Ionization of Rydberg H atoms at band-gap metal surfaces via surface and image states

Surface states and image states are closely related, having energies within an energy gap of the band structure associated with the direction of the surface normal ($\vec{z}$). Their electronic wavefunctions tend to zero towards the bulk metal as well as towards the vacuum ($\Psi \to 0,{z}_{e}\to \pm \infty$, where z_e is the electron position), and they are therefore mainly confined to the metal–vacuum interface. Intrinsic surface states arise from the cleavage of the bulk metal (translational symmetry perpendicular to the surface is lost), and are localized mainly in the surface atomic layer [8, 9], with a wavefunction that decays exponentially towards the vacuum. Image states arise from the Coulomb-like attractive image potential for an electron outside the metal surface ($V({z}_{e})=-\frac{1}{4}{({z}_{e}-{z}_{\mathrm{im}})}^{-1}$ where ${z}_{\mathrm{im}}$ is the image plane position) and from the surface barrier created by a gap of available bulk electronic states in the metal.

Image states are localized mainly in the vacuum region of the interface (or more precisely, where the electron position is outside the image plane position [10].) The energies of the image states form a hydrogen-like Rydberg series (setting the usual nuclear charge term Z as 1/4):

$$\begin{eqnarray}&&{E}_{{n}_{\mathrm{img}}}^{\mathrm{IS}}=-\displaystyle \frac{1}{16}\cdot \displaystyle \frac{1}{2{({n}_{\mathrm{img}}+a)}^{2}},\end{eqnarray} \tag{ 1 }$$

where ${n}_{\mathrm{img}}$ is the image-state index and a is the quantum defect parameter for a given surface (for the Cu(111) and Cu(100) surfaces studied below, a is approximately 0.02 and 0.24 respectively [11, 12]). The corresponding 1D wavefunctions have the form [12]:

$$\begin{eqnarray}&&{\Psi }_{{n}_{\mathrm{img}}}^{\mathrm{IS}}({z}_{e})={z}_{e}{R}_{{n}_{\mathrm{img}},l=0}({z}_{e}/4),\end{eqnarray} \tag{ 2 }$$

where ${R}_{{n}_{\mathrm{img}},l=0}({z}_{e}/4)$ is the normalized hydrogenic (s-wave) radial wavefunction. Thus, image states can extend far into the vacuum ($\langle {z}_{e}{\rangle }_{{n}_{\mathrm{img}}}^{\mathrm{IS}}=6{({n}_{\mathrm{img}}+a)}^{2}$ a₀), and in the context of the Rydberg-surface interaction there can be significant overlap with the Rydberg electronic wavefunction at a long distance from the surface. Thus, the resonance effects in charge transfer via image states are expected to be particularly significant, especially when compared with the surface states which reside primarily inside the surface. For conducting metals such as gold, the image states are degenerate with the conduction band and the strong coupling of the image states to the degenerate continuum means that the energies are extremely broad and the resonant states short lived. Thus it is only for band-gap semiconductors (such as the Cu(111) and Cu(100) surfaces considered here) where such states are sufficiently narrow in energy and long lived to observe resonant charge transfer.

For an uncorrugated surface, the electron moves quasi-freely parallel to the surface ($\vec{\rho }$), such that each surface state and image state forms a 2D continuum of states with energy

$$\begin{eqnarray}&&{E}_{{n}_{\mathrm{img}}}^{\mathrm{IS},\mathrm{SS}}({k}_{\parallel })={E}_{{n}_{\mathrm{img}}}^{\mathrm{IS},\mathrm{SS}}+\displaystyle \frac{{k}_{\parallel }^{2}}{2\mu },\end{eqnarray} \tag{ 3 }$$

where the second term is the dispersion energy, with electron momentum parallel to the surface ${k}_{\parallel }$, and an electron of effective reduced mass $\mu \simeq$1 (in atomic units) [14]. Figure 1 shows the projected bulk band structure and the calculated energies of the surface and image states of the Cu(111) and Cu(100) surfaces [13]. Note that the full 3D bulk states are shaded in grey, and the projected band-gap is unshaded. Surface and image states that lie outside the energy gap, and are degenerate with the 3D bulk continuum, are commonly referred to as surface and image resonances (rather than 'states') and are broadened by the coupling to the continuum.

