E8e “Hall Effect”

Fakultät für Physik und Geowissenschaften Physikalisches Grundpraktikum

Tasks

1. Measure the Hall voltage and the longitudinal voltage of a Germanium sample at constant temperature and constant current as a function of the applied magnetic field. Plot both Hall voltage and longitudinal voltage as a function of the magnetic field. Determine the Hall constant as well as the zero field conductivity of the sample. Further determine the carrier density and the mobility. What conclusions can you draw on the charge carrier type?

2. Measure the longitudinal voltage of the Germanium sample at a constant current in zero magnetic field B = 0 as a function of temperature. Determine the band gap.

Literature

Physikalisches Praktikum, 13. Auflage, Hrsg. W. Schenk, F. Kremer, Elektrizitätslehre, 1.0.1, 2.4 Physics, P. A. Tipler, 3rd Edition, Vol. 2, 22-1, 22-2, 22-5, 24-4

Accessories

Germanium sample on an integrated circuit board with heater and thermocouple, Teslameter with Hall field sensor, electromagnet, laboratory power supplies, digital multimeters

Keywords for preparation

- Conduction processes in metals and semiconductors, conductivity, mobility - Moving charges in a magnetic field (Lorentz force) - Hall Effect, Hall voltage, Hall constant - Practical applications of the Hall Effect - Temperature dependence of the conductivity of semiconductors - Thermoelectric effect, thermocouple

Remarks

As shown in Fig. 1, the Germanium sample is mounted on a circuit board with the contacts for the measurement of the longitudinal voltage (2.1, 2.2, 2.3) and the Hall voltage (3) already attached. Further a sample heater (heating coil with contacts 4 embedded in the circuit board) and a thermocouple with contacts (7) are integrated into the circuit board. These boards are comparatively expensive and can be easily destroyed by erroneous electrical wiring. Therefore, electrical connections should be wired up deliberately and carefully. All circuitry may only be set up after some instruction by the supervisor and with all power supplies being switched off. Measurements are only to be started after the circuits were checked by the supervisor. Special care has to be devoted to the heater electrical connections in task 2. Especially, the temperature of the semiconductor must not exceed 150°C and the magnetic field sensor of the Teslameter must not be exposed to a temperature above 50°C, i.e. in case of measurements above 50°C the magnetic field sensor must be removed from the heated sample area.

Fig. 1. Hall circuit board with contacts.

In task 1, the semiconductor sample with dimensions 20 mm x 10 mm x 1 mm that is mounted on the circuit board as shown in Fig. 1, is placed between the removable pole shoes of an electromagnet. Close to the semiconductor sample the magnetic field sensor of the Teslameter should be positioned, in order to obtain an accurate measurement of the local magnetic field. The elements of the circuit board as well as the electromagnet have to be wired up with appropriate cables; care has to be taken when inserting electrical plugs into the sockets on the board, by exerting a certain counter pressure onto the backside. After the wiring is finished, the pole shoes are careful put into place, without touching the

semiconductor sample. In this experiment four digital multimeters are used. One multimeter each is used for the measurement of the Hall voltage across the contacts 3 and the longitudinal voltage at the contacts 2.1 and 2.2, respectively. The third multimeter is used for the measurement of the current through the semiconductor; the respective laboratory power supply is connected to the contacts 2.1 and 2.3. Between contacts 2.2 and 2.3 a compensation circuit is located that serves to compensate the offset voltage between the contacts 3 that is still there in zero magnetic field (B = 0) and that is caused by a slight misalignment of the Hall contacts. The fourth multimeter is used to control the current through the electromagnet (warning: set to measuring range 20 A). After a check of the circuitry by the supervisor a sample current I = 30 mA is set with the laboratory power supply by applying a voltage between 12 V and maximally 30 V to the contacts 2.1 and 2.3. Since the sample resistance weakly depends on the magnetic field, the sample current must be readjusted during measurements. Hall voltage and longitudinal voltage are measured at room temperature as a function of the magnetic field B. The current is supplied to the electromagnet with an adjustable laboratory power supply, never exceeding the maximum current rating of the coils of 4 A. Measure for positive and negative values of the magnetic field (reversing polarity by replugging the cables at the magnet power supply).

