- Email: [email protected]
Nonlinear resonant optical rectification in a coupled quantum well
surface science
ELSEVIER
Surface Science 361/362 (1996) 401-405
Nonlinear resonant optical rectification in a coupled quantum well K. Unterrainer a.,, J.N. Heyman b, K. Craig ", B. Galdrikian ~, M.S. Sherwin ", H. Drexler ~, K. Campman c, P.F. Hopkins c, A.C. Gossard ° • ~umtwn Institute, University of Cal~orn~ at Santa Barbara, Santa Barbara, CA 93106, USA b Department of Physics, Macalester College, Saint Paul, M N 5505, USA e Materials Department, University of CalO~ornia at Santa Barbara, Santa Barbara, CA 93106, USA Received 21 June 1995; accepted for publication 14 September 1995
Almract We have measured the rectification of far-infrared ~dlation resonant with the lowest intersubband transition of an AIGaAs/GaAs asymmetric coupled double-quantum well in which the subband spacing is 11 meV. From these measurements we can extract an intersubband lifetime of 1.2 +0.4 ms at low excitation intensity and T-- 10 K, which appears promising for devices which can operate at low excitation and temperature, such as FIR detectors or mixers. From simultaneous measurements of the optical rectification and of the intersubband absorption coefficient we can determine the intensity-dependent intersubband lifetime, which shows a strong decrease for increasing inteusitim.
Keywords: Ganium arsenide; Many body and quasi-particle theories; Non-linear optical methods; Photoconductivity; Quantum wells
1. Introduction The recent development of a mid-infrared intersubband laser has encouraged study of the possibility of an intersubband laser in the far infrared spectral region, which has a great demand for semiconductor radiation sources [ 1]. The most important parameter for such devices and detectors is the intersubband relaxation time (T1). We have measured this time using a new experimental technique based on the strong nonlinear susceptibility of an asymmetric double quantum welL Simultaneous measurements of the rectification of far-infrared radiation, resonant with the intersubband transition and the intersubband absorption, enable the determination of the intensity-depen-
dent relaxation time. Optical rectification is the static polarization produced by difference-frequency mixing of a harmonic electric field E(c0) with itself through second-order susceptibility. In asymmetric quantum wells this static polarization is observed for intersubband transitions because the expected value of the position of an electron in an excited state and in the ground state are different, i.e. P(2)( O ) = XI~,~)E(o~)E(to)
= -- e(n2 -- n°)(z2,2 -- zl,1),
(1) where n2 is the population in the second subband, n~ is the equilibrium population in the second subband and zt.j is the dipole-matrix element between states i and j.
* Corresponding author. 0039-6028/96/$15.00 Copyright O 1996 Elsevier Science B.V. All rights reserved P l l S0039-6028 ( 9 6 ) 00431-1
If. Untermtner et aL/Surface Science 361/362 (1996) 401-405
402
2. Experimental The coupled GaAs quantum wells in our heterostructure are 85 and 73 ,~ wide, separated by an Alo.aGao.TAS barrier 25J~ wide (see Fig. 1). The expectated value for the electrons in the ground state is larger in the wide well, and that for the electrons in the first excited state is larger in the narrow well (see Fig. 1, 0 V gate bias). The lowest electron subband spacing was measured to be E 2 - E 1 = 11 meV using photolnminescence. Energies of the higher subbands E 3 - E x -- 110 meV and E4--El = 156 meV were obtained from a selfconsistent model described below. The measurements described below were performed near resonance with h v ~ E2--Ex. Under these conditions the heterostructure may be approximated as a twosubband system. At low temperatures (T< 50 K), the zero-bias charge density obtained by Hall measurements is N . = 2 x 1 0 5 X c m -2 and the electron in-plane mobility is/~ = 1 x 105 em2/Ts. Alnminum Schottky gate was evaporated on the surface of the structure, and ohmic contacts were made to the double well. A negative voltage between the gate and ohmic contacts i m p o s e s a DC electric field across the structure and depletes it of charge. Aluminum was also evaporated on the substrate side of the wafer, so that the sample forms a parallel-plate waveguide. In these experiments, polarized FIR radiation is coupled into the cleaved edge of the wafer with polarization parallel
