Electron and hole states in diluted magnetic semiconductor quantum dots
Kai Chang,
*
S. S. Li, and J. B. Xia
NLSM, Institute of Semiconductors, Chinese Academy of Sciences, P.O. Box 912, Beijing 100083, China
F. M. Peeters
Department of Physics, University of Antwerp (Campus Drie Eiken), B-2610 Antwerpen, Belgium
(Received 15 December 2003; published 17 June 2004
)
The electronic structure of a diluted magnetic semiconductor (DMS) quantum dot (QD) is studied within the
framework of the effective-mass theory. We find that the energies of the electron with different spin orientation
exhibit different behavior as a function of magnetic field at small magnetic fields. The energies of the hole
decreases rapidly at low magnetic fields and saturate at higher magnetic field due to the sp–d exchange
interaction between the carriers and the magnetic ions. The mixing effect of the hole states in the DMS QD can
be tuned by changing the external magnetic field. An interesting crossing behavior of the hole ground state
between the heavy-hole state and the light-hole state is found with variation of the QD radius. The strength of
the interband optical transition for different circular polarization exhibts quite different behavior with increas-
ing magnetic field and QD radius.
DOI: 10.1103/PhysRevB.69.235203 PACS number(s): 78.20.Ls, 78.67.Hc, 78.55.Cr
I. INTRODUCTION
The interest in the spin dynamics of carriers in semicon-
ductor structures has increased remarkably because of its im-
portance for basic physics as well as for its potential appli-
cation in spintronic devices. Several proposals
1,2
for quantum
information storage and processing using (electron or
nuclear) spins in semiconductor quantum dots (QD) have
been put forward due to the long spin-coherence time in
semiconductors. Quantum information processing should
preserve the entanglement while the quantum information is
transferred from the photon system to the spin of the electron
in the semiconductor. This process is closely related to the
spin splitting of carriers in semiconductors, i.e., the effective
g factor of the carrier or the exciton. Due to the strong sp–d
interaction between the carriers and the magnetic ions, di-
luted magnetic semiconductor (DMS) structures
3,4
provides
us with a unique flexibility to tailor the spin splitting of
carriers in DMS systems via the external magnetic field.
5
The
external magnetic field induces a magnetization of the mag-
netic ions in the DMS which gives rise to a giant spin split-
ting of the electron and hole band structure via the exchange
interaction. Very recently, incorporation of Mn ions into the
crystal matrix of different II-VI semiconductors, successful
approaches to fabricate DMS quantum dot and magnet/DMS
hybrid structures has been reported.
6–8
Photoluminescence
(PL) signals clearly demonstrated the transition of quasi-
zero-dimensional electron-hole pairs bound to these nanos-
tuctures. Due to the requirement of a quantitative under-
standing of the optical properties of DMS QD, there arises a
fundamental interest in the electronic structure of DMS
quantum dots.
In the case of semiconductor nanostructures, the elec-
tronic structure varies significantly with decreasing size of
the semiconductor nanostructures, especially for the hole
states. In the zinc-blende bulk material, the heavy- and light-
hole are degenerate with vanishing momentum since the Lut-
tinger Hamiltonian describing the hole states becomes diag-
onal with vanishing momentum. In the quantum well case,
the heavy- and light-hole are nondegenerate due to the con-
finement along the growth direction. But the projection of
the angular momentum of the band-edge Bloch state on the
growth direction J
z
is still a constant of motion. In quantum
dot structures, the situation is very different due to the three-
dimensional quantum confinement. J
z
is no longer a good
quantum number due to the band mixing effect: the hole
eigenstates become mixtures of the heavy- and light- hole
states. In a DMS QD, an external magnetic field induces a
magnetization of the magnetic ions, and the strong exchange
interaction between carriers and the magnetic ions provides
us with a unique and interesting flexibility to tailor the elec-
tronic structure of the DMS QD, consequently changing the
optical property of the DMS QD, i.e., the polarization and
energy position of the PL signals. In this paper, we investi-
gate theoretically the electronic structure of a DMS
Cd
1−x
Mn
x
Te/Cd
1−y
Mg
y
Te QD. We show the energies of the
lowest hole states as function of the magnetic field and the
confinement. The energy of the hole decreases rapidly at low
magnetic fields and saturates at high magnetic field. An in-
teresting crossing behavior between the heavy-hole and
light-hole is found with variation of the in-plane confine-
ment. The strength of the interband optical transition for dif-
ferent circular polarization exhibits quite different behavior
with increasing magnetic field.
