Bound states with no classical turning points in semiconductor heterostructures

~

Solid State Communications, Vol. 90, No. 11, pp. 713-716, 1994 Elsevier Science Ltd Printed in Great Britain 0038-1098(94)$7.00+.00 0038-1098(94)E0267-F

,Pergamon

Bound States With No Classical Turning Points in Semiconductor tleterostructures T. A. Weber Dept. of Physics and Astronomy, Iowa State University, Ames, Iowa, USA 50011 (Received 23 March 1994 by E.Mendez) Recent observation of a discrete bound state with energy above the potential barrier in a semiconductor heterostructure appears to be the first observation of bound states at energies with no classical turning points. Such states were first proposed by yon Neumann and Wigner in 1929. In this paper, the transfer matrix method is used to solve the Schr6dinger equation for the electronic states in an effective potential representing the layered semiconductor structure. This simple approach gives a clear picture of the miniband structure along with the bound states above the barrier height. The results are obtained by analytic methods, thus facilitating the investigation of the ranges of parameters necessary to produce such bound states. The Bragg conditions, although leading to a bound state, are found to be overly restrictive.

states produced are highly localized but should be called quasi-bound. in section 2 the transfer matrix method is applied to the determination of the miniband structure and the bound states, in section 3 the results are applied to the experiment of Capasso et al. In section 4 a heterostructure supporting two bound states above the barrier is designed, while in section 5 the method is generalized to a central well of width unrelated to the spacing of the barriers.

1. Introduction Recently, Capasso, Sirtori, Faist, Sivco, Chu and Cho reported the remarkable observation of an electronic bound state above a potential well in a semiconductor heterostructure I. Such bound states at energies with no classical turning points were first proposed in the early days of quantum mechanics by yon Neumann and Wigner 2. They suggested that certain oscillatory potentials with decreasing amplitude could support bound states in the energy continuum by means of coherent diffraction scattering. At one particular momentum determined by the wave length of the oscillation of the potential, constructive interference of the reflected waves can cause the amplitude of the wave function to vanish at large distances thereby producing a normalizable state. Stillinger3 and Herrich 4 proposed that superlattices consisting of ultra-thin semiconductor layers might be used to construct potentials supporting such bound states. Capasso et. al. used the Bragg reflection conditions to carefully construct such a heterostructure. Infrared absorption measurements revealed a narrow isolated transition from a bound state within a quantum well to a bound state at an energy greater than the barrier height, in this paper the bound states are found by a simple analytical method which facilitates the examination of the conditions and ranges of parameters necessary to produce such bound states. it is found that the Bragg reflection conditions, although leading to a bound state, are too restrictive. The model potential is a modified Kronig-Penny potential consisting of a series of equally spaced barriers necessarily cut off by the finite size of the sample. True bound states above the height of these barriers exist only if the potential is infinite in extent. Even so the

2. Method The model potential is a one-dimensional series of equally spaced square barriers extending to :1: infinity. The barriers all have height Vo, width b, and separation a as shown in Fig. 1. This potential represents the heterostructure in the direction perpendicular to its layered structure. Note that the central well of width 2a represents a defect in the periodicity. In the following only the one dimensional motion of the electron perpendicular to the layered structure is considered. The dispersion relation of an electron in the well material is taken to be E = ~/(2m~),

(I)

while in the barrier material it is E = Vo + k' b/(2m~,).

(2)

Here m" is the effective mass of the electron and k b is real for energies above the height of the barrier and imaginary for energies below. The units chosen are such that ~ = c = 1. To correct for band 713

714

BOUND STATES WITH NO CLASSICAL TURNING POINTS

Vol. 90, No. I 1

Since the transfer matrix is unimodular the product of its two eigenvalues,

V(x)l

X±=

(7 + n ) n

_+ (I/2)v/(.+7)2-4.

(II)

is unity. If the discriminant,

,-

2a

~

a

D = (,7 + 7)2-4,

(12)

0

Effective potential consisting of a series of square barriers.

