The measurement of filling the gate barrier well can be done with two different methods.
The first method has the following steps:
a) setting the electrode potentials and switching on the filament and see wether we have current at the pick-up electrode,
b) setting the gate voltage such as we observe a maximum current at the pick-up and registering this value because we will use it to open the gate later,
c) switching off the filament to start with empty source,
d) applying a certain negative potential to the gate and switching on the filament again,
e) the pick-up current will remain at zero and after some time (charging time) will increase because the electrons spill over the barrier,
f) switching off the filament and waiting for some time (saving time), g) opening the gate and measuring the pick-up current.
Once we opened the gate we have seen a peak at the pick-up current as a function of time (see Fig. 5.11). The total amount of saved charge can be calculated from the area under this peak. These steps were repeated for different gate potentials. In this experiment both, an electrometer and storage oscilloscope were used to measure the pick-up current.
The second method can be described in two steps:
a) storing some electrons at the source by having a relatively high gate barrier by applying a certain voltage to the gate electrode(for example -5V),
b) ramping the gate (opening the gate stepwise) from -5V to 11V (1V/5second) and observing the pick-up current (see Fig. 5.13.a).
By applying a gate voltage ofVgate and fill the resulting potential well (see Fig. 5.10):
due to the substrate potential distribution the electron density is not constant, so if we assume a parallel plate capacitor model, where the charge density goes linearly with the applied potentialne is a function of x:
ne(x) = 012V(x)
(1d2+2d1)e = 012K(xmax−x)
(1d2+2d1)e (5.1) where0,1,2,d1,d2 and e are, the vacuum permittivity, relative silicon oxide
permit-Figure 5.10: Schematically, the electron distribution at the source region. Here the source has a triangular area infront of the channel until xmax = 2mm. The electron density increases from left to right.
tivity, relative helium permittivity, silicon oxide thickness, helium film thickness and elementary charge respectively. Due to the potential gradient across the substrate the electron density should increase, from left to right, across the source area as in Fig.
5.10). So in our potential gradient, the electron density increases linearly from left to right(xmax is the point at which the density = 0). In the diagram we plot what essen-tially the electrochemical potential of the electron system is. The underlying potential acts as a positive background which balances the Coulomb energy.
So we can calculate the total number of electrons by integrating from 0 to xmax and
multiplying by a width W(x):
In the case of rectangular source area W(x)=constant=W and so:
Ne = 012KW
forxmax >2mm, W(x)= constant=W, so
Ne=A(Vgate−V0)2−B(Vgate−V0) +C (5.8) where A, B, C are constants which depends on the source, drain potentials and the potential of gate barrier.
Fig. 5.10 can be explained in the following way: the first passage of the electrons which comes from the filament goes directly to the front of the channel where we will have the maximum density and the next passage distributed behind the first electrons and so on until the well is completely full with 2-D electrons. The number of electrons which comes after that spills over the barrier because the well is already full. The reason of why the energy of the electrons should increase in this well is the Coulomb interaction
Figure 5.11: Measurement of filling the resulting well of the gate barrier forVgate= -2V.
t1 is the charging time andt2 is the saving time. SO is spilled over electrons and SE is saved electrons.
between the dipoles. Fig. 5.11 shows the measurement of filling the resulting well of the gate barrier starting from empty source. Here -2V was applied to the gate and the filament was switched on. The charging time t1 = 10s until the electrons spilled over the barrier and then the filament was switched off. After saving time (delay) oft2= 7s the gate was opened and the peak of the saved electrons was observed at the pick-up.
Using one of the triangular source design samples,sample No.8, withVs−d= 2V(Vs = 10V, Vd = 12V) according to our model, the gate barrier vanishes at Vgate = 11V so we have a barrier even for Vgate = 0 because of the potential distribution, and so the total number of electrons is directly proportional to (Vgate−11)3 ifxmax ≤2mm and have the formA(Vgate−V0)2−B(Vgate−V0) +C ifxmax >2mm(see Fig. 5.12 a)and b)).
This measurement was also repeated using the second method of filling the gate barrier well under the same conditions (by applying a certain gate voltage (for example -5V) and opening the gate stepwise from -5V to 11V). The driving potential used is also Vs−d= 2V (Vs= 10V,Vd= 12V). As we can see from Fig. 5.13 a) and b) the number of saved electrons also agree with our model. The experimental results show a good
Figure 5.12: Forsample No.8: a)Pick-up current as a function of time for different gate voltages. b)The number of saved electrons as a function of gate voltage. This graph has two functions which contain the two contributions of the triangular and the rectangular area. The number of saved electrons as a function of gate voltage is in agreement with our model.
agreement with the theoretical calculations.
We did another experiment to understand this model using different design source
Figure 5.13: For sample No.1: a) Pick-up current as a function of time by ramping the gate voltage. b)The total number of saved electron as a function of gate voltage. As mentioned before this graph has two functions which contain the two contributions of the triangular and the rectangular area. The number of saved electrons as a function of gate voltage is in agreement with our model.
area shape, rectangular source design He-FET, sample No.4. In this experiment, the second method of filling the barrier well was used. Some electrons at the source
have been stored by applying a certain negative voltage to the gate (-6V) and then the gate was opened stepwise from -6V to 0V. Vs−d = 1V (Vs = 1V, Vd = 2V) and the barrier vanishes at Vgate = 1.5V, the total number of electrons for this sample is directly proportional to (Vgate−1.5)2 which is in agreement with our model (see Fig.
