Demonstrating Kondo behavior by temperature-dependent scanning tunneling spectroscopy

The Kondo effect describes the scattering of conduction electrons by magnetic impurities, manifesting as an electronic resonance at the Fermi energy with a distinctive temperature evolution. In this letter, we present a critical evaluation of the current methodology employed to demonstrate Kondo behavior in transport measurements, underscoring the limitations of established theoretical frameworks and the influence of extrinsic broadening. We introduce a novel approach for analyzing spectroscopic indicators of the Kondo effect, employing the Hurwitz-Fano lineshape as a model for the Kondo resonance in the presence of extrinsic broadening. Through precise scanning tunneling spectroscopy measurements on an exemplary spin-1/2 Kondo system, phenalenyl on Au(111), we demonstrate the efficacy of our proposed protocol in extracting accurate intrinsic Kondo linewidths from finite-temperature measurements. The extracted linewidths exhibit a robust fit with a recently derived expression for the temperature-dependent intrinsic Kondo linewidth, providing compelling evidence for the validity of the underlying theory.

The Kondo effect is one of the most enigmatic phenomena in condensed matter physics and one of the hallmarks of strong electronic correlations [1][2][3][4].It occurs when a magnetic impurity interacts with the conduction electrons in a metallic host.Below a characteristic temperature, called the Kondo temperature, T K , the impurity spin is effectively screened by the formation of a total spin-singlet state with the surrounding conduction electrons.As the low-temperature dynamics of individual Kondo impurities are governed by the energy scale k B T K , its magnitude has a major influence on the ground states and quantum phase diagrams of strongly correlated materials, such as Kondo lattices and heavy fermion systems [5][6][7][8][9].
The Kondo effect is signaled by the appearance of a sharp resonance at the Fermi level in the density of states of the magnetic impurity.This can be exploited for the detection of magnetic moments in atoms and molecules adsorbed on metallic substrates by scanning tunneling spectroscopy (STS), where it shows up as a zero-bias anomaly in the conductance (dI/dV ) spectra [10][11][12][13][14].However, a clear-cut proof of Kondo behavior requires to discriminate the Kondo resonance from other zerobias anomalies.This can be achieved, for example, by measuring the temperature evolution of the resonance's linewidth [15][16][17][18][19], which shows a characteristic universal behavior in the Kondo regime [20,21].
Attempts to derive an analytic expression for the temperature dependence of the Kondo peak from Fermi liquid theory [15] were shown to be problematic due to the limitation to very low temperatures T ≪ T K and energies ω ≪ Γ K [22,23].As a consequence, empirical expressions for the temperature-dependent Kondo linewidth are often used to fit experimental data, e.g., Γ emp (T ) = (αk B T ) 2 + 2(k B T K,N ) 2 where α was introduced as an additional fitting parameter instead of α = π as originally defined in Ref. [15].Due to the free parameter α and different definitions for the Kondo temperature (e.g.T K,N = 2.77 T K ) [24], various forms for Γ emp (T ) can be found in the literature [16,17,19,[25][26][27].It has also been noted that simple square-root expressions such as Γ emp (T ) cannot capture the universal scaling behavior obtained in accurate numerical renormalization group calculations [28].
Recently, an analytic equation for the temperaturedependent linewidth of the Kondo peak was derived from a novel theoretical Ansatz for the renormalized selfenergy that extends the temperature and energy range beyond the Fermi liquid regime [23]: where a ≡ 1 + √ 3 ∼ 2.732, b ≡ 2 + √ 3 ∼ 3.732 and c ≡ √ 3/2 ∼ 0.866 are constants, τ ≡ πk B T is the temperature parameter and ∆ K the width parameter of the T = 0 Kondo peak, related to the halfwidth via Γ K = 2.542 ∆ K , and to the Kondo temperature via ∆ K = 1.542 k B T K [29].As shown in Ref. 23, Eq. 1 is in excellent agreement with numerical renormalization group calculations [21].On the other hand, experimental data of Kondo linewidths versus temperature measured by STS [16] could not be fitted very well, most likely due to the presence of extrinsic broadening mechanisms in the STS data [30].Post-hoc removal of extrinsic broadening from the experimentally measured widths only slightly improved the agreement.Therefore a clear experimental proof of the theory of Ref. [23] is still lacking.
