Quantum properties of spherical semiconductor quantum dots
aa r X i v : . [ c ond - m a t . m e s - h a ll ] F e b Quantum properties of spherical semiconductorquantum dots B. Billaud ∗ and T.-T. Truong
Laboratoire de Physique Th´eorique et Mod´elisation (LPTM),CNRS UMR 8089, Universit´e de Cergy-Pontoise,2, av. Adolphe Chauvin, F-95302 Cergy-Pontoise Cedex, France.
September 11, 2018
Abstract
Quantum effects at the nanometric level have been observed in many confined structures, and partic-ularly in semiconductor quantum dots (QDs). In this work, we propose a theoretical improvement of theso-called effective mass approximation with the introduction of an effective pseudo-potential. This advan-tageously allows analytic calculations to a large extent, and leads to a better agreement with experimentaldata. We have obtained, as a function of the QD radius, in precise domains of validity, the QD ground stateenergy, its Stark and Lamb shifts. An observable Lamb shift is notably predicted for judiciously chosensemiconductor and radius. Despite the intrinsic non-degeneracy of the QD energy spectrum, we propose a
Gedankenexperiment based on the use of the Casimir effect to test its observability. Finally, the effect of anelectromagnetic cavity on semiconductor QDs is also considered, and its Purcell factor evaluated. This lastresult raises the possibility of having a QD-LASER emitting in the range of visible light.
Keywords spherical Quantum Dot, semiconductor, exciton, Stark effect, Lamb effect,Casimir effect, Purcell Effect.
PACS
Semiconductor quantum dots (QDs), as well as quantum wires or quantum wells, show properties of standardatomic physics, as a result of the restriction of the motion of one to a hundred conduction band electronsor valence band holes to a confined region of space of nanometric size. But, in contrast to atoms, phonons,surface effects and bulk disorder play a crucial role in determining their electronic properties, so that twoQDs are never really identical. A QD may be thought as a giant artificial atom with an adjustable quantizedenergy spectrum, controlled only by its size. It enjoys prospects to serve in quantum optics as sources for asemiconductor LASER [1] or of single photon [2], in quantum information as qubits [3], in micro-electronicsas single-electron transistors [4], or in biology and medicine as fluorophores [5].During the early 1980s, the so-called quantum size effects (QSE), characterized by a blue-shift of theiroptical spectra, has been observed in a large range of strongly confined systems [6–10]. It comes from awidening of the semiconductor optical band gap, due to the increase of the charge carriers confinementenergy [11]. A review of empirical and theoretical results on quantum confinement effects in low-dimensionalsemiconductor structures is given in [12]. Modern approaches to this problem are discussed in [13, 14]. But,despite numerous theoretical and empirical models, to the best of our knowledge, there exists no simple andcomprehensive one, which offers a significant analytic treatment.To apprehend the origin of QSE in spherical semiconductor QDs, we propose to adopt the effective massapproximation (EMA), which assumes parabolic valence and conduction bands [11, 15, 16]. An electronand a hole behave as free particles with their usual effective mass, but confined in a spherical infinite ∗ To whom correspondence should be addressed: [email protected] otential well. Their Coulomb interaction is taken into account through a variational principle. This model,presented in section , allows the introduction of an effective pseudo-potential, which partially removesthe characteristic overestimation of the electron-hole pair confinement energy for small QDs [18]. As anachievement, an analytic expression for the phenomenological function η ( λ ), introduced in [16], is obtainedin good agreement with numerical data [17].Among many fundamental topics, the atom-like behavior of QDs is nowadays intensively investigatedbecause of its potential technological applications. Of particular interest is the interaction with an externalelectromagnetic field. In semiconductor microcrystals, the presence of a constant electric field gives riseto quantum-confinement Stark effects (QCSE) [19–21]. It manifests itself by a characteristic red-shift ofthe exciton photoluminescence [22–26], and leads to a corresponding enhancement of its lifetime [27]. Ifan electric field is applied perpendicularly to the plane of multi-layers quantum wells, exciton energy shiftpeaks were measured and successfully compared to theoretical results [28], obtained by a perturbative methodintroduced in [29], when the electron-hole Coulomb interaction is negligible. But, in spherical QDs, this turnsout to be more important, and cannot be discarded [30]. In section , we propose to use the previous EMAmodel for spherical QDs. It allows the derivation of analytic criterions for choosing the QD radius and theapplied electric field amplitude, as a result of the interplay between electron-hole Coulomb interaction andan additional polarization energy [31].The Lamb shift in atoms, due to the interaction of valence electrons with a quantized electromagneticfield, has provided a convincing experimental check of the validity of quantum electrodynamics, and has been,ever since, a continual subject of research. Effects of the band gap [32], of a electromagnetic mode [33], andits coupling to the QD surface [34] have been notably investigated in semiconductor QDs. However, theLamb effect, which is experimentally well established [35] and theoretically understood [36, 37] in atomsby the end of the 1940s, seems to be unknown in QDs. The purpose of section is to fill this gap. Thetheoretical framework, set up in section , is used to evaluate the Lamb effect in a large range of sphericalsemiconductor QDs. In particular for small QDs, it can be shown that the electron-hole pair ground stateundergoes an observable negative Lamb shift, at least for judiciously chosen semiconductors. Because of theintrisic non-degeneracy of QD energy levels, the problem of its experimental observability is put to question.A Gedankenexperiment , making use of the Casimir effect [38], is proposed to test its existence [39].To close this paper, the Purcell effect is investigated in section . This phenomenon is one of the strikingphenomenon illustrating of cavity quantum electrodynamics [40]. It consists of a significant enhancementof the spontaneous emission rate of quantum systems interacting with a resonant electromagnetic cavitymode [41, 42], which has found many applications, see e.g. [43, 44]. Nowadays, it provides a test bed forquantum optics [45] and quantum information [46]. A validity condition for obtaining Purcell effect inspherical semiconductor QDs is determined, in the presence of the adverse role of Rabi oscillations. Somepredictive numerical results theoretically support the possibility of using the Purcell effect in such confinedstructures as radiative emitters in LASER devices.A concluding section summarizes our main results and indicates some possible research directions. In a standard EMA model, electrons and holes are assumed to be non-relativistic spinless particles, behavingas free particles with their effective masses m ∗ e , h , in a confining infinite spherical potential well, written as,in spherical coordinates ( r, θ, ϕ ) V ( r e , h ) = (cid:26) ≤ r e , h ≤ R, ∞ if r e , h > R. The choice of an infinite potential well at the QD surface induces an overestimation of the electron-holepair ground state energy for small QDs, as compared to real finite potential. This can be usually correctedby restoring a finite potential step of experimentally acceptable height [18]. However, the standard heightof the realistic step potential implementing the confinement of the charge carriers inside the QD may bereasonably described by an infinite potential well. Electrons and holes are then isolated from the insulatingsurrounding of the QD. In this setting, as far as Stark, Lamb, Casimir or Purcell effects are concerned, theelectromagnetic