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Theory of magnetic susceptibility of Bloch electrons: Effect of localized magnetic moments
Volume 115, number 4
PHYSICS LETTERSA
7 April 1986
THEORY OF MAGNETIC SUSCEPTIBILITY OF B L O C H ELECTRONS: EFFECT OF LOCALIZED MAGNETIC M O M E N T S Gouri S. TRIPATHI Department of Physics, Berhampur Unioersity, Berhampur 760007, Orissa, India Received 9 December 1985; accepted for publication 4 February 1986
We derive for the first time a first-principles theory of the magnetic suceptibility (X) of Bloch electrons including the effects of periodic potential, spin-orbit interaction and localized magnetic moments, and believe it to be the most general and thorough treatment which has yet been made of this problem. The importance of the theory in possible applications is discussed.
The theories of magnetic susceptibility (X) of Bloch electrons both in the absence [1-4] and presence [5] of electron--electron interactions are fairly well unserstood, and have been successfully applied to simple metals [6-8] and semiconductors [9,10]. However, these theories are not valid for solids where there are unfilled d- or f-shells in the constituent atoms, as in the case of transition and rare-earth metals and alloys. These unfilled shells give rise to locallsed magnetic moments which interact with the magnetic moment of conduction electrons and, in the process, polarize them. This kind of interaction is popularly known as s-d or s - f hybridization, depending on the type of unfilled shell present in the system. Such hybridizations are known to give rise to many interesting features in high energy photon and electron spectroscopy, itinerant versus localized magnetism, valence instability, dense Kondo behaviour, anisotropic interactions or heavy fermion superconductivity. Recently the effect of localized magnetic moments is considered on the Knight shift (K) in solids [ 11 ]. In this communication, we present for the first time a first principles theory of magnetic susceptibility of Bloch electrons including the effects of periodic potential, spin-orbit interaction and localized magnetic moments, and report some new findings. The exact one-particle propagator G(r, r', B, ltjp ~t) for a system of electrons in the presence of a periodic potential V(r), spin-orbit interaction, applied magnetic field B, satisfies the equation
(~l - H)G(r, r;, B, IIjj, ~t) = $(r -- r ' ) ,
(1)
where ~'I=~-1(2/+ 1)ilr+/a,
I=0,+1,_+2,+3,
(2)
H = (l/2m) [p + (e/c)A] 2 + V(r) + (~/4m2c2)e'VVX [p + (e/c)A] + ~glaoa'B + ~
/
g0 (P]/)'J~B •
(3)
Here, the first four terms describe the well-known interactions in the presence of an applied magnetic field [12]. The last term describes the interactions of the magnetic moment of the. conduction electron and the localized electrons, (IAI/) being the thermal average of the total magnetic moment associated with the localized electron at the/th site and
0.375-9601/86]$ 03.50 © Elsevier Science Publishers B.V.
(North-HollandPhysicsPublishingDivision)
169
Volume 115, number 4
PHYSICS LETTERSA
7 April 1986
Xsn = (l~r*8("s) + [ " ~ / d + 3(*'"s)"s/'? 5 ] ) + 2'7 x (~ + ealc)l~ir~ -- x~ + x/n.
(4)
In eq. (4) x is the electronic momentum operator in the presence of spin-orbit interaction, rj = r - Ri and ~ is the Pauli spin matrix. The terms from left to right form the parts of contact, dipolar and orbital interactions. Tot take care of the lack of the lattice periodicity in the presence of magnetic fields, we define G(r, r', B, It l, ~'l) = exp lift. (r X r')] G(r, r', B, Itl, ~'l)"
(5)
G"possesses the lattice translational symmetry and h = eB/2hc in the symmetric gauge. Using eq. (5) in eq. (1) and following the technique developed earlier [5,11,13] we write eq. (1) in a representation defined by the periodic part of the Bloch function @nk;, where n is the band index, k is the reduced wave vector and p is the spin index, and
[~'l - H(x)] G'(k, ~'l) = S.
(6)
Here H(K) = (l/2m)(p + hK) 2 + V + (t//4m2c2)a • [VVX (p + t/x)] + lglzog.S +HItj(B),
(7)
where
x - - k + ~ x vk.
