Journal of Physics and Chemistry of Solids 73 (2012) 896–901

Contents lists available at SciVerse ScienceDirect

Journal of Physics and Chemistry of Solids

0022-36

doi:10.1

n Corr

E-m

journal homepage: www.elsevier.com/locate/jpcs

Degree of circular polarization in II–VI diluted magnetic semiconductorquantum dots

Shivani Rana a, Pratima Sen a,n, Pranay Kumar Sen b

a Laser Bhawan, School of Physics, Devi Ahilya University, Takshashila Campus, Indore 452 017, Indiab Department of Applied Physics, Shri G.S. Institute of Technology and Science, Indore 452 003, India

a r t i c l e i n f o

Article history:

Received 18 August 2011

Received in revised form

14 December 2011

Accepted 14 February 2012Available online 3 March 2012

Keywords:

A. Semiconductors

A. Nanostructures

A. Magnetic materials

D. Electronic structure

D. Optical properties

97/$ - see front matter & 2012 Elsevier Ltd. A

016/j.jpcs.2012.02.012

esponding author. Tel.: þ91 731 2762 153.

ail address: pratimasen@gmail.com (P. Sen).

a b s t r a c t

Degree of circular polarization (DCP) in II–VI diluted magnetic semiconductor quantum dots (QDs) has

been studied analytically. Energy levels have been calculated using Luttinger–Kohn Hamiltonian and

effective mass approximation. Effects due to application of externa magnetic field have been

investigated, followed by calculation of transition dipole moment and DCP. Numerical estimates made

for Mn-doped CdSe/ZnSe QDs show that DCP in undoped QDs is negligible while transition metal ion

doping yields substantial polarization rotation (��2:20%) even at moderate magnetic fields (� 0:5 T).

& 2012 Elsevier Ltd. All rights reserved.

1. Introduction

Artificially fabricated semiconductor quantum dots (QDs) havebeen identified as potential candidates for applications in spin-tronic and opto-electronic devices. This is possible because thedegrees of freedom existent in terms of spins and charges of thequasi-particles viz, electrons and holes may be manipulatedelectrically or optically [1–4]. This has enabled quantum informa-tion processing and information storage possible on the samematerial system [2]. In the presence of an externally appliedmagnetic field, the spin degeneracy of the energy levels gives riseto spin-split states that paves the way for spintronic applications.Significant spin splitting in the important electronic materialsbelonging to the III–V or II–VI crystal classes can be achieved onlyon application of high magnetic fields (of the order of 10 T). Thisinherent limitation can be overcome by introducing transitionmetal (TM) ions such as Mn2þ [5,6], Ni2þ [7], Co2þ [8] and Fe2þ

[9] into the semiconductors. This leads to the development of newclass of semiconductor materials, known as diluted magneticsemiconductors (DMSs). The DMSs exhibit interesting magnetic,optical and magneto-optical properties such as large Faradayrotation, giant Zeeman splitting, etc. with many practical applica-tions in spin transport, spin filtering, charge-controlled magnet-ism, etc. [2,10]. Such magneto-electronic phenomena in the DMS

ll rights reserved.

QDs is attributed to the exchange interaction between themagnetic moments of the dopant TM ions and electron or holespins of the host semiconductor [11]. Moreover, quantum con-finement effects in semiconductors significantly enhance the sp–dexchange interactions. This enhancement can be ascribed to thestrong overlapping of the carrier wavefunctions in the TM iondoped QDs [12]. As a consequence, the spin-splitting is enhancedmanifold in DMS QDs in the presence of magnetic field [13]. Theelectron and hole energies in DMS QDs are highly sensitive toapplied magnetostatic field and consequently, the transitionenergy can be tuned.

Govorov [14] has reported voltage tunability of ferromagnet-ism in semimagnetic QDs and discussed the validity of the contactexchange interaction. This is possible when the electronic size ofthe QD is tailored to be smaller than the dimension of theacceptor states of the Mn impurities inside the QD. Recently,Oszwaldowski et al. [15] demonstrated the occurrence of brokensymmetry ground state which is neither singlet nor triplet.Interestingly enough, the influence of shape deformations onthe magnetic ordering can lead to piezomagnetism in QDs [16].This suggests that the control of quantum confinement can leadto reversible control of magnetism in DMS QDs, beyond what isfeasible in bulk DMS. In II–VI semiconductor QDs, the opticaltransitions between the electron and heavy hole bands arepolarization sensitive. The spin-up and spin-down electrons inthe conduction band can be created by exciting the QD with rightand left circularly polarized light, respectively. Such propertymakes the TM ion doped semiconductor QDs more suitable for

S. Rana et al. / Journal of Physics and Chemistry of Solids 73 (2012) 896–901 897

realization of spintronic devices with the usage of both spin andoptical properties together [17]. However, a thorough under-standing of the energy level structures and relaxation mechan-isms is essential prior to the design of the device using DMS QDs.

