Final answer:
To determine the temperature at which half of the arsenic dopant atoms in an N-type silicon sample are ionized, one must use the Fermi-Dirac distribution with the known ionization energy of arsenic and relevant constants. The fraction of ionized impurities is set to 1/2 to derive the equation for temperature without actually solving for a numerical value given the complexity and need for additional information about the silicon's intrinsic properties.
Step-by-step explanation:
The temperature at which half the impurity atoms are ionized in an N-type silicon sample with an arsenic dopant density of 1017 cm-3 can be found by considering that arsenic acts as a donor impurity in silicon. Since arsenic has a binding energy of approximately 0.02 eV for the extra electron, the fraction of ionized impurities is given by the Fermi-Dirac distribution at a given temperature. The general approach involves setting the fraction of ionized donors to 1/2 and finding the temperature at which this occurs using the equation that relates the ionization energy, the temperature, and the intrinsic carrier concentration of silicon.
To derive such an equation, consider the concept that the probability of ionization is given by f(E) = 1 / (1 + exp((E_D - E_F)/kT)), where f(E) is the fraction of ionization, E_D is the donor energy level, E_F is the Fermi level, k is the Boltzmann constant, and T is the temperature. Then, when half the donors are ionized, the fraction f(E) equals 0.5, and the equation simplifies to find T, knowing the ionization energy of 0.02 eV.
It should be noted that the calculation of the exact temperature would require additional information such as the intrinsic carrier concentration of silicon, which is temperature-dependent, and the Fermi level position, which might have to be estimated or calculated based on additional inputs. Without solving the actual equation, the goal is to correctly set up the relation that will allow for solving for T given the aforementioned variables and constants.