Ferromagnetism induced in diluted A₁₋xMnxB semiconductors
Theoretical model has been developed for analysis of the peculiarities of new type of magnetism in diluted magnetic A₁₋xMnxB semiconductors. The coherent potential is introduced using the dynamic mean field theory (DMFT) approximation and the Baym- Kadanoff hypothesis valid for the case x < xc (x...
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Інститут фізики напівпровідників імені В.Є. Лашкарьова НАН України
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| Zitieren: | Ferromagnetism induced in diluted A₁₋xMnxB semiconductors / V.P. Bryksa, G.G. Tarasov, W.T. Masselink, W. Nolting, Yu.I. Mazur, G.J. Salamo // Semiconductor Physics Quantum Electronics & Optoelectronics. — 2004. — Т. 7, № 1. — С. 43-51. — Бібліогр.: 22 назв. — англ. |
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| author | Bryksa, V.P. Tarasov, G.G. Masselink, W.T. Nolting, W. Mazur, Yu.I. Salamo, G.J. |
| author_facet | Bryksa, V.P. Tarasov, G.G. Masselink, W.T. Nolting, W. Mazur, Yu.I. Salamo, G.J. |
| citation_txt | Ferromagnetism induced in diluted A₁₋xMnxB semiconductors / V.P. Bryksa, G.G. Tarasov, W.T. Masselink, W. Nolting, Yu.I. Mazur, G.J. Salamo // Semiconductor Physics Quantum Electronics & Optoelectronics. — 2004. — Т. 7, № 1. — С. 43-51. — Бібліогр.: 22 назв. — англ. |
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| container_title | Semiconductor Physics Quantum Electronics & Optoelectronics |
| description | Theoretical model has been developed for analysis of the peculiarities of new type of magnetism in diluted magnetic A₁₋xMnxB semiconductors. The coherent potential is introduced using the dynamic mean field theory (DMFT) approximation and the Baym- Kadanoff hypothesis valid for the case x < xc (xc is the percolation limit).
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43© 2004, V. Lashkaryov Institute of Semiconductor Physics, National Academy of Sciences of Ukraine
Semiconductor Physics, Quantum Electronics & Optoelectronics. 2004. V. 7, N 1. P. 43-51.
PACS: 72.15.Gd; 72.20.My; 73.61.Ga
Ferromagnetism induced in diluted A1�xMnxB
semiconductors
V.P. Bryksa1, G.G. Tarasov2, W.T. Masselink2, W. Nolting2, Yu.I. Mazur3, G.J. Salamo3
1V. Lashkaryov Institute of Semiconductor Physics, NAS of Ukraine, 41, Prospect Nauki, 03028 Kyiv, Ukraine
2Humboldt Universitat zu Berlin, Institut fur Physik, Newtonstrasse 15, 2489 Berlin, Germany
3Department of Physics, University of Arkansas, Fayetteville, Arkansas 7270
Abstract. Theoretical model has been developed for analysis of the peculiarities of new type of
magnetism in diluted magnetic A1�xMnxB semiconductors. The coherent potential is intro-
duced using the dynamic mean field theory (DMFT) approximation and the Baym- Kadanoff
hypothesis valid for the case x < xc (xc is the percolation limit).
Keywords: diluted magnetic semiconductors, ferromagnetism, coherent potential, exchange
interaction.
Paper received 05.03.04; accepted for publication 30.03.04.
1. Introduction
It is reckoned traditionally that the description of diluted
magnetic semiconductors (DMS) is suitable to start with
the Vonsovskii Hamiltonian. Such approach has proved
to be efficient in the case of magnetic semiconductors [1,
2]. However, in the opposite case of strongly diluted
magnetic A1�xMnxB semiconductors, where the electron
scattering caused by the structural disorder is of great
importance, the traditional scheme of investigation [1, 2]
has to be altered.
It has been proposed [3] to use the technique of a co-
herent potential for determination of the averaged elec-
tron Green�s function with the self-energy part that is a
single node like. Then the ensemble over the spin projec-
tions can be introduced and the configuration averaging
in the sub-space of spin projections for the transference
matrix can be performed thus allowing the coherent po-
tential approximation (CPA) to be closed self-consistently
in the zeroth-order approximation. The numerical solu-
tion of the corresponding equations shows a significant
difference in the density-of-states (DOS) for the opposite
directions of the electron spin in the conduction band.
The peculiarities of the electron spectrum of the semi-
conductors with a chaotic distribution over the magnetic
sub-system have been studied by R.N. Bhatt et al [4,5]
introducing the paramagnetism of the semiconductor
rather as an experimental fact. The diluted magnetic semi-
conductors have been considered also as the strongly cor-
related systems [6�8] allowing the principal mechanism
of the ferromagnetism to be the double exchange through
the resonant energy levels and the levels caused by the
broken bonds due to the presence of deep Mn states [9].
In this paper, we derive the equations for the electron
Green�s functions basing on the CPA for the randomly
distributed ions of transition metals in DMS. A number
of simplifications that are not crucial for the formation of
the energy spectrum of DMS are made. It is assumed the
�cluster� scheme with the energy in the conduction band
that corresponds to the anti-binding energy εA whereas
the energy in the valence band is the binding energy εB.
The energy bands in the tight-binding approximation are
formed through the integrals of transfer tα,β (α, β = A, B).