(8)
Using eq. (8), and the relation (/g~S) : xfZBU ,
(9)
where repeated indices (r and/a stand for cartesian components) imply summation, and x f u is the paramagnetic susceptibility tensor of Curie-Weiss type, we write
H(K)
=
(10)
Ho(k ) + H'(k) .
In eq. (10), H 0 is the one-particle hamiltonian in the absence of magnetic field and H' contains the field dependenl terms. Eq. (6) can then be solved by a perturbation expansion c"o +
+
+ ....
(11)
where G O = (~l - H0) -1 and is diagonal in the Bloch representation. The magnetic susceptibility can be calculated from the formula
X v~ = -0212/~BvOBu IB-~O ,
(12)
where ~2, the thermodynamic potential is given by [14,15] I2 =/3-1 Tr ln(-G'~t) = -(2hi) - I tr f ~(~)G'(~') d~. (13) c Here, the contour C encircles the imaginary axis in an anticlock-wise direction, tr is taken over one-particle states and t}(~) = _/3-1 In[1 + e -~(~-u)] .
(14)
From eqs. (11)-(13), we obtain after long algebra (to be published) x ~" = ~ K
+ x~&,
(15)
where ×MK is the Misra-Kleinman susceptibility [4] of Bloch electrons x ~ K = x~ ~ + x~,~ + x~'o,
170
(16)
Volume 115, number 4
PHYSICS LETTERSA
7 April 1986
Xs, ×o and Xso being the effective Pauli, orbital and spin-orbit contributions, respectively, and X~oc is a new contribution and is due to the hybridisation of the conduction and localized magnetic moments. vu = ~/~/~X~U Xloc . I where/~,r is the shift of the electronic-paramagnetic resonance (EPR) frequency at the ]th site.
(17)
e;r=e;:
(18)
where Ps, Po, Pso are the spin, orbital and spin-orbit contributions to the EPR shift and PRKKYiS due to the inter-site interactions of RKKY type [16-18]. pyr = _ ~ IS k
. 2 r ~ fJ [(1/m)laO%ovXlnp,np,ltnp,,mp,,lrmp,,,np/Emn
t T v +,2gldoXptp,np,Onp,np]f'(En) ,
l~]°r = - (e2 / 3 m c 2 ) ~k (l[r])np'naSvrf(En) + (2/~)/a2eaov ~k
(19)
[(~lr#)np'mp'(L~/r?)mp"nP
(20)
P/so
[ ( i l a 2 / m ) e a , v ( - 3 X ; p , n a ' rrno ' ,mp "n4~'ma",no [E2mn + lrno,npXrNa,mp'ngma' mp[E2mn
- 7r~
~
np,np np,mp
,XT
~-2
a
0
Or
Imp ,np/ mn+ lrnp,mp,Trmp,,qp,,X)ap,,,np/EmnEan
a ygr ,, i n p IF +ltnp,raP'"lmp',qp ~ m n ~Fq n - "+I nx,p gr ,mp + 1 ,, , 2 , X r v + ¢
, ,mp'ltrap',qp" ~ re# IF F qp ,, ,npt~mn~qnJ ,
~*o~'ot /no,mo,Omp,,.p ~p,mo,X]mp, .o)/Zmn]f(g.)
(21)
and vr
P]RKKY x], ..r'vAer' .'.j], ,
(22)
=
where A r]]'t ' _-. t'O2 k ~, k , { X ¢nkp,n,k, p, X n¢', k , p,,nkp exp[-i(k - k ' ) . R ] f ]
+ X nr'k p , n ,k, p'~Xn'k,p,,nkp vr exp[i(k
-k')'R//,] )f(Enk)/(gnk -
En,k, ) .
