Technical Note A note on the relation between discrete and resonance energies in quantum structures A. Dargys * , P. Cimmperman Semiconductor Physics Institute, A. Go stauto 11, 2600 Vilnius, Lithuania Received 20 September 2000; received in revised form 19 December 2000; accepted 10 January 2001 Abstract A uni®ed formula for energy levels in a quantum well is presented. It is shown that the formula can be used in evaluating resonances of a symmetric double-barrier structure in a wide range of barrier and well widths. A numerical example for GaAs/Al 0:3 Ga 0:7 As resonant double-barrier structure is presented to explain why the formula yields correct resonance energies even for narrow barriers, when the thickness of both barriers makes up 20% of the well width. Ó 2001 Published by Elsevier Science Ltd. PACS: 68.35.-p; 71.55 Eq; 73.20.Jc Keywords: Quantum well; Resonant double-barrier structure; Resonances; Approximation formula Because of the importance of resonant-tunnelling device application, considerable eorts have been di- verted towards calculation of resonance energies at which electron transmission through the structure be- comes equal to or nearly equal to unity. A review of various methods of calculation of I±V characteristics of resonant-tunnelling diodes can be found in Mizuta's and Tanoue's book [1] and in the recent paper [2]. In general, the methods of determining the resonance energies re- quire extensive numerical calculations. Frequently there is a need, for example in circuit simulation models [3], to have a simple empirical expression to evaluate quickly the resonant energies of a concrete nanostructure. The aim of this note is to show that the following transcen- dental equation m b E n m w V 0 E n s m w V 0 E n m b E n s 2cot 2m w E n p h L w 1 can be used to ®nd resonance energies E n of a symmetric double-barrier structure with high accuracy in case of thick barriers, and less evident in case of thin barriers) when the barrier thickness makes up only a fraction of the quantum well width. In Eq. 1), L w is the well width, m w and m b is the eective electron mass in the well and barrier regions, V 0 is the depth of the well, and h is Planck's constant. In fact, Eq. 1) represents a uni®ed equation for energies of odd and even parity states of a one-dimensional quantum well QW), when the eective mass in the well and barrier regions is taken into account properly and when commonly used matching conditions for the electron wave function in the well w w and barrier w b and for their derivatives with respect to the coordi- nate x the latter is perpendicular to a heterobarrier plane) are used w w w b 2 1 m w ow w ox 1 m b ow b ox 3 When m w m b , formula 1) reduces to a uni®ed equa- tion for odd and even parity energies of Ref. [4], where the formula has been used to compute revivals and superrevivals of motion of a wave packet in a QW. The points in Fig. 1 show resonance energies in a double-barrier structure as a function of barrier thick- ness at three QW widths: 7, 10, and 20 nm. The QW Solid-State Electronics 45 2001) 525±526 * Corresponding author. Fax: +370-2-627-123. E-mail address: dargys@uj.p®.lt A. Dargys). 0038-1101/01/$ - see front matter Ó 2001 Published by Elsevier Science Ltd. PII:S0038-110101)00040-5