- Email: [email protected]
Intersubband relaxation of hot carriers in coupled quantum wells
Solid-State Electronics Vol. 32, No. 12, pp. 1283-1287, 1989
0038-1101/89 $3.00+0.00 Copyright © 1989 Pergamon Press plc
Printed in Great Britain. All rights reserved
Intersubband Relaxation of Hot Carriers in Coupled Quantum Wells
denifer Lary and Stephen H. Goodnick Department of Electrical and Computer Engineering Oregon State University Corvallis, OR 97331
PaoTo LugTi Dipartimento di Ingegneria Meccanica I I Universita' di Roma Via O. Raimondo 00173 Roma, Italy
Danie7 Y. Oberli and Jagdeep Shah AT&T Bell Laboratories Holmdel, N.J. 07733
ABSTRACT
We compare experimental results for the photoluminscence decay in coupled quantum wells with a Monte Carlo simulation of electron and hole dynamics in this structure. The results support the assumption that the measured decay is associated with intersubband relaxation of electrons with l i t t l e effect due to the holes. Quantitative agreement is found between the measured and theoretical time constants for most fields, although the theoretical values are shorter than the corresponding experimental values close to resonance. KEYWORDS
Intersubband scattering; Tunneling; Coupled quantum wells INTRODUCTION
The relaxation of photoexcited hot carriers in coupled quantum wells has been studied experimentally using time resolved photoluminescence by several groups (Oberli, 198ga, 1989b; Norris, 1989). In such experiments, the primary interest is in the relaxation of carriers from one well to the other, so that the decay of the photoluminescence measures an effective tunneling time through the barrier separating the two wells. An alternate way of looking at this real space transfer between wells is in terms of intersubband scattering since the bound states in the coupled well system are orthogonal eigenstates of the Schroedinger equation. The difference between this situation and the case of intersubband scattering in single quantum wells is that subbands are localized primarily in one well or the other, so that the overlap integral of the wavefunctions between wells (off resonance) is exponentially attenuated as discussed later. In the experiments of Oberli and co-workers (1989a, 198gb) with which we compare here, the coupled wells are imbedded in the i n t r i n s i c region of a p-i-n diode so that the electric field across the wells may be continuously tuned in order to s h i f t the relative energies of the subband levels as shown in Fig. I. The AIGaAs barrier separating the two wells is usually less than IOOA so that some overlap of the wavefunctions is expected. The excitation density in these experiments is low in order to minimize the effects of screening of the electric field in these structures. The interpretation of the luminescence spectrum in such systems may be complicated by the fact that both electrons and holes are present, so that the observed relaxation depends on both types of carriers. In order to elucidate the dynamics of intersubband relaxation measured experimentally, we have used an ensemble Monte Carlo simulation of electrons and holes to model the dynamics of intersubband relaxation in GaAs-AIGaAs coupled wells. For low carrier excitation, the dominant intersubband scattering mechanism is that due to polar optical phonons. Carrler-carrier scattering is included in the model, but mainly contributes to the thermalization of the carriers. Only heavy holes are considered here with coupling due to TO and LO modes. Other effects such as hot phonons, degeneracy and the self-consistent solution of Schroedinger's and Poisson's equations, included in our previous works (Goodnick, 1989, IgB8a, 1988b; Lugli, 1987), are neglected here due to the low injected carrier densities 1283
1284
JENIFERLARY
et al.
