Measurement and Data Analysis Methods
3.5. The AC-technique
Insulating oxide layer: The surface of a metallic or semiconducting sub-strate, like silicon, always reacts with atmospheric oxygen and an oxide layer (rust) is grown on the surface of the substrate. Good substrate preparation reduces but does not diminish this layer, which in the case of SiO2has a thicknessLSiO2≈200nm and a dielectric constant SiO2=4.5.
Substrate: The substrate is made of silicon with a dielectric constant Si=11, having a thickness ofLSi=0.5mm.
For serially connected capacitors, the total inverse capacitance for TDES-bottom plate is the sum of the inverse component capacitances
1
where the various subscriptedCs on the right hand side can be substituted according to eq. (3.13), respecting the fact thatandLshould also bear the appropriate subscript.Con the left-hand side is substituted without subscripts.
Then, the quotient/L(or actually its inverse) is obtained, L
Saturated electron densitynscan now be readily calculated from eq. (3.14), substituting/Lfrom the previous equation. Notice thatL/≈LSi/Si, because the thicknessLSiof a substrate is normally several orders of magnitude greater thanLHeandLSiO2.
3.5. The AC-technique
A major difficulty in the experimental study of TDES on liquid helium—
in contrast to electron systems in heterostructures—is that it is impossible to measure their electric properties by making physical contact through wires or otherwise. Metallic electrodes in proper arrangement, placed under the part of the helium surface to be charged with electrons, are used instead. The electrode sides not facing the helium surface are connected with coaxial cables that lead outside the cryostat.
One of the electrodes (thedriving electrode) is then set to an AC-voltage excitation and, as a consequence, the potential on the helium surface over the electrode alternates. This causes electrons to be alternatively drawn in and out of that part of the helium surface, and charge density fluctuations (an AC-current) develop on the electron sheet. A second electrode (thepickup electrode) responds to those fluctuations and collects the AC-current that flows through the electron system. In an effort to reduce the cross talk between driving and pickup electrodes, a third electrode is usually placed between them.
Obviously, the measuring process just described (AC-technique) depends critically on the alternating voltage excitation, because otherwise no charge fluctuations will be induced and the electron system will remain inert.
3.5.1. Signal detection with a lockin amplifier. A lockin amplifier was used in the experiments of this thesis as source and detector of AC-voltage or cur-rent excitations. As a source, the lockin amplifier delivers an AC-excitation with well defined amplitude, phase and frequency to the driving electrode, whereas as a detector it measures the AC-response at thesamefrequency (but eventually different amplitude and phase) from the pickup electrode.
In an ideal setting, any differences in the amplitude and phase of the response would have been caused exclusively from the TDES, and it is then only a matter of theoretical ingenuity to extract information about the state of the electron system.
The lockin amplifier can measure either the AC-voltage at the pickup elec-trode, or the AC-current that flows through the driving and pickup electrodes.
There exist two modes of operation, affecting the way voltage (or current) is displayed and stored:
RTheta-Mode: The amplitudeRand phaseθof the signal (current or voltage)Reiθis measured.
XY-Mode: The in-phaseXand out-of-phaseYcomponent of the signal (current or voltage)X+iYis measured.
The difference between the two modes lies in the coordinate representation, which is polar in(R, θ)-mode and cartesian in the(X, Y)-mode, not in their information content. Therefore, these modes are completely equivalent and one can shift mode by means of the transformation equations
X=Rcosθ,
Y=Rsinθ. (3.17)
The lockin amplifier is characterized by an impedanceZ0due to the resis-tanceR0and capacitanceC0of its internal circuitry. ImpedanceZ0depends also on the cyclic frequencyωof the AC-excitation and it is calculated from
Z0= R0
1+ (ωR0C0)2+ i ωC0
. (3.18)
Ohm’s law, applied for the measuring circuit of the lockin amplifier, states that
Vex=IZ0. (3.19)
A currentIflows through the bipolar element whose unknown impedance Zis to be measured. Ohm’s law, applied for this element, states that
Y= I
V, (3.20)
3.5. THE AC-TECHNIQUE 69
whereYis the admittance8of the element. By definition, admittance is written in terms of conductanceGand capacitanceCas
Y=G+iωC. (3.21)
3.5.1.1. Current mode. Assume that the lockin amplifier is set in xy-mode, measuring the currentI=X+iY. Then, the voltage in eq. (3.20) is actually the excitation voltageVexand by replacingYwith the right-hand side of eq. (3.21), one arrives at the expression
X+iY=VexGx+iVexωCx, (3.22) whereby the conductanceGxand capacitanceCxof the unknown bipolar element that is connected to the lockin amplifier are readily obtained as
Gx= X Vex
, ωCx= Y Vex
. (3.23)
It should be noticed that the resistance and capacitance of the lockin am-plifier (R0,C0) do not enter into the expressions forGxandCx. This is due to the fact that onecommoncurrent flows through the whole circuit, and this is the biggest advantage of operating the lockin amplifier in current mode instead of voltage mode.
3.5.1.2. Voltage mode. If the lockin measures voltageV=X+iYinstead of current, the electric currentIis substituted from eq. (3.19) to eq. (3.20) to obtain
V= Vex
Z0Y. (3.24)
ReplacingZ0andYfrom eqs. (3.18) and (3.21) respectively into eq. (3.24) and after some algebra, one arrives at the expression
V= VexR0
1+ (ωR0C0)2Gx−Vex
C0Cx
!
| {z }
X−component
+i Vex
ωC0Gx+ ωR0Vex
1+ (ωR0C0)2Cx
!
| {z }
Y−component
.
8Admittance is defined as the inverse of impedance and, likewise, conductance is the inverse of resistance.
