Photodetachment of H Near a Dielectric Surface

Commun. Theor. Phys. (Beijing, China) 53 (2010) pp. 898 902 c Chinese Physical Society and IOP Publishing Ltd Vol. 53, No. 5, May 15, 2010 Photodetachment of H Near a Dielectric Surface WANG De-Hua ( Ù) and HUANG Kai-Yun (áô ) College of Physics, Ludong University, Yantai 264025, China (Received August 3, 2009; revised manuscript received September 3, 2009) Abstract Using the closed orbit theory, the photodetachment cross section of H near a dielectric surface has been derived and calculated. The results show that the dielectric surface has great influence on the photodetachment process of negative ion near the ionization threshold. Above the ionization threshold, the photodetachment cross section starts to oscillate. With the increase of the energy, the oscillating amplitude decreases and the oscillating frequency increases. The oscillation in the photodetachment cross section of H in the presence of a dielectric surface is either larger or smaller than the photodetachment of H without the surface. As the photon energy is larger than the critical value E pc, the oscillatory structure disappeared and the cross section approaches to the case of the photodetachment of H without any external fields. For a given detached-electron energy, the photodetachment cross section becomes decreased with the increase of the ion-surface distance. Besides, the dielectric constant has great influence on the photodetachment of H. With the increase of the dielectric constant, the oscillation in the cross section becomes increased. As the dielectric constant increases to infinity, the cross section is the same as the photodetachment of H near a metal surface. This study provides a new understanding on the photodetachment process of H in the presence of a dielectric surface. PACS numbers: 32.80.Gc, 03.65.Sq, 34.35.+a Key words: photodetachment, closed orbit theory, dielectric surface 1 Introduction Recently, the interactions of Rydberg atom and molecules with metallic surfaces have attracted much attention. [1 3] It is found that as the atoms and molecules approach the metallic surface, the Rydberg electron is subjected to fields caused by the presence of image charges in the metal, therefore this physical phenomenon is closely related to the stark effect behavior of atoms and molecules. Over the last decade, many researchers have studied the problem of Rydberg atom near a metal surface. Among them, the closed orbit theory has provided a clear framework to understand the oscillation in the complicated spectra for atoms near a metal surface. [4] Contrary to the many studies of the Rydberg atom or ion near a metal surface, the study about the photodetachment of negative ion near a metal or a dielectric surface is relatively little. Early experiment and theory showed that external fields can influence the photodetachment of the negative ions. It has been found that the photodetachment cross section of H in the presence of external field displays oscillatory structures. [5 8] In those previous studies, they all investigated the photodetachment of H in the external fields without surface. But with the development of the surface physics, the study of the photodetachment of negative ion near surfaces is important for the understanding of a large variety of process, ranging from electron scattering at surfaces to charge transport dynamics across interfaces. In 2006, Yang et al. applied closed orbit theory to study the photodetachment of H near an elastic interface. [9] Later, many authors have studied the photodetachment of H near an elastic interface in different external fields. [10 11] In these early studies, they all considered the interface as an elastic wall, the interaction potential between the electron and the surface is neglected and the collision of the electron with the surface is elastically. Therefore, this system is a real integrable one and the classical motion of the detached electron is analytically, its theoretical treatment is relatively simple. In fact, the elastic wall is only a simple model, which is different from a metal surface. [4] For the photodetachment of negative ion near a metal interface, the method they used in these early studies does not suit. Since the electron is subjected to the fields caused by the presence of image charge in the metal surface after being detached, this system becomes a nonintegrable one. Its theoretical analysis is complicate. Very recently, Yang et al. first investigated the photodetachment of H near a metal surface. [12] As to the photodetachment of H near a dielectric surface, none has given the study. In this paper, by using the semiclassical closed orbit theory, we study the photodetachment of H near a dielectric surface for the first time. The photodetachment process of negative ion near a metal surface mainly depends on the dielectric constant, which is similar to the case of the non-elastic collision with the surface. [13] Our results suggest that the dielectric constant and the ion-surface distance have great influence on the negative ion photodetachment near the ionization threshold. 