Experimental Study on Conduction Properties

of Semiconductors

Rajath S, Vipul Vaidya, Malay Singh

March 18, 2014

Contents

1 Introduction 1

2 Experiment 2

2.1 Aim . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2

2.2 Experimental Setup . . . . . . . . . . . . . . . . . . . . . . . . . 2

2.3 Procedure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3

2.4 Observations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4

2.4.1 p-Ge . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4

2.4.2 n-Ge . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6

2.4.3 Intrinsic Ge . . . . . . . . . . . . . . . . . . . . . . . . . . 8

3 Theory 8

3.1 Hall Eect: Brief Discussion . . . . . . . . . . . . . . . . . . . . . 8

3.2 Conduction Theories . . . . . . . . . . . . . . . . . . . . . . . . . 10

3.2.1 Free Electron Gas Model . . . . . . . . . . . . . . . . . . 10

3.2.2 Band Gap Theory . . . . . . . . . . . . . . . . . . . . . . 12

3.2.3 Fermi-Dirac Statistics . . . . . . . . . . . . . . . . . . . . 14

3.3 Magneto-resistance Eect . . . . . . . . . . . . . . . . . . . . . . 16

4 Analysis and Results 16

5 Bibliography 17

1 Introduction

Materials are classied as Conductors, Insulators and Semiconductors based on

electrical conductivity. Conductors have virtually unlimited number of charge

carriers and provide little resistance to the ow of current. Insulators have

no free charge carriers and do not conduct electricity. Semiconductors, on the

other hand, are slightly more interesting. They come in two varieties: Intrinsic

and Extrinsic. Semiconductors usually occupy Group 14 of the periodic table,

1

which means that they have 4 valence electrons. One distinguishing property

of semiconductors is the presence of two types of charge carriers: Holes and

Electrons. Conduction properties of semiconductors strongly depend on the

temperature and the nature of impurities present.

This report presents a series of experiments for studying the conduction

properties of semiconductors. The results are analyzed using the relevant theory

to determine the parameters of conduction. Several Conduction theories will be

explored in order to better understand the experiment. Hall eect plays a

crucial role in determining the conduction parameters. The connection between

Hall eect and the conduction properties will be discussed in detail.

2 Experiment

The experiment is divided into three parts. The rst part involves performing

a set of experiments described below, on a rectangular p-Ge sample

1

. All the

above experiments are repeated on a rectangular n-Ge sample in the second

part. In the third part, band gap of intrinsic sample of Ge is determined.

2.1 Aim

Observe Hall Eect in n-Ge and p-Ge. Determine the constant of proportionality in the relation between HallVoltage and Current. Observe the transition from extrinsic semiconductor

to intrinsic semiconductor by plotting Hall Voltage against Temperature.

Determine Specic Conductance, Type of charge carrier, Mobility of chargecarriers in the given n-Ge and p-Ge samples by studying the variation of

Hall eect with change in Magnetic Field.

Observe Magneto-resistance eect in n-Ge and p-Ge. Determine the Band Spacing of the p-Ge, n-Ge and intrinsic Ge sam-ple provided by analyzing the variation of Sample Voltage with changing

Temperature.

2.2 Experimental Setup

The list of apparatus:

p-Ge, n-Ge, intrinsic Ge rectangular sample Hall Eect Board(Heating equipment, Ammeter, Current Source, Ther-mometer)

2

Electromagnet(Copper Coil)1

Ge is the symbol for the element Germanium. Atomic Number: 32. Group: 4.

2

This board is provided by PhyWE. The exact circuitry/design is not available.

2

Hall Probe Power Supply Teslameter Multimeter

The use of each of the above component is self-explanatory. DC current is

passed through the copper coils to produce a magnetic eld. The DC Current is

supplied using the power supply, which can be adjusted to obtain the required

magnetic eld. The Power supply also provides AC Voltage of ~12V to the Hall

Eect Board. The Hall Probe is connected to the Teslameter. Hall Probe is

placed very close to the sample through the groove provided in the Hall Eect

board. The Hall Eect board is provided with two sockets: one for Sample

Voltage and the other for Hall Voltage. The multimeter is connected to either

of the sockets depending on the quantity being measured. The display on the

Hall Eect board can be switched between Current Mode and Temperature

Mode. The Button at the back of the board turns the heating apparatus of the

on/o.

