Field emission resonance spectrum
2.7 Atom manipulation
Manipulation was carried out on Co and Ag atoms on large clean Ag(111) terraces. Pa-rameters for manipulation are∼ 5 mV−20 mV, ∼ 60 nA, and2.5 V−2.8 V, ∼ 60 nA for Ag adatom and Co adatom manipulation, respectively. Manipulation of Co atoms on Ag(111) has previously been studied in detail [101]. Manipulation parameters were hand-adjusted depending on tip condition. In some cases manipulation is possible in one crystallographic direction but not others. To manipulate atoms with the STM we per-formed tip crashes until manipulation was successful. An example current trace recorded during successful manipulation of an atom∼6 nmdistance is shown in figure 2.11.
2.8 Analysis 25
Figure 2.11:Current trace from lateral manipulation. Spikes in current indicate the atom is being manipulated across the surface. During this trace, the manipulation mode shifts at∼5 nm, as evidenced by a change in the pattern of tunneling current spikes. Setpoint current is60 pA.
2.8 Analysis
We fit topography data to find corral radii which allows to accurately model w as a function of corral radius r. Atoms are located by pixels where h > 1.5σh where σh is standard deviation of topographyh. These are then used as first guesses to Gaussian fitsh(x, y) = Hexp (−((x0w−x)
x
2 +y0w−y
y
2)/2), where H is Gaussian peak amplitude and (x0, y0)is peak center. Deviation between first-guess location and Gaussian-fit location are on average < 0.5Å (see figure 6.2). The Gaussian fit locations are used to extract corral radii.
To fit Fano resonances we used equation 1.6 with an additional linear background (fig-ure 6.2). Fitting parameters depend strongly on initial parameters and the data range for fitting. We adjust fit range manually until good fit is achieved by eye using Python and Scipyoptimize.curve_fit(). The outputq,w,ϵ0,a,b, andcare then used as first guesses for a second fit using optimize.least_squares(), in which the resid-ual function to be minimized is the sum of squares residresid-ual multiplied by |q|, pushing q towards 0 in agreement with previous measurements on Co/Ag(111) (table 1.2). This assumes surface state modulations do not affectq, although previous studies demonstrate decay ofq as a function of distance from Co atoms on Cu(100) and Cu(111) [35]. To check if broadening of the typically sharp tip DOS band edge affects our fit results we con-volve the Fano function with the Fermi-Dirac distributionf(T, V) = 1/(1 + exp (E−Ek F
bT )) whereT is temperature in the STM at the time of the spectrum acquisition. Since T ∼ 5 K≪TK ∼80 K, neglecting thermal broadening does not greatly alter fit results. In fit-ting, attention was paid to the ratio between fit residuals and Kondo resonance amplitude A. Large fit residual compared toAmay indicate the fit may return a biased value forw.
Code is available at https://github.com/abekipnis/Small-Kondo-Corrals.
3 Results
We intended to construct three corrals (radius2.5 nm,3.8 nm, and4.5 nm) to tune surface state LDOS atEF and measure the effect on the Kondo temperature. In addition to STM topography used to verify corral construction, we intended to collect the following data for each corral:
1. Large range−100 mVto100 mVspectrum on the central Co atom to resolve corral eigenmodes, Co atom bound states and the Kondo resonance.
2. Small range −20 mVto20 mV spectrum on the central Co atom to resolve the Kondo resonance with high energy resolution.
3. Small range−20 mVto20 mVline spectrum across the diameter of the corral.
4. Large range −100 mVto100 mV line spectrum across the diameter of the corral, to resolve corral eigenmodes spatially.
5. dI/dV map at∼ 8 mVto show LDOS in the corral around the range at which the Kondo resonance appears.
In some cases we collected additional data. A line spectrum across an individual Co atom on a large Ag(111) step edge was collected (figure 3.1); this data can be used to extract the surface state scattering phase shift for Co/Ag(111). A−20 mVto20 mVgrid spectrum was acquired for ar ≈ 2.5 nmcorral (figure 6.1). Additional line spectra were measured on unoccupied corrals (figures 6.4). SpatialdI/dV maps at 45 mV, 125 mV, and175 mVon an empty4.5 nmradius corral were measured (figure 6.3). Unless other-wise noted,dI/dV spectra are normalized by dividing each spectrum by the mean value ofdI/dV.
