Exciton wave function localization and exciton diamagnetic coefficient in semiconductor
quantum rings without reflection symmetry
Oleksandr Voskoboynikov,1,*W. T. Chiu,1and L. M. Thu1,2
1Department of Electronic Engineering and Institute of Electronics, National Chiao Tung University, 1001 Ta Hsueh Road, Hsinchu 300, Taiwan, Republic of China
2Department of Physics, Hanoi National University of Education, 136 Xuanthuy Road, Hanoi, Vietnam (Received 19 June 2013; published 12 August 2013)
We theoretically study how reflection asymmetry affects the neutral exciton diamagnetic coefficient in self-assembled InGaAs/GaAs semiconductor wobbled quantum rings. The previously proposed mapping method is used to simulate the exciton wave function and energy in the rings. The description is suited to clarify the important question of the exciton diamagnetic coefficient stability in the rings with broken reflection symmetry. Our simulation results confirm that the exciton wave function of the reflection symmetrical (balanced) wobbled ring is distributed equally over two potential valleys corresponding to the hills in the ring’s shape. At the same time, even a very small reflectional imbalance in the geometry and (or) material content of the wobbled rings destroys the ringlike shape of the exciton wave function and causes the localization of the function in one of the potential valleys of the ring (dotlike shape of the exciton wave function). This leads to a rapid decrease of the exciton diamagnetic coefficient. Our calculation results are in good agreement with recent experimental observations.
DOI:10.1103/PhysRevB.88.085310 PACS number(s): 73.21.La, 78.67.Hc
I. INTRODUCTION
Semiconductor self-assembled quantum rings (SAQRs) are nano-objects of a non-simply-connected topology.1–5 This distinctive characteristic of SAQRs recently has attracted considerable attention. It is assumed that SAQR specific geometry and the absence of crystal defects and impurities can provide experimentalists with the unique opportunity to observe topological quantum effects for charged particles confined in the SAQRs (including the optical Aharonov-Bohm effect).6–15 Impressive progress
in semiconductor nanofabrication technology makes it possible to produce and investigate SAQRs within a wide range of geometrical and material parameters (see, e.g., Refs. 16–28 and references therein). Among them, IncAs1−cGa/GaAs SAQRs have been investigated extensively
both experimentally and theoretically (see, for instance, Refs.3,8,11,13–15,17,26, and28–43). It was found that magnetic and magneto-optical properties of the rings strongly depend on their actual geometrical and material parameters. At the same time, the shape of embedded IncAs1−cGa/GaAs
SAQRs grown on a (001) surface of a GaAs substrate in general does not possess cylindrical symmetry. The height of the IncAs1−cGa/GaAs ring at the rim (and In concentration near the rim) is larger along the [110] direction (x axis in Fig.1) than in the [110] direction (y axis in Fig.1).17,34 This forms two distinct hills in the ring’s shape (and correspondingly two potential valleys for electrons and holes) along the x direction (see Fig.1). The “wobbled” geometry affects the magnitude of the single-electron magnetization oscillation35,36,43
and exciton diamagnetic response13,37–39 of the rings.
The ground-state energy of an exciton in a weak magnetic field (B) can be presented by
Eext(B)= E0ext+ sμBgextB+ αdB2,
where E0
ext stands for the exciton energy at B= 0 T, μB is
the Bohr magneton, gext is the exciton Land´e factor, s= ±1
presents the exciton spin polarization along the magnetic field direction, and αd is the exciton diamagnetic coefficient. The diamagnetic coefficient is obviously connected to the second derivative of the exciton energy with respect to the magnetic field magnitude and for the weak-field limit: αd =
1 2
dEext
dB |B=0.