2.3 Procedure

p-Ge and n-Ge:

1. At constant temperature and external magnetic eld, the Hall Voltage UHis measured as a function of Control Current I.

2. At room temperature and constant control current, specimen Voltage Uis measured as a function of Magnetic Field B.

3. At constant control current and zero external magnetic eld, measure

specimen Voltage U is measured as a function of Temperature T .

3

4. At room temperature and constant control current, Hall Voltage UH ismeasured as a function of Magnetic Field B.

5. At constant control current and constant external magnetic eld, Hall

Voltage UH is measured as a function of Temperature T .

intrinsic Ge:

1. At constant control current and zero external magnetic eld, measure

specimen Voltage U is measured as a function of Temperature T .

2.4 Observations

2.4.1 p-Ge

Task 1:

Hall Voltage UH vs. Control Current I at constant Temperature and MagneticField.

T = Room Temperature. B = 25mT .refer table (1)

Task 2:

Sample Voltage V vs. Magnetic Field B at constant Temperature and ControlCurrent.

T = Room Temperature and I = 30mArefer table (2)

Task 3:

Sample Voltage V vs. Temperature T at zero Magnetic Field and constantControl Current.

I = 30mA and B w 0Trefer table (3)

Task 4:

Hall Voltage UH vs. Magnetic Induction B at constant Control Current andTemperature.

I = 30mA and T = Room Temperaturerefer table (4)

Task 5:

Hall Voltage UH vs. Temperature T at constant Control Current and MagneticField.

B = 250mT and I = 30mArefer table (5)

4

Table 1: Hall Voltage UH vs. Control Current II(mA) UH(V ) I(mA) UH(V )-30 0.073 0 0.001

-25 0.063 5 -0.010

-20 0.051 10 -0.022

-15 0.039 15 -0.031

-10 0.028 20 -0.045

-5 0.016 25 -0.056

30 -0.065

Table 2: Sample Voltage V vs. Magnetic Field BB(mT ) V (V )50 -1.604

75 -1.607

100 -1.611

125 -1.616

150 -1.620

175 -1.625

200 -1.631

225 -1.637

250 -1.644

Table 3: Sample Voltage V vs. Temperature TT (C) V (V ) T (C) V (V )30 -1.74 90 -1.62

40 -1.86 100 -1.29

50 -1.96 110 -0.97

60 -2.03 120 -0.74

70 -2.02 130 -0.56

80 -1.88 140 -0.43

Table 4: Hall Voltage UH vs. Magnetic Induction BB(mT ) UH(V ) B(mT ) UH(V ) B(mT ) UH(V )-300 0.061 -120 0.024 60 -0.027

-270 0.057 -90 0.016 90 -0.033

-240 0.052 -60 0.006 120 -0.041

-210 0.046 -30 -0.006 150 -0.047

-180 0.040 0 -0.013 180 -0.053

-150 0.033 30 -0.020 210 -0.060

240 -0.066

5

Table 5: Hall Voltage UH vs. Temperature TT (C) UH(mV ) T (C) UH(mV )140 6.5 80 -32.9

130 7.2 70 -49.6

120 7.5 60 -61.5

110 5.5 50 -66.5

100 -0.09 40 -68.6

90 -13.2 30 -68.9

2.4.2 n-Ge

Task 1:

Hall Voltage UH vs. Control Current I at constant Temperature and MagneticField.

T = Room Temperature. B = 250mT .refer table (7)

Task 2:

Sample Voltage V vs. Magnetic Field B at constant Temperature and ControlCurrent.

T = Room Temperature and I = 30mArefer table (8)

Task 3:

Sample Voltage V vs. Temperature T at zero Magnetic Field and constantControl Current.

I = 30mA and B w 0Trefer table (9)

Task 4:

Hall Voltage UH vs. Magnetic Induction B at constant Control Current andTemperature.