The optical properties of QDs can be very well understoodfrom the excitonic structures of their energy states. The excitonsformed by spin-up or spin-down electrons and correspondingholes are represented by s7 exciton states. The lifting of the spindegeneracy of the states leads to the splitting of s7 excitonstates. The de-excitations from these states generate circularlypolarized light with distinct phase shifts. As a consequence, thepolarization state of the emitted radiation will carry informationabout the spin of the electrons [18]. Magneto-optical measure-ments made by Besomobes et al. [19] in Mn2þ doped CdTe/ZnTeQDs reveal that the intensity of both the kinds of circularpolarizations can be controlled by Mn2þ spin distribution.

In the light of the above discussions, it appears worthy tostudy the dependence of the electronic energy levels on thedoping concentration and the externally applied magnetic fieldin a TM ion doped II–VI semiconductor QD. This paper presentsthe study of the energy level structure of cylindrical Mn-dopedCdSe/ZnSe QDs. We have calculated the electron and hole energyeigenvalues in the DMS QDs under effective mass approximationconsidering the four-band Luttinger–Kohn Hamiltonian asobtained from the k �p theory. The authors have also obtainedthe transition dipole moment of the bound electron–hole paircreated during the band-to-band optically induced electronictransitions and analyzed the degree of circular polarization(DCP) of the photoemission.

2. Theoretical formulations

A theoretical model is developed to study the polarizationproperties of TM ion doped II–VI semiconductor QDs. The chargecarriers in the QD experience in-plane (x–y) parabolic confine-ment potential and quantum well potential in the growth (z)direction. The QD is assumed to be placed in a transversemagnetic field defined in the symmetric Coulomb gauge asA¼ ð�y,x,0ÞB=2, where B is the magnetic field strength and A isthe vector potential. In the following, we have obtained (i) energyeigenvalues for electron and hole states; (ii) transition dipolemoment and (iii) degree of circular polarization of thephotoemission.

2.1. Energy eigenvalues of the electron and hole states

Under the axial approximation, the quantum dot is visualizedby applying a confinement potential of the form Vi ¼ VJ

i þV?i ,where VJ

i is the lateral confinement potential of parabolic natureand V?i is the perpendicular quantum well potential along thegrowth direction (z-axis) of the QD. Here, the subscript, i¼e or h

for electrons and holes, respectively. The transverse potential V?iis due to the offset between the valence band edges in the welland barrier materials. The confinement potentials are defined as

VJi ðriÞ ¼

12mio2

i r2i ð1Þ

and

V?i ðziÞ ¼DVi, 9zi9ZL=2,

0, 9zi9oL=2:

(ð2Þ

Here ri and zi are the cylindrical coordinates. mi is the effectivemass of electron or hole; oi is the oscillator frequency and DVi

refers to the conduction (i¼e) and valence (i¼h) band-offsets.Also, L is the width of the potential well.

The electron and hole wavefunctions for the above mentionedconfinement potentials can be written within the effective massapproximation as [6]

Cel ðre,ze,fÞ ¼

Xn,s

cnlcn,lðre,fÞf qðzeÞusðre,zeÞ ð3aÞ

and

Chj ðrh,zh,fÞ ¼

Xjz ,n,s

cnsjzcn,jðrh,fÞf qðzhÞujz

ðrh,zhÞ, ð3bÞ

here cnl and cnsjzare the constants of normalization for the

electron in the conduction band and hole in the valence band,respectively with n and s being the principal and spin-quantumnumbers, respectively. Also, j and l are the angular momentumquantum numbers for electron and hole, respectively andjz(¼73/2,71/2) corresponds to the heavy hole and light holespins. usðre,zeÞ and ujz

