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Magneto-optical studies of two-dimensional electrons in MQW heterostructures
486
Surface
Science 142 (1984) 486-491 North-Holland. Amsterdam
MAGNETO-OPTICAL STUDIES OF TWO-DIMENSIONAL IN MQW HETEROSTRUCTURES J.M. WORLOCK ‘, A.C. MACIEL b.cd, A. PETROU R.L. AGGARWAL ‘.‘, M. SMITH h.c, A.C. GOSSARD and W. WIEGMANN ’ Received
15 July 1983; accepted
for publication
11 August
ELECTRONS
h.c, C.H. ’
PERRY
h.c,
1983
The two topics covered in this report are (1) determination of the cross section for Landau Level Raman scattering and a new explanation for this “forbidden” scattering; and (2) new spectra of photoluminescence by MQW electrons in magnetic fields, showing behavior related to electron exchange energy, and competition for dominance in the valence band between confinement and magnetic field effects.
In this report, we discuss two aspects of our studies of two-dimensional (2D) electrons in multiquantum well (MQW) heterostructures. We have performed magneto-Raman and photoluminescence experiments on MQW samples of GaAs-AlGaAs grown by molecular beam epitaxy using the technique of modulation doping to produce clean, well defined 2D electron layers [l]. In the first part, we present the results of measurements of absolute cross sections for Raman scattering by both intersubband (IS) and cyclotron resonance (CR) modes, and compare these with a new theory for interband magneto optical transitions which we believe explains the forbidden CR scattering. In the second part, we show spectra of photoluminescence and their variation with magnetic field, and discuss these qualitatively in terms of many body effects in the electron system, and competition between quantum-well confinement and the magnetic field in determining the valence band states.
il Bell Laboratories, Holmdel, New Jersey 07733, USA h Physics Department, Northeastern University, Boston. Massachusetts 02139. USA. ’ National Magnet Laboratory, Massachusetts Institute of Technology. Cambridge. Massachusetts 02139, USA. d Present address: Clarendon Laboratory. Oxford University. Oxford. UK. ’ Physics Department, Massuchusetts Institute of Technology. Cambridge, Massachusetts 02139. USA ’ Bell Laboratories, Murray Hill, New Jersey 07974, USA.
0039-6028/84/$03.00 0 Elsevier Science Publishers (North-Holland Physics Publishing Division)
B.V.
J. M. Warlock et al. / Magneto-optical studies of 20 electrons
487
Since our first observation of Landau level (or CR) Raman scattering [2], we have searched for an explanation for this forbidden scattering, while continuing to make experiments designed to limit the theoretical search. In this quest, we have measured the absolute cross sections, at resonance, for both IS and CR modes [3,4]. These were obtained by comparison with the known cross section for optical phonons in silicon (51. We concentrated on a high mobility sample, p = 90,000 cm2/V. s with n, = 5 x 10” electrons/cm2. Working at 2 K and 8 T, we determined the following cross sections (cm2 per electron): IS(E,, transition), - 1.5 x 10e2’; and CR, - 1.0 x 10p2’. We have shown [4] that the maximum theoretical cross section (da/dfi),,, for an electron in GaAs, assuming all envelope function overlap matrix elements are unity, is 1.0 x lO_” cm2, using a resonant energy denominator of 10 meV, which is consistent with our resonance measurements. The cross section for CR scattering is thus about 1% of this theoretical maximum, so we cannot tolerate any really small overlap matrix elements. The crucial factor [6] in the matrix element is Z,?.m = CLb>lL,b
+ 124)>,
where f,, and f,, are the Landau oscillator wave functions in conduction and valence band; I2 is the square of the magnetic length, I2 = hc/eH; and q is the component of optical wave vector perpendicular to H. The “forbiddenness” of first order CR scattering comes from the fact that lim,,,, Z,,, = a,.,, i.e., interband optical transitions preserve Landau level index, in the limit of small 4. In our experiments, for the scattered beam, ql is very small, but for the incident laser beam, at 8 T, ql z 0.091, which gives ]Z,,l]2 = 0.004 and ]Z,,,12 = 0.008, when the overlap integral I,,, is evaluated [6] for finite lq. These are a bit small, but the right order of magnitude to explain our measured cross sections. We note that I becomes larger at lower fields, which helps to explain why our cross sections are larger at small fields, in spite of the fact that many electrons hide in the lower Landau levels, unavailable for promotion, and hence do not participate in CR scattering Now we turn our attention to the second half of our paper. MQW samples have long been studied by luminescence, and the literature is far too vast to be reviewed here. Our contribution is to subject these samples to sizable magnetic fields. Our motiviation was to study electron structure at extremely high fields and low temperatures in the hope of contributing to the understanding of the quantum Hall effect. As we shall see, we are still some distance from this goal, but we believe we can show already some interesting phenomena. Fig. la shows spectra of one MQW sample, with a rather high electron concentration n, - 6 X 10” electrons/cm2 in each quantum well of thickness 250 A. The incident illumination is - 30 mW of 5145 A laser light (- 100 W/cm2), linearly polarized, and the crystal is held at 2 K. This is the same MQW sample analyzed by Pinczuk et al. at this conference [7].