5.14 a) and b)).
Figure 5.14: Forsample No.4: a)Pick-up current as a function of time by ramping the gate voltage. b) The total number of saved electrons as a function of gate voltage. The number of saved electrons for this sample is directly proportional to (Vgate−1.5)2 also in agreement with our model.
These results are reproducible only for a limited gate voltage and for a limited number of saved electrons. Beyond these values nonreproducible phenomena are observed which are probably due to a break-through of the electrons and might be related to Wigner crystallization(see chapter 7).
Electron transport measurements
The electrons on liquid helium are well suited not only for investigations of quasi-infinite 2DES, but also for studies of electrons in nanostructured confinement. On the basis of the previous results more systematic measurements are carried out in order to study the electron transport through nanostructures in more detail. For example, the sample which is used by David Rees and Kimitoshi Kono recently [RK10] resembles our sample, and the difference is in the experimental technique. They alway start their measurements from an equilibrium state, i.e. the electron density on both sides, the source and the drain, has the same value, whereas in our case we always start from a certain electron density on one side (source) and zero electron density on the other side (drain). So the two techniques are complementary to each other. In this chapter discuss the results obtained with our technique.
In our experiment the electron detector which measures the pick-up current is a normal electrometer which is not fast enough to resolve the transport of the electrons in time.
So the idea of pulsing the gate is used to study the electron transport on helium films in both the source area and the channel. This study can be done by using a rectangular positive pulse to open the gate for a very short time (with pulse width in the range of 10ns to 10ms) and determine the number of passed electrons through the channel and the remaining electrons on the source. The passed and rest electrons can be easily observed at the pick-up (see Fig. 6.1). In section 6.1 the number of passed and rest electrons at the source after pulsing the gate are defined and discussed. The mobility calculations are discussed in section 6.2.
6.1 Passed and rest electrons
The number of electrons which can pass from source to drain through the channel can be easily observed at the pick-up. We control the number of passed electrons with the opening time of the gate channel. The number of rest electrons at the source after pulsing the gate is also an interesting quantity because the summation of passed and rest electrons should give the total number of saved electrons for a certain charging time as shown before in section 4.2. Fig. 6.1 shows both, the passed and the rest number of electrons at the source for a long channel (heresample No.6was used).
As we can see from Fig. 6.1 a) and b) the passed electrons increases with the pulse width until we have a saturation because then the total number of saved electrons supposed to be passed. Accordingly the rest electrons should decrease because as men-tioned before the total number of passed and rest electrons is constant. The decrease of the rest electrons seems to be an exponential function which can be used to develop a model of relaxation time of the electrons at the source. When we plot the number of rest electrons on a logarithmic scale, we get a more and less linear dependence, see Fig. 6.2.Here one could imagine that the source is a capacitor which is charged with a charge Q = N e, where N is the number of rest electrons at the source, and the channel is a resistor with a constant resistance R. The simple combination of these two components can be illustrated as an RC circuit as shown in figure Fig. 6.3. As we can see from Fig. 6.1 b) the relaxation time (τ = RC) in this case is in the order of milliseconds. This time constant is a relatively long time compared to the transit time of the electrons through the channel which is in the order of tens of nanoseconds as we will see in section 6.2.
This measurement was repeated using a sample which has a short channel, sample No.7, and the result as expected has a smaller time scale than the long channel, corre-sponding to a smaller value of R (see Fig. 6.4).
From the above one can see that the numbers of passed and rest electrons at the source depend on the pulse width and the channel length.
In addition, the they also depend on the driving potential (Vs−d) as expected. Fig. 6.5 a) and b) shows the number of rest electrons at the source as a function of the number of pulses for different pulse widths and different driving potentials between source and drain for sample No.10.
Figure 6.1: For sample No.6: a) The pick-up current (Ip) during pulsing the gate measurement which show the passed and rest electrons. b) The number of passed and rest electrons at the source as a function of pulse width. Here the the channel is long (100µm) and the helium film thickness ≈65nm and the average of the intial number of saved electrons before pulsing the gate is 4.0∗108electrons.
In Fig. 6.5 a) and b), if we make a comparison for the number of rest electrons as a function of number of pulses for two different driving potentials we will find that the number of rest electrons decreases faster when the driving potential increases.
As we discussed in section 5.3, the driving potential depends on the external illumi-nation because the conductivity of the silicon wafer is increased. Using sample No.1 we observed that the number of passed electrons increases and so the rest electrons at the source decreases faster with the number of pulses when there is enough external
Figure 6.2: The logarithmic dependence of the number of rest electrons on the pulse width.
Figure 6.3: Schematically RC circuit which illustrate the model of the relaxation time on the source.
intensity of light (see Fig. 6.6).
Figure 6.4: Forsample No.7: The number of passed and rest electrons at the source as a function of pulse width. Here the the channel is short (2µm), the helium film thickness
≈50nm and the average of the intial number of saved electrons before pulsing the gate is 7.0∗108electrons.
Figure 6.5: Forsample No.10: The number of rest electron at the source as a function of pulse width: a) for Vs−d = 1.5V. b) for Vs−d = 3V. Here the the channel is short (2µm), the helium film thickness ≈50nm and the average of the intial number of saved electrons before pulsing the gate is 4.5∗107electrons.