involved. The subband envelope functions themselves are calculated numerically for the well potential profile shown in Fig. I. 0.5
60 kV/cm 0.4
Ec (eV)
o.a
o.2
E w2
Enl o.1
i Ewl
0
,
p
-20o
o
~
200
40o
DISTANCE (angstroms) Fig. 1. The conduction band p r o f i l e and energy subband f o r the narrow-wide configuration under an e l e c t r i c f i e l d of 60 kV/cm, biased close to the energy resonance. The envelope wavefunctions are also shown• In the present model, we assume a polar optical coupling due to bulk l i k e GaAs phonons and thus neglect the possible effects of confined modes in the A1GaAs b a r r i e r and GaAs wells (Riddoch, 1985). The polar optical phonon emission rate neglecting screening is given by (Riddoch, 1983; Goodnick, 1988a) 2~ rij(k)=eEo/2/~ (n~o+l) f de H i j ( q ) / q 0
(1)
where q is the scattered wavevector, nun is the phonon occupancy, and eEo is the effective f i e l d . By energy conservation, the scattered wavevector is w r i t t e n q=
Ik-k'l
=(2k2-2m*~*/7/ - k(k2-2m*u*/~h)I/2cose ) i / 2
(2)
where'/}u*=?F~^+(E~-E~) is an effective phonon energy (which can be negative) that reflects the • ,u ,i change in klnetlc eHergy of a particle during intersubband scattering. For simplicity, the carrier effective mass, m*, is taken as constant here. The I/q dependence of (I) implies a resonance condition i f the effective phonon energy is zero, which occurs when the separation between subbands i and j equals the longitudinal phonon energy. Evidence for this effect is clear from the experimental results in coupled quantum wells where resonancei s achieved through bias as discussed later. The form factor, Hij(q) results from the overlap integral of the subband envelope functions and is given by Hij(q)=fdzfdz'~ i(z)~j(z)~ i ( z ' ) ~ j ( z ' ) e - q l z ' z ' I
(3)
In asymmetric coupled quantum wells, the form factor, and thus the intersubband scattering rate is generally quite small due to the high degree of localization of the wavefunctions in their respective wells. I f however the wells are biased into resonance such that two levels are coincident, then the wavefunction is delocalized, and scattering down to the bottom of the lower energy subband occurs quite readily. Thus the electric f i e l d due to the applied bias may dramatically alter the POP scattering rate through the dependencies of'/fu* and Hij(q ) on the position of the relative subband energies. NARROW-WIDE STRUCTURES
Two recent time resolved photoluminescence studies by Oberli and co-workers (1989a, Ig8gb) have demonstrated subband energy and phonon resonance effects in double well structures located in p-i-n diodes. The f i r s t type of device structure shown in Fig. I is designed such that the lowest energy subband associated with a narrow 60A well, Enl, can be biased so as to align to the second level of a 88A wide well, Ew2, through a thin 55A barrier. For the material parameters assumed here, our numerical-calculation of the wavefunctions breaks down above 60 kV/cm as the higher lying states become unbound. However, extrapolating to a s l i g h t l y higher f i e l d indicates an energy resonant condition at approximately 65 kV/cm.
Intersubband relaxation of hot carriers
1285
The form factor for polar optical emission, Hi~(q), calculated using Eq. 3 for the coupled well configuration of Fig. I is plotted in Fig. Z a) a function of q for two values of electric f i e l d below resonance, and at 60 kV/cm, close to resonance. As expected, Hji(q) increases substantially as resonance is approached due to the increased overlap of th~ envelope wavefunctions. This increase in H i i is responsible for an order of magnitude increase in the scattering rate from Enl to Ewl as )hown in the inset. O. ~ 6
0.~40.022
~.~ingR~e
0.02
0.018 0.016 ~ c
- ~ / ~ 450O
O.Ol, 0.012
0.1
0.2
e~rgy (e~
O.Ol 0.008 0.006 0.004
50
0.002 0
i
i
o
2 q ( 1 0 7 cm'-l)
Fig. 2. The form factor for intersubband scattering "(Eq. 3), Enl to Ewl, for the structure shown in Fig. I at three fields. The inset shows the corresponding scattering rates. To simulate photoexcitation of carriers during the Monte Carlo simulation, electron-hole pairs are added to the simulation according to the generation rate of the incident pulse and weighted by the effective dipole moment of the optical transition between valence and conduction band states. The effective lifetime of carriers in the subband of interest is determined by an exponential fit to the density versus time data after the end of the initial pulse transient. To quantitatively compare to measured time resolved photoluminescence data, we assume that the measured lifetime is the inverse sum of the recombination lifetime and intersubband lifetime. A constant recombination lifetime of l.ISns is assumed here. 100 • MonteCarlo
90
80
• •
(Obo~¢. ¢.)
