Accurate calculation of atomic resonances near surfaces

.__ g

Nuclear Instruments and Methods in Physics Research B 100 (1995) 336-341

NOMB

Beam Interactions with Materials B Atoms

!!0 ELSEVIER

Accurate calculation of atomic resonances near surfaces Stefan A. Deutscher

*,

aTb, Xiazhou Yang aTb,Joachim Burgdijrfer

aTb

a University of Tennessee, Department of Physics, A.H. Nielsen Bldg. JOI, Knoxcille, TN 37996, USA b Oak Ridge National Laboratory, Oak Ridge, TN 37831, USA

Abstract We present first results of new accurate calculations for atomic resonances near an Al surface. We employ large-scale matrix diagonalization involving 2 1000 basis states and realistic effective single-particle potentials. We implement two complementary techniques - the complex rotation method and the stabilization method. We also investigate strongly perturbed atomic wave functions near the surface and relate their evolution as a function of the distance, d, to the Wigner-von-Neumann non-crossing rule. The distance dependence of the resonance parameters can be analyzed in terms of semiclassical mechanics.

1. Introduction Resonant charge exchange between metal surfaces and ions plays an important role in many surface-diagnostic methods such as Auger electron spectroscopy @ES), ion neutralization spectroscopy (INS), and secondary ion mass spectrometry (SIMS), as well as for emerging technological applications (e.g. surface catalysis, thin film growth, and molecular beam epitaxy (MBE)). On a fundamental level, the interaction of multiply charged ions (MCI) with surfaces has become the focus of interest in the field of atomic physics in strong Coulomb fields. Neutralization and relaxation of MCI represent an intriguing many-body problem which involves transitions of a large number of “active” electrons and leads to the dissipation of large amounts of potential energy (typically N 1 keV). The understanding of the MCI-surface interaction requires detailed information on the behaviour of atomic resonances near and at surfaces. Both positions and widths of atomic resonances provide the input for coupled-states calculations for inelastic processes near metal surfaces [l]. We have implemented two complementary methods for the calculation of resonances. First, we use the method of complex scaling [2-6] which provides an efficient tool for calculating complex eigenenergies (positions and widths) of atomic resonances. Second, the recently developed stabilization method [7,8] allows the direct determination of the resonant wave function, i.e. the local density of states, and permits larger basis sizes since the calculation is performed on the real axis.

* Corresponding author, tel. + 1 615 974 7838, fax + 1 615 974 7843, E-mail: [email protected]. 0168-583X/95/$09.50

In the following progress report we present first benchmark calculations for the simple system of hydrogen near an aluminium surface. Applications to other systems are in progress. Atomic units are used throughout unless stated otherwise.

2. Model We consider a hydrogenic ion in front of a metal surface. The Hamiltonian fi of the problem reads

B=

P(r,d),

where the potential V( r, d) = V”( r.) + VS”rf( r, d)

(2)

consists of the atomic core potential VA(r), and of additional contributions Vsurf (r, d) due to the presence of the surface. The core potential for hydrogenic atoms with atomic number 2 is a Coulomb potential -Z/r. The surface potential of a metal reads VS”rf(r, d) = V?(d)

+ V,“‘(r) + Vi&,

d).

(3)

For the self-image interaction of the electron, Ves’, we use an accurate numerical fit to test-functions of the model potential proposed by Jenning, Jones, and Weinert [9]. It reproduces the self-consistent LDA calculation of Kohn and Lang [lo] near and inside the surface (apart from the Friedel oscillations) while giving the correct asymptotic behaviour lim V,“‘(r) d+m

0 1995 Elsevier Science B.V. All rights reserved

SSDI 0168-583X(94)00850-7

-$i+

= - &,

(4)

S.A. Deutscher et al. /Nucl.