ðrh,zhÞ are the electron and hole Blochfunctions, respectively. In (3), f qðziÞ is the envelope wavefunctionarising due to the quantum well like confinement potential andgiven by [20]

f qðziÞ ¼

ffiffiffi2

L

rsin

qpL

ziþL

2

� �� �, 9zi9rL=2; ð4Þ

L being the width of the well which is taken to be larger than theheight of the QD. In Eq. (4), q (¼1,2,y) represents the band indexnumber arising due to the z-confinement. The envelope functiondue to parabolic confinement potential slowly varying over thenano-dimensions of the QD is defined as [20]

cn,lðr,fÞ ¼ CnlðirÞ9l9e�r2=2a2

i eilfL9l9n

r2

a2i

!, ð5Þ

where

Cnl ¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffin!

pðnþ9l9Þ!

s1

ai

� �9l9þ1

; ai ¼

ffiffiffi2p

acli

ðl4i þ4a4

c Þ1=4

:

In Eq. (5), L9l9n is the generalized Laguerre polynomial. The

application of an external magnetic field on the QD restricts themotion of the electrons within the cyclotron radius ac ¼

ffiffiffiffiffiffiffiffiffiffi_=eB

p.

Accordingly, the in-plane confinement of carriers will be due tomagnetic effect as well as the geometry chosen. The geometricalconfinement length is given by li ¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi_=mioi

p. ai is known as the

effective confinement length parameter that incorporates theeffects of both geometrical and magnetic confinements whichinfluence the electronic spectra. In semiconductor QDs, a simplecalculation shows that magnetic confinement can become com-parable to the geometric confinement only if the applied mag-netic field strength is around a few Tesla. The magneticconfinement has an added advantage of lifting spin degeneracyof the energy states while the geometrical confinement (sym-metric as well as asymmetric) contributes to band mixingphenomenon [21]. The change in the energy level structure ofthe QDs causes polarization rotation [22]. From spintronic appli-cation point of view, it is preferred to choose spin-split statesunder the influence of a large magnetic field.

The practical realization of semiconductor QD based nanopho-tonic devices under the influence of a strong magnetic fieldwarrants special attention. One such practical approach to attainnon-degenerate spin states is to dope the QDs with TM ions.Incorporation of TM ions in the host material of the II–VIsemiconductor causes s–d and p–d exchange interactions whichadds to the Zeeman splittings in the QD. The strengths of s–d andp–d exchange interactions as well as the Zeeman splitting can beinfluenced by the dopant concentration.

The electron Hamiltonian in the TM impurity doped QD in thepresence of the magnetic field B (applied along the z-direction) is

S. Rana et al. / Journal of Physics and Chemistry of Solids 73 (2012) 896–901898

described by

He ¼ðpþeAÞ2

2meþVeðre,zeÞþ Js�dS � re7

1

2gn

emBB, ð6Þ

where A is the vector potential in the symmetric gauge, gne and mB

are the effective g-factor for the electron and Bohr magneton,respectively. S is the spin of the magnetic ion and re is the paulispin corresponding to the spin of the conduction band electron.The second term in Eq. (6) represents the geometrical confine-ment potential, the third term describes the exchange interactionbetween the conduction band electron and the magnetic impurityions while the last term gives the Zeeman splitting.

The Hamiltonian for the valence band holes is defined bytaking into account the multi-band k �p approximation in theLuttinger Hamiltonian in the 9j,jzS basis states as

HL ¼_2

2m0

Hhh c b 0

cn Hlh 0 �b

bn 0 Hlh c

0 �bn cn Hhh

266664

377775, ð7Þ

m0 being the rest mass of the electron. In Eq. (7), we have taken

Hhh ¼ ðg1þ2g2Þk2z þðg1�g2Þðk

2xþk2

y ÞþEz, ð8aÞ

Hlh ¼ ðg1�2g2Þk2z þðg1þg2Þðk

2xþk2

y ÞþEz, ð8bÞ

b¼ 2ffiffiffi3p

g3kzðkx�ikyÞ ð8cÞ

and

c¼ffiffiffi3p½g2ðk

2x�k2

y Þ�2ig3kxky�: ð8dÞ

The solution of Eq. (7) has been obtained by several authorsunder different conditions. The application of magnetic fieldmodifies the electron momentum and gives rise to Zeemansplitting. Accordingly, the operator k¼�i= gets modified tok¼�i=�eA=_ and one has to introduce an additional Luttingerparameter k¼ g3þ2g2=3�g1=3�2=3 that causes a Zeeman split-ting of the hole states with splitting energy Ez ¼�ðe_=m0ÞkBjz

