Magnetic field tuning of the effective g factor in a diluted magnetic semiconductor
quantum dot
Kai Chang, J. B. Xia, and F. M. Peeters
Citation: Applied Physics Letters 82, 2661 (2003); doi: 10.1063/1.1568825
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Magnetic field tuning of the effective
g
factor in a diluted magnetic
semiconductor quantum dot
Kai Chang
a)
NLSM, Institute of Semiconductor, Chinese Academy of Sciences, P. O. Box 912, Beijing 100083, China
and Department of Physics, University of Antwerp (UIA), B-2610 Antwerpen, Belgium
J. B. Xia
NLSM, Institute of Semiconductor, Chinese Academy of Sciences, P. O. Box 912, Beijing 100083, China
F. M. Peeters
b)
Department of Physics, University of Antwerp (UIA), B-2610 Antwerpen, Belgium
共Received 3 February 2003; accepted 24 February 2003兲
The spin interaction and the effective g factor of a magnetic exciton 共ME兲 are investigated
theoretically in a diluted magnetic semiconductor 共DMS兲 quantum dot 共QD兲, including the Coulomb
interaction and the sp–d exchange interaction. At low magnetic field, the ME energy decreases
rapidly with increasing magnetic field and saturates at high magnetic field for high Mn
concentration. The ground state of the ME exhibits an interesting crossing behavior between
⫹
-ME and
⫺
-ME for low Mn concentration. The g
ex
factor of the ME in a DMS QD displays a
monotonic decrease with increasing magnetic field and can be tuned to zero by an external magnetic
field. © 2003 American Institute of Physics. 关DOI: 10.1063/1.1568825兴
The spin of carriers in semiconductor nanostructures has
attracted recently considerable interest because of its impor-
tance for basic physics as well as for its potential application
in spintronic devices. Recently, several quantum computa-
tional approaches
1,2
based on semiconductor quantum dot
systems were proposed due to the long spin-coherence time
in semiconductors.
3
Electron-hole entanglement involving
two magnetoexciton states was recently identified experi-
mentally in a single GaAs quantum dot 共QD兲.
4
The photon
polarization and the spin of carriers in semiconductors are
suitable candidates for quantum information storage and pro-
cessing. The coherent transfer of quantum information
among the different physical systems requires a quantum de-
vice, e.g. a photodetector, which can preserve the entangle-
ment while the quantum information is transferred from the
photon polarization to the exciton spin in the semiconductor.
In order to maintain the entanglement, the quantum device
should absorb equally into up (
⫹
) and down (
⫺
) exciton
spin states, therefore the effective g
ex
factor of the exciton in
the semiconductor has to be zero which should be realized
by adjusting the parameters of the physical system.
The g factor is important in understanding the spin-
dependent optical and transport properties in semiconductor
nanostructures. Since the Coulomb interaction,
5
the quantum
confinement effect,
6
spin-orbit coupling,
7
electron-hole ex-
change interaction,
8
and hyperfine interaction
9,10
can influ-
ence the spin splitting, therefore the g factor of the electron,
the hole and the exciton will provide us with insights in the
spin relaxation and coherence in semiconductors. In a non-
magnetic semiconductor, the exchange interaction between
the carriers and the nuclei of the host semiconductor material
has been demonstrated experimentally although its strength
is rather weak 共⬃
eV兲.
9,10
In the diluted magnetic semicon-
ductors 共DMS兲 the sp–d exchange interaction between the
carriers and the magnetic impurities is strong which leads to
a giant spin splitting, and spin-dependent transport, and op-
tical properties.
11–15
Very recently, a DMS QD was fabri-
cated, using molecular beam epitaxy techniques, which was
found to be more robust against thermal fluctuations. Photo-
luminescence experiments
16–18
clearly demonstrated that the
formation of a zero-dimensional magnetic exciton 共ME兲 in
the DMS QD leads to a suppression of the nonradiative re-
combination process.
In this work we investigate theoretically the g
ex
factor
and the energy of the zero-dimensional ME in
Cd
1⫺ x
Mn
x
Te/Cd
1⫺ y
Mg
y
Te DMS QDs, including the contri-
bution of the Coulomb interaction and the sp–d exchange
interaction. We show how the effective g
ex
factor of the ME
in the DMS QD can be engineered by using the fact that for
low Mn concentration, the spin splitting caused by the sp–d
interaction is comparable to the intrinsic Zeeman splitting.
