Experiment 8 Measuring the band gap of a semiconductor PDF

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EXPERIMENT 8: Measuring the band gap of a semiconductor

Objectives • To determine the gap energy for a semiconductor material by measuring the resistance of a thermistor as a function of temperature. Apparatus Thermistor, oil bath (silicon oil), digital meter, thermometer

Experimental procedure • A thermistor and a very high input impedance digital meter were used for the experiment. • The thermistor was placed in an oil bath (silicon oil). • The temperature and resistance of the thermistor at room temperature was recorded. • The oil was heated gradually, stirring gently. In increasing steps of 10C, the temperature and resistance of the thermistor was recorded up to T=1000C. 1 • The temperature was converted from 0C to K and a graph of lnR against was 𝑇 plotted. • The plot was analysed according to theory. Theory • “Band gap” or “energy gap” refers to an energy range in a solid where no electron states can exist. • It signifies the energy difference between the top of the valence band to the bottom of the conduction band; electrons are able to jump from one band to another. Equivalently it is the energy required to free an outer shell electron from its orbit about the nucleus to become a mobile charge carrier, able to move freely within the solid material. • In order for an electron to jump from a valence band to a conduction band, it requires a specific minimum amount of energy for the transition, termed the band gap energy. • The band gap energy of insulators is large (> 4eV), but lower for semiconductors (< 3eV). The band gap properties of a semiconductor can be controlled by using different semiconductor alloys such as GaAlAs, InGaAs, and InAlAs. Conductors either have very small band gaps or none, because the valence and conduction bands overlap. • The Fermi level or “chemical potential” is the energy that concerns electrons in a semiconductor and is located in the band gap. The probability of the occupation of an energy level is based on the Fermi function. • An intrinsic semiconductor is ideally a perfect crystal. When an electron in an intrinsic semiconductor gets enough energy, it can go to the conduction band and leave behind a hole. This process is called “electronhole pair (EHP) creation”. • Extrinsic semiconductors are made by introducing different atoms, called dopant atoms, into the crystal. Eg

σ = B 𝑒 2𝐾𝐵 𝑇 R= Page | 1

D σ

=

(1)

D

Eg

B 𝑒 2𝐾𝐵 𝑇

(2)

Eg

lnR = C + ln𝑒 2𝐾𝐵 𝑇 lnR = C +

Eg 2𝐾𝐵 𝑇

(3) Eg 1 2𝐾𝐵 T

equivalent to lnR = C + y = c + mx

(4)

Eg: width of the gap B: approximately a constant. It is weakly dependent on temperature. 𝐾𝐵 : Boltzmann's constant = 1.381×10−23J/K = 8.617×10−5 eV/K Equation 1 shows the temperature dependency of the conductivity of a pure (intrinsic) semiconductor. 1 • Equation 3 represents a linear relationship between lnR and from which Eg is found T from the gradient. • Measuring the resistance of the semiconductor over a range of temperatures allows the band gap to be computed. • Point to note: Slight impurities in a semiconductor hugely impacts on the conductivity. This obscures the measuring of the intrinsic gap energy but though it is required in designing a semiconductor. • For intrinsic semiconductors, the resistance is very sensitive to slight changes in temperature. This forms the root for thermistors. Expectations • From equation (4), a linear graph is expected after plotting the points. • The measured resistance expected to change significantly to small changes in temperature. • Therefore a larger gradient slope of the “line of best fit” is expected. • Theory states that the band gap energy for semiconductors is very low (< 3eV), therefore since we are working with a semiconductor we expect to compute our band gap energy to be less than 3eV. • • • •

Results T /C 0

16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 Page | 2

T /K 289.15 290.15 291.15 292.15 293.15 294.15 295.15 296.15 297.15 298.15 299.15 300.15 301.15 302.15 303.15 304.15

R /kΩ 124.4 120.7 113.4 107.4 104.3 93.5 86 83.5 81.2 78.6 74.4 70.7 67.3 64.5 62.1 55.3

𝟏

X= 𝐓 *10-3 3.46 3.45 3.43 3.42 3.41 3.4 3.39 3.38 3.37 3.35 3.34 3.33 3.32 3.31 3.3 3.29

Y= lnR 4.82 4.79 4.73 4.68 4.65 4.54 4.45 4.42 4.4 4.36 4.31 4.26 4.21 4.17 4.13 4.01

T /C 0

59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74

T /K 332.15 333.15 334.15 335.15 336.15 337.15 338.15 339.15 340.15 341.15 342.15 343.15 344.15 345.15 346.15 347.15

R /kΩ 15.7 15.1 14.5 13.9 13.4 12.8 12.3 11.7 11.2 10.7 10.1 9.8 9.4 9 8.7 8.3

𝟏

X= 𝐓 *10-3 3.01 3 2.99 2.98 2.97 2.97 2.96 2.95 2.94 2.93 2.92 2.91 2.91 2.9 2.89 2.88

Y= lnR 2.75 2.71 2.67 2.63 2.6 2.55 2.51 2.46 2.42 2.37 2.31 2.28 2.24 2.2 2.16 2.12

32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58

305.15 306.15 307.15 308.15 309.15 310.15 311.15 312.15 313.15 314.15 315.15 316.15 317.15 318.15 319.15 320.15 321.15 322.15 323.15 324.15 325.15 326.15 327.15 328.15 329.15 330.15 331.15