+ +
7O
ji
+ Photo Luminescence
•m
+
+
.o
÷
4O 3O
+ +
20
+ •
10 0
+ +
0
2'0
'
R"
÷
+ +
+
100
Field (kV/cm) Fig. 3. The electric f i e l d dependence of the simulated time constant for the narrow-wide structure of Fig. I, with the narrow well photoluminescence decay times (Oberli, 198gb). The simulated Monte Carlo results for the time constant of electrons in Enl is shown in Fig. 3 The calculations are compared with the photoluminesence data of Oberli and-co-workers (1989a), who argued that intersubband relaxation is the rate determining mechanism in their data, even at the energy resonance shown in Fig. I. Although the Monte Carlo results indicate that the energy resonance occurs at an electric f i e l d somewhat lower than the measured minimum time constant, the agreement is reasonable. Part of this discrepancy is attributable to space
1286
JENIFER LARY et al.
charge effects due to the excited carriers. As was noted by Oberli (1989a), the bias voltage at which resonance ( i . e . the minimum of the time constant versus f i e l d curve) occurs continues to decrease as the excitation density is reduced. Thus there is a small uncertainty in the electric f i e l d on the order of 10-15%. At intermediate f i e l d values (20-40 kV/cm) the decay time is dominated by the small form factor entering the scattering rate as was shown in Fig 2. The time constant in this range of fields is very sensitive to variations in the barrier width, as well as model parameters such as the electron effective mass since the scattering rate here is primarily determined by the t a i l s of the exponentially decaying wave Functions in the barrier. As the electric f i e l d is increased and Enl rises towards the excited state Ew2, both the experimental and simulated time constants decrease at approximately the same rate. The measured photoluminescencedepends on the product of the optically coupled electron and hole populations, and thus may in fact be affected by both electrons tunneling out of, and holes into, the narrow well. In our simulated results, the population of holes remained essentially constant over the duration of the simulation. This behavior is as expected since the heavier mass of the holes greatly attenuates the overlap of their respective envelope functions between the wells, at least in terms of a simple effective mass model. However, resonances in the hole subbands occur'with bias as well as the electron subbands. As a function of f i e l d , a resonance in the holes is found at 44kV/cm which resulted in a 55ps time constant for holes to transfer from one well to another. WIDE-NARROWSTRUCTURES The conduction band p r o f i l e of the second structure investigated is shown in Fig. 4. The wide and narrow wells in Fig. 4 are separated by a barrier nominally 55A in width. Under bias, the ground state Ewl of the wider well (110A) is above that of the lowest subband E.I of a narrow well(7OA) so tHAt carriers w i l l tend to thermalize to the narrow well after long times. Although there is r e l a t i v e l y l i t t l e overlap of the wavefunctions between Ew] of En], a minimum in the photo-luminescencedecay time is observed and attributed to a phonoB-reson~fice as discussed e a r l i e r in connection with eqs. I and 2. The calculated energy levels versus electric f i e l d For this structure indicate that this phonon resonance should occur at 48 kV/cm. For separations less than the phonon energy, the transfer of carriers via intersubband phonon emission is expected to be greatly reduced. ~¢
,
0.5 48 kV/om
0.4 Ec: (eV)
&
0.3
0.2
Ew2
0.1 1Enl " ~ 0
-150
b
~-'-.4 ~. F
-50
i
50 150 250 DISTANCE (angstroms)
'
350
Fig. 4. The conduction band p r o f i l e and energy subbands for the wide-narrow configuration, biased such that Ewl is a phonon energy above Enl. The envelope wave functions of the subbands are superimposed. Figure 5 plots the measured photoluminescence f or the wide-narrow structure together with the numerical results of the Monte Carlo simulation. Above about 20 kV/cm, the Fir s t level of the wide we l l , Ew|, rises above E . The numerical results are again weighted with the effect of the recombination l i f e t i m e . ~I can be observed From Fig. 5, the agreement between the experimental decay times and the Monte Carlo values is s u r p r i s i n g l y good. The theoretical curves are extremely sensitive to the assumed thickness of the intervening b a r r i e r , with one monolayer difference accounting for as much as 50% error. The exact position of the phonon resonance does not concur between experiment and theory. In Fact, the measured value appears to correspond more c l o s e l y with a phonon resonance associated with AlAs phonons rather than GaAs modes, although more work is needed in order to ascertain t h i s effect.