field amplitude should not exceed some threshold, so that electrons and holes would notacquire sufficient energy to overstep the real confining potential barrier by tunnel conductivity. We shallrefer to this working assumption as the weak field limit. .1 Interactive electron-hole pair EMA model The Hamiltonian of an interactive electron-hole pair confined in a semiconductor spherical QD reads H = H e + H h + V C ( r eh ) , where, in units of ~ = 1, H e , h = − ∇ , h m ∗ e , h + V ( r e , h ) denote the respective confinement Hamiltonians of theelectron and of the hole, and V C ( r eh ) = − e κr eh the electron-hole Coulomb interaction, with κ = 4 πε , ε beingthe semiconductor dielectric constant, and r eh the electron-hole relative distance. Without loss of generality,the semiconductor energy band gap E g may be set equal to be zero for convenience. In absence of Coulombpotential, electron and hole are decoupled particles with wave functions ψ lnm ( r e , h ) = r R χ [0 ,R [ ( r e , h )j l +1 ( k ln ) j l (cid:18) k ln R r e , h (cid:19) Y ml ( θ e , h , ϕ e , h ) , where χ A ( r ) = (cid:26) r ∈ A A ⊆ R + . l ∈ N , n ∈ N r { } and m ∈ [[ − l, l ]] are quantum numbers, labeling the spherical harmonic Y ml ( θ, ϕ ) and the spherical Besselfunction of the first kind j l ( x ). Finally, the wave numbers k ln are defined as the n th non-zero root of thefunction j l ( x ), resulting from the continuity condition at r = R [15]. The respective energy eigenvalues forelectron and hole, expressed in terms of k ln as E e , h ln = k ln m ∗ e , h R , show that the density of states of the semiconductor bulk has an atomic-like discrete spectrum, with increas-ing energy separation as the radius decreases. The analytical diagonalization of the Hamiltonian H seemsto be out of reach because the Coulomb potential explicitly breaks the spherical symmetry. To handle theinterplay of the quantum confinement energy, scaling as ∝ R − , and the Coulomb interaction, scaling as ∝ R − , two regimes are to be distinguished, according to the ratio of the QD radius R to the Bohr radiusof the bulk exciton a ∗ = κe µ , µ being the exciton reduced mass. In the strong confinement regime, corre-sponding to sizes R . a ∗ , the electron-hole relative motion is sufficiently affected by the infinite potentialwell, so that exciton states should be considered as uncorrelated electronic and hole states. In the weakconfinement regime, valid for sizes R & a ∗ , the exciton conserves its character of a quasi -particle of totalmass M = m ∗ e + m ∗ h . Its center-of-mass motion is confined, and should be quantized [16]. In this regime, the Coulomb potential is treated as a perturbation with respect to the infinite confiningpotential well in a variational procedure. The ground state energy of the electron-hole pair can be evaluatedwith the trial wave function φ ( r e , r h ) = ψ ( r e ) ψ ( r h ) φ rel ( r eh ), with φ rel ( r eh ) = e − σ r e , h , where σ is thevariational parameter. The product ψ ( r e ) ψ ( r h ) insures that the confined electron and hole shouldboth occupy their respective ground state in absence of Coulomb potential. The variational part φ rel ( r eh )is chosen so that the electron-hole exhibits the behavior of an exciton bound state, analogous to the groundstate of an hydrogen-like atom with mass µ . Then, it is expected that σ ∝ a ∗− . Integral representations forrelevant diagonal matrix elements in the state φ ( r e , r h ) are analytically expressed in the Fourier transformformalism of relative electron-hole coordinates. To obtain the mean value of the Hamiltonian H , theseexpressions are Taylor-expanded with respect to the parameter σR near zero, up to the second order. Thisyields a value σ = B ′ a ∗ , for which the electron-hole energy is minimized E strongeh = E eh − A e κR − B ′ E ∗ , where E eh = E e + E h , with the compact notation E e , h01 = E e , h , is the electron-hole pair ground stateconfinement energy, and E ∗ = µa ∗ the binding exciton Rydberg energy. This formula has been alreadyobtained with another trial function of the same form as φ ( r e , r h ), but with an interactive part equal to e φ rel ( r eh ) = 1 − σ r eh , instead of φ rel ( r eh ) [16], which obviously comes from the two first terms of the Taylorexpansion of φ rel ( r eh ), in the limit of σ r eh . Ra ∗ ≪
1. Thus, this indicates that, taking first the limit Ra ∗ ≪
1, and then evaluating matrix elements to perform the variational procedure, is equivalent to thereverse method applied here. This is not clear at first sight. All constants appearing in the text and formulas are listed in Appendix A . η ( λ ) values from numerical results of [16] and theoretical ones given by Eq. (1). λ η num ( λ ) 0.73 1.1 1.4 η theo ( λ ) 0.83 1.1 1.5relative error ≈ < ≈ In this regime, electron-hole pair states consist of exciton bound states. The Coulomb interaction contribu-tion to the exciton ground state should no longer be considered as a perturbation to the confinement energy,but is still treatable as a perturbation to the infinite confining potential well. Therefore, the global form ofthe variational function φ ( r e , r h ) should be retained. However, the QD size allows a partial restoration of thelong range Coulomb potential between the charged carriers, so that it is of the same order of magnitude thanthe kinetic energy in the electron-hole relative coordinates. Then, the leading contribution to the groundstate energy of the exciton should be − E ∗ , the ground state energy of a hydrogen-like atom of mass µ .The total translational motion of the exciton, thought as a quasi -particle of mass M , should be restored andcontribute to the exciton total energy by an amount π MR , the ground state energy of a free particle trappedin a space region of typical size R . In a first approximation, the excitonic ground state energy is the sumof these two contributions. To improve phenomenologically its accuracy in regard to numerical simulations,a monotonic increasing function η ( λ ) of the effective masses ratio λ = m ∗ h m ∗ e has been introduced in [16], as E weakeh = − E ∗ + π M ( R − η ( λ ) a ∗ ) . Then, the exciton is preferentially thought as a rigid sphere of radius η ( λ ) a ∗ .Its center-of-mass, whose motion is quantized, could not reach the infinite potential well surface unless theelectron-hole relative motion undergoes a strong deformation [16].To account for all these contributions, this suggests to multiply the trial function φ ( r e , r h ) by the groundstate plane wave φ G ( r G ) = e i πR σ G · r G , where r G is the center-of-mass coordinates and σ G is a plane waveground state quantum number vector satisfying the condition | σ G | = 1. The new trial function shouldthen be ψ ( r e , r h ) = ψ ( r e ) ψ ( r h ) φ rel ( r eh ) φ G ( r G ), leaving unchanged the exciton density of probabilityas well as the Coulomb potential matrix element. The confinement Hamiltonian H e + H h mean value getsthe expected further contribution π MR . A Taylor expansion on the Hamiltonian H mean value is performedin the region of QD radii σR & π , and the value of the variational parameter σ ≈ a ∗− is then computed.Its second and third order terms in a ∗ R . a ∗ R , E weakeh = − E ∗ + π µR + π M ( R − η ( λ ) a ∗ ) , from where an analytical expression for the function η ( λ ) can be extracted η ( λ ) = δ (1 + λ ) λ . (1)This obviously satisfies the electron-hole exchange symmetry λ → λ − . If l ∗ = a ∗ denotes the electron-holerelative distance mean value in the non-confined exciton ground state, the smallest possible radius l ∗ = a ∗ for the excitonic sphere picture should be obtained when λ = 1, so that η (1) ≈ . This matches quite wellwith both numerical and theoretical results, as shown in table 1. We can compare values from numericalsimulations taken by the function η ( λ ) for λ = 1 , , l ∗ should be reached in the infinite hole mass limit λ → ∞ , because the hole is motionlessand located at the electron-hole system center-of-mass. According to table 1, η (5) ≈ , it is reasonable toconclude that Eq. (1) is valid so long as λ .
5, whereas in the infinite hole mass limit is reached for λ & η ( λ ) ≈ . When compared with [16], the exciton ground state energy E weakeh shows a further contribution π µR , whichshould be interpreted as a kinetic energy term in the relative coordinates because of the reduced mass µ . W ( r eh ) and by a finite potential step of height V ≈ · –) [18] and compared to experimental results for spherical CdS microcrystals [47]. Ra ∗ E eh − E g E ∗ E weakeh . To this end, we propose to introduce an additionalpotential W ( r eh ) to the electron-hole pair Hamiltonian H , which should make contributions to the secondorder of the exciton total energy in the weak confinement but not to the third one, the one responsible forthe expression of the function η ( λ ). In this picture, higher order contributions are interpreted as higherorder corrections to π MR . This reinforces the idea that for very large radius, only the quasi -particle pointof view should be responsible for the exciton kinetic energy. Such potential is uniquely determined to be ofthe form W ( r eh ) = − π E ∗ r R e − r e , h a ∗ , while the amplitude of W ( r eh ) is to be fixed to get the correct kinetic energy − π µR , up to the thirdorder in a ∗ R , in the weak confinement regime. It is attractive at distances ≈ a ∗ to promote excitonic statewith typical size around its Bohr radius, repulsive at short distances to penalize excitonic state with smallsize, and exponentially small for large distances not to perturb the long range Coulomb potential. Finally, itcontributes to the exciton ground state energy with second order terms in both weak and strong confinementregimes, but does not change its zeroth and first order terms.The addition of the pseudo-potential W ( r eh ) to the exciton Hamiltonian H implies a significant decreaseof the expected value of the exciton energy in the strong confinement regime by an amount − π CE ∗ ,up to the second order in Ra ∗ . However, this is only valid when R . a ∗ because of the pseudo-potentialexponential dependence. Figure 1 shows that the excitonic energy computed in presence of the pseudo-potential gets a better fit to experimental results in this validity domain, than those calculated withoutthis tool. Nevertheless, the divergence for very small QD size still persists as a consequence of the infinitepotential well assumption. To extend the validity domain, energy expansions may be carried out to a feworders. But, calculations become so involved that the relevance of such an approach cannot be ascertained. The model discussed in the previous section lends itself to an extensive amount of analytical calculations onspherical semiconductor nanostructures interacting with a fixed external electric field. Even if this modelhas some intrinsic limitations and does not fully describe the QD behavior in the absence of electric field forsmall QD radii, it can be still used, since it gives rather satisfactory theoretical predictions on Stark effectin the weak field limit.Contrary to the case of a large range of microstructures, in which QCSE significantly depend on theelectric field direction [25, 26, 29, 48–50], the applied electric field E a , in spherical QDs, is set along the z -direction of a cartesian coordinates system with its origin at the QD center. As the inside semiconductor D dielectric constant ε is larger than the outside insulating matrix dielectric constant ε ′ , the electric field E d inside the QD, different from E a , is E d = ε r E a [22]. Then, the electron and the hole interactionHamiltonians with the electric field E d are W e , h ( r e , h ) = ± e E d · r e , h = ± eE d r e , h cos θ e , h , (2)where E d is the electric field amplitude inside the microcrystal. As h φ | W e ( r e ) | φ i = −h φ | W h ( r h ) | φ i , inpresence of an electric field, a new dependence on the electron and hole space coordinates for the trialwave function is required. The difference between the dielectric constants also implies the existence of apolarization energy term P ( r e , r h ) = e R X l ≥ α l ( ε r ) R l (cid:16) r l e + r l h − r l e r l h P l (cos θ eh ) (cid:17) , where P l ( x ) denotes a Legendre polynomial, ε r the relative dielectric constant, and α l ( ε r ) = ( l − ε r − κ ( lε r + l +1) [11].The polarization energy P ( r e , r h ) will be neglected first, but taken into account later on, to explore in detailsits relative role vs. the Coulomb potential. To apprehend QCSE, we follow the reasoning of [29], and study the interaction between the charge carrierswith the ambient electric field but neglecting their Coulomb interaction. In the weak field limit, the absolutevalue of their interaction energy with the electric field E d is E ele = eE d R . It should be treated as aperturbation compared to their typical confinement energy E e , h . The Stark shift will be computed byperturbation and variational procedures on individual Hamiltonians H ′ e , h = H e , h + W e , h ( r e , h ). The trialfunction to be used is Φ e , h ( r e , h ) = ψ ( r e , h ) ϕ e , h ( r e , h ), where ϕ e , h ( r e , h ) = e ∓ σ e , h2 r e , h cos θ e , h , because itcontains a deformation of the spherical shape along the electric field direction. The minimizing variationalparameters σ e , h are found to be σ , h = C m ∗ e , h eE d R .Both methods lead to the Stark shift − Γ m ∗ e , h e E R , where the proportionality coefficients are Γ pert = π P n ≥ k n ( k n − π ) and Γ var = C , with a relative error of about 2%. Since, the variational function φ ( r e , r h ) describes the electron-hole pair Coulomb interaction, both occupying their respective ground state,and the electric field interaction part ϕ e ( r e ) ϕ h ( r h ) is liable for the individual electron and hole behaviors inthe electric field E d , this suggests to choose a variationnal function in presence of the electric field of theform Φ( r e , r h ) = φ ( r e , r h ) ϕ e ( r e ) ϕ h ( r h ). To describe Stark effects in spherical semiconductor microcrystals without taking into account polarizationeffects, we apply a variational procedure using the trial function Φ( r e , r h ) to the Hamiltonian H Stark = H + W e ( r e ) + W h ( r h ) . Fourier transform techniques lead to integral representation of diagonal matrix elements, valid if and only ifthe variational parameters σ and σ e , h satisfy the inequality 0 ≤ e · σ e , h < σ , where e = exp(1). This relationanalytically expresses the range of acceptable electric field amplitudes. Following previous results of sections and , we respectively expect that σ ∝ a ∗− and σ e , h ∝ m ∗ e , h eE d R , so that E ele ∝ σ e , h σ Ra ∗ E e , h . Then,the charge carriers energy due to their interaction with the electric field should be at the most of the sameorder of magnitude of a first term correction in Ra ∗ to their confinement energy in the strong confinementregime. This corresponds to the order of magnitude of the typical absolute electron-hole Coulomb interaction.Based on the decoupled electron-hole point of view, the Stark shift for the coupled electron-hole pair shouldscale as ∝ ( m ∗ e + m ∗ h ) e E R ∝ E eh R a ∗ . Therefore, to get at least the lowest order contribution to this Starkshift, a Taylor expansion of the Hamiltonian H Stark mean value is performed up to the second order in thevariational parameters. This first contribution is not sufficiently accurate to fit experimental data, becauseit does not account for the electron-hole coupling through the Coulomb interaction. This is the reason whythe expansion up to the third order should be carried out to obtain the first correction in Ra ∗ . Then, the For later purpose, let us give the expression of the Taylor expansion of the Coulomb potential mean value h Φ | V C ( r eh ) | Φ ih Φ | Φ i = − e κR (cid:0) A + B ′ σR + C ′ σ R + C ′ ( σ + σ ) R + C ′ σ e σ h R + O ( σ R ) (cid:1) . tark shift, being identified as with the term scaling as ∝ E , is determined, up to the first order in Ra ∗ , as ∆ E strongStark = − Γ var ( m ∗ e + m ∗ h ) e E R (cid:18) ehvar Ra ∗ (cid:19) , where Γ var appears as a universal constant, while the constant Γ ehvar depends on the semiconductor, i.e. Γ ehvar = C ′ (cid:18) m ∗ e m ∗ h + m ∗ h m ∗ e (cid:19) + C ′ − B ′ C ′′ C .
Furthermore, this model is capable of describing QCSE, when the QD size and the electric field amplitudesatisfy effective constraints, consistent with strong confinement regime and weak field limit Ra ∗ . B ′ + 4 C ′ ) and E ele E eh . π e C (cid:0) C ′ B ′ (cid:1) . As expected, the first contribution to the found shift is simply the sum of the Stark shift contributionsundergone by the ground states of both electron and hole taken individually. Because of the dependence ofΓ ehvar on the effective masses m ∗ e , h , the second contribution to ∆ E strongStark indicates the existence of a dipolarinteraction between the electron and the hole. Until now, the interaction between the electron or the hole withthe external electric field takes place individually, whereas they interact only through the Coulomb potential.Actually, the Hamiltonian interaction part should also be written as W eh ( r eh ) = W e ( r e )+ W h ( r h ) = E d · d eh ,where d eh = e r eh is the exciton electric dipole moment. In the strong confinement regime, the dipolarinteraction point of view expresses the remnant of electron-hole pair states, thought as exciton bound statesunder the influence of the electric field. The dipolar interaction point of view suggests the inclusion of the term P ( r e , r h ) in the Hamiltonian H Stark describing the exciton-electric field interaction [11], which accounts for the polarization energy of the electron-hole pair, due to the difference between the dielectric constants of the semiconductor QD and its insulatingsurrounding. A variational procedure is applied to the new Hamiltonian H ′ Stark = H Stark + P . For this,we keep the variational trial function Φ( r e , r h ), since the polarization energy should not basically modifythe nature of the electron-hole coupling. The polarization energy mean value is expressed as an expansionin the variational parameters, which has the same form as the Coulomb potential mean value, where theconstants A , B ′ , C ′ , C ′ and C ′ are replaced by functions of the relative dielectric constant ε r , as shownin table 7. In this formalism, expressions for any Stark effect quantity, pertaining either to polarizationenergy or to the combined effect of Coulomb interaction and polarization energy are obtained from section . All the appearing constants are replaced either by the corresponding functions of ε r or by the sum ofboth contributions. Figure 2 shows our theoretical predictions vs. experimental data for spherical CdS . Se . microcrystals[19]. The electric field amplitude inside the microcrystal is set at E d = 12 . − . Two exciton peaksare experimentally resolved, which are attributed to the transitions from the highest valence sub-band andfrom the spin-orbit split-off state to the lowest conduction sub-band, with an energy splitting about 0.39eV,independently of the QD radius [19]. The experimental values depicted by crosses in figure 2 consist of meanvalues of the Stark shift of these two types of excitons. They seem to indicate that the Coulomb interactionis sufficient to explain correctly the amplitude of the Stark effects experimentally observed, as we expect, inthe range of validity of QD radii. The values of the variational parameters are also obtained up to the first order in Ra ∗ σ ′ = 4 B ′ (cid:18) C ′ Ra ∗ (cid:19) − CC ′′ ( m ∗ e + m ∗ h ) e E R E ∗ Ra ∗ ,σ , h = 4 C m ∗ e , h eE d R " C ′ m ∗ e , h µ + C ′ m ∗ e , h µ − B ′ C ′′ C ! Ra ∗ . The polarization energy mean value in the state Φ( r e , r h ) should be written as h Φ | P ( r e , r h ) | Φ ih Φ | Φ i = − e κR (cid:8) A ( ε r ) + B ′ ( ε r ) σR + C ′ ( ε r ) σ R + C ′ ( ε r )( σ + σ ) R + C ′ ( ε r ) σ e σ h R + O ( σ R ) (cid:9) . (3) e , hvar ≈ − . e , hvar ≈ − . · –), with Γ e , hvar ≈ − . . Se . microcrystals. R (˚A) × − ∆ E strongStark (meV) 0 10 20 30 40 50 60-1.5-1-0.50-2In the case of CdS . Se . microcrystals, our predictions should lead to acceptable results in regard toexperimental data as long as the cluster radius does not exceed 30˚A if only the Coulomb interaction is takento account, or 50˚A if the polarization is also included. When only polarization is considered, the strongconfinement regime is no longer valid, because σ ′ ≤
0. This means that polarization energy is repulsive andthe interactive part of the trial function should be φ rel ( r eh ) = e | σ ′ | r eh . In the three previous study cases,the hypothesis of the weak electric field limit remains valid as soon as the typical electric dipole interactionenergy E ele does not represent about 12% of the typical exciton confinement energy E eh . The weak fieldlimit seems to be judiciously chosen, because it appears to be independent of the strong confinement regime.When only the Coulomb potential is taken into account, the highest acceptable electric field amplitude,consistent with the weak field limit, is E maxd ≈ − for R = 10˚A, and E maxd ≈ . − for R = 30˚A. Thus, these values show that, for E d ≈ . − and for R . R . R & E maxd are involved. Whereas thestrong confinement regime condition is satisfied, for QD radii 30˚A . R . R ≈ According to the Dirac theory, the levels 2 s and 2 p of the hydrogen atom should be degenerated. But, theyare actually experimentally split by an energy of about 0.033cm − [35]. This phenomenon consists of theso-called Lamb shift. It has been generally attributed to the potential V ( r ) undergone by a spinless massiveparticle of mass m ∗ and charge qe in interaction with a quantized dynamical electromagnetic field, which is ell described by the Pauli-Fierz Hamiltonian in the Coulomb gauge H PF = H free + H em + eH int , where A ( t, r ) is the electromagnetic potential vector [52]. Here, H free = p m ∗ + V ( r ) is the free particleHamiltonian, whose eigenvectors