Direct and Sequential Two-Photon Double Ionization of Two-Electron

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Direct and Sequential Two-Photon Double Ionization of Two-Electron Quantum Dots Henri Bachau, and Lampros A.A. Nikolopulos J. Phys. Chem. A, Just Accepted Manuscript • DOI: 10.1021/acs.jpca.7b11811 • Publication Date (Web): 22 Jan 2018 Downloaded from http://pubs.acs.org on January 22, 2018

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The Journal of Physical Chemistry

Direct and Sequential Two-Photon Double Ionization of Two-Electron Quantum Dots Henri Bachau∗,† and Lampros A. A. Nikolopoulos∗,‡ †Centre des Lasers Intenses et Applications, Université de Bordeaux-CNRS-CEA, F-33405 Talence Cedex, France ‡School of Physical Sciences, Dublin City University , Dublin 9, Ireland E-mail: [email protected]; [email protected] Abstract In this work we study the double ionization yields and kinetic energy spectra of a two-electron spherical quantum dot (QD) exposed in laser fields. The theoretical description is based on an ab initio non-perturbative configuration interaction theory capable to describe the two-electron QD dynamics in THz and mid-IR ultrashort laser fields. The QD’s confinement potential is approximated to have a Gaussian-like spatial dependence. We have found that significant variations of the two-electron kinetic energy patterns and two-photon double ionization yields occur as we vary the QD’s size. For a given laser pulse the double ionization yield increases by orders of magnitude when the dot size is reduced. Also, for a fixed laser wavelength, the size of the QD determines the sequential or direct character of the two-photon double ionization process. Provided that it is energetically allowed the sequential two-photon double ionization process, requiring minimal interelectronic correlations, becomes dominant over the direct one. In sequential regime, the corresponding two-electron kinetic energy spectrum changes from a broadened single-peaked to a doubly-peaked one. Moreover, we also have identified features in the spectrum that are distinctively different than its atomic counterpart.

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Introduction Semiconductor quantum dots (QDs), also known as artificial atoms, bridge the gap between cluster molecules and bulk materials. With typical dimensions in the range of 1-100 nm, these nanocrystals display discrete electronic transitions reminiscent of isolated atoms and molecules. QDs are one of the fundamental components of complex nanoscale devices that have multiple applications, like optoelectronics, solar energy harvesters or digital imaging to cite a few of them 1 . Using sophisticated physical and chemical methods, it is possible to tailor, to large extent, the size, shape and the electronic properties of quantum well systems 2–5 . This particular property of fabricating customized QDs can be used to control their energy spectrum in view of obtaining certain optical properties, thus to identify QDs as a new class of quantum objects with controlled properties. The versatile nature of their potential of applications has attracted a considerable attention in both applied and fundamental research. Regarding the latter, the quantitative description of energy levels and electron correlation effects represent an important factor for a realistic description of their optical properties. The theoretical and numerical approaches employed to describe the QD electronic structure rely to some extent on the extension of concepts and methods used in atomic physics and physical chemistry 6–9 . On the experimental side recent experiments have demonstrated that the use of laser pulses in the infrared and terahertz regime allows the direct measurement of their electronic and geometrical properties 10,11 . Following the pioneering work of Bryant 12 who showed that the QD’s size, combined with the interelectronic interactions, represent decisive parameters of their electronic properties, there has been a tremendous activity devoted to the study of size-dependent structural behavior and correlation effects. In this context, two-electron photo-excitation and photoemission are powerful tools to investigate electron correlations and quantum size effects. In a recent paper 13 we have proposed an ab initio nonperturbative theory to describe the dynamics of both the bound and continuum states of QDs in an external electromagnetic (EM) field. We have shown that the photo-ionization cross sections exhibit structures whose the 2


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shape and position vary strongly with the QD size. Some resonances show deep symmetric and asymmetric profiles of Fano type 14 , which characterizes doubly excited states in atoms. Other resonances, located just above threshold, are associated with bound states crossing the threshold as the QD’s radius decreases, their cross sections exhibit a sharp peak. The aim of the present work is to investigate two-photon double ionization (TPDI) in two-active electron QDs. During the last decade TPDI has been the subject of a vast number of theoretical studies in atomic and molecular physics, mainly on He and H2 (see 15,16 for recent papers, others references therein). The chief obstacle for the theoretical study of a double photoionization process relies on the accurate description of the two-electron continuum states, representing a long-standing challenge in computational atomic and molecular physics. The main reason is that, in contrast with the bound states and the singly-ionized continuum states, the accurate description of the double-continuum with asymptotic conditions pertaining to a particular channel is intractable until now. Incidentally we note that accurate estimations of one-photon double ionization cross sections of helium have been obtained right at the turn of the present century 17,18 . In the case of TPDI, a good convergence between stationary and time-dependent calculations 19 has been realized rather recently 20 for He. The corresponding problem of QD double ionization, has until now received little attention. We note the work of Fominykh et al 21,22 who have investigated the one-photon double emission process in a parabolic potential well. For strong laser fields, TPDI processes are generally classified as direct or sequential. In the latter case DI is enhanced and the resulting electron spectrum shows two peaks related to the ionization thresholds of the neutral QD (or atom) and its parent ion. In the case of direct TPDI the double electron escape process is strongly affected by interelectronic interactions 23 . Nevertheless, there is a fundamental difference between atoms (and molecules) and QDs; in the latter case the Coulomb interaction of the (confining) potential is of short-range, in contrast to the long-range Coulombic character of the electrostatic interactions in atoms and molecules. The consequences on

