Measurement of the band gap of a semi-conductor and calculation of the Boltzmann constant, using two methods.

The primary apparatus for this investigation was a diode laser (basically a glorified LED for the purpose of this experiment). When electrons are excited to the conduction band of a semiconductor (the stuff diodes are made of) and then recombine with holes in the valence band, photons of energy are emitted. The output spectrum of this laser was recorded at constant current through the diode and varying resistances (proportional to temperature in the diode).

The peak frequency of the observed spectrum is the band gap energy of the semiconductor - intensity falls off at greater or lesser frequencies. These intensity values are nonzero because not all carriers exist at the low- energy edge of the band. Defects allow for smaller energy drops, and excitations beyond the edges of the bands allow for larger energy drops. Analyzing the intensity drop-off on the high energy side of the spectrum allows us to calculate the Boltzmann constant, k when temperature is known.

The relationship between voltage and current in a diode is non-Ohmic. When the diode is forward biased, current flows easily. Resistance drops off as the forward bias voltage increases. However, when the diode is reverse biased at voltages (below some breakthrough voltage), resistance is extremely high. The plot of Current vs Voltage is exponentially dependent on qV/kT (q is the elementary charge, k is the Boltzmann constant). This gives us a second means for finding the Boltzmann constant.

The peak frequencies in the laser spectrum allow us to calculate the band gap energy easily.

E = h*f (h is Planck's constant)

Temp (C) | Freq (Hz) | Bgap Energy (eV) |

15 | 4.62*10^14 | 1.909 eV |

25 | 4.61*10^14 | 1.905 eV |

35 | 4.60*10^14 | 1.901 eV |

According to principles of Boltzmann statistics:

Intensity = (Intensityo)*e^(-E/kT)

Ln I = -E/kT + Ln Io

The slope of a graph of Ln(I) vs Energy (eV) is equal to -1/kT. Linear regressions of the data below yield estimates of the slope. These were used to calculate k given some known T in Kelvins.

Temperature (C) | Calculated Boltzmann (eV/K) |

15 | 9.944*10^-5 |

25 | 9.860*10^-5 |

35 | 9.829*10^-5 |

The accepted value for the Boltzmann constant in eV/K is 8.617*10^-5. These calculated values have approximately 15% error.

According to principles of Boltzmann statistics:

Current = (Currento)*e^(-E/nkT) = (Currento)*e^(qV/nkT)

Ln I = -E/nkT + Ln Io = qV/nkT + Ln Io

The slope of a graph of Ln (I) vs Voltage is equal to qV/nkT; if k is in eV/K, the slope is simply 1/nkT. Linear regressions of the data below yield estimates of the slope. These were used to calculate k given some known T in Kelvins.

Given a specific temperature, and knowing the Boltzmann constant, it is possible to solve for n (ada). Ada is a factor indicating the prevalence of recombination in the depletion layer (3). Based on the slope of the linear regression at 25 degrees Celcius, ada was found to be 1.901. This value, and the accepted value for k, were used to project temperatures for the other two cases. These calculated temperatures were compared to measured temperatures.

Calculated T | Measured T | % Error |

290.4295 K | 15 deg C = 288.15 K | 0.79 % |

298.15 K* | 25 deg C = 298.15 K | 0 % |

307.1558 K | 35 deg C = 308.15 K | .32 % |

Calculation of Energy of gap??