and eigenvalues are | n i and E n , labeled by the quantum numbers n . H em is the Hamiltonian of the electromagnetic field, on which the standard second quatization method is tobe applied. Since experimental light sources possess sufficiently weak intensities, the weak field regime isvalid, and the potential vector quadratic terms are discarded ahead of the linear term in H PF , so that theinteraction Hamiltonian is expressed as H int = − q A · p m ∗ . There exists two methods to correctly apprehend the Lamb shift effect. The Bethe approach is a perturbationprocedure applied to the Hamiltonian H PF with respect to the perturbative Hamiltonian H int [36]. TheWelton approach interprets the Lamb shift as a fluctuation effect [37]. Using standard arguments fromquantum electrodynamics, they both lead to the same general expression for the Lamb shift of the particlestate | n i , given in terms of potential Laplacian matrix elements∆ E n = α π q m ∗ log (cid:18) m ∗ κ ∗ (cid:19) h n |∇ V ( r ) | n i , (4)where α is the fine-structure constant, and κ ∗ an IR cut-off, which is identified with the mean value of alllevel differences absolute values h| E m − E n |i [37]. The Bethe approach historically allowed the theoreticalexplaination of the Lamb shift for hydrogen [36]. Even if this two different methods produce the samepredictive result, the Welton approach brings a deeper comprehension for the Lamb shift as a physical phe-nomena, and satisfies to an invariant gauge property. Whereas, the Bethe perturbative argument attributesthe Lamb shift to a weak radiation-matter couplage in Coulomb gauge.This is the reason why we present in more details the Welton approach. It consists of a semi-classicalpoint of view, in which the position of a quantum particle fluctuates around its mean position r withsmall fluctuations ∆ r , due to its interaction with a classical surrounding electromagnetic field. The meansquare oscillation amplitude position of a charged particle coupled to the zero-point fluctuations of theelectromagnetic field can be easily evaluated as h (∆ r ) i = απ q m ∗ log (cid:16) m ∗ κ ∗ (cid:17) . In this picture, κ ∗ is interpretedas the minimal wave pulsation, for which some particle position fluctuations should be observed, and couldbe determined a posteriori by considerations on the particle classical motion. Position fluctuations lead toa modification of the potential, whose the mean value over an isotropic distribution of the fluctuations ∆ r should be computed. This gives rise to a mean effective potential h V ( r +∆ r ) i = n h (∆ r ) i ∇ + . . . o V ( r ).The second order correction term ∆ V ( r ) = h (∆ r ) i ∇ V ( r ) is responsible for the Lamb shift, as its meanvalue ∆ E n in a state | n i gives the energy shift of Eq. (4). Let us consider the previous massive particle with q = ± V ( r )described by eingenfunctions ψ lnm ( r, θ, ϕ ) and energy eingenvalues E ln = k ln m ∗ R . To use Eq. (4), the infinitepotential well Laplacian matrix element h ψ lnm |∇ V ( r ) | ψ lnm i and the IR cut-off κ ∗ in the spherical infinitepotential well are to reckon. The laplacian of V ( r ) cannot be evaluated even in distributions formalism. Allcalculations are then made with an intermediate finite potential step of height V , and the limit V → ∞ istaken at last. The Lamb shift undergone by a state | ψ lnm i of the particle confined by the infinite potentialwell V ( r ) is assumed to be the finite part (only contribution independant from V ) of the expansion inpowers of V of the Lamb shift undergone by the state of the particle confined by the finite potential step,with same quantum numbers l , n and m . By construction, it seems that the IR cut-off κ ∗ = h| E ij − E ln |i is infinitely large if all possible quantum numbers are taken into account. This is not the case, because theconfined particle interacts with the surrounding electromagnetic field, which possesses a finite energy. It isnot appropriate to consider that the particle can access all its energy levels states. There exists a higherenergy level that it can attain by its interaction with the electromagnetic field, which consists of the totalelectromagnetic energy E lim . This allows to define a associated maximal pulsation κ lim by E lim = κ m ∗ R ,interpreted as a UV cut-off for the autorized wave numbers k ln ≤ κ lim . hese considerations lead to expressions for the Lamb shift and the IR cut-off∆ E ln E ln = − α π − λ ∗ R log (cid:18) RR ∗ min (cid:19) , and κ ∗ ( R ) = 7 π m ∗ R , (5)where − λ ∗ = m ∗− is the reduced particle Compton wavelength. Due to the potential well sphericalsymmetry, it is expected that the Lamb shift is independent of the azimuthal quantum number m . Theradius R ∗ min = π q − λ ∗ ≈ . − λ ∗ insures the validity of the non-relativistic point of view. If R ≤ R ∗ min ,the particle should acquire a confinement energy at least of the same order of magnitude that its massenergy m ∗ . If the confined particle is an electron in vacuum, the Lamb shift is not measurable for reasonablepotential well sizes. This will not be the case, if we consider a confined interactive electron-hole pair inspherical semiconductor QDs, at least in the strong confinement regime.The description of the confined electron-pair made in section is integrated to the Welton approachto Lamb shift. Then, the Lamb shift of the exciton ground state presents four contributions of differentkinds. The first two contributions are due to the confinement infinite potential well V ( r e , h ) given by Eqs.(5), where the electron and hole effective masses are used. The second is due to the Coulomb potential V C ( r eh ), and is of the same nature that the Lamb shift observed in real atoms. And, the third comes fromthe pseudo-potential W ( r eh ). Thanks to Welton approach to the Lamb effect coupled to our description of QSE, we achieve to determineanalytical expressions for the Lamb shift undergone by the ground state of a electron-hole pair confined ina spherical semiconductor QD.In the strong confinement regime, we deduce, up to the second order in Ra ∗ , that the Lamb shift undergoneby the electron-hole pair ground state should be written as∆ E strongLamb = ∆ E stronge + ∆ E strongh , where∆ E stronge , h E eh = − α πε − λ ∗ , h R log RR e , hmin ! " − µFm ∗ e , h + 2 A π ! Ra ∗ + µF ′ m ∗ e , h − F ′′ π + 83 ! R a ∗ + O (cid:18) R a ∗ (cid:19) ≤ , and − λ ∗ e , h = m ∗− , h and R e , hmin = π q − λ ∗ e , h are respectively the electron and the hole reduce Comptonwavelengths and minimal radii in the considered semiconductor. Since by definition R ≥ R e , hmin , this Lambshift is negative. This is an outstanding, property predicted for the first time, to the best of our knowledge,because in real atoms, Lamb effect always raise energy levels. Heuristically, the quantized electromagneticfield non-zero ground state energy, often called the zero-point energy, is responsible for spreading of thecharge and mass of the carriers in a sphere of a