51.6 48 46.1 43.5 43.2 42.8 41.2 39.2 37.4 35 33 31.4 30.3 28.8 27.5 26.4 25.2 24.1 23.2 22.1 21.1 20.2 19.3 18.5 17.6 16.9 16.3

3.28 3.27 3.26 3.25 3.23 3.22 3.21 3.2 3.19 3.18 3.17 3.16 3.15 3.14 3.13 3.12 3.11 3.1 3.09 3.08 3.08 3.07 3.06 3.05 3.04 3.03 3.02

3.94 3.87 3.83 3.77 3.77 3.76 3.72 3.67 3.62 3.56 3.5 3.45 3.41 3.36 3.31 3.27 3.23 3.18 3.14 3.1 3.05 3.01 2.96 2.92 2.87 2.83 2.79

75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100

348.15 349.15 350.15 351.15 352.15 353.15 354.15 355.15 356.15 357.15 358.15 359.15 360.15 361.15 362.15 363.15 364.15 365.15 366.15 367.15 368.15 369.15 370.15 371.15 372.15 373.15

8.1 7.7 7.7 7.2 6.9 6.6 6.4 6.1 5.9 5.7 5.5 5.3 5.1 4.9 4.7 4.6 4.4 4.2 4.1 3.9 3.8 3.6 3.5 3.4 3.3 3.2

2.87 2.86 2.86 2.85 2.84 2.83 2.82 2.82 2.81 2.8 2.79 2.78 2.78 2.77 2.76 2.75 2.75 2.74 2.73 2.72 2.72 2.71 2.7 2.69 2.69 2.68

Analysis Graph of lnR against • •

analysis:

number of variables, N = 85; ∑ 𝑥 = 0.258; ∑ 𝑦 = 242.75; ∑ 𝑥 2 = 0.000788; ∑ 𝑦 2 = 789.94; ∑ 𝑥𝑦 = 0.757; ∑𝑥 0.258 x average, x = = 85 = 0.00304; 𝑁 242.75 ∑𝑦 = 𝑁 85 (∑ 𝑥)2

•

y average, y =

•

Sxx = ∑ 𝑥 2 −

•

𝟏 𝑻

Syy = ∑ 𝑦 2 −

𝑁 (∑ 𝑦)2

= 2.86;

= 0.000788 − =

0.2582

= 4.36*10-6;

85 242.752 789.94 − 85 = 96.69; 0.258∗242.75

𝑁 ∑𝑥∑𝑦 = 𝑁 𝑠𝑥𝑦

•

Sxy = ∑ 𝑥𝑦 −

•

slope of line, m =

•

intercept, b = y - mx = 2.86 – 4708*0.00304 = -11.44;

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𝑠𝑥𝑥

0.757 −

=

0.0205

4.36∗10−6

85

= 0.0205;

= 4 708;

2.09 2.04 2.04 1.97 1.93 1.89 1.86 1.81 1.77 1.74 1.7 1.67 1.63 1.59 1.55 1.53 1.48 1.44 1.41 1.36 1.34 1.28 1.25 1.22 1.19 1.16

•

𝑠𝑦𝑦− 𝑚2 𝑠𝑥𝑥

standard deviation about regression line, sr = √

𝑁−2

0.0216; 𝑠𝑟 2

•

standard deviation of the slope, sm = √

•

standard deviation of the intercept, sb =𝑠𝑟 *√ 0.0315;

𝑠𝑥𝑥

∑ 𝑥2

𝑁 ∑ 𝑥 2 −(∑ 𝑥)2

from y = c + mx, c = b = -11.44 and m= 4708. 1 Eg 1 lnR = C + 2𝐾 T, therefore lnR = -11.44 + 4708

•

m=

•        



𝐵

85−2

=

0.02162

= √ 4.36∗10−6 = 10.36;

• •

Eg , 2𝐾𝐵

96.69− 47082 ∗4.36∗10−6

=√

= 0.0216*√

0.000788 85∗0.000788−0.2582

=

T

therefore Eg = m*2𝐾𝐵 = 4 708*2*8.617×10−5 = 0.811 eV. As expected the plot was a linear graph. The gradient was found to be large as well. The calculated gap energy, Eg, of the unknown semiconductor is 0.811 eV. The known, accepted value for Silicon (Si) semiconductor is 1.11 eV and that for Germanium (Ge) is 0.67 eV. Therefore the unknown semiconductor is Germanium. The difference of the computed value from the actual value suggests that there were some errors introduced. Sources of errors in the experiment were possibly due to: o Random errors in the reading of the temperature from the analogue thermometer. o Moments were stirring speed was increased; it was difficult to take accurate resistance in time hence estimates were taken for those cases. o There could have been impurities on the semiconductor to affect the conductivity. Therefore the measured intrinsic gap energy would be slightly distorted. Even though there were some sources of errors, the standard deviation about regression line, which represents the error, is so small that the results can be considered valid.

Conclusions • The gap energy for the semiconductor is 0.811 eV. • The sample semiconductor used is Germanium. References 1) CJ Sheppard, S Jacobs, & B.P. Doyle. (First Semester 2011). PHYSICS 2A PRACTICAL GUIDE. University of Johannesburg. RSA. 2) http://www.chembio.uoguelph.ca/educmat/chm729/band/fl.htm 3) http://www.ece.umd.edu/~dilli/courses/enee313_spr09/files/supplement1_carrierconc .pdf

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