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Electron spin resonance modes in a spin-glass
MB 4
Physica 108B (1981) 771-772 North-Holland Publishing Company
SPIN RESONANCE MODES IN A SPIN-GLASS S.E. Barnes
Physics Department, University of Miami, Coral c~ables, Florida 33124, U.S.A.
The microscopic Hamiltonian for a spin-glass is diagon~lized using the Hol~teinPrimakoff and Bogoliubov transformations. Both an external, Zeesmn, and internal anisotropy field are included. The Zee~an energy enters the resonance condition with an extra factor of one-half which is shown to be of geometrical origin. For single ion anistropy the resonance condition is angular dependent. It is implied that sinale crystal ESR measurements will yield information about the nature of the anisotropy energy in ~ spin-glass. i.
Second, the constant, a=i/2 for an isotropic spin-glass, is of a geometrical origin.
INTRODU~I~
This work is motivated by the ESR results of Shultz et al [1], which may be mmmarized by the resonance Conditions:
Here H
O
2.
= ~(H c +all o) (i) and H are the external and anisotropy C
fields and a&i/2. The different sions correspond to distinct modes. Sanples for which (i) is valid were cooled in zero field and measured in a s~all field H <
C
Saslow [2] and Schultz et al [i] have developed semi-phenomonlogical theories for these ESR modes based upon an assumed form for the macroscopic free energy. Such modes are the q=0 limit of spin-wave branches. More general microscopic theories of spin-waves included those of Edwards and Anderson [3] and Walker and Walstedt [4], but to date these have not been oriented towards ESR.
OUTLINE OF CALCULATION
The calculation is lengthy. tonian is:
The initial Hamil-
H=~ijJijSi'Sj + ~i[YHo'Si+K(Sz) 2]
(3)
The local equilibrium direction of the ith spin is defined by the first two Euler angles (~i,Bi), the third angle T is redundant. ~bllowing closely the treatment of an antiferromagnet [5], the spin operators are replaced by a linearized Holstein-Primakoff transformation. Even when reduced to its biquadratic terms, the resulting Hamiltonian is t~D long to reproduce here. As a central point we observe that only the part of the external field H° which is parallel to the equilibrium direction appears explicitly as a term HoCOS8 i (S-a+ai) (ai the Holstein~Primakoff
Here the main preoccupation is with the adaption of the Edwards and Anderson theory to Lhree d ~ s i o n s and the ESR. Attention is focused upon the microscopic origin of H c and the Constant "a". The present calculation limited to zero temperature.
is
bosons [5] and with ~=0 for simplicity, see below). The other component HosinB i gets ]oSt in the definition of the (ei,Bi) and appears squared only in places where it is s~all compared with H 2. C
The principal new results are, first, if H
c arises from single ion anisotropy (or almost any anisotropy energy which depends upon the angle a spin makes to the crystal axes), then (i) becomes angular dependent. Specifically for single ion anisotropy: = 7 (He + aHocOS%), where ~ is the angle the external field
The calculation proceeds with the choice of three orthogonal spin-wave variables, for H =0 these are: o = (3/N) ~Zi(coseicosBi + isin~ i) aieik" Ri
(2) O
makes with the easy crystal axis (the c-axis for axial material). That (i) corresponds to measurements on polycrystals may be explained in terms of known properties of "powder spectra" but will not be discussed here.
0378-4363/81/0000-0000/$02.50
© North-HollandPublishingCompany
= (3/N) ~Zi(sin~icosSi JCos~ i) aie
ik.R. i
and
771
772
= (3/N)½Eisin~iaieik'Ri
(4)
The spin-wave variables for Ho#0 are obtained from (4) by a symmetric Schmitt orthogonalization proceedure. In order to write the Hamiltonian in terms of spin-wave variables, certain random averages are performed in a manner justified by Edwards and Anderson [3]. This is only vaid for ~values ~mmll ccmpared with the reciprocal lattice vectors. The result is surprisingly simple:
When the field makes an arbitrary angle ¢ to the easy axis, the field is resolved into components; H° cos~(~sBi, the ~ n e n t along the easy axis, leads to the modification a+acos~ above and thereby modifies equation (i) in such a way that it becomes equation (2). The perpem dicular component generates terms which couple the c % and d % branches to the e 9 branch. Since HI
For cubic systems (2) is valid if ~ is
small cxmpared to (7/2).
