experiment19-resistivity and hall effect in p and n germanium

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EXP # 9 – RESISTIVITY AND HALL EFFECT

IN p AND n GERMANIUM

Muhammad

Experiment conducted on 3 November 2014



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Table of ContentsAbstract .........................................................................................................................1

Introduction ...................................................................................................................1

Theory ...........................................................................................................................1

Method ..........................................................................................................................5

Experimental Apparatus Malfunction .......................................................................7

Analysis......................................................................................................................... 7

Conclusion .................................................................................................................. 11

References ................................................................................................................... 12

Table of Figures

Figure 1 - Wiring diagram for resistance vs. temperature measurement ..................... 5

Figure 2 - Wiring diagram for Hall voltage measurement ........................................... 5

Figure 3 - Hall Effect ................................................................................................... 6

Figure 4 – Plot of UH versus B for p doped Ge............................................................ 8

Figure 5 - Plot of UH versus B for n doped Ge ............................................................ 9

Figure 6 - Basic Schematic of Experimental Apparatus to Indicate Orientation in the

B Field .................................................................................................................... 10

http://p/Dropbox/University/Fall%202014/PHYS%20460A/Experiment19.docx%23_Toc404034951




























1

Abstract

The Hall Effect is used to confirm that the predominant charge carriers are positive

holes in p doped Ge, and electrons in n doped Ge. Further voltage-current analysis of the

samples at room temperature, allow us to calculate various properties of the

semiconductor samples such as their resistivity, carrier density and carrier mobility.

Introduction

Semiconductors have many unique and useful properties that make them very useful

for a variety of applications. The properties of a semiconductor can be further controlled

by doping with another material. Different materials have different effects on the

properties of the semiconductor. One of the major effects of doping is on the

determination of the predominant charge carrier in the semiconductor. The Hall Effect is

an important phenomenon that allows us to study the various characteristics of a given

semiconductor in great detail.

Theory

In the Hall Effect, a potential difference is produced across a conductor perpendicular

to both, the direction of current flow and the magnetic field (which has been applied

perpendicular to the current). In this situation, the magnetic field exerts a transverse force

on the moving charge carriers of the current which tends to push them to one side of the

conductor. A buildup of charge at the sides of the conductor will create an electric field

which will balance this magnetic influence, producing a measurable voltage between the

two sides of the conductor (Nationaal instituut voor subatomaire fysica 1988). The sign



2

of the voltage will be dependent on the sign of the charge of the predominant charge

carrier.

For simple metal and doped semiconductors, where there is only one type of charge

carrier (either electrons or holes), the Hall voltage, VH, is given by

VH = / (1.)

Where I is the current through the conductor’s length, B is the magnetic flux density,

d is the thickness of the conductor, e is the electron charge, and n is the charge carrier

density of the carrier electrons.

The Hall Coefficient is defined as

RH = EyjxB (2.)

Where j is the current density of the carrier electrons. In SI units, this becomes

RH = EyjxB = VH/ (3.)

In general, the Hall voltage is not a linear function of the magnetic field applied.

However, it can be easily calculated, if we assume that all carriers have the same drift

velocity. This can be done in two ways (Nationaal instituut voor subatomaire fysica

1988).

By assuming that carriers of only one type are present :

Metals and heavily doped semiconductors are examples of this. One carrier

dominates here. The Hall voltage comes out to be H = / . So the Hall

coefficient is given by RH = EyjxB = VH/ = . When one carrier dominates, the

conductivity of the material is



3

σ = eq (4.)

where µ is the mobility of the charge carriers. Thus

= σ (5.)

Also, for intrinsic conduction, the relationship between the conductivity, σ, and the

temperature, T, is given by:

σ = σ− (6.)

Since, R=ρl/A and ρ=1/

σ,

ln = [l n l n l n σ] + 2 1 (7.)

By assuming that carriers of both types are present :

Intrinsic and lightly doped semiconductors are of this type. The mobility of

both the carriers enter into the calculation, and a weighted average is used

= ℎ

2(ℎ + ) (8.)

