ZnSe/ZnTe(shell/shell) radial quantum-wire heterostructures: the excitonic properties

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2010 J. Phys. D: Appl. Phys. 43 272003

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IOP PUBLISHING JOURNAL OF PHYSICS D: APPLIED PHYSICS

J. Phys. D: Appl. Phys. 43 (2010) 272003 (7pp) doi:10.1088/0022-3727/43/27/272003

FAST TRACK COMMUNICATION

ZnSe/ZnTe(shell/shell) radialquantum-wire heterostructures: theexcitonic propertiesMehran Bagheri

Condensed Matter Group, Laser and Plasma Research Institute, Shahid Beheshti University, G. C., Evin,Tehran 19835-63113, Iran

E-mail: mh-bagheri@cc.sbu.ac.ir

Received 20 April 2010, in final form 26 May 2010Published 23 June 2010Online at stacks.iop.org/JPhysD/43/272003

AbstractThe ground-state characteristics of spatially indirect excitons trapped in radiallyheteronanostructured type-II band alignment ZnSe/ZnTe nanotubes as functions of themagnetic field for nanotubes with a radial size both smaller and larger than the effective Bohrradius are theoretically investigated. In the former case, dominated by the net kinetic energy ofthe electron and hole, the magnetic field modifies the exciton spectrum through the well-knownZeeman splitting, intra-orbital-state Aharonov–Bohm oscillations and inter-orbital-statecrossovers occurring in very strong magnetic field strengths. However, in the latter case,dominated by the electron–hole Coulomb attraction, the magnetic field adjusts the exciton linesonly by means of the Zeeman splitting and inter-orbital-state transitions happening in typicalmagnetic fields. As a result, the angular momentum transitions occurr at lower magnetic fieldswhen the radial size of the nanotube is increased. Most importantly, another consequence isthe substantially unusual exciton oscillator strength in such heteronanostructures. It is shownthat when the exciton is optically active, due to the full cylindrical symmetry of the problem,the exciton oscillator strength shows undamped oscillations. This effect is associated with theperiodic redistribution of the exciton density as the magnetic field is varied. Also, themagnitude of the magnetically induced excitonic persistent current is decreased withincreasing radial size of the nanotube. This study may provide a platform to investigate newphotonic quantum interference as well as polarization-sensitive photodetector and photovoltaicdevices based on the Aharonov–Bohm effect.

(Some figures in this article are in colour only in the electronic version)

1. Introduction

Over the past years, rapid progress has been made inthe growth mechanisms of nanostructures with greatercomplexities. In fact, it has become possible tocrystallize high quality nanostructured semiconductors suchas quantum wires and dots with different band alignmentsand tunable sizes, shapes, compositions, and morphologies[1–8]. Owing to the strong quantum confinement whichleads to unique electronic and optical properties, thefabrication of such low-dimensional nanosystems would

provide new opportunities for enhancing their functionalitiesin devising nanoelectronic [9], nanophotonic [10, 11],photovoltaic [2, 12], quantum information [13], sensing[14] and biological devices [15]. Among nanostructures,due to their rare structural one dimensionality, specificallythose having heterojunctions either radially or axially,semiconductor quantum wires have recently attractedconsiderable attention [1, 3, 6, 16–20].

Since their electronic and optical characteristics can beengineered by the size and shape of the nanocrystals inquestion, nanoscale excitons are of practical interest for

0022-3727/10/272003+07$30.00 1 © 2010 IOP Publishing Ltd Printed in the UK & the USA

J. Phys. D: Appl. Phys. 43 (2010) 272003 Fast Track Communication

the above mentioned applications [21, 22]. So far, manyefforts were mostly made on attributes of excitons trapped intype-I semiconductor nanostructures, whereas fewer attemptsin type-II heteronanostructures are reported. In type-IIsystems electrons and holes forming spatially indirect excitonsare strongly confined within two adjacent quantum layersseparated from others by a given distance and they only interactthrough their Coulomb attraction. Therefore, manipulatingand controlling the electronic and optical characteristics ofsuch type-II excitons using external electric and magneticfields is of particular interest with potential applications.More importantly, following early intimation suggested byauthors of [23, 24] on forecasting the so-called excitonic-Aharonov–Bohm(EAB) effect, excitons in both type-I andtype-II ring-shaped semiconductor heteronanostructures haverecently gained paramount importance for the prediction andobservation of the EAB oscillations. Although the excitonis a charge-neutral correlated electron–hole pair, its nonzeroelectric dipole moment would give rise to the EAB effect. Intype-I ring-like nanostructures where both electron and holeare confined within the same room, the occurrence of theEAB effect is still under debate. However, for type-II circulargeometries owing to a permanent electric dipole momentyielding a significantly promoted exciton polarization the EABeffect is naturally expected to happen [25–43].