Plot the Hall voltage UH as a function of the magnetic field and determine the charge carrier density. Try to draw any conclusions on the type of the charge carriers. Further, plot the magnetoresistance (R-R0)/R0 (R0: sample resistance at B = 0).as a function of magnetic field. Calculate the conductivity from the resistance measured at zero magnetic field; calculate further the mobility. Determine the number of donor/acceptor atoms per Germanium atom using the following -3 -1 values: Ge= 5.32 g cm , MGe= 72.6 g mol .

In task 2 the circuit board is removed from the vicinity of the electromagnet and magnetic field sensor. Under no circumstances the magnetic field sensor of the Teslameter should be close to the hot circuit board (danger of thermal destruction). The semiconductor sample is heated by the integrated sample heater. To this end, the laboratory power supply that was used to supply the magnet current in task 1 is now connected to the heater contacts (not to the thermocouple contacts!). The sample is heated up to a maximum temperature of 150°C, where the sample temperature is measured with the integrated thermocouple. The heater current should be slowly increased with increasing temperature (increasing thermocouple voltage), until the thermocouple voltage reaches 5.5 mV or the temperature is 150°C. The heater current is then decreased in steps and the longitudinal sample voltage is measured on cooling. The sample temperature might be calculated from the thermocouple equation U TTTh , (1) a r

where Tr denotes room temperature and a the Seebeck coefficient of the Cu/CuNi thermocouple (a= 40 V/K). Since this is a standard thermocouple, the conversion from thermocouple voltage to temperature might be done by a digital multimeter using a standard calibration table; in this case chose the setting “°C” on the multimeter. The analysis is done graphically. Plot the logarithm of the reciprocal sample voltage as a function of the reciprocal absolute temperature and determine the band gap from the slope of this graph.

Hall Effect

Fig. 2. Schematic representation of the Hall Effect of a long, thin, rectangular sample. The magnetic field B is applied perpendicular to the large sample face along the z-axis. The Hall electric field is generated between the sample edges perpendicular to the magnetic field.

Figure 2 shows the typical configuration of a Hall Effect measurement. A current I flows through a long, thin, rectangular sample, i.e. the current density j points along the x-axis. Perpendicular to the large sample face and therefore perpendicular to the current density a magnetic field B is applied along the z-axis. The resulting Lorentz force F qv B (2) deflects the charge carriers along the y-axis. The charge carriers accumulate at the sample edge and generate an electric field EH, such that the Lorentz force is just compensated:

qv B qEH . (3) With the relations j qnv I j() wd (4)

UHH E w One immediately obtains the Hall constant U 1 RdH , (5) H IB qn that has the same sign as the charge of the prevailing charge carrier type (electrons or

holes). The equations might be written in a slightly different way. With the definition of the resistivity  by Ej  , (6) where E denotes the longitudinal electric field, and after conversion of Eq. (5) to

EHHH() R B j j (7) one obtains in compact notation

Ejx Ej   H    xx xy  x             . (8) Ejy EHH   0   yx yy  y 

Besides the diagonal component xx the magnetic field generates an off-diagonal component

yx of the resistivity tensor. From Eq. (8) the denotations longitudinal voltage for the sample voltage and transverse voltage for the Hall voltage become clear.

Temperature dependence of the conductivity

In the regime of intrinsic conduction the conductivity has a thermally activated form:

Eg 0 exp , (9) 2kTB where  denotes the conductivity, Eg the band gap, kB Boltzmann’s constant and T the temperature. If the logarithm of the conductivity is plotted against the reciprocal temperature, the band gap might be determined from the slope.

Mobility

The conductivity  is related to the charge carrier mobilities n or p as well as the charge carrier concentrations n or p (n for electrons and p for holes):

ne np pe  . (10) If one charge carrier type dominartes the conduction processes, one obtains the simplified equations:    R  , R . (11) nHne pH pe