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QD bound states have been extensively studied (see 9 for a recent contribution), but little is known about QD double continuum and double emission process. As in the precedent work we consider a model with a spherical potential, well adapted for semiconductor nanocrystal 24–26 . We use a Gaussian-like confinement potential whose the depth and radius can be easily manipulated, having in addition appropriate continuity and boundary conditions. We extend the non-perturbative theoretical approaches presented in 13 to the case of double ionization of QDs. It is based on the resolution of the time-dependent Schrödinger equation (TDSE) on local polynomial basis (B-splines) to describe the electronic states 27–31 . The electron interaction is fully taken into account during the TDSE resolution with special attention on the extraction of the TPDI probabilities. We consider two cases; QD of radius RQ = 3.2 nm and QD with RQ = 4.6 nm. The photon energy of 304.7 meV (2.8 s.a.u.) allows to investigate direct and sequential TPDI of the QD. The total TPDI probabilities are calculated as well as the associated one- and two-electron kinetic energy spectra. Direct and sequential TPDI processes are discussed on the basis of these results. In the followings we use the atomic-Gaussian unit system (~ = me = e = 1/4πε0 = 1) with scaled energy, length and time units, relative to the a.u. system, as Eh? = Eh (µ/k 2 ), a?0 = a0 (k/µ) and τ0? = τ0 (k 2 /µ), with k = 5 characterizing the dielectric constant of the dot and µ = 0.1 the effective mass of the electron [see table 1].

Calculation of double photoionization We consider a model which describes quantum dots built of narrow-gap semiconductor nanocrystal of size RQ . We employ the effective-mass approximation with the periodic crystal potential taken into account through the electron’s effective mass and the dielectric constant 32 . Within this model the electrons are trapped in a spherically symmetric Gaussian potential with depth V0 and interact each other via the electrostatic Coulombic interelectronic potential 33 . We examine the optical properties of this spherical dot under a linearly

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Table 1: Conversion factors from SI to the scaled atomic unit system. For example 1 s.a.u. of intensity corresponds to 4.119×107 W/cm2 . The peak intensity and the electric field amplitude are related as I0 = cE02 /8π. Quantity

QD

µ

0.1

k

5

a?0 (nm)

2.645

Eh? (meV)

108.84

τ0? (ps)

0.006047

I0?

2

(W/cm )

.

4.119× 107

polarized pulse of central carrier frequency ω. In 13 we have employed a scaled TDSE to study the interaction of the laser field with the QD leading to its single-ionization. Since in this work the theoretical approach is described in great detail (Sec. II), as well as the B-spline basis set used to describe the radial part of the orbitals (Sec. III, Table I), here we present only the absolutely necessary formulas to allow a self-contained formulation of the double ionization processes. In addition, more details about the relation between the TDSE expressed in SI units and its scaled counterpart are given in the Appendix section of the aforementioned work.

Two-electron eigenstates The field-free two-electron QD Hamiltonian in scaled atomic units (s.a.u.) is modelled as,

ˆ1 + h ˆ 2 + Vˆ12 , ˆ Q (r1 , r2 ) = h H

(1)

ˆ q (rq ) = − 1 ∇2 − V0 e−βrq2 , h 2 rq

(2)

where Vˆ12 = 1/|r1 − r2 | and

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is the scaled one-electron QD Hamiltonian. The β and V0 parameters define the confinement Gaussian potential. The scaled radius RQ of the QD is controlled by use of the relation p RQ = ln 2/β. Since the confinement potential depends on the RQ parameter the following considerations depend parametrically on the size of the QD. We start first with the calculation of the zero-order LS-uncoupled two-electron, ΦnΛ , characterized by the total energy (EnΛ ) and Λ = (LSML MS ) formed by the angular (L), spin quantum numbers (S) and the magnetic quantum numbers of their projections to the quantization axis (z−axis) ML , MS . Although in the present case we examine only the singlet symmetry, (S, MS ) = (0, 0), for completeness, we keep in the definition of Λ the S, MS quantum numbers. The electronic structure is calculated by consideration of the two-electron field-free Schrödinger equation,

ˆ Q − EnΛ )ΦnΛ (r1 , r2 ) = 0, (H

(3)

ˆ Q is the non-relativistic Hamiltonian given in Eq. (1). For a given set of Λ quantum where H numbers, the solution of Eq. (3) is expanded on a basis of two-electron configurations which are products of one-electron functions φilml (r) = (Pil (r)/r)Ylml (Ω). Pil (r) is the radial part of the basis and Ylml (Ω) is the usual spherical harmonic function. The position vector is expressed in a spherical coordinate system as r = (r, Ω) = (r, θ, φ). l, ml here represent the angular momentum quantum number of the electron and its projection along the quantization axis (z−axis). We assume that the QD is placed on a large spherical box, of radius R >> RQ , by imposing proper boundary conditions Pil (r ≥ R) = 0. On the other hand, a physically acceptable wavefunctions should have finite values at the QD’s center (r = 0), so we take Pil (0) = 0 as well. We calculate these functions by solving the Schrödinger equation ˆ q (rq ) − ilm )φilm (rq ) = 0, (h l l