2 Theory and Quantitative Formula The schematic plot of the system is given in Fig. 1. The H ion sits at the origin. The distance between the Supported by National Natural Science Foundation of China under Grant No. 10604045, the University Science & Technology Planning Program of Shandong Province under Grant No. J09LA02, the Education Department Foundation of Shandong Province under Grant No. J08LI03 and the Discipline Construction Fund of Ludong University Corresponding author, E-mail: jnwdh@suhu.com

No. 5 Photodetachment of H Near a Dielectric Surface 899 dielectric surface and the negative ion is d. The dielectric surface is perpendicular to the z-axis. As in the previous studies, [8 12] H is regarded initially as a one-electron system loosely bound by a short-range, spherically symmetric potential V b (r) of the hydrogen atom, where r is the distance between the active electron and the nucleus. A z-polarized laser is used for the photodetachment process. When a laser is applied to the negative ion near a surface, it may absorb a photon, then the active electron is detached and it moves away from the hydrogen atom. According to the electrostatic image method, [14] each charge has an image inside the dielectric surface but the charge of the image has the opposite sign. The potential acting on the active electron in the ion-surface system can be described as: V = V b + V c + V i. V c is the interaction of the electron with the image nucleus, which is also a short-range potential; V i is the interaction potential between the detached electron and the image electron e = E, which is a Coulomb-like attractive image potential: V i = e 2 /4(d + z), = (ε 1)/(ε + 1) > 0, ε is the dielectric constant. Therefore, the Hamiltonian of the detached electron near a dielectric surface has the following form (in cylindrical coordinates and atomic units): H = 1 2 ( ) Pρ+ 2 L2 z ρ 2 + 1 2 P z 2 +V b (r)+v c (r) 4(d + z) + 4d.(1) The effect of the short-range potential of the nucleus and the image nucleus V b and V c can be ignored after the electron is detached. [8] /4d is an additional term to ensure V (z = 0) = 0, which has no influence on the photodetachment process. Owing to the cylindrical symmetry, the z-component of angular momentum is conserved. For complicity, we only consider the l z = 0 case. By solving the Hamiltonian motion equations, we find the motion in the ρ-direction is free: ρ(t) = R sin θ + k sin θt. Here R is the initial spherical radius, θ is the outgoing angle of the detached electron and k = 2E is the momentum. According to the closed orbit theory, [8] each classical closed orbit of the detached electron that starts and later returns to the ion produces an oscillation in the photodetachment cross section. 2.1 Closed Orbit The photodetachment process of H near a dielectric surface can be described as follows. When H absorbs photon energies E ph, outgoing electron waves are generated. These outgoing waves propagate to large distances. Sufficiently far from the origin, the wave propagates according to semiclassical mechanics and it is correlated with classical trajectories. Due to the attractive image potential of the dielectric interface, some of the waves are turned back by the surface and return to the origin. The returning waves overlap with the outgoing source wave to produce the oscillation structure in the photodetachment cross section. In all the classical trajectories of the photodetached electron emitting from the origin, only those bounced back by image potential to the starting point are called closed orbits. Due to the free motion in ρ direction, the only closed orbit is along the z-axis. Since the interaction potential between the H and the dielectric surface is a Coulomb-like attractive image potential, the only closed orbit is along the +z direction. This closed orbit leaves the origin and moves initially in the +z direction. It is slowed down, stopped, and returned back to the origin by the electrostatic image potential of the dielectric surface. Some of the detached-electron trajectories are given in Fig. 2. The closed orbits are marked by the darker lines. Fig. 2 Some of the detached-electron trajectories near a dielectric surface, the ion-surface distance d = 90 a.u. The closed orbit is denoted by a darker line. Fig. 1 surface. The schematic plot of the H near a dielectric Next, we calculate the period and action of this closed orbit. The detached electron s energy is E = k 2 /2 with the emitting angle θ = 0, then its momentum along the +z axis is: p z = ( 2 E 4d + ). (2) 4(d + z) From this formula, we find that the electron s energy has a critical value E c = /4d. When the detached-electron energy E is larger than E c, the electron s momentum p z is always greater than zero as the electron moves along the +z direction, hence no closed orbit appears. If the energy of the electron is less than E c, there exists a turning point