to the growth direction. The absorption spectrum of this structure in the FIR consists of a single line with an approximately Lorentzian lineshape, and the peak position can be tuned between 14.5 and 10meV (115 - 80 cm -1) with gate bias. At 0 V bias, the peak position is /k51,2=14.3 meV (115 cm -~, and the full-width at half maximum is 0.55 meV. The difference between the energy of the absorption resonance and the subband spacing is due to the depolarization shift. Our optical rectification experiments employ the UCSB free-electron laser (FEL) as a far-infrared pump. The laser was tuned to produce 5 ps pulses of 103 cm- x radiation at a 1.5 Hz repetition rate. The FIR uniformly illuminates the edge of our sample, which is mounted in a variable-temperature cryostat. Optical rectification is voltage the gate the quantum well because the expected value of the position of an electron in the excited state and the ground state are different. The impedance of the detection circuit and of the gate bias voltage source were high, to ensure that the time constant was longer than the pulse width. Radiation transmitted through the sample is focused onto a 4.2 K bolometer, so that intersubband absorption and optical rectification can be measured simultaneously. We investigate resonant effects by t u n i n g the intersubband absorption-resonance through the laser frequency with gate bias.
a laser-induced
between
observedas and
3. Results and diseession 250 r---~
'
'
i
t.22~' o.ov ='200 ~_~.... . . . . -o.~" ° ~ . . . . . . .
~
..........L 6 v ...,
i r.mta °-°1
..-
150 .... "........ ~.
,~ 100 I-
> 52~ , ~ I
0
50
I
100 150 200
z¢A)
Fig. 1. Conduction band diagram for the coupled double well for different gate bias voltages. The envelope wavefunctions for the first and second subband axe shown (their vertical positions are not drawn to scale).
Fig. 2 shows the ampfitude of the optical rectification and the FIR transmission as a function of gate bias, measured at T--10 K and at a laser intensity of 0.5 W cm 2. A resonance in the rectification occurs when the intersubband absorption frequency is tuned through the laser frequency at V o = - 0 . 8 V, indicating that the rectification is associated with the intersubband transition. We have calculated the energies and envelope functions of electrons at the subband minima in out double well by self-consistently solving the Schr6dinger and Poisson equations within the effective-mass approximation. Temperature effects were included by allowing thermal population of
I~ Unterraineret al./SurfaceScience361/362(1996)401-405 ,
,
t
,
I
'
Y-
o.s .
-1.6
-1.2
-0.8
-0.4
and ~ o n
!
0.8
-o
!
i
!
+
+
• 4[
~
i +
0.4
•
J~"
I'?l '/
+
0.4
~.
0.2 .~
+ +
+ +
I
10"
I
I &•
I01o
(
• AI
10-a
lif t
1
Co) t
(z~'2- zl"l)'~"°'2
leTl )-I,
Io'T1
(2)
in the linearr ~ m e
higher subbands. This model, together with the dephasing time determined from the linewidth of the absorption resonance, allows us to calculate the FIR transmission and optical rectification of our sample (Fig. 2, solid lines), and we find a good agreement with the experiment. Fig. (3a) shows the sensitivity (rectified voltage/intensity) and the intersubband absorption at resonance as a function of pump intensity at T = 10 K. At low intensities the sensitivity and the absorption axe constant. The rectified signal starts +
e
1+2
( T = 1 0 K, I = 0 . 5 W cm-2). Solid lines are calcuhted from theory (~ee text for details).