(4)

ˆ q (rq ) given in Eq. (2). The energy index, i, serving to enumerate with i = 1, 2, ..., Nb and h 6


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the basis, is now discretized as a result of the box confinement assumption. Solutions with ilml < 0 are (spatially) exponentially decaying states (bound spectrum) while those with ilml > 0 are oscillating (continuum spectrum). In the particular case of ionization from the QD’s ground state (1S) by linearly polarized light, the dipole selection rules restrict the involved states to singlet spin (S = 0) states with ML = 0. Then, we can drop the channel index Λ and keep only the indices that vary during the ionization process, namely, L and EnL . With this notation the two-electron eigenstates of the QD can be expanded on a basis of two-electron functions, X

ΦnL (r1 , r2 ) =

(nL)

(L)

vil1 ,jl2 φil1 ;jl2 (r1 , r2 ),

(5)

il1 ;jl2

(L)

with the zero-order two-electron basis functions, φil1 ;jl2 (r1 , r2 ), given by, (L)

φil1 ;jl2 (r1 , r2 ) = A12

Pil1 (r1 ) Pjl2 (r2 ) (L0) Yl1 ,l2 (Ω1 , Ω2 ) r1 r2

(6)

(L0)

where A12 is the antisymmetrization operator and Yl1 l2 (Ω1 , Ω2 ) is the bipolar spherical har(L)

monic, with ML = 0. Projection over the φil1;jl2 (r1 , r2 ) states provides the matrix equivalent of Eq. (3). The direct diagonalization of the latter provides the eigenenergies EnL and the (nL)

configuration interaction (CI) expansion coefficients vil1 ,jl2 . Similarly, as in the one-electron case, the bound states are characterized by energies such that EnL < E1 , where E1 is the ground state of the singly ionized QD. Correspondingly, the continuum states are characterized by energies such that EnL > E1 . In passing, we denote the ground state energy as E0 and the double-ionization threshold energy as E2 ≡ 0 [see Figure 1]. The numerical eigenstates are orthonormalized to unity as hΦnL |Φn0 L0 i = δnn0 δLL0 .

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Time-dependent wavefunction in a laser field The next step in our (non-perturbative) calculational procedure is to expand the timedependent total wavefunction on the basis of the calculated CI two-electron eigenstates,

ψ(r1 , r2 , t) =

X

CnL (t)ΦnL (r1 , r2 ),

(7)

nL

ˆ Briefly, denoting the interaction operator as D(t), substitution of the above expression in ˆ Q + D(t)]ψ(t), ˆ the TDSE, i∂t ψ(t) = [H followed by a projection on the ΦnL eigenstates (thus integrating over all spatial variables) we arrive to a system of ordinary differential equations (ODEs) for the CnL (t) coefficients 13 . The calculations are performed in velocity gauge and the ODE system is propagated in time over the total pulse duration τp . An explicit RungeKutta type approach is used to integrate in time the ODE system. The physical initial condition used is, ψ(r1 , r2 , 0) = Φ10 (r1 , r2 ), to take into account that the QD is initially in its ground state, Φ10 . At times larger than τp (the EM field has vanished) the experimental observables of interest (e.g. ionization yield) remain constant. The coefficients |CnL (t ≥ τp )|2 are directly related to the populations of the QD’s bound and continuum states. While in 13 we have discussed in detail the procedure to extract the single ionization yields from the latter coefficients, the case of the doubly ionized quantities requires a special attention. The latter procedure is discussed in the below.

Calculation of the double ionization yield As mentioned, we extract the values of physical observables by use of the values of the timedependent wavefunction, ψ(r1 , r2 , tf ) when the pulse is over, tf ≥ τp . To this end, we need to invoke the asymptotic solutions of the field-free Hamiltonian, HQ . In the present context, ˆ Q − E)φE = 0, where the electrons are far as asymptotic solutions we mean the solution of (H apart each other. Therefore, since, r1 , r2 → ∞ and |r1 − r2 | → ∞ as asymptotic solutions ˆ1 + h ˆ 2 with well defined angular and spin momentum ˆ (0) = h we take the eigenstates of H Q 8


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Figure 1: (Color online) Sketch for the TPDI paths for the QD with V0 ≈ 544.25 meV (5 s.a.u.). The cases of radii RQ ≈ 3.2 nm (β = 0.5 s.a.u.) and RQ ≈ 4.6 nm (β = 0.2 s.a.u.) are shown. The vertical arrows represent the photon of energy ω = 304.7 meV (2.8 s.a.u.). The size-dependent energies, of the neutral QD ground state (E0 ) are denoted by black filled circles while those of the singly-charged QD (E1 ) by red squares. The DI threshold is chosen at the origin of the energy axis (E2 = 0). The energies indicated in the figure (close to the vertical arrows) are in meV.

