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Quasi-particle recombination and spatial ordering of 2D electrons in the extreme quantum limit
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ELSEVIER
Surface Science 305 (1994) 61-66
Quasi-particle recombination and spatial ordering of 2D electrons in the extreme quantum limit I.N. Harris
a, H.D.M.
Davies a, R.A. Ford a, J.F. Ryan a, A.J. Turberfield C.T. Foxon b, J.J. Harris ’
*3a,
aPhysics Department, Clarendon Laboratory, Oxford University, Oxford, UK b Physics Department, The University, Nottingham, UK ’ Semiconductor Research Centre, Imperial College, London, UK (Received
26 April 1993; accepted
for publication
1 June 1993)
Abstract
The inter-band photoluminescence spectrum of an incompressible liquid at an ultra-high mobility GaAs heterojunction may show multiple recombination channels in which different integral numbers of fractionally charged excitations participate. The spectrum at v = l/2 is discussed in terms of the proposed composite fermion model of the fractional quantum Hall effect. Changes in the photoluminescence spectrum at higher field lead to the proposal of an experimental test for magnetic-field-induced crystalline order.
1. Introduction Three important and unresolved issues in the physics of electron-electron interactions in degenerate two-dimensional electron systems (2DES) are: the dispersion of neutral excitations of the incompressible liquid states responsible for the fractional quantum Hall effect (FQHE); the nature of the ground state at even filling factors; experimental determination of the ground state at filling factors v < l/5. We will show here that optical spectroscopy has the potential to address each of these questions. We present measurements of photoluminescence from two ultrahigh-mobility GaAs single heterojunctions G641
* Corresponding
author.
0039-6028/94/$07.00 0 1994 Elsevier SSDI 0039-6028(93)E0682-K
Science
(n, = 10” cm-‘, p = 9 X lo6 cm* V-’ s-l) [l] and G648 (n, = 3 X 10’” cm-*, p = 2 X lo6 cm2 v-r s-l> [2]; w e explore the physics of the recombination process and demonstrate progress towards the resolution of the three important questions listed above. The experimental technique has been described elsewhere [l-3]. Weak photoexcitation below the band gap of the barrier excites low densities of electrons and holes in continuum states in the GaAs layer; photoexcited holes relax to the valence band edge where they recombine with electrons in the potential well at the interface. Recombination is observed both from electrons in the 2DES in the ground subband and from photoexcited electrons that have scattered into higher subbands. Photoexcited electrons are not in thermal equilibrium with the 2DES, whose temperature is estimated to be approximately 100 mK. Both the energy and inten-
B.V. All rights reserved
sity of photoluminescence are sensitive to the effects of correlation in the 2DES; hierarchies of FQHE states are detected [1,2] and the spectrum responds to the onset of a high-field insulating phase at v = l/S [31.
2. Quasi-particle recombination and the determination of FQHE energy gaps The fundamental characteristic property of an incompressible liquid is an excitation energy gap. No measurement of the zero-wavevector gap has yet been possible, although activated transport measurements give an estimate of the energy A of large-wavevector excitations (quasielectron-quasihole pairs). Various anomalies in the energy of photoluminescence at FQH states have been reported which may allow measurement of excitations of the liquid. In the case of acceptor-doped heterojunctions the photoluminescence energy at fractional filling factors v = p/q has been claimed to change discontinuously by qA [4]; in this model the recombination process either annihilates q quasi-electrons (v > p/q> or creates 4 quasi-holes (v
1 "140
Fig. I. Luminescence (b) v = l/2.