+
--4~e
0
c ~ B ~ CV~ Fig.2.Ol~cal ~ f i o .
to saturate (the sensitivity decreases) at ,-, 1 W cm -2, and the absorption coefficient starts to decrease at ~200 W cm -2. The saturation behavior of both the rectified signal and the absorption is described by the rate equations of a two-level system. The intensity dependent rectified signal is then given by A V°P" =
0.6
403
!
I
10° l0 t 102 103 Intensity (Wlcm 2)
t°
104
Fig. 3. (a) S c ~ (rectified voltage divided by intemity) and intersubband absorption co¢fficieat versus pump intensity, measured at T = 10 K. (b) Intm~ubband scatlm4ng rate calculated from this data.
and the intensity-dependent absorption coefficient is • (I)=(an°.2) ~
(
I~T1) -1,
1+2 ~
(3)
where ~ is the linear absorption cross-section, z|n°l,2is the difference between the equilibrium populations of the first and second subbands, I is the intensity of the pump beam, and a is the thickness of the sample. Fitting the magnitude of the rectification voltage (Eq. (2)) to the experiment yields an intersubband lifetime T~=l.2+0.4ns. This simple fit, however, does not explain the intensity dependence for intermediate power levels very well, which indicates that the assumption of a constant relaxation time is not suitable for all intensities. The obtained lifetime is longer than the calculated lifetime of TI =215 ps we obtain from Ferreira and Bastard's simple model for the acoustic-phonon scattering rate in our structure [2]. Since an intersubband excitation is in fact a collective excitation, the relaxation of a collective mode has to be treated in a different way from the single particle relaxation. This many body effect can reduce the relaxation rate [3]. The different saturation behaviors of the rectified signal and of the absorption coefficient indicate that the relaxation time is intensity-dependent. This intensity dependence has not been considered in any other experiment. Moreover, most experimental results axe derived from nearly saturated systems. However, the intensity-dependent relaxation time can be obtained directly from the ratio
between the rectifiedsignal and the absorption
404
I~ Unterrainer et aL/Surface ScienCe 361/362 (1996) 401--405
coefficient (see Eq. (2) and Eel. (3))
T,=
1,'*'(1)
ooJ
Iog(I)
47~e(zz,z--zl,1)a"
III ~[U
(4)
Therefore, simultaneous measurements of optical rectification and intersubband absorption can determine T~ over a wide range of conditions, including low intensities within the linear r e , me. The intersubband lifetime depends strongly on intensity (Fig. 3b). In the linear regime we measure T~= 1.2 ns (consistent with the result of the fit only to the rectified signal), while in the strongly saturated regime ( I = 2 k W cm -2) we lind T1=15 ps. We suggest two mechanisms for the reduction in lifetime at high intensities. First, the electron-electron intersubband scattering rate is predicted to depend strongly on the population in the second subband. If this mechanism controls the relaxation, the lifetime will decrease as the excited state is populated through absorption (at the highest intensities the population in the first excited subband reaches 50% of the total electron concentration). Second, the excitation by the pump should heat the electron gas. In view of the strong temperaturedependence of the scattering rate, a reduction in the intersubband lifetime at high pump intensities is expected due to heating [4]. In order to investigate the effect of carrier concentration on the relaxation time, we have performed the same experiments in a logarithmically graded quantum well [5]. The intersubband energy can be tuned from 35 to 125 cm-1 in this structure. Since we observe a considerable rectified voltage at the resonance, this structure acts as a widely tunable FIR detector. In addition, the carder concentration can be adjusted by a back-gate. As a function of carrier concentration, the absorption coefficient increases as expected. However, the rectified signal decreases with increasing carrier concentration, which indicates that the relaxation time gets shorter. Since the change of the carrier concentration is controlled by the back-gate voltage, the electric field in the well changes. Therefore, the matrix elements have to be evaluated for each cartier concentration. Fig. 4 shows preliminary results for the relaxation rate as obtained from the linear power dependence of the rectified signal.
