Superlattices and Microstructures, Vol. 4, No. 1, 7988 77

EFFECT OF CONTINUUM RESONANCES ON ELECTRONIC TRANSPORT IN

QUANTUM WELLS

Craig S. Lent and Wolfgang Porod

Department of Electrical and Computer Engineering University of Notre Dame

Notre Dame, IN 46556

(Received 17 August 1987)

Abstract

We investigate the influence of continuum resonances on the release of electrons by a quantum welI. We find that the inclusion of resonance effects leads to a decrease in the rate of scattering by non-polar phonons from bound confined states to unbound continuum states.

It is well known that quantum wells, in addition to confined bound states, produce resonant continuum states. These resonant states are a consequence of the reflections, and the consecutive interference, of the electronic wave- functions at the edges of the quantum well. While the ex- istence of these so-called virtual resonant states has been recognized*-3, their influence on electronic transport has so far received little attention4.

The study of electronic transport in quantum wells is a topic of great current interest. If carrier energies are never sufficient to escape the well, only local&d, quasi- two-dimensional (2-D) states are involved in transport. The quantum well can then effectively be treated as hav- ing infinite height. Several model calculations of phonon scattering rates have been performed in this case for a rectangular quantum wells-* and also for a quantum well appropriate in shape for a I-EMT ‘. These studies have shown that the essential feature of 2-D transport, in con- trast to 3-D, is the loss of momentum conservation in the direction normal to the interface due to the momentum uncertainty resulting from spatial quantization. One of the conclusions of these studies is that the magnitudes of the scattering rates in 2-D and 3-D are very similar to each other - a perhaps surprising result considering the drastic difference in the final states available for scattering.

Only a few studies have addressed the escape of car- riers from the quantum well into delocalized continuum ttdtts. One of these models, for example, considered the process of impact ionization in a quantum welllo and an-

0749-6036/88/010077 +04 $02.00/O

other one treated the transfer of hot carriers out of the conducting channel in a HEMT”.

Here, we specifically focus on the problem of the es- cape of carriers from a quantum well. This naturally involves an investigation of the effect of the continuum resonant states on hot electron transport. Our main goal is to isolate and understand those factors which contain the physics essential to the influence of continuum reso- nances on transport. We consider first the matrix elements which are responsible for the transitions from bound 2-D states to unbound 3-D states. We then calculate the corre- sponding scattering rates for non-polar phonon scattering events.

We adopt a single-band spherical effective mass model for the electronic states. As a model potential, we choose a potential well of width 2a with depth, V,. In the following we express energies in dimensionless form by dividing by the natural energy of the problem, E. = (tLZx2)/(2m’(2a)2). This is, in fact, the energy of the lowest bound state, for the case of an infinitely high barrier. As usual, we separate the motion parallel to the interface, which we call the x-y plane, from the mo- tion in the direction perpendicular to the interface, which we denote by the z-direction. Parallel to the interface, the Schr6dinger equation yields plane wave eigenstates which are labeled by momentum quantum numbers k. and Icy. Normal to the interface, the eigenstates are of two types - bound states which can be distinguisnerl by a discrete quantum number I:, and delocalixcd states with a

0 1988 Academic Press Limited

78 Superlattices and Microstructures, Vol. 4, No. 1, 1988

continuous quantum number, I;,. We call “bound’ those states with E, < 0. Those with E, = (?5*k~)/Znl* > 0 are viewed as free or delocalized. The solutions to the Schrodinger equation for this problem are well known from basic quantum mechanics textsr2. In particular, even and odd parity eigenfunctions exist.

Figure 1 schematically depicts the allowed electronic energy states for this model potential as a function of t, and k,. For a bound state, the locus of possible energies forms a thin shell in the shape of a paraboloid of revo- lution. The vertex is at the value of the l-dimensional bound state energy eigenvalue. In the figure only one bound state, the lowest, is shown for the sake of clarity. The locus of allowed energy values for the unbound states forms a family of thin parabolic shells, spaced closely to- gether. Each shell corresponds to a particular value of the kinetic energy, EL = (h’kL’)/2m*, for free motion perpendicular to the quantum well.

For the study of electronic transport, the quantity of interest is the scattering rate of an electron which initially occupies a state labeled by (k,, k,, n) into a state with (k:, kj, kl) . Scattering rates for emission or absorption of a phonon with energy fiw and momentum tlq, within Fermi’s Golden Rule, are given by

W(kz, k,, n) =

xJ(Ek:,k:,,k: - Ek,,k,.n f fim)dNk;,k;.k:.

Here He_nh is the electron-phonon interaction Hamil- tonian and dN,+L,kb,kL is the density of final StateS. The

square of the matrix element for scattering from state (kl, k,, n) to state (k:, kh, k:) has the following form

Here, 6, and cY,~ represent momentum conservation in the x- and y- direction, respectively. The factor of propor- tionality is specific for a particular scattering mechanism. For non-polar scattering events it is just a number, inde- pendent of the momentum of the phonon involved. The momentum uncertainty perpendicular to the interface due to the localization inside the quantum well is represented by the function IGn,k;(qz)12 which is given by’ja

IGn.k:(qz)i2 = ; ~~~;:(z)e”.“~,(~)dr/z,

where the phonon wavefunction is taken to be pro- portional to eiq=‘.