(23)
In eqs. (19)-(23), repeated indices imply summation, e~o v is an antisymmetric tensor of third rank and we follow Einstein summation convention, Emn = E m - E n , f is the Fermi function, and the matrix elements are taken between the periodic parts of the Bloch function. It may be noted that the Knight shift (K) is the counterpart of P in the case of nuclear magnetic resonance (NMR) and is also expressed [11,13] as a sum ofKs,Ko, Kso and KRKKY. The RKKY coupling constant is very much similar in form to the indirect nuclear coupling constant [19]. We would now like to discuss the importance of the present theory in possible applications. The theory could be valid both for non-magnetic and magnetic solids. In the case of simple metals and semiconductors the MK theory, or the modified MK theory [5] in the presence of electron-electron interactions, is quite satisfactory, which 171
Volume 115, number 4
PHYSICS LETTERS A
7 April 1986
is obtained as a limiting case o f our theory. However, our theory could be usefully applied to metals and alloys containing transition and rare-earth elements, and also to magnetic semiconductors. The theory could also have interesting applications in the intermetaUic compounds, namely, V3Si , V3Ga, Nb3Sn and Nb3A1, possessing Al5 crystal structure. These compounds are known to be good'superconductors. Susceptibility and NMR measurements have revealed a close relationship between the superconducting transition temperature (Tc) and their normal state properties. Indeed susceptibility measurements [20] show that the higher the transition temperature, the stronger is the temperature dependence of susceptibility in the normal state. Since XMK is normally temperature independent, this characteristic could possibly be explained by Xloc which depends on the Curie-Weiss susceptibility which is highly temperature dependent. Another interesting class of compounds to which the theory could suitably be applied includes the rare-earth monopnictides ReX (Re may correspond to Ce, Pr, Nd, Sm, Eu, Gd, Tb, Dy, Ho, Er, Tm, Yb and Lu and X may correspond to N, P, As, Sb and Bi) which exhibit metallic behaviour and their magnetic properties are due primarily to the 4f electrons [21 ]. In conclusion, the principal result of the present work is a first principles derivation of magnetic susceptibility o f solids. Although the electron-electron interaction effects have not been discussed, they can be included following the procedures developed earlier [5,13,19]. Finally, the theory outlined briefly in this communication is, we believe, the most general and thorough treatment which has yet been made of this problem.
References [1] E.I. Blount, Phys. Rev. 126 (1962) 1636. [2] L.M. Roth, J. Phys. Chem. Solids 23 (t962) 433. [3] P.K. Misra and L.M. Roth, Phys. Rev. 177 (1969) 1089. [4] P.K. Misra and L. Kleinman, Phys. Lett. A 37 (1971) 132; Phys. Rev. B 5 (1972) 4581. [5] S.K. Misra, P.K. Misra and S.D. Mahanti, Solid State Commun. 39 (1981) 51 ; Phys. Rev. B 26 (1982) 1903. [6] P.K. Misra, S.P. Mohanty and L.M. Roth, Phys. Rev. B 4 (1971) 1794. [7] N.C. Das and P.K. Misra, Solid State Commun. 22 (1977) 667. [8] S.K. Misra and P.K. Misra, Phys. Lett. A 92 (1982) 300. [9] P.K. Misra and L. Kleinman, Phys. Lett. A 40 (1972) 359. [10] S. Misra, G.S. Tripathi and P.K. Misra, J. Phys. C 17 (1984) 869; J. Phys. C, to be published. [ll]G.S. Tripathi, J. Phys. C Lett., to be published. [12] H.J. Zeiger and G.W. Pratt, Magnetic interactions in solids (Oxford Univ. Press, London, 1973). [13] G.S. Tripathi, L.K. Das, P.K. Misra and S.D. Mahanti, Solid State Commun. 38 (1981) 1207; Phys. Rev. B 25 (1982) 3091. [14] J.M. Luttinger and J.C. Ward, Phys. Rev. 118 (1960) 1417. [15] F.A. Buot, Phys. Rev. B 14 (1976) 3310. [16] M.A. Ruderman and C. Kittel, Phys. Rev. 96 (1954) 99. [17] T. Kasuya, Prog. Theor. Phys. Japan 16 (1956) 45. [18] K. Yosida, Phys. Rev. 106 (1957) 893. [19] G.S. Tripathi, Phys. Rev. B 31 (1985) 5143. [20] H.J. Williams and R.C. Sherwood, Bull. Am. Phys. Soc. 5 (1960) 430. [21] A.J. Freeman and J.B. Darby Jr., ed., Aetinides: electronic structure and related properties (Academic Press, New York, 1974)
172
PHYSICS LETTERSA
7 April 1986
THEORY OF MAGNETIC SUSCEPTIBILITY OF B L O C H ELECTRONS: EFFECT OF LOCALIZED MAGNETIC M O M E N T S Gouri S. TRIPATHI Department of Physics, Berhampur Unioersity, Berhampur 760007, Orissa, India Received 9 December 1985; accepted for publication 4 February 1986
We derive for the first time a first-principles theory of the magnetic suceptibility (X) of Bloch electrons including the effects of periodic potential, spin-orbit interaction and localized magnetic moments, and believe it to be the most general and thorough treatment which has yet been made of this problem. The importance of the theory in possible applications is discussed.
