- Email: [email protected]
Exchange coupling and polarization relaxation in self-assembled quantum dots
Physica E 13 (2002) 216 – 219
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Exchange coupling and polarization relaxation in self-assembled quantum dots R. Ferreira ∗ Laboratoire de Physique de la Matiere Condensee, Ecole Normale Superieure, LPMC, 24 rue Lhomond F75005 Paris, France
Abstract We consider the spin-coupling for one electron–hole pair bound in single self-assembled quantum dots. The long-range exchange coupling leads to a vanishing spin splitting of the ground interband transition in cylindrical dots. We evaluate the spin-splitting for di+erent electron–hole levels of non-cylindrical structures. Both the dot shape anisotropy and an in-plane perturbing -eld are considered. The role of a .uctuating environment on the polarization of the bound dot luminescence is also discussed. ? 2002 Published by Elsevier Science B.V. Keywords: Nanostructure; Luminescence; Polarization
The existence of a discrete sequence of narrow transition lines involving the bound levels of a single quantum dot is nowadays well established. Furthermore, a few high accuracy experiments were able to resolve the -ne structure of the interband lines in good quality samples. In particular, c.w. experiments show the existence of an energy splitting between two lines linearly polarized along the directions [ − 1; 1; 0] and [1; 1; 0] (see for instance Refs. [1,2]). Also, time resolved experiments performed in the strict resonant excitation and detection situation show that the linear polarizations of the ground dot interband transitions are conserved at low temperature [3]. The existence of two particular (orthogonal) directions for the light polarization has been attributed to a geometric anisotropy of the dot con-ning potential (or to internal strain -elds) and
∗
Tel.: +33-1-4432-3373; fax: +33-1-4432-3840. E-mail address: [email protected] (R. Ferreira).
the corresponding electron–hole states associated to the exchange-splitted linear combinations of the two bright, circularly polarized | ± 1 ground dot levels ([4 – 6]; see also below). A few experimental results show, however, that in certain cases a circularly polarized luminescence can be observed after a non-resonant excitation [7–9]. This seems to be in contradiction with the previous image of linearly polarized stationary states for the dots. We report here on calculations of the long-range spin-coupling for an electron–hole pair con-ned in a self-assembled quantum dot. We obtain very di+erent energy couplings for the various electron–hole states. In particular, electron–hole states that are optically active from the point-of-view of their envelope functions (including the ground state) have a non-vanishing spin-splitting only if the total potential felt by the bound carriers is not symmetric in the layer plane, whereas non-radiative states can present an important spin-coupling even for a cylindrical dot. Two asymmetric perturbations are considered: the dot
1386-9477/02/$ - see front matter ? 2002 Published by Elsevier Science B.V. PII: S 1 3 8 6 - 9 4 7 7 ( 0 1 ) 0 0 5 2 3 - 9
R. Ferreira / Physica E 13 (2002) 216 – 219
anisotropy and the e+ect of an in-plane electric -eld. Finally, the role of a .uctuating electrostatic -eld on the polarization dynamics of the dot luminescence is discussed, and will permit us to describe how the apparently contradictory observations can be conciliated if we account for the environment around the dot. The dot con-ning potentials for electrons and holes are assumed to be truncated cones with basis radius R0 = 10 nm, height h=2:5 nm and lateral angle ◦ = 30 , .oating on a thin (1 ML thick) wetting layer. The parameters we use are for the InAs=GaAs dots: e+ective masses m∗e = 0:067 m0 ; m∗hh(z) = 0:34 m0 and m∗hh(x; y) =0:11m0 ; o+-set discontinuities Ve =413 meV and Vhh = 288 meV; Egap(InAs) ≈ 400 meV and dielectric permittivity InAs = 14:5. The bound electron and heavy-hole S-like, Px -like and Py -like states are described by separable envelopes (z is the growth axis): R(r) = R(z; ; ) = GR (z − zR )hR ()N FR () with R = Se ; Xe or Ye for the electron states and R = Sh ; Xh or Yh for the hole states. GR (z − zR ) and FR () are gaussians with variational widths and center zR ; hS ()=1; hX ()=cos() and hY ()=sin(); N =0 for S-like states and N = 1 for P-like states. These trials provide a good description of the -rst few bound levels in self-assembled quantum dots. The energies of the electron and hole levels are: E(Se ) ≈ 241 meV; E(Xe ) = E(Ye ) ≈ 302 meV; E(Sh ) ≈ 94 meV and E(Xh ) = E(Yh ) ≈ 134 meV. For dots, which display a cylindrical symmetry, the P-shell states are of course degenerate. However, due to the shape anisotropy, the electron and hole con-ning potentials present an elongation along the [ − 1; 1; 0] direction [1,2]. This elliptical anisotropy is accounted for as follows. By choosing the x and y directions along the major and minor axes of the ellipse, the lowest order corrections a+ecting the P-like states amount to diagonal contributions: ±e ( ≈ 3 meV) and ±hh ( ≈ 2 meV) for the electrons and holes, respectively, with the plus (minus) signals for X-related (Y-related) states. Pair states are formed by combining the di+erent electron and hole levels. For instance, the ground non-interacting pair is the Se (re )Sh (rh ) state (noted Se Sh in the following, for brevity). Among the nine pair states, the coulomb interaction mix only those of the same manifold: {Se Sh ; Xe Xh ; Ye Yh }, {Se Xh ; Xe Sh }, {Se Yh ; Ye Sh } or {Xe Yh ; Ye Xh }. Thus, the ground pair level only couples to the excited
217
Xe Xh and Ye Yh , states of the P-shells. Diagonal corrections (e.g., Se Sh |Vcoul |Se Sh = −25:1 meV) are by far more important than non-diagonal ones (e.g., Se Sh |Vcoul |Xe Xh = −5:7 meV). States of di+erent manifolds only couple in the presence of an external perturbation, which lowers the symmetry of the single particle dot potentials, as e.g. in the presence of an in-plane electric -eld (see below). Within the previous parabolic scheme, the total wave functions for the bound carriers are of the form R(r)uJ (r), where uJ (r) are the Bloch periodic functions at the bulk edges for electrons and heavy holes. These atomic functions are classi-ed as usual according to the projection of their total (orbital + spin) angular momentum Je = ± 12 and Jhh = ± 32 along the growth axis and any single carrier level is twofold “spin” degenerate. As a consequence, any electron–hole level is fourfold degenerate, account to the “spin” (Je = ± 12 and Jhh = ± 32 ) degree of freedom. The atomic functions are in particular responsible for the polarization-related selection rule for the interband transitions, namely: only radiative processes involving pair states with Jpair = Je + Jhh = ±1 are “spin”-allowed. In particular, the ground dot luminescence presents two lines with orthogonal circular polarizations (the so-called bright | ± 1 states) that are degenerate in absence of any spin-coupling. These trends for single particles and for pair states have been discussed in di+erent papers. Here, we present our results on the long-range exchange couplings in quantum dots. We assume the bulk-like perturbation, previously used to study the -ne structure of free and localized excitons in quantum wells and quantum wires. Let us recall two features of this coupling. First, to the lowest order, it couples only the two “spin-allowed” radiative states among the four spin-degenerate ones. Second, it is well known for exciton states weakly bound by interface defects in wells and wires that the spin-coupling re.ects somehow the spatial anisotropy of the envelope in the plane perpendicular to the growth axis (the direction of quantization for the spin and orbital angular momenta). These two features apply also for pair states in strongly con-ning quantum dots. Owing to the -rst point, we discuss here only the non-diagonal long-range coupling of the two degenerate “spin-allowed” levels. The second feature is illustrated by considering the spin-splitting of the ground dot state, as follows.
