a) Using the spreadsheet, find the probability that an allowed state i) at the Fermi level; ii) 0.1 eV above EF; and iii) 0.1 eV below EF is occupied by an electron at a temperature of 300 K (about room temperature). (Hint: You can use sheet Fermi-Dirac Data to find/calculate those numbers by setting/finding E-EF = -0.1, 0, 0.1 eV) You should use the data sheet to get accurate values (donāt just read them off the plot, though you can check them approximately that way). You can set the value of T either using the Options box on the Fermi-Dirac plot, or by entering it directly on the Materials sheet. However, there are two different temperatures on the latter sheet; you must set the āTemperature for Fermi-Dirac plot,ā NOT the āTemperature for gc/gv plot.ā
b) Check at least one of the calculations in part (a) by hand, and show your work in doing so.
c) Repeat part (a) at temperatures of 100 K and 600 K (but hand calculation not necessary).
d) Using the spreadsheet, find the probability that a state 2kBT below EF is NOT occupied by an electron (i.e., is empty). Show your calculation. (Hint: As I showed in the class, you can use E- EF = -2kBT = -0.052 eV to find the 1-f(E) value.)
The Fermi-Dirac Function & the Maxwell-Boltzmann Approximation
Plotted fMB(E)
Index E-EF
1 fFD(E) fMB(E) % Error Plotted x (eV) (eV)
0 -0.3000 -11.6050 1.00E+00 #N/A #N/A #N/A -0.3000 #N/A #N/A #N/A
1 -0.2985 -11.5469 1.00E+00 #N/A #N/A -0.2985
2 3 -0.2970 -11.4889 1.00E+00 #N/A -0.2970 -0.2955 -11.4309 1.00E+00 #N/A #N/A #N/A #N/A #N/A #N/A -0.2955 #N/A #N/A #N/A #N/A #N/A #N/A #N/A
4 5 6 7 -0.2940 -11.3729 -0.2925 -11.3148 -0.2910 -11.2568 -0.2895 -11.1988 -0.2880 -11.1408 1.00E+00 1.00E+00 -0.2940 -0.2925 1.00E+00 1.00E+00 1.00E+00 1.00E+00 -0.2910 -0.2895 -0.2880 -0.2865
8 9 -0.2865 -11.0827 #N/A
10 11 12 13 14 15 16 17 -0.2850 -11.0247 1.00E+00 #N/A #N/A #N/A #N/A #N/A #N/A #N/A -0.2850 -0.2835 -10.9667 -0.2820 -10.9087 -0.2805 -10.8506 -0.2790 -10.7926 -0.2775 -10.7346 1.00E+00 1.00E+00 -0.2835 -0.2820 1.00E+00 1.00E+00 1.00E+00 -0.2805 -0.2790 -0.2775 -0.2760 #N/A #N/A #N/A #N/A #N/A #N/A #N/A -0.2760 -10.6766 1.00E+00 #N/A -0.2745 -10.6185 1.00E+00 #N/A -0.2745