Chapter (1).docx

Chapter-1 INTRODUCTION TO SEMICONDUCTOR MATERIAL Semiconductors are the materials with small band gap energy, typically between 0.5eV to 1eV. Semiconductors are broadly classified as follows:  A-Intrinsic Semiconductors  B-Extrinsic Semiconductors (i)

n-type Semiconductors

(ii)

p-type Semiconductors

The elemental forms of pure Si and pure Ge are intrinsic. They are the pure form of semiconductors and are not useful. Therefore, they doped with specific dopants to make extrinsic semiconductors. Extrinsic forms are directly useful and widely employed in manufacturing in solid state devices. They belong to categories of alloys and compounds. Electronic industries required purity better than 1:109 in pure Si and Ge. At room temperature, resistivity of semiconductor lies between an insulator and a conductor. Semiconductors show negative temperature coefficient of resistivity. This means its resistance decreases with increase in temperature. Both Si and Ge are elements of IV group i.e. both elements have 4 valence electrons. Both form the covalent bond with the neighboring atom. At absolute zero temperature both behave as insulator i.e. the valence band is completely full while conduction band is completely empty. The resistivity of semiconductor varies from 10-5 to 104 ohm-m.as compare to the values from 10-8 to 10-6 ohm-m for conductor and from 107 to 108 ohm-m for an insulator. The semiconductors have following additional characteristic properties:  They have an empty conduction band and filled valance band at 0K.  Semiconductors are formed by covalent bonds.  They have a small energy band gap.  They possess crystalline nature.  The temperature coefficient of resistance of a Semiconductor is negative

Page 1

1.1 CLASSIFICATION: Based on the composition of materials, purity of the material and nature of band gap, semiconductors are classified into three categories.  Elemental Semiconductors (Si and Ge)  Compound Semiconductors (GaAs, InPi, CdTe, ZnSe etc.)  Alloy (GaAsxP1-x, HgCdxTe1-x etc.)

1.2 ADVANTAGES OF SEMICONDUCTING MATERIALS:  They are much smaller in size.  They are light in weight.  Operates at very low voltage.  They consume negligible power.  Highly durable.  They do not develop any creep and edging effect.  They can operate at wide range of temperatures.

1.3 Fermi Energy Level Fermi level is the highest occupied energy level at absolute zero temperature. This concept comes from Fermi-Dirac statistics. At absolute zero they pack into the lowest available energy states. The Fermi level is the surface of that sea at absolute zero where electrons will not have enough energy to raise the surface. The concept of the Fermi energy is a crucially important concept for the understanding of the electrical and thermal properties of solids. In metals, the Fermi energy gives us information about the velocities of the electrons which participate in ordinary electrical conduction.

1.4 Fermi-Dirac Probability Function and Temperature Effect The thermal behavior of electrons in an atom can be explained by Fermi-Dirac probability function P(E) given by P( E ) 

1 1 e

( E  E F ) KT

(2)

Page 2

Where: k=Boltzmann constant T=Absolute temperature EF=Fermi energy

Fig.1 (Fermi energy Vs. Temperature)

Thus the Fermi energy level can also be defined as the level where the probability of occupation is 50% at temperature greater than 00K. The Fermi-Dirac distribution can be concluded by following points:  At 0K, it shows that P (E) =0 for E > Ef and 1 for E<Ef.  At higher temperature more and more electrons occupy energy greater than Fermi energy.

Page 3

CHAPTER-2 Band Theory of Solids The farthest band from the nucleus is filled with valence electrons and is called the valence band. The empty band is called the conduction band. The energy of the highest filled state is called Fermi energy. There is a certain energy gap, called band gap, between valence and conduction bands. In metals the valence and the conduction band overlaps each other. Insulators and semiconductors have completely filled valence band and empty conduction band. The band gap is relatively smaller in semiconductors while it is very large in insulators. The difference between conductors, insulators and semiconductors has been shown in the following diagram. In insulators the electrons in the valence band are separated by a large gap from the conduction band, in conductors like metals the valence band overlaps the conduction band, and in semiconductors there is a small enough gap between the valence and conduction bands. With such a small gap, the presence of a small percentage of a doping material can increase conductivity in semiconductor. An important parameter in the band theory is the Fermi level, the top most available electron energy states at low temperatures. The position of the Fermi level with the relation to the conduction band is a crucial factor in determining electrical properties.

Fig.2 (Energy Band Gap of Materials) Page 4

2.1 Density of state Density of state means the population density of electrons in a given energy state. It has relevance to Fermi- Dirac distribution. In the last article we discussed that the Fermi probability function p(E) determines the probability of occupancy in a given level. It tells us about the energy level but not about the number of electrons in those levels. The density of state N(E) indicates the number of electrons n across the energy band. This number is not uniform across the energy band; rather it is greatest at the center of the band. The product of p(E), N(E) and the number of electrons for semiconductors are related by

This relation is illustrated over a range of band energy at 0 K and temperature above 0 K. It illustrates that only a small fraction of electrons within the energy range of kT can be excited above Fermi level. Here k is Boltzmann constant and T is absolute temperature. The effective density of energy states can be found by employing the quantum mechanics. If the effective density of state in conduction and at the valence bands are Nc and Nv respectively, then,

Fig.4 (variation of density of state with temperature) Page 5

(9)

Where me and mh are the effective mass of an electron and a hole respectively; and h is the Planck’s constant. The number of negative and positive charge carrier’s and nh in their respective bands may be found from

