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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