Since the Mn ion has the stable half-filled d5 shell that
forms the localized spin momentum , an electron can not
be immediately localized in the state. Moreover, the Mn
presence in the A1�xMnxB semiconductor generates the
resonant and broken bonds with the energy εMn [9]. For
the correct description of the electron sub-system of the
semiconductor, it is necessary to take into account the
field of the nearest surrounding. We assume that the crys-
tal lattice (of the zinc blend structure) is composed of two
sub-lattices: one is the sub-lattice of A atoms and the
substitutional atoms (Mn atoms), and another is the sub-
lattice of B atoms. The conduction band is formed basi-
cally from the s states of the A1�xMnx sub-lattice, and the
valence band is formed from the p states of the sub-lattice
B in the A1�xMnxB compound. First the averaged Green�s
function and the DOS function are determined for the
A1�xMnx sub-lattice, the atoms of which are in the crystal
field created by the B atoms. Then the analogous calcu-
lations are performed for the B sub-lattice, the atoms of
which are in the random field of A and Mn atoms. Deter-
mination of the Green�s functions for both sub-lattices
has to be self-consistent. The general form of the Hamil-
tonian for the A1�xMnxB compound is given in Section 2
44
SQO, 7(1), 2004
V.P. Bryksa et al.: Ferromagnetism induced in diluted A1�xMnxB semiconductors
using the technique of projective operators. The electron
propagator is found in Section 3 using the graph tech-
nique in the Hubbard-I approximation. The configura-
tion averaging of the Green�s functions is performed and
standard CPA equations are written in Section 4. Section
5 presents the results of numerical solution of the CPA
equations. The discussion of these results is given in the
concluding Section 6.
2. The model Hamiltonian for general
problem of DMS
In general case the Hamiltonian of strongly diluted mag-
netic semiconductor (SDMS) can be written in the cluster
approximation as
∑ ∑
><
++=
i ji
jiiji aatHH
σ
σσ
,,
ˆˆ , (1)
where the single node part iĤ of the Hamiltonian has
the Kondo-like form [1, 2],
.)(
ˆ
',
''
),(
∑ ∑
∑
++
−+=
+
++
+=
σσ σ
σσσσσσ
σ
σσ
σα
ε
ii
lat
iiiii
ii
i
i
aaLaaS
aaH
rr (2)
Here +
σia and σia are the creation and annihilation op-
erators, respectively, for an electron with the spin σ at the
i-th node.
For the energy at the i-th node one has:
{ }
∑
∈
=
BMnAk
kk
ik
i X
,,
εε
. (3)
The crystal field energy is given by
∑∑∑
≠
=
k l ji
ll
j
kl
ij
kk
i
lat
i XXL λ
(4)
with
rdrRrUr kijlki
kl
ij
rrrrr
)()()(* ϕϕλ ∫ −= .
The )( il RrU
rr
− term is the potential energy of the elec-
tron at the point r
r
near the node defined by the radius-
vector iR
r
. The cluster wave functions )(rki
rϕ are con-
structed from the atom functions localized at the i-th node
the sort of ê and possess the tetrahedral symmetry Td [9].
The transfer matrix is written as
∑=
'
''
'
kk
kk
j
kk
ikkij XXtt . (5)
The exchange interaction constant is defined by
∑ ∑∑
≠
=
k kk j
kk
j
kk
ij
kk
ii XX
'
'''αα , (6)
with
')'()'(
'
)'()( '
*
'
*' rdrdrr
rr
rr jkkijkki
kk
ij
rrrr
rr
rr ϕϕϕϕα ∫ −
= 1
. (7)
The projective operator takes the meaning 1 if the i-
th node is occupied by the k atom (k∈ (A, Mn, B)) and the
meaning 0 in other case.
3. Green�s functions for the A1-xMnx sub-lattice
The Hamiltonian (1) written in terms of projective opera-
tors kk
iX can be rewritten in the interaction representa-
tion considering the A1�xMnx sub-lattice in the crystal
field of the B sub-lattice. In what follows we allow the k
index of projective operators to be { }MnAk ,∈ and as-
sume that the matrix of electronic transfer determines the
conduction band half-width in the A1�xMnxB semicon-
ductor.
Thus the Hamiltonian (1) reduces to the form of
int
ˆˆˆ HHH o += , (8)
where the 0Ĥ term represents a single node part and
.)(~)(~
)(~)(~)(ˆ
,
,
int
ττ
τττ
σσ
σ
σσ
σ
ji
ij
ij
ii
i
lat
i
aat
aaLH
+
><
+
∑
∑
+
+=
(9)
The operators ( )τσia~ are given by 00 H
i
H
i eaea
ˆˆ
)(~ τ
σ
τ
σ τ −= .
In order to find the Green�s functions the averages
like the
)(...)(... Ĥ
aa eSpXZ β−−=>< 1 (10)
have to be calculated. The technique similar to that de-
veloped in Ref. [10�13] is used for calculation of the
Green�s function
,),()(~)(~),(
)(~)(~);(
aiia
aiiX
XaaTX
aaTiiG
021
1
0
21
21
212211
><><−=
=><−=
+−
+
βσττβσ
ττττ
σστ
σστ
σ
(11)
with
−= ∫
β
τ ττβσ
0
dHTX )(ˆexp),( int . (12)
If one expands the operator ),( Xβσ in the power series
with respect to intĤ , using the Hubbard-I approxima-
tion [15, 16, 22], the graphical equation for the Green�s
function can be written as follows
=↑ );( 2211 ττ iiGX + + +... .
(13)
=↓ );( 2211 ττ iiGX + + +... ,
(14)
with
= + + +
... , (15)
and
= + + + ... . (16)
Here the graphical notations are used as follows:
kk
i
k
n
k
iiiniii Xigig 1121121 ∑ ↑↑ =≡ )()( ωδωδ is the elec-
tron propagator with the spin-up expressed through the
V.P. Bryksa et al.: Ferromagnetism induced in diluted A1�xMnxB semiconductors
45SQO, 7(1), 2004
self-energy part )()( nki iω↑Σ of the electron scattering by
the magnetic sub-system.
kk
i
k
n
k
iiiniii Xigig 1121121 ∑ ↓↓ =≡ )()( ωδωδ is the simi-
lar for the propagator with the spin-down.
lat
iL≡ is the crystal field Eqn (4).
