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Band filling and magnetic properties of a semiconductor structure with antiferromagnetic layers
ELSEVIER
Journal of Magnetism and Magnetic Materials 138 (1994) 99-104
Band filling and magnetic properties of a semiconductor with antiferromagnetic layers Victor V. Tugushev
structure
*, Evgeny A. Zhukovsky
Russian Scientific Center ‘Kurchatov Institute’, Moscow 123182, Russia
Received 12 April 1994
Abstract The effective spin-exchange Hamiltonian is obtained for an antiferromagnetic semiconductor layer of the structure formed by interchanging antiferromagnetically ordered and nonmagnetic semiconductor layers in the limit of wide spacer between the magnetic layers. The nearest-neighbour exchange integral obtained by means of perturbation theory based on Anderson’s Hamiltonian is shown to be a decreasing function of the band filling. The RKKY interaction is taken into account and the total effective spin-exchange integral is shown to change sign at definite band filling, which defines the region in which the spin wave velocity remains positive in a long-wavelength limit.
1. Introduction Semiconductor structures formed by interchanging magnetic and nonmagnetic layers appear to be very important from both applied and fundamental standpoints [l-4]. These structures allow one to control their properties by external magnetic and electric fields. As the objects of fundamental physics these structures allow different spatial dimensions of magnetic and electron systems to be obtained by special choice of band schemes of semiconductors of magnetic and nonmagnetic layers. It can result in unusual effects, such as in the transport properties of such structures. With these structures it is also possible to dope the nonmagnetic layers to achieve levels of free carrier concentrations in the magnetic layers
* Corresponding author.
that can not be obtained in usual bulk magnetic semiconductors due to the destruction of magnetic order by defects introduced with the dopant. This problem can be avoided in a structure in which the region doped by impurities is spatially separated from the magnetic system. Thus the problem of the influence of doping on magnetic properties is a matter of importance in such structures and it can be studied without allowing for the destruction of magnetic order due to defects in the crystal structure. Below we consider a structure consisting of thin antiferromagnetic (AF) layers separated by wide (compared with the correlation length in a magnetic system) nonmagnetic interlayers. We assume that the forbidden band of the antiferromagnetic layer EgAF is smaller than the energy gap EBNM of the nonmagnetic layer (Fig. 1). In this case the position of the Fermi level in the AF layer, which determines its electron (hole) concentration, is controlled by the reservoir of nonmagnetic layer carriers.
0304-8853/94/%07.00 0 1994 Elsevier Science B.V. All rights reserved SSDI 0304-8853(94)00411-J
100
V. V. Tugushev, EA. Zhukovsky / Journal of Magnetism and Magnetic Materials 138 (1994) 99-104
known anisotropic RKKY contribution
3D electron system dependent on doping.
with
the
2. Effective spin Hamiltonian for the AF’ layer The system under consideration consists of a magnetic subsystem formed by localized spins and band electrons. We may write the model Hamiltonian of the system in the following way: H=ZZ,
Fig. 1. Band schemes of antiferromagnetic and nonmagnetic semiconductors. Shaded areas correspond to filled states at T = 0; eF is the Fermi energy.