REFERENCES +9
-
}
+ (c~d) + (c+e)
[1] S. Schultz, E. Gullikson, D.R. Fredkin and M. Tovar, Phys. Rev. Lett. 45 (1980), 1508. (5)
where °~o = SJo(2-y(k)) + YHA
(6)
6oI = SJoY (k)
(7)
and y (k)= [Zj (Jij/Jo) cos0ijeik'Rj ]av.
(8)
and where (3.. is the angle between the equilil] brium directions of the ith and jth spins; []av. denotes an average over all spins.
The effec-
tive exchange Jo= [ZjJijcos0ij]av. is the same as enters the Edwards-Anderson theory for Tc. The effective anisotropy field HA is proportional to K and arises from certain angular averages. These averages are the same for the c 9 and d 9 parts of the Hamiltonian but --H A for the e 9 part (the last bracket in (5)) results from a different average and is ~ l l e r (for axial symmetry). The significant HoOOSB i part of the Zeeman energy generates a term ayHo( ~
+ h.c.)~= 0.
The factor a=I/2 arises from an average of ~ s ~ iApart fr(~n the factor of a=i/2, a difference in the y(k) factors, and the presence of the e 9 part, the Hamiltonian (5) is identical to that for an antiferromagnet. Diagonalization of each part is by a Bogoliubov transformation [5]. There is a gap yH c = _
S(YHAJo) ~ in each of the three spin-wave branches.
The field effects only the c 9 and
d 9 branches; diagonalization is by a linear transformation. The result is (I) since up to here we have assumed ~=0.
[2] W.M. Saslow, Phys. Rev. B22 (1980), 1174. [3] S.F. Edwards and P.W. Anderson, J. Phys. F. 6 (1976), 1927. [4] L.R. Walker and R.E. Walstedt, Phys. Rev. Lett. 38 (1977), 514. [5] see C. Kittel: Quantum Theory cf Solids (Wiley, New York, 1963).
Physica 108B (1981) 771-772 North-Holland Publishing Company
SPIN RESONANCE MODES IN A SPIN-GLASS S.E. Barnes
Physics Department, University of Miami, Coral c~ables, Florida 33124, U.S.A.
The microscopic Hamiltonian for a spin-glass is diagon~lized using the Hol~teinPrimakoff and Bogoliubov transformations. Both an external, Zeesmn, and internal anisotropy field are included. The Zee~an energy enters the resonance condition with an extra factor of one-half which is shown to be of geometrical origin. For single ion anistropy the resonance condition is angular dependent. It is implied that sinale crystal ESR measurements will yield information about the nature of the anisotropy energy in ~ spin-glass. i.
Second, the constant, a=i/2 for an isotropic spin-glass, is of a geometrical origin.
INTRODU~I~
This work is motivated by the ESR results of Shultz et al [1], which may be mmmarized by the resonance Conditions:
Here H
O
2.
= ~(H c +all o) (i) and H are the external and anisotropy C
fields and a&i/2. The different sions correspond to distinct modes. Sanples for which (i) is valid were cooled in zero field and measured in a s~all field H <
C
Saslow [2] and Schultz et al [i] have developed semi-phenomonlogical theories for these ESR modes based upon an assumed form for the macroscopic free energy. Such modes are the q=0 limit of spin-wave branches. More general microscopic theories of spin-waves included those of Edwards and Anderson [3] and Walker and Walstedt [4], but to date these have not been oriented towards ESR.
