ISSN 1063-7850, Technical Physics Letters, 2017, Vol. 43, No. 2, pp. 231–234. © Pleiades Publishing, Ltd., 2017. Original Russian Text © P.A. Golovinski, A.A. Drobyshev, 2017, published in Pis’ma v Zhurnal Tekhnicheskoi Fiziki, 2017, Vol. 43, No. 4, pp. 102–109.
Interference in Tunneling Ionization Involving an Electron Bound by Two Short-Range Potentials P. A. Golovinskia, b* and A. A. Drobysheva a b Moscow
Voronezh State University of Architecture and Civil Engineering, Voronezh, 394006 Russia Institute of Physics and Technology (State University), Dolgoprudnyi, Moscow oblast, 141700 Russia *e-mail:
[email protected] Received August 3, 2016
Abstract—The phenomenon of tunneling ionization involving electron bound by two delta-potentials under the action of a constant electric field has been studied. Distributions of the electron-current density for two different initial states are found. Dependence of the emission current on the orientation of potentials relative to the field direction and the distance between their centers is established. Conditions of manifestation of the interference effects are determined. DOI: 10.1134/S1063785017020183
The method of short-range potentials is widely used in atomic and nuclear physics, as well as in the physics of condensed media [1–3]. Important applications of this method include the description of nonlinear ionization, generation of high harmonics in strong laser fields [4–7], and simple model of tunneling effect in atomic microscopy [8]. An analytical solution of the problem of electron polarization and tunneling detachment from a short-range potential of a negative atomic ion has been found and the photodecomposition of negative ions in the presence of constant electric field of various configurations has been examined [9, 10]. A single delta-potential used in these works does not describe the possible of electron delocalization between several regions of attraction that is in fact characteristic of molecular and solidstate systems. In studying the problem of tunneling ionization of neutral molecules, the method of fitting asymptotic wave functions in parabolic coordinates was generalized to the case of spherically nonsymmetric states [11–13]. However, even in this case the theory is still restricted to a description of localized states of separate molecular orbitals. The notion of localized states that does not take into account their possible interference also serves a basis for the description of tunneling effects in nanostructures [14, 15]. On the other hand, it has been theoretically shown that electron delocalization must significantly influence the scattering of attosecond laser pulses on diatomic molecular anions [16]. Therefore, elucidating the possible role of delocalization in the formation of electron current in tunneling ionization requires special consideration.
As is known, the problem of a particle moving in the field of several short-range potentials in the absence of external fields admits a complete analytical description [17]. For relatively close potential wells, the wave functions overlap and form a common extended state. In addition, these systems exhibit a characteristic response to an external field [18]. In the present work, we have considered the pattern of interference of electron waves formed during electron detachment from a two-well potential in the case of different orientations of the field and potential. The problem of motion for a particle in the field of several delta-potentials can be formulated in terms of boundary conditions imposed on the wave function at the points of location of these potentials. For a single s state with l = 0, the wave function ψ ~ r–1exp(–αr) and the boundary condition takes the following form [2]:
d ln(r ψ) = −α, dr r =0
(1)
where α = − 2E 0 and E0 is the binding energy (in the atomic system of units where |e| = m = ℏ = 1). For several potentials, their action is set as the sum of these. The state of a particle in the field of two identical attractive centers and external electric field F is described by the following stationary Schrödinger equation:
(ε + 12 ∇
2
)
+ Fz ψ(r) = (V1(r − r1) + V 2(r − r2 ))ψ(r), (2)
where each delta-component is [1, 3]
231
V (r) = − 2π δ(r) ∂ r. α ∂r
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E
where ρj = r – rj (j = 1, 2). This system can be rewritten as follows [20]:
(a)
F z2
b+ G (−R /2, R /2, ε) ⎞ ⎛ A ⎞ ⎛ ⎜G (R /2, −R /2, ε) ⎟ ⎜ B ⎟ = 0, b− ⎝ ⎠⎝ ⎠
z1 0
b± =
z
ε−
π (Ai'(ζ )Ci'(ζ ) − ξ Ai(ζ )Ci(ζ )) + α, (3) ± ± ± ± ± 1/3 (2F )
8
cos θ , ξ ± = − 2ε ± FR2/3 (2F ) where r1 = R/2, r2 = –R/2, and R = |r1 – r2|. System (3) has two solutions, which determine two wave functions and two values of energy ε± for the corresponding states with certain widths, found from the condition of zero determinant. Note that Eqs. (3) lead to the following relationship between coefficients A and B:
4
A = − 2π G (−R /2, R /2, ε) + b± . B 2π G (R /2, −R /2, ε) + b∓ ε=ε ±
Fz
ε+ (b)
|A/B|
12
0
0
50
100
150 200 θ, deg
250
300
For a weak electric field (F/α3 ≪ 1), the shifts of levels and their widths have been calculated in [18] as functions of angle θ between the line connecting the two centers and the direction of the electric field. An observable quantity in the process of tunneling ionization is density jz of election current that crosses the plane perpendicular to axis z coinciding with the direction of the electric field:
350
Fig. 1. (a) Relative arrangement of centers of the two deltapotentials and (b) dependence of the |A/B| ratio on angle θ (solid and dashed curves correspond to the ground and excited states, respectively).
The solution of this problem, as well as that without external field [19], has the form of a superposition of Green’s functions
ψ(r) = AG (r, r1, ε) + BG (r, r2, ε), where G(r, rj, ε) are the Green’s functions with outgoing wave asymptotics satisfying Eq. (2) with δ(r – rj) in the right-hand part; r1 is the radius vector of the first center; and r2 is that of the second center (Fig. 1a). The Green’s function of a particle moving under the action of a constant force can be expressed via Airy functions Ai(u) and Bi(u) [8]:
G (r, r ', ε) =
1 (Ci(χ )Ai'(χ ) − Ci'(χ )Ai(χ )), + − + − 2| r − r '|
χ ± = −(2F ) −2/3(2ε + F(r + r ') ± F | r − r '|), where Ci(u) = Bi(u) + iAi(u) has an outgoing wave asymptotics at u → ∞. Taking into account boundary conditions of type (1), we arrive at a system of equations for determining complex quasi-energy ε = Reε – iΓ/2 and coefficients A and B:
1 ∂ (ρ ψ) j ρ j = 0 = −α, ρ j ψ ∂ρ j
∂ψ j z = Im ⎛⎜ ψ* ⎞⎟ . ⎝ ∂z ⎠ The spatial distribution of electron current depends on angle θ to the line connecting the centers and the direction of electric field, parameter α|r1 – r2| characterizing the distance between centers in comparison to the decay length of the unperturbed wave function, and parameter F/α3 that characterizes the force of the electric field relative to the force holding the electron in the potential well. Figure 1b shows the results of calculations of the ratio |A/B| as the function of angle θ between vectors R and F for the set of parameters E0 = 1 eV, F = 107 V/cm, and R = 10 au. As can be seen, the ratio of coefficients in the ground and excited states at θ = π/2 is |A/B| = 1, which corresponds to the case of axial symmetry. For the system orientation parallel to the field (θ = 0) in the ground state, we have |A/B| = 0.07, which implies that the electron is in fact localized at the second center. In the excited state at θ = 0, this ratio is |A/B| = 9, which corresponds to electron localization at the first center. There are two limiting cases of orientation of the two-well potential relative to the field: parallel and perpendicular. In the former case (θ = 0), the value of |A/B| depends on the field strength and distance between potential wells. In this case, the external field produces delocalization of the initial states. Calcula-
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INTERFERENCE IN TUNNELING IONIZATION INVOLVING
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10
(a)
(a)
R = 100 a.u. 2 |ψ(r)|2 × 10−10
8
y × 102, a.u.
1 0
6 4 2
−1
0
−2
6 −8
−6 z × 102, a.u.
−4
−2
5
(b)
|ψ(r)|2 × 10−10
−10
2
y × 102, a.u.