Instr. and Meth. in Phys. Res. B 100 (1995) 336-341

where d is the distance from the surface. The potential contains only three parameters which reflect the properties of the metal; the saturation or bulk value of the potential, U,, the width of the interface region, A, and the image plane position, ze, which for aluminium (rs = 2.07) was found to be located at 0.70 a.u. in front of the surface. For the interaction potential between the electron and the ionic image VP:, we apply the dynamical response formulation of de Abajo and Echenique [ll], and derive the zero velocity limit using the plasmon-pole approximation with dispersion. This approach avoids the well-known difficulties with the singular behaviour of V& near the surface [6], and reproduces the correct asymptotic limit as d -+ 00,

Wry

d) = /&

(5)

.

The first term in Eq. (3) describes the interaction of the ion with its own image and equals at large distances V:‘(d) = -Z2/(4d). It will not be taken into account in the following since in the quasistatic limit the energy positions and widths do not depend on this interaction. It plays, however, a crucial role for the image acceleration of the projectile and, hence, for the scattering dynamics [14].

3. Method In order to obtain positions nances we solve the eigenvalue

and widths of these resoproblem

tiIcCI)=EI$Cr),

(6)

where 2 is the Hamiltonian of the problem, E the (possibly complex) eigenenergy and 1+) the corresponding eigenstate. We carry out a large-scale basis expansion to convert Eq. (6) into an N dimensional matrix equation yielding a set of energy eigenvalues (E&f. Hydrogenic basis functions are not well-suited as their continuum part is not square-integrable. To circumvent this problem we choose an expansion in terms of Sturmian functions which give a good description of both bound and continuum states while remaining complete and square-integrable @$,‘m
1 = ; $?(r)Y/m(~,

dJ>,

(7)

where S!$‘(r> are the Coulomb-Sturmian in coordinate representation by

functions given

S::)(r)

=h,,,e-“‘(2ar)‘+1L(,2’+1)(2~r),

(8)

angular momentum, and magnetic quantum numbers, respectively, Yr” the surface spherical harmonics, and Ly’ denotes the generalized Laguerre polynomials. The basis states (Eq. (7)) form a non-orthogonal basis set which results in a generalized eigenvalue problem. The matrix elements can be expressed analytically with exception of (i 1Qsurf] j) which must be computed numerically. Within the method of complex coordinate rotation we use up to 1000 basis states. The stabilization method allows considerably larger basis sizes (we use up to 4000). The standard complex rotation method [l-6] consists of the canonical transformation in the complex plane r + re’“,

(LO)

p + pep’@.

It converts resonant wave functions into square-integrable functions by effectively projecting out the P space (continuum space) portion of the wave function. Consequently, the Hamiltonian becomes dependent on 0:

ri(r,p)-+Ej(r,p,O)=

-~emzi”8+G(rei”).

(11) Converged resonances are characterized by the stability of the complex eigenvalue E with respect to the rotation angle 0, the Sturmian parameter (+, and the basis size N. The analytic continuation of the potential I$ reie) into the complex plane is explicitly known only in simple cases. For realistic potentials we adopt the “passive” rather than the “active” complex rotation, rotating the wave function @Lyh(re-i”) rather than the Hamiltonian. The price to pay ‘for complex rotation is that the physical interpretation of the resonant wave function is less obvious. An attractive, alternative method [7,8] recently developed uses the fact that stabilization diagrams, well-known as a tool for the determination of the positions of resonances, can also reveal information on their widths. One advantage is that the calculation proceeds on the real axis. The spectral density of resonances (the Feshbach Q space complement) is given by

(12) with Ek - ir, as the complex poles of the Green’s function. pQ(E) can be calculated by repeated diagonalization of Eq. (6) on the real axis (i.e. 0 = 0) for a range of (+ between a,, and ommax.The spectral density follows from 1

where pQ(E)

= %aX -

h,,,=

337

/z,

and

n, Z=O, 1, . . .