A2×2system of equations results whose solution, in terms of the unknown conductanceGxand capacitanceCx, is
Gx= The resulting expressions in the case where the lockin amplifier measures voltage, eqs. (3.25a) and (3.25b), are clearly much more complicated than the corresponding expression for current, eq. (3.23), to the extent that one wonders why would someone want to set the lockin amplifier in voltage mode.
An answer is that many older lockin amplifiers do not come with the facil-ity for current measurement. This is the case for the models EG&G-5510 and Ithaco-5219, which were used in some of our experiments.
However, the most important reason is that normally acoupling circuit is used for measuring the conductance of a TDES. In this case, the resulting expressions for the admittance and capacitance are of comparable complexity both for the current and for the voltage mode and the advantage of current mode is lost (see section 3.5.3).
3.5.2. Electrode geometries. A simple electrode geometry that was used already since the earliest transport measurements [101] is the rectangular or Sommer-Tanner arrangement, show in fig. 3.7B. It comprises of three consecutive rectangular electrodes and it proves advantageous in localization experiments, because a preferred direction of current flow from driving to pickup electrode can immediately be defined. On the other hand, a preferred direction also implies a lack of symmetry, a fact that complicates the analysis significantly and is the main disadvantage of this arrangement.
A highly symmetrical electrode geometry is the Corbino arrangement (fig. 3.7A), where the electrodes are three concentric rings. The azimuthal sym-metry is valuable when a magnetic field is present and it reduces the complexity of data analysis. Corbino geometry is used extensively in experiments on weak electron localization and for studying the properties of edge magnetoplasmons (see the relevant chapters in [10].
3.5.3. Coupling circuit. The application of a holding field that would push the electrons toward the helium surface is imperative in experiments. Such a field can be produced either by applying a negative DC-voltage on the top plate, while keeping the bottom plate grounded, or by applying a positive DC-voltage
3.5. THE AC-TECHNIQUE 7 1
(A) Corbino (B) Sommer-Tanner
Figure 3.7. Sketch of a Corbino-type (A) and Sommer-Tanner-type (B) electrode ge-ometries. White circles indicate conducting contact areas, where cables can be soldered.
In (A), the outer ring is divided into 8 independent subelectrodes. In (B), the driving and response electrodes are divided into 3 independent subelectrodes. This is done for better flexibility in measurements.
Rb Rb Rb
Cb Cb Cb
Co
Ro
VDC
VAC
Figure 3.8. Sketch of the three electrodes (top), the coupling circuit and the lockin amplifier. The coupling circuit consists of three resistors (R
b) and three capacitors (C
b), connected to a voltage source that sets the electrodes on a positive offset DC-voltageVDC. The AC-voltage excitationVACof the lockin amplifier is superimposed onVDCwithout being affected thereby.
on the bottom plate, keeping the top plate grounded. It proves experimentally convenient to use the second possibility.
However, the bottom plate is also used for the application and detection of an AC-voltage excitation via the lockin amplifier. A proper electric circuit must then be designed that would allow an AC-excitation voltage to pass through the electrodes and at the same time set its offset from the ground of the lockin amplifier. A circuit with that function is depicted in fig. 3.8 and it will be referred in the following as the “coupling circuit”.
Each of the three electrodes (or, for that matter, the coaxial cables that lead to them) is connected with a resistorRb. The free ends of the resistors are joined together. An offset DC-voltage can thereby be created, if the active pole of a bipolar voltage source is connected to the resistors’ common end (the neutral pole must be at the same ground with the lockin).
Moreover, each electrode is connected with a capacitorCb. The free end of the capacitor on the response electrode is plugged in the excitation output of the lockin amplifier, whereas the respective capacitor of the detecting electrode will ultimately lead to one of the lockin channels. Since capacitors are interrupts in DC-voltage or current, the AC- and DC- components do not interfere with each other and only the AC-component will be measured by the lockin amplifier.
If the helium surface is not yet charged with electrons, AC-current will pass through a (Cb−2Rb−Cb) circuit and it will be measured as background current. Ideally, this current should be zero, or much smaller than the current expected to flow though surface electrons, because otherwise a weak electron signal might not be accurately resolved or even distinguished.
The demand of a background current close to zero must mean a very high impedance of the open coupling circuit that, in turn, suggests a high resistance Rband a low capacitanceCb. ResistanceRband capacitanceCbshould be also higher than the resistanceReand capacitanceCeof the electron layer, usually of the order of108Ωand10−13F respectively.
Following a procedure completely analogous to the procedure that led into eq. (3.23), it is found out that in the presence of a coupling circuit the conductance Gxand the capacitanceCxof a TDES will be given by
Gx= (ωCb)2 VexX−RbX2−RbY2
(VexωCb−ωRbCbX−Y)2+ (X−ωRbCbY)2, (3.26a) ωCx=ω2Cb VexωCbY−X2−Y2
(VexωCb−ωRbCbX−Y)2+ (X−ωRbCbY)2. (3.26b) In voltage mode, the expressions forGxandCxin the presence of coupling circuit, are derived from the respective expressions eq. (3.25a) and eq. (3.25b), by making the substitutions
R−→R0+Rb, 1 C−→ 1
C0
+ 2 Cb
(3.27) It should be noticed that the simplicity of the current mode expressions for GxandCx, exhibited in eq. (3.23), disappears when a coupling circuit is present.
Therefore, a coupling circuit makes both modes of operation similarly complex for the purposes of data analysis.
3.5.4. Temporal resolution. The time constantτe=ReCeof TDES on liquid helium is expected to be about10−5s, a time that can be seen as the maximum temporal resolution of the system. Phenomena that last less than