We consider the situation when the Fermi level lies within one of the allowed bands of the magnetic layer near the band edge, which results in the magnetic semiconductor being degenerate. It is assumed that the free carrier concentration in the magnetic semiconductor is small compared with those in metals, but that it is large enough for the localization of impurities and spin-polaron effects not to appear. The weak overlapping of the electron wavefunctions of neighbouring magnetic layers results in transverse dispersion. It is assumed that the Fermi energy exceeds the width of the band in the transverse direction. In this case, the electron system of the magnetic layer appears to be effectively two-dimensional, and one can consider an isolated magnetic layer in order to describe the effects of doping on the magnetic properties of the structure. To solve this problem it is necessary to obtain the effective spin Hamiltonian on the basis of a microscopic model. The usual Heisenberg model with the nearest-neighbour exchange constant remains insufficient even when the RKKY [5-71 contribution is taken into account. It does not allow the influence of doping on the magnetic properties to be described, since the doping dependence of the RKKY contribution on nearest neighbours is determined by the density of states at the Fermi level, which is constant in weakly doped 2D systems. In the case of a partially filled transverse band we obtain the well
+ZZ, +Z&,
(1)
where ZZ, is a magnetic subsystem Hamiltonian, H, describes band electrons, and Z&, allows for interactions between localized spins and the spin density of itinerant carriers. We write these Hamiltonians as (2) where c(k) is the electron spectrum, and ak’, is the creation operator for an electron with quasimomenturn k and spin u. For Hs,, we have
JimSiS,, (im>
(3)
Hsb = C
where the brackets denote summation over the nearest-neighbour sites, Si is a localized spin at site i, and s, is the spin density of itinerant electrons at site m. For the magnetic system we consider an effective Heisenberg model in the form H,=
xZ(n)SiSj, (ii)
(4)
where the exchange integral filling n. One can write
Z is a function
Z(n) =Z, +Z,(n),
of band
(5)
where Z, contains contributions from the empty and completely filled bands, while Z,(n) is determined by superexchange through the partially filled band. We assume that Z, is a parameter of our model. Below we demonstrate how Z,,(n) can be obtained on the basis of a microscopic model for the case of spin
d^h
P Fig. 2. Elementary
cluster.
V.V. Tug&m.,
EA. Zhukovsky/Journal
ofMagnetism and Magnetic Materials 138 (1994) 99-104
S = l/2 for magnetic layers formed by compounds of d-metals and chalcogenides. Let us consider a cluster (Fig. 2) that appears to be the simplest unit of alternating lattice in which every two neighbouring d-sites are separated by a p-site. For the cluster of Fig. 2 we write the Anderson Hamiltonian [8] as follows:
H,=‘iEdXjW+ j,u
c(2E,+U)X:‘,, i
Hpd = c Va&(Xd_+ (i,j)
aXi,,,)
(6) + h.c.,
fi=
epLH eL,
(7)
where L satisfies the following [H,,
L] +HP,, + [[[H,,
+[[Hpd,
We introduce the following expansion for the anti-Hermitian operator L into a power series over the hybridization L=L,+L,+L,+..., where the subscripts correspond to the powers of V. It can be shown that L, can be put equal to zero. Contrary to Ref. [lo] we do not limit ourselves to the term of only the first order for L, and for L,,, we obtain the following system of equations from (8): [H,,
L,l+H,,=O,
[Ha,
L3] + 1/3[ [ Hra, Li] 3 Lr] = 0.
The transformed
where Ep,d are the energy levels of p- and d-electrons, respectively, a& creates a p-electron with spin u at site i, X$ is the Hubbard operator [9] at site j, U is the Hubbard repulsion, and V is the hybridization. of p- and d-electrons. We assume that the p-band width W,, is sufficiently wide in the direction parallel to the plane of the magnetic layer, so one can drop the site correlations in the p-system. It is also assumed that the hybridization is weak (V <(:Ep - Ed) and that the d-electron correlation is large (U X- E, -Ed). One can drop the dispersion of p-electrons in the case when W,, -K Ep -Ed. Superexchange between the nearest-neighbour spins results from virtual transitions between p- and d-systems, which correspond to the processes of at least fourth order in Hpd in the cluster of Fig. 2. All the processes leading to spin exchange, but involving configurations different from those of Fig. 2, are of higher orders than fourth order. We will not consider them here, but limit ourselves only to the leading order in spin-exchange formation. We now construct a unitary transformation of the Hamiltonian (6), which eliminates all odd terms in V up to third order (see e.g. Ref. [lo]):
condition:
101
Hamiltonian
(9)
(7) has the form
B=H,+H,+H,, H,=l/2[H,,,
Ll],
(10)
H4 = 1/2[ Hpd, L3] - l/24
[[[Hpdt LI],
LI]LI].
The Hamiltonian Hz corresponds to the second order in V and, along with corrections to H,,, contains terms of interaction between localized spins (Si> and delocalized spin density of p-electrons (s,) (see (3)). According to our calculation for the cluster of Fig. 2, Ji, =J=
V*/(E,
-Ed).
