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Photoluminescence and excitation spectroscopy in coupled GaAs-Ga(Al)As quantum wells
498
Surface Science 142 (1984) 498-503 North-Holland. Amsterdam
PHOTOLUMINESCENCE COUPLED GaAs-Ga(Al)As C.DELALANDE, Groupe de Phynque Cedex 05, France
A.C. GOSSARD Bell Lahoralor~es, Received
AND EXCITATION SPECTROSCOPY QUANTUM WELLS
U.O. ZIEMELIS,
G. BASTARD
des Solides de I’Ecole Normale
Supkieure
IN
and M. VOOS *. 24 Rue Lhomond,
F-75231
Parts
and W. WIEGMANN
Murruy
Hill, New Jerse_y 07974,
3 July 1983; accepted
for publication
USA
6 September
1983
We address the subject of virtual bound states (resonances) m a coupled quantum well, GaAs-Ga(Al)As structure grown by molecular beam epitaxy. The behaviour of bound hole states and hole resonances is calculated in the envelope function approach for a double well system with well (GaAs) widths ranging from 0 to 200 A and a fixed barrier (Ga, s,Al, ,,,As) width of 12 A. taking into account band non-parabolicity and the spin-orbit energies of the host materials. Experimental results based on excitation spectroscopy measurements and pertaining to the existence of a transition involving a virtual bound hole state are presented for a system consisting of 40 periods of two 45 A GaAs wells separated by 12 A Ga, ASe ,hAs barriers. We also discuss briefly the evolution of the photoluminescence and excitation spectra in the 2 to 40 K temperature range. K4
The development of techniques such as molecular beam epitaxy has made it possible to grow single crystal structures approaching atomic dimensions and numerous studies of quantum effects in these systems have been carried out [l]. In particular, Dingle et al. [2] have conducted optical absorption measurements in coupled multi-well GaAs-Ga(Al)As structures, which indicate splitting of single well bound states due to inter-well coupling. We report the first observation of this phenomenon based on excitation spectroscopy measurements in a GaAs-Ga(Al)As double well sample (see fig. la). We also present the first calculations of the level structure associated with a coupled well system based on the envelope function approach, which include in a natural way band non-parabolicity and various boundary condition intricacies at the * Laboratoire
associe au CNRS
0039-6028/84/$03.00 0 Elsevier Science Publishers (North-Holland Physics Publishing Division)
B.V.
C. Delalande
et al. /
Photoluminescence and excitation spectroscopy
499
interface. Our calculations indicate that one of the four peaks observed in the excitation spectrum is due to an optical transition between a bound electron state and a light hole virtual bound state. In the envelope function approach [3,4], for zero wavevector in the layer plane, the envelope function!, associated with the S periodic part of the Bloch function is the solution of a non-parabolic hamiltonian of the Kane type, in which the r,, r, and r, edges shift at the GaAs-Ga, _ YAl,YAs interfaces by the amounts V,, VP and V, respectively. Following Dingle [l], we have used Vs = 1060x and VP = Vs = - 197x. Bound electron (e) and hole (h) states occur for energies C, < Vs and E,, < VP respectively. Since we are dealing with identical wells (of width L,), the bound states are odd or even with respect to the midpoint of the centrally located barrier (of width h). For energies corresponding to the continuum spectra (6, > V, or zr, > V,) one searches for transmission resonances. The transmission coefficient of a double well structure is given by: 2 cos k,L,
cos k,h
-1
sin k,L, where
k,
and k,
sin k,h
are the wavevectors
, and 5 is the ratio
45A
CB
12b
of the probability
45B
AS
s
w
SUBSTRATE
1 GaAs
a
2 G%,Al, As
1 “J&!L
b
HEAVY HOLES
LIGHT HOLES
Fig. 1. (a) Schematic illustration of the sample structure. The 236 A thick barriers which separate individual double well periods are sufficiently thick to prohibit inter-period tunneling. Details of sample growth are presented in Ref. [5]. (b) Schematic representation of calculated level structure and allowed transitions for ground state (n = 1) electrons and holes. The transition labels A, B, C and D refer to the excitation spectrum peak assignments (see fig. 3). Symmetric and antisymmetric levels are labelled S and AS respectively. All hole levels continue across both wells - they are separated here simply for the same of clarity. The hole resonance is indicated by a dashed line.