I = 30mA and T = Room Temperaturerefer table (10)

Task 5:

Hall Voltage UH vs. Temperature T at constant Control Current and MagneticField.

B = 250mT and I = 30mArefer table (6)

6

Table 6: Hall Voltage UH vs. Temperature TT (C) UH(mV )140 7.4

130 10.2

120 13.8

110 18.4

100 24.3

90 30.9

80 37.1

70 41.4

60 43.8

50 45.1

40 45.7

Table 7: Hall Voltage UH vs. Control Current II(mA) UH(mV ) I(mA) UH(mV )-30 -54.5 0 0.0

-25 -43.4 5 6.6

-20 -38.9 10 13.2

-15 -27.4 15 23.7

-10 -20.0 20 29.7

-5 -12.1 25 41.5

30 45.4

Table 8: Sample Voltage V vs. Magnetic Field BB(mT ) V (V ) B(mT ) V (V )0 -0.954 120 -0.958

30 -0.955 150 -0.959

60 -0.955 180 -0.961

90 -0.956 210 -0.964

240 -0.966

Table 9: Sample Voltage V vs. Temperature TT (C) V (V ) T (C) V (V )40 -1.014 90 -0.984

50 -1.053 100 -0.860

60 -1.089 110 -0.729

70 -1.100 120 -0.573

80 -1.071 130 -0.450

140 -0.364

7

Table 10: Hall Voltage UH vs. Magnetic Induction BB(mT ) UH(mV ) B(mT ) UH(mV ) B(mT ) UH(mV )-300 -39.7 -120 -13.5 60 17.3

-280 -37.6 -100 -9.6 80 20.3

-260 -35.3 -80 -5.9 100 23.4

-240 -32.7 -60 -1.7 120 26.5

-220 -30.0 -40 1.3 140 29.6

-200 -27.1 -20 4.6 160 32.8

-180 -24.0 0 8.1 180 36.0

-160 -20.8 20 11.1 200 39.3

-140 -17.1 40 14.2 220 42.6

240 45.7

Table 11: Sample Voltage V vs. Temperature TT (C) V (mV ) T (C) V (mV )140 -0.68 90 -0.295

130 -0.86 80 -0.408

120 -0.115 70 -0.596

110 -0.213 60 -0.856

100 -0.208 50 -1.291

40 -1.950

2.4.3 Intrinsic Ge

Task 1:

Sample Voltage V vs. Temperature T at zero Magnetic Field and constantControl Current.

I = 5mArefer table (11)

3 Theory

The connection of Hall Eect with the conduction properties of the material

gives it a special status in the study of semiconductors. It has led to rise and

fall of several conduction theories, having tried to explain the experimental

results of Hall Eect.

3.1 Hall Eect: Brief Discussion

Hall Eect was discovered by E.H. Hall in 1879. If an electric current ows

through a conductor in a magnetic eld, the magnetic eld exerts a transverse

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

8

conductor. Consider a rectangular block of conducting material with a single

charge carrier.

~J is the current density in the direction of ow of current.~E is the Hall Electric eld.Consider the Lorentz force on electron owing inside the conductor,

~FB = e (~V ~B)

This force causes the electron concentration to increase in one side of the con-

ductor setting up an electric eld EHwhich counter-balances the Magnetic forceon the electron.

e ~EH = ~FB = e (~V ~B)

From electrodynamics,

~J = n e ~V

where variables take their usual meanings.

Hall Co-ecient is dened as the ratio of the induced electric eld to the

product of current density and the applied magnetic eld. From this denition,

RH =| ~EH || ~J ~B| =

1n e

Now, from Ohm's Law,

~J = ~E

=n e ~|V |

~|E|

9

Mobility is dened as,

=~|V |~|E|We obtain the most important relation,

RH = Thus, all conduction properties can be obtained by Hall Eect experiment.