[23]. In a TM ion doped semiconductor QD, further modificationsin the Hamiltonian arise due to p–d exchange interaction as wellas the geometrical confinement potential experienced by theholes. Consequently, the hole Hamiltonian Hh in the presence ofthe magnetic field and confinement potential for a DMS QD can bewritten as

Hh ¼_2

2m0

Hhh c b 0

cn Hlh 0 �b

bn 0 Hlh c

0 �bn cn Hhh

266664

377775þ

3Bex 0 0 0

0 Bex 0 0

0 0 �Bex 0

0 0 0 �3Bex

26664

37775

þ1

2

mhho2hhr

2hh 0 0 0

0 mlho2lhr

2lh 0 0

0 0 mlho2lhr

2lh 0

0 0 0 mhho2hhr

2hh

2666664

3777775þV?h ðzhÞ,

ð9Þ

where mhh ¼m0=ðg1�2gÞ, mlh ¼m0=ðg1þ2gÞwith g¼ 2ðg2þ3g3Þ=5under spherical approximation and Bex ¼/Ch

3=29Hex9Ch3=2S.

In Eq. (9), the second term refers to the p–d exchangeinteraction between the carriers of the p-subband of the hostand d-electrons of the transition metal ions. The exchangeinteraction Hamiltonian for electrons as well as holes can beexpressed under the molecular field approximation as

Hex ¼ Js,p�d/SzðXÞSjz: ð10Þ

In Eq. (10), we have taken Js�d ¼ xN0a=2 and Jp�d ¼ xN0b=2; N0

being the number of cations per unit volume and x is the effectiveconcentration of TM ions contributing to the exchange interactionin the quantum dot [24,25]. a and b are the exchange constantsfor s–d and p–d exchange interactions, respectively. The hole spinjz ¼ 71=2,73=2. /SzðXÞS¼ S0BsðXÞ is the thermodynamical aver-age of the spin component along the magnetic field B whereX ¼ SgMnmBB=kBðTþT0Þ; gMn being the effective Lande g-factor fortransition metal ions and kB is Boltzmann’s constant, S is the spinof the localized d-electrons of the TM ions and BsðXÞ is theBrillouin function describing overall paramagnetic behavior ofthe magnetization at temperature T [25]. S0 and T0 are the fittingparameters accounting for magnetism arising due to TM–TM ionicinteraction. The last two terms in Eq. (9) represent the contribu-tions of confinement potential. We have obtained the energyeigenvalue of heavy holes by diagonalizing Hhh using the wave-functions given by Eqs. (3)–(5) as

Ehh ¼/Ch3=29Hhhþ3Bexþ

12mhho2

hhr2hhþV?h ðzhÞ9C

h3=2S: ð11Þ

In order to calculate the energy eigenvalues in DMS QDs withcylindrical symmetry, k2

xþk2y and k2

z occurring in Hhh have beenwritten in the following forms:

k2xþk2

y ¼�@2

@r2þ

1

r2

@2

@f2þ i

1

r@

@rþ

eB

2_

@

@fþ

e2B2

4_2r2 ð12aÞ

and

k2z ¼�

@2

@z2: ð12bÞ

We obtain the final expression for the energy eigenvalues of theheavy-hole states on mathematical simplification as

E7hh ¼ Ev0þ_Ohð2nþ9l9þ1Þ�_ohh

c l

þp2_2n02

2mhhL27

3_eBk2m0

83Bex: ð13Þ

Similarly, the energy eigenvalues of electrons E7e are calculated as

E7e ¼ Ec0þ_Oeð2nþ9l9þ1Þ�_oe

clþp2_2n02

2meL27

1

2gn

emBB83Aex:

ð14Þ

In Eqs. (13) and (14), Bex ¼N0b/SzSx=6 and Aex ¼N0a/SzSx=6

[10]. oeðhhÞc ð ¼ eB=meðhhÞÞ is the electron (heavy hole) cyclotron

frequency and Oi ¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiðoiÞ

2þðoeðhhÞ

c Þ2=4

qis the oscillator frequency

describing the strength of the parabolic confinement due to bothgeometrical and magnetic confinements. It is worth mentioningthat Ehh has been obtained under spherical approximation. Thedeviation from this approximation will yield [26]

Ehh ¼ g1k2�2

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffig2

2k4�3ðg2

2�g23Þðk

2x k2

yþk2x k2

z þk2yk2

z Þ

qð15Þ

in the absence of the magnetic field. The hole energies inmagnetized quantum dots can be calculated by considering thecorresponding changes in the momentum operators.

