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Impurity states and anomalous density of states in dilute alloys
Volume 25A, number 3
P HYSI C S L E T T E R S
c u r v e s s e p a r a t e , but t h i s m a y be due to c h a n g e s in the F e r m i s u r f a c e o v e r l a p s b e t w e e n B r i U o u i n z o n e s such as t h o s e found by M e r r i a m et al. [10, 11] in t h e i r w o r k on the t r a n s i t i o n t e m p e r a ture. We c o n c l e d e that in the alloy r e g i o n w h e r e no sudden cbm~,es o c c u r in the e l e c t r o n i c s t r u c t u r e , K o b e y s t&e e x p r e s s i o n (1) to b e t t e r than 2% in s pit e of wide v a r i a t i o n s in f r e e path d i s t r i b u t i o n . We find s i m i l a r r e s u l t s at a r a n g e of l o w e r t e m peratures. T h i s r e s e a r c h was s u p p o r t e d by a g r a n t f r o m the S c h w e i z e r i s c h e K o m m i s s i o n z u r F o r d e r u n g der wissenschaftlichen Forschung.
IMPURITY
STATES
14 August 1967
i. L.P.Gorkov, Soviet. Phys. J E T P 10 (1960) 998. 2. K. Maki, Physics 1 (1964) 21. 3. C. Caroli, M. Cyrot and R. G. de Gennes, Solid State Commun. 4 (1966) 17. 4. G. Eflenberger, Phys. Rev. 153 (1967) 584. 5. W. van der Mark, J. L. Olsen and F. B. Rasmussen, Proc. 10th Int. Conf. on Low temperature physics, (Moscow 1966), to be published. 6. C. Chlou, R.A. Connell and D. P. Seraphim, Phys. Rev. 129 (1963) 1070. 7. K. Noto, Y. Mute and T. Fulmroi, J. Phys. Soc. Japan 21 (1966) 9.22. 8. G. BrRndli, P. Cotti, E.M. Fryer and J. L. Olsen, Proc. 9th Int. Conf. on Low temperature physics, B, (Plenum Press, 1965) p. 827. 9. J. L. Harden and V. Arp, Cryogenics 3 (1963) 105. 10. M. F. Merriam and M.von Herzen, Phys. Rev. 131 (1968) 637. 11. M. F. Merriam, J. Hagen and H. L. Luo, Phys. Rev. 154 (1967) 424.
AND ANOMALOUS IN DILUTE ALLOYS
DENSITY
OF
STATES
B. N. GANGULY
Department of Physics, University of Toronto, Toronto, Canada Received 3 July 1967
The question of anomalous density of states in some dilute alloys is examined by taking due cognition of the localized impurity states.
There have been a considerable amount of experimental and theoretical studies of the a n o m a lous behaviour of dilute magnetic alloys at low temperature [1]. Recently, Kondo [2] has put forw a r d a theory based on the dynamical character of s-d exchange interaction which gives a satisfactory explanation of the resistance m i n i m u m in dilute magnetic alloys. However, to the authors's knowledge, there is no explanation for anomalous specific heat for certain magnetic alloys and the m e c h a n i s m of observed anomalies is an open question. Geballe et al. [3] have studied the magnetic behaviour of dilute solutions of F e in alloys m a d e f r o m Os, Ir and Pt. They concluded that there exists an anomalous increase in the heat capacity at low temperatures which does not s e e m to have a magnetic origin and is found to correlate with the presence of non-magnetic or feebly magnetic localized states ol Fe. 262
Recently, w e suggested an indirect Coulomb and exchange m e c h a n i s m involving conduction electrons and the localized electrons of the impurity atoms taking into account the changes in the multiplicity of the impurity d-shells [4, 5]. The purpose of the present note is to calculate the change in the density of states due to nonmagnetic l o c a l i z e d s t a t e s in dilute m a g n e t i c a l loys in the light of above m e c h a n i s m . Th e m a g n e t i c a l l o y s a r e c o n s i d e r e d as a s y s t e m of conduction e l e c t r o n s i n t e r a c t i n g with l o c a l i z e d i m p u r i t y e l e c t r o n s v i a g e n e r a l i s e d Coulomb and exchange i n t e r a c t i o n s . The i m p u r i t y a t o m s axe a s s u m e d to c o n s i s t of two l o c a l i z e d e l e c t r o n s in the si n g l et state. The H a m i l t o n i a n f o r the s y s t e m can be w r i t t e n as H = H o + Hint:
Ho
.k.ct,,,ck.+Elia E;i,, cl$
(1)
Volume 25A, number 3
P HY S I C S L E T T E R S
Hint. = k k ' l~ ~i ~ -
li
(l)'
X icr
C k , G, *
C k (;
+ c.c.
