
It is relatively easy to calculate the magnitude of the Hall
voltage U_{Hall} that is induced by the magnetic field B. 


First we note that we must also have an electrical field E parallel to
j because it is the driving force for the current. 


Second, we know that a magnetic field at right angles to a current causes a force
on the moving carriers, the socalled Lorentz
force
F_{L}, that is given by

 



We have to take the drift velocity v_{D} of the carriers,
because the other velocities (and the forces caused by these components) cancel to zero on average. The vector product assures
that F_{L} is perpendicular to v_{D} and B. 


Note that instead the usual word "electron" the neutral term carrier is used, because in principle an electrical current could also be carried by charged
particles other then electrons, e.g. positively charged ions. Remember a simple but important
picture given before! 

For the geometry above, the Lorentz force F_{L}
has only a component in y  direction and we can use a scalar equation for it. F_{y}
is given by 
 
F_{y}  = 
– q · v_{D} · B_{z} 




We have to be a bit careful, however: We know that the force is in ydirection,
but we do longer know the sign. It changes if either q, v_{D}, or B_{z}
changes direction and we have to be aware of that. 

With v_{D} = µ · E
and µ =
mobility of the carriers, we obtain
a rather simple equation for the force 


F_{y}  = 
– q · µ · E_{x} · B_{z} 




It is important to note that for a fixed current density j_{x}
the direction of the Lorentz force is independent of the sign of the charge carriers (the sign of the charge and the sign
of the drift velocity just cancel each other). 

This means that the current of carriers will be deflected from a straight line
in ydirection. In other words, there is a component of the velocity in ydirection and the
surfaces perpendicular to the ydirection will become charged as soon as the current (or the magnetic field)
is switched on. The flowlines of the carriers will look like this: 
 



The charging of the surfaces is unavoidable, because some of the carriers eventually will
end up at the surface where they are "stuck". 


Notice that the sign of the charge for a given surface depends on the sign of the charge of
the carriers. Negatively charged electrons (e^{} in the picture) end up on the surface opposite to positively
charged carriers (called h^{+} in the picture). 


Notice, too, that the direction of the force F_{y} is the same
for both types of carriers, simply because both q and v_{D} change signs in the force
formula 

The surface charge then induces an electrical field E_{y}
in ydirection which opposes the Lorentz force; it tries to move the carriers back. 


In
equilibrium, the Lorentz force F_{y} and the force from
the electrical field E_{y} in ydirection (which is of course simply q ·
E_{y}) must be equal with opposite signs. We obtain 
 
q · E_{y}  = 
– q · µ · E_{x} · B_{z}
  
 E_{y} 
= 
– µ · E_{x} · B_{z} 



The Hall voltage U_{Hall} now is simply the
field in ydirection multiplied by the dimension d_{y} in ydirection. 


It is clear then that the (easily measured) Hall voltage is a direct
measure of the mobility µ of the carriers involved, and that its sign
or polarity will change if the sign of the charges flowing changes. 

It is customary to define a Hall coefficient
R_{Hall} for a given material. 


This can be done in different, but equivalent ways. In the link
we look at a definition that is particularly suited for measurements. Here we use the following definition: 
 
R_{Hall}  = 
E_{y} B_{z} · j_{x} 



In other words, we expect that the Hall voltage E_{y}
· d_{y} (with d_{y} = dimension in ydirection) is proportional
to the current(density) j and the magnetic field strength B, which are, after all, the main
experimental parameters (besides the trivial dimensions of the specimen): 
 
E_{y}  = 
R_{Hall} · B_{z} · j_{x} 



The Hall coefficient is a material parameter, indeed, because we
will get different numbers for R_{Hall} if we do experiments with identical magnetic fields and current
densities, but different materials. The Hall coefficient, as mentioned before, has interesting properties: 


R_{Hall} will change its sign, if the sign of the carriers
is changed because then E_{y} changes its sign, too. It thus indicates in the most unambiguous way
imaginable if positive or negative charges carry the current. 


R_{Hall} allows to obtain the mobility µ of
the carriers, too, as we will see immediately 

R_{Hall} is easily calculated:
Using the equation for E_{y} from above, and the basic equation
j_{x} = s · E_{x}, we obtain for negatively
charged carriers: 
 
R_{Hall} 
= – 
µ · E_{x} · B_{z}
s · E_{x} · B_{z} 
= –  µ
s 



Measurements of the Hall coefficient of materials with a known
conductivity thus give us directly the mobility of the carriers responsible for the conductance.



The – sign above is obtained for electrons,
i.e. negative charges. 


If positively charged carriers would be involved, the Hall constant would be positive. 


Note that while it is not always easy to measure the numerical value of the Hall voltage and
thus of R with good precision, it is the easiest thing in the world to measure the polarity
of a voltage. 

Lets look at a few experimental data: 
 
Material  Li 
Cu  Ag  Au 
Al  Be 
In 
Semiconductors (e.g. Si, Ge, GaAs, InP,...) 
R (× 10^{–24})
cgs units  –1,89 
–0,6  –1,0  –0,8 
+1,136 
+2,7 
+1,774 
positive or negative values,
depending on "doping"  Comments:
1. the positive values for the metals were measured under somewhat special conditions
(low temperatures; single crystals with special orientations), for other conditions negative values can be obtained, too.
2. The units are not important in the case, but multiplying with 9 · 10^{13} yields the
value in m^{3}/Coulomb 
