2 The Triangular Well and 2D Subbands The triangular potential well is an important simplification in order to solve the Schrdinger equation in a uniform time-independent electric field. In 3D semiconductors, the triangular well can also be used to approximate 2D quan- tum wells formed as a result of extremely thin degenerate doping profiles in 2D, known as 8-doping. At high dopant sheet densities, positively ionized donor atoms create an confining electric field for a two-dimensional electron gas (2DEG) within a symmetric, approximately triangular potential well. a) Assuming a uniform 2D sheet of positive charge with density N2D = 3 x 1015 mm-for the ionized donors, estimate the confining electric field using Gauss' theorem. Assume 8-doped GaAs ( = 13). 1 b) Having calculated the well potential, now calculate the first 2 subband edges for electrons in this triangular potential well. Make a sketch of the well and wave functions. Hint: for the correct solutions, you will require the zeros of the zeros of the first derivative Ai' x) of the Airy functions (see Appendix 5 of Davies), Briefly explain why! c) Now consider the formation of 2D subbands as a result of the vertical confinement. Calculate the density of occupied states, both at T = 0 and at room temperature. Assume that only the first two sub bands are occupied when the Fermi energy is 100 mev. d) Is the confined 2D electron gas degenerate or non-degenerate? 2 The Triangular Well and 2D Subbands The triangular potential well is an important simplification in order to solve the Schrdinger equation in a uniform time-independent electric field. In 3D semiconductors, the triangular well can also be used to approximate 2D quan- tum wells formed as a result of extremely thin degenerate doping profiles in 2D, known as 8-doping. At high dopant sheet densities, positively ionized donor atoms create an confining electric field for a two-dimensional electron gas (2DEG) within a symmetric, approximately triangular potential well. a) Assuming a uniform 2D sheet of positive charge with density N2D = 3 x 1015 mm-for the ionized donors, estimate the confining electric field using Gauss' theorem. Assume 8-doped GaAs ( = 13). 1 b) Having calculated the well potential, now calculate the first 2 subband edges for electrons in this triangular potential well. Make a sketch of the well and wave functions. Hint: for the correct solutions, you will require the zeros of the zeros of the first derivative Ai' x) of the Airy functions (see Appendix 5 of Davies), Briefly explain why! c) Now consider the formation of 2D subbands as a result of the vertical confinement. Calculate the density of occupied states, both at T = 0 and at room temperature. Assume that only the first two sub bands are occupied when the Fermi energy is 100 mev. d) Is the confined 2D electron gas degenerate or non-degenerate