The theories of magnetic susceptibility (X) of Bloch electrons both in the absence [1-4] and presence [5] of electron--electron interactions are fairly well unserstood, and have been successfully applied to simple metals [6-8] and semiconductors [9,10]. However, these theories are not valid for solids where there are unfilled d- or f-shells in the constituent atoms, as in the case of transition and rare-earth metals and alloys. These unfilled shells give rise to locallsed magnetic moments which interact with the magnetic moment of conduction electrons and, in the process, polarize them. This kind of interaction is popularly known as s-d or s - f hybridization, depending on the type of unfilled shell present in the system. Such hybridizations are known to give rise to many interesting features in high energy photon and electron spectroscopy, itinerant versus localized magnetism, valence instability, dense Kondo behaviour, anisotropic interactions or heavy fermion superconductivity. Recently the effect of localized magnetic moments is considered on the Knight shift (K) in solids [ 11 ]. In this communication, we present for the first time a first principles theory of magnetic susceptibility of Bloch electrons including the effects of periodic potential, spin-orbit interaction and localized magnetic moments, and report some new findings. The exact one-particle propagator G(r, r', B, ltjp ~t) for a system of electrons in the presence of a periodic potential V(r), spin-orbit interaction, applied magnetic field B, satisfies the equation
(~l - H)G(r, r;, B, IIjj, ~t) = $(r -- r ' ) ,
(1)
where ~'I=~-1(2/+ 1)ilr+/a,
I=0,+1,_+2,+3,
(2)
H = (l/2m) [p + (e/c)A] 2 + V(r) + (~/4m2c2)e'VVX [p + (e/c)A] + ~glaoa'B + ~
/
g0 (P]/)'J~B •
(3)
Here, the first four terms describe the well-known interactions in the presence of an applied magnetic field [12]. The last term describes the interactions of the magnetic moment of the. conduction electron and the localized electrons, (IAI/) being the thermal average of the total magnetic moment associated with the localized electron at the/th site and
0.375-9601/86]$ 03.50 © Elsevier Science Publishers B.V.
(North-HollandPhysicsPublishingDivision)
169
Volume 115, number 4
PHYSICS LETTERSA
7 April 1986
Xsn = (l~r*8("s) + [ " ~ / d + 3(*'"s)"s/'? 5 ] ) + 2'7 x (~ + ealc)l~ir~ -- x~ + x/n.
(4)
In eq. (4) x is the electronic momentum operator in the presence of spin-orbit interaction, rj = r - Ri and ~ is the Pauli spin matrix. The terms from left to right form the parts of contact, dipolar and orbital interactions. Tot take care of the lack of the lattice periodicity in the presence of magnetic fields, we define G(r, r', B, It l, ~'l) = exp lift. (r X r')] G(r, r', B, Itl, ~'l)"
(5)
G"possesses the lattice translational symmetry and h = eB/2hc in the symmetric gauge. Using eq. (5) in eq. (1) and following the technique developed earlier [5,11,13] we write eq. (1) in a representation defined by the periodic part of the Bloch function @nk;, where n is the band index, k is the reduced wave vector and p is the spin index, and
[~'l - H(x)] G'(k, ~'l) = S.
(6)
Here H(K) = (l/2m)(p + hK) 2 + V + (t//4m2c2)a • [VVX (p + t/x)] + lglzog.S +HItj(B),
(7)
where
x - - k + ~ x vk.