The zero field spectrum displays roughly the shape we would expect from the built-in electrons recombining with photo-excited holes: a low energy onset somewhere near the bandgap E,; and an exponential thermal tail beyond the energy E,; + E,. Analysis of the high energy tail gives us an electron temperature of - 35 K. In the presence of a magnetic field. the electron continuum breaks into Landau levels, and we see peaks in our recombination spectrum (a textbook example) corresponding to transitions from electron Landau levels n = 0, 1, 2, . . to some valence band states, occupied according to Boltzmann statistics, by the photo-injected holes. We note also that the luminescent efficiency increases, and that the broadening is reduced as field is increased. The zero field HWHM value of 3 meV reduces to 0.7-0.8 meV at the extreme field of 1.5 T shown in fig. lb. At 15 T. for this sample only one Landau level is occupied,
I
’
PHOTOLUMINESCENT SAMPLE 7% = 6.5 p
1.500
I
I
SPECTRUM
OF MQW
HETEROSTRUCTURE
9-25-6013 x IO+’ cni’
= 93,000
Cd/V
1.510 PHOTON
see
1.520 ENERGY
(ELECTRON
1.530
1.540
VOLTS)
Fig. 1. Photoluminescent spectra of 2D electrons m MQW heterostructure. (a) The electron continuum (H = 0) breaks into Landau levels (H = 2 T) with increase m luminescent efficiency. Characteristic energies are shown. (b) At the highest fields (15 T) luminescence from the lowest Landau level has two circularly polarized components, somewhat narrower than the low-field peaks.
J. M. Warlock et al. / Magneto-optical I
I
I
I
PHOTOLUMINESCENT HETEROSTRUCTURE MAGNETIC FIELD
SAMPLE H = 15T rjs=
I
1
PHOTON
d
=250A
ENERGY
I
MQW
6.5 xlo”cm-2
= 93,ocJo
I
489
9 - 25 - 8013
p
1.524
1.522
1
I
SPECTRUM
studws of 20 electrons
I
cm2/vsec
I
I
(ELECTRON
I 1.528
1.526 VOLTS)
though both spin components are present. The spectra are now circularly polarized and the single (O-O) transition has split into two components. Fig. 2 shows how the first few spectral peaks develop with H. Roughly, but very roughly, they increase in energy as (n + +) AU, where (w,/H z 1.7 meV/T) is the cyclotron frequency for an electron in GaAs, and n is the Landau level index. If the valence band were uncomplicated, and if the electrons’ energies were not affected by exchange, we would expect the recombination peaks to have a common zero field origin at EG + Econfinemen,[7] and then to increase as (n + f)(ttw, + AU,), with w, the hole cyclotron frequency. The only measurement of cyclotron resonance in two-dimensionally confined holes [8] gives a hole mass on the order of 0.4 m,, implying w, - a wC. The expected slopes of the transitions shown would be, in units of AU,; 0.58, 1.74, 2.90, and 4.06. In contrast, the measured slopes are 1.07, 2.10, 3.27, and 4.20. Beyond a few tesla the slopes do decrease to more respectable values, and eventually the single lines split into two polirized components. The following factors will be important in analyzing our spectra. (1) Electron exchange energy must change with field. Indeed oscillations in exchange energy on the order of several meV are expected as the Fermi level passes through Landau levels [9]. (2) The valence band is, in fact, four-fold
