Figure 10.10 displays, in a simplified way, the set-up of an experiment to measure
the Hall effect. In general, a two-dimensional electric conductor, a platelet, is placed
in a magnetic field with the strength B. Through the specimen, in the x-direction,
perpendicular to the magnetic field, flows an electric current I long driven by the
voltage V long . In such an arrangement, it is possible to measure a voltage, the Hall
voltage V Hall , in y direction. In such a system, one can determine the resistance R long
in the x direction, R long ¼ U long /I long . The Hall resistance R Hall is defined as:
R Hall ¼
U Hall
I long
ð10:5Þ
The Hall voltage can be related to the magnetic field and the thickness of the two
dimensional conductor:
U Hall ¼
1
en
B
d
I long
ð10:6Þ
where e is the electrical charge of one electron, n is the number of electrons, and d is
the thickness of the specimen. From Eq. (10.6), one obtains for the Hall conductivity
G Hall :
G Hall ¼ en
d
B
ð10:7Þ
Equations (10.5) and (10.6) make clear that the application of nanoplates is of special
advantage to obtain a significant Hall effect. In the range of ballistic electrical
conductivity, at low temperatures and high magnetic fields, the Hall effect of
nanoplates is quantized and like the ballistic conductivity independent of the
geometry. Under these conditions, the Hall conductivity is given by:
G Hall ¼ nG K ¼ n
G 0
2
ð10:8Þ
where v is an integer number (integer quantum Hall effect) and the G 0 is the
conductance quantum. G K ¼ e
2 /h is the von Klitzing’s constant – a universal
constant used as a standard for electrical resistance worldwide. Additionally, there
exists a fractional quantum Hall effect. For the detailed theoretical background of the
U Hall
x
y
U long
I long
⊗
B
Figure 10.10 General layout to measure the
Hall effect. There is a two-dimensional electric
conductor in a magnetic field of strength B. An
electric current I long , driven by the voltage V long ,
flows in the direction x through the specimen.
Perpendicular to the direction of the current
and perpendicular to the magnetic field in the
direction y, the voltage U Hall , the Hall voltage, is
measured.
276j 10 Electrical Properties of Nanoparticles
the Hall effect. In general, a two-dimensional electric conductor, a platelet, is placed
in a magnetic field with the strength B. Through the specimen, in the x-direction,
perpendicular to the magnetic field, flows an electric current I long driven by the
voltage V long . In such an arrangement, it is possible to measure a voltage, the Hall
voltage V Hall , in y direction. In such a system, one can determine the resistance R long
in the x direction, R long ¼ U long /I long . The Hall resistance R Hall is defined as:
R Hall ¼
U Hall
I long
ð10:5Þ
The Hall voltage can be related to the magnetic field and the thickness of the two
dimensional conductor:
U Hall ¼
1
en
B
d
I long
ð10:6Þ
where e is the electrical charge of one electron, n is the number of electrons, and d is
the thickness of the specimen. From Eq. (10.6), one obtains for the Hall conductivity
G Hall :
G Hall ¼ en
d
B
ð10:7Þ
Equations (10.5) and (10.6) make clear that the application of nanoplates is of special
advantage to obtain a significant Hall effect. In the range of ballistic electrical
conductivity, at low temperatures and high magnetic fields, the Hall effect of
nanoplates is quantized and like the ballistic conductivity independent of the
geometry. Under these conditions, the Hall conductivity is given by:
G Hall ¼ nG K ¼ n
G 0
2
ð10:8Þ
where v is an integer number (integer quantum Hall effect) and the G 0 is the
conductance quantum. G K ¼ e
2 /h is the von Klitzing’s constant – a universal
constant used as a standard for electrical resistance worldwide. Additionally, there
exists a fractional quantum Hall effect. For the detailed theoretical background of the
U Hall
x
y
U long
I long
⊗
B
Figure 10.10 General layout to measure the
Hall effect. There is a two-dimensional electric
conductor in a magnetic field of strength B. An
electric current I long , driven by the voltage V long ,
flows in the direction x through the specimen.
Perpendicular to the direction of the current
and perpendicular to the magnetic field in the
direction y, the voltage U Hall , the Hall voltage, is
measured.
276j 10 Electrical Properties of Nanoparticles