10 H
i
I
I
~ ~ ii ...,-'l-I
b
O
~
101o O O
[] 10 9
I
0
I
I
i
0.5 1 1.5 2 Charge Density (x 10llcm'2)
Fig. 4. Charge-density dependence of the relaxation rate in a logar/thmically graded quantum well The inset shows the conduction band diagram (A]nminum content of the A1GaAa) of the loga~thmically graded well
4. Summary We have observed resonant optical rectification in our coupled quantum well. From simultaneous measurements of the rectified voltage and the absorption, we can extract the intersubband lifetime as a function of intensity. At low intensities, the lifetime is 1.2_0.4ns at T = 1 0 K, which appears promising for devices which can operate at low excitation and temperature, such as FIR detectors or mixers. At high intensities, the lifetime is reduced to 15 ps at 2 kW cm -2, which is still an order of magnitude larger than the lifetime for intersubband energies above the optical phonon energy.
Acknowledgements The authors would like to thank the staff at the Center for Free-Electron Laser Studies, J.R. Allen, D. Enyeart, J.P. Kaminsky, G. Ramian and D. White. Funding for the Center for Free-Electron
I~ thtterra~,r et aL/Surface Science 361/362 (1996) 401-405
Laser studies is provided by the Office of Naval Research.
References [ 1] J. FaJst, F. Capasso, D.L. Sivco, C. Sitori, A.L. Hutchinson and A.Y. Cho, Science 264 (1994) 553-556.
405
[2] R. Ferreira and G. Bastard, Phys. Rev. B 40 (1989) 1074. [3] See,for example, P. Nozieres and C.T. De Dominicis, Phys. Rev. 178 (1969) 1097. [4] J.N. Heyman, K. Unterrainer, IC Craig. B. Galdrik/a,; M.S. Sherwin, IC Campman, P.F. Hopkins and A.C. Go6sard, Phys. Rev. Lett. 74 (1995) 268?_ [5] P.F. Hopkins, ICL. Campman G. Betlomi, A.C. Gouard, M. Sundaxam, E.L. Yuh and E.G. Gwinn, AppL Phys. Lett. (1994) 348.
ELSEVIER
Surface Science 361/362 (1996) 401-405
Nonlinear resonant optical rectification in a coupled quantum well K. Unterrainer a.,, J.N. Heyman b, K. Craig ", B. Galdrikian ~, M.S. Sherwin ", H. Drexler ~, K. Campman c, P.F. Hopkins c, A.C. Gossard ° • ~umtwn Institute, University of Cal~orn~ at Santa Barbara, Santa Barbara, CA 93106, USA b Department of Physics, Macalester College, Saint Paul, M N 5505, USA e Materials Department, University of CalO~ornia at Santa Barbara, Santa Barbara, CA 93106, USA Received 21 June 1995; accepted for publication 14 September 1995
Almract We have measured the rectification of far-infrared ~dlation resonant with the lowest intersubband transition of an AIGaAs/GaAs asymmetric coupled double-quantum well in which the subband spacing is 11 meV. From these measurements we can extract an intersubband lifetime of 1.2 +0.4 ms at low excitation intensity and T-- 10 K, which appears promising for devices which can operate at low excitation and temperature, such as FIR detectors or mixers. From simultaneous measurements of the optical rectification and of the intersubband absorption coefficient we can determine the intensity-dependent intersubband lifetime, which shows a strong decrease for increasing inteusitim.
Keywords: Ganium arsenide; Many body and quasi-particle theories; Non-linear optical methods; Photoconductivity; Quantum wells
1. Introduction The recent development of a mid-infrared intersubband laser has encouraged study of the possibility of an intersubband laser in the far infrared spectral region, which has a great demand for semiconductor radiation sources [ 1]. The most important parameter for such devices and detectors is the intersubband relaxation time (T1). We have measured this time using a new experimental technique based on the strong nonlinear susceptibility of an asymmetric double quantum welL Simultaneous measurements of the rectification of far-infrared radiation, resonant with the intersubband transition and the intersubband absorption, enable the determination of the intensity-depen-
dent relaxation time. Optical rectification is the static polarization produced by difference-frequency mixing of a harmonic electric field E(c0) with itself through second-order susceptibility. In asymmetric quantum wells this static polarization is observed for intersubband transitions because the expected value of the position of an electron in an excited state and in the ground state are different, i.e. P(2)( O ) = XI~,~)E(o~)E(to)
= -- e(n2 -- n°)(z2,2 -- zl,1),
(1) where n2 is the population in the second subband, n~ is the equilibrium population in the second subband and zt.j is the dipole-matrix element between states i and j.
