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Submitted on 1 Jan 1978
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ON THE THEORY OF THE MAGNETIC KAPITZA
CONDUCTANCE BETWEEN LIQUID He3 AND AN
ORDERED FERROMAGNET
M. Cottam
To cite this version:
JOURNAL DE PHYSIQUE Collogue C6, supplement au n° 8, Tome 39, aout 1978, page C6-277
ON THE THEORY OF THE MAGNETIC KAPITZA CONDUCTANCE BETWEEN L I Q U I D H e3 AND AN ORDERED FERROMAGNET
M.G. Cottam
Physios Department, University of Essex, Colchester, England.
Résumé.- Nous présentons des calculs de la conductance deKapitza magnétique entre le liquide He3
et un ferromagnétique de Heisenberg ordonné. Les spins électroniques du ferromagnétique sont
cou-plés aux spins nucléaires de He3 par une interaction dipole-dipole. Le transfert d'énergie entre
les deux systèmes peut être relié aux propriétés d'ondes de spin du ferromagnétique. Nous avons évalué les contributions dues aux ondes de spin de surface et de volume.
Abstract.- Calculations are presented for the magnetic Kapitza conductance between liquid He and an ordered Heisenberg ferromagnet. The electronic spins in the ferromagnet are couplet by
dipole-dipole interactions to the nuclear spins of the He3. The transfer of energy between tht two
sys-tems can be related to the spin wave properties of the ferromagnet, and we evaluate contributions arising from surface spin waves as well as from the bulk spin waves.
We present calculations for the magnetic con-tributions to the Kapitza conductance (thermal boun-dary conductance) across the interface between liquid
He3 and a ferromagnetic solid at T « T , where T
c c is the Curie temperature of the ferromagnet. The first theoretical explanation fo the magnetic Kapitza con-ductance hj, was given by Leggett and Vuorio /l/, who assumed an isotropic contact interaction coupling the two systems. More recently Mills and Beal-Monod /2,3/ have carried out calculations for h for different types of magnetic solids and for various temperature ranges by adopting a more realistic dipole-dipole
in-teraction between the He3 nuclei and the electronic
spins in the solid. In particular for a ferromagnet at T << T they predicted that lu. should be proportional
to T3/2 in the case of zero applied field /3/. The
present work involves extending their calculations by treating the spin wave properties of the ferromagnet near to the interface in more detail. We include con-tributions to n„ which may arise from localised sur-face spin waves as well as from the bulk spin waves as previously /3/. Only an outline of the results is given here; a fuller account will be published else-where /4/.
We adopt a similar formalism to that of Mills
and Beal-Monod J2,3/. This involves evaluating the
dynamical correlation functions (or the corresponding Green functions) between a pair of electronic spin operators in the ferromagnet. By trking a convolution of this with a similar correlation function between the nuclear spins in the He , an expression for h is
K obtained. A slight generalisation of section II of /2/'
where we have a Cartesian co-ordinate system with the z-axis perpendicular to the plane interface (at
z = 0) between the He3 and the ferromagnet. The He3
spins £ occupy the upper half-plane (z > 0) and the electronic spins j3 are in the lower half-plane
( z < 0 ) . Also £ = (q q„) is a two-dimensional wavec-tor parallel to the surface and 0) is a frequency
label. MOOI(q^to) and til ,(qi-w) are related to
spin-PP — ctct — spin correlation functions by
where a^a',6^6' denote Cartesian components. The functions f (q) arise in taking a two-dimensional
otps —
Fourier transform at £ of the dipole-dipole cou-pling between the nuclear spins of moment u and the
n electronic spins of moment p . They are defined by
fzz= -1' fz vT fuz " " ^ ' V <*V/q2 (with „.
v equal to x or y ) . The correlation functions appea-ring in equations (2) and (3) can be conveniently calculated in terms of their corresponding Green functions by use of the fluctuation-dissipation theorem /5/, chich relates the correlation function to the imaginary part of the Green function
p l i e d by a thermal f a c t o r .
