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Calculation of the magnetization for a disordered Heisenberg spin model
Volume 76A, number 3,4
PHYSICS LETTERS
31 March 1980
CALCULATION OF THE MAGNETIZATION FOR A DISORDERED HEISENBERG SPIN MODEL Milos LEV and David E. MILLER’ Fakultdt für Physik, Universitdt Bielefeld, D-4800 Bielefeld, Fed. Rep. Germany Received 2 January 1980
We use the method of double-time thermal Green functions (TGF) in order to calculate the average magnetization of a disordered spin system for the Heisenberg model in an external magnetic field within the random phase approximation (RPA). A model is proposed in which we may calculate explicitly the magnetization as a function of the temperature through a selfconsistent scheme including the disorder.
The quite recent experimental work on amorphous magnetic systems, particularly in relation to the liquid transition metals and their alloys [1,2], has brought much interest in the magnetic structure of systems possessing simultaneously various types of disorder. It is a very interesting and not so easily answered theoretical question concerning how much ferromagnetic order could possibly remain in the liquid state for such systems [3]. In this note we shall propose a greatly simplified model using the ordinary Pauli spin operators in an isotropic Heisenberg model with disorder in the spatial location of the spins (sites), which in the averaging is reflected in the particle correlation functions. Furthermore, we shall in this model avoid the direct use of the electronic band structure for the transition metals, which can certainly make a considerable contribution to the magnetic properties arising from the d-electrons. Thus we want only to investigate with this present simplified model the disorder from the spins as the sole carrier of the ferromagnetic structure, The system of disordered spins {s~’}in the external magnetic field h taken along the z-a’us with a the coordinates (x, y, z) located at the arbitrary sites f may be described using the isotropic Heisenberg harniltonian H= ~ J(i — f)s~s7 lzBh ~ sf, (1) —
1
Also at the Zentrum für Interdisziplinäre Universität Bielefeld.
Forschung des
where J(i —/) is the exchange integral and ~B the Bohr magneton. For the spin-i /2 case it is somewhat more convenient to represent (1) using the Pauli operators b’, b1 and n1 instead of their equivalent ~f f is3’ and 4, respectively. The representation [4] of the retarded and advanced TGF ((A(t) lB (t’ )~r,ais given in terms of the grand canonical ensemble averages of the form 4 2iriO(± r) (EA (t), B (t’)] ~>of the anticommutator or commutator of the time-dependent Heisenberg operators with 0(r) the unit step for r the time difference t — t’. Using the known commutation properties of the Pauli operators [4] we may readily write the equations of motion for these TGF using the commutator with (1). For the problem at hand we shall use both the retarded and the advanced TGF of the types ((b 1l bI>>r,a and ((b7bk b11 bX’r,a, which wifi be denoted after the Fourier transform from r to the energy variable E as, respectively, G11 and ~ikz/•Writing J0 for the sum ~,, J(n), we may write the Fourier transformed equations of motion for G~in the following form: 2PBh — 2J (E— 0)G11 = (1 — 2b7b~)i5,, 2r’lIk/÷2r’k(kI] (2) — 2 k j(, — k) [Gk/ — Furthermore, we could write down equations of motion for r’ 1~~1 and develop a coupled chain of equations of higher order TGF building a hierarchy, which could not easily be resolved. The simplest nontrivial decou.