(10) And

(11) Where EF is the Fermi energy. Now the product of positive and negative charge carriers is

(12) And the term

depends on the band structure of the semiconductor. For a specific material,

the produced,

Page 6

Chapter-3

INTRINSIC SEMICONDUCTORS Semiconductors which are chemically pure, Semiconductors or

or free of impurities,

Undoped Semiconductor or type-I Semiconductor.

intrinsic semiconductors are Silicon (Si) and Germanium (Ge). each in their outer-most i.e. valence shell

are called Intrinsic The most common

Si and Ge have four electrons

these electrons are called valence electrons and

are responsible for the electrical properties of the semiconductors. Pure silicon behaves like an insulator at 0K. As the temperature increases above 0K, some of valance electron acquires sufficient thermal energy to break their covalent bond. These electrons move randomly in the crystal and refer to as conduction electron. Each electron leaves behind an empty space called hole which act as current carrier. So that valance band break and electron hole pair generates and two carrier’s conductivity is produced. This type of generation of free e-h pair in semiconductors is called thermal generation.

At 0K the valance band is completely filled and conduction band is empty. So the semiconductor behaves as an insulator. The electrons present in the valance band do not conduct. As temperature increases some of the valance electron acquires thermal energy and jump into the conduction band, leave behind equal no of holes in valance band. The electrons in the conduction Page 7

band and holes in the valance band behave as a free carrier and increases conductivity of the material.

3.1 Conduction in Intrinsic Semiconductors: The electrons in the conduction band and hoes in the valence band moves in a random fashion within the crystal due to their thermal energy. When an external field is applied to the semiconductor, a drift velocity is superimposed on the random thermal motion of the charge carriers, i.e. electrons and holes. The drift of the electrons in the conduction band and that of holes in the valence band produce an electric current. The electrons move towards the positive electrode, whereas the holes towards the negative electrode. The currents produced by the movement of electrons and holes in opposite directions and since the electron carry a negative charge and the hole a positive charge. Thus, the conventional current flows within the semiconductor from the positive electrode to the negative electrode. The energy of a hole is measured downward from the top of the valence band.

The motion of the electrons in the valance band may be considered to be equvalent to the motion of the holes in the opposite direction. The holes also contibutre to the conductivity. When the e-h pair is thermaly created , a valance electron in a neighbouing atom can have sufficient thermal energy to jump into the position of the hole and reconstruct the covalent bond.In doing so,the electron leaves a hole in its initial position. Effectively, the hole moves from one position to the other position. Thus the holes move in the opposite direction to the valance electrons.

Fig.7 left: conduction of charge carriers, right: Fermi Level for Intrinsic Semiconductor Page 8

3.2 Sailent Features of Intrinsic Semiconductor: 

The number of electron in the conduction band is equal to the number of holes in the valance band,i.s. n=p=ni, where ni is the intrinsic carrier concentration.



Fermi level lies in the energy gap exactly between the valance and conduction bands(Ef=Eg/2).



The contribution of the electrons to the electric current is more than that due to holes.



An atom out of 103 atoms of an intrinsic semiconductor contributes to the conduction.



An electron and a hole can behave as a pair bound to each other. Such bound pair is called exciton. This pair is electrically neutral and so does not take part in electrical conduction.

3.3 Effective Mass: The mass gained by the electron due to application of external electric field and internal periodic field is termed effective mass. It represented by m*. The effective mass is calculated by following formula. m*= -e E / a Where;

(13)

e= Electronic charge

a = Acceleration due to electric current, m*=Effective mass E= Electric field strength

3.4 Concentration of Intrinsic Carriers: The concentration of intrinsic carriers i.e. the number of electrons in conduction band per unit volume is given by the expression. n =2(2πmekT/h2)3/2exp (µ-Eg)/kT

(14)

Page 9

and the concentration of holes in the valance band is given by the expressionp = 2(2πmhkT/h2)3/2 exp (-µ/kT)

(15)

in an intrinsic semiconductors, the concentration of holes and electrons are equal according to mass action law, the product of hole and the electron concentration isequal to the square of the intrinsic concentration, i.e., np= ni2=4(2πkT/h2)3(memh)3/2 exp(-Eg/kT)

(16)

Where me=Effective mass of electron mh=Effective mass of hole k= Boltzmann constant Eg= Band gap µ= Fermi level T=Absolute temperature (K) By multiplying and dividing by m3 in above equation, we get

ni2=4(2π kT /h2)3/2 (me mh)3/4 exp(-Eg/2kT)

(17)

thus, we can conclude that the concentration of intrinsic carrier depend exponentially on Eg/2kT.

Page 10

3.5 Number of Electrons Crossing-Over to Conduction Band If the total number of electrons available in valence band for semi conduction is N, and out of this n numbers cross-over to conduction band, then n/N = p(E) will be . n= Ne (-Eg/2kT)

(18)

3.6 Holes, Mobility and Conductivity If n number of electrons crosses the gap, n sites become vacant in the valence band. These vacant sites are called holes. Thus the number of electrons ne and number of holes nh are equal (ne= nh). Both electrons and the holes take part in semi conduction. Electrons conduct in the conduction band and the holes in the valence band. They move in opposite directions with certain drift velocity Vd under an applied field gradient E . This movement of electrons and holes is known as mobility. Mobility of an electron and of a hole is designated by µe and µh respectively and defined as

µ=Vd/E

(19)

Page 11

Chapter-4

EXTRINSIC SEMICONDUCTORS The semiconductors with intentionally added impurities are called extrinsic semiconductors. This process of adding impurities in minute quantities into the pure semiconductor material under controlled conditions is known as doping. The introduction of impurity atom i.e. doping, is most efficient method of increasing the conductivity of intrinsic. Those intrinsic semiconductors to which some suitable impurity or doping agent or doping has been added in extremely small amounts (about 1 part in 108) are called extrinsic or impurity semiconductors. Depending on the type of doping material used, extrinsic semiconductors can be sub-divided into two classes: 

N-type semiconductors (Penta-valent Impurities)



P-type semiconductors. (Tri-valent Impurities)

4.1 n-Type Semiconductors and Their Energy Diagram This type of semiconductor is obtained when a pentavalent material like antimony (Sb) is added to pure silicon/germanium crystal. As shown in the Figure each antimony atom forms covalent bonds with the surrounding four germanium atoms with the help of four of its five electrons. The fifth electron is superfluous and is loosely bound to the antimony atom. Hence, it can be easily excited from the valence band to the conduction band by the application of electric field or increase in thermal energy. It is seen from the above description that in N-type semiconductors, electrons are the majority carriers while holes constitute the minority carriers. The energy level corresponding to the 5th electron lies in the band gap in below the conduction band, this level is known as donor level. The depth nearly about 0.01eV for Ge and 0.03eV for Si.

Page 12

The electrons are easily transferred to the conduction band leaving behind positively impurities ion. Such impurities are known as donor or n type semiconductor. Here current is carried mainly by electrons which is called majority carrier. The thermally generated holes are known as minority carrier.

4.2 p-type Semiconductors and Their Energy Diagram This type of semiconductor is obtained when traces of a trivalent impurity like boron (B) are added to a pure silicon/germanium crystal. In this case, the three valence electrons of boron atom form covalent bonds with four surrounding germanium atoms but one bond is left incomplete and gives rise to a hole as shown in Fig. below. Thus, boron which is called an acceptor impurity causes as many positive holes in a germanium crystal as there are boron atoms thereby producing a P-type extrinsic semiconductor. In this type of semiconductor, conduction is by the movement of holes in the valence band. E.g.; group III, such as boron, aluminum, gallium, and indium

Page 13

An electron can easily have transferred from valance band to the accepter level by providing the small amount of energy. This creates a hole in valance band which act as mobile charge carrier. The impurities are known as acceptor, and the semiconductor containing such impurity atom is known as p type semiconductor.

4.3 Temperature Dependency of Carrier Concentrations The carrier concentration is a function of temperature. It is evident from the following relations. N0=ni exp (EF-Ei) / kT

(20)

P0=ni exp (Ei-EF ) / kT

(21)

Where, N0=electron concentration in conduction band. P0=hole concentration in conduction band.

Fig.11 (Variation of Conductivity with Temperature in Semiconductors) The following curve shows three different regions: 1. Ionization region- It occurs at large values of 1/T i.e. at low temperatures where donor electrons are bound to donor atoms.

2. Extrinsic region-It occurs when every available extrinsic electron is transferred to the conduction band, hence ni=Nd and n0becomes almost constant. Page 14

3. Intrinsic region-It occurs at smaller values of 1/T i.e. at higher temperatures when ni>>Nd

Amongst the above three regions, the extrinsic region is desirable for the operation of semiconductor devices. It is due to its constant characteristic. The extrinsic range can also be extended beyond the highest temperature at which the device has to operate. This is accomplished by either  Generating the thermal electron-hole pair  Doping Nd

4.4 Effects of Temperature on Mobility of Carriers The mobility of electrons or holes is influenced by scattering. Main sources of scattering in a semiconductor are phonons and ionized impurity atoms. The mobility of electron and hole carriers are governed by following scattering mechanisms, which themselves result from temperature.  Lattice scattering. In this mechanism, a carrier moving through a crystal is scattered by inherent vibration of the lattice, caused by temperature. The frequency of such scattering increases with increase in temperature, therefore the thermal agitation of the lattice also becomes greater. Hence the mobility decreases.  Impurity scattering. In this mechanism, the scattering dominates low temperatures. At lower temperatures, as the atoms are less agitated, therefore the thermal motion of the carriers is slow. Hence, there is an increase in mobility with increasing temperature.

The approximate temperature dependencies for both the above mechanisms are as follows. (i)

For Lattice Scattering µL=a T -1.5

(ii)

For Impurity Scattering µi=b T -1.5

Where a and b are material constant

Page 15

Fig.12 (Effect of Temperature on Carrier’s Mobility)

4.5 Direct and Indirect Energy Band Semiconductors The band structure of GaAs for momentum(K) = 0 has a maximum in valence band and a minimum in conduction band .Therefore an electron making a transition from conduction band to valence band can do so without any change in the value of momentum(K). The band structure for Si has a maximum in valence band at a different value of k than a minimum in conduction band. Therefore, a transition from conduction band to valence band requires some change in the value of momentum (K). Therefore, the energy band in semiconductor has two different classes:

1- Direct energy band 2- Indirect energy band

Page 16

Thus the semiconductors are falling in two categories:

1- Direct energy band semiconductor e.g. GaAs, GaN, InP, CdSe, InSb etc. 2- Indirect energy band semiconductor e.g. Si, Ge, GaP, PbTe, AlAs etc.