In this work, we demonstrate the validity of the theory reported in Ref. 23 by accurate temperature-dependent measurements and analysis of a prototypical spin-1/2 Kondo system, and provide an efficient protocol to experimentally prove the Kondo nature of a zero-bias peak.To this end, we carried out low-temperature STM experiments on phenalenyl molecules deposited on a Au(111) surface [24].The unpaired π z electron of phenalenyl forms an S = 1/2 ground state and is uniformly delocalized over six equivalent positions, as shown in the two insets of Fig. 1(a) where the spin density plot (inset I) and the experimental constant-height map of the Kondo resonance (inset II) are reported (more details in Ref. 19).The high-resolution STS spectra shown in this work were measured with a metal STM tip and acquired as point spectra on one of the six equivalent Kondo lobes, as depicted in inset II of Fig. 1(a).
Two distinct data sets of temperature-dependent dI/dV spectra were acquired on two different molecules, which are similarly adsorbed on the face centered cubic (fcc) region of the Au(111) herringbone reconstruction.We refer to the lower temperature data set as DS1, while DS2 is the data set from 1.56K to 7.5K.For each temperature the Kondo peak was fitted with the Frota-Fano lineshape (example shown in Fig. 1a), taking quantum interference into account via the Fano phase ϕ [31][32][33], F (V ) = F 0 • Re e iϕ / 1 + iV /∆ where F 0 is the amplitude, and ∆ the Frota width parameter related to the halfwidth of the Frota-Fano lineshape by Γ = 3 + √ 12 • ∆ = 2.542 • ∆.The Frota-Fano lineshape yields good fits for all temperatures [24].The obtained halfwidths of the Frota fits are shown in Fig. 1(b) for both data sets.The two data sets overlap neatly in the temperature range where both molecules have been measured, i.e., between 1.5K and 3K.This justifies the merging of both data sets in order to obtain a larger temperature range.Fitting Eq. 1 to the merged data set yields a relatively poor fit as shown by the red line in Fig. 1(b), although somewhat better than the empirical expression Γ emp (T ) with fixed temperature coefficient α = π (grey solid line), similar to the finding in Ref. 23.
We will now see that the mismatch with Eq. 1 is caused by extrinsic broadening mechanisms.In STS, the conductance spectra (dI/dV ) are measured at finite temperatures T where the Kondo peak is broadened, (i) due to the intrinsic temperature dependence of the Kondo peak [15,21,23], and (ii) due to the presence of different extrinsic broadening mechanisms in the STS measurement [30].Thus, in order to obtain the actual intrinsic halfwidth Γ K of the Kondo peak and demonstrate Kondo behavior, one has to first remove the extrinsic contributions from the measured HWHM.Assuming a noise-optimized (electronic and mechanical) experimental setup, the two main sources of extrinsic broadening are caused by Fermi-Dirac (FD) broadening of the tip and the voltage modulation for the lock-in detection.A good fit with Γ emp (T ) can only be obtained by using α as a free fitting parameter (dashed grey line).
In order to properly incorporate the most important broadening mechanisms into the analysis of the STS data we now resort to theory.The experimental situation of STS is depicted schematically in Fig. 2(a): A molecule (M) on a metallic substrate (S) is probed by an STM tip (T).Application of a voltage V to the sample drives a current I from the sample electrode via the molecule to the tip electrode.Typically, in STS the coupling of M to T is much weaker than the coupling of M to S. In this situation, also called ideal STM limit [34], M is effectively in equilibrium with S, and the dI/dV of the current from the tip to the molecule at the substrate is given by the convolution where f ′ (ω) = −β/ 4 cosh 2 (βω/2) is the derivative of the tip's FD distribution, β = 1/k B T with T the temperature at the tip, and A(ω) the spectral function of the molecule [35].We assume a fully thermalized system so that tip and sample have the same temperature T .The convolution of the spectral function A(ω) with the FD derivative f ′ (ω) leads to broadening of the Kondo res- onance, to which we now simply refer as FD broadening.This is demonstrated in Fig. 2(b) which shows the numerically calculated dI/dV according to (2) for different tip temperatures, assuming a Frota-Fano peak for the spectral function, A(ω) ≡ F (ω) with ϕ = 0 and constant ∆ = Γ K /2.542.Clearly, the effect is already considerable at temperatures of the order of In order to obtain the intrinsic linewidth Γ = 2.542 ∆ of the Kondo resonance in the underlying spectral function, while taking into account a finite temperature T at the tip, we could simply fit (2) to the experimental dI/dV spectra.However, this requires to numerically