3.2.1 Topography
For topographic STM images the tunneling current is used to reconstruct a profile of the surface. Basically there are two approaches, the constant current and the constant height mode [155]. In the constant height mode the tip is moved at a constantz-value and records the (exponentially) varying tunneling current to visualize the surface [Fig-ure 3.4 (a)]. In the constant current mode on the other hand a feedback loop is used to keep the tunneling current constant by adjusting thez-position of the tip [Figure 3.4 (b)].
While this approach has the advantage of avoiding tip crashes, we usually have a mixture of both modes in reality controlled by the loop gain value, which determines the speed at which the feedback loop reacts to the change in tunneling currentI. Topography im-ages are recorded using highest possible loop gain parameters and thez-piezo position is used as a height measurez(x,y).
The height measured in STM does not necessarily correlate with the topography of the sample. Therefore it is often calledapparent height. In case ofs-wave tip functions, the matrix element is proportional toψ(~r0) and thus the total tunneling current is the integrated DOS contour [equation 3.5] within the integration rangeEFtoEF+eV. When it comes to more complex tip states, the measured tunneling current depends on deriva-tives of the sample wave function and thus imaging can drastically increase the ampli-tude of the measured topography [152]. Furthermore asymmetric tip orbitals, such as thedzx etc. can lead to an asymmetric imaging of the surface, an effect occasionally ob-served in the STM experiments presented later.
3.2 Measurement modes and experimental considerations 41
Figure 3.4 | Scanning to-pographies with STM. Con-stant height (a)versus constant current (b). In constant current mode different loop gain settings define the accuracy of the traced contour.
When imaging surfaces that consist of strongly varying LDOS, e.g. a graphene flake edge on top of a metal substrate, the STM topography might even show inverted height values compared to real topography. Similarly, features in the STM topographic image might look like a "hole", nonetheless representing an overlying adsorbate but with con-siderably smaller DOS.
3.2.2 I(V)spectroscopy and density of states
With STM the LDOS of the sample can be acquired by positioning the tip at a certain lo-cation (x,y) of the sample surface and performing anI(V) measurement. The tip to sam-ple separation is adjusted using a stabilization voltageVstaband a stabilization tunneling currentIstab. After opening the feedback loop, the tunneling currentI is recorded while sweeping the bias voltageV. A subsequent numerical derivation of the curve would give as a result the differential conductance or dI/dVsignal, which is proportional to the den-sity of states of the sample (under previously discussed assumptions of low bias voltage and low temperature):
dI
dV(V)∝ρsample(EF+eV) (3.12)
Commonly, the dI/dV(V) signal is acquired simultaneously with theI(V) curve making use of the lock-in detection technique.
Lock-in detection
The basic lock-in setup for dI/dV spectroscopy is depicted in Figure 3.5 (a). A modula-tion voltageV(t)=V˜modcosωtis added to the bias voltageV. This leads to a modulation of the detected tunneling currentI(t)=I(V+V˜modcosωt) in the STM tunnel junction. In the lock-in amplifier, the modulation signal is shifted by a phase factorϕand multiplied with the tunneling current and sent through a low pass filter. The lock-in output can be described in the form [158, 145]
1
(a)
Figure 3.5|Lock-in detection. (a)Principle setup for lock-in detection of dI/dVsignals.(b)Example of anI(V) curve and the corresponding dI/dV signal. Parameters for spectroscopy:Vstab=500 mV, Istab=200 pA,Vmod=4 mV,T=10.3 K,τ= 100 ms,fmod= 672 Hz.
with the integration timeτ.
In the case of a small modulation voltage a Taylor expansion of the tunneling current can be performed aroundV [155]:
I(V+V˜modcosωt)=I(V)+ dI
dV(V)·V˜modcosωt+O¡
( ˜Vmodcosωt)2¢
(3.14) Plugging this into the averaging function above, one can see that the output will cancel all signals with periodicity other than the reference frequencyωand the output of the lock-in will be proportional to dI/dV. This is convenient, because we know from the theory of electron tunneling that under certain assumptions the differential conductance dI/dVis proportional to the DOS at an energyE=EF+eV.
The additional phase shiftφadded to the reference signal is of experimental impor-tance, since the complex capacity of the tip-sample system will give rise to a phase shift of the measured tunneling current, which has to be compensated. Details can be found in [158].
An example of an I(V) curve and the corresponding dI/dV(V) signal is depicted in Figure 3.5 (b). The bias voltageV is always given with respect to the sample, such that negative bias voltages give the density of the occupied sample states. The advantage of the lock-in detection compared to a numerical derivation is the higher signal to noise ratio. Since the modulation frequency can be chosen such that strong noise signals are far away from the reference frequency, mechanical and electrical noise sources in the setup can be eliminated [158].
Due to the smearing of the Fermi distribution and the lock-in technique, the exper-imental energy resolution is limited by the two quantities temperatureT and the root mean square (rms) value of modulation voltageVmod=V˜mod/p
2 according to [159]:
∆E≈ q
(3.3kBT)2+(1.8eVmod)2 (3.15)