Bandgap Measurements In Ge  More Resources 

physics education essay research university report assessment newcastle

Bandgap Measurements in Ge PHYS213 Richard Laugesen, 10 April 2001 ABSTRACT Measured the bandgap of an n-type Ge sample using a four probe resistivity technique. A constant current was applied to the sample while measurements of the voltage were taken as the temperature was increased from room temperature to approximately 1750 C and then cooled. The resistivity was calculated as a function of temperature and from that relationship the bandgap was calculated to be 0.728 ± 0.01 eV, an accepted value is 0.66 eV. 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. PHYS213 Bandgap Measurements in Ge, 213.04 19/07/2008 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 wavefunction associated with it and discrete potential energy levels, but in a solid with the many densely packed atoms, the wavefunctions overlap and interact and 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 bandgap Eg separates the valence band and the conduction band. A semiconductor has a bandgap 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 bandgap 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 bandgap to ≈0.05 eV. Four Probe Resistivity Technique A four probe experimental technique was used to measure the bandgap. 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, we vary the temperature while taking measurements, and hence are able to find the resistivity as a function of temperature. If a plot is then made of ln ρ vs. 1/T then the slope will be proportional to the bandgap of the sample. We use this equation to calculate the bandgap Eg ln ρ = Eg 2kT + ln C The slope must be found from the part of the plot where the temperature is high and the intrinsic component of conduction dominates; the slope is linear in this region. 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 technique overcomes these faults by using pressure contacts but a draw back is they tend to be noisy. Richard Laugesen 2 PHYS213 Bandgap Measurements in Ge, 213.04 19/07/2008 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 smaller 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. 2. 3. 4. 5. 6. The resistivity of the sample is uniform in the area of measurement. 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. The surface on which the probes rest is flat. The four probes lie in a straight line. The diameter of the contact between the probes and the sample is small compared to the distance between the probes. 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 small1 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) Diagram of the Experimental Apparatus Direct Current Source Millivoltmeter Thermometer Heater Probes Ge Sample PROCEDURE The Ge sample was placed into the holder of the probe equipment along with a mercury based thermometer. The spring loaded probes were then fastened into place; care was taken to maintain a good contact between the probes and the Ge surface but not to apply too much pressure. The electric connections were made, and a current of 5.99 mA was set. The thermometer was then turned on and readings of the voltage were taken until the temperature reached1750 C, the thermometer was then turned off and more readings were taken as the sample cooled down to approximately 400 C. While the readings were made the current was checked periodically to be sure that it was constant throughout the experiment. RESULTS 1 5.99 mA in out experiment. Richard Laugesen 3 PHYS213 Bandgap Measurements in Ge, 213.04 19/07/2008 From these readings of temperature against voltage it was trivial to calculate the resistivity for each temperature value from the current and voltage, the results are presented in the following plot2 of ln ρ vs. 1/T. The slope of the linear section (where temperature is high and intrinsic conductivity dominates) was then used to calculate the bandgap Eg using the equation derived in the introduction. The calculated bandgap was 0.728 ± 0.01 eV. An accepted value is 0.66 eV3. Four Probe Resistivity Measurement of Ge 2.500 2.000 1.500 1.000 log( ρ ) 0.500 0.000 0.00E+00 -0.500 -1.000 -1.500 1/T (K-1) 5.00E-04 1.00E-03 1.50E-03 2.00E-03 2.50E-03 3.00E-03 3.50E-03 4.00E-03 y = 4222.40x - 10.31 R2 = 0.98 DISSCUSSION The bandgap of Ge was measured as 0.728 ± 0.01 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 bandgap 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 we calculate the bandgap 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 different 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 valance 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 a high impedance. 