* Corresponding author. 0039-6028/96/$15.00 Copyright O 1996 Elsevier Science B.V. All rights reserved P l l S0039-6028 ( 9 6 ) 00431-1
If. Untermtner et aL/Surface Science 361/362 (1996) 401-405
402
2. Experimental The coupled GaAs quantum wells in our heterostructure are 85 and 73 ,~ wide, separated by an Alo.aGao.TAS barrier 25J~ wide (see Fig. 1). The expectated value for the electrons in the ground state is larger in the wide well, and that for the electrons in the first excited state is larger in the narrow well (see Fig. 1, 0 V gate bias). The lowest electron subband spacing was measured to be E 2 - E 1 = 11 meV using photolnminescence. Energies of the higher subbands E 3 - E x -- 110 meV and E4--El = 156 meV were obtained from a selfconsistent model described below. The measurements described below were performed near resonance with h v ~ E2--Ex. Under these conditions the heterostructure may be approximated as a twosubband system. At low temperatures (T< 50 K), the zero-bias charge density obtained by Hall measurements is N . = 2 x 1 0 5 X c m -2 and the electron in-plane mobility is/~ = 1 x 105 em2/Ts. Alnminum Schottky gate was evaporated on the surface of the structure, and ohmic contacts were made to the double well. A negative voltage between the gate and ohmic contacts i m p o s e s a DC electric field across the structure and depletes it of charge. Aluminum was also evaporated on the substrate side of the wafer, so that the sample forms a parallel-plate waveguide. In these experiments, polarized FIR radiation is coupled into the cleaved edge of the wafer with polarization parallel
to the growth direction. The absorption spectrum of this structure in the FIR consists of a single line with an approximately Lorentzian lineshape, and the peak position can be tuned between 14.5 and 10meV (115 - 80 cm -1) with gate bias. At 0 V bias, the peak position is /k51,2=14.3 meV (115 cm -~, and the full-width at half maximum is 0.55 meV. The difference between the energy of the absorption resonance and the subband spacing is due to the depolarization shift. Our optical rectification experiments employ the UCSB free-electron laser (FEL) as a far-infrared pump. The laser was tuned to produce 5 ps pulses of 103 cm- x radiation at a 1.5 Hz repetition rate. The FIR uniformly illuminates the edge of our sample, which is mounted in a variable-temperature cryostat. Optical rectification is voltage the gate the quantum well because the expected value of the position of an electron in the excited state and the ground state are different. The impedance of the detection circuit and of the gate bias voltage source were high, to ensure that the time constant was longer than the pulse width. Radiation transmitted through the sample is focused onto a 4.2 K bolometer, so that intersubband absorption and optical rectification can be measured simultaneously. We investigate resonant effects by t u n i n g the intersubband absorption-resonance through the laser frequency with gate bias.
a laser-induced
between
observedas and
3. Results and diseession 250 r---~
'
'
i
t.22~' o.ov ='200 ~_~.... . . . . -o.~" ° ~ . . . . . . .
~
..........L 6 v ...,
i r.mta °-°1
..-
150 .... "........ ~.
,~ 100 I-
> 52~ , ~ I
0
50
I
100 150 200
z¢A)
Fig. 1. Conduction band diagram for the coupled double well for different gate bias voltages. The envelope wavefunctions for the first and second subband axe shown (their vertical positions are not drawn to scale).