490
J. M. Warlock et al. / Magneto-optical
studies of 20 electrons
degenerate, and two effects are fighting for dominance in breaking this degeneracy: (a) confinement in the quantum wells, which gives hole level spacings on the order of a few meV; and (b) the magnetic field, which alone
I I I I I ENERGIES OF PHOTOLUMINESCENT PEAKS HETEROSTRUCTURE SAMPLE 9-25-6013
0
0
0
I
I MOW
0
0 0
0
O 0
0
0
1.530 l-II0 (3-3)
0
(2;2’
0
0 0
0 0
0
0
0 0
0
O
0
0 0
1 ,520
0
0
0
(0-O)
0
i
E3D G
0
0
0
0
0
%
d
0
0
0
0
I.510 2
0
4
6
6
MAGNETIC Fig.
2. Energies
curvature
towards
of photoluminescent lower slopes,
peaks
and splitting
FIELD versus
10
12
14
(TESLA) magnetic
into polarized
field
doublets.
showing
large
initial
slopes.
J.M.
Warlock et al. / Magneto-optical
studres of 20 electrons
491
gives the complicated but well honored energies derived by Luttinger [lo]. We believe that at high magnetic fields, the holes will finally simplify and follow Luttinger. An indication that this happens is that the luminescence becomes split and circularly polarized. When the hole states are finally understood, it will be possible to use the recombination energy to study simply the electronic behavior in high magnetic fields. We thank P.A. Wolff, A. Pinczuk, and J. Shah for helpful discussions; and L. Rubin and the staff of the National Magnet Laboratory for cooperation and hospitality. The National Magnet Laboratory is supported by the National Science Foundation. This work was also supported in part by the Office of Naval Research, under contract N-00014-81-k-651, and the National Science Foundation, under grant DMR 8121702.
References [l] A.C. Gossard, in: Thin Films: Preparation and Properties, Eds. K.N. Tu and R. Rosenberg (Academic Press, 1981). [2] J.M. Warlock A. Pinczuk, Z.J. Tien, C.H. Perry, H.L. Stormer, R. Dingle, A.C. Gossard, W. Wiegmann and R.L. Aggarwal, Solid State Commun. 40 (1981) 867; Z.J. Tien, PhD Thesis, Northeastern University (1981). [3] J.M. Warlock, A.C. Maciel, C.H. Perry, Z.J. Tien, R.L. Aggarwal, A.C. Gossard and W. Wiegmann, in: Application of High Magnetic Fields in Semiconductor Physics, Ed. G. Landwehr (Springer, Berlin, 1983) p. 186. [4] A.C. Maciel, J.M. Warlock, C.H. Perry, R.L. Aggarwal, A.C. Gossard and W. Wiegmann, Bull. Am. Phys. Sot. 28 (1983) 448; and to be published. [S] M. Cardona, M.H. Grimsditch, and D. Olego, in: Light Scattering in Solids, Eds. J.L. Birman, H.Z. Cummins and K.K. Rebane (Plenum, 1979) p. 249. [6] J.M. Warlock, Solid State Commun. 48 (1983) 1067. [7] A. Pinczuk, J. Shah, H.L. Stormer, R.C. Miller A.C. Gossard and W. Wiegmann, Surface Sci. 142 (1984) 492. [8] H.L. Stormer, Z. Schlesinger, A. Chang, D.C. Tsui, A.C. Gossard and W. Wiegmann, Phys. Rev. letters 51 (1983) 126. [9] Th. Englert, D.C. Tsui, A.C. Gossard and Ch. Uihlein, Surface Sci. 113 (1983) 295. [lo] J.M. Luttinger, Phys Rev. 102 (1956) 1030.
