Lab I Hall Measurement (1)

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Drexel UniversityECE-E302, Electronic DevicesLab I: Hall Measurement

Hall Measurement

Objective

The objective of this experiment is to study the Hall effect in semiconductors. To do this

we will measure the Hall voltage of an Indium Arsenide sample under different conditions, and

examine a typical application of a Hall device.

Theory

The Hall effect can be achieved by inducing a magnetic field perpendicular to the current

flow direction in a semiconductor. Under such conditions, a voltage is developed perpendicular toboth the current and magnetic field. This voltage is known as the Hall voltage. The origin of the

Hall voltage can be seen by considering the forces on a charged carrier in the presence of a

magnetic field (see figure 1):

F = q

E + q

v

B (1)

The first term is due to the total electric field driving the current through the sample. The

second term is due to the Lorentz force on the charged carriers, and tends to deflect the carrier

toward the side of the sample. The direction of the deflection depends on the sign of the carriers

charge.

zy

x

x

x

x

y+

z

Figure 1. The Hall effect device. Current flows in the positive x-direction. The applied magnetic

field is in the positive z-direction. For a p-type sample an internal electric field develops in the

positive y-direction.

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Consider the example illustrated in Figure 1. Lets assume that we have a p-type

semiconductor bar. The applied electric field and the current are in the positive x-direction, the

applied magnetic field is in the positive z-direction. The y-component of the force is:

Fy = qE y qv xB z (2)

This equation implies that unless something happens, all carriers moving in the sample will

experience a force that will drive them toward one side of the sample. In this case, the holes

would move in the negative y-direction.

If a number of holes were to collect at the right side of the sample, that side would take on

a positive charge relative to the left side. This sets up an internal electric field in the +y-direction.

Note that the only applied electric field is in the +x-direction. The force due to the internal

electric field opposes the Lorentz force. To maintain a steady flow of current through the sample,

we must have a balance of forces:E y = vxBz (3)

resulting in no net force on the carriers in the y-direction. The internal field can be set up by

moving the holes only slightly to the right.

The presence of the internal field can be detected by measuring the voltage developed

across the sample:

E y =V

y

w (4)

where w is the width of the sample. This is known as the Hall voltage.

Carriers subject to an electric field move with a velocity called the drift velocity. The hole

current in our sample can be written as

Ip = qpv dA (5)

where +q is the hole charge, p is the hole density in #/cm3, vd is the drift velocity, and A is the

cross sectional area of the sample. If we convert this to an equation for the current density vector,

where the magnitude J = I/A and the direction is parallel to the drift velocity, we have

Jp = qp

vd (6)

The drift velocity is related to the electric field driving it through a proportionality

constant known as the mobility:

v d = p

E for holes, vd = n

E for electrons(7)

Substituting this into the current density equation, we get

Jp = qp p

E for holes, Jn = qn n

E for electrons(8)

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Using this relationship in our equation for the field Ey, we get:

Ey = vxBz =Jx

qpBz = RHJxBz

(9)

where RH =1/qp is called the Hall coefficient. You can also show that RH = 1/qn for n-doped

samples.

We can also extend this model to consider the Hall effect when both electrons and holes

are present, resulting in the following equation (for small fields):

RH =1

q

pp2 n

n2( )

pp

+ nn( )

2

(10)

Applications

1) Doping concentration

Equation (9) can be rearranged as

RH =Ey

JxB=

V y W( )IB WT( )

=TV y

IB (11)

We see that we can use measurements of the Hall voltage, magnetic field, current, and

sample thickness to determine the Hall coefficient for any sample. From the Hall coefficient we

can derive the doping density, p or n. This measurement is a diagnostic tool for determining the

doping level in the sample.

2) Mobility

If a measurement of sample resistance R is made, you can calculate the resistivity

R =L

A, so =

RWT

L=

Vx IxL WT (12)

Since the conductivity = 1/ is equal to qpp, the mobility p is just the ratio of the Hall

coefficient and the resistivity. Measurements of the Hall coefficient and the resistivity over a

range of temperatures yield plots of majority carrier concentration and mobility vs. temperature,

very useful data to have for semiconductors.