Fig. 2 shows the ampfitude of the optical rectification and the FIR transmission as a function of gate bias, measured at T--10 K and at a laser intensity of 0.5 W cm 2. A resonance in the rectification occurs when the intersubband absorption frequency is tuned through the laser frequency at V o = - 0 . 8 V, indicating that the rectification is associated with the intersubband transition. We have calculated the energies and envelope functions of electrons at the subband minima in out double well by self-consistently solving the Schr6dinger and Poisson equations within the effective-mass approximation. Temperature effects were included by allowing thermal population of
I~ Unterraineret al./SurfaceScience361/362(1996)401-405 ,
,
t
,
I
'
Y-
o.s .
-1.6
-1.2
-0.8
-0.4
and ~ o n
!
0.8
-o
!
i
!
+
+
• 4[
~
i +
0.4
•
J~"
I'?l '/
+
0.4
~.
0.2 .~
+ +
+ +
I
10"
I
I &•
I01o
(
• AI
10-a
lif t
1
Co) t
(z~'2- zl"l)'~"°'2
leTl )-I,
Io'T1
(2)
in the linearr ~ m e
higher subbands. This model, together with the dephasing time determined from the linewidth of the absorption resonance, allows us to calculate the FIR transmission and optical rectification of our sample (Fig. 2, solid lines), and we find a good agreement with the experiment. Fig. (3a) shows the sensitivity (rectified voltage/intensity) and the intersubband absorption at resonance as a function of pump intensity at T = 10 K. At low intensities the sensitivity and the absorption axe constant. The rectified signal starts +
e
1+2
( T = 1 0 K, I = 0 . 5 W cm-2). Solid lines are calcuhted from theory (~ee text for details).
+
--4~e
0
c ~ B ~ CV~ Fig.2.Ol~cal ~ f i o .
to saturate (the sensitivity decreases) at ,-, 1 W cm -2, and the absorption coefficient starts to decrease at ~200 W cm -2. The saturation behavior of both the rectified signal and the absorption is described by the rate equations of a two-level system. The intensity dependent rectified signal is then given by A V°P" =
0.6
403
!
I
10° l0 t 102 103 Intensity (Wlcm 2)
t°
104
Fig. 3. (a) S c ~ (rectified voltage divided by intemity) and intersubband absorption co¢fficieat versus pump intensity, measured at T = 10 K. (b) Intm~ubband scatlm4ng rate calculated from this data.
and the intensity-dependent absorption coefficient is • (I)=(an°.2) ~
(
I~T1) -1,
1+2 ~
(3)
where ~ is the linear absorption cross-section, z|n°l,2is the difference between the equilibrium populations of the first and second subbands, I is the intensity of the pump beam, and a is the thickness of the sample. Fitting the magnitude of the rectification voltage (Eq. (2)) to the experiment yields an intersubband lifetime T~=l.2+0.4ns. This simple fit, however, does not explain the intensity dependence for intermediate power levels very well, which indicates that the assumption of a constant relaxation time is not suitable for all intensities. The obtained lifetime is longer than the calculated lifetime of TI =215 ps we obtain from Ferreira and Bastard's simple model for the acoustic-phonon scattering rate in our structure [2]. Since an intersubband excitation is in fact a collective excitation, the relaxation of a collective mode has to be treated in a different way from the single particle relaxation. This many body effect can reduce the relaxation rate [3]. The different saturation behaviors of the rectified signal and of the absorption coefficient indicate that the relaxation time is intensity-dependent. This intensity dependence has not been considered in any other experiment. Moreover, most experimental results axe derived from nearly saturated systems. However, the intensity-dependent relaxation time can be obtained directly from the ratio
between the rectifiedsignal and the absorption
404
I~ Unterrainer et aL/Surface ScienCe 361/362 (1996) 401--405
coefficient (see Eq. (2) and Eel. (3))
T,=
1,'*'(1)
ooJ
Iog(I)
47~e(zz,z--zl,1)a"
III ~[U
(4)