218
R. Ferreira / Physica E 13 (2002) 216 – 219
The coupling vanishes exactly when the ground state is described by Se (re )Sh (rh ), for any electron and hole envelopes S(r) with cylindrical symmetry. The coulombic perturbation mix the {Se Sh ; Xe Xh ; Ye Yh } states: the ground state envelope acquires small P-like components, but its spin coupling still vanishes if we neglect the dot anisotropy. For an elongated dot, the degeneracy of the two bright ground levels is lifted, but we calculate a small energy splitting for the ground transitions (−10:8 eV), as compared to the rather important splittings for the levels that are directly a+ected by the shape anisotropy (484 eV for the states which are mainly Xe Xh and −473 eV for the Ye Yh related ones). It is however important to stress two points: (i) Although weak, this spin-splitting is nevertheless much greater than the radiative broadening of the bright levels (typically, ˜=rad 6 1 eV). (ii) Although weak, the spin-coupling radically changes the nature of the ground interband transition, leading to new lines that are polarized linearly along the ellipse axes (instead of the | ± 1 circularly polarized transitions for cylindrical dots). It is also instructive to consider the splittings for the P-shell states {Xe Xh ; Ye Yh ; Xe Yh ; Ye Xh }. The states Xe Xh and Ye Yh have a non-vanishing spin-splitting, even in absence of shape anisotropy. This appears contradictory with what has been said of the spin-coupling as a measure of the in-plane anisotropy of the dot envelopes, since for a cylindrical dot, we could as well choose a more symmetrical representation P±1 (r) for the single particle envelopes and thus -nish out with vanishing couplings for all the P-shell states. The missing point is the coulombic coupling, which couple the states Xe Xh and Ye Yh (with same spin). As a consequence, we -nd for a symmetrical dot that the P-like states have no spin-splitting, as expected. For elongated dots, on the contrary, Xe Xh and Ye Yh are no more degenerate and have a -nite splitting (see above). In conclusion, in elliptical dots both the fundamental ( ≈ Se Sh ) and the excited ( ≈ Xe Xh or ≈ Ye Yh ) electron–hole states present a -nite spin-splitting and the -ne structure of the ground interband transition is associated to its indirect (coulomb assisted) coupling to the excited P-like levels which are directly a+ected by the in-plane dot anisotropy. Let us now discuss the e+ect of an external electric -eld on the -ne structure of the interband dot transitions (we give below a possible origin for such
a -eld). We take F = F⊥ [cos(F )x + sin(F )y] and neglect the dot anisotropy. (A Fz term leads only to small spin-independent shifts within the truncated basis we are considering.) The lateral -eld mix the states {Se Sh ; Xe Xh ; Ye Yh ; Se Xh ; Xe Sh ; Se Yh ; Ye Sh } and couples in particular the ground dot level to the “hybrid” ones formed by combining electron and hole states of di+erent shells. These crossed states are placed in energy between the S-like (Se Sh ) and the P-like (Xe Xh and Ye Yh ) levels and di+er from them in two important aspects: (i) they correspond to non-radiative transitions of the dot (selection rule on the envelope parts of the total wave function); (ii) they present a -nite spin-coupling, even for a cylindrical dot. Of course, the consideration of the inter-level coulombic couplings does not alter these two features. Note that the crossed levels, although non-radiative, can provide an important path for intra-dot relaxation of photoexcited carriers. With regards to spin, they gain importance when an in-plane -eld is present (or any perturbation with low enough in-plane symmetry): in this case the hybrid transitions acquire a -nite radiative lifetime, because of their coupling to the ground Se Sh level, which one acquires in turn a -nite spin-coupling. In the following, we focus on this contribution to the ground level spin-splitting. To illustrate this e+ect, we use the restricted basis {Se Sh ; Se Xh ; Se Yh } of the three lowest energy levels. The spin-coupling for the ground state (ground ≈ Se Sh ) is given by UND (ground) = UND (Se Xh ) exp{i2F }= {1 + (eSP − eground )=(eSS − eground )}; (1) where UND (Se Xh ) = UND (Se Yh ) is the spin-coupling of the independent Se Xh and Se Yh levels and eAB = E(Ae ) + E(Bh ) + Ae Bh |Vcoul |Ae Bh + Egap is the energy of the pair state Ae Bh (eSP = eSX = eSY ). We calculate |UND (Se Xh )|=1355:6 eV, a value far above the ones -nd for the “direct” (electron and hole envelopes of the same shell) states. The corresponding spin-splitting for the ground transition equals the one due to the shape anistropy (indirect coulomb coupling with Xe Xh and Ye Yh ) for F⊥ ≈ 20 kV=cm. Such a -eld could be due to charged centers located near the dot surface. Note that it leads to small energy shifts for the interband transitions, since the electron and hole electrostatic shifts tend to cancel out.
