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An approximate model in the band theory of ferromagnetism
Volume 39A, number 1
PHYSICS LETTERS
10 April 1972
AN APPROXIMATE MODEL IN THE BAND THEORY OF FERROMAGNETISM S. SZCZEN1OWSKI
Institute of Physics, TechnicalUniversity, Warsaw,Poland and L. WOJTCZAK
Institute of Physics, Universityof ~odz, £odz, Poland Received 14 Febru.ary 1972 A model of ferromagnetism is proposed on the basis of a physical concept of spin waves propagating in the static ground state distribution of electrons.
Within the band theory of ferromagnetism the ferromagnetic state is considered as spontaneous order of the magnetic moments of electrons. The conditions, in which the spontaneous magnetization occurs, are determined by the minimum of the electronic energy given by the thermodynamic average value of the Hamiltonian usually applied in the following one-band approximation [ 1] H
=
' ' ÷ M(,-,)ci,,,q,.
(ff)rn
÷ - ÷ci- q, ci,ci,
(l)
]
where C~m denote respectively the operators of creation and annihilation of an electron with spin m = t(~) at the point ] of the lattice. The known solutions [1] of the Hamiltonian (1) correspond to two kinds of excitations in a crystal: 1) the individual modes, being the one-electron eigenstates determined by the eigenfunctions ffhra = CTm IO) ~]expik]C'~mlO> belonging to the energy eigenvalues Ehm; h denotes here the wave vector of an electron and 10) is the vacuum state. 2) the collective modes interpreted as spin waves and determined by the eigenfunctions ~q = b(h,q) ~h t ~h-q~ belonging to the energy eigenvalues Eq. The average value of (1) should be taken over all the states as well as the individual and collective excitations. However, the question of the exact treatment of such calculations remains unsolved [ 1]. For this reason this paper deals with a suggestion of an approximate way leading to the description of the ferro-
magnetic properties and based on the following physical assumptions. The individual excitations are responsible for the distribution of electronic spins in the sense of the thermodynamic equilibrium state. Such a distribution is a "background" of average magnetic order, a continuum in which the spin waves as the collective modes can propagate. In such a case the effective magnetic moment M is a result of two processes: a static distribution (#) which is caused by the one-electron excitations and the dynamic behaviour of collective modes. So, from the thermodynamic point of view, the variable M is canonically conjugated to the external field H, while the magnetic distribution (/a) is only the initial state corresponding to the thermodynamic equilibrium without spin waves. The magnetization is then calculated by the standard procedure from the free energy F following from the model introduced above as a direct sum of the free energy of free electrons with the spins up and down Fr + F~ and of the free energy FSW of spin waves considered as quasi-particles with the energy eigenvalues determined by the static distribution (/a). In the simplest approximation the energy eigenvalues for (1) take the forms [1] h2
Ehm =~m,h2+(m)[2pH+I(ta)];
m = +1
(2)
and [e.g., 2]
Eq = 2p/-/+ hf2m* Dq 2
(3) 41
Volume 39A, number 1
PHYSICS LETTERS
where D = 1 +2B 3/2 [1--B3/2] -1
4[1-B5/2]
50 ( 1--B 372) 0 = 2I(~)E~ l ; B = 1 2(2td-/+ l(u))k]71 >~ 0,
(4)
and El.. denotes the Fermi level. The parameter (/a) is calculated as a function of H from the condition of thermodynamic equilibrium [l ]. Then the magnetization M is given by M = N {I +(I/2u)3/OH} -2. G q
(
h2 3D ) . l+4m,--~ ~f i /{exp(.£q/kT)1},
(5)
and the spontaneous magnetization is determined by (5) p u t t i n g H : 0, Analyzing the results following from eq.(5) we get the following conclusions. 1) If the interaction I is very strong or the band (EE) is very narrow, so that 21<,u>EF 1 > I in a large temperature interval (the dependence of
42
10 April 1972
spins; moreover, this explains also why the ferromagnetic properties of metals (for which the bands are rather large but the interaction is strong) are satislhctorily described by means of the Heisenberg model. 2) The result (5) lbr the magnetization at low temperatures (for which 3(la)/3H can be neglected t reduces to a power series with respect to tile temperature. Its coefficients correspond to those known in the literature for individual or collective modes, respectively. 3) The above model applied to the investigations of properties of ferromagnetic thin fihns [3] permits us to expect new effects connected with the dependence of the magnetization on the external magnetic field. Evidently, calculations given in [3] show that it is possible to observe the oscillating character of the magnetization as a function of H. The relative inlensity of these oscillations decreases with increasing temperature and film thickness.
References [ 1] A. Blandin, Band magnetism, in Theory of condensed matter, International Atomic Energy Agency, Vienna, 1968 p. 691 (a review article). [2] S. Szczeniowski and L. Wojtczak, Acta Phys. Polon. 35 (1969) 595. [3] S. Szczeniowski and L. Wojtczak, Conf. on Magnetic thin films, Irkutsk, 1968, in Fizika magnitnych plenok, p, 356.