The paper is organized as follows: the model and formal-
ism are presented in Sec. II, in Sec. III we show the numeri-
cal results along with the discussions. A brief conclusion is
given in Sec. IV.
II. MODEL AND FORMALISM
The DMS quantum dot is constructed from a DMS
Cd
1−x
Mn
x
Te/Cd
1−y
Mg
y
Te quantum well with a lateral con-
finement of the carriers through a parabolic well where the z
axis will be taken along the growth direction. The electron
Hamiltonian is
PHYSICAL REVIEW B 69, 235203 (2004)
0163-1829/2004/69(23)/235203(8)/$22.50 ©2004 The American Physical Society69 235203-1
H
e
=
共p + eA兲
2
2m
e
*
+ V
e
共
,z兲 + J
s−d
S ·
e
± g
e
*
B
B/2, 共1兲
where m
e
*
is the effective mass of the electron in units of the
free electron mass m
0
, A=共−y,x,0兲B/2 is the vector poten-
tial in the symmetric gauge. V
e
=V
储
e
+V
⬜
e
is the confining po-
tential of the electron which will be given explicitly, the third
term describes the exchange interaction between electron and
the magnetic ions in the DMS QD, the last term of the above
equation gives the intrinsic Zeeman splitting.
Within the axial approximation, the hole Hamiltonian in
the DMS QD can be written as
9–13
H
h
=
ប
2
2m
0
冢
H
hh
RS
0
R
*
H
lh
0
S
S
*
0
H
lh
− R
0
S
*
− R
*
H
hh
冣
+ V
h
共
,z兲 + J
p−d
S ·
h
,
共2兲
where
H
hh
= 共
␥
1
+
␥
2
兲共k
x
2
+ k
y
2
兲 + 共
␥
1
−2
␥
2
兲k
z
2
+ E
Z
,
H
lh
= 共
␥
1
−
␥
2
兲共k
x
2
+ k
y
2
兲 + 共
␥
1
+2
␥
2
兲k
z
2
+ E
Z
,
R =2
冑
3
␥
3
ik
−
k
z
,
S =
冑
3
␥
k
−
2
.
Here k=−iⵜ −eA/ប, and k
±
=k
x
±ik
y
,
␥
1
,
␥
2
, and
␥
3
are the
Luttinger parameters,
␥
=共
␥
2
+
␥
3
兲/2. E
Z
=−共បe/m
0
兲
Bj
z
de-
scribes the Zeeman splitting of the hole,
is another Lut-
tinger parameter, the confining potential V
h
=V
储
h
+V
⬜
h
, V
储
e,h
is
the lateral confining potential of electron or hole in the DMS
QD,
V
储
e,h
共
e,h
,z
e,h
兲 =
1
2
m
e,h
e,h
2
e,h
2
, 共3兲
and a quantum well potential confinement is assumed in the
z direction,
V
⬜
e,h
共z
e,h
兲 =
再
⌬V
e,h
,
兩z兩 艌 w/2,
0,
兩z兩 ⬍ w/2,
共4兲
where
i
and z
i
denote the cylinderical coordinates of the
electron or hole. w is the height of the DMS QD. ⌬V
i
is the
band offset of electron or hole. Notice that the zero energy
for the electron and hole are at the bottom of the conduction
band and the top of the valence band of Cd
1−x
Mn
x
Te, respec-
tively. The exchange interaction term describes the sp–d ex-
change interaction between the carriers and the magnetic ion
Mn
2+
and can be expressed as within the mean-field approxi-
mation,
V
exch
e,h
= J
sp–d
e,h
具S
z
典J
z
, 共5兲
where J
s–d
=N
0
␣
x
eff
/2, J
p–d
=−N
0

x
eff
/3, and 具S
z
典
=5/2B
J
共Sg
Mn
B
B/k
B
共T+T
0
兲兲, S=5/2 corresponds to the
spins of the localized 3d
5
electrons of the Mn
2+
ions. B
J
共x兲 is
the Brillouin function, N
0
is the number of cations per unit
volume, the phenomenological parameters x
eff
(reduced ef-
fective concentration of Mn) and T
0
accounts for the reduced
single-ion contribution due to the antiferromagnetic Mn–Mn
coupling, k
B
is the Boltzmann constant,
B
the Bohr magne-
ton, g
Mn
=2 is the effective g factor of Mn
2+
ion and J
z
=±1/2,±3/2 is the hole spin. B is the external magnetic
field.