This work concerns the behaviour of magneto-excitonsconfined in radially heteronanostructured type-II bandalignment ZnSe/ZnTe nanotubes. To the best of the author’sknowledge, the excitonic properties of shell/shell quantumwires of type-II band alignment have been considered onlyin III–V AlAs/GaAs heterostructures [44]. However, nosuggestion is recommended on spatially indirect excitonsconfined in ZnSe/ZnTe shell/shell of II–VI heterostructurematerials which are not only interesting owing to theunderlying physical process of dense excitonic systems[45] but also promising for the sake of being nontoxicfor biological and medical applications and environmentallybenign for emitting and photovoltaic species [7]. In ZnSe/ZnTecompound semiconductors, the electrons and holes makingspatially indirect excitons are strongly confined in the quantumlayers ZnSe and ZnTe, respectively [22, 31, 46]. It should beemphasized that in all compounds AlAs, GaAs, ZnSe and ZnTewith zincblende structure the top of the valence band (TVB) issituated at the �-point of the Brillouin zone. However, in AlAsand ZnSe compounds the bottom of the conduction band (BCB)is located at the �-point and �-point of the Brillouin zone ofthe zincblende lattice, respectively. Therefore, in AlAs/GaAsand ZnSe/ZnTe a spatially indirect exciton is indirect and directin K-space, respectively [47].

The paper is organized as follows. In section 2, thetheoretical model is presented. In section 3, using numericaldiagonalization of the Hamiltonian in the effective-massapproximation, the exciton ground-state energy as a functionof the magnetic field whose flux penetrates the nanotube cross-section for nanotube configurations with a radial size bothsmaller and larger than the effective Bohr radius is calculated.It is shown how an external magnetic field can adjust theexciton energy levels through the Zeeman splitting and intra-

and inter-orbital-state Aharonov–Bohm (AB) oscillations. Inthe former case, where the Coulomb attraction energy issmall compared with the whole kinetic energy contribution,the exciton is in the weakly correlated regime. In this case,the AB effect manifests itself through both intra-orbital-stateoscillations and inter-orbital-state transitions. In the latter case,however, where the electron–hole Coulomb energy dominatesthe net kinetic energy fraction of the electron and hole, theexciton is situated in the tightly correlated regime and itsspectrum is modified by the magnetic field through boththe Zeeman splitting and inter-orbital-state crossovers. Inaddition, the influence of the magnetic field on the excitonicbinding energy, optical intensity and magnetically inducedpersistent current (PC) is examined. Our results show that theoptical intensity of the exciton exhibits undamped oscillationsversus the magnetic field when the exciton is in its opticallyactive (bright) state with zero total angular momentum. Weshow that the bright state of the exciton for small (large)nanotubes is stabilized at larger (smaller) magnetic fields.It is found that the magnitude of the magnetically inducedPC is decreased with increasing radial size of the nanotubeas well.

2. Magneto-exciton Hamiltonian

We consider an electron–hole bilayer nanotube based onrolled-up type-II band alignment ZnSe/ZnTe quantum wellsschematically shown in figure 1. For simplicity, we ignorethe radial degrees of freedom of the charge carriers byassuming that they are strongly confined in the vicinity of theouter and inner wide-band-gap dielectrics using a radial bias.Within the framework of the effective-mass approximation, theHamiltonian of the magneto-exciton system can be written as

HEX =∑j=e,h

1

2m∗j

[−ih �∇j − qj

c�Aj(�xj )