11
where ~kG and E/i G represent respectively the energies of the non-interacting Bloch electrons in the states I > and the impurity electron in the state ]liG >. Ck(; (CkG) and CZ~¢;(CliG) are respectively the creation (annihilation) operators for conduction electron in the state I kG > and the impurity electron in the state I liG >. The f i r s t and the second terms in Hint. represent the Coulomb and exchange type i n t e r k ~ o n s re~p.~tively where the matrix element U ~'k and V ~ ' k axe given in ref. 4. In order to obtain the effective electron-electron interaction due to the virtual excitation and de-excitation of impurity electrons, we s u b j e c t the H a m i l t o n i a n H to the u s u a l c a n o n i cal t r a n s f o r m a t i o n and r e t a i n t e r m s up to second o r d e r in m a t r i x e l e m e n t s . Since we a r e i n t e r e s t e d in finding the r e n o r m a l i s e d e n e r g y of the conduction e l e c t r o n s , we s e l e c t f r o m the t r a n s f o r m e d H a m i l t o n i a n only the diagonal t e r m s in the conduction e l e c t r o n o p e r a t o r . Due cognition of the occupancy of i m p u r i t y s t a t e s is taken. Next, in o r d e r to e x p r e s s the d i a g o n a l i z e d H a m i l t o n i a n in the s i n g l e - p a r t i c l e - o p e r a t o r s , we u s e the H a t t r e e - l i k e a p p r o x i m a t i o n , n a m e l y C/~cr Ckcr is r e p l a c e d by the f k + ( C ~ Ck~ - f k ) and the p r o d uct (C~(; Ck¢; - f k ) " (C~¢; Ck,c; - f k ' ) is neg-
14 August 1967
w h e r e the definition of A ~ , Cou and ~ can be found in ref. 4. The laSt ' t e r m of the above equation is neglected as it i s independent of the wave v e c t o r k and gives only a shift in the z e r o of the e n e r g y s p e c t r u m . After p e r f o r m i n g the s u m m a tion o v e r k ' , the above equation can be w r i t t e n in the following compact f o r m Ek - e F ~(e k - EF)/(l+77), where ~.= 3 ] V] 2 ~ Z(1 + ct)/eF(e F + 2 A ~ ) > 0, ct ~ A~e~/kBT >> 1 where k B is the B o l t z m a n constant. In d e r i v i n g the above equation we have a s s u m e d that l e k - e F [ << A ~ , [ e F l > [A~e~land i g n o r e d .,thek - d e p e n d e n c e of exchange m a t r i x e l e m e n t , V~K,l. Z is the n u m b e r of e l e c t r o n s per atom and e F the F e r m i energy. F r o m the above equation it is c l e a r that the d e n s i t y of s t a t e s at the F e r m i s u r f a c e for these alloys is (1 + 77) t i m e s g r e a t e r than that of pure m e t a l , which in t u r n accounts for the a n o m a l o u s ly high value of heat capacity. A rough e s t i m a t e of~7 with V ~ 0 . 5 eV, ~ ~ 2 at.%, Z ~ 2 , e F ~ 5 eV and A ~ ~ 1 eV at T = 10oK gives ~7~2.
lected. After a long but simple caicnlation, the renorma/ized energy, Ek, of the conduction electron is given by
E k =e k + Rsfe~'~t'lCP.$
1. A.J.De]d~er, J.Appl. Phys. 36 (1965) 906.
I%, (uzx,
coo
}Ix'. (3)
2. J. Kondo, Prog. Theor. Phys. 32 (1964) 37. 3. T.H.Geballe, B.T.Matthias, A.M. Clogston, H.J. WflUams, R.C. Sherwood and J. P. Malta, J. Appl. Phys. 37 (1966) 1181. 4. B.N.Ganguly, U.N.Upadhyaya and K.P.Sinha, Phys. Rev. 146 (1966) 317. 5. B.N.Gangtfly, U.N.Upadhyaya and K.P.Sinha, Proc. Phys. Soc. 90 (1967) 445.
