Electron states in diluted magnetic semiconductors

Superlattices and Microstructures 38 (2005) 122–129

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Electron states in diluted magnetic semiconductors

M. Triki a,∗, S. Jazirib

aLaboratoire de Physique de la Matière Condensée, Faculté des Sciences de Tunis, TunisiabLaboratoire de Physique des Matériaux, Faculté des Sciences de Bizerte, 7021 Jarzouna, Tunisia

Received 11 October 2004; received in revised form 17 March 2005; accepted 11 April 2005Available online 27 June 2005

Abstract

One of the remarkable properties of the II–VI diluted magnetic semiconductor (DMS) quantumdot (QD) is the giant Zeeman splitting of the carrier states under application of a magnetic field. Thissplitting reveals strong exchange interaction between the magnetic ion moment and electronic spinsin the QD. A theoretical study of the electron spectrum and of its relaxation to the ground state viathe emission of a longitudinal optical (LO) phonon, in a CdSe/ZnMnSe self-assembled quantum dot,is proposed in this work. Numerical calculations showed that the strength of this interaction increasesas a function of the magnetic field to become more than 30 meV and allows some level crossings. Wehave also shown that the electron is more localized in this DMS QD and its relaxation to the groundstate via the emission of one LO phonon is allowed.© 2005 Published by Elsevier Ltd

Keywords: Quantum dot; Exchange interaction; Giant Zeeman splitting; LO phonon; Relaxation

Since the first report by Xin et al. [1], self-organized CdSe quantum dots on ZnSe havebeen successfully characterized. More recently, there has been considerable interest indiluted magnetic semiconductor (DMS) quantum dots (QDs), for spintronics applications.Because Mn can easily be incorporated in II–VI materials, the electronic and magneto-optical properties of Mn doped into CdSe/ZnSe nanostructures have been explored. In thisclass of materials the magnetic and transport properties are strongly interconnected due tothe s–d exchange interaction between electrons and Mn2+ ions. On applying an external

∗ Corresponding author. Tel.: +216 72591906; fax: +216 72590566.E-mail address: [email protected] (M. Triki).

0749-6036/$ - see front matter © 2005 Published by Elsevier Ltddoi:10.1016/j.spmi.2005.04.002

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M. Triki, S. Jaziri / Superlattices and Microstructures 38 (2005) 122–129 123

Fig. 1. The shape of the quantum dot.

magnetic field, the magnetic ions Mn2+ will be magnetized giving rise to an exchange fieldacting on the electronic spins. As a consequence, a giant Zeeman splitting will appear. Agreat deal of attention has been paid to DMS quantum wells, wires and superlattices [2–5];however, up to now, the reports on fabrication and study of DMS quantum dots havebeen few and their properties are not very well clarified. The spin splitting, due to thelarge exchange interaction between the electrons and magnetic ions, can be measuredfrom the luminescence or Raman shift. Associated with spin flipping of electrons inthe presence of an external magnetic field, when the spin splitting is sufficiently largerthan the thermal energy, the electron can relax, via LO phonons, to a lower energy spinstate.

In this work, we will concentrate our study on research into the spectrum of an electronin a self-assembled CdSe/Zn0.82Mn0.18Se QD and its relaxation to the ground state via theemission of a LO phonon.

In the effective mass approximation, the Hamiltonian of the system can be expressed as

H = P2

2me+ Vconf(�r)+ me

2

(ωc

2

)2ρ2 − i

�ωc

2

∂

∂θ

+ gµB �B.�σ + Js–d

N∑I=1

�S( �RI ).�σ(�r)δ(�r − �RI ). (1)

The first part of the Hamiltonian,He = P2

2me+ Vconf(�r), denotes the motion of the

electron, whereVconf(�r) is the confining potential which is chosen to be zero in the dotand 750 meV [6] outside. The dot shape is modelled by a truncated cone lying on awetting layer (seeFig. 1). The induced cylindrical symmetry along the growth direction(axis z) allows simplification the wavefunction toψn,σ (�r) = einθϕn,σ (ρ, z); σ = ±1

2 isthe electron spin projection of the operator�σ . The introduction of the quantum numbern = 0,±1,±2, . . . gives a natural way to label the wavefunctions (they will be denotedas|Si , σ 〉, |P±

i , σ 〉, |D±i , σ 〉, . . . in the rest of the text, respectively(i = 1,2,3, . . .)). The

engine energies and associated wavefunctions are calculated by numerical diagonalization

124 M. Triki, S. Jaziri / Superlattices and Microstructures 38 (2005) 122–129

on a Fourier–Bessel basis over a large cylinder domain [7,8]. So, we can write

|ψn, σ 〉 =∑m,l

Cn,σm,l |χn,l,m, σ 〉 (2a)

whereχn,l,m(�r) = αnl einθ Jn(

λnl

R ρ) sin(πmZ z); Cn,σ

m,l are the basis coefficients;λnl is thelth

root of thenth-order Bessel function,Jn andαnl are the normalization constants.