Intersubband relaxation of hot carriers
1287
1000 • • MonteCarlo + PhotoLtlldnucence (Obedlet. al.)
8OO
600 + +
4OO
+
+ •
*m ÷
20O
0
o
+
2o
6o Field~V/cm)
8o
ioo
Fig. 5. The simulated time constant for the structure shown in Fig. 4 verses electric f i e l d , with the corresponding wide well photoluminescence decay times of (Oberli, 1989a). ACKNOWLEDGEMENTS
One of us (JL) acknowledges the support of IBM, Essex Junction, VT. This work was partially supported by Office of Naval Research (NOOOI4-8g-J-1894) and the National Center for Supercomputing Applications for supercomputer time. REFERENCES
Goodnick, S. M. (1988a). Effect of electron-electron scattering on nonequilibrium transport in quantum-well systems. Phys. Rev. B, 37, 2578-2588. Goodnick, S. M. (1988b). Influence of electron-hole scattering on subpicosecond carrier relaxation in AlxGal_Y As/GaAs quantum wells. Phys. Rev. B, 38, 10135-10138. Goodnick, S. M. ann PT Eugli (1989). Intersubband relaxation of electrons in AluGa, uAs/GaAs quantum wells durlng photoexcltatlon. Picosecond Electronics and ODtoelectronlcs Technical Diaest, Salt Lake City, Utah. Lugli, P. and S. M. Goodnick (1987). Nonequilibrium longitudinal-optical phonon effects in GaAs-AIGaAs quantum wells. [email protected], Lett., 59, 716-719. Norris, T. B., N. Vodjdani, B. Vinter, C. Weisbuch and G. A. Mourou (1989). Charge-transfer state photoluminescence in asymmetric coupled quantum wells. To be published. Oberli, D. Y., J. Shah, T. C. Damen, R. F. Kopf, J. M. Kuo, and J. E. Henry (1989a). Optical phonon-assisted tunneling in double quantum well structures. Picosecond Electronics and Ootoelectronics Technical Diqest, Salt Lake City, Utah. Oberli, D. Y., J. Shah, T. C. Damen, C. W. Tu, T. Y. Chang, D. A. B. Miller, J. E. Henry, R. F. Kopf, N. Sauer and A. E. DiGiovani (1989b). Direct measurement of resonant and non-resonant tunneling times in asymmetric coupled quantum wells. Phys. Rev. B, Aug. 15, 1989, to be published. Riddoch, F. A. and B. K. Ridley (1983). On the scattering of electrons by polar optical phonons in quasi-2D quantum w e l l s . J. Phys. C, 16, 6971-6982. Riddoch, F. A. and B. K. Ridley (1985). Electron s c a t t e r i n g r a t e s associated w i t h the p o l a r o p t i c a l phonon i n t e r a c t i o n in a t h i n i o n i c slab. Phvsica, 134B, 342-346.