typical radius p h (∆ r ) i . This is the major phenomenonto the Lamb effect. In atoms, these fluctuations induce a screening of the Coulomb potential, resulting in areduction of the binding energy of the electron to the nucleus [53]. The situation is different in a QD, theobserved effect is not due to an electric charge spreading but to a mass spreading. The total energy, initiallyconcentrated in the kinetic energy of the point-like particle, is now transferred to the energy of a spatialmass distribution, which splits into center of mass motion and relative motion. On the basis of total energyconservation, a reduction of the center mass motion energy is then expected, an effect which is opposite tothe one observed in atoms.In the weak confinement regime, following a same reasoning, the Lamb shift of the exciton ground stateis determined, up to the third in a ∗ R , as∆ E weakLamb = ∆ E weake + ∆ E weakh , where ∆ E weake , h E ∗ = 8 α πε − λ ∗ , h a ∗ log m ∗ e , h κ ∗ e , h ! (cid:20) O (cid:18) a ∗ R (cid:19)(cid:21) . This energy shift is independant from the QD raduis, and reveals the excitonic quasi -particle properties ofthe electron-hole pair in the weak confinement regime. In the limit of infinite hole mass, only the electronicterm ∆ E weake contributes to the Lamb shift of the exciton ground state, so that the Lamb shift undergoneby the ground state of an hydrogen-like atom of reduce mass µ and Bohr raduis a ∗ is retrieved. Followingthis analogy, the IR cut-offs κ ∗ e , h are both taken as κ ∗ e , h ≈ . E ∗ , because the IR cut-off associated to thehydrogen atom ground state yields ≈ . E I , where E I ≈ . · –) and the exciton (—) in the strongconfinement regime a. in spherical CdS . Se . or b. in spherical InAs (heavy hole) microcrystals. R (˚A) × − ∆ E strongLamb E eh a. -5-4-3-2-10 0.1 1 10-6 × − R (˚A)∆ E strongLamb E eh b. -3-10 1 10010-2 . Se . or InAs microcrystals a. for R = 10˚A and b. for R = 30˚A, and c. in the weak confinementregime. Semiconductor CdS . Se . InAsheavy hole light hole a. ∆ E strongLamb ( µ eV) -2.05 10 − -9.49 -36.9 b. ∆ E strongLamb ( µ eV) -2.11 10 − -0.148 -0.594 c. ∆ E weakLamb ( µ eV) 7.25 10 − − − us point out here the important fact that the contribution of the pseudo-potential W ( r e , h ) plays a significantrole on the Lamb effect of the exciton ground state in the weak confinement regime. Contributions due tothe Coulomb interaction to the second order in a ∗ R are exactly discarded by contributions due to the pseudo-potential, the presence of the infinite potential well affecting the exciton Lamb shift in the weak confinementregime only by contributions of at least the fifth order. This supports the phenomenological introduction ofthe pseudo-potential W ( r e , h ).Figure 3 show this Lamb shift for CdS . Se . and InAs microcrystals, and suggest the possibility ofobserving it. More precisely, table 2 confirms that the energy order of magnitude involved in the strongconfinement regime in InAs microcrystals are equivalent to those in hydrogen atom. The observation ofthis Lamb shift in the weak confinement regime seems to be out of question for the moment, since excitonRydberg energies in semiconductors are at most of the order magnitude of ten or so meV. We can wiselyconclude that, at least in the strong confinement regime and in a judiciously chosen semiconductor, it seemspossible to observe Lamb effect. The experimental observability of the Lamb effect in hydrogen atom is possible because the s -spectral bandis separated from p -spectral band, while they should stay degenerated in absence of Lamb effect. In quantumsystems displaying no spectral band degeneracy, such as QDs, energy levels are dressed by the quantum zero-point fluctuations of the electromagnetic field, forbidding the detection of the corresponding bare levels, andthen of the Lamb shift. In quantum field theory, the summation of the zero-point energy yields a divergentground state energy, which is usually subtracted off in an additive renormalization scheme. However, acareful analysis on its boundary conditions shows the occurrence of a finite and observable force, known asCasimir force [38, 55]. In vacuum, two parallel perfectly conducting squared plates of linear size L , placedat a separation distance d ≪ L , are subjected to an attractive force, since the vacuum fluctuations are moreimportant outside than inside the plates.The Lamb effect is also attributed to the zero-point fluctuations energy of the electromagnetic field. So,by placing a QD in vacuum and inside a Casimir pair of conducting plates, one would be able to detect anenergy difference between two Lamb shifted levels. This Gedankenexperiment should allow to overcome theneed of degenerate energy levels, or of exactly computed energy levels. There exist some theoretical worksdealing with Lamb effect of real atoms confining in a Casimir device [56, 57]. They predict an additionalshift to the standard Lamb shift, which depends on the separation distance between the mirrors, which goesto zero in the limit d → ∞ . The coupling between the atom and its own radiation field is usually neglected.This assumption should be valid if the coupling of a two-level quantum atom with itself through absorptionand emission of dipolar radiations reflected by the Casimir plates is dominated by the coupling of the two-level atom with the electromagnetic field vacuum fluctuations. For spherical semiconductor QDs in strongconfinement regime, it means that κ d = πd ≤ κ ∗ e , h ( R ). A direct generalization of these works in such contextis possible, and leads to the addition of a new positive term ∆ E Casimire , h ( d ) to the Lamb shift undergone bythe confined electron-hole pair in a Casimir configuration in comparison with the one in vacuum∆ E stronge , h −→ ∆ E stronge , h + ∆ E Casimire , h ( d ) , where ∆ E Casimire , h ( d )∆ E stronge , h = 649 π R − λ ∗ e , h d log − RR e , hmin ! ≥ . This additional positive term calls for a physical explanation. When the separation distance d decreases,the amplitude of the electromagnetic modes inside the Casimir plates increases, while their number is fixed. d = 1 µ m (—), 0 . µ m(– –) or 0 . µ m (– · –). R (˚A)∆ E eCasimir ( d )∆ E e This leads to the reinforcement of the interaction of the quantum system with the quantized electromagneticfield, implying a strengthening of the Lamb effect. Moreover, this relative enhancement does not depend onthe quantum state under consideration. Finally, there is a competition between the typical lengths of theQD: − λ ∗ e , h ≪ R ≪ d , describing the different scales of the problem.If the separation distance d is chosen to be of 0.5 µ m, such that it allows the experimental observationof the Casimir effect, figure 4 shows that modification of the electron-hole Lamb shift between the Casimirplates is of about 5-10% in spherical InAs QDs of radius of the order of magnitude of a few tens nm. It ispossible to enhance the amplitude of this modification, of