THE HALL EFFECT. Theory

Transcription

1 THE HALL EFFECT Theory For a charge moving in a magnetic field, it experiences a force acting at right angles to both its direction and the direction of the magnetic field H. Hence in a semiconductor, instead of travelling in a straight line, the electrons and holes will initially take paths as shown by the dotted lines in the figure below. Note that both holes and electrons will move transversely in the same direction. A space charge builds up at the upper edge and at equilibrium an electric field is set up across the semiconductor crystal such that its magnitude and direction balance the space charge. Thus at equilibrium the charge carriers flow straight through the crystal. If we connect a potentiometer across D and B, the polarity will indicate whether holes or electrons are the majority carrier and the magnitude of the potential difference V will provide a measure of the Hall coefficient R H. For a Hall measurement of and arbitrary shaped wafer with uniform thickness d in a homogeneous magnetic field H applied perpendicular to the wafer, the Hall coefficient is obtained from 1

2 d H R H = ΔR AC, Here the resistance R AC, given by V R AC, = I AC Is charged by an amount ΔR, AC when a magnetic field is applied. For the diamond shape germanium wafer, V = 0 when H=0 from symmetry, hence V R = I Δ AC, AC In a magnetic field; in this case V = V H where V H is the Hall voltage. The above formulation is written in esu units; in practical units, then R 8 V d.10 = cm I H. 3 H / AC coul Where gauss. V is the Hall voltage in volts, d in cm, I in amp and H in In simple semiconductors with one kind of carriers dominating, the interpretation of R H is most straightforward and can be shown to be 1 R H = ne Where e = 1.6 x coul is the electronic charge and n the chargecarrier density per cm 3. Thus RH provides a measure of the important charge-carrier density. 2

3 Procedure The measurement usually requires four readings. The voltage appearing between the Hall probes is not in general the Hall voltage V H alone. There are other glavanomagnetic effects which produce voltage between B and D. in addition, the potential drop due to misalignment of the Hall contact and thermoelectric voltage due to temperature gradient may be present. Thus the hall voltage is measured for both direction of the current and the magnetic field. The four cases are illustrated below. Then V H V = 1 + V3 V2 V4 4 The sign of V H is the sign of R H, which is the sign of the principal charge carriers. EXPERIMENT 1 Connect the germanium wafer unit using the blue current wires and a 1.5 V dc supply in series. Connect the potentiometer to the Hall voltage wires of the unit, as shown schematically in the following figure: 3

4 Before applying the field, zero the potentiometer by adjusting the variable resistor R on the unit; this will overcome to some extent the misalignment of the Hall contact. 1) Measure V H ; for each value of V H four readings must be taken. 2) Plot a graph of V H vs magnetic field and hence deduce the value of R H. 3) Calculate the charge carrier density and determine its sign. EXPERIMENT 2 Using the Linear Hall effect ic ( ), plot the output characteristics as a function of supply voltage. See Fig. 12 of data sheet attached. 4

5 Linear Hall effect ic ( ) A miniature linear output Hall effect sensor in a moulded 4-pin dil plastic package. This device features a differential output stage. One output increases linearly in voltage whilst the other decreases for a linear increase in magnetic flux density over a ±40mT range. Typical applications for this versatile ic include magnetic field investigation in the vicinity of transformers and cables, current sensors with high isolation, linear feedback elements in analogue control systems, etc. The sensor is immune from damage by high values of flux density. Typical linear output characteristics The linear Hall effect ic features differential outputs. One output increases, whilst the other output decreases with and increase in Gauss. Figure 12 Typical output characteristics as a function of supply voltage Absolute maximum ratings Supply Voltage +12 VDC Output current 20mA Operating frequency 100kHz Operating temperature -40 C to C Storage temperature -55 C to +150 C Electrical characteristics Supply Voltage (VDC) Supply Current (ma) Output type 4 to typ. Differential Outputs, linear Output Voltage 1.75 to 2.25V at 5V & 0 Gauss Sensitivity (-400 to 400 Gauss) 0.75 to 1.06mV/Gauss 5

6 Connections for Hall effect IC Experiment. Typical linear output characteristics 5500 The linear Hall effect ic features differential outputs. One output increases, whilst the other output decreases with an increase in Gauss. Figure 12 Typical output characteristics as a function of supply voltage OUTPUT VOLTAGE (VOLTS) 6

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