0038-1101/89 $3.00+0.00 Copyright © 1989 Pergamon Press plc
Printed in Great Britain. All rights reserved
Intersubband Relaxation of Hot Carriers in Coupled Quantum Wells
denifer Lary and Stephen H. Goodnick Department of Electrical and Computer Engineering Oregon State University Corvallis, OR 97331
PaoTo LugTi Dipartimento di Ingegneria Meccanica I I Universita' di Roma Via O. Raimondo 00173 Roma, Italy
Danie7 Y. Oberli and Jagdeep Shah AT&T Bell Laboratories Holmdel, N.J. 07733
ABSTRACT
We compare experimental results for the photoluminscence decay in coupled quantum wells with a Monte Carlo simulation of electron and hole dynamics in this structure. The results support the assumption that the measured decay is associated with intersubband relaxation of electrons with l i t t l e effect due to the holes. Quantitative agreement is found between the measured and theoretical time constants for most fields, although the theoretical values are shorter than the corresponding experimental values close to resonance. KEYWORDS
Intersubband scattering; Tunneling; Coupled quantum wells INTRODUCTION
The relaxation of photoexcited hot carriers in coupled quantum wells has been studied experimentally using time resolved photoluminescence by several groups (Oberli, 198ga, 1989b; Norris, 1989). In such experiments, the primary interest is in the relaxation of carriers from one well to the other, so that the decay of the photoluminescence measures an effective tunneling time through the barrier separating the two wells. An alternate way of looking at this real space transfer between wells is in terms of intersubband scattering since the bound states in the coupled well system are orthogonal eigenstates of the Schroedinger equation. The difference between this situation and the case of intersubband scattering in single quantum wells is that subbands are localized primarily in one well or the other, so that the overlap integral of the wavefunctions between wells (off resonance) is exponentially attenuated as discussed later. In the experiments of Oberli and co-workers (1989a, 198gb) with which we compare here, the coupled wells are imbedded in the i n t r i n s i c region of a p-i-n diode so that the electric field across the wells may be continuously tuned in order to s h i f t the relative energies of the subband levels as shown in Fig. I. The AIGaAs barrier separating the two wells is usually less than IOOA so that some overlap of the wavefunctions is expected. The excitation density in these experiments is low in order to minimize the effects of screening of the electric field in these structures. The interpretation of the luminescence spectrum in such systems may be complicated by the fact that both electrons and holes are present, so that the observed relaxation depends on both types of carriers. In order to elucidate the dynamics of intersubband relaxation measured experimentally, we have used an ensemble Monte Carlo simulation of electrons and holes to model the dynamics of intersubband relaxation in GaAs-AIGaAs coupled wells. For low carrier excitation, the dominant intersubband scattering mechanism is that due to polar optical phonons. Carrler-carrier scattering is included in the model, but mainly contributes to the thermalization of the carriers. Only heavy holes are considered here with coupling due to TO and LO modes. Other effects such as hot phonons, degeneracy and the self-consistent solution of Schroedinger's and Poisson's equations, included in our previous works (Goodnick, 1989, IgB8a, 1988b; Lugli, 1987), are neglected here due to the low injected carrier densities 1283
1284
JENIFERLARY
et al.
involved. The subband envelope functions themselves are calculated numerically for the well potential profile shown in Fig. I. 0.5
60 kV/cm 0.4
Ec (eV)
o.a
o.2
E w2
Enl o.1
i Ewl
0
,
p
-20o
o
~
200
40o
DISTANCE (angstroms) Fig. 1. The conduction band p r o f i l e and energy subband f o r the narrow-wide configuration under an e l e c t r i c f i e l d of 60 kV/cm, biased close to the energy resonance. The envelope wavefunctions are also shown• In the present model, we assume a polar optical coupling due to bulk l i k e GaAs phonons and thus neglect the possible effects of confined modes in the A1GaAs b a r r i e r and GaAs wells (Riddoch, 1985). The polar optical phonon emission rate neglecting screening is given by (Riddoch, 1983; Goodnick, 1988a) 2~ rij(k)=eEo/2/~ (n~o+l) f de H i j ( q ) / q 0
(1)
where q is the scattered wavevector, nun is the phonon occupancy, and eEo is the effective f i e l d . By energy conservation, the scattered wavevector is w r i t t e n q=
Ik-k'l
=(2k2-2m*~*/7/ - k(k2-2m*u*/~h)I/2cose ) i / 2