277
Volume 76A, number 3,4
pling scheme
PHYSICS LETTERS
[51(see particularly p. 263) is the
RPA,
which amounts to setting n• in the higher order TGF equal to its thermal average (ne), so that in eq. (2) the last two terms become, respectively, (n~>Gkjand (nk)Gl/. The properties of such a decoupling for an
ordered system are known to relate to the usual molecular field approximation [5, p. 277], which are well understood for the quantum field theoretical work in magnetism [6]. The use of this decoupling in eq. (2) yields for the general nonlocal TGF kj’ (1—_2(btb.))~.. 1”E—F 2u..J •k k ~1(E)---- E—E 0
31 March 1980
fG11 (R1
,...,
RN; E )P(R11 R1
RN)
Because of the assumed translational invariance of the total system we have that E0 must be independent of the particular site. Thus we may write (G~~)~ by replacing zX~in eq. (4) by its average value ~ Furthermore, we can extend this approximation to all orders of the local TGF in the series (5), so that we include all the (G1)1, (G~)J,etc. with all their (~‘.)j,etc. in the form (4). When we set all the (~y)~ equa~we have a single self-energy ~(E)
=
~
2uJ(i
—
/)(G
11)12uJ(I
where the local magnetization a,, replaces 1 —2(b~b~), which we shall write simply as a, so that we have the form E0(h, a) as 2p~h+ 2.10a. The local TGF may be written in the form G11 = a/(E
—
E0 —
(4)
~
where the term z~.,can be derived from the use of the renormalized perturbation theory by Watson [7]. The use of this local TGF in relationship to the random lattice problem by Anderson [8] gave ~ in the renor-
malized perturbation series =
~20J(i —/)G)12uJ(/
—
j*1
~ 2aJ(i 1*1
—
/)G)1 2uJ(/
—
i)
(5)
k)G~’k2aJ(k i)+
...,
where the higher order local TGF G~,G~’k,etc. have the original form (4) with their respective etc., which are defined by analogous sums to the form (5). In general we know that the local TGF G11(R RN;E) is dependent upon all the spins located at the set of pbints{R 1, ...,RN}. For a spatially disordered system we need only the configuration averaged TGF (G1~)~, for which the site I is separated from the other N 1 sites. Within a self-consistent single site approximation (SSSA) the spin on the site i is treated correctly while all the other sites are contained in a configurational averaging procedure [9]. This type of approximation is performed using the conditional probabilityP(R11R1, ...,RN) or its related distribution function in such a way that the site i remains fixed while all the others may be integrated over so that we may write the (G1~)~ as ~,
~
~,
278
...,
[TdRJ
—
2crJ(i
~
—
—~
i)
j)(G11)j2aJ(j —k)(Gkk)k 2 c~J(k i) —
k (6)
±
This equation gives an approximation for the spin systern analogous to the SSSA [10], which was originally developed for the tight-binding model of the electronic structure of liquid metals [10]. The averaging process for the SSSA in eq. (6) will be carried out by approximating the N-particle distribution function for N> 2 by the product of pair distri-
butions with a chain structure. However, the pair dis-
2g(R tributions are related to the pair correlation functions n 1 R1) at a uniform density n. After using this approximation together with the translational invariance we are able to explicitly express E(E). Through a Fourier transformation to k-space the resulting series may be summed and reduced to a self-consistent form using the properties G(E) in eq. (4), so that 3kofi~ ~k~k d ~(E) = 4na2f 3 LE~_E (2ir) 0 ~2(E)+ 2naJk ~k~k ~k~k 1 +___~__.~(E\ (7) 0 ~ 1 where ~k is the Fourier transformof J(R, — R1) = J(R1 R1)g(R1 — R1). From eqs. (4) and (7) together we can calculate G11(E). This quantity may be used to calculate o(T) from the spectral theorem [4,6] on the time-dependent correlation function (b,~(r’)b~(t)) with t’ = t. Thus we find the average magnetization as proportional to The easiest test case of this model is when we con—
______
—
—
i~
—
—
Volume 76A, number 3,4
PHYSICS LETFERS
sider a totally uncorrelated spin system, which means explicitly g(R, — R1) = 1. For the model exchange
31 March 1980
a 10
integral we set J(R11)
=
V0(1+alR11I)e~”~ii’
0.8
,
where R11 = 1R1 — R11 together with V0 and a as free parameters. Other useful exponential forms [11] have been previously proposed for the exchange integrals in configurationally disordered spin systems. The particular virtue of this form is that it will allow us to compare our results directly with the known results found for the crystalline structure as well as the fact that the calculation can be performed in a straightforward way similar to that of Matsubara and Toyozawa [12] for the electronic structure. In the high density approximation we find that 3 G(E) = a IrE E 2aV a
06
o.~ ~8
6=6
0.455
0.592
8=4
8=2
80
02
I 0
02
0.4
06
08 0.755
10
T
0.916
Fig. 1. The magnetization a of the disordered spin model
as a function of T/Tc for various values of the parameters a (10~
—
0—
L
3 /
—
a
~ aV0~niraV
2a V0 + 0 9nir \1/2 /
)
(8)
aV
0a\1/21—1
I~E—E0— 2aVo+~T,~—)
j
0
which has only one pole on the real axis at E = E0 + 2aV0. Using the spectral theorem [4,6] to get the average magnetization we find that I r2 1 a 1cosh [~I.Lbh + (J(0) + Vo)alj — 1 3 1 r 2 [~bh + (J(0) + Vo)a] j1 — 9nirkT 2aV~a X sinh[~. (9) A nontrivial solution for the spontaneous magnetization with h = 0 is only possible with T < T~where kT = [J(O)+ V ]2/{[J(0)+ V ] — V a3/9nir}. (10) c
0
cm~),where Tc is the Curie temperature of an ordered fcc solid (e.g. cobalt, a = 0).