spectra
I 515
of Gh41 at 100 mK (a) v = l/3:
high energy. This wing shows structure consistent with the presence of two satellite lines shown by the fitted curves. The strong low-energy component is interpreted as arising from the dominant recombination process involving the generation of 3 quasi-holes. In general, recombination of electrons can proceed by creation of n quasi-holes and annihilation of (q - n) quasi-electrons giving rise to (q + 1) spectral lines separated by A. The two satellite peaks present in Fig. la then corrcspond to n = 2 and 1 processes. At v = l/5 we observe a high-energy wing but no clearly resolved structure; this may be simply due to the fact that A is lower than the experimental resolution in this case. The relative efficiencies of these recombination channels are not known, although those involving annihilation of more than one quasi-electron are expected to be weak due to the low probability of quasi-electrons being present
I.N. Harris et al. /Surface Science 305 (1994) 61-66
in the same place, i.e. the location of the valence hole; MacDonald et al. [6] find evidence in their finite-size calculations for only the II = 3 and 2 processes at v = l/3. The line separation observed here is 0.2 meV (2.2 K). This is smaller than the previously published value of N 5 K for the v = l/3 gap energy deduced from activated transport measurements at similar fields [7]; it is probable that this measurement of A is perturbed by interaction between the 2DES and the valence band hole. Apal’kov and Rashba [S] find that at v = l/3 in systems where the electron-electron and electron-valence hole interactions are strongly asymmetric there is an energy minimum in the magnetoexciton dispersion at finite wavevector; recombination from this state requires emission of a magnetoroton to conserve momentum and is therefore lower in energy than recombination from a magnetoexciton at the zone centre by approximately the magnetoroton energy rather than the quasi-particle gap. Halperin et al. [9] have recently extended Jain’s composite fermion model of the FQHE [lo] to the special cases of even-denominator fractional filling, in particular to I, = l/2. Here the average gauge field cancels the applied magnetic field; the picture that results is that of fermions in zero magnetic field with a well defined Fermi surface. The electron system is gapless at these filling factors. Experimental data for I, = l/2 and l/4 are rather limited: the resistivity pXX shows a weak minimum at v = l/2 and is systematically smaller than that at neighbouring even-denominator fractions, as predicted by the model, but the same does not appear to apply at Y = l/4 [3,11] (it should be recalled that there is a dramatic increase in pXX for v < l/4 which may be associated with formation of an electron solid see below). Our discussion of the use of photoluminescence to detect FQHE gaps suggests the possibility of optical detection of the density of states of this exotic quantum system. Fig. lb shows the luminescence spectrum of G641 at u = l/2. The strong E, line is remarkably symmetrical in contrast to the complex, asymmetric lineshape observed when the ground state is incompressible as at v = l/3 and l/5. By analogy with the true zero-field case, the full lumines-
63
cence linewidth may provide a measure of the Fermi energy; the observed value is N 0.5 meV, which corresponds to an effective mass of approximately 0.35m, (the electron density at v = l/2 with B = 11.5 T is estimated to be 7 x 10’” cm-* [3]). This measurement is close to the theoretical estimate of 0.27m, at B = 10 T [91. However, this lineshape analysis makes no distinction between the recombination of composite fermions and of ordinary electrons, nor does it take any account of the enhancement of recombination at the Fermi edge observed at zero field.
3. “Hot” electron probe of spatial ordering The high-field ground state of an ideal degenerate two-dimensional electron system is believed to be crystalline. Nonlinear transport [ll-131, photoluminescence [3,141 and surface acoustic wave propagation [15] have all provided evidence that a solid phase occurs at filling factor v = l/5 for GaAs but the nature and length scale of the order in this phase remains controversial. A clear test for crystalline order is a diffraction measurement, possibly using phonons, but this has not yet been achieved. We describe here a new method which utilises nonequilibrium electrons in higher-energy subbands of the confining potential as a probe of spatial order in the ground subband. When the two-dimensional electron system in the ground subband is liquid its charge density is spatially uniform; the effective interaction potential experienced by other carriers (electrons in higher subbands and in the continuum, and valence band holes) is therefore also uniform. A magnetic-field-induced electron crystal is easily pinned by fluctuations in the interface potential, however, in which case the charge distribution in the ground subband is not uniform but periodically modulated. A transition from a liquid to a pinned crystal in the ground subband is therefore accompanied by the onset of a periodic perturbation experienced by electrons confined in higher subbands of the potential well. More remote carriers in continuum states, including all valence band states, will be more weakly perturbed. Pho-