For low Mn concentration the g
ex
factor of the ME decreases
rapidly with increasing magnetic field at low magnetic fields
and saturates at high magnetic field. The most interesting
phenomenon is that the effective g
ex
factor of the ME can be
tuned to zero by changing the magnetic field. This is ex-
tremely interesting for practical realization of, e.g., quantum
information transfer from a photon system to excitons in a
semiconductor system.
The effective g
ex
factor is determined by the spin split-
ting between the
⫹
and
⫺
polarized transition in DMS
QDs:
g
ex
⫽ ⌬E
ME
/
B
B⫽ g
ex
0
⫹ g
ex
sp–d
, 共1兲
where the Zeeman splitting ⌬E
ME
⫽ E
ME
(
⫹
)⫺ E
ME
(
⫺
)
⫽ ⌬E
sp–d
⫾
⫹ ⌬E
Zeeman
⫾
denotes the total spin splitting caused
by the intrinsic Zeeman effect and the sp–d exchange inter-
a兲
Electronic mail: kchang@red.semi.ac.cn
b兲
Electronic mail: peeters@uia.ua.ac.be
APPLIED PHYSICS LETTERS VOLUME 82, NUMBER 16 21 APRIL 2003
26610003-6951/2003/82(16)/2661/3/$20.00 © 2003 American Institute of Physics
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action for spin-up or spin-down ME, where E
ME
(
⫾
) is the
energy of the spin-up or spin-down ME, i.e., two possible
transitions with different circular polarization 共see lower in-
set of Fig. 1兲. g
ex
0
is the intrinsic g factor, and g
ex
sp–d
the g
factor induced by the sp–d exchange interaction. The en-
ergy of the zero-dimensional ME in a DMS disk-like quan-
tum dot embedded in a nonmagnetic semiconductor material
can be obtained from the Schro
¨
dinger equation
H
ME
⌿
ME
(r
e
,r
h
)⫽ E
ME
⌿
ME
(r
e
,r
h
). Here, the stationary ME
Hamiltonian H
ME
⫾
for the
⫾
ME in the DMS QD is
H
ME
⫾
⫽
1
2m
e
*
关
p
e
⫹ eA
共
r
e
兲
兴
2
⫹ V
e
共
e
,z
e
兲
⫹ g
e
*
B
B
z
⫹
1
2m
h
*
关
p
h
⫺ eA
共
r
h
兲
兴
2
⫹ V
h
共
h
,z
h
兲
⫹ g
h
*
B
Bj
z
⫹ V
exch
⫺
e
2
⑀
兩
r
兩
, 共2兲
where r⫽ r
e
⫺ r
h
⫽ (
,z) denotes the electron-hole relative
coordinates, m
e
(m
h
) is the effective mass of electron 共hole兲,
⑀
is the dielectric constant and
z
⫽⫾1/2 (j
z
⫽⫾1/2, ⫾3/2兲 is
the electron 共hole兲 spin. The band mixing is neglected due to
the strong confinement along the growth direction. V
e
(V
h
)is
the confining potential of electron 共hole兲 in the QD, i.e.,
V
e(h)
关
z
e(h)
兴
⫽0 inside the QD and V
e(h)
关
z
e(h)
兴
⫽ V
e(h)
other-
wise. g
e
*
(g
h
*
) is the effective Lande
˙
g factor of electron
共hole兲. The sp–d exchange interaction term V
exch
between
the carriers and the magnetic ion Mn
2⫹
is treated in a mean-
field approximation
11
V
exch
⫽ J
s⫺ d
具
S
z
典
z
⫹ J
p⫺ d
具
S
z
典
j
z
, 共3兲
where J
s⫺ d
⫽⫺N
0
␣
x
eff
, J
p⫺ d
⫽⫺N
0

x
eff
/3, and
具
S
z
典
⫽ S
0
B
J
关
Sg
Mn
B
B/k
B
(T⫹ T
0
)
兴
, S⫽5/2 corresponds to the
spins of the localized 3d
5
electrons of the Mn
2⫹
ions. B
J
(x)
is the Brillouin function, N
0
is the number of cations per unit
volume, the reduced effective concentration of Mn is given
by the phenomenological parameter x
eff
, and T
0
accounts for
the reduced single-ion contribution due to the antiferromag-
netic Mn–Mn coupling, k
B
is the Boltzmann constant,
B
is
the Bohr magneton, g
Mn
⫽2istheg factor of the Mn
2⫹
ion.
