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Optical singularities of the one-dimensional electron gas in semiconductor quantum wires
Surface Science 263
:‘.,,:.‘.,;::‘:;::::, ..: ., :....
(1992)34-350
‘..
.:...
.:.:
..:
..’
‘surface science
North-Holland
.:
.... .j::: ‘.‘:,.;.,.c .‘,.:.:.,, .j..::j . .,..’ :>... :>,: .>>..’ ,... :.:,... ;.;.i:.“;;;: :::.j:: ,,:...
Optical singularities of the one-dimensional in semiconductor quantum wires J.M. Calleja ‘, A.R. Goiii, B.S. Dennis, J.S. Weiner, K.W. West AT&T Bell Lahorutorie.c, J.F. Miiller
Received
Murray Ilill.
The optical
agreement
for publication
of a one-dimensional
quantum
wires,
GaAs
obtained
quantum
wells.
and emission. They disappear
and sharper
than
26 August
L.N. Pfciffcr,
(ID)
electron
by electron Large
10’11
singularities
at temperatures
comparable
results based on the exact diagonalization
address: Universidad
Autbnoma,
‘(” lYY2 - Elsevier
Madrid,
Spain.
Science Puhlisherr
B.V.
and
subsequent
have been to the Fermi
hole recoil
The fabrication of semiconductor structures where electron motion is restricted to one or zero dimensions has led to the observation of new phenomena in transport and optics [l]. Experimental results were reported for the quasi-onedimensional electron gas (QlDEG) in channel inversion layers [2,3] and in multiple quantum well wires (QWW) [4-71 with many occupied subbands. In these systems the relatively large size ( - 10’ A> of the low-dimensional structures lead to intersubband spacings much lower than the Fermi energy. However. the features of the clectron dynamics specific for ID systems are expected to be distinct in the 1D limit, where only one subband is occupied by free electrons. We have recently reported measurements of photoluminescence (PL) and photolumincsccncc
0039.r,02X/Y2/$05.00
lithography
optical
in 2D systems due to the lack of certain
with theoretical
gas with only one or two occupied
beam
1. Introduction
’ Permanent
S. Schmitt-Rink,
NJ 07Y74. USA
properties
modulation-doped absorption
A. Pinczuk,
gas
and A.E. Ruckenstein
31 May 1991; accepted
semiconductor
electron
effects
observed
at the
energy. The
in ID.
The
subbands have been ytudicd in
low-energy Fermi
ion
ID singularities
experimental
bombardment
Icvel.
hoth
01
in optical
are much larger
data are in qualitative
of finite chains,
excitation (PLE) in GaAs QWW in the ID quantum limit [8]. The quantum limit is obtained by simultaneous vertical and lateral depletion of patterned, modulation-doped GaAs single quantum wells. Strong singularities arc observed for optical transitions at the Fermi level, which are interpreted as Fermi cdgc singularities (FES) resulting from the response of the Fermi sea to a hole in a valence subband [9,1C)]. Interference between the Fermi level and the lowest empty conduction subband [11,12] could also contribute to the enormous strength of the FES observed in emission spectra. In contrast to the 1D FES, the singularities observed in two-dimensional (2D) systems with similar Fermi energies are much weaker and disappear completely for higher electron conccntrations. In fact, some of the previously I-eported results on 2D FES [ 13- 161 showing clear singularities in optical emission arc better understood on the assumption of localized holes with infinite mass. In abscncc of hole localization, the strong FES is a unique characteristic of ID systems and Yamada
Science Foundation.
All right\
rrsetxctl
J.M. Calleja
et al. / Optical
singularities
of the IDEG
in semiconductor
resulting from the lack of certain recoil effects on the response of the Fermi sea. To approach this problem theoretically an exact diagonalization of finite Hubbard chains has been performed assuming an on-site Coulomb interaction between a variable number of electrons and a single hole. The results of this calculation are in good qualitative agreement with the experiments.
quantum
PL 2D -PL
ofit
JL *
2. Sample preparation
‘\
I.’
1 min.
*\
*-_.
I
:
3. Experimental
results
The PL and PLE spectra of sample A taken at 1.7 K and different bombardment times are shown in fig. 1. After short exposures the PL spectrum of the as-grown 2DEG (dotted line) shifts to higher energies without significant changes in the PLE spectrum. For longer bombardment times
1D
-- PLE
d’
Quantum well wires were fabricated on a oneside modulation doped single GaAs quantum well (QW) following the procedu;e described in ref. [6]. The sample is a 250 A thick GaAs with Ga,,.,,Al,,.,,As barriers. Two monolayers of Si in one of the barriers produce a two-dimensional electron gas (2DEG) with electron density and mobility of II = 3.2 x 10” cmm2 and p = 1.1 X lo6 cm2/V. s, respectively. The 1D pattern is created by elecJron beam lithography and consists of lines 1000 A wide with 2000 A period. The width fluctuation is about 10%. The QWW are formed by depletion of the QW in the openings between the lines, using low-energy ion bombardment. The result is a periodic modulation of the electrostatic potential that confines the electrons and holes to separated 1D spatial regions, in a type II multiple QWW [6]. A careful control of the ion bombardment parameters (acceleration voltage and exposure time) was used to obtain the desired value of the 1D Fermi energy and subband spacing. In this way it is possible to produce samples in the 1D quantum limit. We present results from two samples bombarded for different times with oxygen at 300 V and 1.5 X 10m4 A/cm2 (sample A) and 1.0 X 1O-4 A/cm2 (sample B), respectively.
347
wires
: .. :: : : __ : .., : k. ‘.
0
&.L-
1510
0.5
::
::
min.
OSCJrown, __.-._,.’ ‘A._.
;.\
I, I : L-
‘+,
-
1520 ENERGY
1530
1540
(meV)
Fig. 1. Effect of the ion milling time on the PL and PLE spectra of the QWW. The dotted line is the PL emission of the original 2D electron gas. Other curves correspond to the PL (full line) and PLE spectra (dashed line) of sample A after different ion milling times given in minutes.
the emission spectra become asymmetric and the peak intensity shifts from the band edge to the Fermi level. At the longer times the PLE spectra develop strong and sharp peaks at the transitions from the heavy and light hole states to the Fermi level. The peak at 1535 meV in the PLE spectrum of the as-grown sample is the “forbidden” exciton associated with the first heavy-hole state and the second 2D electron band. The temperature dependence of PL and PLE spectra of sample A are shown in figs. 2a and 2b. The Fermi energy E, obtained from the PL spectra is E, = Ef-(1 + m,/m,,V’ = 4.5 f 0.5 meV, where m, and m,, are the electron and hole effective masses. The intersubband spacing determined by electronic Raman scattering is 6E = 5.0 -t 0.5 meV 181. Since the subband spacing is larger than E,, sample A is in the 1D quantum limit. The measured Ftersubband spacing yields a value of L, = 400 A for the wire width at the Fermi level assuming a parabolic potential. The corresponding 1D electron concentration is n = 5.7 X 10” cm-‘. The sharp peaks observed in the
a)
(b) PLE
EABS
PL
J
(b) PLE
c
T(K)
AHH
J
1 525
1515
6 1.515 ENERGY
1.525
1 52
1.52
ENERGY
(eV)
Fig. 2. Emission (a) and absorption the ID quantum
1.53
1 525
1.535
ENERGY
(eV,
“as-grown“
ilJ,) = I.1 X IO”
cm
with an ‘.
(eV)
(b) spectra of sample A in
limit at different
electron concentration
1 53
1 52
(eV)
Fig. 4. Same as fig. 2 for a 2D sample
1
II .51
ENERGY
30
temperatures.
PLE spectra at low temperatures disappear at temperatures (- 30 K) corresponding roughly to the Fermi energy obtained from PL measurements. Similar results are shown in figs. 3a and 3b for sample B. In this case the intersubband spacing is SE = 3.8 + 0.2 meV and the Fermi energy Et, = 4.5 k 0.5 meV so that two conduction subbands are occupied. The estimated values of the wire
(b) PLE I.
width and 1D electron density for the first band are L, = 600 A and H” = 5.7 x lo5 cm-.‘. respectively. The second conduction band has therefore a low Fermi energy (Et: = 0.7 meV) with an electron density H’ = 2.3 x 10’ cm- ‘, The evolution with temperature of the PL and PLE spectra is similar to that of sample A, but the sharp peaks disappear now at a much lower temperature (- 8 K). In order to understand the specific 1D character of the results shown above, WC measured the PL and PLE spectra of a 2D sample with the same well width and comparable Fermi energy (E, = 3.8 meV). The results are shown in figs. 4a and 4b. The PL spectra display the expected 2D behavior [ 13161 with a peak intensity at the band edge. There is also a well defined shoulder at the Fermi level for temperatures between 3 and 10 K. The sharp peak at 1516 meV is due to defects. The PLE spectra shown in fig. 4b have much weaker peaks that also disappear at temperatures around 10 K.
4. Discussion
1
515
/
1 52 ENERGY
1 525
2
(eV)
Fig. 3. Same as fig. 2 for sample B with
1
1.525 ENERGY
(eV)
occupied subbands.
The strong, excitonic-like peaks observed in the PLE spectra of 1D samples are interpreted as Fermi edge singularities (FES) of the 1D electron gas. Their temperature dependence is in agree-
J.M. Calleja et al. / Optical singularities of the 1DEG in semiconductor quantum wires
ment with this interpretation because the singularities disappear at temperatures comparable to the Fermi energy [17]. The temperature dependence of PLE spectra of sample B indicates that, near the 1D quantum limit, the FES is dominated by the last occupied subband. The 1D FES shows two main differences with respect to the 2D case: (a> The singularity is sharper and stronger in 1D and (b) it is also clearly observed in the emission spectra. These differences can be understood if one considers that the PLE threshold for a 2D sample is an indirect transition from the top of the valence band to the Fermi level, together with a zero-energy excitation of the Fermi sea required for momentum conservation. For high electron concentrations the energy difference between this indirect transition and the direct one at the Fermi vector k, ((m,/m,)E,) is larger than the characteristic Coulomb energy and no FES appears [18]. At low electron densities the FES is weak but observable as shown in fig. 4, even in the emission spectra, for the temperature range where there is a noticeable population of holes at k, and yet the Fermi distribution is not smeared out 18,121. In 1D systems there are no zero-energy excitations of the Fermi sea other than k = 0 or k = 2k,, so that indirect transitions cannot take place and the PLE onset is a vertical transition at k, [S]. This means that FES will be present in 1D systems at any electron concentration and for finite hole mass as shown by our results. A sharp peak at the Fermi level has been observed in PL spectra of 2D electron gases [12,14] in ternary systems. It is due to disorder-induced hole localization. In the present case the same weak FES is observed in 2D samples with low electron concentration, both for as-grown or ion bombarded samples. Therefore ion bombardment does not produce any noticeable hole localization in our case, and defects can be excluded as the origin of the sharp FES observed here. A deeper understanding of our experimental results is obtained by a calculation of the absorption spectra of a 1D electron gas as a function of the electron density. This has been done by numerical diagonalization of the Hamihonian for a small array of atoms containing a few electrons
349
and a single hole coupled by an on-site Coulomb interaction 1171.The results are described in detail in ref. [El and show that the absorption onset of the 1D system has indeed a strong FES even for high electron concentrations and a hole mass equal to the electron one. This striking result is in sharp contrast with the 2D case, where the absorption onset is smooth at high electron concentrations. It shows that the hole recoil plays only a minor role in the many-body reaction of the Fermi sea during the absorption of light in 1D systems, in agreement with out experimental results. In conclusion, we observe new optical properties of the 1D electron gas in and near the quantum limit. FES much stronger than in 2D systems are observed, whose temperature dependence is determined by the Fermi energy of the last occupied subband. The increased strength of the FES in lD, its dominance in the optical emission spectra and its persistence at high electron concentrations can be understood by the lack of certain hole recoil effects in 1D.
Acknowledgment
One of the authors (J.M.C.) is indebted to the Spanish CICYT and AT&T Microelectronics of Spain for financial support.
References Physics and Fabrication, Proceedings of [ll Nanostructure the International Symposium, Texas, USA, March 1989, Eds. M.A. Reed and W.P. Kirk (Academic Press, London, 1989). 121W. Hansen, M. Horst, J.P. Kotthaus, U. Merkt, Ch. Sikorski and K. Ploog, Phys. Rev. Lett. 58 (1987) 2586. and HI. [31 K. Ismail, W. Chu, A. Yen, D.A. Antoniadis Smith, Appl. Phys. Lett. 54 (1988) 460. [41 T.P. Smith III, H. Arnot, J.M. Hong, C.M. Knoedler, S.E. Laux and H. Schmid, Phys. Rev. Lett. 59 (1987) 2802. P. Grambow and K. Ploog, 151 T. Demel, D. Heitmann, Phys. Rev. B 38 (1988) 12732. L.N. 161 J.S. Weiner, G. Danan, A. Pinczuk, J. Valladares, Pfeiffer and K. West, Phys. Rev. Lett. 63 (1989) 1641. G. Weimann, T. Demel, D. I71 T. Egeler, G. Abstreiter, Heitmann, P. Grambow and W. Schlapp, Phys. Rev. Lett. 65 (1990) 1804.
[X] J.M. Calleja, A.R. Goiii, B.S. Dennis, J.S. Weiner. A. Pinczuk. S. Schmitt-Rink, L.N. Pfeiffer. K.W. West, J.F. Miller and A.E. Ruckenstein. Solid State Commun. 7Y (1991) 911. [Y] G.D. Mahan. Phys. Rev. 153 (1967) 8X2: lh3 (1967) 612. [IO] P. Nozieres and C.T. de Dominicis. Phys. Rev. 17X ( I 960) 1OY7. [I I] W. Chen, M. Fritze. A.V. Nurmikko, D. Acklcy. C‘. Colvard and H. Lee. Phys. Rev. Lett. 63 (lY90) 2434. [ 121J.F. Miiller, A.E. Ruckenstein and S. Schmitt-Rink. Mod. Phys. Lett. B, in press: J.F. Miiller. Phys. Rev. B 42 (IWO) 111x9. [ 131G. Livescu, D.A.B. Miller. D.S. Chemla, M. Ramaswnmy.
T.Y. <‘hang. N. Sauer, A.<‘. Gossard and J.H. txglish. IEEE J. Quantum Electron. QE-24 (IYXti) 1677. [I31 MS. Skolnick, J.M. Rorison. K.J. Nash. D.J. Mowb~wy. P.R. Tapster. S.J. Bass and AD. Pitt. Php Re\. Ixtt. 5S (19x7) 2130. [IS] J.S. Lee. N. Miura and T. Ando. J. Phys. Sot. Jpn. 50
( 1’)90)2251. [Ih] A. Pinauk. J. Shah. R.C. Miller, A.C. Goasa~-d and W:. Wiegmann. Solid State C‘onimun. 50 (1983) 735. D.S. C‘hrmla ;md [ 171 For it review. wz: S. Schmitt-Rink. D.A.B. Miller, Ad\. Phya. 3X (198Y) XY. [IX] .I. Gavoret, P. No&rcs. B. Roulct and M. (‘omhescot. J. Phys. (Par&) 30 (lY69) 087.
:‘.,,:.‘.,;::‘:;::::, ..: ., :....
(1992)34-350
‘..
.:...
.:.:
..:
..’
‘surface science
North-Holland
.:
.... .j::: ‘.‘:,.;.,.c .‘,.:.:.,, .j..::j . .,..’ :>... :>,: .>>..’ ,... :.:,... ;.;.i:.“;;;: :::.j:: ,,:...
Optical singularities of the one-dimensional in semiconductor quantum wires J.M. Calleja ‘, A.R. Goiii, B.S. Dennis, J.S. Weiner, K.W. West AT&T Bell Lahorutorie.c, J.F. Miiller
Received
Murray Ilill.
The optical
agreement
for publication
of a one-dimensional
quantum
wires,
GaAs
obtained
quantum
wells.
and emission. They disappear
and sharper
than
26 August
L.N. Pfciffcr,
(ID)
electron
by electron Large
10’11
singularities
at temperatures
comparable
results based on the exact diagonalization
address: Universidad
Autbnoma,
‘(” lYY2 - Elsevier
Madrid,
Spain.
Science Puhlisherr
B.V.
and
subsequent
have been to the Fermi
hole recoil
The fabrication of semiconductor structures where electron motion is restricted to one or zero dimensions has led to the observation of new phenomena in transport and optics [l]. Experimental results were reported for the quasi-onedimensional electron gas (QlDEG) in channel inversion layers [2,3] and in multiple quantum well wires (QWW) [4-71 with many occupied subbands. In these systems the relatively large size ( - 10’ A> of the low-dimensional structures lead to intersubband spacings much lower than the Fermi energy. However. the features of the clectron dynamics specific for ID systems are expected to be distinct in the 1D limit, where only one subband is occupied by free electrons. We have recently reported measurements of photoluminescence (PL) and photolumincsccncc
0039.r,02X/Y2/$05.00
lithography
optical
in 2D systems due to the lack of certain
with theoretical
gas with only one or two occupied
beam
1. Introduction
’ Permanent
S. Schmitt-Rink,
NJ 07Y74. USA
properties
modulation-doped absorption
A. Pinczuk,
gas
and A.E. Ruckenstein
31 May 1991; accepted
semiconductor
electron
effects
observed
at the
energy. The
in ID.
The
subbands have been ytudicd in
low-energy Fermi
ion
ID singularities
experimental
bombardment
Icvel.
hoth
01
in optical
are much larger
data are in qualitative
of finite chains,
excitation (PLE) in GaAs QWW in the ID quantum limit [8]. The quantum limit is obtained by simultaneous vertical and lateral depletion of patterned, modulation-doped GaAs single quantum wells. Strong singularities arc observed for optical transitions at the Fermi level, which are interpreted as Fermi cdgc singularities (FES) resulting from the response of the Fermi sea to a hole in a valence subband [9,1C)]. Interference between the Fermi level and the lowest empty conduction subband [11,12] could also contribute to the enormous strength of the FES observed in emission spectra. In contrast to the 1D FES, the singularities observed in two-dimensional (2D) systems with similar Fermi energies are much weaker and disappear completely for higher electron conccntrations. In fact, some of the previously I-eported results on 2D FES [ 13- 161 showing clear singularities in optical emission arc better understood on the assumption of localized holes with infinite mass. In abscncc of hole localization, the strong FES is a unique characteristic of ID systems and Yamada
Science Foundation.
All right\
rrsetxctl
J.M. Calleja
et al. / Optical
singularities
of the IDEG
in semiconductor
resulting from the lack of certain recoil effects on the response of the Fermi sea. To approach this problem theoretically an exact diagonalization of finite Hubbard chains has been performed assuming an on-site Coulomb interaction between a variable number of electrons and a single hole. The results of this calculation are in good qualitative agreement with the experiments.
quantum
PL 2D -PL
ofit
JL *
2. Sample preparation
‘\
I.’
1 min.
*\
*-_.
I
:
3. Experimental
results
The PL and PLE spectra of sample A taken at 1.7 K and different bombardment times are shown in fig. 1. After short exposures the PL spectrum of the as-grown 2DEG (dotted line) shifts to higher energies without significant changes in the PLE spectrum. For longer bombardment times
1D
-- PLE
d’
Quantum well wires were fabricated on a oneside modulation doped single GaAs quantum well (QW) following the procedu;e described in ref. [6]. The sample is a 250 A thick GaAs with Ga,,.,,Al,,.,,As barriers. Two monolayers of Si in one of the barriers produce a two-dimensional electron gas (2DEG) with electron density and mobility of II = 3.2 x 10” cmm2 and p = 1.1 X lo6 cm2/V. s, respectively. The 1D pattern is created by elecJron beam lithography and consists of lines 1000 A wide with 2000 A period. The width fluctuation is about 10%. The QWW are formed by depletion of the QW in the openings between the lines, using low-energy ion bombardment. The result is a periodic modulation of the electrostatic potential that confines the electrons and holes to separated 1D spatial regions, in a type II multiple QWW [6]. A careful control of the ion bombardment parameters (acceleration voltage and exposure time) was used to obtain the desired value of the 1D Fermi energy and subband spacing. In this way it is possible to produce samples in the 1D quantum limit. We present results from two samples bombarded for different times with oxygen at 300 V and 1.5 X 10m4 A/cm2 (sample A) and 1.0 X 1O-4 A/cm2 (sample B), respectively.
347
wires
: .. :: : : __ : .., : k. ‘.
0
&.L-
1510
0.5
::
::
min.
OSCJrown, __.-._,.’ ‘A._.
;.\
I, I : L-
‘+,
-
1520 ENERGY
1530
1540
(meV)
Fig. 1. Effect of the ion milling time on the PL and PLE spectra of the QWW. The dotted line is the PL emission of the original 2D electron gas. Other curves correspond to the PL (full line) and PLE spectra (dashed line) of sample A after different ion milling times given in minutes.
the emission spectra become asymmetric and the peak intensity shifts from the band edge to the Fermi level. At the longer times the PLE spectra develop strong and sharp peaks at the transitions from the heavy and light hole states to the Fermi level. The peak at 1535 meV in the PLE spectrum of the as-grown sample is the “forbidden” exciton associated with the first heavy-hole state and the second 2D electron band. The temperature dependence of PL and PLE spectra of sample A are shown in figs. 2a and 2b. The Fermi energy E, obtained from the PL spectra is E, = Ef-(1 + m,/m,,V’ = 4.5 f 0.5 meV, where m, and m,, are the electron and hole effective masses. The intersubband spacing determined by electronic Raman scattering is 6E = 5.0 -t 0.5 meV 181. Since the subband spacing is larger than E,, sample A is in the 1D quantum limit. The measured Ftersubband spacing yields a value of L, = 400 A for the wire width at the Fermi level assuming a parabolic potential. The corresponding 1D electron concentration is n = 5.7 X 10” cm-‘. The sharp peaks observed in the
a)
(b) PLE
EABS
PL
J
(b) PLE
c
T(K)
AHH
J
1 525
1515
6 1.515 ENERGY
1.525
1 52
1.52
ENERGY
(eV)
Fig. 2. Emission (a) and absorption the ID quantum
1.53
1 525
1.535
ENERGY
(eV,
“as-grown“
ilJ,) = I.1 X IO”
cm
with an ‘.
(eV)
(b) spectra of sample A in
limit at different
electron concentration
1 53
1 52
(eV)
Fig. 4. Same as fig. 2 for a 2D sample
1
II .51
ENERGY
30
temperatures.
PLE spectra at low temperatures disappear at temperatures (- 30 K) corresponding roughly to the Fermi energy obtained from PL measurements. Similar results are shown in figs. 3a and 3b for sample B. In this case the intersubband spacing is SE = 3.8 + 0.2 meV and the Fermi energy Et, = 4.5 k 0.5 meV so that two conduction subbands are occupied. The estimated values of the wire
(b) PLE I.
width and 1D electron density for the first band are L, = 600 A and H” = 5.7 x lo5 cm-.‘. respectively. The second conduction band has therefore a low Fermi energy (Et: = 0.7 meV) with an electron density H’ = 2.3 x 10’ cm- ‘, The evolution with temperature of the PL and PLE spectra is similar to that of sample A, but the sharp peaks disappear now at a much lower temperature (- 8 K). In order to understand the specific 1D character of the results shown above, WC measured the PL and PLE spectra of a 2D sample with the same well width and comparable Fermi energy (E, = 3.8 meV). The results are shown in figs. 4a and 4b. The PL spectra display the expected 2D behavior [ 13161 with a peak intensity at the band edge. There is also a well defined shoulder at the Fermi level for temperatures between 3 and 10 K. The sharp peak at 1516 meV is due to defects. The PLE spectra shown in fig. 4b have much weaker peaks that also disappear at temperatures around 10 K.
4. Discussion
1
515
/
1 52 ENERGY
1 525
2
(eV)
Fig. 3. Same as fig. 2 for sample B with
1
1.525 ENERGY
(eV)
occupied subbands.
The strong, excitonic-like peaks observed in the PLE spectra of 1D samples are interpreted as Fermi edge singularities (FES) of the 1D electron gas. Their temperature dependence is in agree-
J.M. Calleja et al. / Optical singularities of the 1DEG in semiconductor quantum wires
ment with this interpretation because the singularities disappear at temperatures comparable to the Fermi energy [17]. The temperature dependence of PLE spectra of sample B indicates that, near the 1D quantum limit, the FES is dominated by the last occupied subband. The 1D FES shows two main differences with respect to the 2D case: (a> The singularity is sharper and stronger in 1D and (b) it is also clearly observed in the emission spectra. These differences can be understood if one considers that the PLE threshold for a 2D sample is an indirect transition from the top of the valence band to the Fermi level, together with a zero-energy excitation of the Fermi sea required for momentum conservation. For high electron concentrations the energy difference between this indirect transition and the direct one at the Fermi vector k, ((m,/m,)E,) is larger than the characteristic Coulomb energy and no FES appears [18]. At low electron densities the FES is weak but observable as shown in fig. 4, even in the emission spectra, for the temperature range where there is a noticeable population of holes at k, and yet the Fermi distribution is not smeared out 18,121. In 1D systems there are no zero-energy excitations of the Fermi sea other than k = 0 or k = 2k,, so that indirect transitions cannot take place and the PLE onset is a vertical transition at k, [S]. This means that FES will be present in 1D systems at any electron concentration and for finite hole mass as shown by our results. A sharp peak at the Fermi level has been observed in PL spectra of 2D electron gases [12,14] in ternary systems. It is due to disorder-induced hole localization. In the present case the same weak FES is observed in 2D samples with low electron concentration, both for as-grown or ion bombarded samples. Therefore ion bombardment does not produce any noticeable hole localization in our case, and defects can be excluded as the origin of the sharp FES observed here. A deeper understanding of our experimental results is obtained by a calculation of the absorption spectra of a 1D electron gas as a function of the electron density. This has been done by numerical diagonalization of the Hamihonian for a small array of atoms containing a few electrons
349
and a single hole coupled by an on-site Coulomb interaction 1171.The results are described in detail in ref. [El and show that the absorption onset of the 1D system has indeed a strong FES even for high electron concentrations and a hole mass equal to the electron one. This striking result is in sharp contrast with the 2D case, where the absorption onset is smooth at high electron concentrations. It shows that the hole recoil plays only a minor role in the many-body reaction of the Fermi sea during the absorption of light in 1D systems, in agreement with out experimental results. In conclusion, we observe new optical properties of the 1D electron gas in and near the quantum limit. FES much stronger than in 2D systems are observed, whose temperature dependence is determined by the Fermi energy of the last occupied subband. The increased strength of the FES in lD, its dominance in the optical emission spectra and its persistence at high electron concentrations can be understood by the lack of certain hole recoil effects in 1D.
Acknowledgment
One of the authors (J.M.C.) is indebted to the Spanish CICYT and AT&T Microelectronics of Spain for financial support.
References Physics and Fabrication, Proceedings of [ll Nanostructure the International Symposium, Texas, USA, March 1989, Eds. M.A. Reed and W.P. Kirk (Academic Press, London, 1989). 121W. Hansen, M. Horst, J.P. Kotthaus, U. Merkt, Ch. Sikorski and K. Ploog, Phys. Rev. Lett. 58 (1987) 2586. and HI. [31 K. Ismail, W. Chu, A. Yen, D.A. Antoniadis Smith, Appl. Phys. Lett. 54 (1988) 460. [41 T.P. Smith III, H. Arnot, J.M. Hong, C.M. Knoedler, S.E. Laux and H. Schmid, Phys. Rev. Lett. 59 (1987) 2802. P. Grambow and K. Ploog, 151 T. Demel, D. Heitmann, Phys. Rev. B 38 (1988) 12732. L.N. 161 J.S. Weiner, G. Danan, A. Pinczuk, J. Valladares, Pfeiffer and K. West, Phys. Rev. Lett. 63 (1989) 1641. G. Weimann, T. Demel, D. I71 T. Egeler, G. Abstreiter, Heitmann, P. Grambow and W. Schlapp, Phys. Rev. Lett. 65 (1990) 1804.
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