Quantum size effects on the work function of metallic thin film nanostructures

Abstract

In this paper, we present the direct observation of quantum size effects (QSE) on the work function in ultrathin Pb films. By using scanning tunneling microscopy and spectroscopy, we show that the very existence of quantum well states (QWS) in these ultrathin films profoundly affects the measured tunneling decay constant κ, resulting in a very rich phenomenon of “quantum oscillations” in κ as a function of thickness, L, and bias voltage, V_s. More specifically, we find that the phase of the quantum oscillations in κ vs. L depends sensitively upon the bias voltage, which often results in a total phase reversal at different biases. On the other hand, at very low sample bias (|V_s| < 0.03 V) the measurement of κ vs. L accurately reflects the quantum size effect on the work function. In particular, the minima in the quantum oscillations of κ vs. L occur at the locations where QWS cross the Fermi energy, thus directly unraveling the QSE on the work function in ultrathin films, which was predicted more than three decades ago. This further clarifies several contradictions regarding the relationship between the QWS locations and the work function.

The work function, the minimum energy required to move an electron from a solid into the vacuum, is the most fundamental material parameter in surface science. It plays a key role, for example, in the photo-electric effect, one of the first phenomena through which quantum mechanics unveiled itself. The work function is the result of a complex interplay between quantum mechanics and forces on the atomic scale. Recent studies suggest the exciting possibility of controlling the work function through quantum engineering of electronic structures at the nanoscale (1–5). Such efforts are, however, still in their infancy, and there are many puzzles and contradictions in the observations so far that hinder further progress (1–5). In this report, we show direct evidences for quantum size effects on the work function in ultrathin Pb films. We further establish the direct correlation of this quantity with the behavior of quantum well states. With the ability to control the growth of metallic thin films with atomic layer precision, one can anticipate the possibility of tuning the work function of metallic thin films through the quantum size effects, thus influencing chemical processes on surfaces.

It should be noted that theoretical investigations of quantum size effects (QSE) on the work function (Φ) were carried out several decades ago (6). By using a jellium model and continuously varying the film thickness, Schulte showed that the QSE leads to work function oscillations as a function of layer thickness. Moreover, it was found that the Φ oscillations are directly related to the energy locations of the quantum well states (QWS). In particular, whenever a QWS channel crosses the Fermi level (E_F), there is a negative cusp in Φ as a function of film thickness (L), which results from the sudden increase of the surface dipole due to the additional electronic density outside the surface contributed by the new QWS (6). Subsequent theoretical investigations using more sophisticated methods, while allowing more realistic electronic structure calculations, did not call into question this basic tenet regarding the correlation of Φ and QWS (7, 8).

Experimental investigations of this effect remained dormant for almost 30 years. The situation, however, changed dramatically with recent advancements in the quantum control of thin metal film growth (9–18), by which epitaxial metallic thin films can be grown on semiconductor substrates with atomic layer precision. One example is shown in Fig. 1A, where an epitaxial Pb film was grown on a Si(111) substrate. The scanning tunneling microscopy (STM) image reveals the smoothness and the thickness uniformity of the Pb film where the steps on the overlayer directly reflect the substrate steps. This sample system, epitaxial Pb thin film on Si(111), is the most widely studied system for the investigation of QSE, and many of its physical quantities have been shown to exhibit so-called “quantum oscillations” as a function of layer thickness (18–20). While such atomically smooth, uniform thickness metal films can be grown with near perfection, it is more convenient for direct investigation of QSE to use a different growth process that leads to the formation of large 2D flat-top mesas on the stepped region. These mesas contain regions of different thicknesses in one mesa (15). An example is shown in Fig. 1B, spanning thicknesses ranging from 9 to 19 monolayers (ML). The lateral extent for each film thickness ranges from 50 to 100 nm, about two orders of magnitude larger than the thickness. Moreover, the experimental measurements using STM are performed at ∼1 nm above the surface. One thus can safely assume that the measurements performed in the local region can be used to represent an extended film of the same thickness. On such flat-top mesas, the variation of work function as a function of thickness can be probed by measuring the tunneling decay constant (κ).

