Solid-State Electronics Vol. 31, No. 3/4, pp. 359 362, 1988 0038-1101/88 $3.00+0.00 Printed in Great Britain Pergamon Journals Ltd

Effect of C o n t i n u u m Resonances on Hot Carrier Transport in Q u a n t u m Wells

Wolfgang Porod and Craig S. Lent

Department of Electrical and Computer Engineering University of Notre Dame

Notre Dame, Indiana 46556

A B S T R A C T

In addition to bound states, quantum wells also produce resonant states in the continuum. While bound states have received considerable attention and are now the basis of device applications, the corresponding virtual states have hardly been studied. Here, we specifically investigate the influence of these resonant states on hot electron transport in quantum wells. We find that the matrix elements which determine scattering rates exhibit structure at the resonant energies. This leads to suppression of scattering by polar optical phonons relative to non-polar optical and acoustic phonon scattering. We discuss the effect that these states have on the capture and release of carriers by the quantum well.

K E Y W O R D S

Quantum well; resonance; real-space transfer; hot electron transport.

I N T R O D U C T I O N

The study of electronic transport in quantum wells is a topic of great current interest. Transport of carriers in such quantized states holds the promise for applications in high speed devices. While much attention has focussed on the behavior of confined carriers, the influence of extended states with energies above the well and the transfer from confined states to these extended states has been somewhat neglected.

If carrier energies are never sufficient to escape the well, only localized, quasi-two-dimensional (2D) states are involved in transport. The quantum well can therefore be treated as having infinite height. Prominent models are those by Hess(1979), Ridley(1982) and Price(1981) for a rectangular well, and the model of Polonovski and Tomizawa(1985) for a quantum well appropriate in shape for a HEMT. These studies have shown that the essential feature of 2D transport, in contrast to 3D, is the loss of momentum conservation in the direction normal to the interface due to the momentum uncertainty resulting from spatial quantization. These studies have established the fact that the scattering rates in 2D and 3D are very similar - a perhaps surprising conclusion considering the drastic difference in the final states available for scattering. Only a few models have addressed the escape of carriers from the quantum well into delocalized continuum states, though their presence and potential importance has been recognized (Jaros and Wong,1984: Brum and Bastard, 1986). Examples are the model calcluation by Chuang and Hess(1986) for impact ionization in a quantum well and the Monte Carlo analysis of transport in a HEMT by Ravaioli and Ferry(1986).

Here, we specifically focus on the problem of the capture and the release of carriers by a quantum well. This naturally involves an investigation of the effect of the 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 resonances on transport. This leads us to a discussion of the matrix elements which are responsible for the transitions from bound 2D states to unbound 3D states.

Transport is ultimately determined by the rates for scattering from various mechanisms, most importantly for our purposes, acoustic and optical phonons. Using Fermi's Golden Rule, one calculates a scattering rate from two ingredients: (1) the matrix element for the transition between a particular initial and final state and (2) a sum over those final states which are compatible with energy conservation. Since the summation over the available final states is not influenced by the presence of continuum resonances, we concentrate here on the calculation of the matrix element which is indeed influenced by such virtual states. Scattering rates for acoustic and non-polar optical phonons can be inferred almost immediately from the form of the matrix element. Only multiplication by a density of states factor has to be additionally considered. Calculation of rates for scattering by polar optical phonons proves much less trivial, however, due to the complications involved in the integration over final states (Riddock and Ridley, 1983). Explicit calculations of these rates, as well as results from a Monte Carlo simulation of transport, will be presented in the future.

359

360

M O D E L

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, V0. It is helpful to consider the well to be sandwiched between infinitely high barriers a distance 2L apart (see Fig. 1). This outer, infinite, well is only an artifice and L will be taken to be infinite presently. As usual, we separate the motion parallel to the interface from the motion in the direction normal to the interface. Parallel to the interface, the Schr6dinger equation yields plane wave eigenstates labeled by momentum quantum numbers k~ and ky. Normal to the interface, the eigenstates are of two types - bound states which can be labeled by a discrete quantum number n, and delocalized states with a continuous quantum number, k:. We call "bound" those states with E~ < 0. Those with E~ = (h2k2)/2ra * > 0 are viewed as free or delocalized. The solutions to the Schr6dinger equation for this problem are well known from basic quantum mechanics . Even and odd parity eigenfunctions for free states are given below.

