Scaling of exciton binding energy and virial theorem in semiconductor quantum wells and wires

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Recent numerical calculations [F. Rossi, G. Goldoni, and E. Molinari, Phys. Rev. Lett. 78, 3527 (1997)] have revealed a shape-independent hyperbolic scaling rule for the exciton binding energy versus the exciton Bohr radius in semiconductor quantum wires, and an enhancement in the exciton binding energy in a quantum wire with respect to a quantum well for a given exciton Bohr radius. These findings were attributed to the existence of a constant (shape- and/or size-independent) virial theorem value (potential- to kinetic-energy ratio), respectively, for the wires and wells, and its value was found to be larger (=4) for wires than (=2) for wells. In order to elucidate the physics underlying the above results, we reexamine this subject by calculating the exciton binding energy and the corresponding virial theorem value in quantum wells and wires with infinite confinement barriers. We find the following. (i) The virial theorem value is nonconstant but approaches 2 from above when reducing the finite extension of the electron and hole wave functions in the confined directions. This is because the origin of the virial theorem value of 2 lies in the inverse square Coulomb force being the only interaction seen by the electron and hole. (ii) The scaling rule is nonhyperbolic, because the virial theorem value is not a constant. (iii) The virial theorem value and the exciton binding energy are larger in a wire than in a well for a given exciton Bohr radius, because the wire exciton has a smaller kinetic energy in the nonconfined direction. (iv) The origin of the shape-independent scaling rule for wires lies in the close similarity of the effective Coulomb potentials for wires with different shapes and widths. The virial theorem value being or not being a constant is irrelevant to the scaling rule. (v) There exists a more fundamental and practically more useful shape-independent scaling rule.