For semiconductors, contributions from both holes and electrons must be in-

cluded in the calculations. If we let each charge carrier have a Hall Coecient:

Re (for electrons) and Rh (for holes) as derived before, and we assume the con-ductivity = h + h, the expression for total Hall Coecient can be derivedas:

RH =2hRh +

2eRe

(h + h)2

3.2 Conduction Theories

3.2.1 Free Electron Gas Model

This is the earliest of conduction theories. It was proposed by Paul Drude and

Hendrik Lorentz. This model proposed that, valence electrons of constituent

atoms move about freely through the volume of the solid. This model could

mostly explain all the conduction properties that depend on the kinetic prop-

erties of electrons, especially for metals. The solution for Schrdinger equation

in this case is as follows:

Free Particle Schrdinger equation in three dimensions

3

:

~22m

(52k) = kk

If the solid is a cube of length L

4

, the solution to above equation is:

n(~r) = A sin(pinxx

L) sin(

pinyy

L) sin(

pinzz

L)

nx, ny, nz are positive integers.

~k = kxx+ ky y + kz z =2nXpi

Lx+

2nypi

Ly +

2nzpi

Lz

Linear momentum ~p = ~ ~k, therefore particle velocity in orbital ~k is given by,

~v =~ ~km

3

Time Independent Schrdinger Equation

4

Electrons are conned to move inside the volume of the solid. So, it is more like a three

dimensional potential well.

10

~v is called Fermi velocity. When the solution is substituted back in the Schrdingerequation, we obtain Fermi Energy kas,

=~2

2m k2

For dierent choice of nx, ny, nz triplet, we obtain dierent energies correspond-ing to dierent orbital. There is one distinct choice for a triplet, for the volume

element ( 2piL )3. Thus, total number of orbitals N in a volume 4pi3 k

3f is,

2 4pi3 k

3f

( 2piL )3=

V

3pi2k3f = N

the factor 2 comes from Pauli exclusion principle

5

.

Therefore,

f =~2

2m(3pi2N

V)(

23 )(1)

This relates Fermi energy to Electron concentration. Density of states, dened

as number of orbitals per unit energy range is given by,

D() dNd

=V

2pi (2m

~2)32 12 (2)We can derive all the familiar conduction properties such as, Ohm's law from

these results:

F = md~v

dt= h

d~k

dt= e ( ~E + ~v ~B)

Collisions of electrons counter-balance the force on electrons resulting in steady

state. Continuing the derivation yields,

~J = ne~V = ne2 ~E

m

, where is the average collision time of electrons. Similarly, Hall Eect resultscan be derived using this theory.

Although this theory was reasonably successful, it led to some perplexing

results. The failures of this theory are:

Valance electrons are present in all the solids. But, only a few solids con-duct electricity. This theory failed to explain the existence of insulators.

Some solids have more than one valance electrons. But, experimentallyit is shown that not all valance electrons participate in conduction. This

question remained unanswered by this theory.

5

A maximum of two electrons of dierent spins, can occupy a single orbital.

11

Figure 1: Hall Eect Discrepancies

Results of Hall Eect showed the possibility of a positive charge carrier insome elements. This theory assumes outright that electrons are the only

charge carriers.

In some elements in Silicon, dierent samples yielded dierent results.This had no explanation in this theory.

Observe the Figure(1) below to see the discrepancies in Hall Eect result.

3.2.2 Band Gap Theory

In the previous model, we assumed that the electron is not under the inuence

of any potential when moving inside the volume of a solid. But, generally it is

not true. In fact, it is subject to a potential arising from the periodic lattice of

the solid. The Schrdinger equation is rewritten by perturbing the potential.

There are several models for the potential arising from the lattice. The common

feature of all these models is that the potential is periodic. We shall look at

it in a more qualitative way. The atoms in a solid are tightly bound to each

other in a regular geometric lattice

6

. The atoms are so close to each other

in a solid that their valance electron wave functions overlap. When two wave

functions combine, it splits into a symmetric and an antisymmetric part of lower

6

Amorphous solids lack denite lattice structure and the atoms are arranged irregularly.

But, important fact is that the atoms lie very close to each other.