(11)
The Hamiltonian H4 is of fourth order in V. Along with the fourth-order corrections to H,,, H2 and terms describing the interactions between charge densities, it contains terms of spin exchange between localized spins. We denote them as H,, for which we have H, in the form of (4,5), where
b(n)
=
3(EpTEd)‘1-n)+i
1*
(12) Here II is the p-site filling per spin. When obtaining (12) we used a mean field approximation over the ground state of Ho for the product of operators a,+ap, i.e.
L], L], L]/31
L], L]/21=0,
where the square brackets denote commutators.
(8)
According to (12), the partially filled band contribution to superexchange is of antiferromagnetic type
(Z,(n) > 0).
102
V.V. Tugushev, EA. Zhukovsky / Journal of Magnetism and Magnetic Materials 138 (1994) 99-104
Thus we have demonstrated for the example of the cluster of Fig. 2 how I,(n) can be obtained. One can consider more complex clusters, but the qualitative picture remains the same: while W,, < E, - E,, the dependence of the partially filled band contribution to the exchange integral is linear in n because nonlinear terms in perturbation theory up to fourth order in V do not appear, but the fourth order is enough to establish superexchange.
3. Magnon spectrum The existence of band carriers opens up the possibility of RKKY exchange between localized spins, which results from the second order in HSb processes. In the frames of our model for nearest neighbours (or, to be more exact, for wavevectors close to the vector of the AF structure) it possesses a ferromagnetic character. The destruction of AF long-range order (LRO) in the frame of the considered model appears because of the competition between AF superexchange with integral I(n) and ferromagnetic indirect exchange of RKKY type. The AF LRO as a ground state has a corresponding Goldstone mode sound spin wave. The loss of AF LRO will manifest itself in the spectrum of collective excitations through the destruction of the Goldstone mode, which can be expressed as the loss of positiveness of spin wave energy (or spin wave velocity c) in the long-wavelength limit. Using the criterion c > 0 it is possible to find the band filling range in which the AF state exists. First, we calculate the spin wave dispersion law taking into account many-body effects due to interaction HSb (1). The renormalized spectrum 0(k) can be obtained in the random phase approximation for a spin wave Green function from the equation: 0(k)
=o(k)
+P@C)n(k,
C!(k)),
where w(k) is the initial dispersion
(13)
U( k, w) =
c P
b(P + k) -h(P) w-~(p+k)
Vk
=
-sgny,[([l-y,2]P1’2-1)/2]1’2,
+iS’
where n,(p) is the Fermi function, 6 is a small positive value, the summation in yk(Jk) runs over the lattice sites occupied by delocalized spins (p-electrons) of the first coordination sphere of some localized spin, a is a lattice constant in the spin system, 1= a/2 for the cluster of Fig. 2. Here 17(k, w) is a polarization operator constructed with the help of the Green functions of band electrons. In the long-wavelength limit (k -xpF, pF is the Fermi momentum) for n(k, w) we have:
where N(E~) is the density of states at the Fermi level per spin, and k is counted from the AF structure wavevector. In a 2D system for band filling near the band edge, where the dispersion law is parabolic, the density of states is known to be independent of the Fermi energy, when the Fermi level lies within the allowed energy band. When the transitions of electrons between neighbouring magnetic layers is allowed for, weak transverse dispersion appears in the electron spectrum:
4
.$p)=-+WWI(l-cos
PLd)>
2rnll
(15)
where P,,,~ are the longitudinal and transverse momenta, respectively, W, is the width of the band in the transverse direction, m,, is the effective mass of electron for motion along the layer (m,, N W,: * ), and d is the distance between AF layers. The density of states for the spectrum (15) has the form
law,
J’
+c(p)
Xarccos
W, --E w,
1 ’ m,,b2 27F
(16)
where E = 0 corresponds to the bottom of the band, O(E) is a step function, and b is the in-plane lattice
V. V. Tqushev,
Ed. Zhukovsky / Journal of Magnetism and Magnetic Materials 138 (1994) 99-104
103
n nc Fig. 3. Effective exchange to EF = W, .