course, by reducing the separation distance d untilthe order of a few tenth parts µ m, or more simply by acting on the Casimir configuration geometry. Forexample, the use of a sphere of large radius instead of one of the Casimir plates increases the Casimir force,and then the modification of the Lamb shift, by a factor π [55]. The combination of these two effects almostleads to the doubling of the Lamb shift in free space, which seems significant enough to be observable. It has been observed that coupling a magnetic moment to a resonating circuit of volume V and qualityfactor Q at radio frequencies of wavelength λ significantly enhances its spontaneous emission by a factor F = π Qλ V [40]. This effect can be understood in a simple way. A two-level quantum atom, built from twoeigenstates | n i and | m i of a quantum charged particle Hamiltonian H free , with respective energy eigenvalues E n < E m , is fit into a resonant electromagnetic cavity at a frequency ω close to the Bohr frequency ω mn = E m − E n , with a quality factor Q . It interacts with a single dynamical confined electromagneticcavity mode, also named quasi -mode, characterized by its effective volume V mode . In this picture, the quasi -mode is not only coupled to the two-level quantum atom but also to the continuum of other electromagneticfield modes. To compute spontaneous emission rates, the coupling between the two-level quantum systemand the quasi -mode is treated as a perturbation when compared to the coupling between the confined modeand the continuum of external modes. As shown in [51], in this weak coupling regime, the confined modeis characterized by a normalized Lorentzian energy density distribution ρ ( E ) = Qπω Q (1 − Eω ) +1 , of width | ω − ω mn | = ωQ . In the electric dipole approximation, the perturbation Hamiltonian is the standard dipolarinteraction Hamiltonian W ( t, r ) = − d · E ( r )Θ( t ) , where Θ( t ) is the Heaviside step function, d the particle dipole moment, and E ( r ) the quantized quasi -modeelectric field, which reaches its maximal amplitude at the origin of a cartesian coordinates system. .1 General considerations The general spontaneous emission transition rate A mn associated to the radiative transition | m i → | n i withemission of a photon of pulsation ω ≈ ω mn is given by the Fermi golden rule between the two quantum states | m , i and | n , i of the Jaynes-Cummings Hamiltonian [59], describing the two-level quantum atom- quasi -mode coupled system, H JC = H free + H em + W ( t, r ) (cid:12)(cid:12) {| n i , | m i} , where H free (cid:12)(cid:12) {| n i , | m i} = E n | n ih n | + E m | m ih m | is the two-level atom Hamiltonian, and H em = (cid:0) a † a + (cid:1) ω ,is the quantized electromagnetic mode Hamiltonian, a and a † being its annihilation and creation operators,with [ a, a † ] = 1. In the electric dipole approximation, the electric field variation along the typical particlesize is negligible, and the particle is assumed to be at an electric field maximum value. The dipole momentis thus oriented along the direction of the electric field, therefore A mn = 2 |h m | d | n i| QV mode . From thedefinition A mn = F mn A mn and from the spontaneous emission rate in absence of electromagnetic cavity A mn = ( ω mn ) π |h m | d | n i| [60], the Purcell factor is found to be F mn = Q π ( λ mn ) V mode .In practice, the effective cavity mode volume V mode is experimentally measured. In [58], a review ofelectromagnetic microcavities of different geometries, built by different methods, but characterized by aquality factor Q and by an effective volume of the form V mode ≈ βλ is given. Here, λ is the quasi -modewavelength at resonance, close to the wavelength λ mn associated to the Bohr angular frequency ω mn and β a pure number of order unity. Then, for such cavities, the Purcell factor becomes independent from theradiative transition, and F = Q π ≈ .
51 10 − Q if β = 5 and Q ≥ i.e. to have F ≥
1, the quality factor should be larger than a lower bound of Q min ≈
66. Thus, in commonelectromagnetic cavities, the Purcell effect is generally observable and measurable.The validity criterion for the weak coupling regime is obtained by comparing the characteristic time scaleof the coupling of the two-level system to the electric field confined mode and the coupling of the electric fieldmode to the continuum of external electromagnetic modes. The second coupling dominates if the emittedphoton during the transition escapes the two-level quantum system and the electromagnetic cavity, withoutbeing re-absorbed. The associated photon relaxation time for the radiative transition is defined as τ mn = Qω mn .Moreover, the Purcell effect must face the adverse working of Rabi oscillations. A sufficient condition forthe validity of the weak coupling is then simply τ mn Ω mn ≪ , where Ω mn = q ω V mode |h m | d | n i| is the Rabi angular frequency of the related radiative transition. In thestrong coupling regime, defined by τ mn Ω mn ≫
1, only the interaction between the two-level quantum systemand the confined mode is to be considered. As the dimensionless quantity τ mn Ω mn scales as ∝ Q , the previouscondition imposes an upper bound on Q . In fact, the higher Q is, the smaller is the resonance disagreement | ω − ω mn | , which is responsible for the Rabi oscillations evanescence. Therefore, Rabi oscillations can bemaintained in the electromagnetic cavity, inhibiting the Purcell effect. | ω − ω mn | should be then sufficientlysmall to insure the validity of the resonant approach, but it should not be too small not to promote Rabioscillations, unfavorable for the Purcell effect. In a dielectric medium, the dielectric permittivity ε is related to the refraction index η by ε = η > , the confinement is implemented by a infinite potential well. In the following,electron tunneling is discarded, and only electronic radiative transitions involving energy levels lower thanthe maximum amplitude of the real finite potential step are to be considered.Einstein spontaneous emission coefficients with or without electromagnetic cavity should be computedbetween two electronic eigenstates | ψ lnm i and | ψ l ′ n ′ m ′ i such that E e ln < E e l ′ n ′ as A l ′ n ′ m ′ lnm = 64 α η (cid:18) πλ l ′ n ′ ln (cid:19) ( I ll ′ nn ′ ) J mm ′ ll ′ R For more details, one can refer to [61]. red
QD-LASER: the pumping is realized between a ground state level | g i and a excitedstate | e i , higher than the highest level | i i of the LASER transition, the intermediate state . | i i| e i LASER TransitionΓ ≈ | g i γ ≈ ′ ≈ A l ′ n ′ m ′ lnm = F A l ′ n ′ m ′ lnm , where I ll ′ nn ′ and J mm ′ ll ′ are respectively the radial and the angular integrals occurringin the matrix element h ψ lnm | r | ψ l ′ n ′ m ′ i complex modulus. Since λ l ′ n ′ ln ∝ ( E e l ′ n ′ − E e ln ) − ∝ R , Einsteinspontaneous emission coefficients are large for small QD radius. This shows a typical quantum behaviorfor small QDs through the spontaneous emission enhancement, even in the absence of an electromagneticcavity. As explained before, semiconductor QDs Purcell effect could be used, instead of real atoms, as efficientradiation emitters in LASER devices over quite wide wavelength ranges. The general theory of LASER andthe so-called population inversion is well known, see e.g. [63,64]. Here, we expose the possibility of exploitingthe Purcell effect to produce red-light
