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Submitted on 1 Jan 1981
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EFFECT OF PHONONS ON THE SPIN
SUSCEPTIBILITY
D. Fay, J. Appel
To cite this version:
JOURNAL DE PHYSIQUE
CoZZoque C6, suppl6ment au n o 12, Tome 42, d6cembre 1981 page C6-490
EFFECT OF PHONONS ON THE S P I N S U S C E P T I B I L I T Y
D. Fay and J. Appel*
AbteiZung fiir T h e o r e t i s c h e F e s t k c r p e r p h y s i k , U n i t l e r s i t a t Hamburg, Hamburg,
F. R . G .
* I . I n s t i . t u t fiir T h e o r e t i s c h e Physik, Universit2Et Hamburg, Hamburg, F.R.G.
Abstract.- That phonons do not affect the Pauli spin susceptibility seems to be a standard, well-accepted result. Since the recent suggestion by Enz and Matthias that phonons might play a decisive role in the ferromagnetism of ZrZnq, this question has been "reopened". We review both the old and new work on this problem and try to clarify the present situation.
The suggestion by Enz and ~atthiasl that phonons might play an important role in the paramagnetic to ferromagnetic transition in ZrZnZ has led to renewed interest in the question of to what extent phonons affect the spin susceptibility. This is really an old question and has been (briefly) discussed in many books and articles. A good discussion was given by erri in^^ and a detailed model calculation was done by ~ u i n n ~ . Quinn considered the change in energy AE due to a spin polarization induced by an external magnetic field and wrote AE = AEd
+
AEint where AEd is the "displacement" energy caused by transferring spin-down electrons from below the Fermi surface to empty spin-up states above the Fermi surface and AEint is the change in the interaction energy arising from phonon exchange. Here AEd is propor- tional to l/m*,
where m* is the phonon-induced mass enhancement. Quinn showed that these two contributions to AE cancel exactly to leading order in X , i.e., O ( X ) , where A = O(1) is the usual electron-phonon mass renormalization parameter. The correction terms were not estimated and would presumably enter first at orderh (%/EF) % l (m/M) ^ 0.01 where UD is the Debye energy and m and M are the electron and ion masses, respectively. This is in contrast to the O(A) correction proposed by Enz and Matthias.
Fay and ~ppel* considered the diagrammatic expansion in which, using the W A - contact interaction model for the Coulomb part I of the exchange interaction, the susceptibility can be written as
where q s ($,q ) and the first few terns in the expansion of Xphon are shown in 0
Fig. l.
In order to estimate the relative importance of the phonons we would now like to cast Eq. (1) into the simple RPA form X = X /(l
-
IXo) with I = Ic + Iel-ph where X. is the susceptibility of the non-interacting electron system. Clearly, one cannot in general define a constant effective I,I-~~. However, since we areprimarily interested in the effect of phonons on the ferromagnetic transition, it is sufficient to consider lim(q+O)lim(qo+O)X(q)
.
In Ref.4 it was argued that all contributions to Xphon-
X are O(u /E ) or higher, except diagrams (a) and0 D F
(b) of Fig.1. Our procedure is to evaluate these two diagrams for q+O and identify by writing
lel-ph Xphon = .X + Iel-ph~: and making use of the relation Xo(0)=N(EF) f
-
For I el-ph N(E )I << 1, X now has the desired RPA form. In Ref.4 we foundF el-ph
-
I = E(W /E ) A .
el-ph D F (2)
Here E is O(1) but unfortunately diagram (b) has not yet been evaluated accurately enough to determine the sign. The appearance of the small factor ( m /E ) can be
D F
considered as a manifestation of Migdal's theorem as discussed in Ref.4. Our procedure of course really only determines the effect of phonons on the static, uniform susceptibility, i.e., on the Stoner factor. Indeed, only for qo+O does it seem reasonable to neglect the retarded nature of the phonon-induced electron- electron interaction and to simulate the effect by a constant I
el-ph' A consistent treatment at finite q would be much more difficult and has not yet been attempted. We have also neglected spin fluctuation effects which could be important near the ferromagnetic transition. These corrections could be included in our calculation of the Stoner factor in the manner of ~ o r i y a ~ . At finite qo, additional phonon- spin fluctuation interaction effects might occur.
6
Enz has suggested that under certain circumstances (eg., anisotropic Fermi surface, l-dimensional soft phonon) it may be possible to "beat" Migdal's theorem and have
I
% A.