500
C. Delalande et al. / Photoluminescence
and excltatmn
spectroscopy
currents in the well and the barrier. 5. k, and k, are easily deduced from the dispersion relations of the host materials. Resonances (twice degenerate) occur for: k, L, = pn,
p
integer,
coskuL,cosk,,h-i As k, + 0 resonances of the first kind converge towards the even bound states at the same energy (6, = I$, c,, = I’,), whereas resonances of the second kind converge in the same limit towards the odd bound states. Note that resonances (3) would correspond to a superlattice state with wavevector 7r/2( L, + h) if the (L,, h) basis was infinitely repeated. On the other hand, the condition k, L, = pr is that obtained for a single well structure. Each reasonance can be described as a virtual bound state corresponding to an accumulation of density probability in the double well structure. The present analysis shows that the double well structure exhibits virtual bound states either clamped on a single well or delocalized over the whole double well. The calculations predict, for the structure under investigation (L, = 45 A, h = 12 A, x = 0.16), that two electronic levels and two heavy hole levels are bound, however only a single light hole level (the symmetric one) is predicted to occur within the valence band barrier (V, = 29.9 meV). The light hole resonance, which occurs in the continuum at 4 meV from V, is calculated to be relatively narrow (5 meV). This resonant level is the continuation of the antisymmetric (AS,) light hole bound level after it has merged with the valence continuum. As the well width is increased (x and h being fixed), AS, becomes bound for L, > 50 A. Fig. 2 shows the predicted behaviour of the bound and virtual bound hole states as a function of L, for the system of interest (h = 12 A, x = 0.16). The calculated level structure and the allowed transitions for ground state (n = 1) electrons and holes are shown in fig. lb. Since the conduction and valence band edges have opposite parities, electric dipole optical transitions are allowed only between states with envelope functions having the same symmetries with respect to the midpoint of the central barrier. Fig. 3 shows the observed excitation and photoluminescence spectra at 2 and 40 K. A dye (LD) 700 laser, pumped by the all-lines (red) output of a cw Kr+ laser, was used as a tunable excitation source. The photoluminescence was analysed with a l/4 m monochromator and detected by a cooled photomultiplier (SI photocathode) using conventional lock-in techniques. The calculated transition energies agree well with the energies of the excitation spectrum peaks and allow the immediate assignment of a specific transition to each peak (see figs. lb and 3). Peaks A, B and C correspond to transitions between bound electron and bound hole states. Peak D is assigned to an optical transition between a light hole virtual bound state (resonance) and the asymmetric
C.
Delalande
et al.
/
Photoluminescence
and
excitation
spectroscopy
501
GaAs-Ga(AL)As X ~0.16
0
50
150
L, Jo
Fig. 2.. Calculated energies for light hole bound and resonance states as a function of well thickness in the double well GaAs-Ga,,s4Al,,,, As system with h =12 A. The hatched horizontal line indicates the valence continuum edge. Bound states are labelled according to symmetry: S, symmetric; AS, antisymmetric; subscripts indicate n values. Resonance states are labelled according to type: SL, superlattice (eq. (3)) or pv, “single well” (eq. (2)).
electron state. We have varied the L, and x parameters in the theoretical calculations over ranges greater than the growth error ranges of L, (45 + 2.5 A) and x (0.16 f 0.02) and we find that the light hole level of interest remains unbound. The peak assignments for the excitation spectrum were verified by circular polarization measurements of the photoluminescence signal (fixed energy) generated by circularly polarized excitation (scanned in energy) [6,7]. At 2 K, the photoluminescence associated with the coupled wells consists of a single peak (a - full width at half maximum: 16 meV) shifted by 13 meV from the lowest energy peak (A) of the excitation spectrum. As the sample temperature is increased from 2 to 40 K, a second line (p) appears in the photoluminescence spectrum on the high energy side of CX,separated from the latter by approximately 10 meV. p dominates (Y in intensity for temperatures greater than about 30 K. Over the range of temperatures studied, the excitation spectra associated with peaks LYand j3 are identical, confirming that both peaks are associated with the double well structure. Circular polarization measure-
a /\
A II
GaAs-Ga 0.*4A'0.16AS DOUBLE WELLS L,=45i
h=12/i
2K
1630
PHOTON
ENERGY
1730
(meV)
Fig. 3. Excitation full line and photoluminescence (broken line) spectra of the two-coupled-well sample at 2 and 40 K. The monochromator set energies were 1565 and 1555 meV for the 2 and 40 K excitation spectra respectively. The sharp rise in intensity below peak A is due to the coincidence of the dye laser and monochromator set energies. Theoretically predicted peak positions are indicated by arrows; calculated peak energies are given in meV. Both photoluminescence spectra were excited with 1630 meV light.