15,40 In the strong confinement regime10,41 (when
the magnetic field is applied along the system growth direction, zaxis), the coefficient can be evaluated by using the effective radii (the characteristic confinement lengths) of the electron ρe and hole ρh in the plane perpendicular to the magnetic field,44 αd= e 2 8 ρ2 e me + ρ2h mh ,
where e is the electronic charge and me(h) stands for the
electron (hole) effective mass. Therefore, the actual value of the exciton diamagnetic coefficient can be used for a preliminary estimation of the exciton confinement length. In some recent experiments13,38 (and earlier),8 it has been
found that for some asymmetrical SAQRs, the diamagnetic coefficient for neutral excitons can be surprisingly small (considerably smaller than that expected from the conventional theory). In Ref. 38, it was proposed that the lack of the perfect rotation symmetry (for wobbled rings) is a reason for the hole localization in one of the potential valleys of the rings (which corresponds to one of the hills in the rings shape). Due to the electron-hole Coulomb interaction, the hole’s localization should lead to a localized exciton wave function (in the same potential valley), which decreases the neutral exciton effective confinement length and diamagnetic coefficient. However, the geometrical model of the wobbled ring in Ref. 38still possesses the reflection symmetry with respect to the reflection in the y-z plane. According to a general symmetrical consideration, the ground-state wave functions
FIG. 1. (Color online) Cross section of the ring structure for different values of the parameter dh and xp= R(1 + ξR):
(a) Cross section by the (x,yp,z) plane (inset: general shape of the
IncGa1−cAs/GaAs wobbled SAQR); (b) zoom of the cross section by the (x,yp,z) plane near x= xp; (c) cross section by the (xp,y,z)
plane. Remark: type-A ring is presented; see descriptions below and Table.I.
of the electron and hole confined in the ring consequently should possess the same reflection symmetry.45The symmetry
cannot be broken by the electron-hole Coulomb interaction in a dielectric structure of the same reflection symmetry. Therefore, the ground-state exciton wave function has to be distributed symmetrically between two symmetrical potential valleys. This fact raises the question about the actual reason for the decrease of the neutral exciton confinement length and diamagnetic coefficient in the wobbled rings.
In this paper, we theoretically study the impact of the broken reflection symmetry on the diamagnetic coefficient of the ground state of the neutral exciton confined in IncAs1−cGa/GaAs SAQRs. Using our mapping method,43,46
we can efficiently reproduce three-dimensional geometrical shapes and material compositions of the rings and simulate excitonic properties of the rings with the reflection symmetry and when the symmetry is broken. We demonstrate that for the rings with reflection symmetry with respect to reflection in the (110) plane (y-z plane in Fig.1), the ground-state exciton wave function is equally distributed (balanced) between two potential valleys of the rings. But, a small imbalance in geometrical or material characteristics of the rings along the [110] direction (x axis in Fig.1) leads to the localization of the exciton wave function in one of the potential valleys of the ring, which causes a significant decrease of the exciton diamagnetic coefficient.
II. THEORETICAL MODEL
To simulate IncAs1−cGa/GaAs wobbled SAQRs with and without the reflection symmetry, we adopt the model of the ring’s height profile from Ref. 36with some modifications. This model suggests that the bottom of the ring is perfectly flat and parallel to the x-y plane. The height of the ring h(x,y) along the z direction (Fig. 1) can be written as
follows:36,43 h(x,y) = h0+ [hM(x,y)− h0]γ0(x,y)2 R(x,y)2+ γ 0(x,y)2 ×R(x,y) 2− R(x,y)2 R(x,y)2 , x2+ y2 R(x,y), h(x,y) = h∞+[hM(x,y)− h∞]γ∞(x,y) 2 R(x,y)2+ γ ∞(x,y)2 , x2+ y2> R(x,y), (1) with hM(x,y)= hM 1+ ξhx 2− y2 x2+ y2 Fh(x,y), (2) R(x,y)= R 1+ ξRx 2− y2 x2+ y2 , (3) R(x,y)=x2+ y2− R(x,y), (4) γ0(x,y)= γ0 1+ ξγx 2− y2 x2+ y2 , (5) γ∞(x,y)= γ∞ 1+ ξγx 2− y2 x2+ y2 , (6) Fh(x,y)= 1 + dhexp −(x− xp)2+ (y − yp)2 b2 , (7) where R is the ring’s rim average radius; h0, hr, and h∞,