12

Figure 2: Energy Bands in Sodium Atom

and higher energies respectively. The greater the number of interacting atoms,

the greater the number of energy levels produced by mixing of their valance

wave functions. In a solid, because there are as many energy levels as there are

interacting atoms, the levels are so close together that they form an energy band

that consists of virtually continuous spread of permitted energies.

An electron in a solid can only have energies that fall within its energy bands.

The various outer energy bands in a solid may overlap, in which case its valence

electrons have available a continuous distribution of permitted energies. In other

solids the bands may not overlap, and the intervals between them represent

energies their electrons cannot have. Such intervals are called forbidden bands

or band gaps. The energy bands, the gaps between them and the extent to

which they have been lled governs the electrical behavior of the solid. As

shown in the above picture, the valance band and the conduction band overlap

in Sodium. The 3s energy band of sodium is partly lled. Therefore, when

a potential dierence is applied, the valance electrons can pick up additional

energy while remaining in their original band. The additional energy is in

the form of Kinetic energy, and the drift of the electrons constitute the electric

current. In carbon, the valance band is fully lled and at least 6eV of additionalenergy is required for an electron to attain conduction band where it can move

about freely. In silicon, the situation is similar, but the band gap is only about

13

Figure 3: Energy Bands in Carbon and Silicon

1eV . Both thermal energy and electric potential are insucient to promote theelectrons to the conduction band. Therefore, carbon is an insulator. In silicon,

a small number of electrons have enough thermal energy to attain conduction

band. These electrons, though few are still enough to allow a small current to

ow when an electric eld is applied. Therefore, silicon is an semiconductor.

Whenever an electron is promoted from valance band to the conduction band, a

hole is created. Holes are positively charged and participate in the conduction

process. In an intrinsic semiconductor, number of holes is equal to the number

of electrons. The intrinsic conductivity and charge carrier concentration are

largely controlled by Eg/kT , the ratio of the band gap to the temperature.When this ratio is large, the intrinsic charge carrier concentration will be low

and the conductivity will be low.

Impurities can be added to the semiconductors to boost the conductivity.

This process is called doping. Doping introduces extra holes or electrons which

participate in conduction. When the majority charge carriers are holes, it is a

p-type semiconductor. When the majority charge carriers are electrons, it is an

n-type semiconductors.

3.2.3 Fermi-Dirac Statistics

Fermi-Dirac Statistics describes the distribution of identical particles satisfying

Pauli Exclusion principle in a system in thermodynamic equilibrium.

f() =1

ekT + 1

where, = energy of the particle and = chemical potential of the particle.

14

In an electric eld, at zero temperature, is the Fermi energy plus thepotential energy per electron.

The quantity is interpreted as the average free energy per particle.f() is the probability that the orbital with energy is occupied.At temperatures of interest, we suppose that the kT . Therefore,the function reduces to:

f() = ekT

The energy of an electron in the conduction band is given by:

k = Ec +~2k2

2me

where, Ec is the energy at the conduction band edge. From equation 2,

De() =1

2pi2

(2me~2

) 32

( Ec) 12

The concentration of electrons in conduction band is,

n =

Ec

De() f() d = 2(mekT

2pi~2

) 32

eEckT

For holes, fh = 1 fe and Ec is replaced by Ev . We obtain,

p =

Ev

Dh() fh() d = 2(mhkT

2pi~2

) 32

eEvkT

where p is the concentration of holes in the valance band.

np = 4

(kT

2pi~2

)3(memh)

32 e

EgkT

we obtain the equilibrium relation independent of , with Energy gap Eg =Ec Ev.Because the product of of electron and hole concentrations is constant inde-

pendent of impurity concentration at a given temperature, the introduction of

a small proportion of a suitable impurity to increase n must decrease p and viceversa. Therefore, we have established how carrier concentration is aected by

the change in temperature.

The electrical conductivity is the sum of electron and hole contributions:

= nee + peh

where, e =eemeand h =

ehmh.

There are dierent models for relaxation time / mean collision time. The

most common and the easiest one is taken as follows:

= T p

where p is usually small. Therefore, the temperature dependence of conduc-

tivity will be dominated by the exponential term in carrier concentration.