The second, third and fourth expressions in Eq. (1) describe the diamagnetic, the orbitalZeeman and the spin splitting effects, respectively, whereme = 0.13mo, ωc = eB

me,

µB = ehmoc and g are the electron effective mass, the cyclotron frequency, the Bohr

magneton and the effective Landé factor, respectively;mo is the free electron mass. Thelast term in Eq. (1) represents the exchange interaction between Mn2+ ions at �RI and theelectron spin at�r ; I labels a magnetic ion. In our analysis, we assume that the magneticions are distributed homogeneously in the diluted magnetic semiconductors (ZnMnSe).The extended nature of the electronic wavefunction spanning a large number of magneticions and lattice sites leads the use of the molecular field approximation [9] to replace themagnetic ion spin operator�S with its average〈S〉 taken over all the ions. This approachwas applied successfully in previous theoretical works [10,11]. On the other hand, in theabsence of the magnetic field the spins of the localized 3d5 electrons of the Mn ions donot have any preferred direction. If an arbitrary field is on, the whole pseudo-spins remaindirected along the external field and so the only spin component different from zero is〈Sz〉.Finally, the exchange interaction can be written as

Js–d

N∑I=1

�S( �RI ).�σ(�r)δ(�r − �RI ) = −Noαxeff〈Sz〉.σ (2b)

where〈Sz〉 = SBS(SgMnµB B/KB(T + To)), Noα = 260 meV [12] is the s–d exchangeenergy for the electron;No is the number of Zn sites per unit volume,α is the s–d exchangeconstant,BS(x) is the Brillouin function;S = 5/2 is the ion spin andgMn = 2. Thephenomenological parameter,xeff = 0.135 [13], replaces the fractional occupancy of Znsites by Mn ionsx , andTo = 3.6 K [14] has been added to the experimental temperatureT = 4 K, to take account of the antiferromagnetic interaction between the nearest Mn2+ions pairs.

In order to investigate the exchange field effects on the electron spectrum, we firstcalculate the electron spectrum in the absence of the magnetic field. InFig. 2a, we plotthe electronic energies in a CdSe/ZnMnSe truncated cone versus its radius. It is found thatthe quantum dot displays discrete levels(Ee < 660 meV). The same results are obtainedfor most quantum dots [15]; for that reason they are usually called artificial atoms. Energylevels in the wetting layer are too close to each other(660 meV< Ee < 750 meV);the other levels form a continuum corresponding to the bulk states of the ZnMnSe barrier(Ee > 750 meV). We note the presence of two localized states in the dot forr = 5 nm.The other states are in the wetting layer or in the bulk. The dotted, the short dashed andthe long dashed curves refer to theS-like (n = 0) the P-like (n = ±1) and theD-like(n = ±2) symmetries, respectively.

In Fig. 2b, we exploit the magnetic field effect on the DMS CdSe/Zn0.82Mn0.18Se QD.For comparison with nonmagnetic quantum dots, some electron states of CdSe/ZnSe QDs


Fig. 2a. The energy spectrum of the electron in the CdSe/ZnMnSe quantum dot as a function of its radius. Thedotted, the short dashed and the long dashed curves refer to theS-like (n = 0), the P-like (n = ±1), and theD-like (n = ±2) symmetries, respectively. The two first states have symmetryn = 0 andn = ±1, respectively.

Fig. 2b. The electron spectrum in the CdSe/Zn0.82Mn0.18Se quantum dot, with a radiusr = 10 nm and a highh = 1.4 nm, as a function of the magnetic field. A giant Zeeman splitting lifts the degeneracy of the spin upand spin down electron band states further. The solid lines, the dashed lines and the dotted lines have symmetryn = 0, n = −1 andn = +1, respectively.

without Mn in the barrier are plotted inFig. 2c. Upon inclusion of the giant Zeeman term,added to the Zeeman splitting, all the doublets are also split so that all the degeneracy iscompletely lifted. Because of the inherently small values of the gyromagnetic ratiog for thesemiconductor quantum dots, for realistic magnetic fields the Zeeman splittings are small(1–2 meV) in comparison to the characteristic energy separation between the levels inducedessentially by the magnetic term in the giant Zeeman term (>30 meV forB = 12 T). Thisgiant splitting lifts the degeneracy of the spin up and spin down electron band states further,which can provide us with interesting experimental data, e.g. on magnetoconductivity, spintransport, magnetization, paramagnetic resonance and photoluminescence.Fig. 2bshowsthe presence of some assisted crossings of the electron levels with different spins, forexample, the crossing atB = 11 T which is between the level|S1,−〉 and the level|P−

1 ,+〉. In such crossings a spin flip can happen and this influences the magnetization.


Fig. 2c. Some electron levels in the CdSe/ZnSe quantum dot as a function of the magnetic field. There is a smallspin splitting between the spin up and the spin down electron states.

Fig. 2d. The lowest electron energies, in a self-assembled CdSe/ZnMnSe quantum dot, as a function of themagnetic field.