evaluate the convolution (2) several times during the fitting procedure, leading to considerable computational effort, especially when lock-in modulation is also taken into account (see below).Instead, we now derive an analytic expression for the lineshape of the Kondo resonance in the dI/dV taking into account FD broadening, which allows to process the data very efficiently.A lineshape for the Kondo peak at finite temperature in the spectral function was recently given in Ref. 23.This lineshape reduces to a Frota-Fano lineshape in the T → 0 limit, but can also be well approximated by a Frota-Fano lineshape at finite T [21].We therefore describe the Kondo resonance in the spectral function at some finite temperature T by the Frota-Fano lineshape, A(ω) ≡ F (ω).The Frota width parameter ∆ is determined by the intrinsic halfwidth of the Kondo peak Γ at temperature T according to (1), i.e., ∆ = Γ(T )/2.542.The resulting lineshape for the dI/dV can be expressed analytically in terms of the Hurwitz ζ-function [24]: with the temperature parameter τ ≡ πk B T , as above.The HF lineshape (3) can now be fitted [36] directly to the Kondo resonances in the finite-temperature dI/dV spectra recorded by STS, as shown in Fig. 3(a).We find that it fits slightly better the experimental data than the corresponding Frota fits [24].Fitting the spectra with the HF lineshape yields smaller mean squared errors and the obtained width parameter ∆ is also more consistent with varying the fit range [24].Importantly, the Frota width parameter ∆ resulting from each fit now yields the intrinsic halfwidth Γ(T ) = 2.542∆ of the Kondo resonance for a given temperature T .As a result, the extracted halfwidths do follow the predicted intrinsic temperature broadening and can be fitted well by Eq. 1, as shown by the transparent red line in Fig. 3(b).This is in stark contrast to the halfwidths extracted from the corresponding Frota fits in Fig. 1(b).
Another important contribution to extrinsic broadening in STS experiments comes from the lock-in modulation of the bias voltage.The effect on the spectra can be described by a further convolution of the differential conductance G(V ) [27,30] as where the lock-in function χ m is given by χ rms is the amplitude of the bias modulation.Thanks to the analytic solution (3) of the first convolution (2), this second convolution can be computed numerically efficiently enough to be used in the fitting procedure [36].The resulting Frota width parameter ∆ yields the Kondo width without the extrinsic broadening due to FD smearing or lock-in modulation.Thus, Γ = 2.542∆ now essentially yields the intrinsic halfwidth of the Kondo resonance at a given temperature T since the two most important STM inherent contributions of extrinsic broadening have been removed.Fig. 3(b) shows in blue and green circles the intrinsic Kondo halfwidths extracted in this way, which can be fitted even better by Eq. 1, resulting in a mean square error of MSE HL = 1870 µV 2 compared to MSE H = 2474 µV 2 for the fits using just the HF lineshape.In order to visualize the effect of FD broadening in our experimental dataset (DS1 and DS2), in the Supplemental Material [24] we compare the linewidths from Fig. 1b (Frota fit) with the linewidths from Fig. 3b (Hurwitz fit), emphasizing the importance of removing extrinsic FD broadening.
So far we have shown the advantage of using the HF lineshape in order to directly obtain the intrinsic halfwidth of a Kondo resonance, as well as the validity of Eq. 1 to describe the correct temperature-dependent broadening of the intrinsic halfwidth.The combination of these two expressions is crucial for the verification of a Kondo resonance by its temperature dependence, as illustrated in Fig 3(c) for a simulated data set where a Frota lineshape of constant halfwidth (Γ = 3 mV) is broadened by FD and lock-in modulation (V m = 0.4 mV).Fitting this data set with a Frota-Fano lineshape (blue dots) [24] yields an increasing halfwidth, whose temperature dependence can be fitted well with the empirical expression Γ emp (T ), thus giving the false impression of Kondo behavior.Also the often employed temperature correction of the linewidth Γ corr.= Γ 2 − (1.75k B T ) 2 [16,19,25] does not properly compensate for the FD broadening, as shown by the blue circles in Fig 3(c).Using the HF lineshape, on the other hand, yields the correct constant linewidth of the underlying peak (red circles), revealing its non-Kondo nature.Similar results are obtained for a FD-broadened Lorentzian peak with a constant linewidth [24].The empirical expression Γ emp (T ) is therefore not a suitable proof for Kondo resonances and can only be used as an empirical way to extrapolate the Kondo temperature from such data sets if the presence of the Kondo effect can already be assumed.