0
0
If we set a = 0 and take into consideration only th~
nearest neighbor exchanges, we get the exact form of the Curie temperature known for crystalline structures [5, p. 263; 6]. The dependence of T~on a can be seen in fig. 1, which gives the numerical evaluation of eq. (9) for the spontaneous magnetization a. We see that the parameter a takes the place in this model of the disorder parameter [13]. The structure of the temperature dependence has the same general form as in the mean field approximation [5, p. 263; 6]. Thus the analytical solution of eq. (9) near T c T ~ Tc, . gives, of course the classical critical exponents that is 13 has the value 1/2. The formal similarity between spin models and elec,
tronic systems in the tight-binding approximation has already been noted by Kirkpatrick [14] for disordered systems in relation to the percolation problem. ~4nother type of model for disordered magnetic systems has the structure of a real gas with an additional spin coupling [15]. This sort of model has recently been developed by Hemmer Imbro [16] using the structure of about the van der Waalsand phase transition to make statements the possibility of liquid ferromagnetism. Further work on the application of such a model to liquid metals and alloys [17] extending the above methods offer a number of cases of physical interest. We want to thank Dr. B. Movaghar for discussions on
this problem. References [1] For a general review see: G. Busch and H.-J. Günthero~, Electronic properties of liquid metals and alloys, in: Solid state Physics, Vol. 29, eds. H. Ehrenreich et al. (Academic Press, New York 1974). [2J S. Steeb and R. Bek, Z. Naturforsch. 31a (1976) 1348; Phys. Chem. Liq. 6 (1977) 113. [3J M. Shinoji, Liquid metals (Academic Press, New York, Ch. 5. Soy. Phys. Usp. 3 (1960) 320 see especially [4] 1977) D.N. Zubarev, 7 for ferromagnetism. [51 section R.A. Tahir-Kheli, Heisenberg ferromagnet in the Green’s function approximation, in: Phase transitions and critical
279
Volume 76A, number 3,4
[6]
[7] [8] [9] [101 [111 [12]
280
PHYSICS LETTERS
phenomena, Vol. SB, eds. C. Domb and M.S. Green (Academic Press, London, 1976), see particularly p. 263. S.V. Tyablikov, Methods in the quantum theory of magnetism (Plenum, New York, 1967) see Ch. 8 for applications of the Green function techniques. K.M. Watson, Phys. Rev. 105 (1957) 138. P.W. Anderson, Phys. Rev. 109 (1958) 1492. B. Movaghar et al., J. Phys. F4 (1974) 687; F5 (1975) 261. L.M. Roth, Phys. Rev. Lett. 28 (1972) 1570; Phys. Rev, B7 (1973) 432. J.S. H~yeand G. Stell, Phys. Rev. Lett. 36 (1976) 1569. T. Matsubara and Y. Toyozawa, Prog. Theor. Phys. 26 (1961) 739.
31 March 1980
[13] C.G.Montgomery, J.I. Krugler and R.M. Stubbs, Phys. Rev. Lett. 25 (1970) 669. [14] S. Kirkpatick, Rev. Mod. Phys. 45 (1973) 574. [151 H. Braeter, R. Gruner and G. Heber, Z. Naturforsch. 23a (1968) 648. [16] P.C. Hemmer and D. Imbro, Phys. Rev. A16 (1977) 380; Ann. Israel Phys. Soc. 2 (1978) 540. [17] D.E. Miller and M. Lev, Magnetic model for liquid metals and alloys, Bielefeld preprint, BI-TP 79/28 (August, 1979), to be published in Proc. Conf. on Soft magnetic materials 4 (Münster, September, 1979).
