Question

(i) Consider a system where N electrons are constrained to move in two dimensions in a region of area A. What is the density of states for such a system? Obtain an expression for the Fermi energy of such a system as a function of the surface density n = N/A. (ii) The electrons in a GaAs/AlGaAs heterostructure (this just means they act as though they are in 2D!) have a density of 4 × 10^11 cm^-2. The electrons act as free particles, but their interaction with the lattice means that they “appear” to have a mass of only 15% of their normal mass (don’t worry about this — this property is something you will come across in the Solid State course). What is the Fermi energy of the electrons? (iii) Derive an expression for the density of states and Fermi energy for electrons confined in 1D. (iv) There are certain long-chain molecules which contain mobile electrons. The electrons can move freely along the chain, and the system is a 1D organic conductor, with n electrons per unit length. A typical molecule of this type has a spacing of 2.5! between the atoms that donate electrons, and “on average” each atom donates 0.5 electrons. What is the Fermi energy of this system?

          (i) Consider a system where N electrons are constrained to move in two dimensions in a region of area A. What is the density of states for such a system? Obtain an expression for the Fermi energy of such a system as a function of the surface density n = N/A.
(ii) The electrons in a GaAs/AlGaAs heterostructure (this just means they act as though they are in 2D!) have a density of 4 &times; 10^11 cm^-2. The electrons act as free particles, but their interaction with the lattice means that they “appear” to have a mass of only 15% of their normal mass (don’t worry about this — this property is something you will come across in the Solid State course). What is the Fermi energy of the electrons?
(iii) Derive an expression for the density of states and Fermi energy for electrons confined in 1D.
(iv) There are certain long-chain molecules which contain mobile electrons. The electrons can move freely along the chain, and the system is a 1D organic conductor, with n electrons per unit length. A typical molecule of this type has a spacing of 2.5! between the atoms that donate electrons, and “on average” each atom donates 0.5 electrons. What is the Fermi energy of this system?

(i) Consider a system where N electrons are constrained to move in two dimensions in a region of area A. What is the density of states for such a system? Obtain an expression for the Fermi energy of such a system as a function of the surface density n = N/A.
(ii) The electrons in a GaAs/AlGaAs heterostructure (this just means they act as though they are in 2D!) have a density of 4 times; 10^11 cm^-2. The electrons act as free particles, but their interaction with the lattice means that they “appear” to have a mass of only 15% of their normal mass (don’t worry about this — this property is something you will come across in the Solid State course). What is the Fermi energy of the electrons?
(iii) Derive an expression for the density of states and Fermi energy for electrons confined in 1D.
(iv) There are certain long-chain molecules which contain mobile electrons. The electrons can move freely along the chain, and the system is a 1D organic conductor, with n electrons per unit length. A typical molecule of this type has a spacing of 2.5! between the atoms that donate electrons, and “on average” each atom donates 0.5 electrons. What is the Fermi energy of this system?

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