Mads
Engelund‡
*a,
Szymon
Godlewski‡
b,
Marek
Kolmer
b,
Rafał
Zuzak
b,
Bartosz
Such
b,
Thomas
Frederiksen
cd,
Marek
Szymonski
b and
Daniel
Sánchez-Portal
ac
aCentro de Física de Materiales (CFM), CSIC-UPV/EHU, Paseo Manuel de Lardizabal 5, 20018, Donostia-San Sebastián, Spain. E-mail: mads.engelund@gmail.com
bCentre for Nanometer-Scale Science and Advanced Materials (NANOSAM), Faculty of Physics, Astronomy and Applied Computer Science, Jagiellonian University, Krakow, Poland
cDonostia International Physics Center (DIPC), Paseo Manuel de Lardizabal 4, 20018, Donostia-San Sebastián, Spain
dIKERBASQUE, Basque Foundation for Science, E-48013, Bilbao, Spain
First published on 20th June 2016
Dangling bond (DB) arrays on Si(001):H and Ge(001):H surfaces can be patterned with atomic precision and they exhibit complex and rich physics making them interesting from both technological and fundamental perspectives. But their complex behavior often makes scanning tunneling microscopy (STM) images difficult to interpret and simulate. Recently it was shown that low-temperature imaging of unoccupied states of an unpassivated dimer on Ge(001):H results in a symmetric butterfly-like STM pattern, despite the fact that the equilibrium dimer configuration is expected to be a bistable, buckled geometry. Here, based on a thorough characterization of the low-bias switching events on Ge(001):H, we propose a new imaging model featuring a dynamical two-state rate equation. On both Si(001):H and Ge(001):H, this model allows us to reproduce the features of the observed symmetric empty-state images which strongly corroborates the idea that the patterns arise due to fast switching events and provides an insight into the relationship between the tunneling current and switching rates. We envision that our new imaging model can be applied to simulate other bistable systems where fluctuations arise from transiently charged electronic states.
The ability to manipulate DBs, however, comes at a cost. Often images recorded on DB arrays are difficult to interpret because the underlying tunneling processes are complicated and because the DBs interact in a complex way with the scanning tip.32 This is due to the presence of metastable states, which can be switched dynamically during imaging.14,21,32–34 This situation is not unique to DB arrays and other studies have explored microscopy and spectroscopy under similar conditions.35–38
Recently the dynamic behavior of DB dimers (DBD) on Ge(001):H, when empty states are imaged by STM, was reported.15 A DBD can be created by the selective removal of two H atoms from the hydrogenated Ge(001)39 or Si(001)12 surfaces. As shown in Fig. 1A and B, this results in a buckled ground-state geometry with one atom at a higher position over the surface than the other. This configuration is degenerate with the mirror-symmetric situation, i.e., the system is bistable. Despite this geometric asymmetry it was found that empty-state STM imaging of DBDs on Ge(001):H15 and Si(001):H39 results in a symmetric pattern.
Such symmetric empty-state DBD images are shown in Fig. 1C and D for two different surfaces. Due to the appearance of three characteristic lobes (two side-lobes and a central lobe separated by sharp dips) we call this pattern “the butterfly”. On Si(001):H the side-lobes are dominant to an extent that the central lobe is only visible slightly off the dimer axis. The situation is reversed for Ge(001):H, where instead the central lobe is dominant. These detailed features have so far eluded a precise explanation, although it has been proposed that rapid oscillations15 or characteristic excited states11 are decisive in the butterfly-like appearance.
On Ge(001):H the behavior is somewhat different for filled-state imaging. Here static, asymmetric features can be observed for sample voltages around VS = −0.5 V while the onset of switching events occurs at VS ∼ −0.7 V.15 In the latter fluctuation regime it is reasonable to expect that integration over a sufficiently long time scale would give rise to symmetric filled-state STM images. Thus, if similar switching also takes place under empty-state imaging conditions, this could be invoked to explain the observed symmetric STM images in Fig. 1C and D. We find that understanding the origin of the butterfly shape of the experimental STM images requires a model that goes beyond a simple average of the contributions from two configurations. The description of this model is the main focus of the present work.
