Particular cases of resonant tunneling in multibarrier ... - Springer Link

PARTICULAR CASES OF RESONANT TUNNELING IN MULTIBARRIER QUANTUM STRUCTURES G. F. Karavaev and N. L. Chuprikov

UDC 537.311.322--530.145

An investigation was made of the resonance conditions in systems of one-dimensional potential barriers of a general kind. The types of systems are identified and the conditions determined which are necessary for observing a broad resonance in which thefirst three derivatives of the energy transmission coefficient are equal to zero. The conditions are found for observing an isolated resonance with complete transparency on a background of resonances with incomplete transparency.

Using the matrix transfer method [2, 3], the present authors previously investigated [ 1] resonant tunneling under conditions of complete transparency when the transmission coefficient was equal to unity. The conditions for complete transparency were obtained, were given a clear interpretation and, in addition, a class of one-dimensional multibarrier structures was identified in which a resonance with complete transparency was possible. In the work now reported a number of questions are analyzed which are related to resonant tunneling in multibarrier quantum structures of a general kind and which, in our view, are not dealt with in the literature. Among these, a discussion is given of the conditions for the appearance of a resonance with incomplete transparency, of broad resonances, and of individual resonances with complete transparency in structures of a general kind. It was mentioned in [1] that any multibarrier structure can be represented in the form of a two-barrier structure, each of the two barriers being able to be represented by a fairly complex structure. The transmission coefficient T(I2) of the two-barrier structure is given by the formula (see [2, 3]) T(121= [1 + G + H • cos2F(~2)]-~,

(1)

where G=

T(I~ TI2)

]'d~ T,2)

FII2>= (J V0 (here and in what follows V0 is the height of the potential barriers) if the condition

1 )2 =

=eta=(n ÷ --~ E = 2ml =~,

!/0 ÷ -

,~'-'M 2rnd 2

K:,

(21)

as satisfied, where l is the barrier separation (the width of the "well"); d is the width of the barriers; x = ~ 2 m ( E - V o ) //~ 2; n,~: ~1 . . . . . In a sa'nilar way, one may expect broad resonances also to appear for superbarrier transmission in the case of systems of the special kind consisting of three or more rectangular barriers. However, there is greater interest in the possibility of the appearance of broad resonances in such structures for subbarrier transmission. This, for example, in the case of a three-barrier structure of the special kind (the width of the inner barrier being twice that of the outer barriers and the barrier heights being identical [1]), it is possible to observe a broad subbarrier resonance if the condition E=Vo/2; ×0t=~(Iv+l/'2)"

.,c=0, - t ....

(22)

is satisfied. These conditions are quite attainable, although there are still greater possibilities for observing broad resonances with four-barrier structures of the special kind. These are structures with identical rectangular barriers of width d, identical outer quantum wells of width l 1, and an internal quantum well of width lo. In this case there will be a broad resonance for E < Vo for the conditions 0