ments of the photoluminescence indicate that both (Y and /I arise from the recombination of electrons with heavy holes. The behaviour of the photoluminescence with increasing sample temperature suggests that one or both of the recombining states responsible for a can be associated with a shallow trap; the p luminescence is then interpreted as due to the recombination of free excitons. Under this hypothesis, the appearance of /? is due to the thermal de-trapping of the trapped species. Work is continuing to test this hypothesis and identify the trap or binding centre associated with the (Yluminescence.
References (Advances in Solid State [l] R. Dingle, in: Festkijrperprobleme Queisser (Pergamon/Vieweg, Braunschweig, 1975) p. 21. [2] R. Dingle, A.C. Gossard and W. Wiegmann, Phys. Rev. Letters.
Physics),
Vol.
34 (1975) 1327.
15. Ed.
H.J.
C. Delalande et al. / Photoluminescence [3] [4] [5] [6] [7]
und excitntron spectroscopy
503
S. White and L. Sham, Phys. Rev. Letters 47 (1981) 879. G. Bastard, Phys. Rev. B24 (1981) 5693. R. Dingle, W. Wiegmann and C.H. Henry, Phys. Rev. Letters 33 (1974) 827. R.C. Miller, D.A. Kleinman, W.A. Norland, Jr. and A.C. Gossard, Phys. Rev. B22 (1980) 863. C. Weisbuch, R.C. Miller, R. Dingle, A.C. Gossard and W. Wiegmann. Solid State Commun. 37 (1981) 219.
Surface Science 142 (1984) 498-503 North-Holland. Amsterdam
PHOTOLUMINESCENCE COUPLED GaAs-Ga(Al)As C.DELALANDE, Groupe de Phynque Cedex 05, France
A.C. GOSSARD Bell Lahoralor~es, Received
AND EXCITATION SPECTROSCOPY QUANTUM WELLS
U.O. ZIEMELIS,
G. BASTARD
des Solides de I’Ecole Normale
Supkieure
IN
and M. VOOS *. 24 Rue Lhomond,
F-75231
Parts
and W. WIEGMANN
Murruy
Hill, New Jerse_y 07974,
3 July 1983; accepted
for publication
USA
6 September
1983
We address the subject of virtual bound states (resonances) m a coupled quantum well, GaAs-Ga(Al)As structure grown by molecular beam epitaxy. The behaviour of bound hole states and hole resonances is calculated in the envelope function approach for a double well system with well (GaAs) widths ranging from 0 to 200 A and a fixed barrier (Ga, s,Al, ,,,As) width of 12 A. taking into account band non-parabolicity and the spin-orbit energies of the host materials. Experimental results based on excitation spectroscopy measurements and pertaining to the existence of a transition involving a virtual bound hole state are presented for a system consisting of 40 periods of two 45 A GaAs wells separated by 12 A Ga, ASe ,hAs barriers. We also discuss briefly the evolution of the photoluminescence and excitation spectra in the 2 to 40 K temperature range. K4
The development of techniques such as molecular beam epitaxy has made it possible to grow single crystal structures approaching atomic dimensions and numerous studies of quantum effects in these systems have been carried out [l]. In particular, Dingle et al. [2] have conducted optical absorption measurements in coupled multi-well GaAs-Ga(Al)As structures, which indicate splitting of single well bound states due to inter-well coupling. We report the first observation of this phenomenon based on excitation spectroscopy measurements in a GaAs-Ga(Al)As double well sample (see fig. la). We also present the first calculations of the level structure associated with a coupled well system based on the envelope function approach, which include in a natural way band non-parabolicity and various boundary condition intricacies at the * Laboratoire
associe au CNRS
0039-6028/84/$03.00 0 Elsevier Science Publishers (North-Holland Physics Publishing Division)
B.V.