(2)
where'/}u*=?F~^+(E~-E~) is an effective phonon energy (which can be negative) that reflects the • ,u ,i change in klnetlc eHergy of a particle during intersubband scattering. For simplicity, the carrier effective mass, m*, is taken as constant here. The I/q dependence of (I) implies a resonance condition i f the effective phonon energy is zero, which occurs when the separation between subbands i and j equals the longitudinal phonon energy. Evidence for this effect is clear from the experimental results in coupled quantum wells where resonancei s achieved through bias as discussed later. The form factor, Hij(q) results from the overlap integral of the subband envelope functions and is given by Hij(q)=fdzfdz'~ i(z)~j(z)~ i ( z ' ) ~ j ( z ' ) e - q l z ' z ' I
(3)
In asymmetric coupled quantum wells, the form factor, and thus the intersubband scattering rate is generally quite small due to the high degree of localization of the wavefunctions in their respective wells. I f however the wells are biased into resonance such that two levels are coincident, then the wavefunction is delocalized, and scattering down to the bottom of the lower energy subband occurs quite readily. Thus the electric f i e l d due to the applied bias may dramatically alter the POP scattering rate through the dependencies of'/fu* and Hij(q ) on the position of the relative subband energies. NARROW-WIDE STRUCTURES
Two recent time resolved photoluminescence studies by Oberli and co-workers (1989a, Ig8gb) have demonstrated subband energy and phonon resonance effects in double well structures located in p-i-n diodes. The f i r s t type of device structure shown in Fig. I is designed such that the lowest energy subband associated with a narrow 60A well, Enl, can be biased so as to align to the second level of a 88A wide well, Ew2, through a thin 55A barrier. For the material parameters assumed here, our numerical-calculation of the wavefunctions breaks down above 60 kV/cm as the higher lying states become unbound. However, extrapolating to a s l i g h t l y higher f i e l d indicates an energy resonant condition at approximately 65 kV/cm.
Intersubband relaxation of hot carriers
1285
The form factor for polar optical emission, Hi~(q), calculated using Eq. 3 for the coupled well configuration of Fig. I is plotted in Fig. Z a) a function of q for two values of electric f i e l d below resonance, and at 60 kV/cm, close to resonance. As expected, Hji(q) increases substantially as resonance is approached due to the increased overlap of th~ envelope wavefunctions. This increase in H i i is responsible for an order of magnitude increase in the scattering rate from Enl to Ewl as )hown in the inset. O. ~ 6
0.~40.022
~.~ingR~e
0.02
0.018 0.016 ~ c
- ~ / ~ 450O
O.Ol, 0.012
0.1
0.2
e~rgy (e~
O.Ol 0.008 0.006 0.004
50
0.002 0
i
i
o
2 q ( 1 0 7 cm'-l)
Fig. 2. The form factor for intersubband scattering "(Eq. 3), Enl to Ewl, for the structure shown in Fig. I at three fields. The inset shows the corresponding scattering rates. To simulate photoexcitation of carriers during the Monte Carlo simulation, electron-hole pairs are added to the simulation according to the generation rate of the incident pulse and weighted by the effective dipole moment of the optical transition between valence and conduction band states. The effective lifetime of carriers in the subband of interest is determined by an exponential fit to the density versus time data after the end of the initial pulse transient. To quantitatively compare to measured time resolved photoluminescence data, we assume that the measured lifetime is the inverse sum of the recombination lifetime and intersubband lifetime. A constant recombination lifetime of l.ISns is assumed here. 100 • MonteCarlo
90
80
• •
(Obo~¢. ¢.)
+ +
7O
ji
+ Photo Luminescence
•m
+
+
.o
÷
4O 3O
+ +
20
+ •
10 0
+ +
0
2'0
'
R"
÷
+ +
+
100
Field (kV/cm) Fig. 3. The electric f i e l d dependence of the simulated time constant for the narrow-wide structure of Fig. I, with the narrow well photoluminescence decay times (Oberli, 198gb). The simulated Monte Carlo results for the time constant of electrons in Enl is shown in Fig. 3 The calculations are compared with the photoluminesence data of Oberli and-co-workers (1989a), who argued that intersubband relaxation is the rate determining mechanism in their data, even at the energy resonance shown in Fig. I. Although the Monte Carlo results indicate that the energy resonance occurs at an electric f i e l d somewhat lower than the measured minimum time constant, the agreement is reasonable. Part of this discrepancy is attributable to space
1286
JENIFER LARY et al.