integral as a function of band filling: n, is the critical doping at which the AF LRO is violated,
constant for the p-electron system lattice formed by clusters of Fig. 1 low we consider only this case>. It is ing that in the case of or B- 2W, situation for iV( or):
(for a square b = a/&; beworth mentionwe obtain 2D
N(C,$g, while when filling have an anisotropic
(17) is so small that .er -K W, , we 3D model with the usual N(E~): (18)
Introducing (14), (17) and (18) into (13) we obtain for the in-plane spin wave velocities in the 2D and 3D regimes, %D
=
cOzeff(
‘1
J2m,,a2
=C 0 zm+zb(n) - ~ 4JjsT i
I ’
Er z+- 2w,
=
cOzdf(
=c
EF
0
(19)
‘1
I
z, +Z,(n)
-=Zw,
z (~)“2nl/3)) -J2Y1ipz
,
respectively, where co is a dimensional dependent on the type of lattice.
Expression (19) corresponds to the case of a two-dimensional system of spins and a two-dimensional electron band, while for (20) the electron band is anisotropic three-dimensional. If we consider the case of an AF ground state at n = 0, which corresponds to Z,,(O) > 0, then we obtain two possible situations, n being increased, depending on the value of Z,,(O) (Fig. 3); when the RKKY contribution is large enough the AF LRO can be violated in the 3D electron picture (Fig. 3a) and in the 2D one (Fig. 3b). It should be mentioned that our description is limited to the range of band filling IZ< n, (n, is the band filling, which limits the region of AFM order), especially in the 2D regime, because when considering the region of large band filling one should allow for nonparabolicity of the electron spectrum, which will manifest itself in N(E~), and also in the vicinity of n, the mean field theory for Z,(n) is not valid due to strong fluctuations.
)
and ‘3D
n I corresponds
(20) constant
4. Conclusions In this paper we have demonstrated how the dependence of the exchange integral in the effective spin Hamiltonian on the filling of electron band can be obtained. We assume that the band filling is large enough for the processes of strong localization and spin-polaron effects to be neglected. Really we bear in mind the carrier concentrations of 1011-1012 cme2 for AFM layers lo-100 A thick in semiconductor superlattices of e.g. MnSe/ZnSe or MnSe/CdSe.
104
V. V. Tugushev, Ed. Zhukovsky / Journal of Magnetism and Magnetic Materials 138 (1994) 99-l 04
We have made calculations in a one-band approximation for itinerant electrons and for the case of S = l/2 for localized spins. When interband processes in the delocalized system are allowed for, there appears a linear contribution to Z(n) from interband transitions, but qualitatively the Z versus n dependence remains the same in the limit W,, < E, -
Acknowledgement
Ed.
References
We have considered the case of T = 0. It is well known that the AF LRO in 2D system of spins is possible only at zero temperature [ill. The weak overlapping between electron wavefunctions of neighbouring AF layers in the multilayered structure makes magnetic and electron subsystems only quasi2D, which shifts the long-range ordering temperature to a finite value TLRo. The region of validity of the spin Hamiltonian derived in this paper is determined by temperatures T -=c TLRo, where TLRo is the NCel temperature. In the region T - TLRo our approach can not be applied because of strong critical fluctuations. The temperature region of short-range order TIA0 -c T < TSR0 (TSR0 is the crossover temperature) will be considered elsewhere.
Research was made possible in part by Grant No. MC8000 from the International Science Foundation and by Grant 94-02-03506 (No. 566) from the Russian Foundation for Fundamental Research.