Surface
Science 142 (1984) 486-491 North-Holland. Amsterdam
MAGNETO-OPTICAL STUDIES OF TWO-DIMENSIONAL IN MQW HETEROSTRUCTURES J.M. WORLOCK ‘, A.C. MACIEL b.cd, A. PETROU R.L. AGGARWAL ‘.‘, M. SMITH h.c, A.C. GOSSARD and W. WIEGMANN ’ Received
15 July 1983; accepted
for publication
11 August
ELECTRONS
h.c, C.H. ’
PERRY
h.c,
1983
The two topics covered in this report are (1) determination of the cross section for Landau Level Raman scattering and a new explanation for this “forbidden” scattering; and (2) new spectra of photoluminescence by MQW electrons in magnetic fields, showing behavior related to electron exchange energy, and competition for dominance in the valence band between confinement and magnetic field effects.
In this report, we discuss two aspects of our studies of two-dimensional (2D) electrons in multiquantum well (MQW) heterostructures. We have performed magneto-Raman and photoluminescence experiments on MQW samples of GaAs-AlGaAs grown by molecular beam epitaxy using the technique of modulation doping to produce clean, well defined 2D electron layers [l]. In the first part, we present the results of measurements of absolute cross sections for Raman scattering by both intersubband (IS) and cyclotron resonance (CR) modes, and compare these with a new theory for interband magneto optical transitions which we believe explains the forbidden CR scattering. In the second part, we show spectra of photoluminescence and their variation with magnetic field, and discuss these qualitatively in terms of many body effects in the electron system, and competition between quantum-well confinement and the magnetic field in determining the valence band states.
il Bell Laboratories, Holmdel, New Jersey 07733, USA h Physics Department, Northeastern University, Boston. Massachusetts 02139. USA. ’ National Magnet Laboratory, Massachusetts Institute of Technology. Cambridge. Massachusetts 02139, USA. d Present address: Clarendon Laboratory. Oxford University. Oxford. UK. ’ Physics Department, Massuchusetts Institute of Technology. Cambridge, Massachusetts 02139. USA ’ Bell Laboratories, Murray Hill, New Jersey 07974, USA.
0039-6028/84/$03.00 0 Elsevier Science Publishers (North-Holland Physics Publishing Division)
B.V.
J. M. Warlock et al. / Magneto-optical studies of 20 electrons
487
Since our first observation of Landau level (or CR) Raman scattering [2], we have searched for an explanation for this forbidden scattering, while continuing to make experiments designed to limit the theoretical search. In this quest, we have measured the absolute cross sections, at resonance, for both IS and CR modes [3,4]. These were obtained by comparison with the known cross section for optical phonons in silicon (51. We concentrated on a high mobility sample, p = 90,000 cm2/V. s with n, = 5 x 10” electrons/cm2. Working at 2 K and 8 T, we determined the following cross sections (cm2 per electron): IS(E,, transition), - 1.5 x 10e2’; and CR, - 1.0 x 10p2’. We have shown [4] that the maximum theoretical cross section (da/dfi),,, for an electron in GaAs, assuming all envelope function overlap matrix elements are unity, is 1.0 x lO_” cm2, using a resonant energy denominator of 10 meV, which is consistent with our resonance measurements. The cross section for CR scattering is thus about 1% of this theoretical maximum, so we cannot tolerate any really small overlap matrix elements. The crucial factor [6] in the matrix element is Z,?.m = CLb>lL,b
+ 124)>,
where f,, and f,, are the Landau oscillator wave functions in conduction and valence band; I2 is the square of the magnetic length, I2 = hc/eH; and q is the component of optical wave vector perpendicular to H. The “forbiddenness” of first order CR scattering comes from the fact that lim,,,, Z,,, = a,.,, i.e., interband optical transitions preserve Landau level index, in the limit of small 4. In our experiments, for the scattered beam, ql is very small, but for the incident laser beam, at 8 T, ql z 0.091, which gives ]Z,,l]2 = 0.004 and ]Z,,,12 = 0.008, when the overlap integral I,,, is evaluated [6] for finite lq. These are a bit small, but the right order of magnitude to explain our measured cross sections. We note that I becomes larger at lower fields, which helps to explain why our cross sections are larger at small fields, in spite of the fact that many electrons hide in the lower Landau levels, unavailable for promotion, and hence do not participate in CR scattering Now we turn our attention to the second half of our paper. MQW samples have long been studied by luminescence, and the literature is far too vast to be reviewed here. Our contribution is to subject these samples to sizable magnetic fields. Our motiviation was to study electron structure at extremely high fields and low temperatures in the hope of contributing to the understanding of the quantum Hall effect. As we shall see, we are still some distance from this goal, but we believe we can show already some interesting phenomena. Fig. la shows spectra of one MQW sample, with a rather high electron concentration n, - 6 X 10” electrons/cm2 in each quantum well of thickness 250 A. The incident illumination is - 30 mW of 5145 A laser light (- 100 W/cm2), linearly polarized, and the crystal is held at 2 K. This is the same MQW sample analyzed by Pinczuk et al. at this conference [7].