]2− e2

ε|�xe − �xh| , (1)

where qe = −e and qh = e, with e as the electron charge.m∗

e and m∗h are the effective masses of the electron and hole

for ZnSe and ZnTe, respectively. In cylindrical coordinates(ρi, ϕi, zi), where the indices i = e, h correspond to theelectron and hole, ρe and ρh (ρe > ρh) are the radii ofthe electron and hole shells, respectively. A magnetic field�B = zB is applied parallel to the nanotube axis alongthe z-direction which corresponds to the vector potential�Ae/h = ϕBρe/h/2 in the symmetric gauge. Also, ε is the

static dielectric constant. By transferring into the centreof mass and relative coordinates of the exciton, we definezcm = (m∗

ezze + m∗

hzzh)/(m

∗ez

+ m∗hz

) and its relative coordinate,zrel = ze − zh, in the z-direction, and introduce the total massM∗

z = m∗ez

+m∗hz

and the reduced mass µ∗z = m∗

ezm∗

hz/M∗

z . The

Hamiltonian of the exciton can then be decoupled as HEX =Hcm + Hrel in which Hcm = −(h2/2M∗

z )(∂2/∂z2cm). Thus,

we write the exciton wavefunction as �(zcm; zrel, ϕe, ϕh) =ψ(zrel, ϕe, ϕh)φk(zcm), where φk(zcm) = 1/

√2πeikzcm is the

centre of mass wavefunction and ψ(zrel, ϕe, ϕh) is the relativewavefunction of the exciton. By taking into account theFourier expansion of the electron–hole Coulomb attraction,

2

J. Phys. D: Appl. Phys. 43 (2010) 272003 Fast Track Communication

Figure 1. (a) A type-II shell/shell bilayer nanotube of ZnSe/ZnTesubject to an axial magnetic field B. The outer and inner thickcylindrical layers (brown) are wide-band-gap dielectrics. Todiminish the electron–hole tunnelling, the electron (orange/minussign) and hole (cyan/plus sign) are strongly confined in the vicinityof these dielectric layers through a radial bias. Here, due to theelectron–hole charge separation, the electron and hole form apolarized neutral exciton. This polarization is responsible for theAB effect. (b) The band alignment of ZnSe/ZnTe type-IIheterojunction and the spatially indirect exciton formed by theelectron in ZnSe (Ebulk

g ≈ 2.82 eV at 6 K) and hole in ZnTe(Ebulk

g ≈ 2.35 eV at 300 K) [47]. (Colour online.)

i.e. −e2/

√ρ2

e + ρ2h − 2ρeρh cos(ϕe − ϕh) + z2

rel, in terms of

toroidal functions, the Hamiltonian Hrel describing the relativemotion of the magneto-exciton takes the form [44, 48]

Hrel = − h2

2µ∗z

∂2

∂z2rel

+h2

2m∗exy

ρ2e

(i

∂

∂ϕe+

�e

�0

)2

+h2

2m∗hxy

ρ2h

(i

∂

∂ϕh− �h

�0

)2

− e2

π√

ρeρh

∞∑m=−∞

eim(ϕe−ϕh)Q|m|− 12(χ), (2)

where Q|m|− 12

is the odd-half-integer degree Legendre function

of the second kind, with χ = (ρ2e + ρ2

h + z2rel)/(2ρeρh).

The quantities �e = Bπρ2e and �h = Bπρ2

h serve as theelectron and hole fluxes, respectively, so �0 = hc/|e| is

the magnetic flux quantum. In equation (2), owing to thecylindrical symmetry of the nanotube, one would present therelative eigenfunctions ψn,l , where n and l = le + lh denotethe principal and total orbital angular momentum quantumnumbers of the exciton, respectively, in terms of the normalizedeigenfunctions {(1/2π)ei(leϕe+lhϕh)} in the form

ψn,l(zrel, ϕe, ϕh) = 1

2π

∞∑le=−∞

∞∑lh=−∞

f nle,lh

(zrel)ei(leϕe+lhϕh),

l = le + lh. (3)

By introducing the effective Rydberg energy Ryd∗ =h2/2µ∗

za∗2 and the effective Bohr radius a∗ = εh2/µ∗

ze2

as units of energy and length, respectively, we scale allenergies in the Hamiltonian by Ryd∗ and all lengths by a∗.Therefore, we define the dimensionless quantities zrel =zrel/a

