Advanced Lab - Hall Effect:Analysis of Lab Data

3. Analysis of Lab Data

This part describes how to analyize and present the results of data collected in Part 2.

3.1 Hall Voltage vs. Magnetic Field

For each of the four probe currents (0.1 A, ..., 0.4 A), make a plot of V_Hall versus B. On each plot, show three curves, one for each of the three temperatures.

Draw "best fit" straight lines through your data points. These lines should pass through the origin, while fitting the data pretty well. Failure to show a linear fit, or failure of a good linear fit to pass through the origin, probably indicates defective data taking.

3.2 Hall Voltage vs. Probe Current

(A) Show the data taken in the experiment by preparing tables that show the measured Hall voltage for each value of probe current. Do this for each temperature and each magnetic field strength. A nice way to do this is to make three big tables (one for each temperature), each containing six sub-tables (one for each magnetic field strength).

(B) Plot the Hall voltage versus the probe current for the data shown in the tables in (A). Make three plots, one for each temperature, each containing six sets of points, one for each value of magnetic field strength. Be sure to indicate which data correspond to a particular magnetic field value. Draw "best fit" straight lines through the data. Again, the fits should be nearly linear and pass through the origin.

3.3 Magneto Resistance Data

Make a plot of G (the conductance in the Hall probe) versus B². Conductance is the inverse of resistance, so G = 1/R and G is measured in ohm^-1 (also called "mho"). Plot points for all three temperatures on the same plot, but make sure they are distinguishable. For each temperature, draw a smooth curve through the data points. Although the points with B=0 are important, they should not be given special weight in drawing the smooth curves through the data points. Thus, your judgment of a best fit curve may miss the B=0 data point.

3.4 Direction of Fields

Present the results of your observations in part (1C) as a diagram showing the directions of: (a) magnetic field, (b) Hall probe current, and (c) Hall electric field, where the latter is the electric field associated with the polarity you observed for the Hall voltage.

The results can be interpreted to yield the sign of the charge carriers in the Hall probe. State your conclusion as to the sign of the charge carriers, and show carriers on your diagram with a charge that is consistent with your observations.

Note the effect on the polarity of the Hall voltage of reversal of the probe current.

3.5 Hall Resistance and Hall Coefficient

The Hall resistance is defined as R_Hall = V_Hall/I_Probe.

This definition may cause some conceptual confusion. Although the Hall resistance has units of ohms, as for the usual definition of resistance, it is not a resistance in the sense of that defined in accordance with Ohm’s "Law". In Ohm’s law, R=V/I, the electric field associated with the potential difference V is parallel to the direction of the current I. In the Hall resistance, the electric field associated with the potential difference V_Hall is perpendicular to the current. For the Hall probe, the resistance analogous to that of Ohm’s law is sometimes referred to as the "longitudinal" resistance (R_Probe = V_Probe/I_Probe); it is this longitudinal resistance that is measured in the magneto resistance section of this laboratory.

The Hall coefficient is defined as:

Eq. (1)

with units of m³/C. Note that d is the dimension of the probe, which was measured parallel to the applied magnetic field during the experiment.

Prepare a table that includes the following information, which refers to the six curves plotted in part (2B). For each curve list the temperature, magnetic field, and the slope of the curve of V_Hall versus I_Probe; the latter is the Hall resistance and should be designated R_Hall in your table. For each of the six cases, compute the Hall coefficient R₀.

Verify that the units for R₀ are consistent with the units you have employed for the various factors in the computation of R_Hall and R₀. Also, as a further check on possible blunders in computations, choose one case, from the six, and recompute the Hall coefficient, R₀, for that single case from your original laboratory data, rather than by use of a curve plotted in part (2B).

It is expected that the Hall coefficient, at a given temperature, will be independent of probe current and magnetic field. Comment on your results in this context; independence of probe current is manifested here as a linear plot for part (2B).

Comment on what dependence, if any, you find for the Hall coefficient on temperature.

3.6 Density of Current Carriers

The density of current carriers is related to the Hall coefficient according to 1/(eR₀), where e is the electronic charge, i.e., the charge carried by one carrier.

Use average values for R₀, from the results of part (5), to compute the current carrier density for each of the three temperatures.

For comparison, compute the density of Indium atoms (in number of atoms per m³). For this you need Avogadro’s number, the molecular weight of InAs and the density of InAs (5.67 grams/cm³). You should find a lot fewer current carriers than Indium atoms.

3.7 Drift Velocities

The drift velocity of the charge carriers is related to the Hall voltage according to v_D = V_Hall/wB, where the distance w is measured between the Hall voltage leads; thus, V_Hall/w is the magnitude of the electric field due to the Hall effect.