R. Ferreira / Physica E 13 (2002) 216 – 219
It is worth to stress the exp{i2F } phase in Eq. (1). As an important consequence, the spin-splitted interband lines are polarized linearly along the -eld direction, in the same way as the states splitted by the dot anisotropy coupling are linearly polarized along the ellipse axes. The important point now is that, contrarily to the dot shape perturbation, the one due to the charging=decharging of a distribution of trap centers around the dot should evolve in time, because of the random nature of their occupancy. As a consequence, the resulting in-plane -eld felt by the electron–hole pair bound in the dot would also evolve in time, both in amplitude and orientation. When both couplings are simultaneously considered, a particular axis for the polarization orientation appears only when the e+ect of the dot anisotropy dominates, whereas no preferred (stationary) orientation exists whenever the e+ect of the environment .uctuation dominates. In conclusion, the existence of a preferential orientation for the dot luminescence polarization appears to be sample dependent and rather sensitive to the existence of external perturbations which breaks the cylindrical symmetry of the dot. Another important consequence of the phase exp{i2F } concerns the dynamics of the luminescence polarization. Actually, for the two cases considered above (dot anisotropy and in-plane -eld), we can apply the well known picture for the spin-coupling as resulting from the action of an e+ective magnetic -eld on the spins, the amplitude and orientation of which being peculiar to the particular spin-dependent perturbation. The dot anisotropy perturbation leads to a static e+ective -eld acting on the electron– hole “spins”: the spin-splitted stationary levels are linearly polarized and, in the time domain, the polarization oscillates among the | ± 1 bright levels. On the contrary, the e+ective -eld due to the exterior perturbations evolve in time, both in
219
amplitude and direction, and its time average vanishes (because of the exp{i2F } phase). If in addition, its characteristic .uctuation time is much faster than the intrinsic spin-.ip, we recover the well known “motional narrowing” regime for mobile carriers and excitons in disordered bulk and quantum well samples. Here, however, the electron–hole pair is strongly con-ned in the dot and it is the environment that .uctuates in a random way nearby the dot. This -nally leads to a suppression of the coherent coupling between the | ± 1 states and to a corresponding irreversible evolution (a relaxation) for the circular polarization. This regime should not be diMcult to reach in actual dots, account to the small energy splittings evaluated above. It could, in particular, explain the existence of a circular polarization observed in some experiments after a non-resonant excitation. We thank fruitful discussions with G. Bastard, S. Cortez, J.M. Gerard, R. Grousson, T. Guillet, K. Kheng, X. Marie, V. Voliotis. The Laboratoire de Physique de la Matiere Condensee is “Unite associee au CNRS (UMR8551) et Universites Paris 6 et Paris 7”. The work was supported by a E.U. project (SQID: IST-1999-11311) and a New Energy and Industrial Technology Development Organization Grant (NEDO, Japan). References [1] [2] [3] [4] [5] [6] [7] [8] [9]
M. Bayer, et al., Phys. Rev. Lett. 82 (1999) 1748. D. Gammon, et al., Phys. Rev. Lett. 76 (1996) 3005. M. Paillard, et al., Phys. Rev. Lett. 86 (2001) 1634. S.V. Gupalov, et al., JETP Lett. 86 (1998) 388. D. Larousserie, et al., Phys. Rev. B 60 (1999) 1892. T. Takagahara, Phys. Rev. B 62 (2000) 16,840. V.K. Kalevich, et al., Phys. Solid State 41 (2001) 789. H. Gotoh, et al., Appl. Phys. Lett. 72 (1998) 1341. S. Cortez, et al., Physica E (2002), these Proceedings.