(L)

(L)

quantum numbers, denoted as φk1 l1 ;k2 l2 (r1 , r2 ). φk1 l1 ;k2 l2 (r1 , r2 ) are the zero-order functions given in Eq. (6). Within this assumption, we obtain the partial doubly differential DI yields by projecting these (uncorrelated) states to the final time-dependent wavefunction as, (L)

SL;l1 l2 (ε1 , ε2 ) ≡

dPl1 l2 (L) = |hφk1 l1 ;k2 l2 |ψ(tf )i|2 . dε1 dε2

(8)

k1 , k2 are the electron’s wavenumbers of continuum states determined by i = ki2 /2. It is also assumed that all positive-energy one-electron radial functions are normalized on the

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energy scale. SL;l1 l2 (ε1 , ε2 ) is the probability distribution for the ejection of two electrons, with the electron i = 1, 2 having energy between (εi , εi + dεi ) and angular momentum li , and at a state of angular symmetry L. Here it is important to notice that the zero-order (L)

states φk1 l1 ;k2 l2 (r1 , r2 ) are not orthogonal to the QD two-electron eigenstates, ΦnL (r1 , r2 ). The (L)

latter property has two important consequences (a) the projection of φk1 l1 ;k2 l2 (r1 , r2 ) on the QD bound and continuum states with negative eigenenergies (EnL < 0) should be removed before the analysis is performed and (b) the quantity dPlL1 l2 /dε1 dε2 varies with tf , the time at which the projection is performed. The wave packet must therefore be propagated after the laser-QD interaction until the density of probability reaches a constant value. The validity of this procedure is discussed in details in 15,20,34 in the context of helium TPDI . From the above expression for SL;l1 l2 (ε1 , ε2 ) we can derive various observables by summing/integrating over the electron’s quantum observables, (ε1 l1 ; ε2 l2 ). For example, the total population in this channel is obtained by integrating over all kinetic energies, Z PL;l1 l2 =

dε1 dε2 SL;l1 l2 (ε1 , ε2 )

(9)

Moreover, substitution of Eqs. (6), (5) and (7) in Eq. (8), followed by summation over the total angular momenta, L, results to,

Sl1 l2 (ε1 , ε2 ) =

X L

>0 2 nL X EX (nL) SL;l1 ,l2 (ε1 ε2 ) = CnL (tf )vk1 l1 ;k2 l2 . n

(10)

L

Further summation of the above formula over the l1 , l2 provides the doubly-differential DI P probability (or 2-D PES) as, S(ε1 , ε2 ) = l1 ,l2 Sl1 ,l2 (ε1 , ε2 ). The total double ionisation probability is obtained by the double integration of the latter quantity,

PDI =

X

Z PL;l1 l2 =

dε1 dε2 S(ε1 , ε2 ).

(11)

L,l1 ,l2

The partial-wave singly-differential DI probability is obtained by integrating the 2-D 10


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Table 2: Ratios of partial DI yields to the total DI yield, PL;l1 l2 /PDI for RQ = 3.2 nm (β = 0.5 s.a.u.). Total DI yield PDI = 5.37 × 10−8 . (l1 , l2 ) / L (1,1) (2,2) (0,2) (0,0) (1,3) (3,3) (0,1) (1,2) (2,3) (0,3) Sum

0 0.2816 0.0246 0.0195 3.09 [-3] 0.3288

1 2 3 Sum 0.617 0.898 0.011 0.0356 0.0328 0.033 0.0195 9.0 [-3] 9.0 [-3] 1.54 [-3] 4.64 [-3] 1.62 [-4] 1.63[-4] 1.06 [-4] 9.6 [-6] 1.16 [-4] 1.14 [-5] 3.02 [-7] 1.17 [-5] 3.9 [-7] 3.9 [-7] 2.8 [-4] 0.67 1.03 [-5] 1

PESs over the energies of one of the electrons, Z Sl1 ,l2 (i ) =

dεj Sl1 ,l2 (εi , εj ),

i, j = 1, 2, j 6= i

(12)

Finally, the singly-differential DI probability (1-D PEs) is obtained by,

S(i ) =

X

Sl1 ,l2 (εi ),

i = 1, 2

(13)

l1 ,l2

We have tested the calculations with two different types of basis sets (a) the one presented here and (b) an expansion on a non-orthogonal polynomial basis set (B-splines) 13 . All the calculations show excellent agreement between the two approaches.

Results and discussion For the results shown below, to obtain the QD structure we followed the procedure described in detail in 13 and consider the case of a QD studied in the latter paper and in 35 . We consider a pulse of peak intensity I0 = 6.397 × 105 W/cm2 , sin2 envelope and of

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Figure 2: (Color online) Two-electron partial kinetic energy probability distributions, Sl1 l2 (ε1 , ε2 ), for R = 3.2 nm (β = 0.5 s.a.u.). Note that the colormap for the (1,1) channels is multiplied by 10−6 while those of the (0,0),(2,2) and (0,2) channels by 10−9 .

central carrier frequency ω = 304.7 meV (2.8 s.a.u.) 13 :

E(t) = E0 sin2 (

πt ) sin ωt, τp

0 ≤ t ≤ τp .