Fig. 1.

(A and B) STM topography images of a globally flat 2 ML Pb film and a flat-top Pb mesa on Si(111), respectively. The images were taken with sample bias V_s = 2 V, and tunneling set current I₀ = 100 pA. (C) An illustration of a one-dimensional tunneling junction for the STM configuration. The effect of the image potential is shown as a dashed curve between the sample and tip. Φ_s and Φ_t are the work functions for the sample and tip, respectively. Φ_eff is the effective tunneling barrier height with an applied bias V_s across the junction.

In an STM configuration the tunneling current between the two metallic electrodes (tip and sample) decays exponentially with respect to the tip-to-sample distance, z, and can be expressed as I ∝ e^-2κz. From this general expression, the tunneling decay constant κ can be defined operationally as κ ≡ -d ln I/2dz which is an experimentally measured quantity. One can further define an effective barrier height simply as Φ_eff = (ℏ²κ²)/2 m. Often, the tunnel junction is approximated with a one dimensional trapezoidal tunneling barrier (see Fig. 1C). Within this approximation, if the density of states (DOS) is smoothly varying in energy, then the effective barrier can be expressed as

[1]

where Φ_s and Φ_t are the work functions of the sample and tip respectively and |V_s| is the bias applied across the tunnel junction. Shown in Fig. 1C is the configuration with a negative sample bias V_s (relative to the tip electrode), where the tunneling current would be dominated by the tunneling channel between the highest occupied states of the sample, namely E_F,s, and the unoccupied states of the tip electrode at E_F,t + e|V_s| (labeled by the largest horizontal arrows). Note that the same expression (1) applies for the case with a positive sample bias, where the dominating tunneling channel occurs between the highest occupied states of the tip electrode, namely E_F,t and the unoccupied states of the sample electrode at E_F,s + eV_s. Thus, when the approximation (1) is valid, the sample work function can be extracted if the tip work function is known. This is the basic principle behind work function measurements using STM.

While this approximation ignores the effect of the image potential, which can lower the effective barrier (as shown by the dashed curve in Fig. 1C), it is expected that the correction should be relatively minor. The method just described has been routinely applied ever since the invention of STM and has largely yielded good information about the sample work function, although it was also recognized early that the effective barrier height is often influenced by detailed band structures (21). Furthermore, if one focuses primarily on the work function difference on different surfaces, the correction is canceled.

Presumably, to unravel the QSE on the work function of ultrathin films, one can simply measure the tunneling decay constant, κ, as a function of the layer thickness. Earlier works by Qi et al. (3) and Ma et al. (4) used precisely this approach and claimed to have observed QSE on the work function of ultrathin films. However, as we show in this paper, the very existence of QWS in these ultrathin films has a profound effect on the measurements of κ itself, resulting in a very rich phenomenon of quantum oscillations in κ as a function of thickness, L, and also as a function of sample bias voltage, V_s. The phenomenon is rooted in the sharp resonances of QWS in these thin films, which renders approximation (1) inapplicable for most of the bias range except for the limit where V_s approaches zero. As a result, κ(V_s; L) shows markedly different behaviors in three different regimes: (i) empty states (V_s > 0.2 V), (ii) filled states (V_s < -0.2 V), and (iii) the very small bias regime (|V_s| < 0.03 V). We further show that indeed in regime iii the measurements yield correct information about the work function, thus allowing us to unravel the QSE on the work function of this prototypical materials system.

Results and Discussion

(i) Empty-State (V_s > 0.2 V) Regime.