The zero field spectrum displays roughly the shape we would expect from the built-in electrons recombining with photo-excited holes: a low energy onset somewhere near the bandgap E,; and an exponential thermal tail beyond the energy E,; + E,. Analysis of the high energy tail gives us an electron temperature of - 35 K. In the presence of a magnetic field. the electron continuum breaks into Landau levels, and we see peaks in our recombination spectrum (a textbook example) corresponding to transitions from electron Landau levels n = 0, 1, 2, . . to some valence band states, occupied according to Boltzmann statistics, by the photo-injected holes. We note also that the luminescent efficiency increases, and that the broadening is reduced as field is increased. The zero field HWHM value of 3 meV reduces to 0.7-0.8 meV at the extreme field of 1.5 T shown in fig. lb. At 15 T. for this sample only one Landau level is occupied,
I
’
PHOTOLUMINESCENT SAMPLE 7% = 6.5 p
1.500
I
I
SPECTRUM
OF MQW
HETEROSTRUCTURE
9-25-6013 x IO+’ cni’
= 93,000
Cd/V
1.510 PHOTON
see
1.520 ENERGY
(ELECTRON
1.530
1.540
VOLTS)
Fig. 1. Photoluminescent spectra of 2D electrons m MQW heterostructure. (a) The electron continuum (H = 0) breaks into Landau levels (H = 2 T) with increase m luminescent efficiency. Characteristic energies are shown. (b) At the highest fields (15 T) luminescence from the lowest Landau level has two circularly polarized components, somewhat narrower than the low-field peaks.
J. M. Warlock et al. / Magneto-optical I
I
I
I
PHOTOLUMINESCENT HETEROSTRUCTURE MAGNETIC FIELD
SAMPLE H = 15T rjs=
I
1
PHOTON
d
=250A
ENERGY
I
MQW
6.5 xlo”cm-2
= 93,ocJo
I
489
9 - 25 - 8013
p
1.524
1.522
1
I
SPECTRUM
studws of 20 electrons
I
cm2/vsec
I
I
(ELECTRON
I 1.528
1.526 VOLTS)
though both spin components are present. The spectra are now circularly polarized and the single (O-O) transition has split into two components. Fig. 2 shows how the first few spectral peaks develop with H. Roughly, but very roughly, they increase in energy as (n + +) AU, where (w,/H z 1.7 meV/T) is the cyclotron frequency for an electron in GaAs, and n is the Landau level index. If the valence band were uncomplicated, and if the electrons’ energies were not affected by exchange, we would expect the recombination peaks to have a common zero field origin at EG + Econfinemen,[7] and then to increase as (n + f)(ttw, + AU,), with w, the hole cyclotron frequency. The only measurement of cyclotron resonance in two-dimensionally confined holes [8] gives a hole mass on the order of 0.4 m,, implying w, - a wC. The expected slopes of the transitions shown would be, in units of AU,; 0.58, 1.74, 2.90, and 4.06. In contrast, the measured slopes are 1.07, 2.10, 3.27, and 4.20. Beyond a few tesla the slopes do decrease to more respectable values, and eventually the single lines split into two polirized components. The following factors will be important in analyzing our spectra. (1) Electron exchange energy must change with field. Indeed oscillations in exchange energy on the order of several meV are expected as the Fermi level passes through Landau levels [9]. (2) The valence band is, in fact, four-fold
490
J. M. Warlock et al. / Magneto-optical
studies of 20 electrons
degenerate, and two effects are fighting for dominance in breaking this degeneracy: (a) confinement in the quantum wells, which gives hole level spacings on the order of a few meV; and (b) the magnetic field, which alone
I I I I I ENERGIES OF PHOTOLUMINESCENT PEAKS HETEROSTRUCTURE SAMPLE 9-25-6013
0
0
0
I
I MOW
0
0 0
0
O 0
0
0
1.530 l-II0 (3-3)
0
(2;2’
0
0 0
0 0
0
0
0 0
0
O
0
0 0
1 ,520
0
0
0
(0-O)
0
i
E3D G
0
0
0
0
0
%
d
0
0
0
0
I.510 2
0
4
6
6
MAGNETIC Fig.