21iit≡ is the matrix of transfer Eqn (5).
The sets (13) and (14) after summation reduce to
= =++ ↑↑↑ ...)()()( ni
lat
iniiiniii igLigig ωωδωδ 11121121
∑
↑
↑
−
=
k
k
i
klat
ii
kk
i
k
i
ii
gL
Xg
111
11
21
1 )(
δ , (17)
and
=++= ↓↓↓ ...)()()( ni
lat
iniiiniii igLigig ωωδωδ 11121121
∑
↓
↓
−
=
k
k
i
klat
ii
kk
i
k
i
ii
gL
Xg
111
11
21
1 )(
δ (18)
respectively.
Now we can write the Dyson equations for the elec-
tron propagator kk
i
k
n
k
ini Xigig ∑= )()( ωω σσ :
= + ↑S , (19)
and for the electron propagator )( n
k
i ig ωσ =
)()( nkikn ii ωεω σΣ−−
= 1
:
= + . (20)
One can notice that the self-energy part of the propa-
gator σ
iΣ occurs to be a single nodal following the
DMFÀ[22]. The calculation of such local self-energy
parts is performed in Appendix I.
4. Averaging of the Green�s functions
over configurations
In order to perform the averaging over the Õ operators
the electron DOS operator Xρ̂ can be written as
[ ]∏ ∏ +−==
i i
MnMn
i
AA
i
i
XX xXXx)(ˆ 1ρρ , (21)
where the A and Mn concentrations are determined
through the averages of the Õ operators
k
i
X
kk
iXX
kk
i cXSpX ==>< )( ρ , (22)
with xcxc MnA =−= ,1 .
For the averaged electron propagator Xig >< σ one gets
Mn
iMn
A
iAXiXXi gcgcgSpg σσσσ ρ +≈=>< )ˆ( . (23)
Equation (23) is the approximate one that corresponds
to the alloy approximation and does not take into ac-
count the correlation effects.
Averaging the Green�s function we use the cumulant
decompositions [10�13]
∑
>+
+ >==<
0
2121
mn
mn
nm
mn
Lg M
mn
eS
lat
ii
,
)( ),
!!
][][
exp( σςςσ ςςσ
(24)
where 0
21
21 ==
∂
∂
∂
∂= ςς
σσ
ςς
|
)(SM
m
m
n
n
nm .
The node corresponding to the nmM σ cumulant ga-
thers the n + m separate parts in the electron propagator
)( ni ig ωσ , e.g.
10
↑M =
= Xig >< ↑ ;
20
↑M =
�
>−−<− ↑
↑
↑
↑ ))(( Mn
i
A
i gMgM 1010 , (25)
>=<= ↓↑ lat
iLMM 0101 = .
Resulting from the configuration averaging of the
Green�s functions >< σ
iG given by the graphical expres-
sions (13) and (14) one gets the following diagrams
1 2 3 4
;
;
;
;.... (26)
In this study we consider solely the diagrams for which
the configuration averaging embraces the electron propa-
gators and the crystal field separately (diagrams 1 and 2,
see (26)). Such averaging is equivalent to the statement of
the coherent potential independence on the self-consist-
ent field. The Green�s functions become renormalized,
but new poles do not arise. The configuration averaging
of each line can be performed in different ways (the clus-
ter approximation, the averaging of independent field,
and so on). Therefore it is suitable to introduce the spe-
cial notation of such renormalized Green�s function like
>=< )( ni ig ωσ
= < > =
= [ ]∑ >
−
< −
k lat
in
k
i
kk
i
Lig
X
1
)( ωσ
. (27)
In the cluster approximation one has:
[ ]
[ ]∑
=
−
−
−−−
=
>=
−
≡<
l
r
kk
ij
kk
ijn
k
i
lat
in
k
i
n
k
i
rrlig
rw
Lig
iD
0
1
1
1
,
)()(
)(
)(
)(
'λλω
ω
ω
σ
σ
σ
(28)
where .,',;;
)!(!
!
)( ' MnAkkkkcc
rlr
l
rw rl
k
r
k =≠
−
= − l is the
number of the nearest neighbors; i, j denote the nearest
nodes of the A1-xMnx sub-lattice.
The further averaging over the transfer matrices leads
to the equation for the averaged Green's function,
46
SQO, 7(1), 2004
V.P. Bryksa et al.: Ferromagnetism induced in diluted A1�xMnxB semiconductors
.)(
)(
,
∑ ><Ξ+
+Ξ=><
43
244331
2121
ii
iiiiinii
niiiii
n
n
Gti
iG
ω
σσ
σ
ω
σ
ω
ω
(29)
Here the σ
21iiΞ term is the self-energy part of the aver-
aged Green�s function (irreducible by transfer) for whole
crystal that has the following diagram representation:
σ
21iiΞ =
+
+
+ + � .
(30)
Let us retain the diagrams in the σ
21iiΞ expression (30)
of the single node nature solely (e.g. 1, 2, 3 and so on).
Such procedure corresponds to the DMFT approxima-
tion or to the approximation of self-returning paths [14-
16]. Then equation (30) can be rewritten in the form
(
+
+ + + �). (31)
σJ≡ is the sum of all diagrams beginning and termi-
nating at the same node and having no common
cumulants. The set (31) can be calculated analytically
using equation (25). Thus,
( ) .σ
σ
σ
JDcDc
JDDDcDc
Mn
Mn
A
A
MnAMn
Mn
A
A
i
+−
−+=Ξ
1
(32)
On the other hand, the sum Jσ can be determined if
one finds the sum of all diagrams beginning and termi-
nating at the same node and having the common
cumulants:
s s s
i i i
Ξ Ξ Ξ
+ + ... =
[ ] =+Ξ+Ξ ∑∑ ...)()(
k
nik
k
nik
it
N
it
N r
r
r
r
2211 ωω σσ
(33)
=
[ ]
[ ]∑
−Ξ
−
k kni
k
ti
t
N r
r
r
1
2
1
)( ωσ
,
where ∑ −−=
ij
ij
RRk
k
tet ji )(
rrr
r is the Fourier transform of the
transfer matrix.