[l] T.M. Giebultowicz, P. Klosowski, N. Samarath, H. Luo, J.K. Furdyna and J.J. Rhyne, Phys. Rev. B 48 (1993) 12817. [2] P. KJosowski, T.M. Giebultowicz, N. Samarath, H. Luo, J.K. Furdyna and J.J. Rhyne, J. Magn. Magn. Mater. 104-107 (1992) 1795. [3] T.M. Giebultowicz, N. Samarath, H. Luo, P. Klosowski, J.K. Furdyna and J.J. Rhyne, Phys. Rev. B 46 (1992) 12076. [4] K. Ando, K. Takahashi, T. Okuda and M. Umihara, Phys. Rev. B 46 (1992) 12289. [5] M.A. Ruderman and C. Kittel, Phys. Rev. 96 (1954) 99. [6] T. Kosuya, Prog. Theor. Phys. 16 (1956) 45. [7] K. Yosida, Phys. Rev. 106 (1957) 893. [8] P.W. Anderson, Phys. Rev. 124 (1961) 41. [9] J. Hubbard, Proc. R. Sot. A 285 (1965) 542. [lo] D. Ihle and M. Kasner, Phys. Rev. B 42 (1990) 4760. [ll] N.D. Mermin and H. Wagner, Phys. Rev. Lett. 17 (1966) 1133.
Journal of Magnetism and Magnetic Materials 138 (1994) 99-104
Band filling and magnetic properties of a semiconductor with antiferromagnetic layers Victor V. Tugushev
structure
*, Evgeny A. Zhukovsky
Russian Scientific Center ‘Kurchatov Institute’, Moscow 123182, Russia
Received 12 April 1994
Abstract The effective spin-exchange Hamiltonian is obtained for an antiferromagnetic semiconductor layer of the structure formed by interchanging antiferromagnetically ordered and nonmagnetic semiconductor layers in the limit of wide spacer between the magnetic layers. The nearest-neighbour exchange integral obtained by means of perturbation theory based on Anderson’s Hamiltonian is shown to be a decreasing function of the band filling. The RKKY interaction is taken into account and the total effective spin-exchange integral is shown to change sign at definite band filling, which defines the region in which the spin wave velocity remains positive in a long-wavelength limit.
1. Introduction Semiconductor structures formed by interchanging magnetic and nonmagnetic layers appear to be very important from both applied and fundamental standpoints [l-4]. These structures allow one to control their properties by external magnetic and electric fields. As the objects of fundamental physics these structures allow different spatial dimensions of magnetic and electron systems to be obtained by special choice of band schemes of semiconductors of magnetic and nonmagnetic layers. It can result in unusual effects, such as in the transport properties of such structures. With these structures it is also possible to dope the nonmagnetic layers to achieve levels of free carrier concentrations in the magnetic layers
* Corresponding author.
that can not be obtained in usual bulk magnetic semiconductors due to the destruction of magnetic order by defects introduced with the dopant. This problem can be avoided in a structure in which the region doped by impurities is spatially separated from the magnetic system. Thus the problem of the influence of doping on magnetic properties is a matter of importance in such structures and it can be studied without allowing for the destruction of magnetic order due to defects in the crystal structure. Below we consider a structure consisting of thin antiferromagnetic (AF) layers separated by wide (compared with the correlation length in a magnetic system) nonmagnetic interlayers. We assume that the forbidden band of the antiferromagnetic layer EgAF is smaller than the energy gap EBNM of the nonmagnetic layer (Fig. 1). In this case the position of the Fermi level in the AF layer, which determines its electron (hole) concentration, is controlled by the reservoir of nonmagnetic layer carriers.
0304-8853/94/%07.00 0 1994 Elsevier Science B.V. All rights reserved SSDI 0304-8853(94)00411-J
100
V. V. Tugushev, EA. Zhukovsky / Journal of Magnetism and Magnetic Materials 138 (1994) 99-104
known anisotropic RKKY contribution
3D electron system dependent on doping.
with
the
2. Effective spin Hamiltonian for the AF’ layer The system under consideration consists of a magnetic subsystem formed by localized spins and band electrons. We may write the model Hamiltonian of the system in the following way: H=ZZ,
Fig. 1. Band schemes of antiferromagnetic and nonmagnetic semiconductors. Shaded areas correspond to filled states at T = 0; eF is the Fermi energy.