(9)

They contain the Sturmian parameter (T which allows for variational optimization. n, I, and m represent the radial,

umin

am’Xdop,(E), / mmln

(13)

where

P,(E) =

Cfj(Ej(fl) -E).

(14)

X.4. Deutscher et al. /Nucl.

338

Instr. and Meth. in Phys. Res. B 100 (1995) 336-341

Eq. (13) is now readily evaluated as 4000 1 n=l

Thus pQ(E) can be obtained by binning all energies E,( u ) for a,i, I u I C&ax into a histogram and then fitting a smooth function. Fig. 1 displays a typical stabilization diagram for hydrogen in front of an aluminium surface. The resulting histogram from which the spectral density pQ(E) can be determined is shown in Fig. 2. Eq. (1.5) can be visualized as a projection of the line density of the stabilization diagram onto the energy axis. From the fit of the “raw” spectrum of Eq. (1.5) to a sum over Lorentzian lines, both positions and widths can be extracted with high accuracy. Moreover, the structure of the resonant wave functions can be directly determined. The local spectral density of the corresponding resonance wave function 1!PF(I;E,)~~ at the energy E, is analogous to Eq. (1.5) given by

-1

1

=

_ u,i” %I,

Cl+j(',

a)l2 *

j

E,b)=E,

(16) The local spectral density assists in the phJ IS ical interpretation of the properties of the shape and Feshbach resonances.

4. Result The level positions and widths calculated as outlined above show the same qualitative behaviour that has previ0.001

0.01 f S u;’

0.1

1.0

1.5

2.0

2.5

3.0 3.5 2z/o

4.0

4.5

5.0

Fig. 1. The figure shows a typical stabilization diagram for a hydrogen atom at d = 11 a.u. in front of an aluminium (rS = 2.07) surface. Here, E is the energy and (T the Sturmian parameter. Note the stabilized horizontal “lines” which correspond to resonances (e.g. near E = -0.1 a.“. for n = 2). Resonances up to n = 8 can be clearly distinguished.

z5

3000 _

d 2

2000 -

g a

1000

0 -0.5

u n=2 n=4.

-0.4

-0.3

-0.2

-0.1

0

E (a.u.) Fig. 2. Density of states (DOS) extracted from the data shown in Fig. 1. Here, n denotes the principal quantum number. For n = 2 the two parabolic substates can be distinguished. Note the obvious difference of their respective widths which is explained in the text.

ously been obtained by means of complex rotation [6] and by means of the nonperturbative coupled angular mode (CAM) method [12] if the same sur@e potentials are used. Differences due to the use of other, more realistic potentials are more pronounced for the ground state (n = 1) than for excited states (n 2 2) and will be discussed in detail elsewhere [13]. We focus in the following on the behaviour of the wave functions. As pointed out previously [14], the classical dynamics near the top of the barrier is crucial for the structure of resonances and for the charge transfer process. Properties of the classical phase space structure are reflected in the wave functions. Figs. 3 and 4 show the resonant n = 2 wave functions from the m = 0 subspace near the surface. At large distances when both states form narrow thick-barrier resonances, they closely resemble parabolic states of hydrogen (Fig. 3). As soon as the energy level lies in the vicinity of the saddle point of the potential barrier, resonances undergo structural changes which are reflected in the shape of the wave function, as well as in the position and width of the resonance. The “up-hill” resonance, pointing away from the surface, maintains its approximate parabolic character well above the saddle point energy and the continuum part of the wave function (the leakage current) is small (Fig. 4). This is because the inner classical turning point of the corresponding orbit lies far away from the saddle. Consequently, a fast over-barrier transition, while energetically possible, remains dynamically blocked. Therefore, narrow, high-lying resonances exist close to the surface. In contrast, as soon as the “downhill” resonance, facing the surface, touches the saddle, the resonant wave function experiences a morphological change and the delocalized continuum part represents a significant fraction of the wave function. In other words, the electron undergoes “over-barrier transitions”. Nevertheless, the resonance can be clearly identified even in the

S.A.