15

3.3 Magneto-resistance Eect

We have explored all the theory behind all the experiments except magneto-

resistance eect. We will have a brief introduction in this section. Magneto-

resistance is a property of the material to change the value of its electrical

resistance when an external magnetic eld is applied. In a simple model, sup-

posing the response to the Lorentz force is the same as the electric eld, the

carrier velocity ~v is given by,

~v = ( ~E + ~v ~B)where is the carrier mobility.Solving for ~v, we obtain:

~v =

1 + (B)2 ( ~E + ~E ~B)

where the reduction in mobility due to

~B is apparent. Electric currentwill decrease with increasing magnetic eld and hence the resistance of the

device will increase. In a semiconductor with a single charge carrier type, the

magneto-resistance is proportional to 1 + (B)2.We have covered all the relevant theory. Now, we shall analyze the experi-

mental data based on it.

4 Analysis and Results

Analysis of observations is done by plotting various quantities on a graph and

a correlation between the two variables is then established using the statistical

tools.

According to the theory established in the previous section, Hall Voltage

UH is directly proportional to the Control Current I. This fact directly comesfrom the Lorentz force law. Figure 4 shows the plot between UH and I for arectangular block of p-Ge crystal. The graphs conrms the linear relationship.

On using linear regression to nd the best t line, we obtain the equation:

UH = 2.3549 I + 3.2307. The same analysis for n-Ge crystal yields, theequation: UH = 1.6887 I 2.7846. See Figure 10 for the plot for n-Ge.Figure 5 and Figure 11 demonstrates the Magneto-resistance eect, for p-Ge

and n-Ge respectively. There is a clear change in resistance when the external

magnetic eld is changed. The graph can be approximated to a quadratic func-

tion which satises the theory of magneto-resistance for a semiconductor with

a single charge carrier type.

In the previous section, the relationship between conductivity and T1

is shown to an exponential one. This is conrmed by the linear nature of the

graphs plotted between ln() and T1(Figure 7 and Figure 13). The slope ofthese graphs have the value Eg/2k. Therefore, the band gap of p-Ge andn-Ge semiconductors are determined from the graph. Figure 6 and Figure 12,

shows both regions of intrinsic and extrinsic conduction. The linear t is drawn

16

only for intrinsic conduction case by reducing the data points in Figure 7 and

Figure 13. The same technique is used to determine the band gap of intrinsic

Ge semiconductor. Refer to Figure 16 and Figure 17 for the graphs. The slopes

are as follows: p-Ge: 2809.7 , n-Ge: 2691.8, intrinsic Ge: 4344.5 . Theerror is calculated from the standard deviation of the slopes. The band gap

values are: p-Ge: 0.4849eV 0.0539eV ,n-Ge: 0.4646eV 0.0416eV , intrinsicGe: 0.7499eV 0.0050eV .Graphs in Figure 8 and Figure 14 explore the relationship between Hall

Eect and the Magnetic Field. As expected, the plot is linear in nature. The

slope of this graph is useful in calculating properties such as Hall Coecient,

Carrier Mobility, Charge Carrier Type and the Charge Carrier Concentration.

The slope of p-Ge is: 0.25058 and n-Ge is 0.16326. Hall Voltage UH = | ~EH |dis related to Hall Coecient RH , by the relation:

RH = UHB

dI

This is directly obtained by substituting the value of Hall Voltage UH in theHall Coecient RH equation. For the given sample, d = 10

3m. Therefore, thevalue of RH for p-Ge and n-Ge samples are (0.0083520.000143)m3A1s1 and(0.005442 0.000037)m3A1s1 respectively. From part 2 of the experiment,the resistance of p-Ge and n-Ge samples are determined. The Conductivities

of the samples are as follows: p-Ge : 37.031m1 and n-Ge: 62.601m1.Mobilities: p-Ge : 0.308m2V 1s1 and n-Ge: 0.34m2V 1s1. Charge Carrierconcentration: p-Ge: 7.5 1020m3 and n-Ge: 11.57 1020m3. The sign ofRH conrms the type of charge carriers in both n-Ge and p-Ge samples.The last part of the experiment, i.e, Figure 9 and Figure 15, show the rela-

tionship between Hall Voltage UH and the temperature T . Since, both currentI and magnetic eld B are kept constant in the experiment, the only factorthat changes the Hall Voltage is the charge carrier concentration. The general

trend of the graph is that the Hall Voltage falls as the temperature rises which

is explained by the inverse relationship between charge carrier concentration

and Hall Voltage. As temperature rises, charge carrier concentration increases

due to thermal excitation of more and more valance electrons. This results in

decrease of Hall Voltage.