respectively, stand for the height at the center of the ring, at the rim, and far outside of the ring; γ0 and γ∞, respectively,
determine the inside and outside slopes near the ring’s rim. The wobbling parameters ξh, ξR, and ξγ describe the anisotropy (circular asymmetry) of the ring height on the x-y plane. The reflection asymmetry in the ring hills’ heights (the ring shape’s reflection asymmetry) is described by the function Fh(x,y), where{xp = ±R(1 + ξR),yp = 0} stands for the position of the appropriate ring’s profile maximum (top of the ring’s hills; see Fig.1), and the range of the reflection asymmetry in the wobbling is presented by a parameter b. According to Eq.(1), in our model deviations from the reflection symmetry in the ring’s shape are controlled by a unitless parameter dhas
dh= h(xp,yp)|dh=0− h(xp,yp)|dh=0 h(xp,yp)|dh=0
. (8)
In Fig. 1, we present the shape of the structure and cross sections of the ring by (x,yp,z) and (xp,y,z) planes for different dhwhen the ring height deviations are concentrated at the position of the maximum at the positive xp = R(1 + ξR) and yp = 0.
We use h(x,y) to describe the corresponding three-dimensional smooth confinement potential for electrons (holes) by the shape and composition-dependent profiles of the local conduction (e) and valence (h) band offsets:43,46
where T(r)= 1 4 1+ tanh z a 1− tanh z− h(x,y) a , (10) FV(r)= 1 + dV exp −(x− xp)2+ (y − yp)2 b2 × exp −(z− zp)2 b2 z . (11)
r= (x,y,z) is the three-dimensional radius vector, E0e(h)=
Eout
c(v)− Ecin(v) is the overall conduction (valence) band offset
between the inner and outer semiconductor materials in the InGaAs/GaAs heterostructure, and superscripts “in” and “out” denote the actual material parameters inside and outside the ring. The slope and range (the degree of smoothness) of the potential change at the boundaries of the ring are controlled by a parameter a [when the parameter a goes to 0, Eq.(9) de-scribes a rigorous full three-dimensional hard-wall potential]. In addition to the ring’s shape asymmetry (controlled by dh), the reflection asymmetry in the ring’s potential (the reflection asymmetry in the ring’s material content) is presented by the function FV(r), where rp= (xp,yp,zp) refers to the position of the appropriate potential valley, and the range of the asymmetry in the z direction is presented by a parameter bz. Deviations from the reflection symmetry in the ring’s potential are controlled by a unitless parameter dV, which, according to Eq.(9), can be presented as
dV =Ve(h)(rp)|dV=0− Ve(h)(rp)|dV=0 E0 e(h) + O a zp . (12) This approximate relationship becomes exact when a→ 0. Following the actual three-dimensional shape and content of the ring, Eq.(9)represents the three-dimensional confinement
FIG. 2. (Color online) Projections of the electron (Ve) and hole
(−Vh) confinement potentials (with the actual energy gap in between)
on the (x,yp,z) plane [dh= dV = 0.1 and xp= R(1 + ξR)]. Ee
and−Eg
in− Ehdenote the electron and hole ground-state energies.
Remark: type-A ring is presented; see descriptions below and TableI.
FIG. 3. (Color online) Electron (e⇔ Ve) and hole (h⇔ −Eing− Vh) confinement potential profile along different directions in the
ring and for different values of the parameter dV [dh= 0.1 and
xp= R(1 + ξR)]: (a) [x,yp,zp] direction; (b) [xp,y,zp] direction;
(c) [xp,yp,z] direction. Remark: Type-A ring is presented; see
descriptions below and Table.I.
potential for electrons (holes) with the smooth changes of the material parameters across the boundaries of the ring.46 Deviations of the parameter dV obviously mimic the material content deviations at the hill locations of the wobbled ring. The local band offsets for electrons and holes grow when dV is positive and increases. The offsets locally decrease when dV is negative and its absolute value increases. In Figs. 2
and3, we demonstrate some projections of the asymmetrical confinement potential defined by Eq.(9).
According to the available experimental data, two parame-ters dhand dV can be correlated: the height of a wobbled ring at the rim reflects the profile of the actual In concentration in the ring.17,34However, it is very difficult to derive this correlation. Therefore, to make this theoretical study more self-contained, we consider independently and cumulatively the impacts of the ring’s shape reflection asymmetry and content reflection asymmetry on the exciton diamagnetic coefficient. When the parameters have the same sing, we can consider their changes as “correlated.” When the signs are opposite, the parameters’ changes can be considered as “anticorrelated.”
Using Ve(h)(r), we define the mapping function Me(h)(r) for
electrons (holes) which accumulates experimental information about the ring’s shape and local material content43,46 as
follows:
Me(h)(r)= 1 −
Ve(h)(r)
Ee(h)
. (13)
According to Eq.(13), the mapping functions for electrons and holes can be different only when FV(r) is different for these particles. An appropriate modification obviously requires extra parameters to be introduced to our model of the ring. To keep our study within certain frames and to concentrate on the reflection symmetry issue, in this paper we confine ourselves to a description with the equivalent for electron and hole function FV(r). Therefore, in this study,
M(r)= Me(r)= Mh(r)= FV(r)T (r). (14) We present the position-dependent effective mass me(h)(r) of
electrons (holes), band gap Eg(r), and dielectric constant (r) of the system as follows:
me(h)(r)= mine(h)M(r)+ moute(h)[1− M(r)], (15)
Eg(r)= EginM(r)+ Egout[1− M(r)], (16) (r)= inM(r)+ out[1− M(r)]. (17) The Hamiltonian of the single neutral exciton within the effective position-dependent mass approximation reads30,33,40,47 ˆ Hext= i=e,h ˆ i 1 2mi(ri)ˆi+ i=e,h Vi(ri)+ VC(re,rh), (18)
where ˆe(h)= −¯h∇re(h)− qe(h)A(re(h)) is the electron (hole)
momentum operator, ∇re(h) is the spatial gradient in the
electron (hole) coordinate system, qe(h)= −(+)e, and A(r)
is the vector potential of the magnetic field, B(r)= ∇r×
A(r). VC(re,rh) presents the attractive electron-hole Coulomb interaction, which can be written as
VC(re,rh)= −e2G(re,rh), (19) where G(re,rh) is the Green’s function of the Poisson equation,48,49
0∇r(r)∇rG(r,r)= −δ(r − r). (20)
For systems without reflection symmetry, the conventional Coulomb cylindrically symmetric gauge for the vector po-tential A(r) and uniform magnetic field B [A(r)= 12B× r] leads to unphysical behavior of the magnetic coupling part of the energy of a particle (see, for instance, Refs. 50–52
and references therein). This is an “origin-dependent gauge,” which suggests at the very beginning that the expectation value of the position of a particle in the ground state (¯r) is exactly at the origin of the coordinate system (¯r= 0), which is not correct in general for the particle ground state in a system without reflection symmetry. For the results to be reliable, it is essential that the system model is rigorously invariant with respect to the gauge origin.50–52 To satisfy this condition (according to
the principle of the minimal magnetic coupling)53–55 we use
agauge-origin-independent definition for the vector potential:
A(r)=12B× (r − ¯r). (21)
Using this gauge and Eq.(18)for the weak-field limit (when the magnetic field is applied along the z direction), we can
TABLE I. Geometrical parameters of two types of rings. Type R (nm) hm(nm) h0(nm) h∞(nm) ξh ξR ξγ
A 9 3.6 1.6 0.4 0.2 0.0 −0.25
B 7 2.1 1.0 0.4 0.15 0.07 0.0
write the exciton diamagnetic coefficient as αd = e2 8 (ρe− ¯ρe)2 me(re) + (ρh− ¯ρh)2 mh(rh) B=0 , (22)
whereρ = (x,y) is the two-dimensional radius vector in the x-y plane,f stands for the expectation value of a quantity f in the exciton ground state,
f =
G∗(re,rh)f (re,rh) G(re,rh)dredrh (23) [ G(re,rh) is the exciton ground-state wave function], and
¯
ρe(h)= ρe(h) is the expectation value of the electron (hole)
position in the x-y plane.
We use the Hartree approximation when the exciton wave function is presented by a product of the electron ψe(re) and hole ψh(rh) wave functions: G(re,rh)= ψe(re)ψh(rh). The functions have to be obtained by solving self-consistently the following system of equations:48
[ ˆHe(re)+ Vh(re)]ψe(re)= Eeψe(re), (24) [ ˆHh(rh)+ Ve(rh)]ψh(rh)= Ehψh(rh), (25)
FIG. 4. (Color online) Contour plot of the ground-state wave function of the noninteracting electron and hole in the (x,yp,z) plane
FIG. 5. (Color online) Three-dimensional contour plot of the ground-state wave function of the noninteracting electron and hole for different configurations of the type-A ring.
0∇r(r)∇rVe(h)(r)= e2|ψe(h)(r)|2, (26)
where ˆHe(h) is the effective one-band Hamiltonian for the
electron (hole) confined in the ring: ˆ He(h)(r)= ˆe(h) 1 2me(h)(r) ˆ e(h)+ Ve(h)(r). (27)
The exciton ground-state energy in this approximation is defined as Eext= Eing + Ee+ Eh−1 2 |ψe(r)|2Vh(r)dr + |ψh(r)|2Ve(r)dr . (28)
III. SIMULATION RESULTS AND DISCUSSION We compute the exciton ground-state wave function, energy, and diamagnetic coefficient for two types of asym-metrical IncAs1−cGa/GaAs SAQRs: A and B. Geometrical parameters of the type-A ring are close to those of the rings discussed in Refs.13,37, and39, and the parameters for the type-B ring are chosen according to the data from Ref. 38. TableIpresents basic geometrical parameters for both types of rings. Other relevant geometrical parameters are taken as xp = R(1 + ξR), yp = 0, zp= 0.6hm, b= xp, bz= hm,
a= 0.4 nm, and γ0= γ∞= 3 nm. In Fig. 1, we show
(for the type-A ring with and without the reflection sym-metry) the ring’s height profiles along the x axis [h(x,0)] and along a direction which is parallel to the y axis [h(xp,y)]. Material parameters for IncAs1−cGa/GaAs het-erostructures we take from Refs.48and56and adjust them according to the actual composition and strain inside the rings:43,49,57Eg
InAs= 0.842 eV, meInAs= 0.044m0, mhInAs =
0.074m0, InAs= 15.1, EgGaAs= 1.52 eV, meGaAs= 0.067m0,
mhGaAs= 0.5m0, GaAs= 12.9 (m0is the free-electron mass).
A material parameter Din (out)for the IncGa
1−cAs compound is
obtained according to the linear interpolation Dc= cDInAs+
(1− c)DGaAs. The In concentration inside rings is taken to
be c= 0.55 and 0.7 for types A and B, respectively. In addition, we take 70% of the heterostructure gap difference to be the conductance band offset and 30% to be the valence band offset in the rings. The energy and wave function of the exciton confined in the SAQRs are obtained numerically form the self-consistent solution of Eqs. (24)–(26) by the iterative method using theCOMSOL MULTIPHYSICSpackage.58
We use the wave function to simulate the exciton diamagnetic coefficient [Eq. (22)] for SAQRs with the reflection symmetry and when the symmetry is broken.