The summation over all final states for the determina- tion of the scattering rate involves an integration over the phonon momentum q and over the final electron momen- tum, rc’l. It is useful to separate these two integrations and to calculate an effective matrix element by performing only the integration over the components of the phonon momentum. The scattering rate is then the integral of this effective matrix element over all final electronic states.

Energy

A

kx 1’ /’ k ,

,/ ’ Y

*,’

FIG. 1 Allowed energy states. The locus of allowed bound state energies is shown schematically as a func- tion of the momentum components parallel to the inter- face. The solid paraboloid of revolution represents the continuum of unbound states. Transitions out of the well due to scattering by optical phonons (open circle) and acoustic phonons (solid circle) are indicated.

The integrations over qz and qy are trivial because of the 6 - functions in the matrix element. For non-polar scat- tering, all the qz dependence is in G,+ so the effective matrix element is simply,

lM(n>E:)/’ = /+- IGn.k:(qz)/2dq,. ---a

We plot this function in Figure 2 for the lowest sub- band, n = 1, in the well. The figure is plotted for the case V. = 4Eo. The initial and final wavefunctions are calculated by solving the Schrodinger equation for the well. Notice the structure at the resonant energies. The most important contribution to the oscillatory stmcture observed is the change in the amplitude of the final state wavefunction above the well. This amplitude oscillates with final state energy because of quantum reflections at the well edges. This effect is exactly the same as that which produces resonance oscillations in the transmis- sion coefficient for plane waves passing over the well.

Superlattices and Microstructures, Vol. 4, No. 1, 1988 79

0.8

M2 0.6

0.4

0.2

0 I I I I ; I

0 2 4 6 8 10 12 14 16 18 20

FIG. 2 The effective matrix element jM(n,E~)j2 is shown as a function of the kinetic energy El for sym- metric (thick curve) and anti-symmetric (thin curve) final states. Continuum resonances reveal themselves as the structure in the curves, which would be equal to unity identically if the resonances were neglected.

Thus, the similarity between the graph in Figure 2 and the transmission coefficient oscillations plotted in stan- dard quantum mechanics texts l2 is easily understood.

Note that in the three dimensional case (again for non- polar scattering) jM(n, E:)12 would be unity identically because all three components of momentum would be conserved. For the bound to unbound transitions hem, on the other hand, the function “turns on” gradually and then oscillates toward a limiting bulk value. It is important to notice the horizontal scale in Figure 2. The oscillations have a period of several times Ea. For small quantum wells and typical potentials this can be several electron Volts. The result is that at the relevant energies close to the top of the quantum well, it is not the oscillatory character which is important, but the gradual turning on of the sloping leading edge. This results in a reduction of scattering rates as we shall see.

For the calculation of the scattering rate by non-polar phonons, the additional integration over the z-components of the final state momentum has to be performed. This is necessary because a particular final total energy can conespond to different values of the kinetic energies, EL, for motion perpendicular to the quantum well. The non- polar scattering rate out of the well is then,

W,,(E) 0: lE JM(n,E:)12(EI)-“3dE:

The factor of (EL)-*/’ corresponds to the one- dimensional density of states for energy associated with

0.1 0.2 0.3 0.4

Energy (eV)

FIG. 3 Non-polar scattering rates. Shown are results for two quantum wells with thicknesses of 20A and 40A, respectively. The solid curves include the effect of con- tinuum resonances, whereas the light curves do not. Note that the inclusion of resonance effects leads to a reduction of the rate up to a factor of 2.

motion in the z-direction. We have evaluated this scatter- ing rate for 20 x and 40 x wide model quantum wells which are 0.2 eV in depth. Phonon dispersion relations are assumed to be constant in the optical case and linear in the acoustic case. The results are shown in Figure 3. Note that the energy range is such that only the mono- tonically increasing part of the effective matrix element is sampled. We have compared these results, which in- corporate the resonant structure in the wavefunctions, to results of calculations where the final state wavefunctions were approximated by simple plane waves, thereby ignor- ing the effect of the well on the final state wavefunctions. These results are shown by the thin curves in Figure 3. As can be seen, neglecting the effect of the resonances leads to overestimation of the scattering rate by as much as a factor of two.

The rate for scattering out of the well by non-polar pho:lon scattering is suppressed due to the presence of continuum resonances above the well. Their effect is to make it more difticult for the electron to scatter from the well. The well is, in this sense, effectively deeper. The rates are typically smaller by a factor of two. It is worth noting that the oscillatory structure seen in the one- dimensional transmission coefficient, while present in the effective matrix element, is lost in the scattering rate. The reason is that, because the final state is three-dimensional, one must integrate over all the. ways in which the final state energy can be distributed between the motion par- allel and perpendicular to the interface.

In conclusion, we have examined the effect of con- tinuum resonances above a quantum well on the matrix elements and non-polar phonon scattering rates relevant

80 Superlattices and Microstructures, Vol 4, No. 7, 7988

for carrier transfer out of the well. The presence of res- onances can significantly suppress scattering out of the well. Extension of these results to polar scattering and a Monte Carlo study of their effects is underway. Device designs exploiting these resonances have already been proposed*3.

Acknowledgements - The authors would like to thank Drs. U. Ravailoi and S.M. Goodnick for discussions. The work af one us (W.P.) was supported in part by a grant from SDIO/IST (managed by ONR).

References

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[2] J. A. Brum and G. Bastard, Phys. Rev. B 32, 1420 (1986).

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