Fig.13 (Energy Momentum Diagram for Direct Band Gap Semiconductors e.g.GaAs)

Fig.14 (Energy Momentum Diagram for Indirect Band Gap Semiconductors e.g. Pure Si)

Page 17

Chapter-5 HALL EFFECT The measurement of electrical conductivity is merely not sufficient to determine the carrier’s concentration or type of the conductivity. These measurements will not reveal whether the conductivity is due to the electrons and holes. Therefore it is very difficult to distinguish between P type and N type semiconductor. The Hall Effect is used to overcome these problems. When a semiconductor material carrying a current is placed perpendicular to a magnetic field, potential difference is developed in the conductor in the direction perpendicular to both the electric and magnetic fields. This phenomenon is known as Hall Effect.

Let Ix be the current flowing through the specimen along the X direction and Bz be the transverse magnetic field applied along the z direction. An electric field Ey is induced in a direction perpendicular to both the current and the magnetic field. This phenomenon is known as HALL EFFECT, and the voltage develop is called hall voltage.

Fig.1.a. (Flow of Current in Semiconductor in Absence of Magnetic Field)

Fig.1.b. (Flow of Current in Semiconductors in Presence of Magnetic Field e.g. Hall Effect) Page 18

5.1 Explanation of the Phenomenon Consider a conducting slab as shown in the following figure, with length L in the x direction, width w in the y direction and thickness t in the z direction.

Fig.2(Diagram Showing Hall Effect)

Assume the conductor to have charge carrier of charge q (can be either positive or negative or both, but we take it to be of just one sign here), charge carrier number density n (i.e., number of carriers per unit volume), and charge carrier drift velocity V x when a current flows in the positive x direction. The drift velocity is an average velocity of the charge carriers over the volume of the conductor; each charge carrier may move in a seemingly random way within the conductor, but under the impudence of applied electric field there will be a net transport of carriers along the length of the conductor. The current Ix is the current density Jx times the cross-sectional area of the conductor. The current density Jx is the charge density nq times the drift velocity Vx. In other words

(1) The current Ix is caused by the application of an electric field along the length of the conductor E. In the case where the current is directly proportional to the field, we say that the material obeys Ohm’s law which may be written

Page 19

(2) Where σ as is the conductivity of the material in the conductor. Now assume that the conductor is placed in a magnetic field perpendicular to the plane of the slab. The charge carriers will experience a Lorentz force qvB that will deflect them toward one side of the slab. The result of this deflection is to cause an accumulation of charges along one side of the slab which creates a transverse electric field Eyi that counteracts the force of the magnetic field (Recall that the force of an electric field on a charge q is qE). When steady state is reached, there will be no net flow of charge in the y direction, since the electrical and magnetic forces on the charge carriers in that direction must be balanced. Assuming these conditions, it is easy to show that

(3) Where Ey is the electric field developed in the material, and is called the Hall field in the y direction and B the magnetic field in the z direction. In an experiment, we measure the potential difference across the sample the Hall voltage Vz which is related to the Hall field by H

(4) After solving the equations (1), (3) and (4), we will get

(5) The term in parenthesis is known as the Hall coefficient:

RH=1/nq

Page 20

It is positive if the charge carriers are positive and negative if the charge carriers are negative. In practice, the polarity of the hall coefficient determines the sign of the charge carriers. Note that the SI units of the Hall coefficient are [M3/C] or more commonly stated [M3/A-s].

5.2 Significance of Hall Effect: The idea of Hall Effect is utilized to determine:a. Types of an unknown semiconductor; whether n type or P-type b. Mobility of the semiconductor, c. Conductivity d. Resistivity of the semiconductor. e. Hall voltage and drift current. f. Hall Effect is also used in magnetically activated electronic switches. g. Hall effect semiconducting devices are used as sensor to sense Magnetic fields.

5.3 Hall Effect experiment 5.3.1 Objectives: 1. To study Hall Effect and to determine, (i)

Hall voltage VH

(ii)

Hall coefficient

2. To determine the type of majority carrier’s i.e. whether the semiconductor crystal is of n-type or p-type. 3. To determine the charge carrier density or carrier concentration per unit volume in the Semiconductor crystal. 4. To determine the magnitude of Pointing Vector. 5. To determine the Hall angle. 6. Mobility of charge carriers. 7. Resistivity of the sample.

Page 21

5.4 Hall Effect Experiment Setup: Apparatus:

INDOSAW SK006 Hall Effect apparatus

It consists of: I.

Power supply for electromagnet, i. Specifications: 0-16 V, 5 Amps.

II.

Power supply, i. Specifications: 0-20 mA

III.

Gauss meter with Hall Probe

1. Semiconductor p-type Ge crystal mounted on PCB i. Specification: P-type Ge crystal Thickness 0.5 mm Width 4.0mm Length 6.0 mm

2. Multimeter with mV scale for measuring Hall voltage.

3. Hand held Multimeter with mm scale for measuring current through the sample.

4. Hall Effect apparatus (electromagnet, pole, pieces and pillars) consists Of two 500 turns coil.

Page 22

Fig.3 Hall Effect set-up (CIPET Bhubaneswar)

Fig.4 (Block Diagram for Experimental Setup)

Page 23

5.5 Observations: Width of the specimen,

b =4mm =10-3 m

Length of the specimen,

L = 6mm =10-3m

Thickness of specimen,

t = 0.0005 m

Least count for Ammeter

= 0.01 mA

Least count for voltmeter

= 0.1 mV

Least count for gauss meter = 0.1 Gauss

Fig.5 (PCB with p-type Ge crystal)