FIG. 6. (Color online) The expectation value of the position of the noninteracting electron (a) and hole (b) in the ground state on the x axis (type-A ring). The parameter dC represents simultaneous
changes of dhand dV for different configurations: (c1) dC= dh=
dV 0; (c2) dh= 0, dC = dV 0; (c3) dC= dh 0, dV = 0; (c4)
dC = |dh| = dV, dh 0, dV 0; (c5) dh= 0, dC= |dV| ; dV 0;
(c6) dC= |dh| ; dh 0, dV = 0; (c7) dC= dh= |dV| , dh 0, dV
First we present the ground-state wave functions of the noninteracting electron and hole for the type-A ring at zero magnetic field. We independently consider the impacts of the ring’s shape reflection asymmetry and content reflection asymmetry on the electron and hole wave functions inside the ring. In Fig.4, we show the contour plot of the x-y plane cross section of the electron and hole ground-state wave functions. When two parameters dhand dV change from 0 (reflectional symmetrical ring) to 0.1, the wave functions change from a reflectional symmetrical ringlike function (extended around the ring volume) to reflection nonsymmetrical quantum dotlike wave functions (localized in one of the potential valleys of the ring near x= xp). Note that the wave functions are localized in one of the potential valleys when the imbalance in the ring potential profile is relatively small. Both the electron and hole wave functions are more sensitive to the variations of the parameter dV. At the same time, the hole wave function is generally more sensitive to the imbalance in the reflection symmetry, which can be understood as a consequence of the particles’ effective-mass difference: as the hole has a larger effective mass, the effect of the imbalance is stronger.
Figure 5 shows that for negative values of the parameters dh and dV, the electron and hole wave function mean localization positions move in the direction of x= −xp. However, for the cases in which the parameters have different signs (“anticorrelation”), the position of the localization is controlled by the sign of dV. Notice that the change of the parameters’ signs in our description does not generate a simple symmetrical
FIG. 7. (Color online) The effective lateral radius of the nonin-teracting electron (a) and hole (b) (type-A ring). The configurations c1, . . . ,c8 are described in the caption to Fig.6.
reflection in the y-z plane for the wave functions (compare, for instance, the cases{dh= 0.05, dV = 0.05} and {dh= −0.05, dV = −0.05}). This can be understood from Figs.1and2. It is clear that the transformations dh⇒ −dh and dV ⇒ −dV cannot be presented as C2rotations of the ring.
For a better understanding of the impact of the broken reflection symmetry in the wobbled rings on the actual localization positions and characteristic confinement lengths of the electron and hole, we study the expectation value of the ground-state electron (hole) position ¯ρe(h)= ( ¯xe(h),y¯e(h))
in the x-y plane and the mean (effective) lateral electron (hole) radius
ρe(h)=
(ρe(h)− ¯ρe(h))2
for the type-A ring at zero magnetic field. For all configura-tions, the expectation value of the electron (hole) position on the y axes remains unchanged: ¯ye(h)= 0. Figure6shows ¯xe(h)
of the noninteracting electron and hole in the ground states for different values and combinations of dh and dV. When the absolute values of the parameters dhand dV increase, the electron mean position gradually moves from the center of the ring toward one (appropriate) of the unbalanced potential valleys. At the same time, the hole mean position is almost stabilized in the valley when parameters dh or dV are only
FIG. 8. (Color online) Three-dimensional contour plot of the electron and hole components of the exciton ground-state wave function for different configurations of the type-A ring.
FIG. 9. (Color online) Three-dimensional contour plot of the electron and hole component of the exciton ground-state wave function for different configurations of the type-A ring for dC=
|dh| = |dV| = 0.05 (see Fig.6).
about 0.1 in the absolute value. Furthermore, from Fig.7we see that the effective lateral radii of the electron and hole follow the same tendency: the electron’s radius decreases smoothly and the hole’s radius shrinks rapidly.
The interparticle interaction drastically changes the sen-sitivity of the particles’ wave functions to the imbalance in
FIG. 10. (Color online) The expectation value of the position of the electron (a) and hole (b) on the x axis for the exciton in the ground state (type-A ring). The configurations c1, . . . ,c8 are described in the caption to Fig.6.