In the presence of the magnetic field, the energy levels in theplane perpendicular to the magnetic field get quantized resultinginto the formation of degenerate Landau levels. It is clear fromEqs. (13) and (14) that the degeneracy of Landau levels is lifted bythe spin degrees of freedom of the carriers as well as by theexchange interaction due to the presence of TM ions in the QDlattice. Thus the spin splitting of energy levels depends significantlyon the applied magnetic field, effective g-factor, TM ion concentra-tion, exchange interactions and the QD size. These equations furtherreveal that the energy splitting arising due to the application of themagnetic field through gmBB is to oppose the splitting due to theexchange interaction term through Aex (s–d exchange interaction)

S. Rana et al. / Journal of Physics and Chemistry of Solids 73 (2012) 896–901 899

and Bex (p–d exchange interaction). The possibility of change in thesign of the spin splitting may occur due to this opposing nature [10].

2.2. Optically induced transition dipole moment

It is well established that the optical properties of semiconductorQDs are significantly affected by the excitons. The optically inducedband-to-band transitions of electrons from the valence band to theconduction band cause the formation of excitons in the semicon-ductor QD. We have restricted ourselves only to the excitons formedby the electron–heavy hole pairs. The electronic transitions from theheavy hole states (top of the valence band, jz ¼ 73=2) to theconduction band states (jz ¼ 71=2) are of interest to us due totheir polarization selective properties. Optically induced electronictransitions in QDs suggest that excitations by s7 circular polariza-tions of the photons create the excitons with preferential orientationof spins of energy states E7ex [18] and de-excitations from theseexcitonic states generate the photons of the corresponding polariza-tions. The transition energies between ground and exciton states canbe obtained by using Eqs. (13) and (14) as _o7 ¼ E7

e �E7hh�Eexð0Þ

where Eexð0Þ is the exciton binding energy. Restricting our calcula-tions to only very small semiconductor QDs under strong confine-ment regime, we treat the confinement potential (1/R2 dependence)to be much more dominant than the Coulombic potential (1/Rdependence) [27]. In our calculations made for II–VI semiconductorQDs, the excitons being of Wannier–Mott type, the excitonic bindingenergies are significantly small and has been neglected withoutsacrificing much accuracy. The transition probabilities of de-excita-tion from the spin degenerate excitonic states will remain same andmay lead to the emission of linearly polarized photons in the steadystate regime. The experimental observation of luminescence polar-ization from the QDs is well reported since 1992 [28] and has been amatter of intensive investigations as it can unveil the physics of spindegeneracy and spin relaxation phenomenon of the exciton system.The knowledge of these parameters has significant implications inspintronic devices [29]. So far, the luminescence polarization of theQDs has been explained by considering the band mixing [21], sizeasymmetry [22], intrinsic structural asymmetry [30], etc. The liftingof spin degeneracy of the excitonic states will give rise to distinctrates of transitions from these states and may result in the rotationof polarization plane of the light transmitting through the QD. Fromthe device making point of view, it is desirable that the exciton spinsplitting and the consequent polarization rotation should be con-trollable. Although the size asymmetries and band mixing mayexplain the luminescence polarization, these effects cannot controlthe exciton spin splitting. However, in a TM doped magnetized QDsuch possibility can be explored. The probability of transitions fromthe exciton state 9E7ex S to the ground state 90S is given by [31]

W 7 ¼2p_o

VE 9m7 92dðo7�oÞ, ð16Þ

here the subscript 7 corresponds to spin-up and spin-down states.V is the mode volume, o is the frequency of the incident radiationand the induced transition dipole moment is given by

/m7S¼e_pcv

m0o7

ZCe

l ðre,ze,fÞChj ðrh,zh,fÞ dt: ð17Þ

In Eq. (17), pcv ¼/usðre,zeÞ9p9ujzðrh,zhÞS is the interband

momentum matrix element. According to the quantum analysis,the expectation value for the phasor amplitude l of the dipolemoment induced in the QD by the applied field will be given by [32]

l¼ constant � ½mn

x mn

y mn

z � �

mnx

mny

mnz

264

375�

mx

my

mz

264

375: ð18Þ

For the electron " heavy hole transitions, mz ¼ 0. The aboveequation indicates that the induced response of the atoms willexhibit the same polarization property as that of the transitiondipole matrix elements irrespective of the polarization propertyof the incident radiation [32].