On the other hand, if no proof of Kondo behaviour is required, there is no need to perform temperaturedependent measurements in order to determine T K .According to (1) the intrinsic halfwidth Γ at a certain temperature depends only on ∆ K and the (known) experimental temperature.The relation can therefore be rewritten as: where 423 are constants and τ ≡ πk B T the temperature parameter.Using this relation, the Kondo temperature T K = ∆ K /1.542k B = Γ K /3.92k B [29] can be determined accurately from the intrinsic halfwidth of a single spectrum, taken at finite temperature.Fig. 4(a) displays the T K obtained from the individual spectra of data sets DS1 and DS2 in dark blue and green circles, respectively.There is less than 3% of variation between the individually determined Kondo temperatures and the T K obtained before by fitting Eq. 1 to the linewidths as a function of temperature.Similarly consistent Kondo temperatures are found by applying this method to temperature-dependent data sets reported elsewhere [24].The slight deviations from the expected constant behavior of the extracted T K versus temperature may be attributed to experimental reasons, such as incomplete thermalization at higher temperatures [30].
We now apply the developed methodology to investigate the dependence of the Kondo temperature on the adsorption site of phenalenyl on the herringbone reconstruction of the Au(111) surface.Along with data sets DS1 and DS2, acquired on two molecules in the fcc region of the Au(111) reconstruction close to the herringbone ridge (adsorption site labeled as FCC*), we also carried out dI/dV measurements for phenalenyl adsorbed in the middle of the fcc (light blue) and hcp (orange) regions.By fitting each spectrum with a lock-in broadened HF lineshape and calculating T K using (5), we find that different adsorption geometries result in different Kondo temperatures, as clearly shown in Fig. 4(a).This observation can be rationalized by slight changes in the hybridization between molecule and metal substrate and the exponential dependence of the Kondo temperature on the hybridization [37].The robustness of the singlepoint T K determination, here shown for different adsorption sites and on an entire temperature series, justifies the use of this method as a fast and reliable route to estimate T K of a Kondo system [38].
To conclude, we have performed accurate STS measurements of a pure spin-1/2 Kondo system, and developed tools for an efficient and accurate data analysis that properly take into account extrinsic broadening of dI/dV spectra in STS.A key element of the methodology is an analytic expression for the Fermi-Dirac broadened Kondo lineshape in terms of the Hurwitz ζ-function.Fitting the spectra with this Hurwitz-Fano lineshape yields the intrinsic width of the Kondo peak which fits very well with the recently derived expression (1) for the Kondo width as a function of temperature, proving the validity of the theory [23].The procedure developed here allows to unequivocally prove Kondo behavior of a system probed by STS [36].In contrast the established methodology of fitting the empirical expression Γ emp (T ) to linewidths extracted from Frota fits to STS data may give a false impression of Kondo behavior.Finally, our methodology also allows to obtain the intrinsic Kondo width at T = 0 and corresponding Kondo

I. METHODS
The temperature-dependent measurements in this work were carried out in an Unisoku USM-1300 STM with a 320 mK base sample temperature.The hydrogen-passivated phenalenyl precursors were deposited via flash deposition from a silicon wafer onto a clean Au(111) surface held at room temperature.The precursor molecule synthesis and dehydrogenation by atomic manipulation into the target spin-1/2 nanographene is described in Ref. 1. Nanonis electronics were used for STM control and data collection.The tunneling current signal was amplified with a variablegain FEMTO DLPCA-200 preamplifier and low-pass filters were used in all electronic inputs to the STM.STS spectra were measured with standard lock-in detection using modulation voltages of V m = 400 µV (data set DS2) and 500 µV (DS1, HCP and FCC) and, if not stated otherwise, the Kondo resonances in the spectra were fitted with a fit range of ±15 mV.Data sets DS1, HCP and FCC were measured with the same tip apex, whereas DS2 was measured with a different tip.The data was processed with Wavemetrics Igor Pro software, whereas the fit procedures were implemented both in Python and Igor programming language.The fit procedures are open-source and will be made available on GitHub [2].