Fig. 1.

nonparabolicity, energy dependent effective masses will be used in the applications. The stationary-state Schr6dinger equation for either material, [ _ .i

42 .v] ¢ - E ¢ ,

(3)

2m" dx 2 is solved in the usual way by matching the solutions at the well-barrier interfaces. The wavefunction itself is taken to be continuous, but since the effective mass is discontinuous the appropriate condition on the derivative of the wavefunction, ¢', is that ~'/m" be continuous 5. This condition is easily found by integrating the Schr6dinger equation across an interface. The well-known transfer matrix method 6 gives the solution at the discrete points x = N(a + b) where N is an integer. Let the solution be represented by a two component column matrix consisting of ~ and ~b'/m'. Since the potential is symmetric about the origin the bound states will have even or odd parity. Therefore, at the origin

(ql.~m.)° . A(1)0

(4)

for the even solution and

for the odd solution where A and B are constants. The operator that relates the solution across the interval a + b is given by the matrix,

is negative, the eigenvalues are complex conjugates of unit modulus associated with traveling waves at energies within the minibands. For D greater than zero, the real eigenvalues are denoted by k L and ks where [ k L [ > 1 > [ •s [ • The associated energies give the minigaps. Of course, all of the above band structure is well-known from the study of the stable and unstable regions of the Mathieu equation ~. A bound state will occur in a minigap for an eigenfunction of T,.b with eigenvalue As since the amplitude of the wave function will be down by a factor X~s at x = N(a+b). As N -- ~ the amplitude decreases exponentially thus leading to a normalizable state. It is apparent that an even state will be an eigenstate of To÷b with real eigenvalue 3' provided a = 0, and similarly the odd state will have real eigenvalue provided p = 0. To determine if a bound states exists the magnitudes of these eigenvalues must be found. First consider states below the height of the potential barrier for which the momentum in the barrier material is pure imaginary. The condition a = 0 gives an even state with eigenvalue 7 of magnitude less than one. Similarly, p = 0 gives an odd state with eigenvalue T/of magnitude also less than one. As expected, one obtains alternating even and odd bound states within the well, the number depending on the depth and width of the well. For possible even bound states above the barrier height, the eigenvalue conditions a = 0 gives for the square of the eigenvalue,

, +COS 2 kwa I ~2~m" (E-v o)

which is always greater than one except under unusual circumstances. Therefore, in most cases, no even bound states exist above the barrier height. For possible odd bound states, the eigenvalue condition p= 0 gives for the square of the eigenvalue,

~2=m:m~(E-V°)E m:m;(E-VO) IE ÷ c°sZ kwa[ 1

where 7 = c o s k~a cos k~b - (kJm:,)(m ~/k~) sin k~a sin kbb,(7) p = ( m ' J k , , ) sin k,,a cos kbb+(m~/kb) cos k~a sin kbb,(8 ) a= - ( k ~ m ' ) c o s k,~a sin kbb-(k,,,/m") sin k.,a cos kbb,(9 ) ,7= -(m~,/k,,)(k~/m~)sin L,a sin khb+cos kbb cos k~a.(10)

, (13) m" (~-Vo)

' (14)

which is always less than one provided m" < m~, and cos 2 k~a ;~ 1. Except for unusual circumstances then, odd bound states will exist at energies above the barrier potential. For a bound state at a given energy, T/2 will have a minimum if the parameter a is chosen such that cos k,,a = 0 or k~a = (n + 'A..)f where n is an integer or zero. Furthermore, p = 0 implies that cos kbb must also be zero or kbb = (m + 'A)~r with m an integer or zero.