PHYSICS LETTERS
31 March 1980
CALCULATION OF THE MAGNETIZATION FOR A DISORDERED HEISENBERG SPIN MODEL Milos LEV and David E. MILLER’ Fakultdt für Physik, Universitdt Bielefeld, D-4800 Bielefeld, Fed. Rep. Germany Received 2 January 1980
We use the method of double-time thermal Green functions (TGF) in order to calculate the average magnetization of a disordered spin system for the Heisenberg model in an external magnetic field within the random phase approximation (RPA). A model is proposed in which we may calculate explicitly the magnetization as a function of the temperature through a selfconsistent scheme including the disorder.
The quite recent experimental work on amorphous magnetic systems, particularly in relation to the liquid transition metals and their alloys [1,2], has brought much interest in the magnetic structure of systems possessing simultaneously various types of disorder. It is a very interesting and not so easily answered theoretical question concerning how much ferromagnetic order could possibly remain in the liquid state for such systems [3]. In this note we shall propose a greatly simplified model using the ordinary Pauli spin operators in an isotropic Heisenberg model with disorder in the spatial location of the spins (sites), which in the averaging is reflected in the particle correlation functions. Furthermore, we shall in this model avoid the direct use of the electronic band structure for the transition metals, which can certainly make a considerable contribution to the magnetic properties arising from the d-electrons. Thus we want only to investigate with this present simplified model the disorder from the spins as the sole carrier of the ferromagnetic structure, The system of disordered spins {s~’}in the external magnetic field h taken along the z-a’us with a the coordinates (x, y, z) located at the arbitrary sites f may be described using the isotropic Heisenberg harniltonian H= ~ J(i — f)s~s7 lzBh ~ sf, (1) —
1
Also at the Zentrum für Interdisziplinäre Universität Bielefeld.
Forschung des
where J(i —/) is the exchange integral and ~B the Bohr magneton. For the spin-i /2 case it is somewhat more convenient to represent (1) using the Pauli operators b’, b1 and n1 instead of their equivalent ~f f is3’ and 4, respectively. The representation [4] of the retarded and advanced TGF ((A(t) lB (t’ )~r,ais given in terms of the grand canonical ensemble averages of the form 4 2iriO(± r) (EA (t), B (t’)] ~>of the anticommutator or commutator of the time-dependent Heisenberg operators with 0(r) the unit step for r the time difference t — t’. Using the known commutation properties of the Pauli operators [4] we may readily write the equations of motion for these TGF using the commutator with (1). For the problem at hand we shall use both the retarded and the advanced TGF of the types ((b 1l bI>>r,a and ((b7bk b11 bX’r,a, which wifi be denoted after the Fourier transform from r to the energy variable E as, respectively, G11 and ~ikz/•Writing J0 for the sum ~,, J(n), we may write the Fourier transformed equations of motion for G~in the following form: 2PBh — 2J (E— 0)G11 = (1 — 2b7b~)i5,, 2r’lIk/÷2r’k(kI] (2) — 2 k j(, — k) [Gk/ — Furthermore, we could write down equations of motion for r’ 1~~1 and develop a coupled chain of equations of higher order TGF building a hierarchy, which could not easily be resolved. The simplest nontrivial decou.