It is clear that the sum (33) is equal to
[ ]
[ ] { })()()(
)(
nini
k
ni
kni
k iJiiJ
ti
t
N
ωωω
ω
σσσ
σ
Ξ−=
−Ξ
∑ − 1
1
1
2
r
r
r
.
(34)
Let us rewrite the expressions (29), (33), and (34) in
the traditional form of the equations for the coherent po-
tential as follows:
[ ]∑
∑
−Ξ
=
>=<>=<
−
k kni
k
nkni
tiN
iG
N
iG
r
r
r
r
,
)(
)()(
1
11
1
ω
ωω
σ
σσ
(35)
Then the coherent potential )( ni iJ ωσ takes the form
[ ] [ ] .)()()(
11 −−
><−Ξ= ninini iGiiJ ωωω σσσ (36)
Thus, solving the single node problems (19) and (20)
(see Appendix I) one gets the solution of the self-consist-
ent problem for the A1�xMnx sub-lattice.
5. Numerical results
In order to complete the ÑÐÀ equations ((35) and (36))
the explicit expressions for the local single node self-en-
ergy parts )()( ωσ ikiΣ in the Dyson equations (19) and (20)
must be derived. These parts define the peculiarities of
the exchange interaction between an electron with the
spin σ and the localized magnetic moment of the Mn ion.
They can be found by projecting the Hamiltonian (1) on
the Andersson-like Hamiltonian, the type of [17],
ξ
σ
σσσσ ξξ HaaVHH
i
ii
new ˆ)(ˆˆ +++= ∑ ++
11110 , (37)
where +
σσ ξξ 11 , are the operators of annihilation and crea-
tion of an electron with the spin σ in the non-magnetic ion
outside the ³-th cluster, ξĤ is the Hamiltonian of vacuum.
The Hamiltonian 0Ĥ corresponds to the cluster con-
sisting of the energies of magnetic (1) and non-magnetic
(2) ions as shown in Fig. 1. We distinguish between the
electron transfer over the properly non-magnetic atoms
and that occurring within the cluster between the non-
magnetic and magnetic atoms. This scenario seems to be
reasonable in the case of strongly diluted magnetic semi-
conductors (SDMS). Thus the 0Ĥ part of the Hamilto-
nian (37) takes the explicit form:
.).()/(
)(
ˆ
',
''
∑
∑
∑∑
+++
++
++=
+
+
++
σ
σσ
σσ
σσσσ
σ
σσ
σ
σσ
σα
εε
i
ii
iii
i
iiMn
i
iiA
ccaaSztS
aaS
aaaaH
12
22
22110
12
rr
(38)
The self-energy parts for the Dyson equations (19)
and (20) are defined by relations:
)(
1
|
)(
11
ωεω σωσσ
AiA
aa
Σ−−
=>><< + , (39)
)(
1
|
)(
22
ωεω σωσσ
MniMn
aa
Σ−−
=>><< + , (40)
and the coherent potential Jσ(ω) is determined through
the Baym-Kadanoff field [18] as follows [17]:
ωσσσ ξξπω >><<= +
11
2 |2)( VJ . (40)
The ωσσ >><< +
11 aa | and ωσσ >><< +
22 | aa Green�s
functions finding is given in Appendix I. Starting the SDMS
investigation one can use the spin-polaron approximation
of the single node Hamiltonian developed in Ref [19]. Ne-
V.P. Bryksa et al.: Ferromagnetism induced in diluted A1�xMnxB semiconductors
47SQO, 7(1), 2004
glecting transverse components of the classical spins the
single node Hamiltonian can be written in the form [19]:
.).)(/1()12/( 12
22110
∑
∑∑
++++
++=
+
++
σ
σσ
σ
σσ
σ
σ
σσ εε
i
ii
z
i
i
ii
i
iiA
ccaaSSSztS
aaaaH
(41)
For the sake of simplicity, we introduce the common
notation σ both for the pseudo-spin, if the electron is at
the magnetic atom (2), and for the spin, if it is at the non-
magnetic atom (1). 221 // SMn αεε σ −== is the energy of
the spin-polaron with the pseudo-spin σ = 1/2 and
2121 /)(/ ++=−= SMn αεε σ is the energy for the pseudo-
spin –σ = 1/2. Introducing the Hamiltonian (10) we modify
somewhat the effect of magnetic field on the A1�xMnxB
semiconductor similarly to the effect of magnetic field in
the transverse Ising model.
In what follows, we calculate the electron spectrum
basing on the self-consistent solution of the set of equa-
tions (32), (35), (36), and (À.9), (À.19) for the DFTA tech-
nique [22], that reduces equation (35) to the form:
[ ] [ ]∑ ∫
−Ξ
=
−Ξ
−−
k nikni itiN r
r
max
min
11
)(
)(
)(
11
ε
ε
σσ εω
ερ
ω
. (42)
The DOS function for free electron states is approxi-
mated with 22
2
2
)( ε
π
ερ −= W
W
in case of the semi-
elliptic Bethe lattice, and with
2
1
)(
−
= We
W
ε
π
ερ in
case of the cubic lattice.
The iteration procedure is exploited in order to get
the solution. At the first stage the zero approximation for
the coherent potential Jσ(ω) is taken and the expression
)(ωσΞ is calculated from the equation (32). The single
node local propagators )(ωσ
AD and )(ωσ
MnD are defined
using the expressions (À.9) and (À.19) if the crystal field
(28) is neglected. Then using the relation between the
averaged Green�s function and the Green�s function of
whole crystal (35), and the model DOS functions (5.7)
the averaged Green�s function )(ωσG is calculated.
