Physios Laboratory I, H.C. 0rsted Institute, University of Copenhagen, Denmark.

Résumé.- Nous présentons et comparons les mesures et les calculs de la résistivité électrique du Tb ferromagnétique dans l'intervalle de température 1,5-50 K.

Abstract.- We present and compare measured and calculated values of the electrical resistivity of ferromagnetic Tb in the temperature range 1.5-50 K.

The interaction between magnons and conduction electrons in a ferromagnetic metal contributes to its electrical resistivity and causes a broadening of the magnon states. We present here theoretical and experimental results for the electrical resis-tivity of ferromagnetic Tb in the temperature region from I.5 to 50 K, and relate the spin wave resisti-vity to the spin disorder resistiresisti-vity of the high-temperature paramagnetic phase. Our model employs realistic magnon energies in the simplest model pos-sible, that of a spherical Fermi surface.

The electron-magnon resistivity was first mea-sured in several rare earth metals by Legvold and his collaborators /l/. Mackintosh / 2 / showed that it exhibits the characteristic low temperature expo-nential behaviour e " , expected from the presen-ce of a magnon energy gap , and suggested that the temperature dependence of the resistivity in a me-tallic ferromagnet with a gap is p <\> T2e "

m

Our variational solution of the Boltzmann equation shows that p at low temperatures is proportional

—A/k T m

to Te B and allows us to make a quantitative

comparison between the measured and calculated ma-gnetic resistivity in a ferromagnet with a gap (for Tb A ^ 19 K)'.

We employ the customary variational approach to the solution of the Boltzmann equation, which describes the competing effect of the electric field and the electron-magnon scattering on the distribution function for the electrons. When ordi-nary impurity scattering dominates the total resis-tivity, the use of an energy independent trial func-tion becomes exact within our model. The spin wave resistivity p then becomes p = m/ne2T, where n = k^3/3ir2 is the number density and the transport time x is given by

r F rd « . OH-/k T

i-MKO) l q

3

d q U a | g - |

2

2_J

( 1 )

T Jo k l J417 q 4sinh2 (Ho^/2k„T)

r q 15 Here 2k represents the maximum wavevector transfer,

r

g*- the electron-magnon coupling constant and J4w+ the q _,. i magnon energy for a given wavevector q.

N(0) = mk /2ir2]d2 is the density of states per spin

at the Fermi surface and the electron energies e, are given by £ = li2k2/2m.

The electron-magnon coupling constant g->- is obtained from the electron-ion exchange interaction H = -A(g-l)6(r-R)s.J, where r(R) is the position coordinate of the conduction electron (ion). s and J are the spin and total angular momentum operators for the electron and the ion respectively (s =±1/2) A is the strength of their interaction, and g the gyromagnetic factor. If we assume the magnon dis-persion to be isotropic, #«•*- = A + K2q2/2m , the low temperature electron-magnon resistivity p be-comes

„ _ _J -A/k

B

T TA ,. „ V + . (2)

m 4 ( J + ] ) kfl 2 A s d

B m

where k^Q = tf2k,2/2m and p , is the constant spin

B m T? o sd r

disorder resistivity. The leading low temperature behaviour is seen to be Te B .

The magnon dispersion in Tb is strongly ani-sotropic, differing considerably in the direction of the c-axis and in the basal plane. The observed magnon energies are well presented by

JlaH- = A + aq2 + b q2 + cq (3)

1

z

J. _L

with a = 20 meV A2, b = 16 meV A2, c = 10 meV A

and A = 1.65 meV. We have calculated numerically the electron-magnon resistivity from equations (1) and (3), the result being exhibited in figure 1

(dashed line)/ In the calculation we have included

5 10 15 20 25 30 35 LO

65

50

TEMPERATURE T

( K l

Fig. 1 : Measured and calculated resistivity of Tb as a function of temperature. The filled ( 0 ) and open

(0) circles represent measurements with the current along the a- and c-axis.

the effect on of the additional Bogoliubov

9

transformation necessaryto remove the part of the crystal field and magnetoelastic interaction, which is not diagonal in the Holstein-Primakoff operators. The measured resistivity obtained by a standard four-point method has been plotted in figure 1 in the temperature region 1.5-50 K. To compare the da-

ta with our theoretical result for p based on

m

equations (1) and (3) we must also consider the

meters. Physically the small value of kF may be un-

derstood in a rough sense as a measure of the ave-

rage radius of curvature of those pieces of Fermi surface which contribute to the conduction. At low temperatures small angle scattering is dominant, and it becomes crucial for a quantitative estimate to

know the distortion of t h ~ free electron surface.at

zone boundaries, that is the details of the Fermi surface. There is very little experimental informa-

electron-phonon contribution to the resistivity. tion available on the Fermi surface of Tb, but dl

With the electron-phonon matrix element g? given Haasvan Alphen measurements on the neighbouring

q

~~

element Gd

/4/

indicate that our effective kF-value

-

X A,

where X is a numerical constant

by

-

2N(O) is reasonable.

of brder unity, one obtains from equation(1) the We have performed a similar calculation of

well known

h loch-Griilieisen

result the temperature independent magnon-electron lifeti-

me and related it to the high temperature spin di- 1

"

k ~ O T

-

=

-A-(-)

J5 (20/T) (4) sorder resistivity. The order of magnitude of the

T 2

M

0 calculated lifetime agrees well with the observed

e-P

with temperature defined from low temperature magnon broadening 151- A more de-

the sound velocity c as kBO =

Hk~'s

.

The theoreti-

tailed account of our results will appear elsewhere

/6/. References

cal curve (solid line) in figure 1 represents the

_-

sum of the electron-phonon resistivity = (m/ne2) / I / Legvold,S., Magnetic Properties of Rare Earth Metals

P Edited by R.J. Elliott (Plenum Press) 1972 p.335

obtained from equation(4) and the spin wave

/2/ Mackintosh,A.R., Phys. Lett.

5

(1963) 140

contribution p The parameters

%

and

A

are chosen

m ' /3/ De Gennes, P.G. and Friedel,J., J. Phys. Chem.

to be 0.37

i-'

and 0.155 respectively, the longitu- Solids ( 1 958)

71

dinal sound velocity being equal to 3 x

lo5

cm/s. _{/ 4 /} Mattocks,P.G. and Young,R.C., J. Phys.

5

(1977)

The agreement between the calculated and measured 1219

resistivity is remarkably good, even considering /5/ Mackintosh,A.R. and Bjerrum Mdller,H.,

to be published the fact that we have used kF and X as free para-