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Modern Physics Letters B
c
° World Scientific Publishing Company
OPTICAL CONDUCTIVITY OF (III,Mn)V DILUTED MAGNETIC
SEMICONDUCTORS
HOANG ANH TUAN
∗
Max-Planck-Institut f¨ur Physik komplexer Systeme
N¨othnitzer Strasse 38, 01187 Dresden, Germany
hatuan@mpipks-dreden.mpg.de
LE DUC ANH
Hanoi University of Education, Xuan Thuy Str. 136, Hanoi, Vietnam
anh@grad.iop.vast.ac.vn
Received (Day Month Year)
Revised (Day Month Year)
The coherent potential approximation (CPA) is used to a minimal model of diluted
magnetic semiconductors (DMS), where the carrier feels a nonmagnetic potential at a
magnetic impurity site, and its spin interacts with the localized spins of the magnetic
impurities through exchange interactions. The CPA equations for one particle Green
function are derived and the optical conductivity is investigated in dependence on the
system parameters and temperature. For illustration, the case of Ga
1−x
Mn
x
As is con-
sidered and compared with experimental data.
Keywords: Optical conductivity; Diluted magnetic semiconductors; Coherent potential
approximation
1. Introduction
The discovery of ferromagnetism in Ga
1−x
Mn
x
As and In
1−x
Mn
x
As has recently
attracted much interest. With transition temperature in excess of 100 K these sys-
tems become one of the most promising materials for spintronic applications
1
. At
present the ferromagnetism in DMS is not well understood and parameters control
of the magnitude of Curie temperature is still open question. To explain ferromag-
netism in DMS , various model and approaches have been proposed
2,3,4,5,6,7,8
.
An overview of the theory of ferromagnetic (III,Mn)V semiconductors was recently
presented in Ref. 9. The models differ from each other in detail, however, they all
agree that the ferromagnetism in DMS is carrier mediated. Currently there is no
common understanding on the origin of the carrier induced ferromagnetism. Dietl
∗
Permanent address: Institute of Physics & Electronics, PO Box 429, Bo Ho, Hanoi 10 000,
Vietnam, email: hatuan@iop.vast.ac.vn
1
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2 A. T. Hoang, D. A. Le
et al.
2,3
referred to the Ruderman-Kitel-Kasuya-Yosida (RKKY) interaction as the
origin of the ferromagnetism. Though their theory can give a Curie temperature
in agreement with exp eriment, the RKKY model is questionable because the lo-
cal coupling between the carrier and the impurity spin is much larger than Fermi
energy and cannot be treated perturbatively
4
. In contract to the RKKY picture,
Yagi and Kayanuma
5
assumed a system where p holes move around interacting
with localized spin at impurity sites through the antiferromagnetic exchange inter-
action as a model for I II-V-based DMS. Their model, however, does not contain
nonmagnetic (Coulombic) potential asising from magnetic dopant, and therefore
does not reproduce the magnetism of (Ga,Mn)As in a reasonable way
6
. Recently,
Takahashi and Kubo
10,11
showed that the nonmagnetic attractive potentials at Mn
sites strongly assist the ferromagnetism in (III,Mn)V DMS. Their theory is in good
agreement with experimental results for magnetism in (Ga,Mn)As. Based on the
minimal model proposed by Takahashi and Kubo, and by using CPA in Ref.12 we
have investigated transport properties of DMS system. We note that the impurity
band model has been used in many theoretical works on (III,Mn)V DMS, since
it is usually believed that such a description can still catch the essential physics
13,14,15,16
. As well known that the optical spectrum contain important information
about the physics of DMS, the purpose of this paper is to calculate the optical
conductivity of DMS and to study its dependence on the system parameters and
temperature.
2. Model and Theoretical Formulation
We employ the minimal model of (A
1−x
Mn
x
)B-type DMS which includes the ex-
change interaction and nonmagnetic attractive potential. The Hamiltonian reads
H =
X
ijσ
t
ij
c
+
iσ
c
jσ
+
X
i
u
i
, (2.1)
where t
ij
is the hopping matrix element between the site i and j, u
i
depends on
the ions species occupying the i site:
u
i
=
E
A
P
σ
c
+
iσ
c
iσ
, i ∈ A
E
M
P
σ
c
+
iσ
c
iσ
+ ∆
P
σ
(σS
i
) c
+
iσ
c
iσ
, i ∈ Mn.