2.3. Degree of circular polarization

The intensity of emitted radiation is proportional to thetransition rate and depends on the square of the induced dipolemoment. Accordingly, the degree of circular polarization (DCP) Yhas been obtained using the standard relation as

Y ¼Iþ�I�

Iþ þ I�

� �¼

m2þ�m2

�

m2þ þm2

�

, ð19Þ

with Iþ ðI�Þ as the intensity of sþ ðs�Þ emissions for transitionsfrom the Eþex ðE�exÞ energy state to the ground state. In our forth-coming discussions, we have calculated Y in percentage rangingbetween 0% and 100%.

3. Results and discussions

The theoretical framework developed so far has the generalityintact and is applicable to all magnetic impurity doped II–VIsemiconductor QDs. The present results have been applied to aspecific QD i.e., CdSe/ZnSe QD doped with Mn2þ ions; for numericalestimation of various physical parameters such as the transitionenergies and DCP. We have chosen Mn2þ ions as the transition metalions for doping because of its isoelectronic nature with the II–VIsemiconductors. The material parameters used are mc ¼ 0:13m0 [33],the conduction (valence) band offset DVc(DVv) is 0.84 eV (0.23 eV)[33]. Luttinger parameters under spherical approximation are takenas g1 ¼ 3:33, g2 ¼ 1:11 and g3 ¼ 1:45 [34] while the exchangeconstants N0a (N0b) are 0.26 eV (�1.24 eV) [35], Lande g-factor forelectron ge¼1.2 [36] and for manganese gMn ¼ 2:0, spin quantumnumber of 3d electrons S¼5/2. We have considered very small DMSQDs in the strong confinement regime with both magnetic field andgeometrical confinements influencing the optical properties of theQDs. The dimensions of the cylindrical QDs are taken to be of 1.25 nmradius and 1.5 nm height (under strong confinement limit).

Using these parameters in Eqs. (13) and (14), the variations in theelectron and hole energies are plotted as functions of magnetic fieldfor various dopant concentrations in Figs. 1 and 2, respectively. FromFig. 1, it can be seen that the heavy hole energy of undoped QDsremain unaffected with increasing magnetic field while it increases/decreases monotonously for spin-up/spin-down states in the dopedQDs even for very small amount of dopants. The spin splitting in theabsence of magnetic field (B� 0) is evident in doped QDs from thesame figure. The reason for such splitting can be ascribed to the p–dexchange interaction in the absence of applied magnetic field (B¼0).The splitting energy under such circumstance is approximately 6Bex

with Bex being defined earlier as Bex ¼N0b/SzSx=6. If we consider anequivalent magnitude of Zeeman splitting for holes, then the intrinsicmagnetic field which can generate such amount of Zeeman splittingcan be found to be Hi ¼ 2Bexm0=e_k. Since Bex is directly proportionalto the dopant concentration x, the magnitude of intrinsic magneticfield can be seen to increase with increasing Mn-dopant concentra-tion. The numerical analysis further reveals that the exchangesplitting of hole dominates over the Zeeman splitting.

Similar feature of energy splitting of electrons can be seenfrom Fig. 2. Here, the magnitude of splitting is found to be lessthan that of the hole splitting because, s–d exchange energy forelectrons is smaller than the p–d exchange energy correspondingto the holes. Another interesting observation from Fig. 2 is theoccurrence of change of sign of the splitting energy which was not

0.0 1.00.5 1.5 2.01.254

1.255

1.256

1.257

1.258

1.259

Ene

rgy

E± e (

in e

V) E

+e(x = 0.0)E

+e(x = 0.01)

E-e(x = 0.0)

E-e(x = 0.01)

E-e(x = 0.05)

Magnetic Field B (in Tesla)

E+e(x = 0.05)

Fig. 2. Electron energy as a function of applied magnetic field for undoped QDs

(x¼0.0) and Mn-doped CdSe/ZnSe quantum dots (x¼0.01 and 0.05).

0.0 0.5 1.0 1.5 2.0

1.64E+015

1.64E+015

1.65E+015

1.65E+015

1.66E+015

1.66E+015

1.67E+015

1.67E+015

ω −(x = 0.0)

ω −(x = 0.05)

ω +(x = 0.0)

ω +(x = 0.01)

ω −(x = 0.01)

ω +(x = 0.05)

Magnetic Field B (in Tesla)

Tra

nsit

ion

freq

uenc

y (

in s

ec-1

)

Fig. 3. Variation of the transition frequency with magnetic field in undoped

(x¼0.0) and Mn-doped CdSe/ZnSe quantum dots for x¼0.01 and 0.05.