2 3 Tabulated form of results are in the appendix. Accepted value of Ge from http://www.applied-epi.com/pages/lattice.htm Richard Laugesen 4 PHYS213 Bandgap Measurements in Ge, 213.04 19/07/2008 CONCULSION The bandgap of a Ge was measured using a four probe resistivity technique and found to be 0.728 ± 0.01 eV. The accepted value is 0.66 eV. The discrepancy is most probably due to the assumptions made for the four probe technique. APPENDIX Probe and Sample Dimensions Probe Spacing: 2 mm Sample width: 0.5 mm Ge sample has a nonconducting bottom surface and small thickness compared to the probe spacing. Uncertainties Voltage: 0.3 mV Temperature: 0.5 0C Current : +0.05 % The uncertainty in current would not effect individual measurements as it is a constant of the experiment. Tabulated form of measurements Constants of Experiment I S G7(W/S) 5.99E-03 2.00E-03 5.89 A m Uncertainties δT (K) 0.5 δP D (V) 0.0003 δI (A) 3.00E-06 T (K) 300 302 304 306 309 311 313 313 314 314 316 316 318 318 1/T (K-1) 3.33E-03 3.31E-03 3.29E-03 3.27E-03 3.24E-03 3.22E-03 3.19E-03 3.19E-03 3.18E-03 3.18E-03 3.16E-03 3.16E-03 3.14E-03 3.14E-03 PD (V) 0.1755 0.1763 0.1772 0.1781 0.1790 0.1795 0.1800 0.1817 0.1802 0.1818 0.1808 0.1820 0.1810 0.1812 ρ (Ω.cm) 6.251 6.279 6.311 6.344 6.376 6.393 6.411 6.472 6.418 6.475 6.440 6.482 6.447 6.454 δρ (Ω.cm) 0.01381 0.01383 0.01384 0.01386 0.01387 0.01388 0.01389 0.01392 0.01389 0.01392 0.01391 0.01393 0.01391 0.01391 ln(ρ) 1.833 1.837 1.842 1.847 1.852 1.855 1.858 1.867 1.859 1.868 1.862 1.869 1.864 1.865 T (C) 27 29 31 33 36 38 40 40 41 41 43 43 45 45 Richard Laugesen 5 PHYS213 Bandgap Measurements in Ge, 213.04 19/07/2008 323 323 328 328 333 333 338 338 343 343 348 348 353 353 358 358 363 363 368 368 373 373 378 378 383 383 388 388 393 393 398 398 402 403 408 408 413 413 418 418 423 423 428 428 3.10E-03 3.10E-03 3.05E-03 3.05E-03 3.00E-03 3.00E-03 2.96E-03 2.96E-03 2.92E-03 2.92E-03 2.87E-03 2.87E-03 2.83E-03 2.83E-03 2.79E-03 2.79E-03 2.75E-03 2.75E-03 2.72E-03 2.72E-03 2.68E-03 2.68E-03 2.65E-03 2.65E-03 2.61E-03 2.61E-03 2.58E-03 2.58E-03 2.54E-03 2.54E-03 2.51E-03 2.51E-03 2.49E-03 2.48E-03 2.45E-03 2.45E-03 2.42E-03 2.42E-03 2.39E-03 2.39E-03 2.36E-03 2.36E-03 2.34E-03 2.34E-03 0.1801 0.1795 0.1778 0.1746 0.1731 0.1679 0.1660 0.1571 0.1567 0.1449 0.1452 0.1314 0.1320 0.1180 0.1188 0.1040 0.1058 0.0918 0.0936 0.0802 0.0821 0.0695 0.0714 0.0610 0.0614 0.0533 0.0538 0.0466 0.0468 0.0410 0.0510 0.0365 0.0317 0.0353 0.0308 0.0275 0.0273 0.0243 0.0242 0.0215 0.0215 0.0192 0.0189 0.0170 6.415 6.393 6.333 6.219 6.165 5.980 5.913 5.596 5.581 5.161 5.172 4.680 4.702 4.203 4.231 3.704 3.768 3.270 3.334 2.857 2.924 2.475 2.543 2.173 2.187 1.898 1.916 1.660 1.667 1.460 1.817 1.300 1.129 1.257 1.097 0.979 0.972 0.866 0.862 0.766 0.766 0.684 0.673 0.606 0.01389 0.01388 0.01385 0.01379 0.01377 0.01368 0.01364 0.01348 0.01348 0.01327 0.01327 0.01303 0.01304 0.01279 0.01280 0.01254 0.01257 0.01232 0.01235 0.01211 0.01215 0.01192 0.01196 0.01177 0.01178 0.01163 0.01164 0.01152 0.01152 0.01142 0.01159 0.01134 0.01125 0.01131 0.01123 0.01118 0.01117 0.01112 0.01112 0.01107 0.01107 0.01103 0.01102 0.01099 1.859 1.855 1.846 1.828 1.819 1.788 1.777 1.722 1.719 1.641 1.643 1.543 1.548 1.436 1.443 1.309 1.327 1.185 1.204 1.050 1.073 0.906 0.933 0.776 0.783 0.641 0.650 0.507 0.511 0.379 0.597 0.262 0.121 0.229 0.093 -0.021 -0.028 -0.144 -0.149 -0.267 -0.267 -0.380 -0.396 -0.502 50 50 55 55 60 60 65 65 70 70 75 75 80 80 85 85 90 90 95 95 100 100 105 105 110 110 115 115 120 120 125 125 129 130 135 135 140 140 145 145 150 150 155 155 Richard Laugesen 6 PHYS213 Bandgap Measurements in Ge, 213.04 19/07/2008 433 433 438 438 443 443 448 2.31E-03 2.31E-03 2.28E-03 2.28E-03 2.26E-03 2.26E-03 2.23E-03 0.0168 0.0153 0.0147 0.0138 0.0129 0.0125 0.0117 0.598 0.545 0.524 0.492 0.459 0.445 0.417 0.01098 0.01096 0.01095 0.01093 0.01092 0.01091 0.01089 -0.514 -0.607 -0.647 -0.710 -0.778 -0.809 -0.875 160 160 165 165 170 170 175 Richard Laugesen 7