toluminescence spectroscopy is sensitive to the densities of states for both electrons and holes and, in the range of filling factors in which crystallisation is expected, luminescence from nonequilibrium electrons in the first excited subband is observed. Spatial order in the ground subband might therefore be detected through its effect on this component of the luminescence spectrum. In a strong magnetic field the effect of a periodic potential is to broaden the Landau level density of states and split it into bands in a pattern that depends on the symmetry and on the number of flux quanta penetrating a unit cell of the potential [16,17]. (The term Hofstadter band will be used to distinguish these fragments of a Landau level, corresponding to in-plane motion, from the subbands of the potential well confining the electron system.) When the filling factor for a crystalline 2DES in the ground subband is a rational fraction v =p/q then q/p flux quanta penetrate a unit cell and the density of states for electrons in the first excited subband is split into q bands. Relatively large gaps in the density of states occur at simple filling factors, where there are few bands, and persist over a range of magnetic fields [18]. To confirm the presence of spatial order in the ground subband it is this characteristic pattern of gaps that must be detected. In the crystalline phase we expect the spectrum of luminescence from photoexcited electrons in the first excited subband to broaden and split into a field-dependent pattern of transitions from Hofstadter bands. It is reasonable to assume that all bands are populated by photoexcitation. The most easily identified feature in this spectrum should be the principal gap which, for l/5 < v < l/10, is approximately half the total width of the spectrum in magnitude and divides it into a narrow high-energy line and a broader collection of transitions from bands at lower energies. The characteristic energy scale for this structure, the magnitude of the periodic potential, depends on the in-plane charge distribution in the ground subband and on the form of both ground and excited subband wavefunctions normal to the interface; we estimate this to be of order 1 meV. The excitonic character of luminescence from the upper subband [l] will reduce the broadening of
1.521
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1.519
,Y 2 i : w
1.518
1.516
I-1. 6
.7
6
9 B
10
11
12
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Fig. 2. Upper subband (E, ) luminescence (3648. Representative spectra are inset.
peak
energies
for
the spectrum as the sign of the perturbation experienced by a valence hole is reversed. Fig. 2 shows the magnetic-field-dependent energy of E, luminescence peaks (corresponding to recombination from states in the lowest Landau level of the first excited subband) for sample G648. Inset spectra show the full luminescence lineshape with E, transitions marked; the peak - 0.5 meV lower in energy corresponds to E,, recombination from the degenerate 2DES in the lower subband. At filling factors v > l/5, where the 2DES is liquid, the E, peak is featureless (spectrum a). Between v = l/4 and v = l/5 the E, peak broadens from 0.3 meV FWHM to 0.6 meV and at v = l/5, the suggested threshold for crystallisation, a high-energy shoulder appears (spectrum b); the transition develops into a wellresolved doublet at lower filling factors (spectra c and e). The lower-energy E, peak loses intensity as the filling factor is reduced below v = l/7 and at v = l/10 the higher-energy E, peak dominates
I.N. Harris et al. /Surface
the spectrum. This characteristic development of the luminescence spectrum correlates with changes in transport properties associated with crystallisation and has been observed previously over a smaller range of filling factors in the higher-density heterojunction G641 [3]. Over a range of filling factors around v = l/7 the E, spectrum reverts to the single line (spectrum d) characteristic of higher filling factors v > l/5. Changes in the width and structure of the E, luminescence transition for v _< l/5 suggest corresponding changes in the density of states of electrons in the upper subband that are consistent with the effect of crystallisation of the 2DES in the ground subband. The width of the E, luminescence transition for v < l/5 is consistent with our estimate of the periodic perturbation due to the electron crystal. The minimum luminescence linewidth measured from this sample is N 0.3 meV so we cannot expect to resolve small gaps in the density of states; the shape of the spectrum for v < l/5 is consistent with the calculated density of states if only the principal gap is resolved. The reversion of the spectrum to its low-field form around u = l/7 may indicate a reversion to a liquid ground state of the 2DES: a re-entrant liquid phase at this filling factor has been inferred from measurements of a pXX minimum [13,19]; luminescence spectra of an acceptor-doped system have been interpreted as showing coexisting solid and liquid phases over all filling factors v < l/5 [14]. Changes in the luminescence spectrum are not due solely to the orbital wavefunction: the spin states of the E, electrons respond differently to spin and spatial ordering of the 2DES. There may be enhancement of the spin splitting in the upper subband due to exchange with the degenerate 2DES in the ground subband; if this were responsible for the structure observed at low filling factors then Fig. 2 shows that there must be a rapid threshold for this interaction at v = l/5. Measurements of the polarisation of the luminescence do not support this interpretation: at v = l/5 both E, and E, are predominantly left circularly polarised (LCP) with ZJZ, = 3 [2]; the higher-energy E, peak that develops for v < l/5 becomes more strongly LCP at lower filling fac-
Science 305 (1994) 61-66
65
tors whereas the lower-energy E, peak becomes distincly unpolarised. A detailed interpretation of the measurements requires a complete model for electron-valence hole recombination, but the magnitude of the splitting and the polarisation data are not consistent with a simple spin-splitting picture. It would be interesting, however, to investigate whether a transition from spinpolarised liquid to a solid could enhance the magnitude of the exchange interaction with upper subband electrons.