C. Delalande
et al. /
Photoluminescence and excitation spectroscopy
499
interface. Our calculations indicate that one of the four peaks observed in the excitation spectrum is due to an optical transition between a bound electron state and a light hole virtual bound state. In the envelope function approach [3,4], for zero wavevector in the layer plane, the envelope function!, associated with the S periodic part of the Bloch function is the solution of a non-parabolic hamiltonian of the Kane type, in which the r,, r, and r, edges shift at the GaAs-Ga, _ YAl,YAs interfaces by the amounts V,, VP and V, respectively. Following Dingle [l], we have used Vs = 1060x and VP = Vs = - 197x. Bound electron (e) and hole (h) states occur for energies C, < Vs and E,, < VP respectively. Since we are dealing with identical wells (of width L,), the bound states are odd or even with respect to the midpoint of the centrally located barrier (of width h). For energies corresponding to the continuum spectra (6, > V, or zr, > V,) one searches for transmission resonances. The transmission coefficient of a double well structure is given by: 2 cos k,L,
cos k,h
-1
sin k,L, where
k,
and k,
sin k,h
are the wavevectors
, and 5 is the ratio
45A
CB
12b
of the probability
45B
AS
s
w
SUBSTRATE
1 GaAs
a
2 G%,Al, As
1 “J&!L
b
HEAVY HOLES
LIGHT HOLES
Fig. 1. (a) Schematic illustration of the sample structure. The 236 A thick barriers which separate individual double well periods are sufficiently thick to prohibit inter-period tunneling. Details of sample growth are presented in Ref. [5]. (b) Schematic representation of calculated level structure and allowed transitions for ground state (n = 1) electrons and holes. The transition labels A, B, C and D refer to the excitation spectrum peak assignments (see fig. 3). Symmetric and antisymmetric levels are labelled S and AS respectively. All hole levels continue across both wells - they are separated here simply for the same of clarity. The hole resonance is indicated by a dashed line.
500
C. Delalande et al. / Photoluminescence
and excltatmn
spectroscopy
currents in the well and the barrier. 5. k, and k, are easily deduced from the dispersion relations of the host materials. Resonances (twice degenerate) occur for: k, L, = pn,
p
integer,
coskuL,cosk,,h-i As k, + 0 resonances of the first kind converge towards the even bound states at the same energy (6, = I$, c,, = I’,), whereas resonances of the second kind converge in the same limit towards the odd bound states. Note that resonances (3) would correspond to a superlattice state with wavevector 7r/2( L, + h) if the (L,, h) basis was infinitely repeated. On the other hand, the condition k, L, = pr is that obtained for a single well structure. Each reasonance can be described as a virtual bound state corresponding to an accumulation of density probability in the double well structure. The present analysis shows that the double well structure exhibits virtual bound states either clamped on a single well or delocalized over the whole double well. The calculations predict, for the structure under investigation (L, = 45 A, h = 12 A, x = 0.16), that two electronic levels and two heavy hole levels are bound, however only a single light hole level (the symmetric one) is predicted to occur within the valence band barrier (V, = 29.9 meV). The light hole resonance, which occurs in the continuum at 4 meV from V, is calculated to be relatively narrow (5 meV). This resonant level is the continuation of the antisymmetric (AS,) light hole bound level after it has merged with the valence continuum. As the well width is increased (x and h being fixed), AS, becomes bound for L, > 50 A. Fig. 2 shows the predicted behaviour of the bound and virtual bound hole states as a function of L, for the system of interest (h = 12 A, x = 0.16). The calculated level structure and the allowed transitions for ground state (n = 1) electrons and holes are shown in fig. lb. Since the conduction and valence band edges have opposite parities, electric dipole optical transitions are allowed only between states with envelope functions having the same symmetries with respect to the midpoint of the central barrier. Fig. 3 shows the observed excitation and photoluminescence spectra at 2 and 40 K. A dye (LD) 700 laser, pumped by the all-lines (red) output of a cw Kr+ laser, was used as a tunable excitation source. The photoluminescence was analysed with a l/4 m monochromator and detected by a cooled photomultiplier (SI photocathode) using conventional lock-in techniques. The calculated transition energies agree well with the energies of the excitation spectrum peaks and allow the immediate assignment of a specific transition to each peak (see figs. lb and 3). Peaks A, B and C correspond to transitions between bound electron and bound hole states. Peak D is assigned to an optical transition between a light hole virtual bound state (resonance) and the asymmetric
C.