OUTLINE OF CALCULATION
The calculation is lengthy. tonian is:
The initial Hamil-
H=~ijJijSi'Sj + ~i[YHo'Si+K(Sz) 2]
(3)
The local equilibrium direction of the ith spin is defined by the first two Euler angles (~i,Bi), the third angle T is redundant. ~bllowing closely the treatment of an antiferromagnet [5], the spin operators are replaced by a linearized Holstein-Primakoff transformation. Even when reduced to its biquadratic terms, the resulting Hamiltonian is t~D long to reproduce here. As a central point we observe that only the part of the external field H° which is parallel to the equilibrium direction appears explicitly as a term HoCOS8 i (S-a+ai) (ai the Holstein~Primakoff
Here the main preoccupation is with the adaption of the Edwards and Anderson theory to Lhree d ~ s i o n s and the ESR. Attention is focused upon the microscopic origin of H c and the Constant "a". The present calculation limited to zero temperature.
is
bosons [5] and with ~=0 for simplicity, see below). The other component HosinB i gets ]oSt in the definition of the (ei,Bi) and appears squared only in places where it is s~all compared with H 2. C
The principal new results are, first, if H
c arises from single ion anisotropy (or almost any anisotropy energy which depends upon the angle a spin makes to the crystal axes), then (i) becomes angular dependent. Specifically for single ion anisotropy: = 7 (He + aHocOS%), where ~ is the angle the external field
The calculation proceeds with the choice of three orthogonal spin-wave variables, for H =0 these are: o = (3/N) ~Zi(coseicosBi + isin~ i) aieik" Ri
(2) O
makes with the easy crystal axis (the c-axis for axial material). That (i) corresponds to measurements on polycrystals may be explained in terms of known properties of "powder spectra" but will not be discussed here.
0378-4363/81/0000-0000/$02.50
© North-HollandPublishingCompany
= (3/N) ~Zi(sin~icosSi JCos~ i) aie
ik.R. i
and
771
772
= (3/N)½Eisin~iaieik'Ri
(4)
The spin-wave variables for Ho#0 are obtained from (4) by a symmetric Schmitt orthogonalization proceedure. In order to write the Hamiltonian in terms of spin-wave variables, certain random averages are performed in a manner justified by Edwards and Anderson [3]. This is only vaid for ~values ~mmll ccmpared with the reciprocal lattice vectors. The result is surprisingly simple:
When the field makes an arbitrary angle ¢ to the easy axis, the field is resolved into components; H° cos~(~sBi, the ~ n e n t along the easy axis, leads to the modification a+acos~ above and thereby modifies equation (i) in such a way that it becomes equation (2). The perpem dicular component generates terms which couple the c % and d % branches to the e 9 branch. Since HI
For cubic systems (2) is valid if ~ is
small cxmpared to (7/2).
REFERENCES +9
-
}
+ (c~d) + (c+e)
[1] S. Schultz, E. Gullikson, D.R. Fredkin and M. Tovar, Phys. Rev. Lett. 45 (1980), 1508. (5)
where °~o = SJo(2-y(k)) + YHA
(6)
6oI = SJoY (k)
(7)
and y (k)= [Zj (Jij/Jo) cos0ijeik'Rj ]av.
(8)
and where (3.. is the angle between the equilil] brium directions of the ith and jth spins; []av. denotes an average over all spins.
The effec-
tive exchange Jo= [ZjJijcos0ij]av. is the same as enters the Edwards-Anderson theory for Tc. The effective anisotropy field HA is proportional to K and arises from certain angular averages. These averages are the same for the c 9 and d 9 parts of the Hamiltonian but --H A for the e 9 part (the last bracket in (5)) results from a different average and is ~ l l e r (for axial symmetry). The significant HoOOSB i part of the Zeeman energy generates a term ayHo( ~
+ h.c.)~= 0.
The factor a=I/2 arises from an average of ~ s ~ iApart fr(~n the factor of a=i/2, a difference in the y(k) factors, and the presence of the e 9 part, the Hamiltonian (5) is identical to that for an antiferromagnet. Diagonalization of each part is by a Bogoliubov transformation [5]. There is a gap yH c = _
S(YHAJo) ~ in each of the three spin-wave branches.
The field effects only the c 9 and
d 9 branches; diagonalization is by a linear transformation. The result is (I) since up to here we have assumed ~=0.
[2] W.M. Saslow, Phys. Rev. B22 (1980), 1174. [3] S.F. Edwards and P.W. Anderson, J. Phys. F. 6 (1976), 1927. [4] L.R. Walker and R.E. Walstedt, Phys. Rev. Lett. 38 (1977), 514. [5] see C. Kittel: Quantum Theory cf Solids (Wiley, New York, 1963).