(8)
Using eq. (8), and the relation (/g~S) : xfZBU ,
(9)
where repeated indices (r and/a stand for cartesian components) imply summation, and x f u is the paramagnetic susceptibility tensor of Curie-Weiss type, we write
H(K)
=
(10)
Ho(k ) + H'(k) .
In eq. (10), H 0 is the one-particle hamiltonian in the absence of magnetic field and H' contains the field dependenl terms. Eq. (6) can then be solved by a perturbation expansion c"o +
+
+ ....
(11)
where G O = (~l - H0) -1 and is diagonal in the Bloch representation. The magnetic susceptibility can be calculated from the formula
X v~ = -0212/~BvOBu IB-~O ,
(12)
where ~2, the thermodynamic potential is given by [14,15] I2 =/3-1 Tr ln(-G'~t) = -(2hi) - I tr f ~(~)G'(~') d~. (13) c Here, the contour C encircles the imaginary axis in an anticlock-wise direction, tr is taken over one-particle states and t}(~) = _/3-1 In[1 + e -~(~-u)] .
(14)
From eqs. (11)-(13), we obtain after long algebra (to be published) x ~" = ~ K
+ x~&,
(15)
where ×MK is the Misra-Kleinman susceptibility [4] of Bloch electrons x ~ K = x~ ~ + x~,~ + x~'o,
170
(16)
Volume 115, number 4
PHYSICS LETTERSA
7 April 1986
Xs, ×o and Xso being the effective Pauli, orbital and spin-orbit contributions, respectively, and X~oc is a new contribution and is due to the hybridisation of the conduction and localized magnetic moments. vu = ~/~/~X~U Xloc . I where/~,r is the shift of the electronic-paramagnetic resonance (EPR) frequency at the ]th site.
(17)
e;r=e;:
(18)
where Ps, Po, Pso are the spin, orbital and spin-orbit contributions to the EPR shift and PRKKYiS due to the inter-site interactions of RKKY type [16-18]. pyr = _ ~ IS k
. 2 r ~ fJ [(1/m)laO%ovXlnp,np,ltnp,,mp,,lrmp,,,np/Emn
t T v +,2gldoXptp,np,Onp,np]f'(En) ,
l~]°r = - (e2 / 3 m c 2 ) ~k (l[r])np'naSvrf(En) + (2/~)/a2eaov ~k
(19)
[(~lr#)np'mp'(L~/r?)mp"nP
(20)
P/so
[ ( i l a 2 / m ) e a , v ( - 3 X ; p , n a ' rrno ' ,mp "n4~'ma",no [E2mn + lrno,npXrNa,mp'ngma' mp[E2mn
- 7r~
~
np,np np,mp
,XT
~-2
a
0
Or
Imp ,np/ mn+ lrnp,mp,Trmp,,qp,,X)ap,,,np/EmnEan
a ygr ,, i n p IF +ltnp,raP'"lmp',qp ~ m n ~Fq n - "+I nx,p gr ,mp + 1 ,, , 2 , X r v + ¢
, ,mp'ltrap',qp" ~ re# IF F qp ,, ,npt~mn~qnJ ,
~*o~'ot /no,mo,Omp,,.p ~p,mo,X]mp, .o)/Zmn]f(g.)
(21)
and vr
P]RKKY x], ..r'vAer' .'.j], ,
(22)
=
where A r]]'t ' _-. t'O2 k ~, k , { X ¢nkp,n,k, p, X n¢', k , p,,nkp exp[-i(k - k ' ) . R ] f ]
+ X nr'k p , n ,k, p'~Xn'k,p,,nkp vr exp[i(k
-k')'R//,] )f(Enk)/(gnk -
En,k, ) .