900 WANG De-Hua and HUANG Kai-Yun Vol. 53 that limits the vertical expansion of the electron wave. For energy less than E c, as the electron moves in the +z axis direction, its momentum decreases and reaches zero. It then moves toward and finally returns back to the origin to form a closed orbit. By setting p z = 0, we get the maximum distance the electron can travel along the z-axis: z max = 4Ed 2 /( 4Ed). The period of the closed orbit along the z-axis can be obtained by the following formula: T = 2 zmax 0 1 p z dz = 4d2 2E 4Ed + 2d 2d arctan ( 4Ed) 3/2 4dE 4Ed, 0 < E E c. (3) The action of the closed orbit along the z-axis is: S = 2 zmax 0 p z dz = 2d 2E + 2 2d ( 8dE ) 4Ed arccos, 0 < E E c. (4) 2.2 Photodetachment Cross Section According to the closed orbit theory, the photodetachment cross section of H near a dielectric surface can be split into two parts: σ(e) = σ 0 (E) + σ osc (E). (5) The first part is the smooth background term without any external fields: [7] σ 0 (E) = 16 2π 2 B 2 E 3/2 (E b + E) 3. (6) The second part in Eq. (5) is the oscillating term, which corresponds to the contribution of the returning wave traveling along the closed orbit: σ osc (E) = 4π c (E + E b)im Dψ i ψ ret, (7) where ψ i ( q ) = Be kbr /r is the initial wave function of the detached electron. E b = kb 2 /2 is the binding energy, which is approximately 0.754 ev. [15] B is a normalization constant and is equal to 0.315 52, D is the dipole operator. [16] For the z-polarized light, D = z. ψ ret is the returning part of the detached electron wave function which represents the electron wave propagates outward into the external region first, then is attracted back by the dielectric interface and finally returns to the vicinity of the ion core along the closed orbit. In order to obtain the returning wave function associated with each closed orbit, we draw a sphere of radius R 1.0a 0. The outgoing wave on the surface of this sphere is then: [10] ψ 0 (q) = 4Bk2 cosθei(kr π) (kb 2 + k2 ) 2. (8) kr When this wave propagates out from the surface and travels along the closed orbit, it changes phase and amplitude. In the semiclassical approximation, the wave outside this sphere is a sum of the above outgoing wave: ψ(q) = i ψ 0 (q)a i e i[si µiπ/2], (9) where S i = pdq is the action along the i-th trajectory, µ i is the Maslov index and A i is the amplitude: A i (ρ, z, φ) = J i(ρ, z, 0) 1/2, J i (ρ, z, t) J i (ρ, z, t) is the Jacobian. Due to the classical motion of the detached electron, the amplitude is given by A i (ρ, z, φ) = P z0 1/2 R, (10) P zti R + kt i in which t i is the period for the electron going out from the source region and back to the origin. P z0 is the z component of the initial momentum and P zti is the momentum at time t i. If there is no dielectric surface, the detached electron will propagate away from the source region near the nucleus as a spherical wave and never return. Nevertheless, when there is a dielectric surface, some of the associated waves will be attracted back by the surface. According to the energy conservation, as the detached electron returns to the nucleus, P z0 = P zti. When the returning wave comes back close to the nucleus, it can be approximated by a plane wave traveling in the z-direction: (ψ) i ret (q 0) = N ie ikz, (11) in which N i is a constant. According to the general method given by Du and Delos, [17] we have N i = A i e i(si π/2) ψ 0 (q) (R, θ = 0). (12) Substituting Eq. (8) into the the above formula, N i can be described as: 4Bk N i = iãi(ρ, z, φ) (k 2 + kb 2)2 ei(si π/2), (13) in which Ãi = 1/kt i. The total returning wave is the sum of each returning wave. The overlap integral of the returning waves with the source wave function Dψ i gives the oscillation in the photodetachment cross section. By substituting the above formula into Eq. (7), we get σ osc (E) = 8π2 B 2 2E c(e b + E) 3 cos(s), (14) T where T and S are the period and action of the closed orbit along the +z direction, which are given by Eqs. (3) (4). If the energy of the electron is larger thane c, there is no closed orbit, the photodetachment cross section of H near a dielectric surface is the same as no surface exists. Therefore the total photodetachment cross section can be described as: σ 0 (E) + 8π2 B 2 2E σ(e)= c(e b + E) 3 T cos(s), 0 < E E c, (15) σ 0 (E), E > E c.