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0 2 4 6 8 10 12 14 nm
100 50 0 50 100
mV
0 5 10 15
nm 0
5 10
0.0 0.2 0.4 0.6
Å
(a) (b)
Figure 3.1: (a) Single Co atom line spectrum. The line across the Co atom (bright con-trast peak) in (b) marks locations at which spectra shown in (a) were acquired. At dis-tances≳2.5 nmfrom the atom, the Ag(111) surface state onset appears as an increase in dI/dV around−67 mV. A bound state appears below−67 mV as the Co atom reduces tunneling from the STM tip into surface state [35]. The surface state electron scattering phase shift for Co/Ag(111) can be extracted [10] from this data to compare with litera-ture [3, 38] and may inform fitting equation 1.10 to our corralr-dependent measurements of Kondo widthw(figure 3.10). The Kondo resonance is a dip (dark contrast) indI/dV at
∼8 mVon top of the Co atom. Current setpoint: 300 pA, lock-in modulation amplitude:
4 mV. (b) Constant current topography of Co adatom, bias92 mV, current125 pA.
0 2 4 6 8
Figure 3.8: 4.65 nm radius corral constructed from 14 atoms. (a): Large range spectra
−100 mV−100 mV, current: 1254 pA, bias voltage: 100 mV, lock-in modulation am-plitude: 1 mV, lock-in frequency: 746 Hz, bias offset: −0.23 mV. (b): Constant current topography, bias100 mV, current790 pA. (c): Small range spectra, −20 mV −20 mV.
Current: 1249 pA, bias voltage: 20 mV, lock-in modulation amplitude: 0.5 mV, lock-in frequency: 746 Hz, bias offset: −0.8 mV. (d): Constant current topography, bias20 mV, current1 nA.
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Figure 3.2: 2.44 nm radius corral constructed from 14 atoms. (a): Large range spec-tra,−100 mV-100 mV, with current405 pA, bias voltage100 mV, modulation amplitude 1 mV, lock-in frequency526 Hz, bias offset−0.34 mV (b): constant current topography, bias20 mV, current500 pA. (c): small range spectra,−20 mV-20 mV. Current:1280 pA, bias voltage:20 mV, modulation amplitude:0.5 mV, lock-in frequency:526 Hz, bias off-set0.32 mV. (d): constant current topography, bias20 mV, current500 pA.
0 1 2 3 4 5
Figure 3.3: 2.59 nm radius corral constructed from 14 atoms. (a): Large range spec-tra, −100 mV-100 mV. Current: 405 pA, bias voltage: 100 mV, modulation amplitude:
1 mV, lock-in frequency: 526 Hz, bias offset:−0.34 mV. (b):
0 1 2 3 4 5
Figure 3.4:2.48 nmradius corral constructed from 8 atoms. (a): Line spectrum. Current:
500 pA, bias voltage: 100 mV, lock-in modulation amplitude: 2 mV, lock-in frequency:
602 Hz, bias offset: −0.77 mV. (b): Constant current topography, bias300 mV, current 500 pA.
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Figure 3.5: 2.38 nmradius corral constructed from 14 atoms. (a): 2.5 nm radius corral line spectrum,300 mVto -200 mV. Current: 1590 pA, bias voltage: 299.98 mV, lock-in modulation amplitude: 1 mV, lock-in frequency: 526 Hz, bias offset: −0.33 mV. Insta-bility occurred in two of the spectra acquired near the Co atom, possibly due to the large current set point (1.59 nA) or some instability in the STM tip. (b):
0 1 2 3 4 5 6 7 8
Figure 3.6: 3.81 nm radius corral constructed from 14 atoms. (a): Large range spectra,
−100 mV−100 mV. Current: 1580 pA, bias voltage: SI100.8mV, lock-in modulation amplitude: 1 mV, lock-in frequency: 746 Hz, bias offset: −0.40 mV. (b): Constant cur-rent topography, bias1 V, current500 pA.
0 1 2 3 4 5 6
Figure 3.7:3.65 nmradius corral constructed from 14 atoms. (a): Current:1585 pA, bias voltage: 20 mV, lock-in modulation amplitude: 0.5 mV, lock-in frequency: 746 Hz, bias offset:0.05 mV. (b): Constant current topography, bias1 V, current500 pA.
20 15 10 5 0 5 10 15 20
Figure 3.9: Representative small-range (−20 20 mV) and large-range (−100 mV-100 mV)dI/dV spectra showing the Kondo resonance on top of the Co atom at the center of quantum corrals made from Ag adatoms on Ag(111). The spectra are normalized by the value at8 mVin order to better compare the shape of the Kondo resonance.
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4 Discussion
We measure a change in Kondo resonance widthw as a function of corral radiusr due to changes in surface LDOS at EF. We use the w(r) dependence and equation 1.10 to estimate Js and Jb which determine surface and bulk state contributions to TK (fig-ure 3.10). Our results ((Jb, Js) = (0.52 meV,0.12 meV)) are similar to those from [3]
((Jb, Js) = (0.53 meV,0.21 meV)). We note consistencies and discrepancies between our data and those previously reported for Co spectra in Ag(111) corrals. Our measure-ments of w ≈ 5 meV on r ≈ 4.6 nm corrals contradict measurements of w reported in [3], wherew = 20 meVforr = 4.5 nm. Background spectra vary across corrals, thus contributions from background surface LDOS could be affecting the Kondo resonance.