The total magnetic field is
B⫽ B
ME
⫾ B
ext
, 共4兲
where B
ext
is the external magnetic field, and the carrier-
induced exchange field
16
inside ME is proportional to the
squared wave function of the carriers, i.e., B
ME
⬇(1/3
B
g
Mn
)

j
z
兩
h
(r)
兩
2
. The ME wave function with total
angular momentum L is constructed as a linear combination
of the single particle eigenstates
⌿
ex
L
共
r
e
,r
h
兲
⫽
兺
n,k,l
1
,l
2
,l
1
⫹ l
2
⫽ L
a
nk
l
n,l
1
e
共
e
,z
e
兲
k,l
2
h
共
h
,z
h
兲
.
共5兲
The single-particle eigenstates of the electron (
n,l
1
e
(
e
,z
e
)
兴
and the hole
关
k,l
2
h
(
h
,z
h
)
兴
are obtained by solving the sta-
tionary Schro
¨
dinger equation using the finite difference
method.
19
More detailed information about the nonuniform
space grid can be found in Ref. 19. In our calculation, the
QD is modeled as a disk 共see the left upper inset in Fig. 1兲
and we take n⫽10 and k⫽10 关see Eq. 共5兲兴 which leads to an
accuracy for the ground state energy better than 1%. The
transition energy of ME for both
⫹
and
⫺
excitation is
E⫽ E
g
(T)⫹ E
ME
⫾
, where E
g
(T)⫽ E
g
(0)⫺ aT
2
/(b⫹ T) is the
semiconductor band gap which depends on the temperature,
the parameters a⫽0.346 and b⫽15.059 are obtained by fit-
ting the band gap E
g
(T) at low temperature, and dE
g
(T)/dT
is obtained from these parameters at T⫽77 K, which agrees
well with the previous experimental results 共see Ref. 20兲,
dE
g
(T)/dT⬇⫺ 3⫻ 10
⫺ 4
eV/K兲. E
ME
⫹
(E
ME
⫺
) is the ME en-
ergy for the
⫹
(
⫺
) transition, respectively. The parameters
used in our calculations are m
e
*
⫽ 0.096m
0
, m
h
⫽ 0.6m
0
, m
0
is the free electron mass. x
eff
⫽0.045, g
Mn
⫽2, N
0
␣
⫽0.22 eV,
N
0

⫽⫺0.88 eV, S
0
⫽1.32, T
0
⫽3.1 K,
16
g
e
*
⫽⫺1.47,
g
h
*
⫽⫺0.24,
21
g
ex
0
⫽ (g
e
*
⫹ 3g
h
*
)⬇2.2,
⑀
⫽10.6, E
g
⫽ 1.586
⫹ 1.51x 共eV兲 for Cd
1⫺ x
Mn
x
Te, and E
g
⫽ 1.586⫹ 1.705y 共eV兲
for Cd
1⫺ y
Mg
y
Te.