The inapplicability of approximation (1) in the presence of sharp QWS resonances can be understood by looking at the schematic diagram shown in Fig. 2A. Considering first a positive sample bias that lines up the tip Fermi level (E_F,t,1) with the lowest unoccupied QWS (LUQWS) above the sample Fermi level (E_F,s), the effective tunneling barrier height is marked by a vertical arrow and labeled as Φ_eff,1. When the bias is raised so that E_F,t,2 is above the first QWS, from approximation (1), one would expect a lowering of the effective barrier height. However, the effective barrier height actually increases (as shown by the dashed vertical arrow, labeled Φ_eff,2), because the tunneling is still dominated by the LUQWS. This trend would continue until the tip Fermi level reaches the next unoccupied QWS. The tunneling will then be dominated by the second QWS, and one expects a sudden drop in effective barrier height. As a result, the decay constant (κ) should show oscillatory behavior as a function of bias, and the valleys in κ oscillation should coincide with the QWS peaks. Shown in Fig. 2 B and C are spectra of dI/dV and κ vs. V_s respectively for 9 and 10 ML films. Indeed, the oscillations in κ vs. V_s are correlated with the QWS positions with the valleys in κ vs. V_s coinciding with the peaks in dI/dV (labeled by the arrows with different colors for different thicknesses).

Fig. 2.

(A) A schematic illustration of the STM tunneling process when probing the empty states of the sample (i.e., when V_s is positive). The Fermi levels of the sample and tip are labeled by E_F,s and E_F,t, respectively. E_F,t,1 and E_F,t,2 show the energy level of E_F,t at two different applied biases V_s,1 and V_s,2. The unoccupied QWS are represented by the curved lines in the sample side. The vertical arrows represent the effective barrier height, Φ_eff, at different bias conditions. (B) Differential conductance spectra (dI/dV) for 9 and 10 MLs. The peak positions in the dI/dV spectra correspond to the locations of QWS in the empty states for each thickness. (C) Measured κ vs. V_s for 9 and 10 MLs. All κ images used to extract κ vs. V_s were taken under same tunneling set current I₀ = 100 pA without changing the tip or sample. (D) Theoretical simulation of the bias dependence of κ for 9 and 10 ML with an assumption of for the solid lines and 3.92 eV for the dashed line of 10 ML. (E and F) κ images taken at V_s = 1.2 V and 0.7 V respectively for the same island shown in Fig. 1B. Odd numbers labeled on the images indicate the number of underlying ML. The apparent contrast of each layer clearly exhibits a bilayer oscillation behavior and the contrast of E is completely reversed in F.

This correlation leads to very interesting behaviors of bias-dependent mapping of the decay constant κ. Shown in Fig. 2 E and F are images of κ of an island, containing local regions with thicknesses ranging from 9 to 19 MLs, at two different bias voltages, V_s = +1.2 V and +0.7 V, respectively. One obvious feature in the κ-image is the contrast between different thicknesses. For example, in Fig. 2E all odd layers have higher κ values than the neighboring even layers. On the other hand, in Fig. 2F, the contrast is exactly reversed: All even layers have higher κ values than the odd layers. The odd-even bilayer oscillations, often referred to as quantum oscillations in the literature (10–17), are the consequence of nearly half integer phase matching between the Fermi wavelength and the interlayer spacing for Pb (111) ultrathin films (7, 8). The contrast reversal in κ, on the other hand, is a consequence of the onset of different QWS at the imaging bias: At V_s = +0.7 V, the bias is close to the peak positions of the QWS for the odd layers (thus depressions in κ for odd layers), while at V_s = +1.2 V it is close to the QWS peaks for even layers. Although this phenomenon itself is interesting, such a bias-dependent behavior of κ vs. L and its correlation with the QWS locations (see also the top panel of Fig. 4A) raise serious questions about using measurements of effective barrier heights to extract the work function because the contrast in work functions at different thicknesses should be a fixed quantity. This effect is explored further using theoretical simulations.

Fig. 4.

(A) Measured κ as a function of Pb thickness with various sample biases for 9–23 ML. (B) The locations of QWS obtained from the peak positions in dI/dV spectra. The horizontal dashed line at 0 eV is a guided line for the Fermi level. The red dots represent the highest occupied QWS for each thickness and the black dashed curves indicate a series of energy subbands. (C) Comparison between measured κ at ± 0.01 V (upper) and the work function simulation (lower). The thickness-dependence of the work function is well reflected on measurements of κ vs. L at ± 0.01 V.