We consider the situation when the Fermi level lies within one of the allowed bands of the magnetic layer near the band edge, which results in the magnetic semiconductor being degenerate. It is assumed that the free carrier concentration in the magnetic semiconductor is small compared with those in metals, but that it is large enough for the localization of impurities and spin-polaron effects not to appear. The weak overlapping of the electron wavefunctions of neighbouring magnetic layers results in transverse dispersion. It is assumed that the Fermi energy exceeds the width of the band in the transverse direction. In this case, the electron system of the magnetic layer appears to be effectively two-dimensional, and one can consider an isolated magnetic layer in order to describe the effects of doping on the magnetic properties of the structure. To solve this problem it is necessary to obtain the effective spin Hamiltonian on the basis of a microscopic model. The usual Heisenberg model with the nearest-neighbour exchange constant remains insufficient even when the RKKY [5-71 contribution is taken into account. It does not allow the influence of doping on the magnetic properties to be described, since the doping dependence of the RKKY contribution on nearest neighbours is determined by the density of states at the Fermi level, which is constant in weakly doped 2D systems. In the case of a partially filled transverse band we obtain the well
+ZZ, +Z&,
(1)
where ZZ, is a magnetic subsystem Hamiltonian, H, describes band electrons, and Z&, allows for interactions between localized spins and the spin density of itinerant carriers. We write these Hamiltonians as (2) where c(k) is the electron spectrum, and ak’, is the creation operator for an electron with quasimomenturn k and spin u. For Hs,, we have
JimSiS,, (im>
(3)
Hsb = C
where the brackets denote summation over the nearest-neighbour sites, Si is a localized spin at site i, and s, is the spin density of itinerant electrons at site m. For the magnetic system we consider an effective Heisenberg model in the form H,=
xZ(n)SiSj, (ii)
(4)
where the exchange integral filling n. One can write
Z is a function
Z(n) =Z, +Z,(n),
of band
(5)
where Z, contains contributions from the empty and completely filled bands, while Z,(n) is determined by superexchange through the partially filled band. We assume that Z, is a parameter of our model. Below we demonstrate how Z,,(n) can be obtained on the basis of a microscopic model for the case of spin
d^h
P Fig. 2. Elementary
cluster.
V.V. Tug&m.,
EA. Zhukovsky/Journal
ofMagnetism and Magnetic Materials 138 (1994) 99-104
S = l/2 for magnetic layers formed by compounds of d-metals and chalcogenides. Let us consider a cluster (Fig. 2) that appears to be the simplest unit of alternating lattice in which every two neighbouring d-sites are separated by a p-site. For the cluster of Fig. 2 we write the Anderson Hamiltonian [8] as follows:
H,=‘iEdXjW+ j,u
c(2E,+U)X:‘,, i
Hpd = c Va&(Xd_+ (i,j)
aXi,,,)
(6) + h.c.,
fi=
epLH eL,
(7)
where L satisfies the following [H,,
L] +HP,, + [[[H,,
+[[Hpd,
We introduce the following expansion for the anti-Hermitian operator L into a power series over the hybridization L=L,+L,+L,+..., where the subscripts correspond to the powers of V. It can be shown that L, can be put equal to zero. Contrary to Ref. [lo] we do not limit ourselves to the term of only the first order for L, and for L,,, we obtain the following system of equations from (8): [H,,
L,l+H,,=O,
[Ha,
L3] + 1/3[ [ Hra, Li] 3 Lr] = 0.
The transformed
where Ep,d are the energy levels of p- and d-electrons, respectively, a& creates a p-electron with spin u at site i, X$ is the Hubbard operator [9] at site j, U is the Hubbard repulsion, and V is the hybridization. of p- and d-electrons. We assume that the p-band width W,, is sufficiently wide in the direction parallel to the plane of the magnetic layer, so one can drop the site correlations in the p-system. It is also assumed that the hybridization is weak (V <(:Ep - Ed) and that the d-electron correlation is large (U X- E, -Ed). One can drop the dispersion of p-electrons in the case when W,, -K Ep -Ed. Superexchange between the nearest-neighbour spins results from virtual transitions between p- and d-systems, which correspond to the processes of at least fourth order in Hpd in the cluster of Fig. 2. All the processes leading to spin exchange, but involving configurations different from those of Fig. 2, are of higher orders than fourth order. We will not consider them here, but limit ourselves only to the leading order in spin-exchange formation. We now construct a unitary transformation of the Hamiltonian (6), which eliminates all odd terms in V up to third order (see e.g. Ref. [lo]):
condition:
101
Hamiltonian
(9)
(7) has the form
B=H,+H,+H,, H,=l/2[H,,,
Ll],
(10)
H4 = 1/2[ Hpd, L3] - l/24
[[[Hpdt LI],
LI]LI].