263
P HYSI C S L E T T E R S
c u r v e s s e p a r a t e , but t h i s m a y be due to c h a n g e s in the F e r m i s u r f a c e o v e r l a p s b e t w e e n B r i U o u i n z o n e s such as t h o s e found by M e r r i a m et al. [10, 11] in t h e i r w o r k on the t r a n s i t i o n t e m p e r a ture. We c o n c l e d e that in the alloy r e g i o n w h e r e no sudden cbm~,es o c c u r in the e l e c t r o n i c s t r u c t u r e , K o b e y s t&e e x p r e s s i o n (1) to b e t t e r than 2% in s pit e of wide v a r i a t i o n s in f r e e path d i s t r i b u t i o n . We find s i m i l a r r e s u l t s at a r a n g e of l o w e r t e m peratures. T h i s r e s e a r c h was s u p p o r t e d by a g r a n t f r o m the S c h w e i z e r i s c h e K o m m i s s i o n z u r F o r d e r u n g der wissenschaftlichen Forschung.
IMPURITY
STATES
14 August 1967
i. L.P.Gorkov, Soviet. Phys. J E T P 10 (1960) 998. 2. K. Maki, Physics 1 (1964) 21. 3. C. Caroli, M. Cyrot and R. G. de Gennes, Solid State Commun. 4 (1966) 17. 4. G. Eflenberger, Phys. Rev. 153 (1967) 584. 5. W. van der Mark, J. L. Olsen and F. B. Rasmussen, Proc. 10th Int. Conf. on Low temperature physics, (Moscow 1966), to be published. 6. C. Chlou, R.A. Connell and D. P. Seraphim, Phys. Rev. 129 (1963) 1070. 7. K. Noto, Y. Mute and T. Fulmroi, J. Phys. Soc. Japan 21 (1966) 9.22. 8. G. BrRndli, P. Cotti, E.M. Fryer and J. L. Olsen, Proc. 9th Int. Conf. on Low temperature physics, B, (Plenum Press, 1965) p. 827. 9. J. L. Harden and V. Arp, Cryogenics 3 (1963) 105. 10. M. F. Merriam and M.von Herzen, Phys. Rev. 131 (1968) 637. 11. M. F. Merriam, J. Hagen and H. L. Luo, Phys. Rev. 154 (1967) 424.
AND ANOMALOUS IN DILUTE ALLOYS
DENSITY
OF
STATES
B. N. GANGULY
Department of Physics, University of Toronto, Toronto, Canada Received 3 July 1967
The question of anomalous density of states in some dilute alloys is examined by taking due cognition of the localized impurity states.
There have been a considerable amount of experimental and theoretical studies of the a n o m a lous behaviour of dilute magnetic alloys at low temperature [1]. Recently, Kondo [2] has put forw a r d a theory based on the dynamical character of s-d exchange interaction which gives a satisfactory explanation of the resistance m i n i m u m in dilute magnetic alloys. However, to the authors's knowledge, there is no explanation for anomalous specific heat for certain magnetic alloys and the m e c h a n i s m of observed anomalies is an open question. Geballe et al. [3] have studied the magnetic behaviour of dilute solutions of F e in alloys m a d e f r o m Os, Ir and Pt. They concluded that there exists an anomalous increase in the heat capacity at low temperatures which does not s e e m to have a magnetic origin and is found to correlate with the presence of non-magnetic or feebly magnetic localized states ol Fe. 262
Recently, w e suggested an indirect Coulomb and exchange m e c h a n i s m involving conduction electrons and the localized electrons of the impurity atoms taking into account the changes in the multiplicity of the impurity d-shells [4, 5]. The purpose of the present note is to calculate the change in the density of states due to nonmagnetic l o c a l i z e d s t a t e s in dilute m a g n e t i c a l loys in the light of above m e c h a n i s m . Th e m a g n e t i c a l l o y s a r e c o n s i d e r e d as a s y s t e m of conduction e l e c t r o n s i n t e r a c t i n g with l o c a l i z e d i m p u r i t y e l e c t r o n s v i a g e n e r a l i s e d Coulomb and exchange i n t e r a c t i o n s . The i m p u r i t y a t o m s axe a s s u m e d to c o n s i s t of two l o c a l i z e d e l e c t r o n s in the si n g l et state. The H a m i l t o n i a n f o r the s y s t e m can be w r i t t e n as H = H o + Hint:
Ho
.k.ct,,,ck.+Elia E;i,, cl$
(1)
Volume 25A, number 3
P HY S I C S L E T T E R S
Hint. = k k ' l~ ~i ~ -
li
(l)'
X icr
C k , G, *
C k (;
+ c.c.