Consider that the sample DOS, D(E), is largely reflected in the dI/dV spectra through the relation (see SI Text). This relationship allows us to extract D(E) from the experimental dI/dV spectra in Fig. 2B. Using the extracted D(E), the tunneling current versus a small change in barrier width z is calculated to obtain κ. Shown in Fig. 2D are theoretical simulation curves of κ vs.V_s for 9 and 10 MLs, respectively. Note that all calculations are done for infinite flat surfaces. The solid lines are the theoretical results with a fixed work function for both 9 and 10 MLs. One can see that the oscillatory behavior of κ vs. V_s is well reproduced and the phases of oscillations for 9 and 10 MLs are complementary, as we have observed, except at very low sample bias. We notice that if the value of Φ is lowered by 0.08 eV for 10 ML in the simulation, then the low bias behavior is also well reproduced (dashed line), suggesting the existence of a true work function difference between 9 and 10 MLs. Nevertheless, such an inference is not a direct experimental measurement.

(ii) Filled-State Regime (V_s < -0.2 V).

The behavior of κ vs. V_s and κ vs. L in the filled states (i.e., negative sample biases), on the other hand, is very different from the behavior in the empty states. As illustrated by the schematic shown in Fig. 3A, at a negative sample bias, whenever the tip Fermi level is below the highest occupied QWS (HOQWS), the tunneling process is dominated by this HOQWS (labeled by the black arrow), regardless of the presence of other occupied QWS. Thus, the effective barrier height will be primarily determined by this tunneling channel. Indeed, this is observed experimentally. Shown in Fig. 3B are dI/dV and κ vs. V_s for 9 ML film probed at negative sample biases. While the dI/dV spectrum reveals the clear signature of QWS at -0.45 V (albeit with smaller contrast), the oscillation in κ vs. V_s is now absent. Instead, a monotonic decrease of κ is observed as the negative sample bias varies from low to high. Nevertheless, above the location of the highest occupied QWS (HOQWS), one observes a tapering of slope in the κ vs. V_s curve.

Fig. 3.

(A) A schematic illustration of the STM tunneling process when probing the sample filled states. E_F,s and E_F,t represent Fermi levels of the sample and tip, respectively. E_F,t,1 and E_F,t,2 show different energy levels of E_F,t at two different applied biases. (B) Differential conductance spectra (red) and κ (blue) for 9 ML. (C–E) Special mappings of κ taken at large negative biases: V_s = -1.0 V, -1.5 V, and -2.0 V, respectively, for the same island shown in Fig. 1B.

The thickness-dependence of the tunneling decay constant (κ vs. L) continues to exhibit quantum oscillations. This can be seen in the spatial imaging of the tunneling decay constant, κ, shown in Fig. 3 C–E. Most interestingly, here the contrast of spatial imaging of κ remains identical, as long as V_s < -0.5 V (namely, the tip Fermi level is below the HOQWS). This can be seen in the middle panel of Fig. 4a, in contrast to empty-state κ vs. L behavior where the phase of the oscillation changes drastically at different biases (upper panel of Fig. 4A). In fact, for V_s < -0.5 V, κ vs. L is anticorrelated with the location of the HOQWS (red dots in Fig. 4B): That is to say, the closer the HOQWS is to E_F,s, the lower the measured value of κ (with an exception at 10 ML). This can be easily understood as we discussed above: At a negative sample bias, whenever E_F,t is below the HOQWS, the tunneling process is dominated by this HOQWS, regardless of the presence of other occupied QWS, thus determining the effective barrier height measurements. Such an antiphase relationship between κ and HOQWS for Pb (111) thin films indeed was reported previously but was wrongly understood as direct evidence for the QSE on the work function (3, 4).

At the bias between 0 and -0.5 V, κ vs. V_s deviates from the above behavior. When the negative sample bias is decreased toward zero, the κ vs. L oscillation continues to undergo a profound phase change (lower panel of Fig. 4A), especially in the crossover region (shaded region in Fig. 4A). The question that arises is whether any of these measurements tells us about the work function.

(iii) Very Low Bias Regime (-0.03 V < V_s < 0.03 V).