III. DERIVATION OF THE HURWITZ-FANO LINESHAPE
In this section we calculate the integral of Eq (2) in the main text assuming a Frota peak for the spectral function A(ω) ≡ F (ω).The integral can be written as where β = 1/k B T .Choosing z = 1 + iω/∆ and defining a = ∆β 2 , the integral can be rewritten as along C, see Fig. (S2).Since the integrand is analytic within the region enclosed by the contour C and the composite contour 2) is equal to . The first and third integrals of Eq. ( 3) vanish due to the exponential decay of sech(z) for large real values of z when k → ∞, given their contours of integration.The finite contribution can be written again as a real integral making the change x = −iaz with G = Re(e iϕ I 1 )/2.The contribution along negative x can be expressed as where it has been used that sech(−z) = sech(z) and e ∓iπ/4 = (±i) −1/2 .Therefore with d ± = ∓ia ± aeV /∆.On the other hand, since sech FIG. S2.Illustration of complex contours.The black line, labeled as C, represents a direct path from 1 − i∞ to 1 + i∞.The blue contour comprises three segments: C1 begins at 1 − i∞ and moves leftwards to reach the imaginary axis at −i∞ , C2 ascends along the imaginary axis to i∞ , and C3 proceeds rightwards to conclude at 1 + i∞.
Making use of the well-known complete Fermi-Dirac integral [3,4] where Li s (z) is the polylogarithm function, identifying j = −1/2, t = −2d ± , and taking into account that , we can express the integral as and using Making use of the relation between the polylogarithm and the Hurwitz zeta function we can finally write IV. DEFINITIONS OF THE KONDO TEMPERATURE In the literature many different definitions of the Kondo temperature can be found.In this work we are using Wilson's thermodynamic definition T K [5], corrected by Wiegman and Tsvelick [6], in terms of the magnetic susceptibility χ imp (T ) = 0.68(gµ B ) 2 /4k B (T + √ 2T K ) in the crossover regime (0.5T K < T < 16T K , see Eq. 4.60 in Ref. 7): where g is the electronic g-factor, and µ B the Bohr magneton.On the other hand, in the low-temperature strongcoupling regime (T ≪ T K ), the magnetic susceptibility becomes constant: χ imp (T ) = w (gµ B ) 2 /4k B T K where w is Wilson's number w = 0.4128 (see Eq. 4.58 in Ref. 7).In this regime one often uses the strong-coupling definition of the Kondo temperature T 0 (sometimes also denoted by T L ) instead of T K which is related to Wilson's T K by T K = w T 0 .Thus the magnetic susceptibility in the strong-coupling regimes becomes: In Ref. 8 Wilson's T K is related to the intrinsic T = 0 halfwidth Γ K of the Kondo peak by In experimental work the Kondo temperature is often defined by the empirical formula used to fit the temperaturedependent broadening of the Kondo resonance, where one of two definitions are most often used: First, in the equation for the temperature-dependent width of the Kondo peak originally derived from Fermi liquid theory by Nagaoka et al. in Ref. 9 with α being either fixed (α = π) or used as fitting parameter, yields Γ K = √ 2k B T K,N .Second, a variation of that equation used in several experimental works [1,[10][11][12][13] is given by and thus yields Γ K = k B T emp , i.e. the Kondo temperature in that definition is identical with the intrinsic width Γ K of the Kondo peak at T = 0.The analysis reported in Fig. S3 essentially reveals that the Hurwitz-Fano lineshape fits overall better than the Frota-Fano function, here shown for data set DS2. Panel (c) shows that the mean squared error (MSE), which is a value representing the discrepancy of the fit with respect to the experimental data, is always smaller for the Hurwitz fit than the Frota fit, and that this difference becomes larger for higher temperatures.Notably the dI/dV spectrum acquired at 1.56K, despite being the lowest temperature of data set DS2, was affected by an overall higher noise level, which explains its high MSE value.Panels (a,b) reveal the bias dependency of the residuals, clearly showing that the Frota function fails to fit the raw experimental data in the close proximity of the Fermi level.