We note also the increase of the electron energy levels with a spin down, whereas thosewith a spin up decrease. The decrease of the ground state|S1, +〉 allows the electron to bemore localized in the dot.

Although electronic charge effects are subject to phonon decoherence, the acousticor optical phonon coupling is the main intrinsic mechanism for electron transitions inquantum dots. In the following we will concentrate our study on the lowest electron energy.In fact, we remark that for some values of the magnetic field, the level separation is about alongitudinal optical LO phonon (seeFig. 2d), which induces the possibility of the electronrelaxation to the ground state via the emission of one LO phonon. Recently, there has beenmuch controversy in the literature over the phonon mediated relaxation process. The term‘phonon bottleneck’ [16,17] has been coined for the argument explaining why, due to thediscrete nature of the QD energy spectrum, LO phonons cannot lead to carrier relaxationif the LO phonon energy does not coincide with the energy level splitting. In the literature,there is still much debate regarding the relevance of this argument [18–20]. In a series of


Fig. 3a. The probability of single-phonon relaxation from the state|S1,−〉 to the fundamental state|S1,+〉 as afunction of the magnetic field.

papers by Verzelen and co-authors [21–23] it is claimed that electrons and LO phononsare in the strong coupling regime leading to polaronic effects. When considering polaronicstates, the arguments leading to the ‘phonon bottleneck’ effect are completely irrelevant,since LO phonons lead to electronic intra-band mixing in the QD. In the model given byVerzelen and co-authors there is still a relaxation process but it is due to the finite lifetimeof optical phonons which they introduce. In our work, we consider the electron as a smallsystem interacting with a thermostat which comprises the optical phonons. At thermalequilibrium, the thermostat imposes its temperature on the system. The probability that anelectron relaxes to the ground state|S1,+〉 by emitting an LO phonon can be expressedusing Fermi’s Golden Rule:

Γ = 2π

�

∑K ,�q,σ

A2(�q)|〈S1,+|e−i �q.�r |K , σ 〉|2

× (nB(�q)+ 1)δ(ES1,+ − EK ,σ + �ωLO(�q)) (3)

where A2(�q) = 4π(�ωLO(�q))3/2(�2 /2me)1/2

V αF , EK ,σ (ES1,+) is the energy of an excited(ground) state|K , σ 〉 (|S1,+〉). The sum extends over all possible initial electron levelsK , the phonon wave vector�q and the two possible spin situations ((K = Si , P±

i , . . .

andσ = ±) with, (K , σ ) �= (S,+)); αF and V denote the Fröhlich constant and thedot volume, respectively. The LO phonon energy dispersion is given by:�ωLO(�q) =�ωLO − β �q2; �ωLO = 26.75 meV. The average occupation of a phonon state is given

by the Bose–Einstein distribution:nB(�q) = [e�ωLO(�q)

kB T −1]−1; kB is the Boltzmann constantand the temperature is taken as equal to 4 K. For a certain specific case (for a givenquantum dot size), the probability of single-phonon relaxation to the fundamental stateis shown, as a function of the magnetic field, inFigs. 3aand 3b. It is seen that thereare bands of the magnetic field where the phonon mediated relaxation is allowed, i.e. for5 T < B < 8 T (plotted inFig. 3a) and for B > 14 T (plotted inFig. 3b). In fact, peaksin the phonon relaxation rate as a function of the magnetic field are obtained where theresonance condition is given by energy conservation.


Fig. 3b. The probability of electron relaxation from the excited state|P−1 ,+〉 to the ground state|S1,+〉 via the

emission of an LO phonon.

From comparison toFig. 2d, we conclude that the first band corresponds to the spin fliptransition between the Zeeman sublevels of the ground state:|S1,−〉 and|S1,+〉. So, wecan say that this relaxation does not change the symmetry of the electron wavefunctionbut affects only its spin. The probability of this relaxation is about 0.14 ps−1, whichcorresponds to a relatively short relaxation time, in the range of a few ps. The secondrelaxation is from the excited state|P−

1 ,+〉 to the ground state|S1,+〉. The probabilityof this relaxation has a sharp maximum atB = 16 T, where the electron–phonon matrixelement saturates.

In conclusion, we have performed a detailed investigation of the electron spectrum of aCdSe/Zn0.82Mn0.18Se diluted magnetic semiconductor quantum dot. In particular, we haveshown a giant Zeeman splitting of the electronic levels. This effect, characteristic of dilutedmagnetic semiconductor quantum dots, is a consequence of the strong exchange interactionbetween the carriers and the d electrons localized on Mn ions. As a result, we show thatthe electron becomes more localized in this magnetic environment; its spin can be modifiedand its relaxation, via the emission of a LO phonon, to the ground state is allowed. We hopethat our solution can be used in future research for obtaining further interesting results onthe giant Zeeman effect in semimagnetic quantum dots.

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Electron states in diluted magnetic semiconductors