Delalande
et al.
/
Photoluminescence
and
excitation
spectroscopy
501
GaAs-Ga(AL)As X ~0.16
0
50
150
L, Jo
Fig. 2.. Calculated energies for light hole bound and resonance states as a function of well thickness in the double well GaAs-Ga,,s4Al,,,, As system with h =12 A. The hatched horizontal line indicates the valence continuum edge. Bound states are labelled according to symmetry: S, symmetric; AS, antisymmetric; subscripts indicate n values. Resonance states are labelled according to type: SL, superlattice (eq. (3)) or pv, “single well” (eq. (2)).
electron state. We have varied the L, and x parameters in the theoretical calculations over ranges greater than the growth error ranges of L, (45 + 2.5 A) and x (0.16 f 0.02) and we find that the light hole level of interest remains unbound. The peak assignments for the excitation spectrum were verified by circular polarization measurements of the photoluminescence signal (fixed energy) generated by circularly polarized excitation (scanned in energy) [6,7]. At 2 K, the photoluminescence associated with the coupled wells consists of a single peak (a - full width at half maximum: 16 meV) shifted by 13 meV from the lowest energy peak (A) of the excitation spectrum. As the sample temperature is increased from 2 to 40 K, a second line (p) appears in the photoluminescence spectrum on the high energy side of CX,separated from the latter by approximately 10 meV. p dominates (Y in intensity for temperatures greater than about 30 K. Over the range of temperatures studied, the excitation spectra associated with peaks LYand j3 are identical, confirming that both peaks are associated with the double well structure. Circular polarization measure-
a /\
A II
GaAs-Ga 0.*4A'0.16AS DOUBLE WELLS L,=45i
h=12/i
2K
1630
PHOTON
ENERGY
1730
(meV)
Fig. 3. Excitation full line and photoluminescence (broken line) spectra of the two-coupled-well sample at 2 and 40 K. The monochromator set energies were 1565 and 1555 meV for the 2 and 40 K excitation spectra respectively. The sharp rise in intensity below peak A is due to the coincidence of the dye laser and monochromator set energies. Theoretically predicted peak positions are indicated by arrows; calculated peak energies are given in meV. Both photoluminescence spectra were excited with 1630 meV light.
ments of the photoluminescence indicate that both (Y and /I arise from the recombination of electrons with heavy holes. The behaviour of the photoluminescence with increasing sample temperature suggests that one or both of the recombining states responsible for a can be associated with a shallow trap; the p luminescence is then interpreted as due to the recombination of free excitons. Under this hypothesis, the appearance of /? is due to the thermal de-trapping of the trapped species. Work is continuing to test this hypothesis and identify the trap or binding centre associated with the (Yluminescence.
References (Advances in Solid State [l] R. Dingle, in: Festkijrperprobleme Queisser (Pergamon/Vieweg, Braunschweig, 1975) p. 21. [2] R. Dingle, A.C. Gossard and W. Wiegmann, Phys. Rev. Letters.
Physics),
Vol.
34 (1975) 1327.
15. Ed.
H.J.
C. Delalande et al. / Photoluminescence [3] [4] [5] [6] [7]
und excitntron spectroscopy
503
S. White and L. Sham, Phys. Rev. Letters 47 (1981) 879. G. Bastard, Phys. Rev. B24 (1981) 5693. R. Dingle, W. Wiegmann and C.H. Henry, Phys. Rev. Letters 33 (1974) 827. R.C. Miller, D.A. Kleinman, W.A. Norland, Jr. and A.C. Gossard, Phys. Rev. B22 (1980) 863. C. Weisbuch, R.C. Miller, R. Dingle, A.C. Gossard and W. Wiegmann. Solid State Commun. 37 (1981) 219.