11
where ~kG and E/i G represent respectively the energies of the non-interacting Bloch electrons in the states I > and the impurity electron in the state ]liG >. Ck(; (CkG) and CZ~¢;(CliG) are respectively the creation (annihilation) operators for conduction electron in the state I kG > and the impurity electron in the state I liG >. The f i r s t and the second terms in Hint. represent the Coulomb and exchange type i n t e r k ~ o n s re~p.~tively where the matrix element U ~'k and V ~ ' k axe given in ref. 4. In order to obtain the effective electron-electron interaction due to the virtual excitation and de-excitation of impurity electrons, we s u b j e c t the H a m i l t o n i a n H to the u s u a l c a n o n i cal t r a n s f o r m a t i o n and r e t a i n t e r m s up to second o r d e r in m a t r i x e l e m e n t s . Since we a r e i n t e r e s t e d in finding the r e n o r m a l i s e d e n e r g y of the conduction e l e c t r o n s , we s e l e c t f r o m the t r a n s f o r m e d H a m i l t o n i a n only the diagonal t e r m s in the conduction e l e c t r o n o p e r a t o r . Due cognition of the occupancy of i m p u r i t y s t a t e s is taken. Next, in o r d e r to e x p r e s s the d i a g o n a l i z e d H a m i l t o n i a n in the s i n g l e - p a r t i c l e - o p e r a t o r s , we u s e the H a t t r e e - l i k e a p p r o x i m a t i o n , n a m e l y C/~cr Ckcr is r e p l a c e d by the f k + ( C ~ Ck~ - f k ) and the p r o d uct (C~(; Ck¢; - f k ) " (C~¢; Ck,c; - f k ' ) is neg-
14 August 1967
w h e r e the definition of A ~ , Cou and ~ can be found in ref. 4. The laSt ' t e r m of the above equation is neglected as it i s independent of the wave v e c t o r k and gives only a shift in the z e r o of the e n e r g y s p e c t r u m . After p e r f o r m i n g the s u m m a tion o v e r k ' , the above equation can be w r i t t e n in the following compact f o r m Ek - e F ~(e k - EF)/(l+77), where ~.= 3 ] V] 2 ~ Z(1 + ct)/eF(e F + 2 A ~ ) > 0, ct ~ A~e~/kBT >> 1 where k B is the B o l t z m a n constant. In d e r i v i n g the above equation we have a s s u m e d that l e k - e F [ << A ~ , [ e F l > [A~e~land i g n o r e d .,thek - d e p e n d e n c e of exchange m a t r i x e l e m e n t , V~K,l. Z is the n u m b e r of e l e c t r o n s per atom and e F the F e r m i energy. F r o m the above equation it is c l e a r that the d e n s i t y of s t a t e s at the F e r m i s u r f a c e for these alloys is (1 + 77) t i m e s g r e a t e r than that of pure m e t a l , which in t u r n accounts for the a n o m a l o u s ly high value of heat capacity. A rough e s t i m a t e of~7 with V ~ 0 . 5 eV, ~ ~ 2 at.%, Z ~ 2 , e F ~ 5 eV and A ~ ~ 1 eV at T = 10oK gives ~7~2.
lected. After a long but simple caicnlation, the renorma/ized energy, Ek, of the conduction electron is given by
E k =e k + Rsfe~'~t'lCP.$
1. A.J.De]d~er, J.Appl. Phys. 36 (1965) 906.
I%, (uzx,
coo
}Ix'. (3)
2. J. Kondo, Prog. Theor. Phys. 32 (1964) 37. 3. T.H.Geballe, B.T.Matthias, A.M. Clogston, H.J. WflUams, R.C. Sherwood and J. P. Malta, J. Appl. Phys. 37 (1966) 1181. 4. B.N.Ganguly, U.N.Upadhyaya and K.P.Sinha, Phys. Rev. 146 (1966) 317. 5. B.N.Gangtfly, U.N.Upadhyaya and K.P.Sinha, Proc. Phys. Soc. 90 (1967) 445.
263