Where ℎ and are the mobilities of holes and electrons and p, n are the

carrier densities of holes and electrons. Since the mobilities are a function of

temperature, the Hall coefficient is also a function of temperateure. Infact, the

Hall coefficient can even become zero or change signs with changes in

temperature.

Explicitly, the temperature dependence of the Hall Coefficient for the p doped

sample can be fitted to (Dept. of Physics n.d.)



4

()= 1

+ √ 4 + − 2 ) ( 1 )

+ √ 4 + − 2 )(1+) (9.)

Where = /ℎ , N0 is the effective density of states and is the

density of holes.

Similarly, the temperature dependence of the Hall Coefficient for the n doped

sample can be fitted to (Dept. of Physics n.d.)

()= 1 + √ 4 + − 2 )(1 1) + √ 4 + − 2 )(1+ 1)

(10.)

Where ns is the density of carrier electrons.



5

Method

The experiment was to be conducted in 3 parts.

The aim of part 1 was to measure the conductivity in undoped Ge to determine the

band gap. To accomplish this, the experimental apparatus was to be wired as shown in

Figure 1. After activating the data acquisition software to monitor the resistance, the

heater circuit is activated, and the resistance and temperature data collected.

Figure 2 - Wiring diagram for Hall voltage measurement

Figure 1 - Wiring diagram for resistance vs. temperature measurement



6

The next part of the experiment was to observe the Hall Effect, and calculate the Hall

coefficient and conductivity at room temperature. For this purpose the p and n doped

samples were used (i.e. this step was repeated twice; once for each sample). The the

experimental apparatus was wired as shown in Figure 2. After ensuring that a current of

approximately 1mA was passing through the circuit, the apparatus was placed within a

magnetic field, such that the face of the semiconductor strip was perpendicular to the

magnetic field Figure 3. The Hall Voltage, UH, was then measured for varying magnetic

fields.

The final part of the experiment was to study the effect of temperature on the Hall

Effect. For this part, the experimental apparatus was setup as in Figure 2. However, this

time, the magnetic field was set at a constant amount, while the temperature was varied

by activating the heater element of the circuit. The data acquisition software was used to

measure the temperature and Hall voltage, as the semiconductor was heated, and then

cooled. This part was to be repeated twice; once for the p doped material, and once for

the n doped material.

Figure 3 - Hall Effect



7

Exper imental Ap paratus Mal funct ion

Unfortunately, the electronic circuitry of the heater element of the experimental

board malfunctioned and was not working. Thus, I was unable to perform the first and

last parts of the experiment (as use of the heater element is an essential part of those

steps).

Professor Martin was kind enough to provide me with data from a similar experiment

that he had performed before. Unfortunately, on a detailed study of his data, I discovered

that it only contained resistance versus temperature data for the p and n doped samples

(Data Set 1), and Hall voltage versus magnetic field data for the p and n doped samples

(Data Set 2). Both these data sets, were not of any direct use for my purposes:

For the first part of the experiment I required resistance versus temperature data

for the undoped sample (whereas Data Set 1 was for the doped samples).

The data of Data Set 2 covered the same data that I was able to collect in the

second part of the experiment, hence I did not use it.

The third step required Hall Voltage versus Temperature data for the p and n

doped sample, which was not covered in the supplied Data Sets.

Analysis

As mentioned previously, due to a malfunction in the heater circuit, I was unable to

perform the first part of the lab. If I had been able to obtain the Resistance (R) versus

Temperature (T) data for the undoped Ge sample, I would have used it to plot ln(R)

versus 1/T for the undoped sample. From Equation (7.) it is expected, that for an undoped

sample, such a plot would be a straight line, whose gradient, m, may be used to determine



8

the band-gap of the undoped Ge sample,

= 2 (11.)

⇒ = 2 (12.)

As I was unable to experimentally determine a value for Eg, I will be using the

commonly accepted value of 0.66eV for further analysis (Bar-Lev 1984).

Figure 4 – Plot of UH versus B for p doped Ge



9

The data collected in the second part of the experiment, was used to plot Figure 4 and

Figure 5. Using the gradient, m, of these plots, and Equation 2:

= (13.)