LASER emission from spherical InAs QDs. To make contact withexperimental results, we assume that R = 25nm [65], for which the strong confinement regime is valid. Athree-level LASER is built with the previous QD states, where the transition is found at 755nm. To have aconcrete idea of the working of the LASER mechanism described by figure 5, we have collected in table 3the relevant numerical data.The spontaneous emission is the only phenomenon to be considered. Stimulated emission and absorptionshould be discarded, and non-radiative effects should be omitted. The non-radiative effects in QDs leadto energy dispersion by phonon creation, when inelastic collisions between electrons and the potential welloccur. These only shorten the lifetime of excited states. So, non-radiative effects do not matter in thequalitative arguments for non-LASER transitions. In this context, we shall keep radiative effects, especiallyspontaneous emission effects which can be enhanced by the Purcell effect in LASER transition, and therebyinitiate LASER oscillations.We assume that the decay | e i → | i i , governed by the relaxation rate Γ ′ , is faster than the decay | i i → | g i governed by the relaxation rate Γ, i.e. the intermediate state should be metastable as compared to theexcited, and Γ ′ ≫ Γ. In a stationary regime and in the case of weak pumping ω ≪ Γ ′ , the populationinversion holds only for ω ≥ Γ. This implies that Γ ′ ≫ Γ, which means that the excited state is almostempty, and the intermediate state is the most populated state. They can be analytically evaluated as J mm ′ ll ′ = δ l ′ l − (cid:26) ( l + m )( l + m − l + 1)(2 l − δ m ′ m +1 + 2 ( l + m )( l − m )(2 l + 1)(2 l − δ m ′ m + ( l − m )( l − m − l + 1)(2 l − δ m ′ m − (cid:27) + δ l ′ l +1 (cid:26) ( l + m + 2)( l + m + 1)(2 l + 3)(2 l + 1) δ m ′ m +1 + 2 ( l + m + 1)( l − m + 1)(2 l + 3)(2 l + 1) δ m ′ m + ( l − m + 2)( l − m + 1)(2 l + 3)(2 l + 1) δ m ′ m − (cid:27) , and I ll ± nn ′ = − k ln k l ± n ′ (cid:16) k l ± n ′ − k ln (cid:17) . The vacuum transition rates A l ′ n ′ m ′ lnm are non-vanishing only if the selection rules l ′ = l ± m ′ = m, m ± n and n ′ . red LASER presented in figure 5in comparison with He-Ne LASER.LASER Transition Wavelength (nm) A mn (MHz)QD-LASER | e i → | i i | i i → | g i
755 0.617He-Ne LASER 632 ≈ As Γ ′ ≥ A ei ≈ A ig ≈ . Q kHz, the assumptions for having a red-light emitting three-level LASER are met, if the Purcell factor is about F ≈ i.e. for a quality factor about Q ≈ ω ≈ τ ig Ω ig ≈ , . − ≪
1. Table 3 suggests that spontaneous emissionrates are of the same order of magnitude than those of the He-Ne LASER transition, even if He-Ne LASERare built upon a four-level system. So, the Purcell effect coupled to the artificially tailored spectrum of the QDs allows the possibility toobserve LASER emission in a QD-LASER, working with poor quality factor electromagnetic cavities. But,the present treatment, even in the strong confinement regime, based on the particle-in-a-sphere model, isintrinsically limited. To fully describe the Purcell effect, it is essential to exactly diagonalize the totalHamiltonian H . However, at the moment, the inclusion of electron-hole Coulomb interaction, even in thepresence of an infinite confinement potential well, turns the problem into a challenge to be met. In this paper, a new approach to some interesting atom-like properties of spherical semiconductor QDs ispresented. It is based on an improved EMA model to which is added an effective pseudo-potential. Thisallows extensive analytic calculations of physical quantities yielding a better agreement with empirical datafor QSE and QCSE. The Lamb shift in spherical semiconductor QDs is also calculated in this theoreticalframework. It turns out to be negative and in principle observable, at least in the strong confinementregime. A
Gedankenexperiment , based on a modification of the electromagnetic field vacuum fluctuationsenvironnement for the QD, created by a Casimir configuration, is proposed. A modification of the QD Lambshift should be observable, as compared to the one existing in free space. Finally our study also illustratesthe utility of the Purcell effect, predicted for atoms, for QD-LASER emission in the visible part of thespectrum.These wide ranging theoretical results are encouraging for further investigation of QDs structure, basedon this improved EMA model. In view of a full description of phenomena involving radiative transitionsbetween two energy levels of the QD, like Purcell effect or LASER emission, it seems relevant to develop thegeneral theory of the confined interactive electron-hole pair states.Another fundamental outlook consists of investigating the description of the charged carriers confinementpotential by an finite potential step instead of the infinite potential well used here, in order to fully describeQDs of particularly small radius.
A Constants
In the following tables, we sum up all appearing constants and give their approximate values. The func-tion Si( x ) = R x tt sin( t ) denotes the standard sine integral. Table 4 presents analytical expressions andapproximate values of constants occurring in section .Table 5 presents analytical expressions and approximate values of constants occurring in section ,when only the Coulomb potential is taken into account. We are able to provide exact expressions for allthe constants occurring in section , when the polarization energy is also taken into account, except for δ ′′′ , γ ′′′ and γ ′′′′ . For these quantities, we obtain integral representations, which cannot be analyticallycomputed at the moment. Their approximate values are evaluated numerically. As exact expressions forother constants are quite cumbersome, we give only their approximate values. Table 6 presents approximate The mechanism of a four-level LASER is described in [63]. Its principal advantage consists of, contrary to a three-level LASER,that there is no population inversion condition on the pumping frequency, only the LASER cavity losses fixes the threshold forcoherent light emission. .Name Expression Value Name Expression Value S Si(2 π ) − Si(4 π )2 0.6720 A − Sπ B − π B
29 + 1324 π + S π B B + B B ′ AB − δ π (cid:26) − πS (cid:27) .Name Expression Value Name Expression Value C − π C ′ A ( B − C ) − B C ′ B − AC
12 -0.0339 C ′ B
18 0.0160 D − π + 14764 π D − π − π D
225 + 37120 π − π − S π D D + 10 D + D
30 0.2539 D ′ D + 4 D + D D ′′ D − D
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12 -0.0082 δ ( ε r ) ε r −
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