Deutscher

et at. /Nucl.

fnstr.

and Meth.

over-barrier region. Its destruction (i.e. promotion to the smooth continuum) can be related to the semiclassical dynamics of over-barrier reflections [13]. It is closely related to the “avoided crossings” of resonances in the complex plane. Unlike proper bound states, resonances of the same exact symmetry do not satisfy the Wigner-vonNeumann non-crossing rule. Curves of the positions of resonances E,(d) as a function of the distance, d, from the surface can therefore cross. A generalization of the Wigner-von-Neumann non-crossing rule can be derived

in Phys.

Rex

B 100

(1995)

336-341

339

for the complex energy plane [15]. Typically, the trajectories of resonance energies avoid “collisions” in the complex plane. Fig. 5 illustrates the trajectories for the n = 2 resonances with d as the continuous parameter. While the position curves (projection of E,,Jd) onto the real axis) cross, the trajectories in the complex plane do not since the widths repel each other. Extension to higher n shells shows [13] that the suppression of “collisions” in the complex plane is generic. The point to be noted is that collisions imply the coalescence of complex eigenenergies

d = 10.0 a.“.

“V02wf161.dlOO” “V02wf161.dlOO” -

0.15 -

03

d = 10.0 au.

“V02wf162.d100”

-

“V02wf162.dlOO” -

Fig. 3. The behaviour states. The resonance already a very small origin was chosen to

of the two n = 2, m = 0 wave functions at a distance of d = 10 a.u. Both wave functions closely resemble parabolic pointing away from the surface (a) is nearly unperturbed whereas the one pointing towards the surface (b) shows leakage current. The surface is located at I = d parallel to the p and perpendicular to the z axis, respectively. The be the position of the hydrogenic ion.

S.A. Deutscher et al. / Nucl. Instr. and Meth. in Phys. Res. B 100 (I 995) 336-341

340

for a given d. Crossings of trajectories belonging to different parameter values, d, are however possible.

5. Conclusions We have presented first results on atomic resonances near the surface employing a new stabilization technique.

(4

It allows the detailed analysis of the structure of these resonances. A close correspondence to the classical overthe-barrier dynamics can be established. This correspondence is of crucial importance for highly charged ion surface interactions and for the behaviour of resonances very close to the surface at distances d smaller than the orbital radius, d I (I-).

d = 5.0 a.u

“VOZwf177.dO50” ~ “VOZwf177.dO50” -

lb)

d = 5.0 a.“.

“V02wf181.dO50” “V02wf181.dO50” -

Fig. 4. The behaviour of the n = 2, m = 0 wave functions at a distance of d = 5 a.“. The resonance pointing away from the surface (a) shows already a noticeable leakage current while still somewhat resembling a parabolic state. The other resonance (b) pointing towards the surface, however shows a significant leakage and shows already characteristics of a spherical wave function.

S.A. Deutscher et al. /Nucl.

Instr. and Meth. in Phys. Res. B 100 (1995) 336-341

341

References

Re(E)

(a.u.)

Fig. 5. Trajectory of the n = 2 resonances in the complex energy plane. Re(E) and lm(E) denote the real and imaginary parts of the complex eigenenergy obtained with complex rotation. The figure shows that in accordance with the Wigner-von-Neumann non-crossing rule, generalized as outlined in the text, the n = 2, m = 0 states do not cross. The diamonds correspond to different distances d of the hydrogen atom from the aluminium surface ranging from 24 to 5 or 0.5 a.u., respectively.

Acknowledgements We would hold,

supported Energy contract Energy

like

and Jiirg

to thank

Miiller

Lars

for many

by US Department Sciences, no.

Division

of Energy, of Chemical

DE-AC05-840R21400

Systems,

Inc.

Anderson, helpful

with

Carlos

discussions. Office Sciences, Martin

ReinWork

of Basic under Marietta

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