No. 5 Photodetachment of H Near a Dielectric Surface 901 3 Numerical Results and Discussions Using Eq. (15), we calculate the photodetachment cross section of H near a GaAs dielectric surface with the ion-surface distance d = 60 a.u., see Fig. 3. The dashed line is the photodetachment cross section of H without dielectric surface, which is plotted for comparison. For the GaAs dielectric, the dielectric constant ε = 13.18, then = 0.86. [18] For this system, the critical energy of the detached electron is E c = 0.097 ev and the critical photon energy E pc = E c +E b = 0.851 ev. The results show as the photon energy is above the ionization threshold but less than E pc, the photodetachment cross section becomes oscillate. The oscillation corresponds to the closed orbit of the detached electron and is the signature of the interference between the returning electron wave and the initial outgoing electron wave. For a fixed distance d between the ion and the dielectric surface, as the photon energy increases and approaches E pc, the oscillating amplitude becomes decreased and the oscillating frequency becomes increased. As the photon energy is larger than E pc, the oscillatory structure disappeared. The cross section is the same as the photodetachment of H without the dielectric surface, the influence of the dielectric surface vanished. = 1.0, the photodetachment cross section approaches to the case of the photodetachment of H near a metal surface. [12] The reasons can be analyzed as follows: when the dielectric constant is small, the electrostatic image potential acting on the detached electron is very small and can be neglected. With the increase of the dielectric constant, the influence of the image potential acting on the electron is also increased. Therefore, the effect of the dielectric surface becomes significant. As the dielectric constant ε, = 1.0, the dielectric surface can be considered as a metal surface, its influence on the detached electron is great and the oscillation in the photodetachment cross section becomes obvious. Fig. 4 The oscillating cross section σ osc of H near a GaAs dielectric surface versus the ion-surface distance, the electron s energy E = 0.046 ev, = 0.86. Fig. 3 The solid line is the photodetachment cross section of H near a GaAs dielectric surface, the distance between the H and the surface is d = 60 a.u., = 0.86. The dashed line is the photodetachment cross section of H without any external fields. The photon energy E p = E + E b. The vertical dash-dotted line denoted the critical photon energy E pc = E c + E b. Next, we calculate the oscillating cross section σ osc of H at different ion-surface distances, the detachedelectron energy E = 0.046 ev, = 0.86, see Fig. 4. From this figure we find with the increase of the ion-surface distance, the amplitude of the oscillation in the cross section becomes diminished. As the ion-surface distance is very large, the influence of the dielectric surface can be neglected. Figure 5 shows the influence of the dielectric constant on the photodetachment of H near a dielectric surface. The results show: as is small, the influence of the dielectric surface on the photodetachment of H can be neglected. With the increase of the value of, its influence becomes dominate, oscillatory structure appears. As Fig. 5 The photodetachment cross section of H near a dielectric surface for different dielectric constants. The distance between the H and the surface is d = 60 a.u. Finally, we compare the photodetachment of H near a dielectric surface with the photodetachment of H in a uniform electric field. [17] Since the interaction potential between the active electron and the dielectric surface is an attractive potential, the photodetachemnt of H near a dielectric surface is similar to the photodetachment of H in a uniform electric field. [17] Near the threshold, the motion of the detached electron is limited to a small region of space near the hydrogen atom. For z d, the