Then the coherent potential Jσ(ω) is re-defined using the
relation (36). This re-defined potential serves again as
the zero approximation potential, and the next loop of
the calculation is fulfilled. In this scheme it is of great
importance to find an appropriate expression of the co-
herent potential for the zero approximation. To this end
the particular algebraic equation has been solved. This
equation was derived in the semi-elliptic case and in case
of the alloy approximation for the Ξσ(ω) corresponding
to the simplified expression (28):
)()()()( ωωω σσ
σ MnA
i xDDx +−=Ξ 1 . (43)
Similar approximation was used in Ref. [3].
The dependences of the band centers on the εMn en-
ergy for non-magnetic and magnetic atoms calculated
from the equations (À.8) are depicted in Fig. 2. It can be
seen that in case of a deep position of the Mn2+ energy
levels the positions of the conduction band center for dif-
ferent spins are very close, whereas the centers of bands
which are formed with the Mn2+ participation are split
by the spin-polaron energy αS. If the manganese levels
move upper the complicated scheme of splitting is ob-
served [9]. While the analysis of the cluster with the Ham-
iltonian (41) is performed in the mean field approxima-
tion for the magnetic sub-system, the effect of additional
magnetization 0>≠< zS reduces to the additional shift
of the band centers in Fig. 2 due to the change of the
electron hopping rate between magnetic and non-mag-
netic atoms interior the cluster (À.2).
The edges of the formed bands demonstrate more
complicated behavior. Indeed, the semi-elliptic band in
the alloy approximation (43) that is the partial case of
expression (28) has been considered in Ref. [3]. It was
shown the existence of the induced ferromagnetism due
to a complicated structure of the band edge that is sensi-
tive to the change of the magnetization SSz / of the
magnetic sub-system. This structure manifests itself not
only in the edge transformation, but in the formation of
new bands also. The value of the ratio αS/W, αS/W < 1
for the case of DMS, is of importance for the develop-
ment of the structure. It has been shown that for magnetic
atoms the narrow bands with different spins arise in the
spin-polaron approximation whereas for the conduction
bands with different spins the DOS functions nearly coin-
cide. It follows that the conduction band is significantly
narrower due to the correlation effects (if x → 0,
6WW → ) [14�16, 20, 21]. If x increases, the conduc-
tion band width (determined by non-magnetic atoms) has
to be smaller, whereas the spin-polaron states (related to
magnetic atoms) broaden out. Besides the additional
sub-bands arise near the conduction band as well as the
resonant states appear interior the conduction band due
to more complicated scattering processes amplified by
correlation.
In order to fall outside the limits of the alloy approxi-
mation (43), the more complicated cumulants basing on
Fig. 1. The �cluster� consisting of two atoms, magnetic and non-
magnetic. At the magnetic atom there are the spins up and down
corresponding to the energies
2
S
d
α
ε − and
2
)1( +
+
S
d
αε respec-
tively. The level εA is degenerate by spin at the non-magnetic
atom.
spin
e
x
x
e a+ ( + 1)/2
S
e a� /2
S
A
s
s s
pseudo-spin
non-magnetic atom
magnetic atomvacuum
+
J ( )w
48
SQO, 7(1), 2004
V.P. Bryksa et al.: Ferromagnetism induced in diluted A1�xMnxB semiconductors
the equation (32) have to be taken into account. Fig. 3
shows the electron DOS dependences on energy for non-
magnetic atoms if three different approximations are con-
sidered. In case of the elliptical DOS (Fig. 4), the corre-
lation squeezing of the band decreases and the additional
sub-band forms the edge of the conduction band beyond
the alloy approximation. In the case of the gauss DOS
(Fig. 5), the band undergoes even smaller correlation
squeezing. Comparing the results for elliptical and gauss
densities we find that the edge of the conduction band
behaves itself very similarly in both cases. However, the
elliptic band becomes narrower in comparison with the
gauss band, thus providing better facility to see the addi-
tional sub-bands. Besides, the more complicated struc-
tures arise near the edges of the conduction band (Fig.5).
Such structures are absent for spin-up states.
3
4 6
2
2
1
0
0
�1
�2
�2
�3
�4
�4
�5
�6
�6
e = 0
a
,
b
e , eV
Mn
W = 2 eV
< > = 0, S e
< > = 0, S e
< > = 0, S e
< > = 0, S e
< >/ = 1, S S e
< >/ = 1, S S e
< >/ = 1, S S e
< >/ = 1, S S e
z
s =↓
s =↓
s =↑
s =↑
s =↑
s =↑
s =↓
s =↓
z
z
z
z
z
z
z
a
a
a
a
b
b
b
b
Fig. 2. Dependences of the band centers βα
σε , on the energies
εMn in the case of the fixed value εA = 0. These dependences are
found from (À.8) for two limiting cases: paramagnetic 0/ =SSz
and saturated ferromagnetic 1/ =SSz .
�4.8
0.0
0.1
0.2
0.3
�4.7 �1.5
w, eV
r
(
w
)
↑
elliptic
elliptico
gauss
Fig. 3. Dependences of the electron DOS on the energy for the
states with the spin up calculated in the alloy approximation (43)
and the elliptic band, taking into account the higher cumulants
(32) and the elliptic band, and assuming the higher cumulants
contribution and the gauss density for the set of parameters:
W = 2 eV, x = 0.1, εA = 0, εMn = �2W, α = 0.45, 0/ =SSz (non-
magnetic atoms).