For t h e temperature range 3$T,$100 mK t h e ~ e ~ system can be t r e a t e d a s a normal Fermi f l u i d , and on t h i s b a s i s t h e q u a n t i t y mm, (q.-w) h a s been e v a l u a t e d by M i l l s and Beal-Monod 121. On employing t h e i r r e s u l t f o r mm,(q.-w) i t may be v e r i f i e d u s i n g e q u a t i o n s (1)
-
(3) t h a t\
may be w r i t t e n i n terms of t h e Green f u c t i o n < < s + ( z ) ; S - ( z l ) > > T h i s Greenq.w'
function f o r a semi-inf i n h e ~ e i s e n b e r g ferromagnet h a s r e c e n t l y been e v a l u a t e d by Cottam 161, and t h e r e s u l t i n t h e c a s e of a simple c u b i c s t r u c t u r e (with l a t t i c e parameter a ) may be expressed a s
where we assume lqal << 1 ( s i n c e t h e i n t e g r a l i n q u a t i o n (1) i s dominated by t h e behaviour a t small
q ) . The parameters X and Q depend on t h e model em- ployed t o d e s c r i b e t h e s u r f a c e p r o p e r t i e s of t h e f e r - romagnet. We d i s c u s s h e r e a simple c a s e where t h e exchange i n t e r a c t i o n s couple only n e a r e s t neighbours having t h e v a l u e Js i f both s p i n s a r e i n t h e s u r f a c e o t h e r w i s e , whereu- pon =
-
S J (5) f o r s p i n S, and QZ i s r e l a t e d t o t h e frequencyw
by w = gv H B+
S J ~ ~ ( ~ ~ + Q:) (6) H and Ha denote t h e a p p l i e d magnetic f i e l d and t h e s u r f a c e l a y e r a n i s o t r o p y f i e l d (the pinning f i e l d ) r e s p e c t i v e l y .A s d i s c u s s e d i n /6/ t h e bulk s p i n waves cor- respond t o QZ t a k i n g t h e v a l u e s of a r e a l wavevector component i n t h e z - d i r e c t i o n , whereupon e q u a t i o n (6) g i v e s t h e bulk s p i n wave d i s p e r s i o n r e l a t i o n . The v a n i s h i n g o f thedenomitor i n t h e second term of equa- t i o n ( 4 ) o c c u r s f o r iQZa=X, and s u b s t i t u t i o n of t h i s i n t o e q u a t i o n (6) produces t h e d i s p e r s i o n r e l a t i o n f o r a c o u s t i c s u r f a c e sp'in waves.Equation(4)thus c o n - t a i n s a d e s c r i p t i o n of both s u r f a c e and b u l k s p i n waves. The f i r s t term i n e q u a t i o n (4) depends on
12-2'
I
and may b e i d e n t i f i e d a s t h e Green f u n c t i o n f o r t h e i n f i n i t e c r y s t a l , w h i l s t t h e second term, which depends on ( Z + Z t ) , c o n t a i n s t h e e f f e c t s due t ot h e s u r f a c e . The c a l c u l a t i o n s of M i l l s and Beal-Monod
/3/ f o r a n ordered ferromagnet a r e e q u i v a l e n t t o in- c l u d i n g j u s t t h e bulk s p i n waves, t o g e t h e r with a zero s l o p e boundary c o n d i t i o n f o r t h e bulk s p i n waves.
On employing e q u a t i o n (4) we e v e n t u a l l y f i n d f o r t h e Kapitza conductance
\
( i n t h e c a s e of zero ap- p l i e d magnetic f i e l d )hK = b Z 3 l 2 + b l ~ 3 / 2 + b2(T) (7) T h i s i s t h e form a p p r o p r i a t e t o t h e l i m i t i n g s i t u a - t i o n s of e i t h e r v e r y small pinning f i e l d ( g p H<<k T)
B a B o r l a r g e pinning f i e l d (gpBHa>> %T). The c o n t r i b u - t i o n boT 3 1 2 , coming from t h e f i r s t term i n equation ( 4 ) , i s equal t o 314 of t h e r e s u l t i n / 3 / , and t h e a d d i t i o n a l two c o n t r i b u t i o n s come from t h e f i n a l term i n e q u a t i o n ( 4 ) . One o f t h e s e c o n t r i b u t i o n s i s a l s o p r o p o r t i o n a l t o T ~ / ~ , and a r i s e s from t h e mo- d i f i c a t i o n of t h e bulk s p i n wave s p e c t r a l i n t e n s i t y c l o s e t o t h e s u r f a c e . The c o e f f i c i e n t bl h a s t h e va-
1 1
l u e s -b and
-
-b i n t h e l i m i t s of small and l a r g e 3 0 3 0pinning f i e l d s r e s p e c t i v e l y . T h i s change of s i g n of b, between t h e two l i m i t s i s a s s o c i a t e d w i t h a si- m i l a r change i n s i g n of t h e amplitude c o e f f i c i e n t of r e f l e c t i o n f o r a bulk s p i n wavemeeting t h e s u r f a c e . The f i n a l term bp(T) i n e q u a t i o n (7) comes from t h e s u r f a c e s p i n waves, and i s p r o p o r t i o n a l t o T~ i n t h e l i m i t of small pinning f i e l d . For t h i s c a s e we e s t i - mate t h a t , e x c e p t i n a v e r y low temperature l i m i t , t h e T2 term due t o s u r f a c e s p i n waves provides a n important c o r r e c t i o n t o t h e o v e r a l l T'/Z dependence due t o t h e bulk s p i n waves. For l a r g e r v a l u e s of t h e pinning f i e l d t h e c o n t r i b u t i o n b2(T) i s reduced i n magnitude. I f t h e a p p l i e d f i e l d H i s non-zero, t h e r e s u l t s f o r
\
a r e more complicated, but we a g a i n f i n d c a s e s where t h e s u r f a c e s p i n wave c o n t r i b u t i o n t o\
p r o v i d e s a n important c o r r e c t i o n t o t h e b u l k s p i n wave c o n t r i b u t i o n , even i f t h e pinning f i e l d i s non-zero.I n / 4 / we g i v e f u r t h e r d e t a i l s , including the- r e s u l t s i n t h e c a s e of i n t e r m e d i a t e pinning f i e l d s and f o r non-zero a p p l i e d f i e l d s . Also o t h e r models d e s c r i b i n g t h e s u r f a c e c h a r a c t e r i s t i c s of t h e f e r r o - magnet a r e c o n s i d e r e d .
References
/ l / Legget, A . J . and Vuorio, M., J.Low Temp.Phys. 3 (1970) 359.
-
/ 2 / M i l l s , D.L. and Beal-Monod, M.T.,Phys.Rev.A
2
(1974) 343./ 3 / M i l l s , D.L. and Beal-Monod, M.T.,Phys.Rev.A
2
(1974) 2473./4/ Cottam, M.G., to be published.
/ 5 / Zubarev, N . , Sov.Phys. Uspekhi