∗ and E = E/Ryd∗. In equation (3), the Fouriercomponents f n

le,lh(zrel)s can be determined by the coupled-

channel Schrodinger equation

d2

dz2rel

f nle,lh

(zrel, �e, �h) =∞∑

m=−∞[V m

le,lh(χ, �e, �h) − δm,0k

2]

×f nle+m,lh−m(zrel, �e, �h), k2 = En,l(�e, �h), (4)

with the orbital states l = 0, ±1, ±2, . . . and n = 0, 1, 2, . . ..The matrix elements of the potential function, V m

le,lh, as

functions of the electron and hole fluxes can be expressed by

V mle,lh

(χ, �e, �h) = − 2

π√

ρeρhQ|m|− 1

2(χ)

+

[(le + m +

�e

�0

)2 µ∗z

Ie+

(lh − m − �h

�0

)2 µ∗z

Ih

]δm,0,

(5)

where Ie = m∗exy

ρ2e and Ih = m∗

hxyρ2

h are moments ofinertia of the electron and hole, respectively, with ρi =ρi/a

∗ for i = e, h. The numerical solution of equation (4)for a fixed value of the total angular momentum l ofthe exciton provides the eigenvalues Enls and eigenvectorsf n

le,lhs [49]. It should be emphasized that for ψn,l(zrel, ϕe, ϕh)

to be single-valued, it must be periodic both in ϕe andϕh with period 2π , i.e. ψn,l(zrel, ϕe, ϕh) = ψn,l(zrel, ϕe ±2π�e/�0, ϕh ±2π�h/�0) which compels the transformationf n

le,lh(zrel) −→ f n

le,lh(zrel) exp[i2π(∓le�e/�0 ∓ lh�h/�0)]

preserving the periodic redistribution of the exciton density.

3. Results and discussion

In this section, we compute the ground-state spectrum, bindingenergy, optical intensity and PC of the exciton as functions ofthe magnetic field. For this purpose the system of coupledequation (4) is diagonalized for each orbital state l separately.To meet precise exciton eigenvalues and eigenfunctions, thebasis given by equation (3) is truncated in such a way thatthe numerical accuracy is satisfied by 10−5. Also, for thesake of simplicity, we ignore the anisotropy of the electronand hole effective masses and the parameters chosen areε = (εZnSe + εZnTe)/2 = 9.45, m∗

ez= m∗

exy= 0.16me and

3

J. Phys. D: Appl. Phys. 43 (2010) 272003 Fast Track Communication

Magnetic field (T)

Opt

ical

inte

nsity

(arb

.uni

ts)

0 10 20 30 40 50 60

0.08

0.16

0.24

0.32

0.4

(c) l=0

Magnetic field (T)

Gro

und-

stat

een

erg

y(m

eV)

0 100 200 300 400-40

0

40

80

120

160

200l=0

l=-1

l=1

l=-2

l=2

l=-3

l=3

l=-4

l=4

B≈61 T

ρe≈39.34Åρh≈11.80Å(a)

↓ ↓

Magnetic field (T)

Bin

din

gen

erg

y(m

eV)

0 100 200 300 40027

28

29

30l=0

l=-1

l=-2

l=-3

l=-4

(b)

Magnetic field (T)

Per

sist

entc

urre

nt(m

A)

0 10 20 30 40 50 600

0.1

0.2

0.3

0.4

0.5

0.6

0.7

(d) l=0

Figure 2. (a) The exciton ground-state energies E0,l (B)s as functions of magnetic field for different orbital states l = 0, ∓1, ∓2, ∓3, ∓4.The dashed vertical line (thick green) represents the first crossing point at B ≈ 61.616 T. When B � 61.616 T no angular momentumcrossover happens. (b) The associated exciton binding energy versus magnetic field for orbital states l = 0, −1, −2, −3, −4. (c) The opticalintensity of the exciton as a function of the magnetic field for the orbital state l = 0 (0.1 T step). (d) The negative exciton PC, −Ipc, againstthe magnetic field for the orbital state l = 0 (0.1 T step). Geometrical parameters of the nanotube are ρe = 39.34 Å and ρh = 11.8 Å.(Colour online.)