4. Conclusions Structure in the spectrum of luminescence from an incompressible liquid reveals multiple recombination channels which involve different numbers of quasi-electrons and quasi-holes and allow measurement of the quasi-particle gap A. The spectral lineshape at Y = l/2 is anomalous and may provide information about the compressible ground state at even fractions. Spatial order in the 2DES may be detected by measuring luminescence from photoexcited electrons in a higher subband. Our measurements are consistent with crystallisation at filling factor v N l/5 and suggest a reentrant liquid phase around u = l/7.
5. References [l] A.J. Turberfield, S.R. Haynes, P.A. Wright, R.A. Ford, R.G. Clark, J.F. Ryan, J.J. Harris and C.T. Foxon, Phys. Rev. Lett. 65 (1990) 637. 121 A.J. Turberfield, R.A. Ford, I.N. Harris, J.F. Ryan, C.T. Foxon and J.J. Harris, Phys. Rev. B 47 (1993) 4794. 131 R.G. Clark, R.A. Ford, S.R. Haynes, J.F. Ryan, A.J. Turberfield, P.A. Wright, C.T. Foxon and J.J. Harris, in: High Magnetic Fields in Semiconductor Physics III, Ed. G. Landwehr (Springer, Berlin, 19921 p. 231; A.J. Turberfield, S.R. Haynes, P.A. Wright, R.A. Ford, R.G. Clark, J.F. Ryan, J.J. Harris and CT. Foxon, Surf. Sci. 263 (1992) 1. [4] H. Buhmann, W. Joss, K. von Klitzing, I.V. Kukushkin, G. Martinez, A.S. Plaut, K. Ploog and V.B. Timofeev, Phys. Rev. Lett. 65 (1990) 1056. [5] I.V. Kukushkin, N.J. Pulsford, K. von Klitzing, K. Ploog and V.B. Timofeev, Surf. Sci. 263 (1992) 30. [6] A.H. MacDonald, E.H. Rezayi and D. Keller, Phys. Rev. Lett. 68 (1992) 1939.
06
I. N. Hurric
et ul. /
SurfuwScirncr
[7] .I. Wakabayashi. S. Kawaji, J. Yoshino and H. Sakaki, J. Phys. Sot. Jpn. 55 (1986) 1319. [8] V.M. Apal’kov and E.I. Rashba, Pis’ma Zh. Eksp. Tear. Fiz. 55 (1992) 38 [JETP Lett. 55 (1992) 371. [9] B.I. Halperin. P.A. Lee and N. Read, Phys. Rev. B 47 (1993) 7312. [IO] J.K. Jain, Adv. Phys. 41 (1992) 105. [II] H.W. Jiang, R.L. Willett, H.L. Stormer, D.C. Tsui, L.N. Pfeiffer and K.W. West, Phys. Rev. Lett. 65 (1990) 633. [12] V.J. Goldman, M. Santos, M. Shayegan and J.W. Cunningham, Phys. Rev. Lett. 65 (1990) 2189. [I31 F.I.B. Williams. P.A. Wright, R.G. Clark, E.Y. Andrei. G. Deville. D.C. Glattli. 0. Prohst, B. Etienne, C. Dorin. C.T. Foxon and J.J Harris. Phys. Rev. Lett. 66 (1991) 3285.