In this paper, we take advantage of the strong and distinctive butterfly pattern to reveal the underlying physics. At sufficiently low sample voltages we are able to resolve in time a bistable switching and characterize how the switching rate grows rapidly with larger voltages. We demonstrate that this low-voltage behavior can be extrapolated to explain the symmetric and feature-rich, empty-state images. The experimental DBD images can be reproduced by solving a rate equation for each and every tip position to find the appropriate time-averaged current over the two ground-state configurations. While previous studies have proposed similar models to compute the current through a tunnel junction,40–42 to the best of our knowledge, this is the first time that such a model has been used to reproduce STM images. Furthermore, our imaging model may have a wider applicability to other bistable systems where fast fluctuations are induced by the transiently charged electronic states that are the intermediates in the overall tunneling processes.
The nominally undoped Ge(001) samples (TBL Kelpin crystals, n-type, 45 Ωcm) were prepared by several cycles of argon ion sputtering and annealing.39 The passivation by a hydrogen layer was performed by exposure of clean samples to atomic hydrogen obtained from a home built hydrogen cracker.39 Note that although nominally undoped, the finally prepared Ge(001):H samples tend to be significantly p-doped regardless of intrinsic doping – even to the extent that n-doped samples tend to have inversion layers.43
Si(001):H samples were provided by CEA-LETI Grenoble.12 They contained two hydrogen terminated samples prepared in a reduced-pressure chemical vapor deposition (RP-CVD) reactor and bonded together by van der Waals forces. In order to obtain the hydrogen terminated reconstructed surface the 200 nm diameter starting wafers undergo a multistep preparation process including wet chemical cleaning and thermal treatment.12 The Si samples were slightly p-doped with a resistivity of 10.5 Ωcm corresponding to the doping level of 1015 at cm−3. The samples were de-bonded in situ under UHV conditions providing high quality hydrogenated Si(001):H surfaces for atomic scale experiments. The preparation procedure delivers samples that could be characterized by STM at liquid helium temperature (5 K) as demonstrated in our previous studies.12,14,16
The dI/dV data were obtained by averaging of several I(V) curves followed by a numerical differentiation. The switching rates were determined from I(t) traces exhibiting a two-level character registered for various tunneling current and bias voltage settings. For STM image processing the SPIP and WSxM44 software packages were applied.
To compute the STM images an accurate description of the decay into vacuum of the defect state wave-function, and in particular its nodal planes, proved crucial.47,48 Therefore, we followed the surface integration technique of Paz and Soler.49 When the total electron density falls below 0.0001 Å−3, all wavefunctions, ψn, are propagated into the vacuum assuming a constant exponential decay, κn. Here κn was determined using the energy eigenvalue en relative to the vacuum level, . We used the Tersoff–Hamann approximation50 in our simulations. In this method the current is assumed to be proportional to the local density of states (LDOS) evaluated in the center of curvature of the tip. Since the constant-current experimental images in Fig. 1C and D are acquired using different current values for Si and Ge, we think that a consistent and convincing theory–experiment comparison requires the use of the same proportionality constant a in order to relate the LDOS and the current for both substrates. We chose a = 0.01 Å3A for this ratio. With this value of a, which was determined by inspection, we can get a good agreement between theoretical and experimental STM images for both substrates. However, it is worth noting that the value of a is not critical and can be changed in a relatively wide range. Finally, in order to mimic the effect of the tip in reducing the resolution of the STM images, we have convoluted our currents with a Gaussian kernel K(r,r′) = (πσ2)−3/2e−|r−r′|2/2σ2 with σ = 0.5 Å.
The calculated unoccupied gap states are shown in Fig. 2A and B for the Si(001):H and Ge(001):H surfaces along with the corresponding calculated constant-current STM topographies in Fig. 2C and D. Evidently, these unoccupied gap states are asymmetric with respect to the surface rows and therefore cannot alone explain the symmetric butterfly pattern observed in the experiments. Additionally, we observe that these gap states are rather similar for both surfaces – the main difference is a stronger asymmetry for Ge than Si, which is seen most clearly in the calculated STM topographies. This difference probably arises from the fact that the Ge DBD has a larger charge-polarization than its Si counterpart, which in turn causes the unoccupied electronic states to have a greater weight over the positively charged (lower) site and moves the nodal plane in the opposite direction towards the negatively charged (upper) site. For Ge the electronic asymmetry overcompensates for the geometric one and the lower site appears brighter for empty-state imaging. Curiously, for Si the geometric and electronic asymmetry compensate each other to a large extent. We will see later that the degree of asymmetry plays a crucial role in the appearance of the final STM image.