m∗hz

= m∗hxy

= 0.6me, where me is the free-electron mass[47]. So, the values of the effective energy and length areRyd∗ 19.41 meV and a∗ 39.34 Å, respectively. Basedon the strength of the electron–hole Coulomb attraction weconsider two separate regimes weakly and tightly correlated.The former corresponds to small nanotubes with ρe(h) � a∗

where the net kinetic energy contribution is bigger than thestrength of the Coulomb interaction. In the latter case, whereρe(h) � a∗, the electron–hole Coulomb attraction overcomesthe whole kinetic energy fraction of the electron and holein such a way that they move together over the nanotube.When the exciton is subjected to an external static magneticfield its spectrum is often split into further sublines owing tothe Zeeman effect. As can be inferred from equation (5),both the linear and quadratic terms in the magnetic fieldcontribute to the exciton spectrum. However, the weight ofthe quadratic quota depends crucially upon the radial sizeof the nanotube. In fact, the linear coefficient given bythe expression γ1(le, lh) = 2µ∗

zπa∗2(le/m∗exy

− lh/m∗hxy

)/�0

depends explicitly on the sign and module of the total orbitalangular momentum of the exciton, yet is independent of thenanotube size. On the other hand, the quadratic coefficientexpressed by γ2(ρe, ρh) = µ∗

zπ2a∗2(ρ2

e /m∗exy

+ ρ2h/m∗

hxy)/�2

0is strongly dependent upon the radial size of the nanotube.In the following we show how the physical quantities of thenanotube are established on new scales, when the radial sizeof the nanotube is increased.

3.1. Weakly correlated regime: small nanotubes (ρe,h � a∗)

Figure 2(a) exhibits our numerical results for the excitonground-state energy as a function of the magnetic field forvarious orbital states l = 0, ∓1, ∓2, ∓3, ∓4 for a smallnanotube with ρe = 39.34 Å and ρh = 11.8 Å. As shown,in order to observe the orbital-state transitions we need avery strong magnetic field. When B � 400 T the ABeffect manifests itself through the Zeeman splitting, intra-orbital-state oscillations and inter-orbital-state crossovers. We

4

J. Phys. D: Appl. Phys. 43 (2010) 272003 Fast Track Communication

ascribe the intra-orbital-state oscillations to the loosely boundelectron–hole pair orbiting around the nanotube. Becausethe strength of the Coulomb attraction is weaker than the netkinetic energy of carriers, the single-particle picture of theelectron and hole plays the main role in such a way that theymove over the nanotube independently. More importantly, theground state of the exciton alternates among various orbitalstates. Therefore, we obtain a hybrid ground state composedof different ground states E0,0, E0,−1, E0,−2 and E0,−3 joining atcrossing points. As can be seen, angular momentum transitionshappen at very strong magnetic field strengths. Such inter-orbital-state crossovers from the orbital state l = 0 to statesl = −1, l = −2 and l = −3 are actually a clear signature ofthe excitonic AB effect which originates from the fact thatthe exciton has been trapped in the type-II structure. Thereason is that the magnetic flux threading the region betweenthe electron and hole cylindrical sheets, i.e. �� = �e −�h =πB(ρe + ρh)(ρe − ρh), is proportional to the radial dipolemoment of the exciton, D = −e(ρe − ρh) = −ed whichis responsible for the magneto-interference effect when theexciton travels in the nanotube. In type-I systems (single-walled nanotubes), where ρe = ρh and the electron–holetunnelling is of particular importance, �� = 0. Consequently,no AB effect is expected to be observed. Nevertheless, somestudies have reported the signature of AB effect in type-I ring-like nanostructures [40]. Moreover, when B � 61.616 T asdepicted in figure 2(a) by a dashed vertical line (thick green),the magnetic field adjusts the exciton energy level only throughthe Zeeman splitting and no intra-orbital-state AB oscillationas well as angular momentum crossovers occur. In other words,the ground state always preserves its total angular momentumquantum number l = 0 for B � 61.616 T. Locations of twotypical crossing points connecting two neighbour orbital statesare denoted by the arrows. As long as B � 61.616 T themagnetic field dependence of the exciton lines is nearly linearand monotonic. This is because, in this case the diamagneticcoefficient depending sensitively on the size of the nanotubeis quite small. In figure 2(b), the exciton binding energyfor B � 400 T is plotted for different orbital states l =0, −1, −2, −3, −4. All curves exhibit sharp peaks which arecompatible in period with those intra-orbital-state oscillationsdepicted in figure 2(a). It is worth mentioning that for smallnanotubes the amount of exciton binding energy is larger thanthe thermal energy at room temperature (∼25 meV).

We now examine the influence of magnetic field on theoptical intensity of the exciton by evaluating the excitonicoscillator strength which for the state α is determined byJα(�e, �h) = ∫

d�x�α(�xe = �x, �xh = �x, �e, �h) [50].Because of the cylindrical symmetry of our problem, it isstraight to show that only the orbital state l = 0 is opticallyactive and has a contribution to this integral which in our case issimply reduced to J (B) = (1/

√2π)

∑le,lh

f 0le,lh

(zrel = 0, B).One would compute the optical intensity via I = |J |2. Asdepicted in figure 2(c), it exhibits undamped fluctuationscaused by the AB effect. This means that, due to the fullcylindrical symmetry of the problem, the exciton oscillatorstrength in such a heteronanostructure is substantially different.This effect is associated with the periodic redistribution of the

exciton density through the magnetic-field-dependent Fouriercoefficients f 0

−µ,µ(0, B)s. These continual fluctuationsindicate that the state l = 0 behaves coherently. Notethat the oscillator strength of the exciton in other ring-likesystems would be either increased or decreased as the magneticfield is varied [39]. When 0 � B � 61.616 T, where theexciton preserves its zero angular momentum, the exciton isin the bright phase with a finite lifetime. On the other hand,when B � 61.616 T, where the exciton has nonzero angularmomenta, the exciton is always in the dark phase with a verylong lifetime. So, the magneto-exciton undergoes a bright(l = 0) to dark (l �= 0) transition at the first crossing pointas the magnetic field increases. As a result, when the excitonis in the bright phase, oscillations in its intensity profile canbe tuned by the magnetic field. After having obtained theoptical intensity one would compute the absorption spectrumusing the expression D(ω, B) = πIδ[ω − E0,0(B)]. Suchtunable oscillations may lead to an experimental proposal thatelectron–hole bilayer nanotubes of type II could be utilized asmagneto-optical switching devices based on the AB effect.

In figure 2(d), the magnetically induced PC for the orbitalstate l = 0 (optically active state) which can be expressed byIpc(B) = −∂E0,0(B)/∂�e −∂E0,0(B)/∂�h is plotted. The PChas a monotonic behaviour and no oscillation is observed as themagnetic field is increased. The exciton PC is closely relatedto the exciton magnetization through the relation M = IpcAwhere A is the effective nanotube cross-section defined by A =πρ2

X, with ρ2X = ρ2

e ρ2h/(ρ2

e + ρ2h ) [39]. The positive (negative)

magnetization expresses the fact that both the induced excitonmagnetic moment and the applied magnetic field are parallel(antiparallel). Most importantly, the magnitude of the PC as afunction of the magnetic field is meaningful and it can be easilymeasured (milliampere) in nanotubes of a particular size.

3.2. Strongly correlated regime: large nanotubes (ρe,h � a∗)

With increasing radial size of the nanotube, the system istransformed from the weakly to strongly correlated regimewhere the electron–hole pair is tightly bound. Figure 3(a),which is associated with the nanotube with ρe = 511.42 Åand ρh = 483.89 Å, depicts that the magnetic field affectsthe exciton ground-state level through the Zeeman splitting aswell as angular momentum crossovers. As can be seen, sinceelectron and hole are tightly bound here no intra-orbital-stateoscillations happen. The positions of a few crossing pointsare denoted by the arrows. In this case, the most importantissue is clearly the significant response of the crossing pointsto ρh. As a result, when the radial size of the nanotube isincreased the crossing points joining two adjacent orbital statesoccur at lower values of the magnetic field. The computedbinding energies of different orbital quantum numbers l =0, −1, −2, −3, −4 are shown in figure 3(b). The states withpositive orbital quantum numbers also have a similar variation.All binding energies have oscillatory behaviours caused bythe AB effect and they all show an overall downward trendas the magnetic field is increased. Figure 3(c) shows theoptical intensity of the exciton. As expected, it again exhibitsundamped fluctuations made possible by the AB effect over a