Fig. 4. Electron DOS dependences on the energy in the A1�xMnxB
semiconductor calculated assuming the higher cumulants con-
tribution and the elliptic band at W = 2 eV, x = 0.1; õ = 0.2; õ =
= 0.3; õ = 0.4; õ = 0.6, εA = 0, εMn = �2W, α = 0.45 and 0/ =SSz .
�4.8 �4.0 �3.2 �0.5 0.0 0.5
< >/ = 0S S
z
x = 0.1
x = 0.2
x = 0.3
x = 0.4
x = 0.6
w, eV
w, eV
r w( )
r w( )
r w( )
r w( )
↓
↓
↑
↑
0.00
�4.8 �4.0 �3.2 0�1
< >/ = 0S S
z
x = 0.1
x = 0.2
x = 0.3
x = 0.4
w, eV
w, eV
r w( )
r w( )
↓
↑
0.00
r w( )
r w( )
↓
↑
Fig. 5. Electron DOS dependences on the energy in the A1-xMnxB
semiconductor calculated subject to the higher cumulants (32)
and the gauss band at W = 2 eV, x = 0.1; õ = 0.2; õ = 0.3; õ = 0.4,
εA = 0, εMn = �2W, α = 0.45 and 0/ =SSz .
�4.8 �4.0 � .0 5 0 5.0
< >/ = 0.5S S
z
x = 0.1
x = 0.2
x = 0.3
x = 0.4
w, eV
w, eV
r w( )
r w( )
↓
↑
0.00
r w( )
r w( )
↓
↑
Fig. 6. Electron DOS dependences on energy in the A1�xMnxB
semiconductor calculated assuming the higher cumulants con-
tribution (32) and the gauss band at W = 2 eV, x = 0.1; õ = 0.2;
õ = 0.3; õ = 0.4, εA = 0, εMn = �2W, α = 0.45 and 0/ =SSz .5.
V.P. Bryksa et al.: Ferromagnetism induced in diluted A1�xMnxB semiconductors
49SQO, 7(1), 2004
All dependences given above have been calculated for
the case ./ 0=>< SS z It is clear that the single node prob-
lem solved in Appendix I is oversimplified to be used for
the investigation of the spin correlations. Under such ap-
proach the introduction of 0≠>< SS z / could result in a
shift of the band centers (À.8) (see, Fig. 2), that in turn can
magnify the band edges for smaller x with respect to the
case 0=>< SS z / (Fig. 6). Thus, the principal effect of
0≠>< SS z / is the DOS change near the band edges,
whereas the band profile does not change (Fig. 7).
6. Conclusions
There exist a number of points differing the class of SDMS
the type of A1�xMnxB from other classes of magnetic ma-
terials (spin glasses, diluted Heisenberg magnets, weakly
ferromagnetic semiconductors, etc.): i) the peculiarities
of SDMS are determined by the peculiarities of the elec-
tron band structure. The magnetism disposed experimen-
tally in this class of semiconductors belongs to the inte-
rior type of magnetism making it contiguous to the
Hubbard class of materials; ii) since at high x concentra-
tions of the transition metal the formation of clusters is
unavoidable, the theory has to be developed for the range
of concentrations cxx < . The percolation concentration
xc is the constant depending on the space dimensionality
d and on the type of the crystal lattice, i.e. for d = 3,
0.1 < x < 0.3. For the A1�xMnxB semiconductors at the
random distribution of the magnetic component over the
crystal lattice is of importance that makes the investiga-
tion of these systems basing upon the diluted Heisenberg
model inadequate. In this model the formation of the Curie
temperature Tc occurs due to the long-range order whereas
for the SDMS the Tc formation is contributed by the short-
range ordering magnified by magnetic fluctuations; iii)
the magnetic momentum of the Mn2+ ion in AB lattice
behaves itself similarly to the case of free Mn2+ ions. Thus
the magnetic sub-system of the strongly diluted A1�xMnxB
semiconductors can be described using the Hamiltonian
like the Vonsovskii Hamiltonian.
The self-consistent formalism developed in this paper
takes into account all above-mentioned points. It can be
considered as a first step to the problem of magnetism in
the SDMS. In our consideration of the model Hamilto-
nian we do not use the great canonical distribution. It
means that the electronic sub-system is considered at zero
temperatures, whereas the temperature effect on the mag-
netic sub-system is taken in the mean field approxima-
tion (À.2). The conduction band plays here the formal
role, while rather the single-band model of semiconduc-
tor is used. The development of the multiple band theory
based on the coherent potential is in progress. It is clear
that the cluster approximation has to be extended to seek
the Green�s functions of the spin operators [1, 2]. In the
developed model the correlation band squeezing is ob-
served in contrast to the model [3] where the band edges
follow the Hubbard model in the alloy approximation. It
is of importance to take into account the Coulomb repul-
sion of the electrons with opposite spins at the magnetic
atom [6�8]. This repulsion can substantially change the
behavior of SDMS at least at certain relations between
three parameters W, U, αS available in such models.
Appendix 1. Single-node Hamiltonian for the
A1�xMnxB compounds
The Hamiltonian in the node representation can be writ-
ten as
ξ
σ
σσσσ ξξ HaaVHH
i
ii
new +++= ∑ ++ )(ˆˆ
11110 , (À.1)
where
∑
∑∑
++++
++=
+
++
σ
σσ
σ
σσ
σ
σ
σσ εε
i
ii
z
i
i
ii
i
iiA
ccaaSSSztS
aaaaH
).)(/()/(
ˆ
12
22110
112
and
221 // SMn αεε σ −== , 2121 /)(/ ++=−= SMn αεε σ .