In the axial approximation the hole Hamiltonian is rota-
tionally invariant around the z axis, therefore the projection
f
z
of the total angular momentum F=L+J on the z axis is a
good quantum number, where J is the angular momentum of
the band-edge Bloch function and L is the envelope angular
momentum. The single-particle eigenstates of the electron
and hole with, respectively, the angular quantum number l
and f
z
are expanded in the basis of the two-dimensional har-
monic oscillator function
l
e
=
兺
n,s
c
nl
⌿
n,l
共
,
兲f
s
共z兲兩1/2,
z
典共6a兲
and
f
Z
h
=
兺
j
z
,n,s
c
nsj
Z
⌿
n,f
Z
−j
Z
共
,
兲f
s
共z兲兩3/2,j
z
典共6b兲
where f
s
共z兲, s=1,2... is a convenient basis for the subband
part,
f
s
共z兲 =
冑
2
L
sin
冋
s
L
冉
x +
L
2
冊
册
, 兩z兩 艋 L/2, 共6c兲
and zero otherwise. L is allowed to be larger than the height
w of the DMS QD. The number of terms in the summation s
is determined by the convergence of the subband energy.
兩1/2,
z
典 and 兩3/2,j
z
典 are the band-edge Bloch functions of
electron and hole, the oscillator function ⌿
n,l
共
,
兲 is
⌿
n,l
共
,
兲 = C
nl
共i
兲
兩l兩
e
−
2
/2a
2
e
il
L
n
兩l兩
共
2
/a
2
兲, 共7兲
where C
nl
is the normalization constant and L
n
兩l兩
is the gener-
alized Laguerre polynomial, a
e,h
is the length related to the
magnetic length a
c
=共ប/eB兲
1/2
and the confinement length
l
e,h
=共ប/m
e,h
*
0
兲
1/2
by a
e,h
=
冑
2l
e,h
a
c
/共l
e,h
4
+4a
c
4
兲
1/4
.
The dipole optical transition probability for photoemis-
sion is proportional to the optical transition matrix element
G =
2
m
0
兩
⑀
· 具
l
e
兩P兩
f
Z
h
典兩
2
,
where
⑀
is the unit vector along the direction of the electric
field component and
l
e
共
f
Z
h
兲 is the electron (hole) wave
fucntion. When the light propagates in the direction of the
magnetic field, i.e., z axis, the
±
polarizations [electric field
polarization
⑀
±
=共1/
冑
2兲共
⑀
x
±i
y
兲] is of interest. Therefore the
corresponding matrix elements between the spin-down elec-
tron state and the hole states with J
z
=1/2,−3/2 components
are
G
+
=
2P
2
m
0
兩具
−1/2
e
兩
−3/2
h
典兩
2
, 共8a兲
KAI CHANG, S. S. LI, J. B. XIA, AND F. M. PEETERS PHYSICAL REVIEW B 69, 235203 (2004)
235203-2
G
−
=
2P
2
3m
0
兩具
−1/2
e
兩
1/2
h
典兩
2
, 共8b兲
and the matrix elements between the spin-up electron and the
hole states with J
z
=−1/2,3/2 components are
G
+
=
2P
2
m
0
兩具
1/2
e
兩
3/2
h
典兩
2
, 共9a兲
G
−
=
2P
2
3m
0
兩具
1/2
e
兩
−1/2
h
典兩
2
, 共9b兲
P=具s兩p
x
兩X典, s and X are the Bloch functions at the bottom of
the conduction band and the top of the valence band, respec-
tively.
