MIT OpenCourseWare http://ocw.mit.edu 3.23 Electrical, Optical, and Magnetic Properties of MaterialsFall 2007For information about citing these materials or our Terms of Use, visit: http://ocw.mit.edu/terms.Homework # 7 Novemb er 1, 2007 Homework is due on Thursday November 8st, 5pm 1 Intrinsic semiconductor Let us consider a 2 band intrinsic semiconductor. From the microscopic point of view this semiconductor is caracterized by a valence density of states, Dv (E), a conduction density of states, Dc(E), and a band gap Eg = Ec −Ev. Using those density of states and the Fermi factor f(T, E) = (E−1 µ)/kB T , we can 1+eeasily express the volumic density of electrons in the conduction band, nc(T, µ), and the volumic density of holes in the valence band, pv(T, µ): nc(T, µ) = � +∞ f(T, E)Dc(E)dE and pv(T, µ) = � Ev (1 − f(T, E))Dv(E)dEEc −∞Those formulas are extremely general and always applicable in practice. Now we will use the hypothesis of non-degeneracy, meaning that the chemical potential is far away in terms of thermal energy from the top of the valence band and the bottom of the conduction band: µ − Ev >> kB T and Ec − µ >> kB T non-degeneracy conditions For intrinsic semiconductors like Silicon, Germanium and Galium Arsenide those conditions of non-degeneracy are very well satisfied. Using this hypothesis, we see that the Fermi factor can be simplified to a Maxwell-Boltzmann factor (here in the case where E > µ): f(T, E) = (E−1 µ)/kB T becomes f(T, E) ≈ e−(E−µ)/kB T 1+e1.1 Some general results for any Dc(E) and Dv(E) and both intrinsic and extrinsic semiconductors 1) Using the non-degeneracy conditions, express the densities of electrons and holes, nc(T, µ) and pv(T, µ), as follows: • nc(T, µ) = Nc(T )e−(Ec−µ)/kB T • pv(T, µ) = Pv(T )e−(µ−Ev )/kB T Find an integral expression for both Nc(T ) and Pv (T ). What happens when one multiplies nc(T ) by pv (T )? 1 Nicolas Poilvert & Nicola MarzariImportant Remark: This relation is called the law of mass action. It is satisfied for both intrinsic and extrinsic semiconductors. All the doping does is to introduce some states inside the band gap, and shift the chemical potential, but this does not change the densities of states Dv(E) and Dc(E) so it does not change the law of mass action. 2) Now if we consider an intrinsic semiconductor, what is the relationship between nc(T, µ) and pv (T, µ)? Use this relationship to express the chemical potential as a function of T only. What is the chemical potential at 0K? What is the order of magnitude of the chemical potential shift at finite T from its value at T = 0K? In which direction does the chemical potential shift from its value at T = 0K when one increases the temperature? 1.2 Introducing the parabolic approximation for the va-lence and conduction bands 3) By using the following results from the parabolic approximation of the valence and conduction bands: 1 c 1 vDc(E) = 2π2 ( 2m∗ )3/2√E − Ec and Dv(E) = 2π2 ( 2m∗ )3/2√Ev − Eh¯2 h¯2 calculate explicitly Nc(T ) and Nv(T ). Then use this to calculate the chemical potential µ as a function of T , m∗ c , m∗ v, Ev and Ec. By taking the origin of the energy axis in the middle of the band gap, calculate the chemical potential shift in eV at 300K for GaAs (m∗= 0.063 m and m∗= 0.505 m, where m is the mass c v of a free electron). 2 p-doped semiconductors We will now consider a semiconductor homogeneously doped with acceptors. Those acceptors are what we call ”shallow impurities” in the sense that their presence inside the host semiconductor is responsible for the appearence of al-lowed energy states for holes right above the top of the valence band. The con-centration of acceptors is Na and the binding energy of the holes is �a = Ea −Ev. 2.1 qualitative description of the physics with increasing temperature 4) Describe qualitatively what happens to the majority carriers (in this p-doped semiconductor the majority carriers are the holes) as one increases slowly the temperature T from 0K to high temperature. Use the terms : freezing-out regime, saturation regime and intrinsic regime to specify the different behavior regimes of the doped semiconductor with temperature. 2.2 quantitative calculation of the majority carrier density with temperature 25) Let us now construct the curve ln(pv(T )) as a function of 1/kB T . We will study the different regimes that you outlined above and find the expression for the chemical potential and the volumic density of holes in the valence band at different temperatures. To do this we will establish a general balance equation for the charges in the system. We denote by P (Ea, T ) the probability for a hole to occupy an acceptor level at temperature T. Show that we have the following equation between the total density of electrons in the conduction band nc(T, µ), the total density of holes in the valence band pv(T, µ) and the concentration of acceptor impurities Na: nc(T, µ) + Na(1 − P (Ea, T )) = pv(T, µ) balance equation 6) At very low temperature, which term in the balance equation can we safely neglect and why? The probability P (Ea, T ) is given by P (Ea, T ) = µ−1 Ea . In the low temperature limit the chemical potential sits between 1 kB T 2 e +1 the top of the valence band Ev and the acceptor level Ea. Simplify the term 1 − P (Ea, T ) to a single exponential term up to a constant in front, and then use this simplified expression to show that the chemical potential can be written as follows (to obtain this expression you need to use the general expression for pv(T, µ) found in the first part of the problem set): µ(T ) = Ea+Ev − 1 kB T ln( Na )2 2 2Pv (T ) 7) Using the expression for the chemical potential in 6), express ln(pv(T, µ)) as a function of T only. What is the coefficiant of the linear dependance of ln(pv (T, µ)) with 1/kB T ? This regime is called the freezing-out regime. 8) At intermediate temperatures, where all the impurities have been ionized but no electrons have been promoted from the valence to the conduction band, simplify the balance equation and find an expression for the chemical potential. Show that in this regime called the saturation regime, ln(pv(T,