The Hall effect was discovered by E.H. Hall in 1879 [105] and has become one of the most widely used characterisation techniques to investigate the electrical transport properties of materials, as well as finding many applications in magnetic sensors [106]. Principally, Hall effect measurements are used to obtain the bulk carrier concentration and mobility of a material. A schematic of a typical Hall effect experiment is shown in figure 2.12. When a current flows through a material placed in a magnetic field, a force is generated on the carriers perpendicular to both the magnetic field and the direction of current flow due to the Lorentz force (F)
F=q(E+v×B), (2.17)
whereq is the charge, E is the electric field, vis the velocity and Bis the magnetic
field. The Lorentz force causes carriers to drift in the y direction, but once they
reach the edge of the sample they can no longer flow, and accumulate at the edges. This continues until the build up of charge generates an electric field opposing this,
x y z vx ev×B Ex Bz Ey jx + + + + + + + + + + + + A VH
Figure 2.12. Schematic showing the Hall effect using the Hall bar geometry. An
electric field (Ex) is applied along the bar, the motion of carriers then sets up and
electric field (Ey) generating a Hall voltage (VH) measured across the bar.
resulting in a equilibrium given by
Ey =vxBz. (2.18)
As the applied magnetic field (Bz) and applied current density (jx) are set by the
experiment, and the electric field is measured, it is useful to define a quantity known as the Hall coefficient (RH)
RH = Ey jxBz = vx jx . (2.19)
If the conductivity of the sample is dominated by electrons, then vx = µeEx and jx = nµeeEx, where n is the carrier density and µE is the electron mobility. This
allows the Hall coefficient to be expressed as
RH =
µeEx nµeeEx
= 1
ne, (2.20)
and from this the carrier density can be determined directly from the Hall coefficient. In semiconductors however, the situation can be more complicated as there may be conduction by both electrons and holes. In this case the Hall Coefficient is given by
RH = Ey jxBz = pµ2 p−nµ2e e(pµp +nµe)2 , (2.21)
1 2 3 4 1 2 3 4 d A V A V Thin film Substrate (a) (b)
Figure 2.13. Diagram of the Van der Pauw geometry for thin film Hall effect
measurements. (a) shows the geometry used for Hall measurement, while (b) shows the geometry used for resistance measurement. The thin film of thickness d must
be on an insulating substrate.
wherepandµp are the hole concentration and mobility respectively. However, as all
the CdO samples considered in this thesis are degeneratelyn-type it can be assumed
that pn in all cases and therefore equation 2.20 is applicable.
In the actual experiment the applied current (I) and measured Hall voltage (VH)
are known rather than the current density (jx) and electric field strength (Ey).
Therefore the experiment yields a sheet carrier density (ns), which can be converted
to the bulk carrier density (n) through the sample thickness (d)
n=ns/d. (2.22)
2.2.1
Van der Pauw Geometry
A limiting factor in Hall effect measurements, as described above, is that the sam- ple needs to be in the Hall bar geometry to carry out the experiment. The CdO samples discussed in this thesis are thin films on an insulating substrate, therefore an alternative geometry known as the Van der Pauw geometry is used [107]. Four contacts are made to the top surface of the sample, as shown in figure 2.13, allowing both Hall measurements (shown by panel (a)), and resistance measurements (shown by panel (b)) to be made.
To minimize the error in the Hall voltage that may be present due to non- perpendicular current flow in the Van der Pauw geometry, it is common to measure with positive (B), negative (−B) and zero (0) magnetic field. Applying this
methodology the Hall coefficient is given by [107, 108]
RH = [ V1,3(B)−V1,3(0)]d I2,4B = [V1,3(B)−V1,3(−B)]d 2I2,4B , (2.23)
where dis the film thickness, the voltage is measured across contacts 1 and 3 (V1,3)
and the current across contacts 2 and 4 (I2,4). The current and voltage contacts
can then be switched, allowing repeat measurements and averaging to improve the accuracy of the final result.
For resistivity measurements, a current can be passed between contacts 1 and 2 (I12)
while the voltage is measured across contacts 3 and 4 (V34), as shown in figure 2.13
(b). The resistance is then given by [109]
R12,34 =
|V34|
I12
. (2.24)
The contacts can then be switched to allow a measurement ofR23,41 to be obtained.
From these two measurements the resistivity can be calculated using
ρ= πd
ln(2)
R12,34+R23,41
2 f, (2.25)
where f depends on the ratio R12,34/R23,41 via
R12,34−R23,41 R12,34+R23,41 =farccosh expln 2 f 2 . (2.26)
In practice, R12,34/R23,41 > 2 suggests poor contacts or an inhomogeneous sample.
through equation 2.23, the carrier mobility can be determined by the equation
µ= 1 neρ =
RH
ρ . (2.27)
The van der Pauw geometry provides many benefits over a Hall bar geometry, es- pecially in the examination of thin film samples, and allows several complementary measurements to be taken and averaged in order to reduce uncertainties. For these reasons all the Hall effect measurements in this thesis are carried out in the van der Pauw geometry.
2.2.2
Ecopia HMS-3000 Hall Effect System
The Hall effect measurements reported in this thesis were obtained using an Ecopia HMS-3000 Hall effect measurement system, operating in the van der Pauw sample geometry. The system can apply currents from 1 nA to 20 mA and measure carrier concentrations from 107 cm−3 to 1021 cm−3. A 0.55 T permanent magnet is used
to apply the magnetic field. The system allows I-V measurements to be made
across all of the contact pairs to ensure Ohmic contacts have been made to the sample before carrying out a Hall measurement. The data acquisition software automatically performs the Hall measurements, including repeat measurements and field reversal. The system can also be operated at 77 K using liquid nitrogen if low temperature measurements are required.