(14)

The pulse duration τp = 12 × 2π/ω ≈ 163 fs (12 cycles), unless stated otherwise. We recall that the peak intensity and the electric field amplitude are related as I0 = cE02 /8π (here E0 = 0.053376 s.a.u.). As it is shown in Figure 1 we can see that both the ionization potentials of the neutral and singly-ionized QD are increasing with the QD’s radius. This means that single-photon double ionization requires higher photon energies with increasing the radius. Accordingly, in the present context the same is true for a two-photon double ionization

12


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1e-07 sum (0,0) (1,1) (2,2) (0,2)

9e-08 8e-08 Partial 1-D PES (1/s.a.u.)

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


7e-08 6e-08 5e-08 4e-08 3e-08 2e-08 1e-08

0

0.2

0.4

0.6

0.8 1 1.2 1.4 1.6 electron kinetic energy (s.a.u.)

1.8

2

2.2

Figure 3: (Color online) Doubly and singly differential PES for R = 3.2 nm (β = 0.5 s.a.u.). Top: Total DI 2-D PES, S(ε1 , ε2 ). Bottom: Partial 1-D PESs, Sl1 l2 (ε1 ) and their sum.

process. Nevertheless, the possibility of double ionization with two photons gives rise to qualitatively (and quantitatively) two different ionization mechanisms. More specifically, 13



2.5e-07 8 cycles 12 cycles 16 cycles

2e-07 Partial 1-D PES (1/s.a.u.)

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

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1.5e-07

1e-07

5e-08

0 0

0.25

0.5

1 0.75 1.25 1.5 electron kinetic energy (s.a.u.)

1.75

2

2.25

Figure 4: (Color online) Singly differential PESs, Sl1 l2 (ε1 ), for β = 0.5 s.a.u. and ω = 2.8 s.a.u. for pulses of different duration, 8,12,16 cycles corresponding to 0.108, 0.162 and 0.217 ps, respectively.

let’s take the present case of a radiation field with ω = 304.7 meV (2.8 s.a.u.). If we irradiate a QD with RQ ∼ 4.6 nm (β = 0.2 s.a.u.) we have the following ionization processes (a) oneand two-photon single ionization of the neutral [since ω > E1 (0.2) − E0 (0.2) = 248.4 meV (2.282 s.a.u.)] and (b) TPDI [since 2ω > E2 (0.2) − E0 (0.2) = 582.8 meV (5.355 s.a.u.)]. The situation, though, is different if the QD radius decreases to RQ ∼ 3.2 nm (β = 0.5 s.a.u.). Then, the above ionization paths, (a) and (b), are still present but now a third ionization mechanism becomes energetically possible, namely, (c) the two photon are absorbed, the first from the neutral QD, leading to its ionization, and the second from the newly created singlycharged ion. This ionization path is energetically allowed since ω > E2 (0.5) − E1 (0.5) = 232.9 meV (2.14 s.a.u.). This sequential photon absorption process, leaving the QD doubly ionized (one-electron from the neutral QD and the other electron from its singly-charged 14


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QD), is known as sequential double ionization (SDI) in contrast to the process (b) which is known as direct double ionization (DDI) [see Figure 1]. These two different double ionization mechanisms have been observed in atomic and molecular systems and have been the subject of theoretical and experimental studies for two decades now (see 20 , other references therein). Amongst the most important properties of the SDI and DDI is the dominance of the SDI (when present) over the DDI. It is relatively straightforward to reach the latter conclusion if one thinks the SDI as a two one-photon one-electron ionization processes and the DDI as one two-photon two-electron ionization process. The relative contribution of SDI and DDI and the pertinence of these concepts have also been the subject of considerable literature in atomic and molecular physics 29 . The SDI, in principle, is a process where no interaction between the two-ejected electrons takes place (during ionization), while, in contrast, DDI electron-electron interactions play a predominant role. Our calculations show some of the features of the SDI and DDI processes by examining the respective yields and the ejected electron’s kinetic energy spectra patterns. One important difference compared with similar studies in the atomic case is that the dominance of the SDI mechanism, over the DDI, can be controlled by varying the size of the QD. The equivalent for atomic and molecular systems is to changing to a system with different atomic number Z. In practice, in the atomic (and molecular) case to ensure the presence of the SDI channel one instead varies the central photon frequency of the field.

Sequential double ionization regime RQ = 3.2 nm (β = 0.5) The maximum total angular momentum L was 3, while the partial waves channels (l1 , l2 ) are shown in table 2 together with the respective ionization yields (ratios relative to the total DI yield). Total DI yield is PDI = 5.37 × 10−8 . In this table we see that the dominant ionization channels are the pp partial waves both for 1S and total 1D symmetries totalling to ∼ 89.8% of the DI yield. Following these channels the most important contribution comes from the dd (3.56%), sd (3.3%) and ss (1.9%) channels. In terms of the total angular momentum 15



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Table 3: Ratios of partial DI yields to the total DI yield, PL;l1 l2 /PDI , for RQ = 4.6 nm (β = 0.2 s.a.u.). Total DI yield PDI = 2.287 × 10−11 . (l1 , l2 ) / L (1,1) (2,2) (0,2) (0,0) (1,3) (3,3) (1,2) (0,1) (2,3) (0,3) Sum