22
Figure 1 depicts the variation of the g
ex
factor and the
energy 共the inset of Fig. 2兲 of the ME with magnetic field for
low Mn concentrations. The g
ex
factor shows a monotonic
increase with increasing magnetic field for different Mn con-
centrations. It is important to notice that the g
ex
factor can be
tuned to zero by changing the magnetic field, i.e., the ener-
gies of
⫾
ME become degenerate at these magnetic fields
B
c
. This behavior can be understood from the inset which
shows the ME energy as a function of magnetic field. From
the inset, one finds that the variation of the ME energy versus
magnetic field is different from that in DMS QD with high
Mn concentration 共see Ref. 16兲. For the DMS QD with low
Mn concentration, the spin splitting ⌬E
sp–d
induced by the
sp–d interaction is comparable to the intrinsic Zeeman split-
ting ⌬E
Zeeman
. Due to the competition between these inter-
actions, the ground state of the ME energy changes from
⫹
ME to
⫺
ME, i.e., the polarization of the emission light
changes from
⫹
circular polarization to
⫺
circular polar-
ization with increasing magnetic field. This interesting cross-
ing behavior is only possible when the sign of the spin split-
ting ⌬E
sp–d
is opposite to the intrinsic Zeeman splitting
⌬E
Zeeman
. The significant variation of the g
ex
factor with
FIG. 1. The effective g
ex
factor and the ME energy 共inset, the same line
conventions are used as in the main figure兲 vs magnetic fields for low Mn
concentrations. The quantum disk has R⫽4nm,h⫽1 nm, and the results are
shown for T⫽2 K. The magnetic field where g
ex
⫽0 defines the critical
magnetic field B
c
.
2662 Appl. Phys. Lett., Vol. 82, No. 16, 21 April 2003 Chang, Xia, and Peeters
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146.175.11.111 On: Mon, 09 Dec 2013 14:56:13
magnetic field provides us with a freedom to tailor the ME
transition energies, and its polarization, in semiconductors.
In Fig. 2 we plot the g
ex
factor as a function of the
effective Mn concentration x
eff
in a DMS QD for different
magnetic fields. Two characteristics can be found in this fig-
ure: 共i兲 the g
ex
factor of ME is initially equal to g
ex
0
⫽2.2, i.e.,
the intrinsic g
ex
factor, decreases almost linearly with in-
creasing Mn concentration, and 共ii兲 decreases with increasing
magnetic field. The first characteristic arises from the fact
that the spin splitting induced by the sp–d exchange inter-
action increases linearly with increasing effective Mn con-
centration 关see Eq. 共3兲兴. The second characteristic can be
understood from the dependence of the ME energy on the
magnetic field 共see Fig. 1兲. Because of the competition be-
tween the intrinsic Zeeman effect and the sp–d exchange
interaction a crossing occurs between the
⫹
ME and the
⫺
ME, i.e., the spin splitting of the
⫾
ME decreases with
increasing magnetic field. The inset shows the critical mag-
netic field B
c
versus the effective Mn concentration x
eff
. The
line indicates the zero g
ex
factor at critical magnetic fields,
below the line g
ex
⬍0 and above the line g
ex
⬎0.
Recent experiments have shown that a magnetic field
parallel to the hole quantization axis 共Faraday geometry兲
suppresses strongly the magnetic fluctuations in DMS QD.
23
Therefore, the mean-field approximation adopted in our
calculation
11
in which the fluctuation effect of the magnetic
ions is neglected, is expected to be a good first-order ap-
proximation. The effect of fluctuation will be enhanced for
low Mn concentration and higher temperature, which well
lead to a higher critical magnetic field B
c
. Nevertheless the
underlying physics picture of the magnetic field tuning of g
ex
will stay valid.
In summary, we investigated the spin interaction and the
effective g
ex
factor of a ME in a DMS QD for different
magnetic fields and Mn concentrations. The effective g
ex
fac-
tor of the ME in a DMS QD can be tuned to zero by a
magnetic field for low Mn concentration. The variation of
g
ex
and the
⫾
ME energy with magnetic field are quite
different for different Mn concentrations. The exchange in-
teraction in the DMS QD provides us with a freedom to
tailor the spin interaction and the effective g
ex
factor of the
semiconductor quantum dot by varying the external magnetic
field and the Mn concentration.
This work was partly supported by the NSF of China,
the special fund for Major State Basic Research Project No.
G001CB3095 of China, Nano Science Foundation from
CAS, the Flemish Science Foundation 共FWO-Vl兲, and the
Bilateral Cooperation program between Flanders and China.
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FIG. 2. The effective g
ex
factor and the critical magnetic field B
c
共inset兲 vs
the effective Mn concentration for a quantum disk with R⫽4nm,h⫽1nm
at a temperature T⫽2K.
2663Appl. Phys. Lett., Vol. 82, No. 16, 21 April 2003 Chang, Xia, and Peeters
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