3) Current measurement

Another real-world application of Hall effect devices is as a sensor for current

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measurement. A current (dc or ac) passing through a wire (figure 2) generates a magnetic field:

B =oIw2 r (13)

where Iw is the current flowing in the wire, and r is the radial distance from the wire. If a

Hall device is placed near the wire and a constant current Is is passed through it, the magnetic

field generated by the wire will induce a Hall voltage Vy in the device.

B-field

Wire

Hall senso

Figure 2. End view of a current carrying wire. Current is flowing into the page, so using the right-

hand-rule, the magnitic field flux circles the wire in a clockwise sense.

Solving equation 11 for B, we see

B =TV y

IsRH (14)

Setting equations 13 and 14 equal and solving for Iw,

Iw =2rV yT

oIsRH (15)

Thus, if we know the details of the sensor and the distance of the sensor from the wire, and

measure the Hall voltage we can make a non invasive determination of the current flowing in

the wire.

Experiments

A commercial Hall device will be used in this lab. This removes the need to solder two or

three contacts to a semiconductor bar. The Hall device has four leads as shown in figure 3, and is

constructed of Indium Arsenide. The Hall device used in this experiment has an impedance of

around 50 across the Ic contacts, and the maximum Ic is 25 mA.

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(W)

(G) (G)

Ic

A

V

(W)

V

+

Figure 3. Measurement of Hall voltage

Notice: The outside two leads of the Hall device (G in the above figure) are used to apply control

current Ic. The inner two leads (W in the above figure) are used to measure Hall voltage.

Procedure

1. Hall Offset Voltage (without magnetic field)

Setup the circuit as in Fig 3. Place a 500 resistor in series with the DC power supply to

limit the control current Ic. With no magnetic field present, measure Hall Offset voltage VH-off

on the DVM as you vary the control current Ic from zero to 25 mA (DC) in 5 mA steps by

changing DC supply voltage.

2. Hall Voltage under DC magnetic fielda) Place the Hall generator in between the pole and the keeper of the small horseshoe

magnet. Adjust the screw so the keeper almost touches the Hall device. Vary the control

current Ic from zero to 25 mA in 5 mA steps and measure the corresponding Hall voltage VH-

mag.

b) Without changing the screw adjustment, measure the magnet's strength B on the Gauss

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meter (1 K Gauss = 0.1 Tesla).

c) Determine the Hall Coefficient RH = VH*T/ (B * Ic), where VH = VH-mag VH-off , T

is the thickness of the Hall device and T = 0.1 mm = 10-4 m .

d) Determine your Hall generator's "product sensitivity" Kp = VH / (B * Ic).

e) Find the doping concentration of the device p = 1 / (q * RH ), where q=1.6E-19 Coulomb.

Be careful with units: VH (volt), Ic (Amp), B (Tesla), T (meter), Kp (volt/Tesla.Amp), p (m-3)

3. Application of Hall device as a Gauss meter

Now that you know the sensitivity of your Hall device, you can use it as a sensor you

can measure unknown magnetic fields.

An electromagnet can be used to produce a voltage controlled magnetic field. Use your

Hall device to measure the magnetic field produced by the electromagnet. The electromagnet

used in this lab consists of hundreds of turns of magnet wire wound on a one-inch torroid. The

coil has a resistance less than 50 and its MAXIMUM CURRENT is 150 mA.

Set the control current Ic to 15 mA. Vary the magnet current Im from 150 mA

down to 0 in at least seven steps, including at least one value of Im less than 10 mA. For each Im,

measure the Hall voltage VH. Use the value of Kp found in Part 2 (Kp at Ic = 15 mA) to calculate

the field strength B = VH / (Ic * Kp) at various magnet currents Im used.

Report

1. Introduction: Objective and Theory

2. Data sheet and calculations:

1) Plot of Hall voltage VH ( = VH-mag VH-off ) vs. control current Ic

2) Determination of Hall coefficient RH, Hall device sensitivity Kp and doping density p

3) Plot of (calculated) coil magnetic field B vs. coil current Im

3. Discussion of the results

4. Questions

1) Can the Hall effect be observed in a metal?

2) Can the Hall effect be used to determine experimentally what type of doping (n- or p-

type) of a sample?

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Lab I Hall Measurement (1)