FIG. 11. (Color online) The effective lateral radius of the electron (a) and hole (b) for the exciton in the ground state (type-A ring). The configurations c1, . . . ,c8 are described in the caption to Fig.6. the reflection symmetry of the ring. We show in Fig. 8 the electron and hole components of the ground-state exciton wave function [self-consistent solutions of Eqs. (24)–(26)]. Now even very small (∼0.01) nonzero dhand (or) dV generate the simultaneous localization of the electron and hole wave functions (the ground-state exciton wave function) in one of the potential valleys. The reason is that the Coulomb interaction makes the electron and hole move in the same direction of the position of the hole (dotlike wave function). The actual mean position ¯xe(h)is controlled by the sign of dV. Figure9
shows that a further increase of dhand dV magnitudes does not considerably affect the wave functions which are well localized in one of the potential valleys.
The high sensitivity of the holes to the imbalance in the reflection symmetry of the ring is a “triggering factor” in the “one-valley localization effect” for the exciton. We demonstrate in Fig. 10 that the dependency of the electron mean localization position on dh and dV becomes very similar to that for the hole when we impose the electron-hole interaction. Moreover, the Coulomb interaction makes the electron and hole effective lateral radii both very sensitive to the reflection asymmetry of the ring: the radii simultaneously and rapidly decrease when a small imbalance appears, as is shown in Fig.11(compare with Fig.7).
With the obtained actual positions of the electron and hole components of the ground-state exciton wave function, we examine now the effect of the reflectional asymmetry on the exciton diamagnetic coefficient αd [Eq. (22)]. In
FIG. 12. (Color online) Exciton diamagnetic coefficient (a) and ground-state energy (b) for the type-A ring. The configurations c1, . . . ,c8 are described in the caption to Fig.6.
Fig.12(a), we show the dependence of the exciton diamagnetic coefficient on the parameters dh and dV. It is clear that the above-described sensitivity of the electron and hole wave functions’ localizations and distributions to the imbalance in the reflection symmetry leads to a rapid decrease of the exciton diamagnetic coefficient (about two times) already for small values of dhand dV (about 0.01 in magnitude). However, the coefficient decreases only gradually when the ring’s geometry and material content become more unbalanced along the x direction.
Figure12(b)shows our simulation results for the ground-state energy of the exciton confined in the ring. Clearly, accord-ing to the actual profiles of the energy gap and confinement potential in the structure (see Fig.3), the exciton energy can decrease, increase, and remain unchanged when the absolute values of the parameters dhand dV increase. Notice that the positive nonzero dV always leads to an effective decrease of the distance between the bottom of the conduction band and the top of the valence band [Eg(rp)], which causes the exciton energy to decrease when dV is growing. When the parameter dV is taken to be negative and (or) we impose dh= 0, the effective band gap (the smallest distance between the bottom of the conduction band and the top of the valence band) does not change. Some minor growth of the self-consistent energies of the electron and hole (because of their dotlike confinement in one of the valleys) is mainly compensated by an increase of the electron-hole Coulomb interaction. As a result of those factors’ combination, the exciton energy can decrease
FIG. 13. (Color online) Exciton diamagnetic coefficient (a) and ground-state energy (b) for the type-B ring. The configurations c1, . . . ,c8 are described in the caption to Fig.6.
slightly, increase, or even remain unchanged for dV <0 [see Fig.12(b)].