Fig. 2 (A and B) Isosurfaces of the computed unoccupied gap state wave function superimposed on the structural model of the DBD on Si(001):H and Ge(001):H, respectively. The two different colors of the isosurface represent opposite signs of the isovalue (±0.003 Å−3/2). (C and D) Calculated local density of states (LDOS) isosurface of the unoccupied gap state (color scale range is 1.5 Å; iso-LDOS value is 10−10 Å−3). Within the Tersoff–Hamann approximation,50 this can be regarded as a constant-current STM image. The unoccupied DBD gap state can be loosely described as an anti-bonding combination of a p-orbital on the lower site and an sp3 hybridized orbital on the upper site. |
As recently shown by Wojtaszek et al.,43 STM measurements can be performed at very small sample voltages VS on the Ge(001):H substrate, even with only several tens of mV applied, both in the empty- and filled-state imaging modes. This is in marked contrast to the Si(001):H substrate, where the larger band gap hinders measurements for VS between approximately −1 V and +1 V. For VS = ±1 V, the recorded images on Si(001):H are perfectly stable and smooth (no fluctuations can be resolved). At these voltages the switching rates of the DBD are likely to exceed the apparatus time resolution. A detailed analysis of DBD switching behavior is therefore limited to Ge(001):H.
The disparity between the two substrates is likely connected to the substantially smaller gap of Ge. For negative bias voltages the different energetic position of the occupied DBD state with respect to the bulk states probably also plays an important role. As can be seen in Fig. 1E and F, the occupied DBD state on Ge(001):H is well below the edge of the valence band, in contrast to that of Si(001):H. Any hole created by tip extraction of an electron is therefore expected to have a longer lifetime in the Si-DBD state than in the Ge-DBD resonance, which increases the efficiency of energy transfer to vibrations and consequently the switching rate.
In Fig. 3A we show how the STM image of a Ge-DBD evolves from being stable at low negative voltage into being increasingly dominated by discrete jumps, which create the streaky image at VS = −1 V. The bistable character of the jumps resulting in a two-level telegraph noise of the current I versus time t traces is shown in Fig. 3B. By characterizing the frequency of these jumps presented in Fig. 3C and D we observe that the DBD switching on Ge(001):H can be recorded over a large current and voltage range for negative VS. The dependence of the switching rate on voltage (Fig. 3C) shows that the switching starts around VS = −0.6 V, which is where the tail of the filled-state resonance begins. The switching rate then grows quickly at larger voltages, indicating that the switching is efficiently initiated when electrons are extracted from the DBD state.
In Fig. 3D we analyze further the switching rate as a function of the tunnel current I (in the low current state), and find it to be proportional to IN with an exponent N ∼ 1, indicating that DBD switching is essentially a one-electron process. For the smaller voltage, i.e., VS = −0.7 V, close to the threshold voltage, the rate increases slightly faster than being linear with the exponent reaching N = 1.29. This may suggest that here also two-electron processes play a role.35
Now we turn our attention to the switching rates in the positive low-bias regime on Ge(001):H. Due to experimental difficulties in this regime, the recorded maximal switching rate and the tunneling current are substantially lower than those of negative voltages. The positive low-voltage measurements proved difficult, since these measurements are performed within the surface band gap.51 In order to maintain the desired tunneling current within the band gap the feedback loop has to approach the tip close to the substrate. A further increase of the tunneling current leads to the unstable behavior due to the significant reduction of the tip altitude, close approach to the surface and increased interaction between the tip apex and surface atoms.