5 Bibliography

1. Concepts of Modern Physics, Arthur Baiser

2. Introduction to Solid State Physics, Kittel

3. Hall Eect and Semiconductor Physics, E H Putley

17

Figure 4: Linear t of Hall Voltage vs. Current : p-Ge

-30 -20 -10 0 10 20 30I (mA)

-50

0

50

U (m

V)

Figure 5: Resistance vs. Magnetic Field: p-Ge

0 50 100 150 200 250 300B (mT)

-54.6

-54.4

-54.2

-54

-53.8

-53.6

-53.4

R (O

hm)

18

Figure 6: Natural Log of Conductivity vs. Inverse of Temperature: p-Ge

0.0024 0.0026 0.0028 0.003 0.00321/T (1/K)

-0.5

0

0.5

1/V

(1/V

)

Figure 7: Linear t of Natural Log of Conductivity vs. Inverse of Temperature,

Intrinsic Conduction: p-Ge

0.0024 0.0026 0.0028 0.003 0.00321/T (1/K)

-0.5

0

0.5

1/V

(1/V

)

19

Figure 8: Linear t of Hall Voltage vs. Magnetic Field: p-Ge

-300 -200 -100 0 100 200 300B (mT)

-60

-40

-20

0

20

40

60

U (m

V)

Figure 9: Hall Voltage vs. Temperature: p-Ge

0 50 100 150 200T (degree C)

-60

-40

-20

0

U (m

V)

20

Figure 10: Linear t of Hall Voltage vs. Current : n-Ge

-30 -20 -10 0 10 20 30I (mA)

-60

-40

-20

0

20

40

U (m

V)

Figure 11: Resistance vs. Magnetic Field: n-Ge

0 50 100 150 200 250 300B (mT)

31.8

31.9

32

32.1

32.2

32.3

R (O

hm)

21

Figure 12: Natural Log of Conductivity vs. Inverse of Temperature: n-Ge

0.0024 0.0026 0.0028 0.003 0.00321/T (1/K)

0

0.5

1

1.5

2

1/V

(1/V

)

Figure 13: Linear t of Natural Log of Conductivity vs. Inverse of Temperature,

Intrinsic Conduction: n-Ge

0.0024 0.0026 0.0028 0.003 0.00321/T (1/K)

0

0.5

1

1.5

2

1/V

(1/V

)

22

Figure 14: Linear t of Hall Voltage vs. Magnetic Field: n-Ge

-300 -200 -100 0 100 200 300B (mT)

-40

-20

0

20

40

U (m

V)

Figure 15: Hall Voltage vs. Temperature: n-Ge

50 100 150 200Temperature (degree C)

0

10

20

30

40

50

U (m

V)

23

Figure 16: Natural Log of Conductivity vs. Inverse of Temperature: intrinsic

Ge

0.0024 0.0026 0.0028 0.003 0.00321/T (1/K)

0

1

2

3

ln(si

gma)

Figure 17: Linear t of Natural Log of Conductivity vs. Inverse of Temperature,

Intrinsic Conduction: intrinsic Ge

0.0024 0.0026 0.0028 0.003 0.00321/T (1/K)

0

1

2

3

ln(si

gma)

24

IntroductionExperimentAimExperimental SetupProcedureObservationsp-Gen-GeIntrinsic Ge

TheoryHall Effect: Brief DiscussionConduction TheoriesFree Electron Gas ModelBand Gap TheoryFermi-Dirac Statistics

Magneto-resistance Effect

Analysis and ResultsBibliography