Let us now consider the quantum ring of the type B. The impact of the broken reflection symmetry on the characteristics of this ring is similar to that for the type-A ring (see Fig.13). The mean localization positions and effective radii of the noninteracting and interacting electron and hole follow the general dependencies described above for the type-A ring. Nevertheless, the rim radius of the type-B ring is smaller than that for the type-A ring (see Table I). This obviously leads to a smaller value of the exciton diamagnetic coefficient for the reflection symmetrical ring (dh= dV = 0). When the symmetry is broken, the exciton diamagnetic coefficient decreases rapidly and it remains gradually decreasing when the absolute values of the parameters dhand dV increase up to 0.2 [Fig.13(a)]. Note that in the type-B ring, the distance between potential valleys is smaller than that in the type-A ring. Therefore, even when the exciton wave function is mainly localized in one of the potential valleys, the small distance between the valleys causes some “relict” penetration of the function into the opposite valley. By increasing the absolute values of the parameters dhand dV, the electron and hole wave functions gradually shrink near the mean localization position. This results in the gradual decrease of the exciton diamagnetic coefficient, which is more visible than that for the type-A ring. In Fig.13(b), we show the ground-state exciton energy for the type-B ring. Comparing with Fig.12(b), we notice that
the dependence of the energy on the parameters dhand dV in general has the same explanation as for the type-A ring.
Our results for both types of rings agree well with exper-imental data.13,38Furthermore, by comparing Figs.12(a)and 13(a)with the data from Refs.13,38, and39, we can suggest that for the type-A ring, the reflection asymmetry can be characterized by|dh| |dV| ≈ 0.1 (αd ≈ 10 μeV/T2), while for the type-B ring it can be characterized by|dh| |dV| ≈ 0.2 (αd ≈ 7μeV/T2). This is consistent with our results for the
exciton ground-state energy. The best agreement we obtain with the experimental data for the exciton energy is when the parameters dhand dV are chosen to be in between+0.1 and +0.15 (type-A ring), and in between +0.15 and +0.2 (type-B ring). Notice that for both types of quantum rings, our results suggest a strong “correlation” between dhand dV.
IV. CONCLUSION
In this paper, using the mapping method and Hartree approximation, we calculated the diamagnetic coefficient for the neutral exciton confined in IncAs1−cGa/GaAs wobbled SAQRs. We have systematically investigated the impact of the rings’ reflection asymmetry on the exciton ground-state wave function localization, energy, and diamagnetic coefficient. Two types of the rings known from experimental works were considered. Deviations from the reflection symmetry caused by the imperfect ring shape and material content are controlled by the parameters dh and dV, respectively (the case dh= dV = 0 stands for the reflection symmetrical ring). We have found that for the noninteracting particles, the electron wave function is more stable to the unbalance in the ring reflection symmetry than the hole wave function. This originates from the difference in the particles’ effective masses. The Coulomb interaction causes the localization of the exciton wave function in one of the potential valleys of the wobbled ring for very small deviations of the parameters dh and dV
from zero. Moreover, the effective lateral radii of the electron and hole are shrinking rapidly when the absolute values of the parameters dh and dV are growing. This results in the rapid decrease of the neutral exciton diamagnetic coefficient. After the rapid drop, the diamagnetic coefficient only gradually decreases with the further increase of the magnitudes of parameters dh and dV. In addition, we have found that a “correlative” imbalance in the ring geometry and material content has a recognizable impact on the ground-state energy of the neutral exciton. Our results are in good agreement with experimental observations. Moreover, they can give a very useful insight into the actual magneto-optical properties of the self-assembled semiconductor quantum rings. We note that a strong sensitivity of the magneto-optical properties to the actual imbalance in the reflection symmetry (which is very likable in experiments) and the rapid transition from the ringlike exciton wave function to the dotlike exciton wave function can be a reason for the optical Aharonov-Bohm effect suppression in IncAs1−cGa/GaAs wobbled rings. At the same
time, the wave function of an independent election is less sensitive to the lack of the reflection symmetry, which helps to explain the observation of the Aharonov-Bohm magnetization oscillation even in quantum rings with broken reflection symmetry.
More generally, our approach can be used for the realistic modeling of the magneto and magneto-optical characteristics of semiconductor nano-objects with realistic and nonsymmet-rical geometry and material content.
ACKNOWLEDGMENTS
This work is supported by the National Science Council of the Republic of China under Contracts No. NSC 101-3113-P-009-004 and No. NSC 101-2112-M-009-010, and Aiming for the Top University Program of the National Chiao Tung University.
*Author to whom all correspondence should be addressed: [email protected]
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