(2.2)
Here c
+
iσ
(c
jσ
) is the creation (annihilation) operator for a carrier at site i with spin
σ. The localized spins are approximated as Ising spins and S
i
(= ±1) denotes the
direction of localized spin at site i, ∆ = JS/2 is the effective coupling constant,
E
A
(E
M
) represents a nonmagnetic local potential at an A (Mn) site. The difference
of the nonmagnetic potential on the impurity atom (Mn) from that on the host atom
(A), E
M
− E
A
, acts as an attractive potential in III-V-based DMS.
Denote by N
+
and N
−
the number of localized up−spin and the down−spin sites,
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Optical Conductivity of (III,Mn)V DMS 3
respectively, then the average magnetization of localized spin is given by M =
(N
+
−N
−
)/N
s
, where N
s
= xN . Here N is the number of lattice sites and x is the
mole fraction of Mn atom. For the average magnetization M, each site is occupied
by an A atom (denoted as 0 site) with probability p
0
= 1 − x, by a Mn atom with
up−spin (denoted as + site) with probability p
+
= x(1 + M)/2, by a Mn atom
with down−spin (denoted as − site) with probability p
−
= x(1 −M)/2. According
to CPA, the local Green function at α-site (α = +, −, 0) for carriers with σ-spin
G
α
σ
(ε) is determined by
G
α
σ
(ε) = G
σ
(ε) + G
σ
(ε)T
α
σ
(ε)G
σ
(ε), (2.3)
where T
α
σ
(ε) is the single site T −matrix for carriers with σ−spin at α−site, G
σ
(ε)
is the Green function for the effective medium which is given by
G
σ
(ε) =
Z
ρ
0
(z)dz
ε − z −
P
σ
(ε)
(2.4)
Here ρ
0
(z) denotes the unperturbed density of states (DOS),
P
σ
(z) denotes the
coherent potential which is determined self−consistently.
The CPA demands that the scattering matrix vanishes on average over all pos-
sible configuration of the random potential hT
α
σ
(ε)i = 0. This is equivalent to
G
σ
(ε) =
D
(G
−1
σ
(ε) +
X
σ
(ε) − u
α
σ
)
−1
E
, (2.5)
where u
α
σ
−
P
σ
(ε) being the scattering potential for carriers with σ−spin at α−site,
and u
α
σ
is equal to E
A
, E
M
+ ∆σ, E
M
−∆σ for α = 0, +, −, respectively. By using
the semi-elliptical DOS ρ
0
(z) =
2
π W
2
√
W
2
− z
2
, where W is the half-width of the
band, Green function (2.4) takes the form
G
σ
(ε) =
2
W
2
[ε −
X
σ
(ε) −
r
(ε −
X
σ
(ε))
2
− W
2
]. (2.6)
Eliminating
P
σ
(ε) from Eq. (2.6) we get
X
σ
(ε) = ε −
W
2
4
G
σ
(ε) −
1
G
σ
(ε)
. (2.7)
Substituting u
α
σ
and
P
σ
(ε) into Eq. (2.5) we obtain
G
σ
(ε) =
1 − x
ε − wG
σ
(ε) − E
A
+
x(1 + M)/2
ε − wG
σ
(ε) − E
M
− ∆σ
+
x(1 − M)/2
ε − wG
σ
(ε) − E
M
+ ∆σ
,
(2.8)
where w = W
2
/4 and σ = ±1.
The Eq. (2.8) is easily transformed into a quartic equation for G
σ
(ε) and it is solved
analytically. The total DOS ρ
σ
(ε) is then obtained by
ρ
σ
(ε) = −
1
π
ImG
σ
(ε). (2.9)
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4 A. T. Hoang, D. A. Le
The free energy F per site is given by
F = nµ − k
B
T
∞
Z
−∞
[ρ
↑
(ε) + ρ
↓
(ε)] ln[1 + e
µ−ε
k
B
T
]dε −
T S
N
, (2.10)
where n is the density of the carriers, which is considered as an independent input
parameter, µ is the chemical potential for the carriers, S is the entropy due to the
localized spins given by
S = k
B
ln
N
s
!