Page 24

5.6 Table for Hall Voltage:

Current S.No I (mA) 1 0.5 2 1

B=280 ,VH 0.4 0.8

Vh/I 0.8 0.8

3 4 5

1.5 2 2.5

1.1 1.5 1.9

6

3

7 8

B=240, VH 0.5 0.8

Vh/I 1 0.8

B=200, VH 0.5 0.7

B=160, VH 0.5 0.6

Vh/I 1 0.7

0.73333333 1.1 0.75 1.4 0.76 1.7

0.733333333 0.7 0.68

1 1.3 1.6

0.666666667 0.8 0.65 1.1 0.64 1.3

0.533333 0.55 0.52

2.2

0.73333333 2.1

0.7

1.9

0.633333333 1.5

0.5

3.5 4

2.6 3

0.74285714 2.4 0.75 2.7

0.685714286 0.675

2.2 2.5

0.628571429 1.8 0.625 2.1

0.514286 0.525

9 10

4.5 5

3.4 3.8

0.75555556 3 0.76 3.4

0.666666667 0.68

2.8 3.1

0.622222222 2.3 0.62 2.6

0.511111 0.52

11

5.5

4.1

0.74545455 3.8

0.690909091

3.4

0.618181818 2.9

0.527273

12

6

4.5

0.75

4.1

0.683333333

3.7

0.616666667 3.1

0.516667

13

6.5

4.9

0.75384615 4.5

0.692307692

4.1

0.630769231 3.4

0.523077

14

7

5.3

0.75714286 4.9

0.7

4.4

0.628571429 3.6

0.514286

15 16

7.5 8

5.8 6.1

0.77333333 5.3 0.7625 5.6

0.706666667 0.7

4.7 5

0.626666667 3.9 0.625 4.1

0.52 0.5125

17

8.5

6.6

0.77647059 5.9

0.694117647

5.3

0.623529412 4.4

0.517647

18

9

7

0.77777778 6.2

0.688888889

5.5

0.611111111 4.6

0.511111

19 20

9.5 10

7.5 7.9

0.78947368 6.6 0.79 7

0.694736842 0.7

5.8 6.1

0.610526316 4.9 0.61 5.1

0.515789 0.51

Vh/I 1 0.6

Page 25

A graph was plotted between Hall voltage (VH) Vs current (I) as follow:

Hall Voltage Vs. Current 9

8

Hall Voltage (mV)

7 6 5

B = 2800 Gauss

4

B = 2400 Gauss

3

B = 2000 Gauss

2

B = 1600 Gauss

1 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 5.5 6 6.5 7 7.5 8 8.5 9 9.5 10 Appled Current (mA)

5.7 Calculation: 1. Mean value of VH / I= 0.668270614 ohms 2. Mean Hall coefficient RH= VH t/ I x B = 0.00154557 V m / A T

3 Carrier concentration n = 1/(1.6 x 10-19 x RH) = 4.04381555 x 1023

Page 26

5.8 Table for Resistivity:

S.N

1 2 3 4 5 6 7 8 9 10

Current Distance between two points I (mA) between which potential difference is measured L (m) 0.1 0.2 0.3 0.4 0.5 2.9 x 10-3 0.6 0.7 0.8 0.9 1.0

VL (mV) VL / I

0.34 0.64 0.97 1.27 1.58 1.91 2.21 2.53 2.83 3.14

ρ = VLBt /L (10-3)

3.400 2.346 3.200 2.208 3.230 2.229 3.175 2.191 3.160 2.181 3.167 2.185 3.157 2.178 3.163 2.183 3.144 2.169 3.140 2.167 Mean ρ = 2.2037x10-3Ω m

Resistivity is calculated from the Tabulated Data as above is about ρ = 2.2037 x 10-3 Ω m. 4 Mobility of carriers µm = RH / ρ = 0.00154557 /2.2037 x 10-3 = 7.0135227m2 V-1 s-1 5 Hall angle tanϴH= µm x B =7.0135227 x0.76 =5.330277252 ϴH =79.37439462 0

Results: From the above analysis we obtained the following important results 1. Mean Hall coefficient RH = 0.00154557 V m / A T 2. Carrier concentration n = 4.04381555 x 1023 3. Mobility of carriers µm = 7.0135227m2 V-1 s-1

Page 27

Chapter-6

FOUR PROBE EXPERIMENT: THEORY: This technique is also known four terminals sensing, four wire sensing. This method is used to measure the resistivity of semiconductors. In its useful form, the four probes are collinear. The error due to contact resistance, which is especially serious in the electrical measurement of semiconductors, is avoided by the use of two extra contacts (probes) between the current contacts. in this arrangement the contacts resistance may all be high compare to the sample resistance, but as long as the resistance of the sample and contact resistances are small compared with the effective resistance of the voltage measuring device (potentiometer, electrometer, or electronic voltmeter), the measured value will remain unaffected. Because of the pressure contacts, this is also especially use full or quick measurement on different samples or sampling different parts of the same sample.

Apparatus: SK012- FOUR PROBE APPARATUS 6.1 Description of experimental setup: This setup consists a four probe arrangement with a p-type Ge sample, oven. The basic unit contains digital display for voltmeter current and temperature, is has in built power supply for oven. Following Figure shows various parts of apparatus.

Specifications: 1-Probe ArrangementIt has four individually spring loaded probes. The probes are collinear and equally spaced. The probes are mounted in a Teflon bush, which ensure a good electrical insulation between the probes. A Teflon spacer near the tips is also provided to keep the probe at equal distance. The whole arrangement is mounted on a suitable stand and leads Page 28

are provided for the voltage measurement. The probe distance is measured approximate s = 2.4 mm.

Fig.6 (Four probe Circuit Connection)

Fig.7 (Four Probe Arrangement for Mounting Sample).

Page 29

3- Sample: P-type Ge crystal, 10x8x0.2 mm. 4- Oven: It is a small oven for the variation of the temperature of the crystal from the room temperature to about 2000C (max). 5- Four Probe Set-up: The Set up consists of three units in the same cabinet. (1) Digital Voltmeter In this unit, intersil 3 ½ digit single chip A/D converter has been used. it has a high accuracy like auto zero to less than 10µm, zero drift of less than 1µV/0C, input bias current of 10pA max. Specification: Display: 7segment LED (12.5mmhight) with auto polarity and decimal indication, Range: X1 (0-200.0mV) DC & X10 (0-2.00V) DC Since the use of internal reference causes the degradation in the performance due to internal heating. Impedance: 1ΩM. Accuracy: ±0.1% of reading ± digit. (2) Constant Current Generator: It is an IC regulated current generator to provide a constant current to the outer probe irrespective of changing resistance of the sample due to change in temperatures. The basic scheme is to use the feedback principle to limit the load current of the supply to preset maximum value. The supply is a highly regulated and practically ripple free DC source. The current is measured by the digital panel meter. Specification: Open circuit Voltage: 18V Current range: 0 - 20mA Resolution: 10µA Page 30

Load regulation: 0.03% for 0 to full load. Accuracy: ±0.25% of the reading ±1 digit. Display: 3 ½ digit, 7 segment LED (12.5mm height).

(3) Oven Power supply: A suitable voltage for the oven is obtained through a step down transformer with a provision for low and high rates of heating. A glowing LED indicates power supply is “ON”.

Fig.8 (Four Probe Apparatus)

6.2 OBVERSATIONS: (1) Current, I = 8.0 mA (constant). (2) Distance between probes, s = 0.24 cm. (3) Thickness of sample, w = 0.05 cm. Page 31

6.3 Calculations: (1) The temperature was converted from 0C to K with help of following expression K = 273.15 + 0C

(1)

(2) The correction factor f(w/s) is calculated from the following table:

w/s

f(w/s)

0.100

13.863

0.141

9.704

0.200

6.931

0.333

4.159

0.500

2.780

1.000

1.504

1.414

1.223

2.000

1.094

3.333

1.023

5.000

1.007

10.00

1.0005

The value of w/s, =0.208, by the interpolation method:

(0.208-0.200)/(0.333-0.200) = (f(w/s)-6.931)/(4.195-6.931)

f (w/s) = 6.762736. (3) The different values of ρ0 was calculated at different values of V, by using following formula: Page 32

ρ0 = V x 2πs/I,

(2)

(4) The values of resistivity (ρ) is calculated at different values of

ρ0 at different

temperatures, by using following formula and an observation table is made as below:

ρ = ρ0 / f (w/s),

(3)

6.4 TABLE FOR RESISTIVITY Vs TEMPERATURE:

S.N Temperat ure (0C)

Temperat ure T(K)

Voltage (mV)

T-1 *103

ρ (Ω cm)

Log10ρ

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17

393 388 383 378 373 368 363 358 353 348 343 338 333 328 323 318 313

7 8.6 10.1 11.8 13.6 15.7 18 20.3 22.7 24.7 26.4 27.6 28.1 28.2 27.9 27.4 26.6

2.544529262 2.577319588 2.610966057 2.645502646 2.680965147 2.717391304 2.754820937 2.793296089 2.83286119 2.873563218 2.915451895 2.958579882 3.003003003 3.048780488 3.095975232 3.144654088 3.194888179

0.97515528 1.198047915 1.40700976 1.643833185 1.8945874 2.187133984 2.507542147 2.827950311 3.162289264 3.440905058 3.677728483 3.844897959 3.914551908 3.928482697 3.886690328 3.81703638 3.705590062

-0.01093 0.078474 0.148297 0.215858 0.277515 0.339875 0.399248 0.451472 0.500002 0.536673 0.56558 0.584885 0.592682 0.594225 0.58958 0.581726 0.568857

120 115 110 105 100 95 90 85 80 75 70 65 60 55 50 45 40

(5)The temperature(T) and resistivty (ρ) are converted in T-1 x103 and log10ρ respectively. (6)A graph is plotted between (T-1x103) and (log10ρ) as below: The slope of the graph is calculated : slope = 1.677582918

Page 33

0.7 0.6 0.5

log10(Rho)

0.4 0.3 0.2 0.1 0 2.54 2.58 2.61 2.65 2.68 2.72 2.75 2.79 2.83 2.87 2.92 2.96 3.00 3.05 3.10 3.14 3.19 -0.1

(1/T)*10^3

Fig.9 (Variation of Resistivty of p-type Ge Crystal)

(7) The energy band gap of p-type Ge is calculated by following formula: Eg =(( 2 K x 2.3026 x log10 ρ) / T-1 ) eV

Where K=8.6 x 10-5 eV / deg, thenEg = 2 x 8.6 x 10-5 x 2.3026 x slope x 103 e V Eg =0.396 x slope Eg = 0.396 x 1.677582918 Eg = 0.664322835 eV

Page 34

Chapter-7 APPLICATION OF HALL EFFECT: 7.1 Hall Effect Sensor: The Hall Effect is an ideal sensing technology. The Hall element is constructed from a thin sheet of conductive material with output connections perpendicular to the direction of current flow. When subjected to a magnetic field, it responds with an output voltage proportional to the magnetic field strength. The voltage output is very small (µV) and requires additional electronics to achieve useful voltage levels. When the Hall element is combined with the associated electronics, it forms a Hall Effect sensor. The heart of every MICRO SWITCH Hall effect device is the integrated circuit chip that contains the Hall element and the signal conditioning electronics. Although the Hall Effect sensor is a magnetic field sensor, it can be used as the Principle component in many other types of sensing devices (current, temperature, pressure, position, etc.). Hall Effect sensors can be applied in many types of sensing devices. If the quantity (parameter) to be sensed incorporates or can incorporate a magnetic field, a HallSensor will perform the task.

Fig.1 (Block diagram of general sensor based on Hall Effect)

Page 35

7.2 Basics Hall Effect sensors The Hall element is the basic magnetic field sensor. It requires signal conditioning to make the output usable for most applications. The signal conditioning electronics needed are amplifier stage and temperature compensation. Voltage regulation is needed when operating from an unregulated supply. Fig.2 illustrates a basic Hall Effect sensor. If the Hall voltage is measured when no magnetic field is present, the output is zero. However, if voltage at each output terminal is measured with respect to ground, a non-zero voltage will appear. This is the common mode voltage (CMV), and is the same at each output terminal. It is the potential difference that is zero. The amplifier shown in Fig.3 must be a differential amplifier so as to amplify only the potential difference – the Hall voltage.

Fig.2 (Hall element orientation)

Fig.3 (Differential Amplifier)

Page 36

7.3 Analog output sensors The sensor described in Fig.3 is a basic analog output device. Analog sensors provide an output voltage that is proportional to the magnetic field to which it is exposed. Although this is a complete device, additional circuit functions were added to simplify the application. The sensed magnetic field can be either positive or negative. As a result, the output of the amplifier will be driven either positive or negative, thus requiring both plus and minus power supplies. To avoid the requirement for two power supplies, a fixed offset or bias is introduced into the differential amplifier. The bias value appears on the output when no magnetic field is present and is referred to as a null voltage. When a positive magnetic field is sensed, the output increases above the null voltage. Conversely, When a negative magnetic field is sensed, the output decreases below the null voltage, but remains positive. This concept is illustrated in Fig.4.

The output of the amplifier cannot exceed the limits imposed by the power supply. In fact, the amplifier will begin to saturate before the limits of the power supply are reached. This saturation is illustrated in Fig.4. It is important to note that this saturation takes place in the amplifier and not in the Hall element. Thus, large magnetic fields will not damage the Hall Effect sensors, but rather drive them into saturation. To further increase the interface flexibility of the device, an open emitter, open collector, or push-pull transistor is added to the output of the differential amplifier. Fig.5 shows a complete analog output Hall Effect sensor incorporating all of the previously discussed circuit functions. The basic concepts pertaining to analog output sensors have been established. Both the manner in which these devices are specified and the implication of the specifications follow. 7.4 Output vs. power supply characteristics Analog output sensors are available in voltage ranges of 4.5 to 10.5, 4.5 to 12, or 6.6 to 12.6 VDC. They typically require a regulated supply voltage to operate accurately. Their output is usually of the push-pull type and is ratio metric to the supply voltage with respect to offset and gain.

Page 37

Fig.4 (Null Voltage Concept)

Fig.5 (Simple analog output sensor)

Page 38

Fig.6 (Ratio metric linear output sensor) Fig.6 Illustrates a ratio metric analog sensor that accepts a4.5 to 10.5 V supply. This sensor has a sensitivity (mV/Gauss) and offset (V) proportional (ratio metric) to the supply voltage. This device has “rail-to-rail” operation. That is, its output varies from almost zero (0.2 V typical) to almost the supply voltage (Vs - 0.2 V typical).

7.5 Digital output sensors The preceding discussion described an analog output sensor as a device having an analog output proportional to its input. In this section, the digital Hall Effect sensor will be examined. This sensor has an output that is just one of two states: ON or OFF. The basic analog output device illustrated in (Fig.3) can be converted into a digital output sensor with the addition of a Schmitt trigger circuit. Fig.7 illustrates a typical internally regulated digital output Hall Effect sensor. The Schmitt trigger compares the output of the differential amplifier (Fig.7) with a preset reference. When the amplifier output exceeds the reference, the Schmitt trigger turns on. Conversely, when the output of the amplifier falls below the reference point, the output of the Schmitt trigger turns off.

Page 39

Fig.7 (Digital output Hall Effect sensor)

Hysteresis is included in the Schmitt trigger circuit for jitter-free switching. Hysteresis results from two distinct reference values which depend on whether the sensor is being turned ON or OFF.

7.6 Transfer function The transfer function for a digital output Hall Effect sensor incorporating hysteresis is shown in Fig.8. The principal input/output characteristics are the operate point, release point and the difference between the two or differential. As the magnetic field is increased, no change in the sensor output will occur until the operate point is reached. Once the operate point is reached, the sensor will change state. Further increases in magnetic input beyond the operate point will have no effect. If magnetic field is decreased to below the operate point, the output will remain the same until the release point is reached. At this point, the sensor’s output will return to its original state (OFF). The purpose of the differential between the operate and release point (hysteresis) is to eliminate false triggering which can be caused by minor variations in input. As with analog output Hall Effect sensors, an output transistor is added to increase application flexibility. This output transistor is typically NPN (current sinking). The fundamental characteristics relating to digital output sensors have Page 40

been presented. The specifications and the effect these specifications have on product selection follows.

Fig.8 (Transfer function hysteresis digital Hall Effect sensor)

7.7 Power supply characteristics Digital output sensors are available in two different power supply configurations - regulated and unregulated. Most digital Hall Effect sensors are regulated and can be used with power supplies In the range of 3.8 to 24 VDC. Unregulated sensors are used in special applications. They require a regulated DC supply of 4.5to 5.5 volts (5± 0.5 v). Sensors that incorporate internal regulators are intended for general purpose applications. Unregulated sensors should be used in conjunction with logic circuits where a regulated 5 volt power supply is available. 7.8 Input characteristics The input characteristics of a digital output sensor are defined in terms of an operate point, release point, and differential. Since these characteristics change over temperature and from sensor to sensor, they are specified in terms of maximum and minimum values. Maximum Operate Point refers to the level of magnetic field that will insure the digital output sensor turns ON under any rated condition. Minimum Release Point refers to the level of magnetic field that Page 41

insures the sensor is turned OFF.Figure-7 shows the input characteristics for a typical uni polar digital output sensor. The sensor shown is referred to as uni polar since both the maximum operate and minimum release points are positive (i.e. South Pole of magnetic field). A bipolar sensor has a positive maximum operate point (South Pole) and a negative minimum release point (North Pole). The transfer functions are illustrated in Fig.9. Note that thereare three combinations of actual operate and release points possible with a bipolar sensor. A true latching device, represented as bi polar device 2, will always have a positive operate point and a negative point.

Fig.9 (Uni polar Input Characteristics, Digital Output Sensor )

7.9 Output characteristics The output characteristics of a digital output sensor are defined as the electrical characteristics of the output transistor. These include type (i.e. NPN), maximum current, breakdown voltage, and switching time.

Page 42

Fig.10 (Bipolar input characteristics Digital output sensor)

7.10 Why use the Hall Effect sensors? The reasons for using a particular technology or sensor vary according to the application. Cost, performance and availability are always considerations. The features and benefits of a given technology are factors that should be weighed along with The specific requirements of the application in making this decision. General features of Hall Effect based sensing devices are: 

True solid state



Long life (30 billion operations in a continuing keyboard module test program



High speed operation - over 100 kHz possible



Operates with stationary input (zero speed)



No moving parts



Logic compatible input and output



Broad temperature range (-40 to +150°C)



Highly repeatable operation

Page 43

Conclusion: By using the Hall Effect apparatus and Four Probe apparatus we have calculated following parameters which are as follow, Mean Value of (VH/I) = 0.668270614Ω, Hall Resistivity (RH) =0.00154557Vm/AT, Carrier concentration (n) = 4.04381555 x 1023m-3, Resistivity ρ = 2.202483x10-3Ωm Mobility µB = 7.0135227 m2V-1s-1 and Hall angle ӨH = 79.37439462 0, Band Gap = 0.664322835 eV and resistivity at different temperatures of the samples. These results are useful in so many applications such as Hall Effect based Motion Sensing Devices, Linear Hall ICs, Hall Effect Fan Motor Drivers, and Hall Effect Switches And the work that can be done in the future on our project is as follows;

1-

Facilitating easy installation of monitoring device by using alternatives like non-invasive

CT Sensor. 2-

Implementing the Control Objective on every possible electrical appliance.

3-

Blood Pressure Monitoring Devices.

4-

Produce a digital signal

5-

Used for current sensor

6-

Automotive fuel level indicator

7-

Spacecraft propulsion

8-

Smart phones use hall sensor to determine if the flip cover accessory is closed.

Page 44

References: 1. Material Science by S.L.KAKANI, AMIT KAKANI; 5; 155-167, 15; 488-500. 2. Materials Science by V Rajendran; 7; 146-157, 10; 241-264. o The Science and Engineering of Materials by Donald R. Askelandi, Pradeep P. Fully, Wendelin J. Wright, Kandelin Balani; 20; 763-785. 3. Materials Science and Engineering an Introduction by William D.Callister, Jr. David G. Rethwisch; 17; 623-659. 4. Advance Electrical and Electronics materials Processes and Application by K.M.Guptai and Nishu Gupta.6; 185-229. 5. Semiconductor Material and Device Characterization, By Dieter K. Schroder 3rd Edition (Arizona State University Tempe AZ, IEEE Press New Jersey 2006) chapter 2 page 94. 6. Electrons in Metals and Semiconductors, by R.G.Chambers (Chapman and Hall, London 1990) chapter 1 and 2. 7. Introduction to Solid State Physics, by Charles Kittle, Seventh edition (Wiley, New York 1996) chapter 6 and 7. 8. Band theory and electronic properties of Solids, by John Sinletoni (oxford university press, 2001) chapter 1. 9. Electronics: Theory and Application by S.L.Kakani and K.C.Bhandr (3rdEdition: 2004). 10. S.L.Kakani and C.Hemrajanii, Solid State Physics(4th Edition:2004). 11. Rolf T.Hunnel, Electronic property of materials.(Springer, New York, 1993) 12. Hall E H 1879 Am. J. Sci (3rd), 20;161-186. 13. Hall E H 1880 Am. J. Mat. 22; 87-92. 14. Van Der Pauw, L.J (1958), “A method of measuring specific resistivity and Hall Effect of arbitrary shape- Philips Research Report”, 13;1-9.

Page 45