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Exchange coupling and polarization relaxation in self-assembled quantum dots R. Ferreira ∗ Laboratoire de Physique de la Matiere Condensee, Ecole Normale Superieure, LPMC, 24 rue Lhomond F75005 Paris, France
Abstract We consider the spin-coupling for one electron–hole pair bound in single self-assembled quantum dots. The long-range exchange coupling leads to a vanishing spin splitting of the ground interband transition in cylindrical dots. We evaluate the spin-splitting for di+erent electron–hole levels of non-cylindrical structures. Both the dot shape anisotropy and an in-plane perturbing -eld are considered. The role of a .uctuating environment on the polarization of the bound dot luminescence is also discussed. ? 2002 Published by Elsevier Science B.V. Keywords: Nanostructure; Luminescence; Polarization
The existence of a discrete sequence of narrow transition lines involving the bound levels of a single quantum dot is nowadays well established. Furthermore, a few high accuracy experiments were able to resolve the -ne structure of the interband lines in good quality samples. In particular, c.w. experiments show the existence of an energy splitting between two lines linearly polarized along the directions [ − 1; 1; 0] and [1; 1; 0] (see for instance Refs. [1,2]). Also, time resolved experiments performed in the strict resonant excitation and detection situation show that the linear polarizations of the ground dot interband transitions are conserved at low temperature [3]. The existence of two particular (orthogonal) directions for the light polarization has been attributed to a geometric anisotropy of the dot con-ning potential (or to internal strain -elds) and
∗
Tel.: +33-1-4432-3373; fax: +33-1-4432-3840. E-mail address: [email protected] (R. Ferreira).
the corresponding electron–hole states associated to the exchange-splitted linear combinations of the two bright, circularly polarized | ± 1 ground dot levels ([4 – 6]; see also below). A few experimental results show, however, that in certain cases a circularly polarized luminescence can be observed after a non-resonant excitation [7–9]. This seems to be in contradiction with the previous image of linearly polarized stationary states for the dots. We report here on calculations of the long-range spin-coupling for an electron–hole pair con-ned in a self-assembled quantum dot. We obtain very di+erent energy couplings for the various electron–hole states. In particular, electron–hole states that are optically active from the point-of-view of their envelope functions (including the ground state) have a non-vanishing spin-splitting only if the total potential felt by the bound carriers is not symmetric in the layer plane, whereas non-radiative states can present an important spin-coupling even for a cylindrical dot. Two asymmetric perturbations are considered: the dot
1386-9477/02/$ - see front matter ? 2002 Published by Elsevier Science B.V. PII: S 1 3 8 6 - 9 4 7 7 ( 0 1 ) 0 0 5 2 3 - 9
R. Ferreira / Physica E 13 (2002) 216 – 219
anisotropy and the e+ect of an in-plane electric -eld. Finally, the role of a .uctuating electrostatic -eld on the polarization dynamics of the dot luminescence is discussed, and will permit us to describe how the apparently contradictory observations can be conciliated if we account for the environment around the dot. The dot con-ning potentials for electrons and holes are assumed to be truncated cones with basis radius R0 = 10 nm, height h=2:5 nm and lateral angle ◦ = 30 , .oating on a thin (1 ML thick) wetting layer. The parameters we use are for the InAs=GaAs dots: e+ective masses m∗e = 0:067 m0 ; m∗hh(z) = 0:34 m0 and m∗hh(x; y) =0:11m0 ; o+-set discontinuities Ve =413 meV and Vhh = 288 meV; Egap(InAs) ≈ 400 meV and dielectric permittivity InAs = 14:5. The bound electron and heavy-hole S-like, Px -like and Py -like states are described by separable envelopes (z is the growth axis): R(r) = R(z; ; ) = GR (z − zR )hR ()N FR () with R = Se ; Xe or Ye for the electron states and R = Sh ; Xh or Yh for the hole states. GR (z − zR ) and FR () are gaussians with variational widths and center zR ; hS ()=1; hX ()=cos() and hY ()=sin(); N =0 for S-like states and N = 1 for P-like states. These trials provide a good description of the -rst few bound levels in self-assembled quantum dots. The energies of the electron and hole levels are: E(Se ) ≈ 241 meV; E(Xe ) = E(Ye ) ≈ 302 meV; E(Sh ) ≈ 94 meV and E(Xh ) = E(Yh ) ≈ 134 meV. For dots, which display a cylindrical symmetry, the P-shell states are of course degenerate. However, due to the shape anisotropy, the electron and hole con-ning potentials present an elongation along the [ − 1; 1; 0] direction [1,2]. This elliptical anisotropy is accounted for as follows. By choosing the x and y directions along the major and minor axes of the ellipse, the lowest order corrections a+ecting the P-like states amount to diagonal contributions: ±e ( ≈ 3 meV) and ±hh ( ≈ 2 meV) for the electrons and holes, respectively, with the plus (minus) signals for X-related (Y-related) states. Pair states are formed by combining the di+erent electron and hole levels. For instance, the ground non-interacting pair is the Se (re )Sh (rh ) state (noted Se Sh in the following, for brevity). Among the nine pair states, the coulomb interaction mix only those of the same manifold: {Se Sh ; Xe Xh ; Ye Yh }, {Se Xh ; Xe Sh }, {Se Yh ; Ye Sh } or {Xe Yh ; Ye Xh }. Thus, the ground pair level only couples to the excited
217
Xe Xh and Ye Yh , states of the P-shells. Diagonal corrections (e.g., Se Sh |Vcoul |Se Sh = −25:1 meV) are by far more important than non-diagonal ones (e.g., Se Sh |Vcoul |Xe Xh = −5:7 meV). States of di+erent manifolds only couple in the presence of an external perturbation, which lowers the symmetry of the single particle dot potentials, as e.g. in the presence of an in-plane electric -eld (see below). Within the previous parabolic scheme, the total wave functions for the bound carriers are of the form R(r)uJ (r), where uJ (r) are the Bloch periodic functions at the bulk edges for electrons and heavy holes. These atomic functions are classi-ed as usual according to the projection of their total (orbital + spin) angular momentum Je = ± 12 and Jhh = ± 32 along the growth axis and any single carrier level is twofold “spin” degenerate. As a consequence, any electron–hole level is fourfold degenerate, account to the “spin” (Je = ± 12 and Jhh = ± 32 ) degree of freedom. The atomic functions are in particular responsible for the polarization-related selection rule for the interband transitions, namely: only radiative processes involving pair states with Jpair = Je + Jhh = ±1 are “spin”-allowed. In particular, the ground dot luminescence presents two lines with orthogonal circular polarizations (the so-called bright | ± 1 states) that are degenerate in absence of any spin-coupling. These trends for single particles and for pair states have been discussed in di+erent papers. Here, we present our results on the long-range exchange couplings in quantum dots. We assume the bulk-like perturbation, previously used to study the -ne structure of free and localized excitons in quantum wells and quantum wires. Let us recall two features of this coupling. First, to the lowest order, it couples only the two “spin-allowed” radiative states among the four spin-degenerate ones. Second, it is well known for exciton states weakly bound by interface defects in wells and wires that the spin-coupling re.ects somehow the spatial anisotropy of the envelope in the plane perpendicular to the growth axis (the direction of quantization for the spin and orbital angular momenta). These two features apply also for pair states in strongly con-ning quantum dots. Owing to the -rst point, we discuss here only the non-diagonal long-range coupling of the two degenerate “spin-allowed” levels. The second feature is illustrated by considering the spin-splitting of the ground dot state, as follows.