The Hamiltonian Hz corresponds to the second order in V and, along with corrections to H,,, contains terms of interaction between localized spins (Si> and delocalized spin density of p-electrons (s,) (see (3)). According to our calculation for the cluster of Fig. 2, Ji, =J=
V*/(E,
-Ed).
(11)
The Hamiltonian H4 is of fourth order in V. Along with the fourth-order corrections to H,,, H2 and terms describing the interactions between charge densities, it contains terms of spin exchange between localized spins. We denote them as H,, for which we have H, in the form of (4,5), where
b(n)
=
3(EpTEd)‘1-n)+i
1*
(12) Here II is the p-site filling per spin. When obtaining (12) we used a mean field approximation over the ground state of Ho for the product of operators a,+ap, i.e.
L], L], L]/31
L], L]/21=0,
where the square brackets denote commutators.
(8)
According to (12), the partially filled band contribution to superexchange is of antiferromagnetic type
(Z,(n) > 0).
102
V.V. Tugushev, EA. Zhukovsky / Journal of Magnetism and Magnetic Materials 138 (1994) 99-104
Thus we have demonstrated for the example of the cluster of Fig. 2 how I,(n) can be obtained. One can consider more complex clusters, but the qualitative picture remains the same: while W,, < E, - E,, the dependence of the partially filled band contribution to the exchange integral is linear in n because nonlinear terms in perturbation theory up to fourth order in V do not appear, but the fourth order is enough to establish superexchange.
3. Magnon spectrum The existence of band carriers opens up the possibility of RKKY exchange between localized spins, which results from the second order in HSb processes. In the frames of our model for nearest neighbours (or, to be more exact, for wavevectors close to the vector of the AF structure) it possesses a ferromagnetic character. The destruction of AF long-range order (LRO) in the frame of the considered model appears because of the competition between AF superexchange with integral I(n) and ferromagnetic indirect exchange of RKKY type. The AF LRO as a ground state has a corresponding Goldstone mode sound spin wave. The loss of AF LRO will manifest itself in the spectrum of collective excitations through the destruction of the Goldstone mode, which can be expressed as the loss of positiveness of spin wave energy (or spin wave velocity c) in the long-wavelength limit. Using the criterion c > 0 it is possible to find the band filling range in which the AF state exists. First, we calculate the spin wave dispersion law taking into account many-body effects due to interaction HSb (1). The renormalized spectrum 0(k) can be obtained in the random phase approximation for a spin wave Green function from the equation: 0(k)
=o(k)
+P@C)n(k,
C!(k)),
where w(k) is the initial dispersion
(13)
U( k, w) =
c P
b(P + k) -h(P) w-~(p+k)
Vk
=
-sgny,[([l-y,2]P1’2-1)/2]1’2,
+iS’
where n,(p) is the Fermi function, 6 is a small positive value, the summation in yk(Jk) runs over the lattice sites occupied by delocalized spins (p-electrons) of the first coordination sphere of some localized spin, a is a lattice constant in the spin system, 1= a/2 for the cluster of Fig. 2. Here 17(k, w) is a polarization operator constructed with the help of the Green functions of band electrons. In the long-wavelength limit (k -xpF, pF is the Fermi momentum) for n(k, w) we have:
where N(E~) is the density of states at the Fermi level per spin, and k is counted from the AF structure wavevector. In a 2D system for band filling near the band edge, where the dispersion law is parabolic, the density of states is known to be independent of the Fermi energy, when the Fermi level lies within the allowed energy band. When the transitions of electrons between neighbouring magnetic layers is allowed for, weak transverse dispersion appears in the electron spectrum:
4
.$p)=-+WWI(l-cos
PLd)>
2rnll
(15)
where P,,,~ are the longitudinal and transverse momenta, respectively, W, is the width of the band in the transverse direction, m,, is the effective mass of electron for motion along the layer (m,, N W,: * ), and d is the distance between AF layers. The density of states for the spectrum (15) has the form
law,
J’
+c(p)
Xarccos
W, --E w,
1 ’ m,,b2 27F
(16)
where E = 0 corresponds to the bottom of the band, O(E) is a step function, and b is the in-plane lattice
V. V. Tqushev,
Ed. Zhukovsky / Journal of Magnetism and Magnetic Materials 138 (1994) 99-104
103
n nc Fig. 3. Effective exchange to EF = W, .