Typical STM images showing switching events are displayed in Fig. 4A for three different bias voltages. As demonstrated in the inset in Fig. 4B, in this regime the dimer on Ge(001):H switches between asymmetric states at a rate roughly proportional to the STM current, which provides an indication that the positive-bias switching is dominated by one-electron processes.
The effective excitation of frequent switching for Ge(001):H is demonstrated in Fig. 4B, where the switching rate starts growing rapidly beyond a voltage of VS ∼ 0.3 V. The switching rate exceeds the time resolution limit of the apparatus already at VS = 0.7 V leading to completely smooth symmetric STM images of the defect at higher voltages15 as shown in Fig. 1C (see also ESI,† Fig. S2 for further details). It is worth noting that at higher temperatures thermal excitations may also lead to switching of the DBD between the two configurations. However, as discussed in more detail in the ESI,† thermal fluctuations would give rise to STM images characterized by a different symmetric pattern than that discussed here. This gives additional support to the current-induced switching mechanism presented here.
An interesting feature of the data presented in Fig. 5 is the existence of stable spin-polarized solutions for configurations close to the transition state. We find a ferromagnetic (FM) state that exhibits a local minimum in the symmetric atomic configuration, and it is even more stable than the unpolarized and anti-ferromagnetic (AFM) solutions in the case of Ge. Since it has a different overall spin than the ground state DBD, it could be a relatively long-lived state. However, in all our current traces for both positive and negative voltages, we observe only two distinct levels of the current, suggesting that only the two-fold degenerate asymmetric ground states are at play. In other words, additional levels that could be attributed to other charge and/or spin-polarized states were absent on the time-scale of our instrumental resolution of ∼1 ms.
The absence of long-lived charged states in DBDs is consistent with the available experimental information, in marked contrast to the case of single DBs. For single DBs it is typically possible to generate long-lived charged states that cause bright/dark halos in the STM images around the defects depending on the charge state and imaging conditions. These charge related halos have been consistently reported for single DBs,19,21 but are completely absent when imaging DBDs. This clearly indicates the shorter lifetimes of the charged states in DBDs, which are thus well below the instrumental resolution. However, the charge states are likely to play the dominant role in switching (as mentioned above, the only requirement being that their lifetimes should be at least in the order of hundreds of femtoseconds). The observed high switching rate, as well as its linear dependence on the current, strongly indicates that single-electron tunneling events can cause the observed switching. Furthermore, our calculations show that this picture is fully consistent with the energetics of the system.
To go beyond the above estimates and actually calculate the switching rates is a highly nontrivial task, particularly in the described physical regime, as it would involve an accurate calculation of both electronic and vibrational lifetimes. The electronic lifetimes are expected to be highly dependent on the tip-induced field and therefore on the precise doping conditions of the substrate as well as on the imaging voltage. Rather than embarking on this task, we seek a way to model the STM images without the need for explicitly computing the switching rates.
This assumption leads to the following simple two-level model, similar to earlier works explaining dI/dV spectra.40–42 For each tip position r the time-averaged current 〈I〉 is expressed in terms of the state occupations n1(2)(r) and the corresponding instantaneous currents I1(2)(r),
〈I(r)〉 = n1(r)I1(r) + n2(r)I2(r), | (1) |
ṅ1(r) = n2(r)Γ2→1(r) − n1(r)Γ1→2(r), | (2) |
ṅ2(r) = n1(r)Γ1→2(r) − n2(r)Γ2→1(r), | (3) |
Γ1→2(r) = αINDB,1(r), | (4) |
Γ2→1(r) = αINDB,2(r), | (5) |
The model leads to the following simple expression for the time-averaged current,
(6) |
Another observation, which has important consequences for the appearance of the STM image, is that if the DB current dominates the total current and if the current for a given (off-center) tip position is much larger than the other, i.e., I1,2 ≈ IDB,1,2 and IDB,1 ≪ IDB,2, then 〈I(r)〉 ≈ IDB,1[1 + (IDB,1/IDB,2)N−1], e.g., for N = 1, one has 〈I(r)〉 ≈ 2IDB,1. Thus, the time-averaged current is dominated by the low-current configuration for each tip position for N > 0. This observation explains the presence of the pronounced nodal features which, assuming a simple linear average (N = 0), would disappear (see ESI† Section 1 for further details).