The parameters used in our calculation are m
e
*
=0.13m
0
,
␥
1
=4.02,
␥
2
=1.37,
␥
3
=1.64,
=0.617, x
eff
=0.045, g
Mn
=2,
N
0
␣
=0.27 eV, N
0

=−1.31 eV, and T
0
=3.6 K.
III. NUMERICAL RESULTS AND DISCUSSIONS
In this section we present our numerical results on the
electron and hole states in Cd
1−x
Mn
x
Te/Cd
1−y
Mg
y
Te DMS
QDs. The magnetic field is applied perpendicular to the x–y
plane. The energies of the lowest eight electron and hole
states in the DMS QD are plotted in Figs. 1 and 2(a)–2(d) as
a function of magnetic field. From Fig. 1, it is apparent that
at small magnetic field the energies of spin-down (spin-up)
electron states decrease (increase) with increasing magnetic
fields. But at high magnetic fields they all increase. At small
magnetic field the energies are determined by the exchange
interaction term [Eq. (5)] for spin-up and spin-down states.
At large magnetic field the exchange term approaches to a
constant and the energies are determined mainly by the mag-
netic confinement term. Due to the rotational symmetry
around the z axis, the projection of the total angular momen-
tum f
z
is a constant of motion, the hole eigenstate S
f
z
is
labeled by its total angular momentum f
z
and the dominant
term in Eq. (6b) is usually the term with the smallest 兩l兩. The
behavior of the energies of the hole states [see Figs.
FIG. 1. The energy of the lowest eight electron states in a DMS
QD versus magnetic field. The QD radius is 10 nm and the thick-
ness of the QD is 10 nm. The solid lines and the dashed lines
denote the energies of the spin-down and the spin-up states,
respectively.
FIG. 2. The same as Fig. 1 but
now for the different hole states
S
f
z
.
ELECTRON AND HOLE STATES IN DILUTED… PHYSICAL REVIEW B 69, 235203 (2004)
235203-3
2(a)–2(d)] is much more complicated, they decrease rapidly
with increasing magnetic field at small magnetic field, and
saturate at high magnetic fields. The hole states are more
complicated than the electron states, this is because the wave
function of the hole state consists of four components, each
component has a different J
z
and l, and they mix with each
other due to the contribution of the off-diagonal terms in the
hole Hamiltonian [Eq. (2)]. The exchange interaction term
and the Zeeman term cause splitting of the energies of the
four components at small magnetic field. At large magnetic
field the hole energies increase slightly due to the magnetic
confinement, but is different for different states S
f
z
, i.e., dif-
ferent orbital momentum l. Since the spacing of the bound
states is comparable with that of the magnetic energy level in
the x–y plane and the heavy- and light-hole mix with each
other which is induced by the contribution of the off-
diagonal terms in the hole Hamiltonian [Eq. (2)], therefore
there appear many crossings and anticrossings between en-
ergy levels as shown in Figs. 3(c) and 3(d). Notice that the
S
3/2
and S
−3/2
(S
1/2
and S
−1/2
) states are degenerate at B=0
and the energy of the S
3/2
and S
1/2
ground state exhibits a
local maximum at small magnetic fields. This arises from a
crossover from the heavy-hole component 兩3/2,3/2典 with l
=0 to 兩3/2,−3/2典 with l=3, the latter becomes dominant
with increasing magnetic fields due to the off-diagonal terms
in the hole Hamiltonian (see Fig. 2). The slight increase of
the energies of the lowest states with different angular mo-
mentum (S
3/2
, S
−3/2
, S
1/2
, S
−1/2
) at high magnetic fields is
caused by the magneto-confinement effect and the intrinsic
Zeeman effect.