N
+
!N
−
!
. (2.11)
By minimizing F with respect to µ and M we obtain the following equations
n =
∞
Z
−∞
[ρ
↑
(ε) + ρ
↓
(ε)] f(ε)dε, (2.12)
M = tanh
1
x
∞
Z
−∞
[
∂ρ
↑
∂M
+
∂ρ
↓
∂M
] ln[1 + e
µ−ε
k
B
T
]dε
, (2.13)
where f (ε) =
³
1 + e
ε−µ
k
B
T
´
−1
is the Fermi distribution function. Eqs. (2.8)−(2.9)
and (2.12)−(2.13) form a set of self−consistent equations for µ and M for a given
set of parameter values x, n, ∆, E
A
, E
M
and T . If these equations have nontrivial
solution M 6= 0, the system has a magnetic order. The Curie temperature T
C
is determined by differentialing the both sides of Eq. (2.13) with respect to M
at M = 0. Note that in contract to
11,16
where the local moment magnetization
M is left as an input parameter, this value is determined self-consistently in our
calculations.
To derive the optical conductivity of a disordered one-particle system in CPA, we
follow the procedure described in Ref. 17, which gives
σ(ω) = σ
0
Z
f(ε − ω) − f(ε)
ω
Y (ε, ω)dε, (2.14)
with σ
0
being the Mott minimal metallic conductivity, and
Y (ε, ω) =
4πW
3
X
σ
W
Z
−W
(1 −
z
2
W
2
)
3/2
A
σ
(ε, z)A
σ
(ε − ω, z) dz, (2.15)
A
σ
(ε, z) = −
1
π
Im
1
ε − z −
P
σ
(ε)
. (2.16)
The static conductivity is found from Eqs. (2.14)−(2.15) in the limit ω → 0.
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Optical Conductivity of (III,Mn)V DMS 5
0.0
0.2
0.4
0.6
0.8
1.0
-1 0 1 2 3 4
ρ(ω)
ω
∆ = 0.2W
∆ = 0.4W
∆ = 0.6W
Fig. 1. The calculated majority spin DOS for various coupling constants for x = 0.05, T = 0 and
E
M
= −0.3W .
3. Numerical Results and Discussion
Through this work we take E
A
as the origin (= 0) and W as the unit of energy.
We assume the effective coupling constant ∆ to be positive since the result does
not depend on the sign of ∆. From Eq. (2.8), it is easily seen that for fixed x and
M the DOS is determined by combined coupling E
M
± ∆, and not solely by the
exchange coupling ∆. In Fig. 1, examples of the calculated majority spin DOS are
shown for parameter values x = 0.05 (we focus on the doping of x = 0.05 associ-
ated with the highest T
c
in Ga
1−x
Mn
x
As), T = 0, E
M
= −0.3W and three values
of ∆. If the combined coupling is not strong (|E
M
− ∆| = 0.5W ) the impurity
band is not formed, at the intermediate combined coupling (|E
M
− ∆| = 0.7W )
the impurity band is formed but not well separated from the main band; however,
when the combined coupling is relatively large (|E
M
−∆| = 0.9W), we find a sepa-
rated impurity band below the main band. As well known that the carrier density
is much smaller than the impurity concentration due to the heavy compensation,
therefore the chemical potential µ is located in the lower impurity band or the lower
band edge. Thus the key physics issue is, obviously, whether the combined potential
E
M
± ∆ is week or strong, and all physics properties are determined in the lower
energy band edge.
Fig. 2 displays the change of the conductivity with the change coupling ∆ for fixed
values of x, E
M
, n and temperatute T . This optical conductivity has two main
features: i) a zero-frequency or Drude peak corresponding to motion within the
impurity band or the lower energy band, and ii) a finite-frequency broad peak