30.7 (19941 61-M
[I41 H. Buhmann. W. Joss. K. von Klitzing. 1.V. Kukushkin. AS. Plaut, G. Martinez. K. Ploog and V.B. Timofcev. Phys. Rev. Lett. 66 (1991) 92h. [15] M.A. Paalanen, R.L. Willett, P.B. Littlewood. K.W. West, L.N. Pfeiffer and D.J. Bishop. Phys. Rev. B 45 (1992) I 1342. [lo] D.R. Hofstadter, Phys. Rev. B 14 (1976) 2239; A.H. MacDonald, Phys. Rev. B 28 (1983) 6713. [I71 F.H. Clara and G.H. Wannier. Phys. Rev. B I9 (1979) 6068. [1X] Eq. (5) and Fig. 1 of Ref. [I71 give the expected result for a hexagonal potential. [IY] V.J. Goldman, M. Shayegan and D.C. Tsui, Phys. Rev. Lett. 61 (1988) 881.
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ELSEVIER
Surface Science 305 (1994) 61-66
Quasi-particle recombination and spatial ordering of 2D electrons in the extreme quantum limit I.N. Harris
a, H.D.M.
Davies a, R.A. Ford a, J.F. Ryan a, A.J. Turberfield C.T. Foxon b, J.J. Harris ’
*3a,
aPhysics Department, Clarendon Laboratory, Oxford University, Oxford, UK b Physics Department, The University, Nottingham, UK ’ Semiconductor Research Centre, Imperial College, London, UK (Received
26 April 1993; accepted
for publication
1 June 1993)
Abstract
The inter-band photoluminescence spectrum of an incompressible liquid at an ultra-high mobility GaAs heterojunction may show multiple recombination channels in which different integral numbers of fractionally charged excitations participate. The spectrum at v = l/2 is discussed in terms of the proposed composite fermion model of the fractional quantum Hall effect. Changes in the photoluminescence spectrum at higher field lead to the proposal of an experimental test for magnetic-field-induced crystalline order.
1. Introduction Three important and unresolved issues in the physics of electron-electron interactions in degenerate two-dimensional electron systems (2DES) are: the dispersion of neutral excitations of the incompressible liquid states responsible for the fractional quantum Hall effect (FQHE); the nature of the ground state at even filling factors; experimental determination of the ground state at filling factors v < l/5. We will show here that optical spectroscopy has the potential to address each of these questions. We present measurements of photoluminescence from two ultrahigh-mobility GaAs single heterojunctions G641
* Corresponding
author.
0039-6028/94/$07.00 0 1994 Elsevier SSDI 0039-6028(93)E0682-K
Science
(n, = 10” cm-‘, p = 9 X lo6 cm* V-’ s-l) [l] and G648 (n, = 3 X 10’” cm-*, p = 2 X lo6 cm2 v-r s-l> [2]; w e explore the physics of the recombination process and demonstrate progress towards the resolution of the three important questions listed above. The experimental technique has been described elsewhere [l-3]. Weak photoexcitation below the band gap of the barrier excites low densities of electrons and holes in continuum states in the GaAs layer; photoexcited holes relax to the valence band edge where they recombine with electrons in the potential well at the interface. Recombination is observed both from electrons in the 2DES in the ground subband and from photoexcited electrons that have scattered into higher subbands. Photoexcited electrons are not in thermal equilibrium with the 2DES, whose temperature is estimated to be approximately 100 mK. Both the energy and inten-
B.V. All rights reserved
sity of photoluminescence are sensitive to the effects of correlation in the 2DES; hierarchies of FQHE states are detected [1,2] and the spectrum responds to the onset of a high-field insulating phase at v = l/S [31.
2. Quasi-particle recombination and the determination of FQHE energy gaps The fundamental characteristic property of an incompressible liquid is an excitation energy gap. No measurement of the zero-wavevector gap has yet been possible, although activated transport measurements give an estimate of the energy A of large-wavevector excitations (quasielectron-quasihole pairs). Various anomalies in the energy of photoluminescence at FQH states have been reported which may allow measurement of excitations of the liquid. In the case of acceptor-doped heterojunctions the photoluminescence energy at fractional filling factors v = p/q has been claimed to change discontinuously by qA [4]; in this model the recombination process either annihilates q quasi-electrons (v > p/q> or creates 4 quasi-holes (v
1 "140
Fig. I. Luminescence (b) v = l/2.