5

J. Phys. D: Appl. Phys. 43 (2010) 272003 Fast Track Communication

Magnetic field (T

0 10 20 30 40 50-21.18

-21.12

-21.06

-21

-20.94

-20.88

-20.82

-20.76

l=0

l=-1

l=1

l=-2

l=2

l=-3

l=3

l=-4

l=4

ρe≈511.42Åρh≈483.89Å

Magnetic field (T

0 10 20 30 40 50-21.18

-21.12

-21.06

-21

-20.94

-20.88

-20.82

-20.76

l=0

l=-1

l=1

l=-2

l=2

l=-3

l=3

l=-4

l=4

ρe≈511.42Åρh≈483.89Å

Magnetic field (T)

Bin

ding

ener

gy(m

eV)

0 5 10 15 20 25 30 35 4021.54

21.56

21.58

21.6

21.62

21.64

l=0

l=-1

l=-2

l=-3

l=-4

(b)

Magnetic field (T)

Opt

ical

inte

nsity

(arb

.uni

ts)

0 1 2 31.5

3

4.5

6

7.5

9

10.5

(c) l=0

Magnetic field (T)

Gro

und-

stat

een

erg

y(m

eV)

0 10 20 30 40 50-21.18

-21.12

-21.06

-21

-20.94

-20.88

-20.82

-20.76

l=0

l=-1

l=1

l=-2

l=2

l=-3

l=3

l=-4

l=4

ρe≈511.42Åρh≈483.89Å

(a)

↓ ↓↓

Magnetic field (T)

Per

sist

entc

urre

nt(n

A)

0 1 2 30

0.06

0.12

0.18

l=0(d)

Figure 3. (a) The exciton ground-state energies E0,l (B)s versus magnetic field for different orbital states l = 0, ∓1, ∓2, ∓3, ∓4. Thelocations of some crossing (orbital-state transition) points are shown by arrows. (b) The associated exciton binding energy as a function ofmagnetic field for orbital states l = 0, −1, −2, −3, −4. (c) is the optical intensity of the exciton as a function of the magnetic field for theorbital state l = 0 (0.01 T step). (d) The negative exciton PC, −Ipc, against the magnetic field for the orbital state l = 0 (0.01 T step).Geometrical parameters of the nanotube are ρe = 511.42 Å and ρh = 483.89 Å. (Colour online.)

shorter interval of the magnetic field compared with the smallnanotube (0 � B � 3.405). As a result, with increasingradial size of the nanotube the magnetic interval where theexciton maintains its zero angular momentum is decreased.An issue of particular interest is the plot of the exciton PCversus the magnetic field. As shown in figure 3(d), the PCexhibits a stair-like behaviour which is superimposed by tinyoscillations caused by the AB effect. However, as a result,a quantitative comparison between the magnitudes of PCs ofthe weakly and strongly correlated regimes reveals that in thelatter case the PC is much smaller than that of the former by afactor of 106.

4. Conclusions

In conclusion, with numerical diagonalization of theHamiltonian of magneto-excitons confined in a type-II band

aligned ZnSe/ZnTe (shell/shell) bilayer nanotube we haveinvestigated the excitonic ground-state spectrum, bindingenergy, oscillator strength and PC for nanotubes with a radialsize both smaller and larger than the effective Bohr radius. Inthe first case, where the net kinetic energies of the electron andhole overcome the electron–hole Coulomb attraction energy,the AB effect manifests itself through both intra- and inter-orbital-state oscillations which occur in very strong magneticfields. On the other hand, in the second case, where theelectron–hole Coulomb energy dominates the whole kineticenergy of carriers, the magnetic field affects the exciton energylevels through inter-level angular momentum transitions whichhappen in the usual magnetic fields. Additionally, the influenceof magnetic field on the binding energy, optical intensity andPC of the exciton has been investigated. Our calculation showsthat the oscillator strength in this heterosystem has radicallydifferent profiles as the magnetic field is altered.

6

J. Phys. D: Appl. Phys. 43 (2010) 272003 Fast Track Communication

Acknowledgments

This work was supported by the research council of ShahidBeheshti University through project No 600/4415.

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