While the localized spin moments follow the para-
magnetic gas behavior we use the mean field approxima-
tion assuming that the magnetic field influences the clus-
ter changing only the rate of hopping between the mag-
netic and non-magnetic ions. In this case the expression
reduces to the form
,).(
ˆ
∑
∑∑
+Ω+
++=
+
++
σ
σσ
σ
σσ
σ
σ
σσ εε
i
ii
i
ii
i
iiA
ccaa
aaaaH
12
22110
(À.2)
with the )/)(/( SSSztS z ><++=Ω 112 .
In fact the Hamiltonian newĤ is composed in such
way that for the averaged Green function determined by
the CPA technique the following equality takes place
)ˆexp(ˆexp(
)(
eff
Hnew HH ββ ξ −=>−< . (À.3)
Within the CPA technique the transition to the statis-
tic operator is performed as follows:
Fig. 7. Electron DOS dependences on the energy in the A1�xMnxB
semiconductor calculated subject to the higher cumulants (32)
and the gauss band at W = 2 eV, x = 0.1, εA = 0, εMn = �2W, α =
0.45 and /SSz = 0.1; /SSz = 0.3; /SSz = 0.5; /SSz = 0.7;
/SSz = 0.9.
< >/ = 0.1S S
< >S / = 0.3S
< >S / = 0.5S
< >S / = 0.7S
< >/ = 0.9S S
z
z
z
z
z
x = 0.1
w, eV
w, eV
r w( )
r w( )
↓
↑
0.00
r w( )
r w( )
↓
↑
50
SQO, 7(1), 2004
V.P. Bryksa et al.: Ferromagnetism induced in diluted A1�xMnxB semiconductors
.)()()(exp
ˆ
ˆˆ
−−=
=→
∫ ∑∫ +−
−−
β
σ
σσσ
β
β
ββ
ττττττ
0
211121
0
21
0 aaJdde
ee
H
HH eff
(À.4)
In terms of Green functions this procedure generates
the self-consistent set of equations
[ ]∑∑
−Ξ
>=<>=< −
k kk
k
tN
G
N
G
r
r
r
r ,
)(
)()(
1
111
ω
ωω
σ
σσ
[ ] [ ] 11 −−
><−Ξ= )()()( ωωω σσ
σ GJ , (À.5)
where the ωσσσ ξξπω >><<= +
11
22 |)( VJ is the dynamic
Baym�Kadanoff field. Under the assumption of the
)(ωσΞ function, independent on the k
r
vector, the set
(A.5) gives also the DMFT set [15, 16].
Thus the (A.5) set determines a dependence of the av-
eraged Green function of the coherent potential,
))(()( ωω σ
σ JfG >=< . (À.6)
The transformation
σ
σ
σ
σ
σ
σ
σ
σ
σ
σ
β
θ
α
θ
β
θ
α
θ
iii
iii
a
a
+
−=
+
=
2
cos
2
sin
,
2
sin
2
cos
2
1
(À.7)
reduces the 0Ĥ to the diagonal form
∑∑ ++ +=
σ
σσ
β
σ
σ
σσ
α
σ ββεααε
i
ii
i
iiH0
ˆ . (À.8)
Here ( )σ
σσα
σ θθεεεε sinsin)( Ω−
−−=
2
2
AA ,
( )σ
σσσβ
σ θθεεεε sinsin)( Ω+
−+=
2
2
A ,
and ( ) σσ εε
θ
−
Ω−=
A
tg
2
.
In order to find the Green functions ωσσ >><< +
11 aa |
and ωσσ >><< +
22 aa | , one needs to find the Green func-
tions ωσσ αα >><< +| and ωσσ ββ >><< +| . Indeed,
( )
,|sin
)||(
sin
|cos|
ωσσ
σ
ωσσωσσ
σ
ωσσ
σ
ωσσ
ββθ
αββαθ
ααθ
>><<
+
+>><<+>><<+
+>><<
=>><<
+
++
++
2
2
2
2
2
11 aa
(A.9)
( )
.|cos
)||(
sin
|sin|
ωσσ
σ
ωσσωσσ
σ
ωσσ
σ
ωσσ
ββθ
αββαθ
ααθ
>><<
+
+>><<+>><<−
−>><<
=>><<
+
++
++
2
2
2
2
2
22 aa
Write the equations of motion for the operators α and β:
[ ]
[ ]
[ ]
[ ] +++
+++
−−=
−−=
+=
+=
σ
σ
σ
β
σσ
σ
σ
σ
α
σσ
σ
σ
σ
β
σσ
σ
σ
σ
α
σσ
ξθβεβ
ξθαεα
ξθβεβ
ξθαεα
2
2
2
2
sinˆ,
,cosˆ,
,sinˆ,
,cosˆ,
VH
VH
VH
VH
new
new
new
new
(À.10)
The generalized Wick theorem can be used to find the
Green functions ωσσ αα >><< +| and ωσσ ββ >><< +| .
This theorem in terms of the irreducible Green functions
[17, 22] is formulated as follows
[ ]
[ ] .sinˆ,
,cosˆ,
β
σ
σ
σ
β
σσ
α
σ
σ
σ
α
σσ
θβεβ
θαεα
ZH
ZH
new
new
+=
+=
2
2
(À.11)
Here, β
σ
α
σ ZZ , , and ++ β
σ
α
σ ZZ , are the irreducible parts of
the Green functions ωσσ αα >><< +| and ωσσ ββ >><< +|
respectively, and they can be written explicitly as
σ
β
σσ
α
σ ξξ VZVZ == , . (À.11′)
In order to rewrite the equations of motion in more
compact form, the relations are used
{ } [ ] ωω π
ω >><<+><=>><< BHABABA new ˆ|ˆ,ˆˆ,ˆˆ|ˆ
2
1
and
{ } [ ] ωω π
ω >><<−><=>><< newHBABABA ˆ,ˆ|ˆˆ,ˆˆ|ˆ
2
1
,
(À.12)
where σσ βα ,ˆ =A ; ++= σσ βα ,B̂ and 1=h .