PHYSICS LETTERS
10 April 1972
AN APPROXIMATE MODEL IN THE BAND THEORY OF FERROMAGNETISM S. SZCZEN1OWSKI
Institute of Physics, TechnicalUniversity, Warsaw,Poland and L. WOJTCZAK
Institute of Physics, Universityof ~odz, £odz, Poland Received 14 Febru.ary 1972 A model of ferromagnetism is proposed on the basis of a physical concept of spin waves propagating in the static ground state distribution of electrons.
Within the band theory of ferromagnetism the ferromagnetic state is considered as spontaneous order of the magnetic moments of electrons. The conditions, in which the spontaneous magnetization occurs, are determined by the minimum of the electronic energy given by the thermodynamic average value of the Hamiltonian usually applied in the following one-band approximation [ 1] H
=
' ' ÷ M(,-,)ci,,,q,.
(ff)rn
÷ - ÷ci- q, ci,ci,
(l)
]
where C~m denote respectively the operators of creation and annihilation of an electron with spin m = t(~) at the point ] of the lattice. The known solutions [1] of the Hamiltonian (1) correspond to two kinds of excitations in a crystal: 1) the individual modes, being the one-electron eigenstates determined by the eigenfunctions ffhra = CTm IO) ~]expik]C'~mlO> belonging to the energy eigenvalues Ehm; h denotes here the wave vector of an electron and 10) is the vacuum state. 2) the collective modes interpreted as spin waves and determined by the eigenfunctions ~q = b(h,q) ~h t ~h-q~ belonging to the energy eigenvalues Eq. The average value of (1) should be taken over all the states as well as the individual and collective excitations. However, the question of the exact treatment of such calculations remains unsolved [ 1]. For this reason this paper deals with a suggestion of an approximate way leading to the description of the ferro-
magnetic properties and based on the following physical assumptions. The individual excitations are responsible for the distribution of electronic spins in the sense of the thermodynamic equilibrium state. Such a distribution is a "background" of average magnetic order, a continuum in which the spin waves as the collective modes can propagate. In such a case the effective magnetic moment M is a result of two processes: a static distribution (#) which is caused by the one-electron excitations and the dynamic behaviour of collective modes. So, from the thermodynamic point of view, the variable M is canonically conjugated to the external field H, while the magnetic distribution (/a) is only the initial state corresponding to the thermodynamic equilibrium without spin waves. The magnetization is then calculated by the standard procedure from the free energy F following from the model introduced above as a direct sum of the free energy of free electrons with the spins up and down Fr + F~ and of the free energy FSW of spin waves considered as quasi-particles with the energy eigenvalues determined by the static distribution (/a). In the simplest approximation the energy eigenvalues for (1) take the forms [1] h2
Ehm =~m,h2+(m)[2pH+I(ta)];
m = +1
(2)
and [e.g., 2]
Eq = 2p/-/+ hf2m* Dq 2
(3) 41
Volume 39A, number 1
PHYSICS LETTERS
where D = 1 +2B 3/2 [1--B3/2] -1
4[1-B5/2]
50 ( 1--B 372) 0 = 2I(~)E~ l ; B = 1 2(2td-/+ l(u))k]71 >~ 0,
(4)
and El.. denotes the Fermi level. The parameter (/a) is calculated as a function of H from the condition of thermodynamic equilibrium [l ]. Then the magnetization M is given by M = N {I +(I/2u)3/OH} -2. G q
(
h2 3D ) . l+4m,--~ ~f i /{exp(.£q/kT)1},
(5)
and the spontaneous magnetization is determined by (5) p u t t i n g H : 0, Analyzing the results following from eq.(5) we get the following conclusions. 1) If the interaction I is very strong or the band (EE) is very narrow, so that 21<,u>EF 1 > I in a large temperature interval (the dependence of
42
10 April 1972
spins; moreover, this explains also why the ferromagnetic properties of metals (for which the bands are rather large but the interaction is strong) are satislhctorily described by means of the Heisenberg model. 2) The result (5) lbr the magnetization at low temperatures (for which 3(la)/3H can be neglected t reduces to a power series with respect to tile temperature. Its coefficients correspond to those known in the literature for individual or collective modes, respectively. 3) The above model applied to the investigations of properties of ferromagnetic thin fihns [3] permits us to expect new effects connected with the dependence of the magnetization on the external magnetic field. Evidently, calculations given in [3] show that it is possible to observe the oscillating character of the magnetization as a function of H. The relative inlensity of these oscillations decreases with increasing temperature and film thickness.
References [ 1] A. Blandin, Band magnetism, in Theory of condensed matter, International Atomic Energy Agency, Vienna, 1968 p. 691 (a review article). [2] S. Szczeniowski and L. Wojtczak, Acta Phys. Polon. 35 (1969) 595. [3] S. Szczeniowski and L. Wojtczak, Conf. on Magnetic thin films, Irkutsk, 1968, in Fizika magnitnych plenok, p, 356.