Prepare a table that gives the following information: Hall probe current, temperature, drift velocity, and average thermal velocity. Include data for only four conditions: probe currents of 0.1 A and 0.4 A, and temperatures of room temperature and (approximately) 100°C. The four values of V_Hall/B that are needed in the computation of the drift velocities should be obtained as the slopes of curves plotted in part (1), where you plotted V_Hall versus B for a variety of temperatures and probe currents. The average thermal velocities for each temperature can be computed from <KE> = 3kT/2.

The linearity of the curves plotted in part (1) is indicative of independence of drift velocity and magnetic field strength. Compare the drift velocities at the two temperatures and compare the ratio of drift velocities (at a given temperature) at different probe currents with the ratio of the probe currents. Comment on your findings.

You should find the drift velocities to be much smaller than the thermal velocities.

3.8 Type of Charge Carriers

For the following questions consult Dr. Gersch's analysis.

If only one type of current carrier were present, i.e., the conductivity was predominantly n-type or p-type, how would you expect G to depend on the magnetic field?

If the two types of current carriers were present in nearly equal numbers, what value would you expect for the parameter d₂.

Are your experimental results consistent with either of the above two special cases.

3.9 Conductivity of the Hall Probe

For your room temperature data only, compute the conductivity of the Hall probe for the smallest and for the largest magnetic fields that you used. For a sample of length L and cross sectional area A, the conductivity and conductance are related by G = σ A/L, where σ is the conductivity. (Resistivity is the reciprocal of conductivity.) The conductances you want here are obtained from the "longitudinal" resistance measurements, i.e., the magneto resistance values as plotted in part (3).

3.10 Mobility

Mobility is defined according to v_D = μE. v_D is the drift velocity, as computed in part (7). μ is the mobility, and E is the electric field that causes the probe current to flow. This equation gives a better picture of the significance of the drift velocity than the equation of part (7), which related the drift velocity to the Hall voltage. When an electric field is applied to a sample that contains charge carriers, a drift of the carriers along the direction of the field is superimposed on their usual random (thermal) motion. Part (7) showed that the velocity associated with this drift is much smaller than the velocity associated with the random motion. The drift will be parallel or anti-parallel to the electric field, depending on the sign of the charge carriers.

Prepare a table that treats the same four cases as analyzed in part (7), viz. room temperature and about 100°C, and probe currents of 0.1 and 0.4 A. Enter the following information in the table: probe current, temperature, drift velocity (from part (7)), conductance for magnetic field B=0, conductance for the largest magnetic field that was measured, and μ for the largest magnetic field measured.

The conductances should be read from the smooth curves drawn in part (3), based on your measurements of the longitudinal resistances. The conductances will be temperature dependent, but you use the same conductances for both probe currents at a given temperature.

In order to compute the mobilities, you must use the following analysis to relate the electric fields, along the direction of the probe current flow (these are not the same as the Hall effect electric fields), to your data. ΔV is the potential difference between the current leads to the probe, which was not measured but may be computed from quantities that were measured or are known.

Eq. (2)

Verify that the units for μ reduce to tesla^-1. Comment on the observed dependence of μ on probe current, temperature, and magnetic field.

3.11 Density of Current Carriers, Revisited

The picture of the probe current being carried by charge carriers that drift in the direction of the applied electric field is consistent with the equations:

and

Eqs. (3),(4)

where j is the current density (in A/m²), n is the density of charge carriers (in number of carriers per m³, as in part (6)), and e is the charge of the carriers.

For the four cases treated in part (9) (i.e., probe currents of 0.1 A and 0.4 A and temperatures of room temperature and about 100°C), compute the density of current carriers. These results should be consistent with those found in part (7).

3.12 Dr. Gersch's Analysis

B : magnetic field = B_z only
E : electric field; E_y drives probe current; E_x is Hall
q : charge on current carrier (q = electronic charge) j : current density, j_y = σE_y is probe current (σ is conductivity)
v : velocity of charge carriers
n : concentration of charge carriers
m : effective mass of charge carriers
J : relaxation time of charge carriers
indices: + holes, - conduction electrons

Force on current carrier = Lorentz force bue to B and a retarding (resistive or damping) force. At equilibrium the net force = 0.

holes

electrons

Other relaxations. j = σE defines conductivity

j₊

₊

v₊

j₊

Put B = B_z

z components require j_z = j_z+ + j_z_ = q(n₊v_z+-n_)

E_z=0

then v_z+ = v_z_ = 0

x components

		Eq. (5)
		Eq. (6)

require j_x = j_x+ + j_x_ = 0 so:

Eq. (7)

y components

		Eq. (8)
		Eq. (9)

observed probe current is j_y = j_y+ + j_y_ or

Eq. (10)

6 equations in 6 unknowns: E_x, v_x+, v_x_, v_y+, v_y_, j_y

Eliminate E_x from (5) and (6):

Eq. (11)

Use (3) to eliminate v_x_ from (6), (9), (11). This gives 3 equations in 3 unknowns: v_x+, v_y+, v_y_. These solutions are put into (10) to get j_y and into (1) to get: