Four Probe Method

K.SURESH SENANAYAKE

PS/2004/225

PHYS 43115 ADVANCED PHYSICS LABORATORY

INVESTIGATIONOFBANDGAPOFASEMICONDUCTORCRYSTAL

INVESTIGATION OF BAND GAP OF A SEMICONDUCTOR CRYSTAL

Investigation of Band Gap of a Ge Crystal by using Four Probe Method

Four Probe Method PS/2004/225

Department of Physics University of Kelaniya

1

INDEX INDEX ......................................................................................................................................1

INTRODUCTION..................................................................................................................2

APPARATUS ..........................................................................................................................4

THEORY ..................................................................................................................................5

EXPERIMENTAL PROCEDURE ........................................................................................7

OBSERVATIONS ..................................................................................................................8

RESULTS ...............................................................................................................................10

ERROR ANALYSIS.............................................................................................................15

CONCLUSION.....................................................................................................................16

DISCUSSION .......................................................................................................................17

REFERENCES .......................................................................................................................18



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INTRODUCTION This introduction will be broken into two parts; a description of the Ge crystal structure, and a detailed description of the four probe Resistivity experimental technique that was used. Crystal Structure In solids, atoms are arranged in an ordered, dense packing, forming a crystal. Silicon, germanium and carbon crystals have identical structure, called diamond structure. A single atom has a wave function associated with it and discrete potential energy levels, but in a solid with a many densely packed atoms, the wave functions overlap and interact and the rather than a single energy level it is split and spread. A solid of N atoms has N times as many levels as each atom. Because of the enormous number of atoms in a crystal (1023 cm-3) this results in bands of energy levels. The band gap Eg separates the valence band and the conduction band. A semiconductor has a band gap in the order of 1 eV. At absolute zero temperature the valence band is completely filled and the conduction band is empty, thus no conduction takes place. As the temperature increases some electrons gain energy greater than the band gap and move to the conduction band, leaving a hole in the valence band. The electron in the conduction band and the hole in the valence band hence contribute to conduction in an applied electric field. This is called intrinsic conduction. The net electrical conductivity is thus the sum of both the negative (electrons) and the positive (holes) charge carriers. To enhance the conduction, and other properties, a pure material may be doped with n-type or p-type impurity elements (resulting in extrinsic and well as intrinsic conduction). At room temperature, conduction is mostly by the extrinsic dopant, but as the temperature is increased, the intrinsic conduction mechanism dominates. The effect of the dopant is to greatly reduce the band gap to 0.05 eV. Four Probe Resistivity Technique A four probe experimental technique was used to measure the band gap. This involves four small sharp probes in a straight line placed on a flat surface of the sample to be measured, a constant current is passed between the outer contacts this flows in a certain pattern in the body of the material, and the resulting floating potential between the inner contacts is measured. The resistivity is calculated from Ohms Law and the geometry of the sample. If the temperature is increased, more electrons will be excited into the conduction band so the conduction will increase, therefore the resistivity will decrease. Since the resistivity is highly dependant on the temperature, by varying the temperature while taking measurements, and hence are able to find the resistivity as a function of temperature. The motivation for using the four probe technique is that many conventional methods for measuring resistivity such as soldering contacts to the sample are unsatisfactory for semiconductors. Soldering directly onto the sample can disrupt the properties of the sample through heating and carrier injection. Metal-semiconductor contacts are usually rectifying in nature and there tends to be minority carrier injection by one of the current carrying contacts. An excess concentration of minority carriers will affect the potential of other contacts and modulate the resistance of the material, hence giving an unsatisfactory result. The four probe



3

technique overcomes these faults by using pressure contacts but a draw back is they tend to be noisy. The four probe method also allows measurements to be made on a smaller area; hence there will be less variation in the properties of the crystal over the small area. Another benefit over a soldered contact based technique is the precision at which the probe spacing can be measured. In order to use the four probe technique certain assumptions need to be made:

1. The resistivity of the sample is uniform in the area of measurement. 2. Any minority carriers injected into the sample from the current-carrying electrodes

recombine near the electrodes so that their effect of the conductivity is negligible. 3. The surface on which the probes rest is flat. 4. The four probes lie in a straight line. 5. The diameter of the contact between the probes and the sample is small compared to

the distance between the probes. 6. The boundary between the current carrying probes and the sample is hemispherical or

small in diameter. It is also assumed that the current through the sample is small enough not to cause heating, and that the voltage drop at the contacts is low so that injection is minimized (this is somewhat recovered in assumption 2 above).



4

APPARATUS Hot plate Thermometer

Four Probe Apparatus (Constant Current generator and Digital Voltmeter)

Ge sample Connecting wires



5

THEORY For the current density J in a body under the influence of an electric field E Ohms law states; EJ .= The proportionality factor is called electric conductivity. Since this quantity strongly depends on the material, it is common to classify materials with regard to their conductivity. Semiconductors, for example, are solids that do not conduct electric currents at low temperatures, but show a measurable conductivity at higher temperatures. The reason for this temperature dependence is the specific band structure of the electronic energy levels of a semiconductor. The valence band, i.e. the highest band that is completely or partially populated in the ground state, and the conduction band, i.e. the next unpopulated band, are separated by a band gap Eg (Ge: Eg 0.7 eV). The region between the two bands is not populated by electrons in an undoped, pure semiconductor and is called the forbidden zone. At higher temperatures, more and more electrons are thermally activated from the valence band into the conduction band. They leave holes in the valence band which move like positive charged particles, thus contributing to the current density J as do the electrons.

The conduction which is made possible by the excitation of the electrons from the valence band into the conduction band is called intrinsic conduction. Since under conditions of thermal equilibrium the number of holes in the valence band and of electrons in the conduction band are equal, the current density in the case of intrinsic conduction can be written in the form; pn vnevneJ ...).( +=e: elementary charge, n: concentration of electrons or holes respectively. vp: average drift velocity of the electrons. vn: average drift velocity of the holes. The average drift velocities vn and vp of the electrons and the holes are proportional to the field strength E. Then; Ev nn .= and Ev pp .= Where the mobility n and p are chosen to be positive quantities, ( )EneJ pn ... += By comparing this equation with the equation of Ohms law, The conductivity, ( )pnne += ..


University of Kelaniya

6

Apart from the elementary charge e, all quantities in the above equation depend on the temperature T. The concentration of intrinsic conduction n is given by;

( ) kTEg

ePNn .221

..= k: Boltzmann constant, Eg: band gap of the semiconductor. And N and P are the effective state densities in the conduction band and in the valence band.

23

2

..2.2

=

hkTmN n And 2

3

2

..2.2

=

hkTm

P p

h: Planck constant, mn: effective electron mass, mp: effective hole mass. The mobility n and p also depends on the temperature. At low temperatures, the

proportionality 23

T holds roughly, as does the proportionality 23

T at high temperatures. Because of the predominance of the exponential function, conductivity is well approximated and represented by;

kTgE

e .20

=

kTEg.2

lnln 0 = The resistivity can be written as inverse of the conductivity. Then,

The resistivity, 1=

By substituting this in the above equation,

0ln.2ln +=

kTEg

With the object of confirming above equation and determining the band gap Eg, the resistivity of undoped germanium is determined as a function of the temperature T in the experiment. At a constant current; cbJI ..= Where b: breadth of the crystal, c: thickness of the crystal. And voltage drop; aEV .= a: length of the crystal. By combining above equations the conductivity can be written as;

VI

cba ..

= IV

acb ..=

Then it can rearrange the equation as the graph of ln vs 1\T as follows;

0ln1.

.2ln +

=

TkEg

Department of Physics Y = m . X + C



7

EXPERIMENTAL PROCEDURE

Digital Voltmeter & Constant current generator

Probe equipment with the Ge sample

Oven

Digital Voltmeter & Constant current generator

1. The probe equipment with Ge sample was placed on the hot plate and then the thermometer was kept into its holder. (In this set up, the hot plate was used instead of the oven because it was not worked as well.)

2. The electric connections were made, and a current of 3.0 mA was set. 3. The hot plate was turned on and readings of the voltage were taken until the

temperature reached 140 0C, the hot plate was then turned off and more readings were taken as the sample cooled down to approximately 70 0C.

(At the low temperatures the semiconductor resistivity was rapidly decreased. Therefore to get the linear variation of ln vs 1/T, the slightly high temperatures were selected to plot the graphs). 4. The same procedure was repeated for the currents 5.0, 7.0, mA. 5. The graphs of ln vs 1/T were plotted to calculate the band gap.



8

OBSERVATIONS

I = 310-3 A

T / K Vi

(Increasing)/V Vd

(Decreasing)/V 35 0.232 0.231 40 0.218 0.213 45 0.200 0.193 50 0.181 0.170 55 0.160 0.149 60 0.142 0.129 65 0.123 0.112 70 0.106 0.094 75 0.094 0.082 80 0.082 0.070 85 0.071 0.060 90 0.054 0.051 95 0.048 0.044

100 0.042 0.039 105 0.037 0.033 110 0.033 0.029 115 0.029 0.026 120 0.026 0.023 125 0.024 0.020 130 0.021 0.018 135 0.018 0.017 140 0.016 0.016

I = 510-3 A

T / K Vi

(Increasing)/V Vd

(Decreasing)/V 35 0.382 0.382 40 0.350 0.350 45 0.313 0.308 50 0.277 0.269 55 0.243 0.231 60 0.211 0.198 65 0.178 0.171 70 0.150 0.148 75 0.129 0.129 80 0.110 0.109 85 0.094 0.092 90 0.080 0.079 95 0.068 0.066

100 0.058 0.057 105 0.050 0.048 110 0.043 0.041 115 0.038 0.036 120 0.033 0.032 125 0.029 0.028 130 0.025 0.025 135 0.021 0.020 140 0.018 0.018



9

I = 710-3 A

T / K Vi

(Increasing)/V Vd

(Decreasing)/V 35 0.517 0.480 40 0.470 0.426 45 0.417 0.371 50 0.366 0.320 55 0.315 0.275 60 0.269 0.238 65 0.232 0.208 70 0.198 0.178 75 0.167 0.153 80 0.140 0.130 85 0.120 0.111 90 0.103 0.095 95 0.089 0.082 100 0.077 0.071 105 0.066 0.062 110 0.057 0.053 115 0.050 0.047 120 0.043 0.040 125 0.038 0.036 130 0.033 0.032 135 0.029 0.028 140 0.026 0.025

Crystal Dimensions Length = 1010-3 m Width = 1010-3 m Thickness = 0.510-3 m The space between two probes = 210-3 m Room temperature: 30 0C Uncertainties Voltage: 0.310-3 V Temperature: 0.5 0C


RESULTS

Department of Physics University of Kelaniya 10

I = 310-3A T (0C)

T (K) 1/T (K-1) Vi (V) Vd (V) V (V) ln

70 343 0.002915 0.106 0.094 0.1000 0.01667 -4.09434 75 348 0.002874 0.094 0.082 0.0880 0.01467 -4.22218 80 353 0.002833 0.082 0.070 0.0760 0.01267 -4.36878 85 358 0.002793 0.071 0.060 0.0655 0.01092 -4.51746 90 363 0.002755 0.054 0.051 0.0525 0.00875 -4.73870 95 368 0.002717 0.048 0.044 0.0460 0.00767 -4.87087 100 373 0.002681 0.042 0.039 0.0405 0.00675 -4.99821 105 378 0.002646 0.037 0.033 0.0350 0.00583 -5.14417 110 383 0.002611 0.033 0.029 0.0310 0.00517 -5.26553 115 388 0.002577 0.029 0.026 0.0275 0.00458 -5.38533 120 393 0.002545 0.026 0.023 0.0245 0.00408 -5.50084 125 398 0.002513 0.024 0.020 0.0220 0.00367 -5.60847 130 403 0.002481 0.021 0.018 0.0195 0.00325 -5.72910 135 408 0.002451 0.018 0.017 0.0175 0.00292 -5.83731 140 413 0.002421 0.016 0.016 0.0160 0.00267 -5.92693

0ln1.

.2ln +

=

TkEg



11

According to the graph, (By using the Origin 6.0 software) The gradient of the graph, m1, = 3786.7795 K

According to the equation, k

Em g

21=

Therefore, 7795.378621

==k

Em g

Where, k = Boltzmann constant =1.3810-23JK-1Therefore,

eVE

JE

JE

g

g

g

19

19

19

23

10602.1100452.1

100452.1

1038.127795.3786

==

=

0.65eVEg =

I = 510-3A T (0C)

T (K) 1/T (K-1) Vi (V) Vd (V) V (V) ln

70 343 0.002915 0.150 0.148 0.149 0.02483 -3.69557 75 348 0.002874 0.129 0.129 0.129 0.02150 -3.83970 80 353 0.002833 0.110 0.109 0.1095 0.01825 -4.00359 85 358 0.002793 0.094 0.092 0.093 0.01550 -4.16692 90 363 0.002755 0.080 0.079 0.0795 0.01325 -4.32376 95 368 0.002717 0.068 0.066 0.067 0.01117 -4.49482 100 373 0.002681 0.058 0.057 0.0575 0.00958 -4.64773 105 378 0.002646 0.050 0.048 0.049 0.00817 -4.80769 110 383 0.002611 0.043 0.041 0.042 0.00700 -4.96185 115 388 0.002577 0.038 0.036 0.037 0.00617 -5.08860 120 393 0.002545 0.033 0.032 0.0325 0.00542 -5.21827 125 398 0.002513 0.029 0.028 0.0285 0.00475 -5.34961 130 403 0.002481 0.025 0.025 0.025 0.00417 -5.48064 135 408 0.002451 0.021 0.020 0.0205 0.00342 -5.67909 140 413 0.002421 0.018 0.018 0.018 0.00300 -5.80914



12

0ln1.

.2ln +

=

TkEg



Em g

22=


==k

Em g


eVE

JE

JE

g

g

g

19

19

19

23

10602.1101784.1

101784.1

1038.126883.4269

==

=

0.74eVEg =


I = 710-3A T (0C)

T (K) 1/T (K-1) Vi (V) Vd (V) V (V) ln

70 343 0.002915 0.198 0.178 0.188 0.03133 -3.46307 75 348 0.002874 0.167 0.153 0.16 0.02667 -3.62434 80 353 0.002833 0.140 0.130 0.135 0.02250 -3.79424 85 358 0.002793 0.120 0.111 0.1155 0.01925 -3.95024 90 363 0.002755 0.103 0.095 0.099 0.01650 -4.10439 95 368 0.002717 0.089 0.082 0.0855 0.01425 -4.25100 100 373 0.002681 0.077 0.071 0.074 0.01233 -4.39545 105 378 0.002646 0.066 0.062 0.064 0.01067 -4.54063 110 383 0.002611 0.057 0.053 0.055 0.00917 -4.69218 115 388 0.002577 0.050 0.047 0.0485 0.00808 -4.81795 120 393 0.002545 0.043 0.040 0.0415 0.00692 -4.97382 125 398 0.002513 0.038 0.036 0.037 0.00617 -5.08860 130 403 0.002481 0.033 0.032 0.0325 0.00542 -5.21827 135 408 0.002451 0.029 0.028 0.0285 0.00475 -5.34961 140 413 0.002421 0.026 0.025 0.0255 0.00425 -5.46084

Department of Physics University of Kelaniya 13

0ln1.

.2ln +

=

TkEg



14



Em g

23=


==k

Em g


eVE

JE

JE

g

g

g

19

19

19

23

10602.1101211.1

101211.1

1038.121123.4062

==

=

0.70eVEg =

The average value of the band gap eVeV 697.03

70.074.065.0 =++= The band gap of the Ge is 0.697 eV.



15

ERROR ANALYSIS The band gap, mkEg .2= ( ) mkEg ln2lnln +=

By taking partial differentiation, mm

EE

g

g =

gg EmmE =

For I = 310-3A

gg EmmE

1

1 = By the statistical method, (by using Origin 6.0 software), m1 = 45.2263 K

eVEg 65.07795.37862263.45 =

eVEg 008.0=

For I = 510-3A

gg EmmE

2


eVEg 74.06883.42695528.41 =

eVEg 006.0=

For I = 710-3A

gg EmmE

3


eVEg 70.01123.40627484.12 =

eVEg 002.0=

The average value of the error of the band gap = eVeV 01.0006.03

002.0006.0008.0 =++



16

CONCLUSION Experimental value The band gap of the Ge is 0.697 0.006 eV. Standard value The band gap of the Ge is 0.67 eV.



17

DISCUSSION The band gap of Ge was measured as 0.700.01 eV using a resistivity method. A four probe experimental technique was used to avoid the use of soldering contacts to the sample. This method has its benefits but also makes assumptions about the characteristics of the sample. The accepted band gap for Ge is 0.66 eV, the discrepancy between the measured value and the accepted value could be explained by the assumptions of the four probe method, the difference in voltage readings for cooling and heating stages of the experiment, and/or the doping effect of the sample. The impurities in the Ge sample would produce an extrinsic component of the conductivity, and although the band gap was calculated from measurements when the temperature is high and the intrinsic component dominates; the extrinsic component would still have a residual effect. During the cooling phase of the experiment the measured values of resistivity were slightly lower than in the heating phase, this is due to the thermometer and/or sample cooling at difference rates. The observed difference can be explained if the thermometer is cooling quicker than the sample. A digital thermometer would have produced more accurate results. At absolute zero temperature there would be no thermal excitation of the electrons from the valence band to the conduction band; the valence band would be completely full and the conduction band completely empty, hence there would be no conduction of electrons and the semiconductor would behave as an insulator. In this experiment it would be important for the millivoltmeter to have high impedance. There is a contact resistance in the semiconducting crustal. This is a serious matter for electrical semiconductors and to avoid these errors they used the extra probes between the current contacts. It should not be exceed the temperature over the 1800C then the device can be destroyed. Also for the accurate value the probe spacing should be greater than the thickness of the semiconductor device. In this experiment, the thermo plate was used to heat the semiconductor instead of Oven of four probe apparatus. (It was destroyed by high current). It was not so accurate for the experiment because we cannot keep at constant temperature in longer time like in the oven. This experimental technique has important limitations and assumptions; we assume that the resistivity of the material is uniform in the area of measurement, there is minimum carrier injection from the probes into the material, and a good contact is made with the surface.



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REFERENCES C. Kittel, Introduction to Solid State Physics, (Willey, New York).

www.sestechno.com/pro1/2c.pdf

www.cce.ufes.br/jair/web/gap-Ge.pdf

LEYBOLD Physics Leaflets P7.2.1.5

INVESTIGATION OF BAND GAP OF A SEMICONDUCTOR CRYSTAL.pdfINVESTIGATION OF BAND GAP OF A SEMICONDUCTOR CRYSTALInvestigation of Band Gap of a Ge Crystal by using Four Prob

Four Probe Method.pdfINDEXINTRODUCTIONAPPARATUSTHEORYEXPERIMENTAL PROCEDUREOBSERVATIONSRESULTSERROR ANALYSISCONCLUSIONDISCUSSIONREFERENCES

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Four Probe Method

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