integral as a function of band filling: n, is the critical doping at which the AF LRO is violated,
constant for the p-electron system lattice formed by clusters of Fig. 1 low we consider only this case>. It is ing that in the case of or B- 2W, situation for iV( or):
(for a square b = a/&; beworth mentionwe obtain 2D
N(C,$g, while when filling have an anisotropic
(17) is so small that .er -K W, , we 3D model with the usual N(E~): (18)
Introducing (14), (17) and (18) into (13) we obtain for the in-plane spin wave velocities in the 2D and 3D regimes, %D
=
cOzeff(
‘1
J2m,,a2
=C 0 zm+zb(n) - ~ 4JjsT i
I ’
Er z+- 2w,
=
cOzdf(
=c
EF
0
(19)
‘1
I
z, +Z,(n)
-=Zw,
z (~)“2nl/3)) -J2Y1ipz
,
respectively, where co is a dimensional dependent on the type of lattice.
Expression (19) corresponds to the case of a two-dimensional system of spins and a two-dimensional electron band, while for (20) the electron band is anisotropic three-dimensional. If we consider the case of an AF ground state at n = 0, which corresponds to Z,,(O) > 0, then we obtain two possible situations, n being increased, depending on the value of Z,,(O) (Fig. 3); when the RKKY contribution is large enough the AF LRO can be violated in the 3D electron picture (Fig. 3a) and in the 2D one (Fig. 3b). It should be mentioned that our description is limited to the range of band filling IZ< n, (n, is the band filling, which limits the region of AFM order), especially in the 2D regime, because when considering the region of large band filling one should allow for nonparabolicity of the electron spectrum, which will manifest itself in N(E~), and also in the vicinity of n, the mean field theory for Z,(n) is not valid due to strong fluctuations.
)
and ‘3D
n I corresponds
(20) constant
4. Conclusions In this paper we have demonstrated how the dependence of the exchange integral in the effective spin Hamiltonian on the filling of electron band can be obtained. We assume that the band filling is large enough for the processes of strong localization and spin-polaron effects to be neglected. Really we bear in mind the carrier concentrations of 1011-1012 cme2 for AFM layers lo-100 A thick in semiconductor superlattices of e.g. MnSe/ZnSe or MnSe/CdSe.
104
V. V. Tugushev, Ed. Zhukovsky / Journal of Magnetism and Magnetic Materials 138 (1994) 99-l 04
We have made calculations in a one-band approximation for itinerant electrons and for the case of S = l/2 for localized spins. When interband processes in the delocalized system are allowed for, there appears a linear contribution to Z(n) from interband transitions, but qualitatively the Z versus n dependence remains the same in the limit W,, < E, -
Acknowledgement
Ed.
References
We have considered the case of T = 0. It is well known that the AF LRO in 2D system of spins is possible only at zero temperature [ill. The weak overlapping between electron wavefunctions of neighbouring AF layers in the multilayered structure makes magnetic and electron subsystems only quasi2D, which shifts the long-range ordering temperature to a finite value TLRo. The region of validity of the spin Hamiltonian derived in this paper is determined by temperatures T -=c TLRo, where TLRo is the NCel temperature. In the region T - TLRo our approach can not be applied because of strong critical fluctuations. The temperature region of short-range order TIA0 -c T < TSR0 (TSR0 is the crossover temperature) will be considered elsewhere.
Research was made possible in part by Grant No. MC8000 from the International Science Foundation and by Grant 94-02-03506 (No. 566) from the Russian Foundation for Fundamental Research.
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