218
R. Ferreira / Physica E 13 (2002) 216 – 219
The coupling vanishes exactly when the ground state is described by Se (re )Sh (rh ), for any electron and hole envelopes S(r) with cylindrical symmetry. The coulombic perturbation mix the {Se Sh ; Xe Xh ; Ye Yh } states: the ground state envelope acquires small P-like components, but its spin coupling still vanishes if we neglect the dot anisotropy. For an elongated dot, the degeneracy of the two bright ground levels is lifted, but we calculate a small energy splitting for the ground transitions (−10:8 eV), as compared to the rather important splittings for the levels that are directly a+ected by the shape anisotropy (484 eV for the states which are mainly Xe Xh and −473 eV for the Ye Yh related ones). It is however important to stress two points: (i) Although weak, this spin-splitting is nevertheless much greater than the radiative broadening of the bright levels (typically, ˜=rad 6 1 eV). (ii) Although weak, the spin-coupling radically changes the nature of the ground interband transition, leading to new lines that are polarized linearly along the ellipse axes (instead of the | ± 1 circularly polarized transitions for cylindrical dots). It is also instructive to consider the splittings for the P-shell states {Xe Xh ; Ye Yh ; Xe Yh ; Ye Xh }. The states Xe Xh and Ye Yh have a non-vanishing spin-splitting, even in absence of shape anisotropy. This appears contradictory with what has been said of the spin-coupling as a measure of the in-plane anisotropy of the dot envelopes, since for a cylindrical dot, we could as well choose a more symmetrical representation P±1 (r) for the single particle envelopes and thus -nish out with vanishing couplings for all the P-shell states. The missing point is the coulombic coupling, which couple the states Xe Xh and Ye Yh (with same spin). As a consequence, we -nd for a symmetrical dot that the P-like states have no spin-splitting, as expected. For elongated dots, on the contrary, Xe Xh and Ye Yh are no more degenerate and have a -nite splitting (see above). In conclusion, in elliptical dots both the fundamental ( ≈ Se Sh ) and the excited ( ≈ Xe Xh or ≈ Ye Yh ) electron–hole states present a -nite spin-splitting and the -ne structure of the ground interband transition is associated to its indirect (coulomb assisted) coupling to the excited P-like levels which are directly a+ected by the in-plane dot anisotropy. Let us now discuss the e+ect of an external electric -eld on the -ne structure of the interband dot transitions (we give below a possible origin for such
a -eld). We take F = F⊥ [cos(F )x + sin(F )y] and neglect the dot anisotropy. (A Fz term leads only to small spin-independent shifts within the truncated basis we are considering.) The lateral -eld mix the states {Se Sh ; Xe Xh ; Ye Yh ; Se Xh ; Xe Sh ; Se Yh ; Ye Sh } and couples in particular the ground dot level to the “hybrid” ones formed by combining electron and hole states of di+erent shells. These crossed states are placed in energy between the S-like (Se Sh ) and the P-like (Xe Xh and Ye Yh ) levels and di+er from them in two important aspects: (i) they correspond to non-radiative transitions of the dot (selection rule on the envelope parts of the total wave function); (ii) they present a -nite spin-coupling, even for a cylindrical dot. Of course, the consideration of the inter-level coulombic couplings does not alter these two features. Note that the crossed levels, although non-radiative, can provide an important path for intra-dot relaxation of photoexcited carriers. With regards to spin, they gain importance when an in-plane -eld is present (or any perturbation with low enough in-plane symmetry): in this case the hybrid transitions acquire a -nite radiative lifetime, because of their coupling to the ground Se Sh level, which one acquires in turn a -nite spin-coupling. In the following, we focus on this contribution to the ground level spin-splitting. To illustrate this e+ect, we use the restricted basis {Se Sh ; Se Xh ; Se Yh } of the three lowest energy levels. The spin-coupling for the ground state (ground ≈ Se Sh ) is given by UND (ground) = UND (Se Xh ) exp{i2F }= {1 + (eSP − eground )=(eSS − eground )}; (1) where UND (Se Xh ) = UND (Se Yh ) is the spin-coupling of the independent Se Xh and Se Yh levels and eAB = E(Ae ) + E(Bh ) + Ae Bh |Vcoul |Ae Bh + Egap is the energy of the pair state Ae Bh (eSP = eSX = eSY ). We calculate |UND (Se Xh )|=1355:6 eV, a value far above the ones -nd for the “direct” (electron and hole envelopes of the same shell) states. The corresponding spin-splitting for the ground transition equals the one due to the shape anistropy (indirect coulomb coupling with Xe Xh and Ye Yh ) for F⊥ ≈ 20 kV=cm. Such a -eld could be due to charged centers located near the dot surface. Note that it leads to small energy shifts for the interband transitions, since the electron and hole electrostatic shifts tend to cancel out.