{ A ~ ~'~ cos(Kzz) b~.~'*~'(z) = A~ *'~ cos(/%z)

¢oddt~ = { A;~ d sin(/G~) ~ ' ~ A~y sin(~z~)

for 0 < I~l < a for a < tzl < L (1)

with E~ _ ~2,2,~ - ~82I(z-- V 0 , and lira LA 2 --+ 1. (2) 2rr~* 2m* L~oo

In the following we express energies in dimensionless form by dividing by the natural energy of the problem, F..o = (h2u2)/(2m'(2a)2) . This is, in fact, the energy of the lowest bound state.

Figure 2 depicts schematically the allowed electronic energy states for this potential as a function of kx and k~. For a bound state, the locus of possible energies forms a thin shell in the shape of a paraboloid of revolution. The vertex is at the value of the 1-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 together. Each shell corresponds to a discrete state of the large well of length 2L with the energy separation

between the vertices given by A E = ~f f (h~E) /2rn *. In the limit as L ~ 0% the thin shells merge to form the solid paraboloid of revolution as shown.

0

~Eo -v, I

Energy

f ~ 2 a ~

_ _ I I I

-L 0 +L k

Energy

k Y

Fig. 1. The model potential. The zero of energy is chosen to be the top of the well. The infinite barriers al i L are an artifice to aid in the counting of states. In the calculations the limit L -~ oc is taken.

Fig. 2. Allowed energy states. The locus of allowed bound state energies is shown schemati- cally as a function of the two continuous quan- tum numbers k~ and ky. The solid paraboloid of revolution represents the continuum of unbound states. Transitions out of the well due to scatter- ing by optical phonons (open circle) and acoustic phonons (solid circle) are indicated.

(4)

M A T R I X E L E M E N T S

For transport, the quantity of interest is the scattering rate of an electron which initially occupies a state given by (k~, ku, n) into a state with (k~, Uy, k'z) . Scattering rates for emission or absorption of a phonon with energy hw and momentum hq, within Fermi's Golden Rule, are given by

2~ f f f V W(kx, k u, n) = T ~j~31 < k ' , k;, k'~lH~-phlk~, k~, n > 125(Ek' k',k: -- Ek~,k,,= + h~)dNk,,kb,k,. (3)

Here He_ph is the electron-phonon interaction Hamiltonian and dNk,,kl,,k, is the density of final states. The square of the matrix element for scattering from state (k,, k~, n) to state ( ~, ky, k'~) has the following structure.

' ' k' n 12 5(k'~ - &~ q ~ ) 5 ( k ' u - 1% - % ) Ia=,k,(qz)l 2 I < k~:,ky, z[ e-phlk~:,kv, n > oc

(}.4

The factor of proportionality is specific for a particular scattering mechanism.The delta-functions express conservation of crystal momentum for motion parallel to the interface. The momentum uncertainty perpendicular to the interface due to the localization inside the well is represented by the function IG=,k;(q,)l 2 which is given by (Price, 1981; Ridley, 1982)

t + L . 2

i J-L ¢"(z)e'q~z~bk'(z)dz (5) IG,~,k,(q~)I = "~

where the phonon wavefunction is taken to be proportional to e iq~z. The summation over final states contributes a factor L and which leaves L. ]G,,ki(qz)l 2 finite in the limit o f L ~ ec. The summation over final states also involves an integration over the phonon momentum, which leads to the following function

361

L / a IGI 2 L IMI 2 0 . 5

qza

0.3

0 .2

0.1

0

FO J 6 E 0

0 1 2 3 4 5 6

I I I I I I I I t

0 2 4 6 8 1 0 1 2 1 4 16 18 2 0

E ' / E o

Fig. 3. The quanti ty ~lG~,k,(qz)l 2 is plotted for transit ions from the symmetric n = 1 bound state to continuum states which are resonances and non-resonances. The thick curves are for tran- sitions to symmetric final states with 0 energy (non-resonance) and energy Eo (resonance). The thin curves are for transit ions to anti-symmetric final states with energies 3E0 (non-resonance) and 6E0 (resonance).