Within the one-particle picture, the work function of the metal can be defined as the energy difference between the Fermi level and the vacuum level. This definition suggests that one should use the measurements performed at the bias voltage close to zero because only then are the states near E_F being probed. Moreover, at such very low bias, the results from filled-state and empty-state measurements should be consistent, and this is indeed what we observed (top panel of Fig. 4C). This result would suggest that the measurements of the effective barrier height at very low bias range reflect the true QSE on the work function in these ultrathin films.

According to theoretical calculation using a jellium model (6) with thickness varied continuously, the Fermi level with respect to the vacuum level varies as a function of film thickness. This is due to the variation of the depth of the “effective potential,” which is in turn determined by the number of subbands occupied by the quantized electrons in the film. As the thickness of the film increases, new subbands get occupied one by one. Whenever a new subband becomes occupied at a particular thickness, the theoretical work function, being a function of a continuous thickness variable, exhibits a cusp right at the thickness where a new subband crosses the Fermi level (6, 7). In reality, the film thickness is not continuous but quantized, and the corresponding work function should take the discretized value. So far all theoretical calculations of the L-dependent work function of the Pb (111) surface are carried out only for free standing film and thus cannot be compared directly with our measurements. However, the underlying principle that the QSE on work function is due to the QWS crossing the Fermi level should remain valid. Here the QWS locations as a function of thickness are directly measured. By using these data points, we can interpolate the locations in L (as a continuous variable) where the QWS channels cross the Fermi level (see the crossings of the black curves with the blue dashed horizontal line in Fig. 4B). By using these crossing points, the period and phase of the oscillating work function as a function of continuous variable L are determined. The calculated continuous curve is shown in the lower panel of Fig. 4C, while blue square points on the calculated curve indicate discrete thickness values. It is evident in Fig. 4C that the behavior of the measured κ at low biases of ± 0.01 V indeed agrees with theoretical simulation based on the principle of Fermi level crossing of QWS channels. 17 ML is expected to show the lowest work function value, because one of its QWS is located exactly at the point where the subband crosses the Fermi energy level, as shown in Fig. 4B.

The consistent result of experimentally measured κ vs. L at V_s → 0 with the expected work function oscillations predicted based on the Fermi level crossing of QWS (Fig. 4C) further confirms that such measurements indeed truthfully reflect the QSE of work function in ultrathin films.

It should be emphasized that the underlying physics of the bilayer oscillations in κ at ± 0.01 V and κ at large negative or positive biases are due to totally different origins. The oscillation of κ vs. L measured at a large negative bias and its antiphase relation to the HOQWS are direct consequences of how QWS influence the tunneling process and do not involve work function oscillation. Similarly the rich behavior of κ measured at a positive bias as a function of L and V_s is also a direct manifestation of QWS in the tunneling process. Only when κ is probed at very low sample bias (|V_s| < 0.03 V) does the result truthfully reflect quantum oscillations of work function vs. layer thickness.

Methods

The experiments were conducted in a home-built low temperature STM system with in situ sample preparation chamber. Pb was deposited onto atomically clean Si(111) 7 × 7 surfaces at room temperature with a flux rate of 0.5 ML/ min by using a thermal evaporation technique. The room temperature growth of Pb leads to the formation of (111)-oriented 2D flat-top mesas on the stepped region, which is convenient for STM investigation because one mesa contains regions of different thicknesses. Moreover, the local thickness referenced to the wetting layer can be unambiguously determined. The base pressure of the system was maintained at better than 1.0 × 10^-10 torr during the sample preparation. All STM measurements were done at 78 K with liquid nitrogen cooling.

The derivative of the tunneling current with respect to the tip-to-sample distance, dI/dz, is acquired by using the lock-in amplifier with a z-modulation amplitude of 0.01 nm and a modulation frequency of 2 kHz, which is faster than the feedback time constant. This results in a small error current superimposed on the feedback set current I_o. The tunneling decay constant, κ ≡ -d ln I/2dz, is directly related to the measured quantity dI/dz as κ = -(dI/dz)/2I_o.