VI. FIT RANGE DEPENDENCE OF EXTRACTED HALFWIDTHS
Since the Hurwitz-Fano function describes the lineshape of a dI/dV spectrum measured at finite temperature better than the Frota-Fano function, it is also less dependent on the used fit range.In Fig. S4(a-c) three spectra from data set DS2 were fitted with different fit ranges using both Frota-Fano, as well as lock-in-convoluted Hurwitz-Fano lineshapes.At low temperatures (Fig. S4a) both fits yield similar and consistent results, whereas at higher temperatures (Fig. S4b,c) the Frota fit overestimates the halfwidth, especially for small fit ranges.The Hurwitz fit on the other hand yields a much more constant halfwidth.The small increase of the halfwidth obtained by Hurwitz-Fano at smaller fit ranges (< 10 mV) can be attributed to a small contribution of noise broadening.
Fig. S4(d) displays the halfwidths obtained by Frota and Hurwitz fits applied to a simulated data set of a Kondo resonance with Γ K = 3.0 meV and V m = 0.4 mV.Similar to the experimental results, the Frota fit yields a strong overestimation of the halfwidth and varies strongly with the fit range, whereas the Hurwitz fit yields constant halfwidth.Please note that the obtained halfwidth by the Hurwitz fit is not Γ K but Γ according to Eq. ( 1) of the main text.

B. Lorentzian lineshape
If we consider a typical non-Kondo lineshape, i.e. a Lorentzian, and add, both, the Fermi-Dirac broadening and voltage modulation (V m = 0.4 mV), we obtain the simulated data reported below.If we fit the spectra with a Hurwitz * Lock-in lineshape, the obtained halfwidth will not yield a constant value (as expected).Instead it even even decreases at higher temperature.Therefore it does not follow the expected temperature evolution of a Kondo peak as described by eq. ( 1), and it is obvious that the zero-bias resonance is not a Kondo peak.A fit with a Frota-Fano lineshape on the other hand, does result in an increase of the halfwidth with temperature and can therefore lead to a misattribution as a Kondo peak, even when the often used temperature correction term Γ corr.= Γ 2 − (1.75k B T ) 2 is used.3a of main text).Red solid lines display fits of Γ(T ) (Eq. 1 in main text) to the three adsorption sites.
The experimental dI/dV spectra were first fitted according to Eq. ( 4) (Hurwitz * Lock-in), and the obtained intrinsic HWHMs (shown in Fig. S9a) were then fitted to Eq. ( 1).We also prove the validity of the shortcut method (Fig. S9b), where, for each intrinsic HWHM the corresponding T K is calculated according to Eq. ( 5) and compared to the value (dashed blue line) obtained from the fit to Eq. ( 1).Coincidentally the fitted Kondo temperatures of datasets in Ref. [11] and Ref. [16] are approximately equal.
Figure S9a convincingly demonstrates the validity of Eq.( 1) for four distinct systems and temperature windows with respect to T K .Panel b further justifies the use of the shortcut method to rapidly evaluate T K from a single dI/dV spectrum, and as a quick test to check if the system is properly thermalized during a temperature-dependent measurement.

FIG. 1 .
FIG. 1.(a) dI/dV spectrum of phenalenyl from DS2 at T = 1.56K (black dots) showing the Kondo resonance, fitted by Frota-Fano lineshape (green line).Insets I and II show the theoretical and experimental spatial distribution of the unpaired electron [19].(b) Kondo linewidths extracted from Frota fits to dI/dV spectra of DS1 (blue circles) and DS2 (green circles) versus temperature, and fits to Eq. 1 (red solid line), and to Γemp(T ) with fixed α = π (grey solid line) and with α as fit parameter (grey dashed line, α = 7.45).Error bars show the estimated standard deviation.

FIG. 2 .
FIG. 2. (a) Schematic STM setup for measuring dI/dV spectra of a molecule (M) on a surface (S) by an STM tip (T).(b) Simulated dI/dV spectra assuming a Frota-Fano lineshape in the spectral function at different temperatures.(c) HF lineshapes (red solid lines) according to Eq. 3 at finite temperature T = 0.4 ΓK/kB and corresponding Frota-Fano lineshapes (blue dashed lines) in underlying spectral function for different values of ϕ and fixed halfwidth ΓK = 2.542 ∆K.

FIG. 3 .