Vol. 90, No. 1i

BOUND STATES WITH NO CLASSICAL TURNING POINTS

These Brag~ conditions give the maximum localization for a bound state at a given energy and also furnish a simple physical interpretation for the localization by constructive interference of the waves reflected back toward the origin. Unfortunately no such simple picture exists for the general case p -- 0. It can also be shown that each minigap will contain one odd bound state. However, Eq. 14 indicates that the larger the energy of the bound state, the less localized it becomes. Therefore only the lowest lying bound states will be sufficiently localized to be of interest. More on the relationship between bound states and the widths a and b will be given in the context of the material constants of the next section. Also an example of an even bound state above the barrier height will be given in section 5. 3. Applications Capasso et al. t reported the observation of a bound state in a layered semiconductor with well material Ga0.~ Inej~ As and barrier material Al0.~ lno.sz As. The central well has depth Vo = 0.5 eV and width 2a = 32 ,~ while the barrier widths are b -- 39 ,~. The effective masses are taken to be m~ = 0.043 m° and m b = 0.073 mo. To account for band nonparabolicity, energy dependent effective masses are used according to the scheme of Nelson, Miller and Kleinman*. The energy gaps between the conduction and the light-hole valance bands are taken to be 0.88 eV and 1.49 eV for the well and barrier materials respectively. In the well one finds an even state at 203 meV and a miniband from 307 to 387 meV. An odd hound state is found at 563 meV giving a transition energy of 360 meV. The probability density of the bound state above the barrier is down by a factor ~2 = 0.11 at a distance x = a + b. All the results agree with those of Capasso et al. t to within a few per cent. The energy and localization of the bound state above the barrier is now briefly examined as a function of the widths of the well and barriers. To ensure reasonable localization, the range of variation will be limited so that ~1z <: 0.33. Three cases, 1) a fixed, 2) b fixed, and 3) E fixed, will be considered separately. 1) Fix a at 16 ,~ and vary b. As b increases from 9.9 to 137 ,~ the energy decreases from 690 to 510 meV and ~z decreases monotonically from 0.33 to 0.002. Furthermore, as b -* 0, E -* 1350 meV and ~2 . . 1. 2) Fix b at 39 ,~ and vary a. As a increases from 10.4 to 18.8 ,~ the energy of the bound state decreases from 660 to 510 meV and ~I2 decreases from 0.33 to 0.065. As a -* 0, E -* 783 meV and ~z . . 1. 3) Fix E at 563 meV and vary both a and b. As a increases from 10.3 to 20.5 A, b decrease from 74.3 to 9.0 ,~. The square of the eigenvalue, ,/2 as a function of a is shown in Fig. 2. Note the broad minimum where y/z < 0.11. With the material constants given above, a = 16 ,~ is not precisely at the Brag& conditions which give ,72 = 0.104.

715

0.35,l

t

0.30~

i

F 0.254

020i 0~51

0.10~ 00510

11

12 13

14

15

16

17

'r8

19

20 21

a (Angstrom)

Fig. 2.

The square of the eigenvalue as a function of the barrier separation. The energy of the bound state is held fixed at E = 563 meV. The barrier width, b, decreases from 74.3 to 9.0/~ as a increases from 10.3 to 20.5,~,.

4. Two Bound States Above the Barrier Potential Although it is possible to choose a and b such that the Bragg conditions will hold for two different energies, the result is too restrictive. In order to get reasonably localized states choose the higher energy state at the Brag& conditions, say, k,a = T/2 and khb = 3T/2. The other bound state will be more localized than this state if it is sufficiently close to the top of the barrier. For a = 12.3 ,~ and b --- 60 ,~, and the material constants used in section 3, odd bound states are found at E = 772 meV with T/2 = 0.33 and E = 574 meV with z = 0.20. Within the well there is an even bound state at E = 256 meV with 72 = 0.0012. Minibands exists from 369 to 405 meV and from 591 to 722 meV. More than 2 bound states can be obtained for larger values of b. For example, with b = 120 ,~ there are 4 minigaps between 500 meV and 880 meV; each contains an odd bound state. The eigenvalue squared, 72, ranges from 0.17 to 0.39. 5. Generalization to Central Well of Any Width In the foregoing analysis, the central well has width double the spacing of the barriers. In this section the central well is taken to have width 2c with the barriers again having width b and constant separation a. The wave function at any point x ffi c + N (b + a) can now be obtained by the transfer matrix T~.,T,. The matrix "Is÷. has the same eigenvalues and miniband structure as T , . b since it has the same elements but with 7 and interchanged. But now, by adjusting the width of the central well, even or odd bound states can be placed at various energy levels within the minigaps. The transfer matrix from the origin to the point x = c is