Figures 3(a)–3(e) show the weight factors of the wave
functions of hole ground states with f
z
=−3/2,3/2,1/2,
−1/2, i.e., the relative contribution of the hole components
with angular momentum J
z
, and the average value of the
angular momentum 具J
z
典 in the DMS QD as a function of the
magnetic field. Notice that the magnetic-field dependence of
the weight factor of the hole components with different an-
gular momentum f
z
exhibit very different behavior. For the
S
−3/2
state, the weight factor of the heavy-hole component
兩3/2,−3/2典 is dominant and stays almost constant with in-
creasing magnetic field, but for the S
3/2
state, the dominant
component is the heavy-hole component 兩3/2,3/2典 at small
FIG. 3. The weight factor (a)–(d) and the ex-
pected value of the angular momentum 具J
z
典 (e) of
the hole ground state for different angular mo-
mentum F in a DMS QD versus the magnetic
field. The solid, dashed, dotted, and dashed-
dotted curves in (a)–(d) correspond to the hole
components 兩3/2,−3/2典, 兩3/2,−1/2典, 兩3/2,1/2典,
兩3/2,3/2典, respectively. The QD structure is the
same as in Fig. 1.
KAI CHANG, S. S. LI, J. B. XIA, AND F. M. PEETERS PHYSICAL REVIEW B 69, 235203 (2004)
235203-4
magnetic fields but change rapidly to the component 兩3/2,
−3/2典 at higher magnetic fields. The main relative contribu-
tions of the S
1/2
and S
−1/2
states come from the components
兩3/2,1/2典 and 兩3/2,−3/2典, and the latter becomes dominant
with increasing magnetic field. The expected value of the
angular momentum 具J
z
典 is plotted in Fig. 3(e) as a function
of magnetic field. In QD structures, J
z
is no longer a good
quantum number due to the band mixing effect that exists
even at vanishing momentum. For these four hole ground
states, 具J
z
典 decreases with increasing magnetic fields and
keep constant −3/2 at high magnetic fields. This phenom-
enon can be understood from the weight factors of the dif-
ferent hole components in the hole eigenstates [see Figs.
2(a)–2(d)]. Since the dominant component of the hole eigen-
states at high magnetic field is the spin-down heavy holes
兩3/2,−3/2典, therefore 具J
z
典 of these four hole eigenstates is
equal to −3/2. It is interesting to find that 具J
z
典 of the hole
eigenstate S
−3/2
experiences a sharp change from 3/2 to
−3/2 with increasing magnetic field due to the crossover of
the hole components 兩3/2,3/2典 and 兩3/2,−3/2典, and a simi-
lar behavior can also be found for the S
1/2
state.
In Fig. 4 we plot the overlap factor 兩具
e
兩
fz
h
典兩, i.e., the
interband transition strength for different circular polariza-
tion of the light [see Eqs. (8) and (9)] as a function of mag-
netic field. The overlap factors for different transitions be-
tween the electron states and the hole states exhibit different
behavior as a function of magnetic field. The biggest differ-
ence can be found at small magnetic fields. This difference is
easily understood from the wave functions of the electron
and hole eigenstates. The wave functions of hole ground
states are shown in Figs. 5(a)–5(d) for different magnetic
fields. The wave functions of the S
−3/2
and S
−1/2
states change
slightly for different magnetic fields, but the wave functions
of the S
3/2
and S
1/2
states varies significantly with changing
magnetic field. Since the electron always localizes at the cen-
ter of the DMS QD for different magnetic fields, and the hole
FIG. 6. The energies of the electron and the hole ground states
in DMS QD versus QD radius. The thickness of the QD is 10 nm
and B=1 T.
FIG. 4. The overlap factor of the electron and hole in the DMS
QD versus the magnetic field. The QD structure is the same as in
Fig. 1.
FIG. 5. The wavefunction of
the hole ground states for B=0 T
(the solid curves) and 10 T (the
dotted curves). The dashed curves
correspond to the wave function
of the hole ground states at B
=1 T for S
−3/2
and S
−1/2
, but B
=0.5 T for S
1/2
, and 1.2 T for S
3/2
.
The QD structure is the same as in
Fig. 1.
ELECTRON AND HOLE STATES IN DILUTED… PHYSICAL REVIEW B 69, 235203 (2004)
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