0 1 2 3 0.171 0.482 0.0782 0.05 0.126 0.0849 5.08 [-3] 0.0013 5.04 [-4] 1.29 [-4] 1.98 [-4] 1.76 [-4] 1.28 [-4] 1.24 [-5] 1.1 [-5] 0.335 3.475 [-4] 0.664 2.2 [-4]

Sum 0.653 0.128 0.126 0.0849 5.1 [-3] 1.783 [-3] 3.27 [-4] 1.76 [-4] 1.40 [-4] 1.1 [-5] 1

the DI yields are larger for the 1D (67%) symmetry than the 1S (∼ 32.9%) symmetry, consistent with the two-photon absorption from the ground state of the neutral QD, mainly due to the dominance of the pp channel. The ratio is very close to 2 in agreement with the angular momentum matrix elements properties (due to the Clebsch-Gordan coefficients) which favour, by a factor of two, transitions from 1P to 1D, relative to transitions from 1P to 1S. The value of this ratio implies that, although interelectronic radial correlations do not contribute in the magnitude of the ionization yields, the two outgoing electrons are still angularly correlated. However, the same ratio is not true for the other channels, present to both 1S and 1D symmetries, dd and f f . In contrast, for these channels, the population in the 1

S symmetry is larger than that of the 1D. Finally, DI yields to 1P and 1F are significantly

smaller as these channels are populated through three-photon absorption which is negligible at the intensity considered here. A more detailed information about the populated doubly ionization channels is obtained by calculating the partial 2-D PES [Sl1 ,l2 (ε1 , ε2 )] and 1-D PES [Sl1 ,l2 (ε1 )] for the four dominant partial wave channels, (ss, pp, dd and sd). The plots of the partial 2-D PES are shown in Figure 2. Assuming the SDI as a two-step ionization process, for sufficiently long (not less than 10 cycles) and not too strong pulses, time-dependent perturbation theory (with 16


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respect to the electric field) provides an analytical, semiquantitative, expression 30,36 of the two-electron PES in terms of energies and ionization widths in the limit of infinite time:

S(1 , 2 ) =

γ1 /2π γ2 /2π × , [(E0 + 2ω − E2 ) − (1 + 2 )]2 + (γ1 /2)2 [(E1 + ω − E2 ) − 1 ]2 + (γ2 /2)2

with γ1 and γ2 the ionization widths of the neutral and ionized QD, respectively. On account of the above expression it is straightforward to show that the 2-D PES of the SDI is characterized from strong peaks at 1 = E1 + ω − E2 and 2 = E0 + ω − E1 . Since E0 = −3.35 s.a.u. (-365 meV) E1 = −2.14 s.a.u. (-233 meV) and E2 = 0 we find that 1 = 0.66 s.a.u ( 71.8 meV) and 2 = 1.59 s.a.u. (173 meV). These values are consistent with the findings of the present ab initio calculations as shown in Figure 2. On the other hand, for TPDI the electrons are ejected with kinetic energies up to ε1 + ε2 = E0 + 2ω − E2 = 2.25 s.a.u. (245 meV). Moreover in Figure 3, presenting the 2-D (top) and 1-D (bottom) PESs, we see fundamental differences between the pp and the ss, dd, sd channels. It is only the pp PES that is strongly peaked in contrast to the much broader spectra of the other channels. These distinctly different patterns suggest that the pp channels are produced mainly through a SDI process while for the other channels the DDI competes with the SDI. For the ss, dd, sd channels the concept of SDI becomes blurred and hardly one can assume that the electrons are released from the neutral and the ionized QD independently. Additionally, for a two-photon absorption, this last conclusion can be reached from purely angular momentum rules (within dipole approximation for the transition amplitudes).Provided that we neglect angular correlations and screening with an initial 1s2 state, the continuum state (following one-photon absorption) behaves like an uncorrelated 1sεp state, leading to the result that further photon absorption (and thus a purely sequential process) would lead to a pp final state. In fact, it is concluded that the outgoing electrons of the sd, ss and dd channels are able to acquire these angular momentum values only through interelectronic interactions. Incidentally, we note that, with ω = 2.8 s.a.u., the singly-charged ion can be also left in 1p excited state (located