Since we cannot access the switching rate at the bias voltage of VS = 1.3 V, at which the butterfly pattern is observed, we extrapolate the linear behavior observed at low positive voltages (see Fig. 4) and also assume N = 1 for higher voltages. This extrapolation is questionable, since higher-order processes could contribute at higher voltages. However, the most important features of the topography only have a very low sensitivity to the value of N ≥ 1 (see ESI† Section 2 for further details). The consequence of this low sensitivity is that (1) our reproduction of the main image features does not depend on the actual value of N, but also conversely that (2) we cannot determine N based on the features of the constant-current topography.
Our procedure for simulating the experimental empty-state constant-current images thus amounts to the following steps: (i) we compute the instantaneous tunneling currents IDB,1(2)(ri) and IO,1(2)(ri) with the dimer in each of the two configurations on a 3D grid of tip positions {ri}. Note that with the help of the Tersoff–Hamann approximation,50 this only amounts to computing the LDOS at {ri}. (ii) For each tip position ri we evaluate the corresponding time-averaged current 〈I(ri)〉 according to eqn (6). (iii) Constant-current topographies are determined from interpolation.
In Fig. 6, we directly compare the simulated STM images with the experimental ones, and as one can see, the dominant features are well reproduced. From our model we see that it is the nodal plane of the empty-state wave function which gives rise to the two symmetric crescent-shaped depressions separating the three lobes in the image. Each dip corresponds to the nodal plane position in one of the two asymmetric geometries. We note that, using the method by Paz and Soler,49 the appearance of the simulated STM images is only weakly dependent on the details of the calculations. It is possible to obtain results similar to those presented in Fig. 6 using less stringent computational parameters. We have explictly checked this using slabs containing only 4 bulk layers, the Γ-point only for k-sampling, a single-ζ-polarized basis set, 150 Ry mesh cutoff, and 150 meV energy shift to fix the orbital radii.
Fig. 6 (A and B) Experimental constant-current empty-state images of DBDs on Si(001):H (I = 10 pA; VS = 2.0 V) and Ge(001):H (I = 20 pA; VS = 1.3 V), respectively (same data as in Fig. 1). (C and D) The corresponding simulated STM images using the two-level fluctuation model described according to eqn (6). The range of the color scale, which is the same for theory and experiment, corresponds to 3.5 Å and 2.0 Å for Si and Ge, respectively. |
The central lobe arises due to the asymmetry of the unoccupied DBD state. Since the node of the wavefunction is off-centered compared to the dimer row both the asymmetric positions will give a large current in the center of the dimer. For a weakly asymmetric wave function (Fig. 2A), which occurs in Si, the nodal planes of the two configurations essentially coincide and the central lobe almost vanishes. On the other hand, for a strongly asymmetric wave function (Fig. 2B), which occurs in Ge, the nodal planes are spatially separated and the central lobe becomes broader and more dominant.
In addition to the features of the butterfly motif we can also reproduce the transition as the voltage sweeps the resonance (see ESI† Section 4).
Our model can be applied to low-temperature STM imaging of other systems dominated by current-induced fluctuations between equivalent states arising from a single type of electronic transition, e.g., the silicon quantum dots investigated by Yamazaki et al.38 or the hydrogen transfer reactions studied by Kumagai et al.35 It should be noted that this model cannot be directly applied to well-separated DBs due the existence of non-equivalent long-lived (charged) states. We consider the extension of the model to encompass this regime as a topic of future work. We hope that this work will inspire further efforts to simulate STM images taking into account dynamical effects.
Footnotes |
† Electronic supplementary information (ESI) available: (1) A figure depicting the change in the STM image as the N parameter is varied, (2) a short discussion on the sensitivity of the N parameter and (3) a series of STM topographies depicting the transition from a noisy to a smooth image as the voltage is increased. (4) A short discussion of the low-bias regime. See DOI: 10.1039/c6cp04031d |
‡ These authors contributed equally to this work. |
§ Half the oscillation period of the vibration associated with the reaction coordinate in Fig. 5, calculated to be ∼200 fs and ∼300 fs for Si and Ge, respectively. |
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