(23)
In eqs. (19)-(23), repeated indices imply summation, e~o v is an antisymmetric tensor of third rank and we follow Einstein summation convention, Emn = E m - E n , f is the Fermi function, and the matrix elements are taken between the periodic parts of the Bloch function. It may be noted that the Knight shift (K) is the counterpart of P in the case of nuclear magnetic resonance (NMR) and is also expressed [11,13] as a sum ofKs,Ko, Kso and KRKKY. The RKKY coupling constant is very much similar in form to the indirect nuclear coupling constant [19]. We would now like to discuss the importance of the present theory in possible applications. The theory could be valid both for non-magnetic and magnetic solids. In the case of simple metals and semiconductors the MK theory, or the modified MK theory [5] in the presence of electron-electron interactions, is quite satisfactory, which 171
Volume 115, number 4
PHYSICS LETTERS A
7 April 1986
is obtained as a limiting case o f our theory. However, our theory could be usefully applied to metals and alloys containing transition and rare-earth elements, and also to magnetic semiconductors. The theory could also have interesting applications in the intermetaUic compounds, namely, V3Si , V3Ga, Nb3Sn and Nb3A1, possessing Al5 crystal structure. These compounds are known to be good'superconductors. Susceptibility and NMR measurements have revealed a close relationship between the superconducting transition temperature (Tc) and their normal state properties. Indeed susceptibility measurements [20] show that the higher the transition temperature, the stronger is the temperature dependence of susceptibility in the normal state. Since XMK is normally temperature independent, this characteristic could possibly be explained by Xloc which depends on the Curie-Weiss susceptibility which is highly temperature dependent. Another interesting class of compounds to which the theory could suitably be applied includes the rare-earth monopnictides ReX (Re may correspond to Ce, Pr, Nd, Sm, Eu, Gd, Tb, Dy, Ho, Er, Tm, Yb and Lu and X may correspond to N, P, As, Sb and Bi) which exhibit metallic behaviour and their magnetic properties are due primarily to the 4f electrons [21 ]. In conclusion, the principal result of the present work is a first principles derivation of magnetic susceptibility o f solids. Although the electron-electron interaction effects have not been discussed, they can be included following the procedures developed earlier [5,13,19]. Finally, the theory outlined briefly in this communication is, we believe, the most general and thorough treatment which has yet been made of this problem.
References [1] E.I. Blount, Phys. Rev. 126 (1962) 1636. [2] L.M. Roth, J. Phys. Chem. Solids 23 (t962) 433. [3] P.K. Misra and L.M. Roth, Phys. Rev. 177 (1969) 1089. [4] P.K. Misra and L. Kleinman, Phys. Lett. A 37 (1971) 132; Phys. Rev. B 5 (1972) 4581. [5] S.K. Misra, P.K. Misra and S.D. Mahanti, Solid State Commun. 39 (1981) 51 ; Phys. Rev. B 26 (1982) 1903. [6] P.K. Misra, S.P. Mohanty and L.M. Roth, Phys. Rev. B 4 (1971) 1794. [7] N.C. Das and P.K. Misra, Solid State Commun. 22 (1977) 667. [8] S.K. Misra and P.K. Misra, Phys. Lett. A 92 (1982) 300. [9] P.K. Misra and L. Kleinman, Phys. Lett. A 40 (1972) 359. [10] S. Misra, G.S. Tripathi and P.K. Misra, J. Phys. C 17 (1984) 869; J. Phys. C, to be published. [ll]G.S. Tripathi, J. Phys. C Lett., to be published. [12] H.J. Zeiger and G.W. Pratt, Magnetic interactions in solids (Oxford Univ. Press, London, 1973). [13] G.S. Tripathi, L.K. Das, P.K. Misra and S.D. Mahanti, Solid State Commun. 38 (1981) 1207; Phys. Rev. B 25 (1982) 3091. [14] J.M. Luttinger and J.C. Ward, Phys. Rev. 118 (1960) 1417. [15] F.A. Buot, Phys. Rev. B 14 (1976) 3310. [16] M.A. Ruderman and C. Kittel, Phys. Rev. 96 (1954) 99. [17] T. Kasuya, Prog. Theor. Phys. Japan 16 (1956) 45. [18] K. Yosida, Phys. Rev. 106 (1957) 893. [19] G.S. Tripathi, Phys. Rev. B 31 (1985) 5143. [20] H.J. Williams and R.C. Sherwood, Bull. Am. Phys. Soc. 5 (1960) 430. [21] A.J. Freeman and J.B. Darby Jr., ed., Aetinides: electronic structure and related properties (Academic Press, New York, 1974)
172