277
Volume 76A, number 3,4
pling scheme
PHYSICS LETTERS
[51(see particularly p. 263) is the
RPA,
which amounts to setting n• in the higher order TGF equal to its thermal average (ne), so that in eq. (2) the last two terms become, respectively, (n~>Gkjand (nk)Gl/. The properties of such a decoupling for an
ordered system are known to relate to the usual molecular field approximation [5, p. 277], which are well understood for the quantum field theoretical work in magnetism [6]. The use of this decoupling in eq. (2) yields for the general nonlocal TGF kj’ (1—_2(btb.))~.. 1”E—F 2u..J •k k ~1(E)---- E—E 0
31 March 1980
fG11 (R1
,...,
RN; E )P(R11 R1
RN)
Because of the assumed translational invariance of the total system we have that E0 must be independent of the particular site. Thus we may write (G~~)~ by replacing zX~in eq. (4) by its average value ~ Furthermore, we can extend this approximation to all orders of the local TGF in the series (5), so that we include all the (G1)1, (G~)J,etc. with all their (~‘.)j,etc. in the form (4). When we set all the (~y)~ equa~we have a single self-energy ~(E)
=
~
2uJ(i
—
/)(G
11)12uJ(I
where the local magnetization a,, replaces 1 —2(b~b~), which we shall write simply as a, so that we have the form E0(h, a) as 2p~h+ 2.10a. The local TGF may be written in the form G11 = a/(E
—
E0 —
(4)
~
where the term z~.,can be derived from the use of the renormalized perturbation theory by Watson [7]. The use of this local TGF in relationship to the random lattice problem by Anderson [8] gave ~ in the renor-
malized perturbation series =
~20J(i —/)G)12uJ(/
—
j*1
~ 2aJ(i 1*1
—
/)G)1 2uJ(/
—
i)
(5)
k)G~’k2aJ(k i)+
...,
where the higher order local TGF G~,G~’k,etc. have the original form (4) with their respective etc., which are defined by analogous sums to the form (5). In general we know that the local TGF G11(R RN;E) is dependent upon all the spins located at the set of pbints{R 1, ...,RN}. For a spatially disordered system we need only the configuration averaged TGF (G1~)~, for which the site I is separated from the other N 1 sites. Within a self-consistent single site approximation (SSSA) the spin on the site i is treated correctly while all the other sites are contained in a configurational averaging procedure [9]. This type of approximation is performed using the conditional probabilityP(R11R1, ...,RN) or its related distribution function in such a way that the site i remains fixed while all the others may be integrated over so that we may write the (G1~)~ as ~,
~
~,
278
...,
[TdRJ
—
2crJ(i
~
—
—~
i)
j)(G11)j2aJ(j —k)(Gkk)k 2 c~J(k i) —
k (6)
±
This equation gives an approximation for the spin systern analogous to the SSSA [10], which was originally developed for the tight-binding model of the electronic structure of liquid metals [10]. The averaging process for the SSSA in eq. (6) will be carried out by approximating the N-particle distribution function for N> 2 by the product of pair distri-
butions with a chain structure. However, the pair dis-
2g(R tributions are related to the pair correlation functions n 1 R1) at a uniform density n. After using this approximation together with the translational invariance we are able to explicitly express E(E). Through a Fourier transformation to k-space the resulting series may be summed and reduced to a self-consistent form using the properties G(E) in eq. (4), so that 3kofi~ ~k~k d ~(E) = 4na2f 3 LE~_E (2ir) 0 ~2(E)+ 2naJk ~k~k ~k~k 1 +___~__.~(E\ (7) 0 ~ 1 where ~k is the Fourier transformof J(R, — R1) = J(R1 R1)g(R1 — R1). From eqs. (4) and (7) together we can calculate G11(E). This quantity may be used to calculate o(T) from the spectral theorem [4,6] on the time-dependent correlation function (b,~(r’)b~(t)) with t’ = t. Thus we find the average magnetization as proportional to The easiest test case of this model is when we con—
______
—
—
i~
—
—
Volume 76A, number 3,4
PHYSICS LETFERS
sider a totally uncorrelated spin system, which means explicitly g(R, — R1) = 1. For the model exchange
31 March 1980
a 10
integral we set J(R11)
=
V0(1+alR11I)e~”~ii’
0.8
,
where R11 = 1R1 — R11 together with V0 and a as free parameters. Other useful exponential forms [11] have been previously proposed for the exchange integrals in configurationally disordered spin systems. The particular virtue of this form is that it will allow us to compare our results directly with the known results found for the crystalline structure as well as the fact that the calculation can be performed in a straightforward way similar to that of Matsubara and Toyozawa [12] for the electronic structure. In the high density approximation we find that 3 G(E) = a IrE E 2aV a
06
o.~ ~8
6=6
0.455
0.592
8=4
8=2
80
02
I 0
02
0.4
06
08 0.755
10
T
0.916
Fig. 1. The magnetization a of the disordered spin model
as a function of T/Tc for various values of the parameters a (10~
—
0—
L
3 /
—
a
~ aV0~niraV
2a V0 + 0 9nir \1/2 /
)
(8)
aV
0a\1/21—1
I~E—E0— 2aVo+~T,~—)
j
0
which has only one pole on the real axis at E = E0 + 2aV0. Using the spectral theorem [4,6] to get the average magnetization we find that I r2 1 a 1cosh [~I.Lbh + (J(0) + Vo)alj — 1 3 1 r 2 [~bh + (J(0) + Vo)a] j1 — 9nirkT 2aV~a X sinh[~. (9) A nontrivial solution for the spontaneous magnetization with h = 0 is only possible with T < T~where kT = [J(O)+ V ]2/{[J(0)+ V ] — V a3/9nir}. (10) c
0
cm~),where Tc is the Curie temperature of an ordered fcc solid (e.g. cobalt, a = 0).