Then
000 GAPAGGG ˆˆˆˆˆˆˆ += , (À.13)
where
>><<>><<
>><<>><<= ++
++
ωσσωσσ
ωσσωσσ
ββαβ
βααα
||
||
Ĝ ,
−
−=
β
σ
α
σ
εω
π
εω
π
21
0
0
21
0 /
/
Ĝ ,
=
2
20
0
2
2
σ
σ
θπ
θπ
sin
cos
 ,
and
>><<>><<
>><<>><<= ++
++
β
σ
β
σ
α
σ
β
σ
β
σ
α
σ
α
σ
α
σ
ZZZZ
ZZZZ
P
||
||ˆ
.
Let us transform the matrix equation (A.13) into the
equation of the Dyson type transforming the mass opera-
tor according to
...ˆˆˆˆˆˆˆ ++= MGMMAPA 0 , (À.14)
where
.ˆ
||
||ˆˆ A
ZZZZ
ZZZZ
AM
irrirrirrirr
irrirrirrirr
>><<>><<
>><<>><<= ++
++
β
σ
β
σ
α
σ
β
σ
β
σ
α
σ
α
σ
α
σ
(À.15)
V.P. Bryksa et al.: Ferromagnetism induced in diluted A1�xMnxB semiconductors
51SQO, 7(1), 2004
Thus one gets
GMGGG ˆˆˆˆˆ
00 += . (À.16)
Irreducible parts of the Green functions can be calcu-
lated using the technique of two-time decoupling and the
spectral theorem, e.g.
.)()(
|
><+
−
=
=>><<
+∞
∞−
∞
∞−
−
+
∫ ∫ α
σ
α
σ
ωβω
α
σ
α
σ
πωω
ω
π
ZtZe
dt
e
d
ZZ
ti
irrirr
11
2
1
2
1
1
1
(À.17)
In the case of approximation (A.11′), the matrix
(A.15), using (A.17) and (A.5), can be reduced to the form:
A
JJ
JJ
AM ˆ
)()(
)()(
ˆˆ
=
ω
π
ω
π
ω
π
ω
π
σσ
σσ
2
1
2
1
2
1
2
1
. (À.18)
Solving matrix equations (A.16), the explicit expres-
sions for the Green functions are found.
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( )
−−
−
−−
=>><< +
2
sin)((4
sin)(
2
cos)()(2
1
|
2
22
2
σ
σ
β
σ
σσσ
σ
α
σ
ωσσ
θ
ωεω
θωθ
ωεωπ
αα
J
J
J
,
( )
−−
−
−−
=>><< +
2
cos)((4
sin)(
2
sin)()(2
1
|
2
22
2
σ
σ
α
σ
σσσ
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| id | nasplib_isofts_kiev_ua-123456789-118113 |
| institution | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| issn | 1560-8034 |
| language | English |
| last_indexed | 2025-12-07T15:31:29Z |
| publishDate | 2004 |
| publisher | Інститут фізики напівпровідників імені В.Є. Лашкарьова НАН України |
| record_format | dspace |
| spelling | Bryksa, V.P. Tarasov, G.G. Masselink, W.T. Nolting, W. Mazur, Yu.I. Salamo, G.J. 2017-05-28T17:39:36Z 2017-05-28T17:39:36Z 2004 Ferromagnetism induced in diluted A₁₋xMnxB semiconductors / V.P. Bryksa, G.G. Tarasov, W.T. Masselink, W. Nolting, Yu.I. Mazur, G.J. Salamo // Semiconductor Physics Quantum Electronics & Optoelectronics. — 2004. — Т. 7, № 1. — С. 43-51. — Бібліогр.: 22 назв. — англ. 1560-8034 PACS: 72.15.Gd; 72.20.My; 73.61.Ga https://nasplib.isofts.kiev.ua/handle/123456789/118113 Theoretical model has been developed for analysis of the peculiarities of new type of magnetism in diluted magnetic A₁₋xMnxB semiconductors. The coherent potential is introduced using the dynamic mean field theory (DMFT) approximation and the Baym- Kadanoff hypothesis valid for the case x < xc (xc is the percolation limit). en Інститут фізики напівпровідників імені В.Є. Лашкарьова НАН України Semiconductor Physics Quantum Electronics & Optoelectronics Ferromagnetism induced in diluted A₁₋xMnxB semiconductors Article published earlier |
| spellingShingle | Ferromagnetism induced in diluted A₁₋xMnxB semiconductors Bryksa, V.P. Tarasov, G.G. Masselink, W.T. Nolting, W. Mazur, Yu.I. Salamo, G.J. |
| title | Ferromagnetism induced in diluted A₁₋xMnxB semiconductors |
| title_full | Ferromagnetism induced in diluted A₁₋xMnxB semiconductors |
| title_fullStr | Ferromagnetism induced in diluted A₁₋xMnxB semiconductors |
| title_full_unstemmed | Ferromagnetism induced in diluted A₁₋xMnxB semiconductors |
| title_short | Ferromagnetism induced in diluted A₁₋xMnxB semiconductors |
| title_sort | ferromagnetism induced in diluted a₁₋xmnxb semiconductors |
| url | https://nasplib.isofts.kiev.ua/handle/123456789/118113 |
| work_keys_str_mv | AT bryksavp ferromagnetisminducedindiluteda1xmnxbsemiconductors AT tarasovgg ferromagnetisminducedindiluteda1xmnxbsemiconductors AT masselinkwt ferromagnetisminducedindiluteda1xmnxbsemiconductors AT noltingw ferromagnetisminducedindiluteda1xmnxbsemiconductors AT mazuryui ferromagnetisminducedindiluteda1xmnxbsemiconductors AT salamogj ferromagnetisminducedindiluteda1xmnxbsemiconductors |