charge effects due to the excited carriers. As was noted by Oberli (1989a), the bias voltage at which resonance ( i . e . the minimum of the time constant versus f i e l d curve) occurs continues to decrease as the excitation density is reduced. Thus there is a small uncertainty in the electric f i e l d on the order of 10-15%. At intermediate f i e l d values (20-40 kV/cm) the decay time is dominated by the small form factor entering the scattering rate as was shown in Fig 2. The time constant in this range of fields is very sensitive to variations in the barrier width, as well as model parameters such as the electron effective mass since the scattering rate here is primarily determined by the t a i l s of the exponentially decaying wave Functions in the barrier. As the electric f i e l d is increased and Enl rises towards the excited state Ew2, both the experimental and simulated time constants decrease at approximately the same rate. The measured photoluminescencedepends on the product of the optically coupled electron and hole populations, and thus may in fact be affected by both electrons tunneling out of, and holes into, the narrow well. In our simulated results, the population of holes remained essentially constant over the duration of the simulation. This behavior is as expected since the heavier mass of the holes greatly attenuates the overlap of their respective envelope functions between the wells, at least in terms of a simple effective mass model. However, resonances in the hole subbands occur'with bias as well as the electron subbands. As a function of f i e l d , a resonance in the holes is found at 44kV/cm which resulted in a 55ps time constant for holes to transfer from one well to another. WIDE-NARROWSTRUCTURES The conduction band p r o f i l e of the second structure investigated is shown in Fig. 4. The wide and narrow wells in Fig. 4 are separated by a barrier nominally 55A in width. Under bias, the ground state Ewl of the wider well (110A) is above that of the lowest subband E.I of a narrow well(7OA) so tHAt carriers w i l l tend to thermalize to the narrow well after long times. Although there is r e l a t i v e l y l i t t l e overlap of the wavefunctions between Ew] of En], a minimum in the photo-luminescencedecay time is observed and attributed to a phonoB-reson~fice as discussed e a r l i e r in connection with eqs. I and 2. The calculated energy levels versus electric f i e l d For this structure indicate that this phonon resonance should occur at 48 kV/cm. For separations less than the phonon energy, the transfer of carriers via intersubband phonon emission is expected to be greatly reduced. ~¢
,
0.5 48 kV/om
0.4 Ec: (eV)
&
0.3
0.2
Ew2
0.1 1Enl " ~ 0
-150
b
~-'-.4 ~. F
-50
i
50 150 250 DISTANCE (angstroms)
'
350
Fig. 4. The conduction band p r o f i l e and energy subbands for the wide-narrow configuration, biased such that Ewl is a phonon energy above Enl. The envelope wave functions of the subbands are superimposed. Figure 5 plots the measured photoluminescence f or the wide-narrow structure together with the numerical results of the Monte Carlo simulation. Above about 20 kV/cm, the Fir s t level of the wide we l l , Ew|, rises above E . The numerical results are again weighted with the effect of the recombination l i f e t i m e . ~I can be observed From Fig. 5, the agreement between the experimental decay times and the Monte Carlo values is s u r p r i s i n g l y good. The theoretical curves are extremely sensitive to the assumed thickness of the intervening b a r r i e r , with one monolayer difference accounting for as much as 50% error. The exact position of the phonon resonance does not concur between experiment and theory. In Fact, the measured value appears to correspond more c l o s e l y with a phonon resonance associated with AlAs phonons rather than GaAs modes, although more work is needed in order to ascertain t h i s effect.
Intersubband relaxation of hot carriers
1287
1000 • • MonteCarlo + PhotoLtlldnucence (Obedlet. al.)
8OO
600 + +
4OO
+
+ •
*m ÷
20O
0
o
+
2o
6o Field~V/cm)
8o
ioo
Fig. 5. The simulated time constant for the structure shown in Fig. 4 verses electric f i e l d , with the corresponding wide well photoluminescence decay times of (Oberli, 1989a). ACKNOWLEDGEMENTS
One of us (JL) acknowledges the support of IBM, Essex Junction, VT. This work was partially supported by Office of Naval Research (NOOOI4-8g-J-1894) and the National Center for Supercomputing Applications for supercomputer time. REFERENCES
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