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The details of the implementation of the wavepacket propagation approach for the work presented here are given in [15]. In brief, the initial electronic wavefunction at atom–surface separations of ${D}_{0}\sim 6{n}^{2}$ is found through matrix diagonalization using a Lagrange–Laguerre discrete variable representation (DVR) for the radial coordinate [16], and a Legendre DVR for the angular coordinate, θ (the metal surface is assumed to be flat and isotropic in dimensions parallel to the surface plane, thus the system is cylindrically symmetric and rendered two-dimensional). The radial part of the initial electronic wavefunction is then projected onto a Coulomb wave DVR [17] where the time propagation takes place.

At each time step, the electronic wavefunction at a later time (or distance from the surface) is computed through the action of the time evolution operator, ${{\rm{e}}}^{-{\rm{i}}\hat{H}\Delta t}$:

$$\begin{eqnarray}&&{\Psi }_{\mathrm{el}}({\bf{r}};t+\Delta t)={{\rm{e}}}^{-{\rm{i}}{\hat{H}}_{\mathrm{el}}({\bf{r}};t)\Delta t}{\Psi }_{\mathrm{el}}({\bf{r}};t),\end{eqnarray} \tag{ 4 }$$

where ${\hat{H}}_{\mathrm{el}}({\bf{r}};t)$ is the electronic Hamiltonian which contains the kinetic energy operator, the centrifugal potential, and potential terms due the Coulombic interaction with the ion core, the applied electric field and the interaction of the electron with the surface. The time evolution propagator is approximated by a symmetric split operator [18] and the details are given in [15]. In most of these calculations, the surface is moved towards the H atom at constant velocity and, unless otherwise stated, the velocity is taken to be ${v}_{\perp }=3\times {10}^{-4}\;{\rm{a}}.{\rm{u}}.\;\equiv 656$ m s⁻¹ which is comparable to the mean velocity component perpendicular to the surface in experiments in our laboratory [19, 20].

In order to impose the correct outgoing boundary condition and to remove the spurious reflections of the wavefunction from the edge of the numerical grid, a complex absorbing boundary is included close to the grid edge [21]. To study the extent of the Rydberg-surface charge transfer, the total electron density on the numerical grid bound by the absorbing boundary is monitored as a function of atom–surface separation, and its derivative with respect to atom–surface separation corresponds to the ionization probability at the given distance.

The advantage of using the CWDVR is its suitability for treating the Coulomb potential (in particular the singularity at the origin) and for describing the wavefunction in the metal, with a dense distribution of grid points near the origin and becoming regularly spaced at long range. This, together with favorable scaling of the wavepacket time-propagation (which only involves matrix multiplications, equation (4)), allows calculations over a range of principal quantum numbers to become feasible, albeit still very computationally demanding.

Following the work of [5, 12, 14, 22], the one-electron pseudo-potential of Chulkov et al [11] is used to model the Cu(111) and Cu(100) surfaces in the direction normal to the surface ($\vec{z}$), while the electron is allowed to move freely in the direction parallel to the surface ($\vec{\rho }$). The potential has the analytical form:

$$\begin{eqnarray}{V}_{{ee}}({z}_{e})={V}_{1}({z}_{e})+{V}_{2}({z}_{e})+{V}_{3}({z}_{e})+{V}_{4}({z}_{e}), & \\ {V}_{1}({z}_{e})={A}_{10}+{A}_{1}\mathrm{cos}\left(\displaystyle \frac{2\pi }{{a}_{s}}{z}_{e}\right), & {z}_{e}\lt 0,\\ {V}_{2}({z}_{e})=-{A}_{20}+{A}_{2}\mathrm{cos}[\beta ({z}_{e})], & 0\lt {z}_{e}\lt {z}_{1},\\ {V}_{3}({z}_{e})={A}_{3}\mathrm{exp}[-\alpha ({z}_{e}-{z}_{1})], & {z}_{1}\lt {z}_{e}\lt {z}_{\mathrm{im},}\\ {V}_{4}({z}_{e})=\displaystyle \frac{\mathrm{exp}[-\lambda ({z}_{e}-{z}_{\mathrm{im}})]-1}{4({z}_{e}-{z}_{\mathrm{im}})}, & {z}_{\mathrm{im}}\lt {z}_{e},\end{eqnarray} \tag{ 5 }$$

where a_s is the bulk interlayer spacing, ${A}_{10},{A}_{1},{A}_{2}$ and β are the independent model potential parameters, and the values for the Cu(111) and Cu(100) surfaces are given in table 1. The remaining parameters in equation (5) are determined by requiring the potential and its first derivative to be continuous:

$$\begin{eqnarray}{A}_{20} & = & {A}_{2}-{A}_{10}-{A}_{1}\quad {A}_{3}=-{A}_{20}-\displaystyle \frac{{A}_{2}}{\sqrt{2}},\\ {z}_{1} & = & \displaystyle \frac{5\pi }{4\beta },\quad \alpha =\displaystyle \frac{{A}_{2}\beta }{{A}_{3}}\mathrm{sin}({z}_{1}\beta ),\\ \lambda & = & 2\alpha ,\quad {z}_{\mathrm{im}}={z}_{1}-\displaystyle \frac{1}{\alpha }\mathrm{ln}\left(\displaystyle \frac{-\alpha }{2{A}_{3}}\right).\end{eqnarray} \tag{ 6 }$$

The bulk potential and the width and position of the energy gap is described by V₁ in equation (5), while the metal–vacuum interpolation potential and the energies of the surface and image states are described by V₂ and V₃. The V₄ term describes the long range image-potential. The form of the Cu(111) and Cu(100) pseudo-potential, and the energies and wavefunctions of the surface state and image states calculated from diagonalization of the 1D potential are shown in figure 2 For comparison the calculations for these surfaces are benchmarked against a Jellium surface [24] and the parameters used are those for aluminum.

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Table 1.  Values of the parameters used in the one-electron pseudo-potential given by equation (5) for the Cu(111) and Cu(100) surface [11].

	as (a.u.)	A10 (eV)	A1 (eV)	A2 (eV)	β (a${}_{0}^{-1}$)
Cu(111)	3.94	−11.895	5.14	4.3279	2.9416
Cu(100)	3.415	−11.480	6.10	3.7820	2.5390

The computation of the wavepacket propagation for the electronically structured Cu(111) and Cu(100) surfaces is much more demanding than the Jellium surface. This is due to the periodic nature of the pseudo-potential inside the metal, which requires a large number of radial grid points, and the large components of the electron momentum parallel to the surface in the resonant charge transfer (see below), which requires a large angular momentum basis. Typically for the calculations carried out below, the numerical grid consists of $\sim 1000\times 200$ (radial × angular) grid points. Even so, for the charge transfers that are restricted by the band-gap and unaided by the resonance effects of surface or image states (see below), it is found that high frequency oscillations are superimposed on the ionization probability profiles, and an increase of radial and angular points would alter the form of the profiles. However, it was found that for the results presented below, the calculations are sufficiently converged that the average ionization distance, and the range of atom–surface separations spanned by the ionization probability profiles, do not change significantly with a larger radial or angular basis, and so are acceptable for the comparison carried out below. Furthermore, it is shown below that resonant charge transfer to band-gap embedded surface states and image states can be observed directly from visual snapshots of the wavepacket propagation, confirming that the different onset of ionization distances is not an artifact of the unconverged calculations.

The calculations presented here are for the H atom for which the levels of a given n have a degeneracy (arising from l and m_l) of n². Under the surface potential (assumed cylindrically symmetric) the l-degeneracy of the Rydberg n-manifold is broken, and the Rydberg manifold splits and forms states which are polarized with respect to the surface normal [25] akin to the Stark effect. If a field is present perpendicular to the surface (as typically occurs in experimental measurements [19], then the energy levels are also split at long range from the surface by the normal linear Stark effect, and as the atom approaches the surface from long range these states correlate adiabatically with the surface-polarized states. To focus on the resonance effects of the band-gap embedded surface states and image states, the calculations are performed here on the most surface-oriented state (the most red-shifted state) of the n-manifold, which has the largest overlap with the surface. All the calculations reported in this section used a constant incoming velocity of 656 m s⁻¹ for the Rydberg atom along the surface normal. This velocity lies in the middle of the experimentally accessible range of perpendicular collisional velocities in the densest part of the H atom beam for our experimental setup [19, 20]. The effect of varying the velocity is discussed in section 3.2.

The energies of the unperturbed H atom states are shown in figure 1 as horizontal lines to compare with the energies of Jellium, Cu(111) and Cu(100) surface electronic states. The values of the energies are given in figure 3 to provide a more quantitative comparison and these suggest that there might be two or three resonances between the image or surface states and the H atom levels in the range $n=2-8$. Figure 4 shows a comparison of the ${m}_{l}=0$, $n=2-8$ wavepacket propagation ionization probabilities calculated for a Jellium, Cu(111) and Cu(100) model potential, as a function of scaled atom–surface separation (scaled by ${n}^{-2}$).