1
−4
−2
0 2 y × 102, a.u.
4
(b)
R = 200 a.u.
4 3 2 1
0
0
−1
−4
−2
0 2 y × 102, a.u.
4
Fig. 3. Transverse profiles of electron current from the ground state in two-well potentials with different distances between wells for θ = π/2, E0 = 1 eV, F = 107 V/cm, and 500 au distance to the screen.
−2 −14
−10 −6 z × 102, a.u.
−2
external field in this geometry does not lead to delocalization of the initial states.
Fig. 2. Distribution of electron current from the ground state in two-well potentials with field orientations (a) θ = 0 and (b) θ = π/2 for the set of parameters F = 107 V/cm, E0 = 1 eV, and R = 100 au.
tions show that an increase in the distance and field strength leads to a decrease in the mixing of states so that electron is localized at one of them. In the ground state, the electron localizes at the second center, the tunneling takes place almost without interaction with the first center, and the tunneling-current distribution is analogous to the current of a single point source (Fig. 2a). In the excited state, the electron is localized at the first center and can exhibit scattering on the second center upon tunneling [20]. If the two-well potential is orientated perpendicular to the field (θ = π/2), the ratio of coefficients is |A/B| = 1. This implies that the electron-wave function is a superposition of the wave functions of each deltapotential and the electron is equally localized at both centers. In this case, the two centers behave during ionization as point coherent sources (Fig. 2b). The TECHNICAL PHYSICS LETTERS
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Figure 3 shows the results of calculations of the transverse distribution of electron current for the twowell potentials with different distances between wells. A comparison of these curves shows that, as the distance increases, the pattern approaches the distribution of current in the case of two independent point sources. Thus, the above analysis of the interference of electron waves formed during tunneling ionization of a system with two-well potential showed that this phenomenon strongly depends on the mutual orientation of an external field and the potential, which influences the degree of electron localization at the two centers. For the two-well system oriented perpendicular to the field, the electron-current distribution is equivalent to that from two identical coherent point sources. With increasing distance between centers, the pattern approaches that for the two independent sources. In the case of orientations different from the perpendicular, the initial states exhibit delocalization that grows with increasing field strength and distance between centers.
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Observation of the interference effects described above is possible in diatomic molecular anions [16]. However, such experiments encounter additional problems related to the task of orienting the axis of a molecular ion in a desired direction and taking into account (or fixing) its spatial position. However, modern technologies can ensure the creation of nanoemitters comprising sharp nanopoint arrays [21, 22] and nanopyramids with single-atom apices [23]. When closely spaced, these nanodimensional sources will lead to interference manifestations in the emission current [24, 25]. This makes possible experimental verification of the peculiarities in the spatial distribution of electron current predicted by the proposed model. Our calculations also point to the importance of taking into account the type of binding in molecules during interpretation of the experimental data on tunneling ionization and the need for additional investigations of these phenomena. Acknowledgments. This work was supported in part by the Ministry of Education and Science of the Russian Federation, project no. 2014/19-2881. REFERENCES 1. Yu. N. Demkov and V. N. Ostrovskii, The Method of Zero-Radius Potentials in Atomic Physics (Leningr. Gos. Univ., Leningrad, 1975) [in Russian]. 2. A. I. Baz’, Ya. B. Zel’dovich, and A. M. Perelomov, Scattering, Reactions and Decays in Nonrelativistic Quantum Mechanics (Nauka, Moscow, 1971) [in Russian]. 3. D. A. Kirzhnits, Field Methods of the Theory of ManyParticle (Theoretical Body Systems) (Librokom, Moscow, 2010; State Atom, Moscow, 1963). 4. M. V. Frolov, N. L. Manakov, and A. F. Starace, Phys. Rev. A 78, 063418 (2008). 5. M. V. Frolov, A. V. Flegel, N. L. Manakov, and A. F. Starace, Phys. Rev. A 75, 063407 (2007). 6. S. V. Borzunov, M. V. Frolov, M. Y. Ivanov, et al., Phys. Rev. A 88, 033410 (2013).
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