Therefore, simultaneous measurements of optical rectification and intersubband absorption can determine T~ over a wide range of conditions, including low intensities within the linear r e , me. The intersubband lifetime depends strongly on intensity (Fig. 3b). In the linear regime we measure T~= 1.2 ns (consistent with the result of the fit only to the rectified signal), while in the strongly saturated regime ( I = 2 k W cm -2) we lind T1=15 ps. We suggest two mechanisms for the reduction in lifetime at high intensities. First, the electron-electron intersubband scattering rate is predicted to depend strongly on the population in the second subband. If this mechanism controls the relaxation, the lifetime will decrease as the excited state is populated through absorption (at the highest intensities the population in the first excited subband reaches 50% of the total electron concentration). Second, the excitation by the pump should heat the electron gas. In view of the strong temperaturedependence of the scattering rate, a reduction in the intersubband lifetime at high pump intensities is expected due to heating [4]. In order to investigate the effect of carrier concentration on the relaxation time, we have performed the same experiments in a logarithmically graded quantum well [5]. The intersubband energy can be tuned from 35 to 125 cm-1 in this structure. Since we observe a considerable rectified voltage at the resonance, this structure acts as a widely tunable FIR detector. In addition, the carder concentration can be adjusted by a back-gate. As a function of carrier concentration, the absorption coefficient increases as expected. However, the rectified signal decreases with increasing carrier concentration, which indicates that the relaxation time gets shorter. Since the change of the carrier concentration is controlled by the back-gate voltage, the electric field in the well changes. Therefore, the matrix elements have to be evaluated for each cartier concentration. Fig. 4 shows preliminary results for the relaxation rate as obtained from the linear power dependence of the rectified signal.
10 H
i
I
I
~ ~ ii ...,-'l-I
b
O
~
101o O O
[] 10 9
I
0
I
I
i
0.5 1 1.5 2 Charge Density (x 10llcm'2)
Fig. 4. Charge-density dependence of the relaxation rate in a logar/thmically graded quantum well The inset shows the conduction band diagram (A]nminum content of the A1GaAa) of the loga~thmically graded well
4. Summary We have observed resonant optical rectification in our coupled quantum well. From simultaneous measurements of the rectified voltage and the absorption, we can extract the intersubband lifetime as a function of intensity. At low intensities, the lifetime is 1.2_0.4ns at T = 1 0 K, which appears promising for devices which can operate at low excitation and temperature, such as FIR detectors or mixers. At high intensities, the lifetime is reduced to 15 ps at 2 kW cm -2, which is still an order of magnitude larger than the lifetime for intersubband energies above the optical phonon energy.
Acknowledgements The authors would like to thank the staff at the Center for Free-Electron Laser Studies, J.R. Allen, D. Enyeart, J.P. Kaminsky, G. Ramian and D. White. Funding for the Center for Free-Electron
I~ thtterra~,r et aL/Surface Science 361/362 (1996) 401-405
Laser studies is provided by the Office of Naval Research.
References [ 1] J. FaJst, F. Capasso, D.L. Sivco, C. Sitori, A.L. Hutchinson and A.Y. Cho, Science 264 (1994) 553-556.
405
[2] R. Ferreira and G. Bastard, Phys. Rev. B 40 (1989) 1074. [3] See,for example, P. Nozieres and C.T. De Dominicis, Phys. Rev. 178 (1969) 1097. [4] J.N. Heyman, K. Unterrainer, IC Craig. B. Galdrik/a,; M.S. Sherwin, IC Campman, P.F. Hopkins and A.C. Go6sard, Phys. Rev. Lett. 74 (1995) 268?_ [5] P.F. Hopkins, ICL. Campman G. Betlomi, A.C. Gouard, M. Sundaxam, E.L. Yuh and E.G. Gwinn, AppL Phys. Lett. (1994) 348.