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For n = 2, the charge transfer occurs at the greatest distance for Cu(100), then Jellium, and then for Cu(111) at the shortest distance. Figure 5(a) shows a snapshot of the wavepacket calculation for the three different surfaces; the atom is at a distance of 20.76a₀ from the surface. The amount of electron flux towards the surface clearly follows the order Cu(100) $\gt \mathrm{Jellium}\gt \mathrm{Cu}(111)$. The n = 2 Rydberg energy level is nearly degenerate with the surface resonance of the Cu(100) surface at ${k}_{\parallel }=0$ ($\bar{\Gamma }$ point, see figures 1 and 3), and so efficient charge transfer to the surface can occur via the surface resonance with little momentum transfer parallel to the surface, as illustrated by the directed beam of electron density moving towards the surface. The resonant effect is so strong that the charge transfer is even more efficient than for the free-electron Jellium surface. For the Cu(111) surface, the n = 2 energy is in the energy gap at ${k}_{\parallel }=0$, and so charge transfer to the surface state or the 3D bulk states can only occur with large ${k}_{\parallel }$ component, which is inefficient since the dominant and preferential charge transfer axis is along $\vec{z}$ where the saddle point in the potential lies. Thus little charge transfer has occurred at this point of the trajectory—in accord with the ionization probability profile shown in figure 4.

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Figure 4 shows that for n = 3, the charge transfer occurs much closer to the metal surface for both the copper surfaces compared with the Jellium. This is because the Rydberg energy level is in the energy gap of both Cu(111) and Cu(100) surfaces at the $\bar{\Gamma }$ point (see figures 1 and 3), and so charge transfer occurs relatively inefficiently with non-negligible parallel momentum components. This behavior is also apparent in figure 5(b). For n = 4, the charge transfer distance is similar for the Cu(111) and Jellium surfaces, while it is much smaller for Cu(100) surface. The snapshot of the wavepacket calculation in figure 5(c) for the three different surfaces illustrates that the more significant degree of ionization for the Jellium and Cu(111) surfaces. The n = 4 Rydberg energy level is nearly degenerate with the image state of the Cu(111) surface, and so charge transfer can occur via the population of the image state, which can be clearly seen in the horizontal line of electron density, indicated by the arrow in the snapshot in figure 5(c). For the Cu(100) surface, the Rydberg energy is within the band-gap and so charge transfer is inefficient.

Above n = 4 the H atom energies are above the upper edge of the projected band-gap of the Cu(111) surface, and are degenerate with the 3D bulk states (see figure 1 and 3), and so the calculated ionization distance is similar to the Jellium model, as shown in fig 4 (although resonant transfer to image resonances can still be observed by studying the evolution of the electronic wavefunction, see below). For n = 5, the Rydberg energy is nearly degenerate with the first image state of the Cu(100) surface, and in the projected bandgap, (see figure 1 and 3), and the population of this image state in the charge transfer process can be clearly seen in the snapshot of the wavepacket propagation shown in figure 6(a). The resonant charge transfer to the image state results in a similar ionization distance to the Cu(111) and Jellium model surface. This resonance effect to the first image state of Cu(100) is 'lost' for n = 6 and 7, which involves a large ${{\boldsymbol{k}}}_{\parallel }$ transfer, and so the ionization distance becomes progressively smaller than the Cu(111) and Jellium surfaces. For the Cu(111) surface, ionization of the n = 7 Rydberg state via the close-in-energy image state resonance (see figures 1 and 3) can be observed in figure 6(b). In fact, the effect of the image state resonance is so large that the charge transfer occurs at slightly larger distances than the Jellium surface (figure 4).

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For the Cu(100) surface, the n = 8 Rydberg energy is close in energy with the second image state embedded in the projected band-gap (see figures 1 and 3). The population transferred to the second image state is illustrated in figure 6(c), and the resonance effect is reflected in the ionization probability shown in figure 4 (the ionization distance is similar to the Cu(111) and Jellium model surface).

