Inelastic tunneling

In the fictitious case displayed in figure 3.10, the elastic processes are described by the linear increase of the tunneling current with a bias voltage up to an energyV_B= 8 meV at which point the tunneling electron can loose its energy∆= 8 meV to the system and excite it. An example of such an excitation would be the excitation of a spin system [71] [72] [5] or the onset of vibra-tions or quantized rotavibra-tions of a molecule [73]. At this voltage, the inelastic pathway provides another tunneling path and adds to the overall tunneling current thus increasing its slope (see figure 3.10a). The number of datapoints is important as they can also lead to artificial broad-ening effects as seen for the 30 simulated datapoints used in figure 3.10 where each datapoint corresponds to 1 meV.

Tunneling currentI_T/ nA

Bias voltageV_B/ meV

Figure 3.10: IETS signal simulation in the tunneling current (a) and its two derivatives (b and c) with respect to the bias voltage. The onset of inelastic excitations occurs at 8 meV and adds to the elastic contribution of the tunneling current. In the first derivative this corresponds to a step and in the second derivative to a peak. Here the broadening of the steps and peaks is due to the number of datapoints (one datapoints = 1 meV) that is too small.

A real data example is shown in figure 3.11 where we use 400 datapoints for a total voltage interval of 30 meV with each point corresponding to 0.075 meV. The noise in the tunneling current due to low frequency contributions does not allow for a smooth dI/dV signal if a simple numerical derivative is applied to the tunneling current data. The lock-in signal was simulta-neously recorded and shows a much higher signal to noise ratio than the numerical derivative of the tunneling current. Thus, we are employing a lock-in detection to enhance the signal. A small sinusoidal voltage with an amplitude,V_m, and a frequency, f, of around 600-800 Hz is added to the bias voltage, i.e.: eV_B +eV_msin(2πf t). The amplitude of this voltage is equal to the energy resolution that should be as small as possible compared to the thermal and elec-tronic broadening within the tunneling circuit. The sinusoidal voltage of the lock-in results

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Figure 3.11: IETS signals with different signal to noise ratios for a CoHS = 1 dataset with two inelastic excitations at around 1 meV and 5 meV. The direct numerical derivative of the tunneling current (a) leads to a noisy spectrum as seen in (b). The lock-in signal (c) that directly measures the slope of the tunneling current signal provides a much better signal to noise ratio and is shown in comparison to the same dataset.

in an oscillation of the tunneling current with the same frequency. The resulting amplitude, however, is proportional to the slope,I_T/V_B, as can be seen from the first Taylor expansion term when the upper limit of the integral(eV_B)is replaced by(eV_B+eV_msin(2πf t))in equation 3.11.

The modulation voltage of the lock-in signal is usually phase shifted so that the the signal and its noise contribution can be distinguished from each other by taking the lock-in signal as a reference signal. Therefore the lock-in signal is a direct measurement of the first derivative of the tunneling current with an improved signal to noise ratio. However, also the modulation of the lock-in introduces an artificial broadening of the energy resolution [70] in which a δ-function peak in the DOS is broadened to a half-sphere with a width of2eVm. Also the finite temperature introduces a thermal broadening of such aδ-function peak in the tunneling current to a Gaussian peak with a FWHM of3.2k_BT in the lock-in signal. In terms of inelastic excita-tions the thermal broadening of both the sample and the tip states alone results in a FWHM of 5.4k_BT (0.52 meV for T = 1.1 K⁵) of the peaks in the derivative of the lock-in signal. The peaks in figure 3.12b and c have a FWHM of around 0.8 meV to 1 meV and thus additional effects such as the already mentioned lock-in broadening or the real temperature at the tunnel junction, which is usually hard to measure except with a superconducting sample, must be responsible for this additional broadening of around 0.4 meV.

Another effect that could have an impact on the broadening is the actual lifetime of the ex-cited states of the spin system. For magnetic atoms on surfaces the measured lifetimes are usually in the order of ps to ns, e.g. [74]. According to the Heisenberg uncertainty principle,

5After an upgrade to our pumping line we are consistently at 1.1 K. The datasets presented in the next subsec-tions of this chapter were taken at T = 1.4 K.

an energy uncertainty of up to 2 meV (∆E∆t ≥ ^¯h₂) is possible for 0.1 ps and could therefore lead to quite a dominant contribution to the overall broadening of the inelastic peaks. In our simulations all these broadening effects are included in an effective temperature,Tef f, that is usually around 2 K instead of the measured temperature of 1.1 to 1.4 K in experiments.

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dI/dV/ arb. units

Bias voltage / meV Bias voltage / meV Bias voltage / meV

0

1

a b c

Figure 3.12: Example of the energy step determination of a typical CoH S = 1 dataset and the step broadening in experiment (267 points for a total bias voltage range of 20 meV). The noisy grey curve in (b) is the raw point by point derivative of the lock-in signal in (a). This derivative is smoothed by a Savitzky-Golay method (red curve) in 1st polynomial order taking ten neighboring datapoints into account. (c) Four Gaussian fits (blue curve) to the smoothed curve determine the energy positions of the inelastic excitations and their errorbar. From left to right: -4.16 meV (FWHM = 1.06); -1.22 (FWHM = 0.84); +1.35 (FWHM = 0.87); +4.29 (FWHM = 1). As the excitations are symmetric in energy a voltage offset is accounted for by taking the average out of both, the negative and the positive values. In this case the first excited state has an energy of₂ = 1.29meV and the second excited state an energy of₃ = 4.23meV.

From these values and the energy equations defined in the previous subsections we can deduce the magnetic anisotropy parameters: D=−3.58meV andE = 0.65meV.