2. Energies
curvature
towards
of photoluminescent lower slopes,
peaks
and splitting
FIELD versus
10
12
14
(TESLA) magnetic
into polarized
field
doublets.
showing
large
initial
slopes.
J.M.
Warlock et al. / Magneto-optical
studres of 20 electrons
491
gives the complicated but well honored energies derived by Luttinger [lo]. We believe that at high magnetic fields, the holes will finally simplify and follow Luttinger. An indication that this happens is that the luminescence becomes split and circularly polarized. When the hole states are finally understood, it will be possible to use the recombination energy to study simply the electronic behavior in high magnetic fields. We thank P.A. Wolff, A. Pinczuk, and J. Shah for helpful discussions; and L. Rubin and the staff of the National Magnet Laboratory for cooperation and hospitality. The National Magnet Laboratory is supported by the National Science Foundation. This work was also supported in part by the Office of Naval Research, under contract N-00014-81-k-651, and the National Science Foundation, under grant DMR 8121702.
References [l] A.C. Gossard, in: Thin Films: Preparation and Properties, Eds. K.N. Tu and R. Rosenberg (Academic Press, 1981). [2] J.M. Warlock A. Pinczuk, Z.J. Tien, C.H. Perry, H.L. Stormer, R. Dingle, A.C. Gossard, W. Wiegmann and R.L. Aggarwal, Solid State Commun. 40 (1981) 867; Z.J. Tien, PhD Thesis, Northeastern University (1981). [3] J.M. Warlock, A.C. Maciel, C.H. Perry, Z.J. Tien, R.L. Aggarwal, A.C. Gossard and W. Wiegmann, in: Application of High Magnetic Fields in Semiconductor Physics, Ed. G. Landwehr (Springer, Berlin, 1983) p. 186. [4] A.C. Maciel, J.M. Warlock, C.H. Perry, R.L. Aggarwal, A.C. Gossard and W. Wiegmann, Bull. Am. Phys. Sot. 28 (1983) 448; and to be published. [S] M. Cardona, M.H. Grimsditch, and D. Olego, in: Light Scattering in Solids, Eds. J.L. Birman, H.Z. Cummins and K.K. Rebane (Plenum, 1979) p. 249. [6] J.M. Warlock, Solid State Commun. 48 (1983) 1067. [7] A. Pinczuk, J. Shah, H.L. Stormer, R.C. Miller A.C. Gossard and W. Wiegmann, Surface Sci. 142 (1984) 492. [8] H.L. Stormer, Z. Schlesinger, A. Chang, D.C. Tsui, A.C. Gossard and W. Wiegmann, Phys. Rev. letters 51 (1983) 126. [9] Th. Englert, D.C. Tsui, A.C. Gossard and Ch. Uihlein, Surface Sci. 113 (1983) 295. [lo] J.M. Luttinger, Phys Rev. 102 (1956) 1030.