3.2.1. Varying the constant velocity

The wavepacket calculations presented in the previous section used a constant velocity of 656 m s⁻¹ along the surface normal for the incident H atom. Experimentally we can alter the velocity of the incident H atoms by changing the delay between the pulsed excimer laser that produces H atoms from NH₃ and the excitation laser pulses that produce the Rydberg atoms further down the supersonic beam path [19]; the velocity can also be varied by altering the seeding gas. Using unseeded ammonia as the precursor gas we can access a range of perpendicular collisional velocities for the Rydberg H atoms from 500 to 850 m s⁻¹ while maintaining a usable molecular beam density. Figure 7 shows repeats of the n = 4 WPP calculations for the Cu(100) and Cu(111) surface for different velocities in this range. In general it would be expected that the higher the velocity, the less time the atom has to ionize in a given incremental range of positions $D\to D-\delta D$. Therefore the probability distribution for the ionization distance should be shifted to shorter distances at higher velocities. Second if there are energy level crossings between the Rydberg atom and the surface image states that come into resonance over a specific range of distances, then the crossing would be traversed more quickly at higher velocities and hence the process may be less sensitive to resonance at increasing velocity. The results shown in figure 7 illustrate these expected trends. For the Cu(111) surface in particular, there is a very clear shift of the probability distribution with velocity, with the peak probability shifting from $4.1{n}^{2}{a}_{0}$ at 500 m s⁻¹ to around $3.3{n}^{2}{a}_{0}$ at 850 m s⁻¹. The shift in the probability distribution is much less pronounced for the Cu(100) surface, as this is a non-resonant case for n = 4 H atoms, whereas it is resonant for Cu(111).

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In the experiments described in [19, 20] there is an additional effect of the velocity impacting on the measurements, associated with the probability that the ions produced are extracted and detected after ionization. For greater collisional velocities the minimum distance of ionization from which the ions can be extracted for a given field increases, such that the detection region shifts further from the surface. Coupled with the shift in ionization probability to smaller mean atom–surface separations, the integrated ion detection probability will decrease with increasing collisional velocity.

3.2.2. Taking the acceleration due to image charge attraction into account

The assumption that the Rydberg atom retains its initial velocity at all distances from the surface is only an approximation. In reality, as the Rydberg atom approaches the surface, the attractive interaction with its own image charge causes the atom to accelerate towards the surface. There are effectively three phases of acceleration: initially on approaching the surface the Rydberg atom is accelerated by the van der Waals interaction of the neutral Rydberg atom to the surface, which scales as D⁴. In the second phase of the wavepacket model, as the atom starts to ionize it develops an overall partial charge as the electron flux into the surface is generated, and hence there is a greater acceleration due to the image charge attraction to the surface. Finally the atom becomes fully ionized and there is a classical acceleration of the ion core due to its own image charge. These effects are neglected in the constant velocity calculations but can be included by the addition of a mean field term as shown in our previous work [15, 26]. Here the mean field due to the electron wavepacket is determined by calculating the expectation value of the electronic Hamiltonian on the fly, and then the force acting upon the ion core (which is effectively at the centre of mass of the atom) is calculated. The acceleration will tend to shift the probability of ionization towards lower distances. However, in order to model the experiments a further effect must be included; that of the applied field which is present for extracting the ions. This will oppose the acceleration of the partially charged atom in the second phase of acceleration.

In these calculations we use very large (and experimentally impractical) fields, such as would be necessary to detect ions in experiments with n = 4 Rydberg atoms—in practice experiments are performed with much higher n values (26–34) and hence much lower, more realistic fields ($\lt 2000$ V cm⁻¹) are used. However it is computationally unfeasible to investigate such high-n states using a wavepacket propagation method; thus in these calculations the n = 4 hydrogen Rydberg state, which is degenerate with an image state in the Cu(111) surface and the projected band gap in the Cu(100), is used as a model for assessing the effect of velocity on resonance-enhanced charge transfer.

As shown in figure 8, inclusion of the mean field term, and the resulting acceleration, causes the expected reduction in the mean atom–surface separation for ionization. Additionally the difference between ionization behavior for different incident velocities is reduced, because the acceleration tends to dominate the velocity effects. For the on-resonance case, such as the image charge resonance at Cu(111) for n = 4, the acceleration is less because the ionization occurs further from the surface (and hence image charge effects are less) - thus, the effects of the initial velocity are still more visible.

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The conclusion from the velocity-dependence calculations is that measurements of the velocity dependence of the surface ionization behavior may provide helpful additional indication of resonant behavior. Recent experimental results from our group indicate that this is indeed the case [20].