It is of course true that Migdal's theorem is not a generalel-ph
law that applies in all situations; it is well-known, for example, that it does not hold in the limit ($//qO* 0. In the present case however, we have been able to show rather generally, using Ward identities, that the theorem is valid for the limit q /\3(30,
fd\+
0 which is appropriate here.A good test of the importance of phonons is given by the isotope effect on the Curie temperature. Appel and i?ay7 found
C6-492 JOURNAL DE PHYSIQUE
h a s no phonon r e n o r m a l i z a t i o n t o o r d e r X and t h e O ( w /E ) c o r r e c t i o n i s a l r e a d y D F
accounted f o r by 7
el-ph' Thus, assuming X i s independent of M , i n t h e absence of t h e Migdal f a c t o r t h e r e i s no i s o t o p e e f f e c t a t a l l . Another formula f o r a h a s been given by ~ n z * . U n f o r t u n a t e l y , t h e experiments t h a t could d i s t i n g u i s h among t h e t h e o r e t i c a l p o s s i b i l i t i e s a r e q u i t e d i f f i c u l t , a t l e a s t i n zrzn2. 7'9
Recently ~ i m " h a s c a l c u l a t e d I by a d i f f e r e n t method and f i n d s el-ph
-
lel-ph
=
S(w /E D F ) where S = l / ( l - i ) i s t h e S t o n e r f a c t o r and can compensate t h e s m a l l Migdal f a c t o r . We b e l i e v e t h a t K i m h a s c a l c u l a t e d o n l y p a r t ofi
and t h a t ,el-ph i n t h e p a r t he h a s c a l c u l a t e d , t h e f u l l S t o n e r f a c t o r should n o t appear. K i m
c a l c u l a t e s f el-ph by s e p a r a t i n g t h e f r e e energy i n t o an e l e c t r o n p a r t Fel and a phonon p a r t F = ( 1 / 2 ) 5 h w ( a t T=O)
.
K i m ' si
ph 8 g el-ph a r i s e s from t h e change o f t h e screened phonon frequency wq due t o a s p i n p o l a r i z a t i o n . The s e p a r a t i o n i n t o F
e l
and F i s however an a d i a b a t i c approximation2 and does n o t i n c l u d e t h e non- ph
a d i a b a t i c c o n t r i b u t i o n s which e n t e r a t o r d e r (U /E ) . A t t h i s o r d e r one should a l s o D F
i n c l u d e a term F
el-ph which a r i s e s from t h e electron-phonon p a r t of t h e Hamiltonian i n second o r d e r p e r t u r b a t i o n t h e o r y . We doubt t h a t t h e f u l l S t o n e r f a c t o r o c c u r s i n t h e c o n t r i b u t i o n K i m has c a l c u l a t e d s i n c e t h e diagrams which p r o v i d e t h e s c r e e n i n g i n wq a r e bubble diagrams c o n t a i n i n g p a r t i c l e - h o l e p a i r s i n a s i n g l e t s p i n s t a t e w h i l e t h e Stoner f a c t o r a r i s e s from t h e p a r t i c l e - h o l e t r i p l e t s p i n channel. We conclude t h a t o u r diagrammatic approach4 seems p r e f e r a b l e i n t h a t one can i d e n t i f y more c l e a r l y a l l c o n t r i b u t i o n s .
References
'Enz, C. P. and M a t t h i a s , B. T., Science
201,
828 (1978); Z . Phys.=, 129 (1979). 2Herring, C . , i n Magnetism, e d i t e d by G.T. Rado and H. Suhl ( ~ c a d e m i c P r e s s , New York, 1966), Vol.IV, Chap. X I I .3Quinn, J. J., i n The Fermi S u r f a c e , e d i t e d by W. A. Harrison and M. B. Webb (Wiley New York, 1960), p. 58.
4 ~ a y , D. and Appel, J . , Phys. Rev. B
20,
3705 (1979). 'Moriya, T . , Physica ( U t r e c h t )91,
235 (1977).6Enz, C. P., i n S u p e r c o n d u c t i v i t y i n 8- and f-band Metals, e d i t e d by H . Suhl and M. B. Maple ( ~ c a d e m i c P r e s s , New York, 1980) p. 181.
7
Appel, J. and Fay, D., Phys. Rev. B
22,
1461 (1980). 8 ~ n z , C. P - , Phys. Rev. B 2 , 1 4 6 4 (1980).9
