Pergamon Sohd State Commumcatmns, Vol 91, No 1, pp 39-43, 1994 Elsevier Science Ltd Pnnted in Great Britain 0038-1098/94 $7 00 + 00 0038-1098(94)E0181-A HARTREE-LIKE CALCULATIONS OF ENERGY LEVELS IN QUANTUM WIRES G Bastard Laboratoire de Physique de la Mati6re Condens6e ENS, 24 rue Lhomond F 75005 Paris, France and J Y Marzln France Telecom, CNET PAB, Laboratolre de Bagneux, 196 avenue H Rav6ra F 92220 Bagneux, France (Received 22 February 1994 by M Cardona) We report on approximate energy levels calculations in semiconductor quantum wires using separable wavefunctions These are the solutions of the Hartree-hke integro-differential equations for two one-dimensional carriers which interact through the bl-dlmenslonal confining potential of the wire Surface quantum wires and T-shaped wires are used as examples PROGRESS In nanofabrlcaUon has allowed the growth and patterning [1-5] of semiconductor quantum wires of increasing quality Recent optical experiments undertaken on surface quantum wires [4] or T-shaped wires [5] have evidenced clearly the effect of the lateral patterning or re-growth on the energy levels of the underlying quantum well structure, as evidenced by a sizeable energy shift of the optical absorption or emission edges In the Schrodinger equation for an electron moving m a quantum wire the motion along the wire axis (x) is separable from the one m the wire cross section (y and z) On the other hand the z and y motions are non separable in general Numerical solutions of the two dimensional Schr6dmger equation provide accurate energy levels and wavefunctions They lrmy not be as practical as algebraical solutions, even approximate, if one is interested In evaluating relaxation times, energy loss rate, etc In this communication, we discuss separable solutions of the two dimensional Schr6dlnger equation and apply our considerations to surface quantum wires and T-shaped quantum wires We seek for the solutions of {--hZ/2m*[O2/Oz2 + 02/Oy 2] + V(y, z)}O(y, z) = e~b(y,z), (1) which would be separable in y and z We remark that equation (1) can formally be seen as that of two one dimensional electrons which interact via the potential V(y, z) in lieu of the Coulomblc one Then, we know [6] that the best separable solutxons of equation (1), ~(y, z) = a(y)x(z), IS provided by the set of Hartree equations {-h2/2m* O2/Oz2 + I V(y,z)o~2(y)dy}x(z)= ~zX(Z), (3) with the elgenvalue + ez - J V(y,z)x2(z)c~2(y) dydz (4) ~y The last term in equation (4) is to avoid double counting the "electron-electron" interaction V(y, z) In case where there is no exact solution of the Hartree equations and where one should resort to variational estimates, the total energy e, and not the one particle energies ey, ez, has to be minimized with respect to the variational parameters The Hartree estimate can be contrasted with other approximate approaches like the projection of equation (1) on a basis for the z motion [7, 81 ~b(y, z) = Z °ln(Y)Xn(Z)' (5) n where the X.s are the solutions of any one- dimensional Schr6dmger equation. In practice, since 39