Subtracting empty corral spectra from occupied corral spectra [102, 103] may enhance the Kondo signature. This necessitates unoccupied and occupied corrals be constructed in the same area with high precision. Future experiment with this precision may be at-tempted with automated atom manipulation and will take greater care to measure nearby background surface LDOS.
We measure increasing w with increasing corral radius r for r = 2.44 nmto2.7 nm and decreasing w with increasing r for 3.8 nmto4.6 nm. This suggests that we can fit was a function of corral radiusrwith the same oscillatory dependence (equation 1.8) as in [3]. We obtain similar values for exchange constantsJs andJb which define surface and bulk state contributions to the Kondo effect, but different values for the scattering phase shiftsδ1 andδ2. Our measurements extend to a lower radius range than in [3]; the smallest corral for which they measuredwwas2.88 nm, whereas the smallest corral we measured was2.44 nm. From line spectra we can also show Kondo width as a function of distance from the central Co atom to compare with previously reported measurements on the spatial decay of the Kondo resonance and Fano lineshape parameters [9, 22, 35], as well as inform the consistency of outputs from our fitting.
Multiple eigenstates near EF emerge in figure 3.8 (c), potentially due to broken rota-tional symmetry resulting from the imperfect circular geometry or surface state modula-tion from perturbamodula-tions outside the corral such as step edges (figure 1.4). Extents to which this impacts our spectra is unknown.
Fitting Fano resonances to spectra fromr ≈ 2.5 nmandr ≈ 4.5 nmcorrals is easier than fitting tor ≈ 3.8 nm corrals, as the dip in the lineshapes are more prominent (fig-ure 3.9, orange versus green and blue curves). Fitting data from ther ≈ 3.8 nm corral was difficult as the eigenmode at EF made d2I/dV2 large near the Kondo resonance.
Comparing the robustness of the Fano fit as opposed to a Frota fit [104], for example, is also included in goals for future work.
We built several corrals with similar radii (∼2.5 nm) from 8, 12, and 14 atoms. The ef-fects of wall atom density on surface state confinement in Ag(111) has been studied in re-mote switching of an H2Pc molecule on Ag(111) [58]. For Fe atoms on Cu(111)), "greater electron confinement will not be achieved by simply packing Fe atoms more densely along the borders of the quantum corrals. The borders, themselves, act as ‘absorbers’ and so greater confinement will likely come only form new surface state/absorbate systems that do not couple surface state electrons so strongly to the bulk" [105].
2 3 4 5 6 7 8 9 10 corral radius (nm)
5 10 15 20 25 30
w (mV)
Fit to our data in Co/Ag corrals Fit by Li et. al. for Co/Co corrals
Figure 3.10: wfrom Fano fits to line spectra on quantum corrals of varyingr. The blue curve is fitting equation 1.10 to our data with fixedD = 4.48 eV, k = 0.82, α = 0.88, ρb = 0.27 eV−1,ρs0 = 0.125 eV−1 [3], initial guess (Jb,Js,δ1,δ2,A)=(0.530, 0.210, 0,0, 3.2), and bounds ((0,1), (0,1), (-π, π), (-π, π), (0,10)), the fit results inJb = 0.52 meV, Js = 0.12 meV, δ1 = 0.79, δ2 = 0.69, andA = 1.98. The bounds around the orange curve show the fit withD= (4.48±0.62) eV. A shift ofkorδ1,2 values is not adequate to make the orange curve match our data. wis shown from fits to line spectra rather than individual spectra and thus not all spectra were acquired directly atop the Co. Both small and large range spectra data are shown. Fit data points whereq > 1.5 andd > 0.5 nm where d is the distance from the Co atom are not shown here or used in the fitting of equation 1.10. Due to surface state density modulations away from the Co due to the corral confined modes, differences in LDOS atEF cause spread inwat givenr. The size of the point corresponds to1/dwheredis lateral distance from the Co; larger circles mean the point should carry more weight in the fit of equation 1.10. Discrepancies between our measurements and the fit to measurements made by Li et. al. could be from a few sources.
Spectra used to fit Fano lineshape and equation 1.10 by Li et. al. were acquired using a lock-in amplitude of4 mV, larger than our lock-in amplitude (0.5 mV-1 mV). Our corral walls are nonmagnetic silver atoms rather than magnetic Co atoms as in the case of Li.
et. al, which could affect scattering phase shifts. Currently, all data points carry the same weight in the fitting of the function (blue), which skews the results of the fit. Ideally, we would show a single w value for each r with error bars. This has problems with fitting if the number of data points is less than the number of fit parameters. Further analysis is needed to report and propagate confidence intervals in values gathered from Fano resonance fitting. More data, for example at larger corral radii, would inform better fits to this phenomenological curve.
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