FIG. 3. (a) dI/dV spectra of DS1 and DS2 (black dots) fitted with HF lineshapes according to Eq. 3 (blue and green solid lines).(b) Intrinsic Kondo linewidths obtained from HF fits (transparent blue and green circles) versus temperature, fitted with Eq. 1 (transparent red line) and with Γemp(T ) with fixed α = π (black dashed line).Solid blue and green circles show intrinsic linewidths obtained from fitting with Eq. 4, taking into account both FD and lock-in broadening, fitted with Eq. 1 (solid red line).Vm = 0.5 mV for DS1 and Vm = 0.4 mV for DS2.Error bars show the estimated standard deviation.(c) Halfwidths versus temperature obtained from Frota (blue dots) and HF (red dots) fits for simulated data of Frota peak with constant intrinsic linewidth.Also shown are temperature corrected Frota widths (blue circles) using Γcorr.(see text).Grey lines show fits with Γemp(T ) resulting in α = 5.32 (Frota) and α = 5.02 (Frota corr.)

FIG. 4 .
FIG. 4. (a) Kondo temperatures TK(T ) determined for each dI/dV spectrum individually by HF * Lock-in fits (Eq.4) and using the resulting intrinsic linewidth Γ in Eq. 5. Error bars represent the estimated standard deviations.Red dashed lines indicate TK obtained by fitting the temperature dependence of the Kondo linewidth for each adsorption site with Eq. 1 [24].(b) STM images of phenalenyl molecules on Au(111) with boxes highlighting the investigated molecules.Scanning parameters for top image are V = −2 V, I = 100 pA and V = −0.1 V, I = 50 pA for the bottom image.Ridges of the herringbone reconstruction are marked with dashed white lines.
FIG. S3.Residuals analysis of Frota and Hurwitz fitting of DS2 spectra.(a,b) Residuals of Frota and Hurwitz fits, respectively, of the spectra acquired in the temperature range 1.56 K -7.50 K. (c) The MSE is plotted in the top graph as a function of the temperature for both Frota and Hurwitz fits, while the bottom graph shows their difference as a function of temperature.
FIG. S4. (a-c) Halfwidth of three dI/dV spectra from data set DS2, determined by fitting a Frota-Fano (blue) or lock-inconvoluted Hurwitz-Fano (red) lineshape using different fit ranges.The error bars represent the estimated standard deviation.(d) Fit-range dependent halfwidth of a simulated Kondo spectrum (ΓK = 3.0 meV, Vm = 0.4 mV), obtained by Frota (red) and Hurwitz * Lock-in (blue) fit.
FIG. S5.Simulated temperature-dependent dI/dV spectra of a Frota-Fano lineshape with constant intrinsic halfwidth (black lines).Blue lines in (a) and red lines in (b) display Frota and Hurwitz * Lock-in fits, respectively.

FIG. S6 .
FIG. S6.Simulated temperature-dependent dI/dV spectra of a Lorentzian lineshape with constant intrinsic halfwidth (black lines).Blue lines in (a) and red lines in (b) display Frota and Hurwitz * Lock-in fits, respectively.(c) Halfwidths versus temperature obtained from Frota (blue dots) and Hurwitz (red dots) fits to a simulated data set without intrinsic broadening, as well as the often used temperature correction term Γcorr.= Γ 2 − (1.75kBT ) 2 for the Frota halfwidth (blue circles).Grey lines show fit with Γemp(T ) resulting in α = 4.83 (Frota) and α = 4.51 (Frota corr.) FIG. S7.Kondo linewidths obtained by fitting the joint (DS1+DS2) dataset with a Frota function (in red) and Hurwitz function (in blue) as well as the often used temperature correction term Γcorr.= Γ 2 − (1.75kBT ) 2 for the Frota halfwidth (blue circles).The Figure highlights the importance of removing the Fermi-Dirac broadening, which, for this specific system, has a significant effect for T ≥ 2 K.

TABLE I .
TableIdisplays the different definitions of the Kondo temperature and their numerical conversion factors, including the halfwidth Γ K of the Kondo peak.Numerical conversion factors between different definitions of the Kondo temperature, where TK is Wilson's definition, T0 the strong-coupling definition (sometimes denoted TL), TK,N the definition introduced by Nagaoka et al.[9], ΓK is the halfwidth of the Kondo peak and kB = 8.6173 • 10 −2 meV/K is Boltzmann's constant.The quantities given in the leftmost column can be expressed by the quantities given in the top row multiplied by the conversion factor given by the corresponding table entry, e.g., kBTK = 0.2551 × ΓK.V. COMPARISON OF FROTA AND HURWITZ FITS OF DATA SET DS2