17



at -0.6413 s.a.u.). Therefore electrons in the sd, ss and dd channels can be produced through SDI involving the 1pks and 1pkd continua in the intermediate step. The single-electron PES resulting from the latter process should show two peaks at 0.0877 s.a.u. (9.54 meV) and 2.16 s.a.u. (235 meV). It is clear that none of these peaks are present in Figure 3, the ad hoc assumption of independent electron ejections is therefore inapplicable for the sd, ss and dd channels. Finally, we must remember that the SDI approximation (independent electron ejection) is valid in the limit of weak fields and for infinite duration, so within the regime of perturbation theory. Therefore the SDI dominates for long pulse duration; however, as the pulse duration decreases the assumption of independent electron ejection becomes gradually invalid. To examine the latter, we have calculated the 1-D PES for three different number of cycles (8, 12, 16), corresponding to 0.108, 0.163 and 0.217 ps, respectively. The results are shown in Figure 4, and indeed, we observe that the PES is strongly peaked only for the longer pulse (16 cycles) while for the shorter pulse (12 cycles) the peaks broaden. For the last case, (8 cycles) there is no any structure appearing in the PES signal. Here it is worth noticing that the positions of the peaks are shifted with the pulse duration. The two peaks are approaching each other toward shorter pulses, until they eventually merge to a single broad one (8 cycles). This merge occurs when the pulse duration is roughly of the order of 2π/∆Eint , where ∆Eint is the distance between the two peaks for a very long pulse duration (∆Eint is the total electron-electron interaction in the QD ground state). Here since ∆Eint ≈ 0.93 s.a.u. (101 meV) and ω = 2.8 s.a.u. we expect the merge to occur for pulses durations of few cycles, this is consistent with Figure 4. This behaviour has also been observed in the case of TPDI of He and and extensively studied in the past 37,38 .

Direct double ionization for RQ = 4.6 nm (β = 0.2) We now turn our attention to a similar QD but with larger RQ . Larger QD results to increased ionization potentials for the neutral and ionized QD [see Figure 1]. For instance, 18


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Figure 5: (Color online) Two-electron partial kinetic energy probability distributions, Sl1 ,l2 (ε1 , ε2 ), for R = 4.6 nm (β = 0.2 s.a.u.). The colormaps for the (1,1), (2,2) and (0,2) channels are multiplied by 10−9 while that of (0,0) channel by 10−12 .

for β = 0.2 s.a.u. (RQ ∼ 4.6 nm), we have E2 (0.2) − E1 (0.2) = 334.4 meV (3.072 s.a.u.). We now see that two-photon SDI is not any more energetically possible. Therefore, since ω = 304.7 meV (2.8 s.a.u.), no single-photon ionization of the singly-charged QD can occur. It is worth to point out here that the sequential ionization is possible via two-photon absorption ionization of the singly-ionized QD. Therefore the possible DI ionization channels would be two-photon DDI and three-photon SDI (1-photon ionization of the neutral and two-photon of the singly-ionized QD). But for the peak intensities we have used, three-photon SDI is practically negligible compared to two-photon DDI. Similarly as in the SDI case (β = 0.5 s.a.u.) we have collected in table 3 the partial 19



DI yields (ratios relative the total DI yield). Total DI yield is PDI (0.2) = 2.287 × 10−11 . This yield has been calculated just at the end of the pulse and we have checked that this value changes by a small percent, had the propagation stopped few cycles later. First, we notice that this probability is dramatically lower (by three orders of magnitude) than the DI yield calculated for the case RQ = 3.2 nm (5.37 × 10−8 ). This behavior is explained by the fact that the dipole’s transition amplitudes between the bound and continuum states decrease sharply when the QD radius increases, in agreement with our findings published in Figure 3 of 13 , where the ionization yield show a sharp drop toward extended QDs. We have found a similar behavior here. In table (3) we see that, again, the pp partial waves dominate the ionization signal both for the 1S and total 1D symmetries, however totalling now to only ∼ 65.3% of the DI yield. Compared with the SDI case (β = 0.5 s.a.u.) the remaining channels have now a larger contribution amounting to (12.8%) for the dd, (12.6%) for the sd, and (8.49%) for the ss channels. In terms of the total angular momentum, the DI yields populate less strongly the 1S (∼ 33.5%) and more the 1D (66.4%) symmetries while populations in the 1P and 1F symmetries remain negligible. This confirms that, at the intensity of I0 = 6.397 × 105 W/cm2 , the dominant channel for double ionization is the two-photon absorption. An additional remark, worth noticing, is that the percentages of population to 1S and 1D remained the same with those of the SDI (see table 2). This surmises that, as in SDI, the DDI dominant process mainly populates the pp channel but, due to a stronger CI in the double continuum, a more significant part of the pp population is transferred to dd, sd and ss channels in DDI. Despite the strong electron-electron interactions for the pp channel the contribution of the symmetries, 1S and 1D, to DI is dominated from the (dipole selection) rules governing the interaction with the field (28.16% for 1S and 61.7% and for 1D). Similarly as in the SDI case in Figure 5 we show the partial 2-D PES [Sl1 ,l2 (ε1 , ε2 )] for the four dominant partial wave channels, ss, pp, dd and sd. Simple inspection of these plots and the plots of Figure 2 shows that the 2D PES patterns are distinctly different for all channels.