0
0
If we set a = 0 and take into consideration only th~
nearest neighbor exchanges, we get the exact form of the Curie temperature known for crystalline structures [5, p. 263; 6]. The dependence of T~on a can be seen in fig. 1, which gives the numerical evaluation of eq. (9) for the spontaneous magnetization a. We see that the parameter a takes the place in this model of the disorder parameter [13]. The structure of the temperature dependence has the same general form as in the mean field approximation [5, p. 263; 6]. Thus the analytical solution of eq. (9) near T c T ~ Tc, . gives, of course the classical critical exponents that is 13 has the value 1/2. The formal similarity between spin models and elec,
tronic systems in the tight-binding approximation has already been noted by Kirkpatrick [14] for disordered systems in relation to the percolation problem. ~4nother type of model for disordered magnetic systems has the structure of a real gas with an additional spin coupling [15]. This sort of model has recently been developed by Hemmer Imbro [16] using the structure of about the van der Waalsand phase transition to make statements the possibility of liquid ferromagnetism. Further work on the application of such a model to liquid metals and alloys [17] extending the above methods offer a number of cases of physical interest. We want to thank Dr. B. Movaghar for discussions on
this problem. References [1] For a general review see: G. Busch and H.-J. Günthero~, Electronic properties of liquid metals and alloys, in: Solid state Physics, Vol. 29, eds. H. Ehrenreich et al. (Academic Press, New York 1974). [2J S. Steeb and R. Bek, Z. Naturforsch. 31a (1976) 1348; Phys. Chem. Liq. 6 (1977) 113. [3J M. Shinoji, Liquid metals (Academic Press, New York, Ch. 5. Soy. Phys. Usp. 3 (1960) 320 see especially [4] 1977) D.N. Zubarev, 7 for ferromagnetism. [51 section R.A. Tahir-Kheli, Heisenberg ferromagnet in the Green’s function approximation, in: Phase transitions and critical
279
Volume 76A, number 3,4
[6]
[7] [8] [9] [101 [111 [12]
280
PHYSICS LETTERS
phenomena, Vol. SB, eds. C. Domb and M.S. Green (Academic Press, London, 1976), see particularly p. 263. S.V. Tyablikov, Methods in the quantum theory of magnetism (Plenum, New York, 1967) see Ch. 8 for applications of the Green function techniques. K.M. Watson, Phys. Rev. 105 (1957) 138. P.W. Anderson, Phys. Rev. 109 (1958) 1492. B. Movaghar et al., J. Phys. F4 (1974) 687; F5 (1975) 261. L.M. Roth, Phys. Rev. Lett. 28 (1972) 1570; Phys. Rev, B7 (1973) 432. J.S. H~yeand G. Stell, Phys. Rev. Lett. 36 (1976) 1569. T. Matsubara and Y. Toyozawa, Prog. Theor. Phys. 26 (1961) 739.
31 March 1980
[13] C.G.Montgomery, J.I. Krugler and R.M. Stubbs, Phys. Rev. Lett. 25 (1970) 669. [14] S. Kirkpatick, Rev. Mod. Phys. 45 (1973) 574. [151 H. Braeter, R. Gruner and G. Heber, Z. Naturforsch. 23a (1968) 648. [16] P.C. Hemmer and D. Imbro, Phys. Rev. A16 (1977) 380; Ann. Israel Phys. Soc. 2 (1978) 540. [17] D.E. Miller and M. Lev, Magnetic model for liquid metals and alloys, Bielefeld preprint, BI-TP 79/28 (August, 1979), to be published in Proc. Conf. on Soft magnetic materials 4 (Münster, September, 1979).