The sheet resistance of a thin film can be measured in different ways. In Figure 3.9, some methods are shown, which are all four-point methods. In four-point methods, two points are used to pass through a defined current and two points are used to sense the voltage signal. The advantage of this geometry is that almost no current flows through the voltage contacts, hence, the resistance of the contact itself can
3.3 Methods to determine transport properties of thin films 37
Figure 3.9: Techniques for measuring the sheet resistance of thin films. (a) Four-terminal method for conducting stripes. (b) Four point probe with contact needles.
(c) Van der Pauw method for measuring the sheet resistance of an arbitrarily shaped film. From [75].
be neglected. In the following, several methods will be discussed regarding the advantages and disadvantages for their application on multi-layer systems, like the ones investigated in this thesis.
Four-terminal method (cf. Figure 3.9a)
In the four-terminal method, the conduction layer has a shape as shown in Fig-ure 3.9a, where the film itself forms the outer current and the inner voltage leads.
The principle is similar to transport measurements on bulk. The resistivity can be calculated by [75]:
RS = w l
V
I (3.10)
where l is the distance of the voltage contacts and w the width of the stripe. In the case of a square film (l = w), the sheet resistance is directly determined by RS = V /I
Advantages:
• contact resistance problems are eliminated
• uniform current flow from the left to the right voltage contact
Disadvantages:
• the specific shape of the layer requires patterning before the deposition.
Four-point probe method (cf. Figure 3.9b)
In the four-point probe method, four metal tips forming a line or a square are mechanically pressed to the layer surface. The current passes through the outer tips, the voltage is measured by the inner tips. If the distance of the tips is large enough (relative to the thickness of the film), the sheet resistance can be calculated by [75]:
RS = π ln 2
V
I (3.11)
Advantages:
• very fast method (no patterning or contacting required) Disadvantages:
• uniform current flow between the tips cannot be guarantied
• inappropriate for temperature-dependent measurements in a helium dewar
• dimension of the sample should be large enough to avoid edge effects
Van der Pauw method (cf. Figure 3.9c)
With the method, which was developed in 1958 by L. J. van der Pauw, it is possible to measure the sheet resistance of an arbitrarily shaped sample. The conditions are [76]:
(i) no holes inside the film
(ii) the contacts should be small and located at the edge of the sample The sheet resistance is then calculated by [76]:
RS = π ln 2
RAB,CD+ RBC,DA
2 f RAB,CD
RBC,DA
(3.12)
with RAB,CD = VCD/IAB and RBC,DA = VDA/IBC. f is the correction function, which allows the sample to be shaped arbitrarily. For homogeneous, quadratically-shaped films f ≈ 1.
By applying a magnetic field B perpendicular to the sample surface and measuring
3.3 Methods to determine transport properties of thin films 39
gold contacts l
w
Figure 3.10: Contact layout of a 4-stripe resistivity measurement. The big outer contacts panels are used for the current and the thin inner panels are used to sense the voltage.
the resistance in two diagonal configurations, the Hall coefficient can be extracted:
RH = d B
VAC
IBD (3.13)
where iBD is the applied current, and VAC the measured Hall voltage in a specific magnetic field B. In zero magnetic field no voltage signal is measured because all electrons pass straightly through the sample. If a magnetic field is applied the conduction electrons are deflected by the Lorentz force and the perpendicular voltage signal increases. For a metal, this so-called Hall voltage exhibits a linear dependence on the magnetic field. The Hall coefficient is then determined by the slope of these lines.
From the Hall coefficient the charge carrier density can be determined by [8]:
N = − 1
eRH (3.14)
where e is the elementary charge.
Advantages:
• sheet resistance and Hall constant can be measured at the same time
• no specific shape of the sample is required
• geometric imperfections of the contacts have no influence on the measured signal
Four-stripe method (cf. Figure 3.10)
In this method, four stripes of a conducting material are deposited on the film surface, e.g. by thermal evaporation or sputtering. To minimize the geometric error, the inner stripes, where the voltage V is sensed, should be very thin. The sheet
resistance can be calculated as in the four terminal method by:
RS = w l
V
I (3.15)
where w is the width of the sample, and l the distance between the inner stripes.
Advantages:
• large sample area is probed, which reduces the influences of small defects in the film
• well suited for low temperature measurements Disadvantages:
• precise sample and contact geometry is obligatory
In this thesis, van der Pauw measurements were performed to obtain the sheet resistance and the Hall coefficient at room temperature. For temperature-dependent resistivity measurements, both the 4-stripe method and the van der Pauw method were utilized. A comparison between both methods applied on the same sample resulted in a very good agreement (cf. Section 3.5.2).
For all the transport measurement methods presented above, the only quantity which is obtained directly is the sheet resistance. The resistivity – the quantity of interest – is always calculated by taking into account the thickness d of the film:
ρ = RS· d (3.16)
Due to the fact that the resistivity scales linearly with the film thickness, an accurate determination of this quantity is essential.