⇒ =/ (14.)

For the p doped sample, d = 1mm, I = 0.9977 mA ± 0.01%, and m = 0.0033V/T±

0.03%,

⇒ R H = 3.31 x 10-3 m3/As ± 0.03%,

R H >0 combined with the orientation of the experimental apparatus (Figure 6) ⇒

the predominant charge carriers in p doped are positive holes as expected.

For the n doped sample, d = 1mm, I = 0.9993 mA ± 0.01%, and

m=-0.0045 V/T ± 0.01%,

Figure 5 - Plot of UH versus B for n doped Ge



10

⇒ R H = -4.503 x 10-3 m3/As ± 0.03%

R H <0 combined with the orientation of the experimental apparatus (Figure 6) ⇒

the predominant charge carriers in n doped are (negatively charged) electrons as

expected.

A coil of wire was pulled out of the magnetic, inducing an emf in it which was

measured using a multimeter. The direction of the induced emf, alongside

Faraday’s Law allowed us to confirm the direction of the magnetic field.

Next, using the equation

RH

=

, and that e = 1.602x10-19 C we may very

easily calculate the carrier densities:

For the p doped sample, density of carrier holes, p = 1.39x1021 m-3± 0.03%

For the n doped sample, density of carrier electrons, n = 1.89x1021 m-3± 0.03%

Next, a simple voltage-current was made across the semiconductors, to allow

calculation of their resistivity, ρ, at room temperatures. This resistivity value can then be

used along with the equation = , where = = and A is the cross-sectional

area:

For the p doped sample, l=20mm, w=10mm, d=1mm, I=0.908 mA ± 0.01%,

Figure 6 - Basic Schematic of Experimental Apparatus to Indicate Orientation in the B Field



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V=47.67mV ± 0.02% and R H calculated above

⇒ ρ = 0.0252 Ω ± 0.02% and = 0.126 m2/Vs ± 0.04%

For the n doped sample, l=20mm, w=10mm, d=1mm, I=0.822mA ± 0.01%,

V=30.6mV ± 0.02% and R H calculated above

⇒ ρ = 0.0186 Ω ± 0.02% and = -0.24 m2/Vs ± 0.04%

If I had been able to perform the final part of the experiment, I would have Hall

voltage versus Temperature data for the semiconductors. This data could have been fitted

to Equations 9 (for the p doped sample) and 10 (for the n doped sample), to determine k,

the ratio of the electron to hole motilities, and N0, the effective density of states for each

of the samples.

Conclusion

The Hall Effect provides a very useful means to determine various properties of a

semiconductor such as the nature of the predominant charge carrier, and the charge carrier

density. In the case of p-doped Ge it was confirmed to be positive holes, and in the case

of n-doped Ge it was confirmed to be electrons. We determined, that at room temperature

our sample of p doped Ge has a Hall coefficient of 3.31 x 10-3 m3/As ± 0.03% and a

carrier density of 1.39x1021 m-3 ± 0.03% at room temperature, and our sample of n doped

Ge has a Hall coefficient of -4.503 x 10-3 m3/As ± 0.03% and a carrier density of

1.89x1021

m-3

± 0.03%.



12

ReferencesBar-Lev, Adir. 1984. Semiconductors and Electronic Devices. 2nd. Prentice Hall.

Dept. of Physics. n.d. "Lab Manual PHYS 360/460." University of Waterloo.

LD Didactic GmbH. n.d. LD Physics Leaflets. Huerth: LD Didactic GmbH, 7.2.1.1,

7.2.1.4, 7.2.1.5.

Nationaal instituut voor subatomaire fysica. 1988. December 03.

http://www.nikhef.nl/~h73/kn1c/praktikum/phywe/LEP/Experim/5_3_02.pdf.

Pal, Sambit Bikas. n.d. Indian Institute of Science Education and Research.

http://sam.botcyb.org/reports/temperature_hall_coeff.pdf.

experiment19-resistivity and hall effect in p and n germanium

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