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It is interesting to note that the patterns of pp (dominant) and dd channels exhibit a clear peak around the equal energies positions, ε1 ∼ ε2 (circa at 0.14 s.a.u. for the pp channel and 0.11 s.a.u. for the dd one). Similar pattern is observed for the sd channel, however less clear as comparable probability values, relative to the central peak, show up parallel to the axes ε1 , ε2 ; however this is not the case for the ss channel which exhibits a much broader area covering the lowest parts of the spectrum. As we haven’t a definite explanation of these patterns in terms of the angular and radial correlations experienced by the two electrons we only comment that these patterns are the results of strong electron-electron interactions occurring during their way out of the dot’s core region. Finally, the 2-D and 1-D differential PESs are shown in Figure 6. The bottom figure shows that the pp contribution dominates, in accord with the previous analysis. The doublydifferential 2-D PES (top figure) exhibit a broad maximum in the region of electron energies ranging from 0.08 to 0.12 s.a.u. The structure appears in the singly-differential PES (bottom figure) with a peak around 0.1 s.a.u. while the excess energy in the double continuum (E0 + 2ω) is about 0.25 s.a.u. This is in contrast with the results obtained in direct TPDI of He (with photon energy of 40-46 eV) which show a quasi-flat distribution at 40 eV going to a U-shaped one as the photon energy increases 39 . The question that arises now is whether this single-peaked structure of the 1-D PES is related with the dot’s size or with the fact that we have a non-sequential double ionization process. To answer this question we calculated direct TPDI in the case of RQ = 3.2 nm (β = 0.5 s.a.u.), with a photon energy of 195.91 meV (1.8 s.a.u.) and a pulse duration of 8 cycles. Although the yield is much less now (relative to the DI yield of the 2.8 s.a.u pulse), for the reasons explained above, we have found that the electron distribution also shows a peak close to 0.08 s.a.u. while the excess energy in the double continuum is 0.25 s.a.u.. Therefore, it appears that the peaked structure is a persistent feature and appears that is related with non-sequential TPDI processes in QDs.

21



2e-10 sum (0,0) (1,1) (2,2) (0,2)

1.75e-10 1.5e-10 Partial 1-D PES (1/s.a.u.)

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1.25e-10 1e-10 7.5e-11 5e-11 2.5e-11 0 0

0.1

0.2 0.3 0.4 electron kinetic energy (s.a.u.)

0.5

Figure 6: (Color online) Doubly- and singly differential PES for R = 4.6 nm (β = 0.2 s.a.u.). Top: Total two-electron DI 2-D PES, S(ε1 , ε2 ). Bottom: Single-electron 1-D PESs, Sl1 l2 (ε1 ) and their sum.

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Conclusion We have applied an ab initio configuration interaction approach to describe the double ionization yields of a two-electron semiconductor spherical quantum dot exposed in a laser field. We have investigated the TPDI mechanism with a central carrier frequency of 304.7 meV (2.8 s.a.u.) and an intensity of 6.397 × 105 W/cm2 . We have confirmed that double ionization processes involving more than two photons is negligible at this intensity. The QD is 2

described by a Gaussian spherical potential −V0 e−βr with V0 = 5 s.a.u. (544.2 meV) and β = 0.5, 0.2 s.a.u, corresponding to QD radii of RQ ≈ 3.2 nm and RQ ≈ 4.6 nm, respectively. For RQ = 3.2 nm and for a pulse duration of 0.162 ps (12 cycles), TPDI is dominated by a sequential absorption of two photons, leading to two peaks in the single electron kinetic energy spectrum. The DI channel with the angular pair pp has a relative weight close to 90%, which is consistent with the scheme were the electrons are emitted sequentially and consequently weakly interact each other during QD’s ionization. As the pulse duration shortens the peaks broaden and move towards each other. In the limit of the shorter pulse duration, 108 ps (8 cycles), the double peak structure has disappeared and the concept of sequential ionization looses its pertinence. On the other hand, for the larger dot (RQ = 4.6 nm), sequential double ionization is energetically forbidden and TPDI proceeds through a direct process, where the two-photons are absorbed ’simultaneously’ from the neutral QD’s ground state. The direct TPDI is significantly weaker compared with the smaller dot, RQ = 3.2 nm, a trend already observed in single photon ionization which shows an ionization yield sharply decreasing while the QD radius increases 13 . Again, the dominant doubly ionization channel is the pp one, but less overwhelming now (counts circa 65% of the total DI yield) as there is now a stronger mixing which other angular configurations. This is a clear sign that interelectronic interactions become more important in the direct DI channel. The single electron kinetic energy spectrum of the RQ = 4.6 nm dot, associated with DI, shows a pronounced single peak, in contrast with the doubly-peaked spectrum of the 23



RQ = 3.2 nm dot. It is worth to comment here that this appears to be a characteristic feature of this artificial quantum structure as it is not observed in the case of TPDI of natural quantum systems. One possible explanation about the origin of this difference could be the short-range extension of the QD’s central potential relative the long-range behaviour, 1/r of the electrostatic Coulomb interactions in atoms. In relation to the latter, it would be interesting to compare the QD’s spectrum with the one corresponding from negatively charged systems with the negative hydrogen, H− , being the most representative system of this class. Further investigations are required to fully understand its role in the direct DI channel and more general in QD’s laser ionization.

Acknowledgement This work was supported by the Irish Research Council (IRC), the Ministère des Affaires Étrangères et du Développement International (MAEDI) and the Ministère de l’Education Nationale, de l’Enseignement Supérieur et de la Recherche (MENESR) via the French-Irish ULYSSES 2015 program and by the European COST Action CM1204 (XLIC). The authors also acknowledge the support of the University of Bordeaux for providing access to the Mésocentre de Calcul Intensif Aquitain (MCIA).

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Direct and Sequential Two-Photon Double Ionization of Two-Electron

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