R. Ferreira / Physica E 13 (2002) 216 – 219
It is worth to stress the exp{i2F } phase in Eq. (1). As an important consequence, the spin-splitted interband lines are polarized linearly along the -eld direction, in the same way as the states splitted by the dot anisotropy coupling are linearly polarized along the ellipse axes. The important point now is that, contrarily to the dot shape perturbation, the one due to the charging=decharging of a distribution of trap centers around the dot should evolve in time, because of the random nature of their occupancy. As a consequence, the resulting in-plane -eld felt by the electron–hole pair bound in the dot would also evolve in time, both in amplitude and orientation. When both couplings are simultaneously considered, a particular axis for the polarization orientation appears only when the e+ect of the dot anisotropy dominates, whereas no preferred (stationary) orientation exists whenever the e+ect of the environment .uctuation dominates. In conclusion, the existence of a preferential orientation for the dot luminescence polarization appears to be sample dependent and rather sensitive to the existence of external perturbations which breaks the cylindrical symmetry of the dot. Another important consequence of the phase exp{i2F } concerns the dynamics of the luminescence polarization. Actually, for the two cases considered above (dot anisotropy and in-plane -eld), we can apply the well known picture for the spin-coupling as resulting from the action of an e+ective magnetic -eld on the spins, the amplitude and orientation of which being peculiar to the particular spin-dependent perturbation. The dot anisotropy perturbation leads to a static e+ective -eld acting on the electron– hole “spins”: the spin-splitted stationary levels are linearly polarized and, in the time domain, the polarization oscillates among the | ± 1 bright levels. On the contrary, the e+ective -eld due to the exterior perturbations evolve in time, both in
219
amplitude and direction, and its time average vanishes (because of the exp{i2F } phase). If in addition, its characteristic .uctuation time is much faster than the intrinsic spin-.ip, we recover the well known “motional narrowing” regime for mobile carriers and excitons in disordered bulk and quantum well samples. Here, however, the electron–hole pair is strongly con-ned in the dot and it is the environment that .uctuates in a random way nearby the dot. This -nally leads to a suppression of the coherent coupling between the | ± 1 states and to a corresponding irreversible evolution (a relaxation) for the circular polarization. This regime should not be diMcult to reach in actual dots, account to the small energy splittings evaluated above. It could, in particular, explain the existence of a circular polarization observed in some experiments after a non-resonant excitation. We thank fruitful discussions with G. Bastard, S. Cortez, J.M. Gerard, R. Grousson, T. Guillet, K. Kheng, X. Marie, V. Voliotis. The Laboratoire de Physique de la Matiere Condensee is “Unite associee au CNRS (UMR8551) et Universites Paris 6 et Paris 7”. The work was supported by a E.U. project (SQID: IST-1999-11311) and a New Energy and Industrial Technology Development Organization Grant (NEDO, Japan). References [1] [2] [3] [4] [5] [6] [7] [8] [9]
M. Bayer, et al., Phys. Rev. Lett. 82 (1999) 1748. D. Gammon, et al., Phys. Rev. Lett. 76 (1996) 3005. M. Paillard, et al., Phys. Rev. Lett. 86 (2001) 1634. S.V. Gupalov, et al., JETP Lett. 86 (1998) 388. D. Larousserie, et al., Phys. Rev. B 60 (1999) 1892. T. Takagahara, Phys. Rev. B 62 (2000) 16,840. V.K. Kalevich, et al., Phys. Solid State 41 (2001) 789. H. Gotoh, et al., Appl. Phys. Lett. 72 (1998) 1341. S. Cortez, et al., Physica E (2002), these Proceedings.