The four-terminal measurement is widely used in the cryogenic experiments due to
its high accuracy. By separating the current source and the measurement unit from
shar-ing the same contact, signals containshar-ing the contact resistance can be effectively reduced
as compared to the traditional two-terminal measurement. According to van der Pauw
method, this can reduce the inaccuracy resulted from the contact resistance.
The van der Pauw method have these constriction: First, the shape of the sample must
be flat and uniform; second, the sample cannot have any isolated holes; third, the sample
should be homogeneous and isotropic; fourth, all four contacts must be located at the
edges of the sample; fifth, the area of any individual contact should be at least an order
of magnitude smaller than the area of the entire sample. Based on these constrictions, the
resistance shown in figure 4.7 between M and N can be calculated as follows.
R mn,op
=V mn
I op ,
(4.8)where V
mn
is the measured voltage across the point M and N, Iop
is the current detected bythe current source. By van der Pauw formula, the resistance defined by eq.4.8 will satisfy
e −
πdRmn,op
ρ
+ e−
πdRop,mn
ρ
= 1, (4.9)where d is the thickness of the sample and ρ is the resistivity of the sample. With the
help of four-terminal measurement, which satisfies the van der Paul theorem, our electric
measurement can be much more accurate and efficient.
Figure 4.7: Illustration diagram of van der Pauw electrical configuration. The resistance
R mn,op
is defined as Vmn /I op
[3]Bibliography
[1] Triton 200/400 cycrofree dilution refrigerator Operator’s Handbook (Oxford
Instru-ment, 2012).
[2] F. Pobell, Matter and methods at low temperatures (2007).
[3] S. H. N. Lim, D. R. McKenzie, and M. M. M. Bilek, Rev. Sci. Instru. 80, 075109
(2009).
Chapter 5
Transport through CVD-Grown
Graphene Constrictions
Many researches demonstrated that disorder and doping will lower the mobility of
graphene nano-divices. These devices usually involved hydrogenated doping graphene [1,
2], nitrogen doping graphene [3], fluorinated doping graphene [4], devices of disorder
from the substrate [5], and of the remaining photoresistance [6], etc. Hopping transport
similar to those in reduced graphene oxide devices [7, 8] is pronounced in these kinds of
disordered system. Hence, due to that thermal annealing was not applied to our samples
before the measurement, chemical doping during each process may exist, such as nitrogen
and oxygen adsorbed from atmosphere, substrate random disorder, or the photoresistance
from the pattern process, etc. This allowed us to investigate in the low-mobility graphene
Figure 5.1: SEM picture of all the samples on a chip.
device and its transport behavior.
Our CVD graphene samples were provided by Prof. Yang-Fang Chen and his
re-search group, which were fabricated via CVD method. Graphene was transferred to SiO
2
(300nm)/n+ Si substrate and then followed by the photolithography process as described
in chapter 3. Si substrate was used as the back gate and SiO
2
layer was served as thedi-electric material, where the capacitance of SiO
2
for 300 nm is around 11.6 nF/cm2
. Therewere three different channel widths on a single chip as shown in figure 5.1. The channel
widths of each device were 1.25, 5.90, 25.09 µm respectively and the label of each device
were device 1 (D1), device 2 (D2), device 3 (D3) in sequence. The mobility of our sample
could be calculated by equation 2.12 and is 400 cm
2
/Vs at 2 K.A Four-terminal measurement method was applied to scan the electrical properties
of our sample with a Keithley 2400 General-Purpose Source meter served as the current
Figure 5.2: The configuration of the four-terminal measurement.
source and a Keithley 2000 Multimeter connected to the other two pads to measure the
voltage difference. The back gate voltage is controlled by a Keithley 236 Source Measure
Unit. Also, the sample was connected to a 1 KΩ resistance in series and another Keithley
2000 measured the voltage difference of this resistance as a current monitor. The setup of
four-terminal measurement was ready and a LabVIEW program was used to control these
machines. Following sections will show the electrical measurement of our samples under
low temperature regime.
5.1 Ambipolar Field Effect and Nonlinear I-V Curves
Figure 5.5 gives an overview of the current-voltage (I-V ) curve at different back gate
voltage. By changing the gate voltage from
−2.5V to 17.5V, Fermi level is tuned from the
hole region to the neutrality point (V
BG
= 17.5V). Thus, the slope of the I-V curve, whichis the conductivity, becomes smaller as the back gate voltage approaches 17.5V. On the
10μm
1.25μm
20μm
5.90μm
20μm
25.09μm
Figure 5.3: SEM picture of each device. From top to bottom are label as D1, D2, and D3 in sequence. The channel widths of each device are 1.25, 5.90, 25.09 µm respectively.
Figure 5.4: Resitivity R at low bias field versus , defined as dV /dI, at different back gate voltage. The Dirac point of three devices are around 17.5 V, which means that doping level is the same. At V
BG
lower than 17.5 V is in hole region and on the otherside (higher than 17.5V) is in electron region.other hands, as the back gate voltage increases, Fermi level is then tuned to the electron
region and the slope increases as well. Figure 5.4 shows the resistance R, defined as
dV /dI at the low field regime, varies with the back gate voltage. This figure reveals the
properties of conical band structure in graphene and this is called the ambipolar field effect
as depicted in other controllable-gate devices of graphene sample [4, 9–11].
As the temperature decreases, the I-V curve becomes more nonlinear at low-field
re-gion as shown in figure 5.7. The conductance, the slope in the figure, decreases as the
temperature decreases, which exhibits an insulator behavior. D1 and D2 also show this
nonlinear behavior and insulator behavior, and their I-V curves are included in appendix A.
I focus on this nonlinear behavior in this thesis.
-80 -60 -40 -20 0 20 40 60 80
Figure 5.5: I-V curves for D3 at 30K. (a) and (b) are in the hole and the electron region respectively. The back gate voltage is set at
−2.5, 0, 2.5, 5, 7.5, 8.75, 10, 11.25, 12.5,
13.75, 15, 16.25, 17.5 V from left to right for figure a; 17.5, 18.75, 20, 21.25, 22.5, 23.75, 25, 27.5, 30, 32.5, 35, 37.5 V from right to left for figure b.-150 -100 -50 0 50 100 150 electron region respectively. The back gate voltage is set at
−2.5, 0, 2.5, 5, 7.5, 8.75, 10,
11.25, 12.5, 13.75, 15, 16.25, 17.5 V from left to right for figure a; 17.5, 18.75, 20, 21.25, 22.5, 23.75, 25, 27.5, 30, 32.5, 35, 37.5 V from right to left for figure b.-150 -100 -50 0 50 100 150
= 17.5V (neutrality point)
Figure 5.7: Temperature dependence of I-V curve at the neutrality point (V
BG
= 17.5 V) for D3.Figure 5.6 shows that the nonlinear behavior is strong at the neutrality point. Curves
far from the neutrality point are much more linear. This phenomena has been observed in
other researches[8] and as a solution the VRH model is used to explain it. In this graph,
as the Fermi level is far from the neutrality point, the energy of the carriers is larger than
the potential of the tray hole which induced by the impurities. Thus, carrier transport far
from the neutrality point cannot be affected by the impurities.