1.0

0 .8

0 . 6

0.4

0 .2

Fig. 4. The effective matr ix element L - IM(n, E'z)l 2 is plot ted as a function of final s tate energy E'~. The results for synnnetr ic (thick line) and anti-symmetric ( thin line) finals states are shown. The influence of continuum resonances is clear. This function is equal to unity identically if the resonances are neglected.

(6)

362

This function represents an effective matrix element for scattering to a particular final energy. The integration is over allowed transverse phonon momenta. Note that in the three dimensional case this function would be unity identically because all three components of momentum would be conserved.

We have evaluated these effective matrix elements for the model potential with Vo/Eo = 3. Figure 3 shows the form of IGn,k,(qz)l ~ for transitions originating from the n = 1 bound state for several energies. The symmetrical and anti- symmetrical solutions are shown separately. For simplicity, we have neglected the exponential tails. There are two important qualitative features of this figure. Firstly, the magnitude of the scattering increases as the final state energy moves from "off resonance" to "on resonance". Secondly, notice that in all cases the matrix element favors phonon scattering with large tranverse momentum, qz. In Fig. 4, we show the effective matrix element, IM(n, E'z)l ~, as a function of energy E'~. The structure due to the presence of continuum resonances is evident. This function is directly proportional, to the scattering rate for acoustic and non-polar optical scattering. For polar optical rates, the integration over phonon momenta and final states is more complicated and has to be performed numerically.

D I S C U S S I O N a n d C O N C L U S I O N

The suppression by the effective matrix element of small q~ scattering can be understood from a simple qualitative argument. Resonant continuum states correspond to states which are orthogonal to bound states inside the well. This condition of orthogonality implies that for q~ = 0, the matrix element connecting bound states with resonant states vanishes. Thus the total scattering rate is dominated by the contribution of those phonons with larger momentum components normal to the interface. This is clearly evident in Fig. 3. Polar optical scattering is predominatly small q, forward scattering. We conclude, therefore that the polar optical scattering rate will be suppressed relative to processes not so strongly weighted toward forward scattering. This implies that it is non-polar optical and acoustic phonons that are primarily responsible for carriers scattering out of the well and also their capture.

A point that also deserves discussion is the question of the normalization of the wavefunctions. No conceptual problem arises for those scattering events where the spatial extent of initial and final wavefunctions is comparable, as in transitions between localized states or between extended states. For scattering from a localized (2D) to a delocalized (3D) state, however, care has to be taken. Since the amplitude of a delocalized state decreases as the length L of the box normalization increases, the matrix element for scattering into that particular state decreases. The total scattering rate, however, remains constant because there are more available final states. More specifically, the product of the density of states, which is proportional to L, and the matrix elements, which are inversely proportional to L, has a finite, nonzero limit. This can be graphically inferred from our Fig. 2. The limiting process of L ~ oe can be understood as a denser and denser filling of the paraboloid. In the continuum limit, solid filling with allowed final states is reached, each of which has an in infinitesimally small associated scattering probability.

In conclusion, we have examined the effect of continuum resonances above a quantum well on the matrix elements relevant for carrier escape and capture. The presence of resonances can significantly influence scattering out from, and into the well. We expect the rates for polar optical scattering to be suppressed over those for acoustic and non-polar optical processes so that the latter play a dominant role in escape and capture. We intend to extend our current work to include complete rate calculations and a Monte Carlo study of their effects. Device designs exploiting these resonances have already been proposed (Lent, 1987).

A C K N O W L E D G E M E N T

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

R E F E R E N C E S Brum, J. A. and G. Bastard, Phys. Rev. B 32, 1420 (1986). Chuang, S. L. and K. Hess, J. Appl. Phys. 50, 2885 (1986). Hess, K., Appl. Phys. Lett. 35,484 (1979). Jaros, M. and K. B. Wong, J. Phys. C 17, L765 (1984). Lent, C. S., Superlattices and Microstructures, in press. Polonovski, J.-P. and K. Tomizawa, Jap. J. Appl. Phys. 24, 1611 (1985). Price, P. J., Annals of Physics 133, 217 (1981). Ravaioli, U. and D. K. Ferry, IEEE Trans. Electron Dev. ED-33, 677 (1896). Riddoch, F. A. and B. K. Ridley, J. Phys. C 16, 6971 (1983). Ridley, B. K., J. Phys. C 15, 5899 (1982).