0.0 0.5 1.0 1.5 2.0

-2.0

-1.5

-1.0

-0.5

0.0D

egre

e of

Cir

cula

r Po

lari

zatio

n (in

%)

Magnetic Field (in Tesla)

x = 0.0x = 0.01x = 0.05

Fig. 4. Variation of the degree of circular polarization with magnetic field in

undoped (x¼0.0) and Mn-doped CdSe/ZnSe quantum dots for x¼0.01 and 0.05.

0.0 1.00.5 1.5 2.0

0.372

0.376

0.380

0.384

0.388

0.392

Hol

e E

nerg

y (in

eV

)

E-hh(x = 0.05)

E+hh(x = 0.05)

E-hh(x = 0.01)

E+hh(x = 0.01)

E-hh(x = 0.0)

E+hh(x = 0.0)

Magnetic Field B (in Tesla)

Fig. 1. Heavy hole energy as a function of applied magnetic field for undoped

(x¼0.0) and Mn-doped CdSe/ZnSe quantum dots (x¼0.01 and 0.05).

S. Rana et al. / Journal of Physics and Chemistry of Solids 73 (2012) 896–901900

found in case of holes as seen in Fig. 1. The change in the sign ofenergy splitting arises due to the fact that the exchange splittingopposes the Zeeman splitting and their orders of magnitude aresame for the electrons. Zeeman splitting energy for holes issmaller as compared to that for the electrons due to greater holeeffective mass. The splitting of the hole energies can be ascribedentirely to the p–d exchange effect.

Fig. 3 shows the splitting of sþ and s� exciton states withmagnetic field variations in both the undoped and DMS QDs. Thefigure also reveals that the transition energy for sþ exciton statedecreases while for s� exciton state, it increases with increasingmagnetic field. For large values of magnetic fields, the transitionenergies become equal and one may infer that the s7 statesbehave as spin degenerate states.

In Fig. 4, we have plotted the degree of circular polarization(DCP) as a function of magnetic field B for Mn-doped CdSe/ZnSeQDs with different dopant concentrations. The effect of doping theII–VI semiconductor by TM ions can be distinctly noticed. Thepolarization rotation in undoped QDs (x¼0.0) is vanishingly smalland has its origin only in Zeeman splitting while nearly �2.20%polarization rotation of the emitted radiation can be achieved for5% dopant concentration of Mn at a magnetic field B¼2 T.Experimentally, Worschech et al. [17] observed circularly polarized

photoluminescence of excitons in semimagnetic CdMnSe QDs.They have studied the possibility of tuning the exciton g-factor tozero by the application of magnetic field which leads to no Zeemansplitting. Nonetheless, they have reported 10% degree of circularpolarization of CdMnSe QDs with 2 mol% Mn content. Whilestudying magnetic polarons in type II (Zn, Mn)Te QDs, Sellerset al. [12] observed a very small Zeeman shift but very largecircular polarization in TM ion doped QDs. The theoretical calcula-tions made in the present paper also suggest that undopedsemiconductor QDs, the polarization rotation does not take placein the absence of magnetic field (B¼0) while in case of an impuritydoped DMS QD, a finite rotation is detected even in the absence ofthe field. The theoretical results reported in the paper are sup-ported by the experimental observations of Worschech et al. [17]and Sellers et al. [12] and the origin of large circular polarization indoped QDs can be attributed to the s–d and p–d exchangeinteractions.

4. Conclusions

In the present paper, doping concentration dependent as wellas applied magnetic field dependent energy splitting of electron,

S. Rana et al. / Journal of Physics and Chemistry of Solids 73 (2012) 896–901 901

hole and excitonic states has been studied analytically in TM iondoped II–VI semiconductor QDs. The numerical analysis has beencarried out for manganese (Mn) doped CdSe/ZnSe cylindrical QDsunder strong confinement regime. The change in the sign of theenergy splitting as well as zero magnetic field splitting in TM iondoped QDs is found to occur. The QDs are also expected to exhibitrotation of polarization and the degree of circular polarization isfound to increase with increasing dopant concentration.

Acknowledgment

The authors are grateful to the Department of Science andTechnology (DST), government of India, New Delhi for financialassistance.

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