902 WANG De-Hua and HUANG Kai-Yun Vol. 53 image potential in the Hamiltonian in Eq. (1) can be approximated by: 4(d + 2) + 4d 4d 2 z. (16) Under this approximation, the Hamiltonian in Eq. (1) is identical to that in a uniform electric field, with the strength F /4d 2. In the next calculation, we choose the dielectric constant with = 0.98, the ion-surface distance d = 60 a.u., then the equivalent electric field strength F = 350 kv/cm. Fig. 6 The dashed line is the photodetachment cross section of H in a uniform electric field, the electric field strength F = 350 kv/cm. The solid line is the photodetachment cross section of H near a dielectric surface, the ion-surface distance d = 60 a.u., = 0.98. The vertical dash-dotted line denoted the critical photon energy E pc = 0.865 ev. The result is given in Fig. 6. The solid line is the photodetachment cross section of H in a uniform electric field while the dashed line is the case of H near a dielectric surface. From this figure, we find near the threshold, the oscillation structures of the two systems look similar. But with the increase of the energy, the photodetachment cross section of H near a dielectric surface is different from the photodetachment of H in a uniform electric field. When the energy is larger than the critical energy E pc, the oscillation in the cross section near a dielectric surface is vanished, but the oscillation in the electric field is still present. 4 Conclusions In summary, we have studied the photodetachment of H near a dielectric surface using the closed orbit theory. An analytical formula of the cross section is derived. It is found that the cross section consists of a smooth background term plus a cosine oscillating term, which is quite similar to the case when the negative ion is in an electric field. We find that the dielectric surface has significant influence on the photodetachment process near the ionization threshold. At present, studies of ion-surface interaction become increasingly important in conjunction with technological applications such as the development of ion sources, control of ion-wall interactions in fusion plasma, surface chemistry and analysis, reactive ion etching and semiconductor miniaturization via thin-film deposition, photodetachment microscopy experiments, etc. H is a very simple model to the study of the photodetachment process near a surface, in the future we will study some more complex negative ion systems. We hope that our studies will be useful in guiding the future theoretical and experimental research of the photodetachment processes of negative ions or atoms in the vicinity of interfaces, cavities, and ion traps. References [1] S.B. Hill, C.B. Haich, and Z. Zhou, Phys. Rev. Lett. 85 (2000) 5444. [2] M. Iñrrea, V. Lanchares, J.F. Palaciá, A.I. Pascual, J.P. Salas, and P. Yanguas, Phys. Rev. A 76 (2007) 052903. [3] G.R. Lloyd, S.R. Procter, and T. Softley, Phys. Rev. Lett. 95 (2005) 133202. [4] D.H. Wang, M.L. Du, and S.L. Lin, J. Phys. B 39 (2006) 3529. [5] C. Blondel, C. Delsart, and F. Dulieu, Phys. Rev. Lett. 77 (1996) 3755. [6] A.R.P. Rau and H. Wong, Phys. Rev. A 37 (1988) 632. [7] A.D. Peters and J.B. Delos, Phys. Rev. A 47 (1993) 3020. [8] Z.Y. Liu and D.H. Wang, Phys. Rev. A 55 (1997) 4605. [9] G.C. Yang, Y.Z. Zheng, and X.X. Chi, J. Phys. B 39 (2006) 1855. [10] G.C. Yang, Y.Z. Zheng, and X.X. Chi, Phys. Rev. A 73 (2006) 043413. [11] D.H. Wang, Eur. Phys. J. D 45 (2007) 179. [12] K. Kui and G.C. Yang, Surface Science 603 (2009) 632. [13] G.C. Yang, Y.Z. Zheng, and X.X. Chi, J. Theor. Comp. Chem. 6 (2007) 353. [14] K. Ganesan and K.T. Taylor, J. Phys. B 29 (1996) 1293. [15] K.R. Lykke, K.K. Murray, and W.C. Lineberger, Phys. Rev. A 43 (1991) 6104. [16] M.L. Du and J.B. Delos, Phys. Rev. A 38 (1988) 5609. [17] M.L. Du and J.B. Delos, Phys. Rev. A 38 (1988) 1896. [18] F.A.P. Osoriio, M.H. Degani, and O. Hipolito, Phys. Rev. B 37 (1988) 1402.