Tc -- -

CkwOS kwc sin kvc

~--~sinkvc]. COSkvc ]

(15)

BOUND STATES WITH NO CLASSICAL TURNING POINTS

716

Operating on the even or odd state at the origin gives a column matrix (with elements v, and v,) which, for a bound state, must be an eigenvector of the transfer matrix Tb.. with real eigenvalue hs. Hence

Vol. 90, NO. 11

0.06 while Eq. 17 gives an even hound state at 82 meV with h~ = 0.002.

Conclusion.

v2/vt = a/(ks- "r),

(16)

which when applied to even states gives (k.,,/m:,) tan k~c = -a/(h s - ~,),

(17)

while for odd states, ( ~ m ~ . ) cot k.c = a/(Xs - %

(18)

There is no guarantee, of course, that an energy level within a minigap will satisfy either of these two conditions for a given well width 2e. For example, if c = a ~ then the potential is periodic and no bound states exist. On the other hand, these equations can be used to place a bound state anywhere within a minigap. Note that the variation of c does not alter the miniband structure or the eigenvalues. As an example, an odd bound state can be placed at E = 510 meV. The material constants of section 3 and widths a = 16/~ and b = 39/~ give X~ = 0.08 for this energy. The smallest value of c found from Eq. 18 is 19,~. Furthermore, with this value of c, Eq. 17 gives an even bound state within the well at 173 meV. Similarly, an even bound state can be placed at E = 510 meV for which hl = 0.08. From Eq. 17, one obtains a series of well widths; the smallest two values are 2.5 and 35.5 A.. F o r e = 35.5 ~, Eq. 18 gives an odd bound state within the well at 275 meV with Xsz =

The original proposal of von Neumann and Wigner suggested an oscillating potential with decreasing amplitude. Such potentials support a single bound state with momentum related to the period of the oscillations. For energies slightly different, the usual traveling wave solutions exist. However, for an infinite range potential with oscillation of constant amplitude, it is well known that a series of gaps are obtained; bound states may exist for energies within each gap. Of course, in a realistic physical situation, the potential must be cut off so that the states are not truly bound but are resonances with increasing widths as the range of the potential is decreased. The simple method presented here gives a clear picture of the miniband structure with possible bound states above the potential barrier height in layered semiconductors. Although the Bragg conditions lead to a simple physical picture of the coherent reflections needed to produce such bound states, these conditions are not necessary. Acknowledgement - The author is grateful for the hospitality of the Department of Physics and Astronomy at the University of Glasgow where some of this work was done. Special thanks to F. Hiller for preliminary work on the location of the bound states within the unstable gaps and to Carlo Sirtori for some helpful conlments.

References 1.

. 3. 4. 5.

Frederico Capasso, Carlo Sirtori, Jerome Faist, Deborah L. Sivco, Sung-Nee G. Chu, and Alfred Y. Cho, Nature 358, 565 (1992). J. yon Neumann and E. Wigner, Z. Phys. 30, 465 (1929). F. H. Stillinger, Physica B 85, 270 (1977). D. R. Herrick, Physica B 85, 44 (1977). G. Bastard, Phys. Rev. B 24, 5693 (1981).

6.

7.

8.

E. Merzbacker, Quantum Mechanic's 2nd edition (John Wiley, New York, 1970) pp. 93-105. See, for example, Jon Mathews & R. L. Walker Mathematical Methods of Physics 2nd edition (W. A. Benjamin, New York, 1970) pp. 198-204. D. F. Nelson, R. C. Miller, and D. A. Kleinman, Phys. Rev. B 35, 7770 (1981),