spectra
I 515
of Gh41 at 100 mK (a) v = l/3:
high energy. This wing shows structure consistent with the presence of two satellite lines shown by the fitted curves. The strong low-energy component is interpreted as arising from the dominant recombination process involving the generation of 3 quasi-holes. In general, recombination of electrons can proceed by creation of n quasi-holes and annihilation of (q - n) quasi-electrons giving rise to (q + 1) spectral lines separated by A. The two satellite peaks present in Fig. la then corrcspond to n = 2 and 1 processes. At v = l/5 we observe a high-energy wing but no clearly resolved structure; this may be simply due to the fact that A is lower than the experimental resolution in this case. The relative efficiencies of these recombination channels are not known, although those involving annihilation of more than one quasi-electron are expected to be weak due to the low probability of quasi-electrons being present
I.N. Harris et al. /Surface Science 305 (1994) 61-66
in the same place, i.e. the location of the valence hole; MacDonald et al. [6] find evidence in their finite-size calculations for only the II = 3 and 2 processes at v = l/3. The line separation observed here is 0.2 meV (2.2 K). This is smaller than the previously published value of N 5 K for the v = l/3 gap energy deduced from activated transport measurements at similar fields [7]; it is probable that this measurement of A is perturbed by interaction between the 2DES and the valence band hole. Apal’kov and Rashba [S] find that at v = l/3 in systems where the electron-electron and electron-valence hole interactions are strongly asymmetric there is an energy minimum in the magnetoexciton dispersion at finite wavevector; recombination from this state requires emission of a magnetoroton to conserve momentum and is therefore lower in energy than recombination from a magnetoexciton at the zone centre by approximately the magnetoroton energy rather than the quasi-particle gap. Halperin et al. [9] have recently extended Jain’s composite fermion model of the FQHE [lo] to the special cases of even-denominator fractional filling, in particular to I, = l/2. Here the average gauge field cancels the applied magnetic field; the picture that results is that of fermions in zero magnetic field with a well defined Fermi surface. The electron system is gapless at these filling factors. Experimental data for I, = l/2 and l/4 are rather limited: the resistivity pXX shows a weak minimum at v = l/2 and is systematically smaller than that at neighbouring even-denominator fractions, as predicted by the model, but the same does not appear to apply at Y = l/4 [3,11] (it should be recalled that there is a dramatic increase in pXX for v < l/4 which may be associated with formation of an electron solid see below). Our discussion of the use of photoluminescence to detect FQHE gaps suggests the possibility of optical detection of the density of states of this exotic quantum system. Fig. lb shows the luminescence spectrum of G641 at u = l/2. The strong E, line is remarkably symmetrical in contrast to the complex, asymmetric lineshape observed when the ground state is incompressible as at v = l/3 and l/5. By analogy with the true zero-field case, the full lumines-
63
cence linewidth may provide a measure of the Fermi energy; the observed value is N 0.5 meV, which corresponds to an effective mass of approximately 0.35m, (the electron density at v = l/2 with B = 11.5 T is estimated to be 7 x 10’” cm-* [3]). This measurement is close to the theoretical estimate of 0.27m, at B = 10 T [91. However, this lineshape analysis makes no distinction between the recombination of composite fermions and of ordinary electrons, nor does it take any account of the enhancement of recombination at the Fermi edge observed at zero field.
3. “Hot” electron probe of spatial ordering The high-field ground state of an ideal degenerate two-dimensional electron system is believed to be crystalline. Nonlinear transport [ll-131, photoluminescence [3,141 and surface acoustic wave propagation [15] have all provided evidence that a solid phase occurs at filling factor v = l/5 for GaAs but the nature and length scale of the order in this phase remains controversial. A clear test for crystalline order is a diffraction measurement, possibly using phonons, but this has not yet been achieved. We describe here a new method which utilises nonequilibrium electrons in higher-energy subbands of the confining potential as a probe of spatial order in the ground subband. When the two-dimensional electron system in the ground subband is liquid its charge density is spatially uniform; the effective interaction potential experienced by other carriers (electrons in higher subbands and in the continuum, and valence band holes) is therefore also uniform. A magnetic-field-induced electron crystal is easily pinned by fluctuations in the interface potential, however, in which case the charge distribution in the ground subband is not uniform but periodically modulated. A transition from a liquid to a pinned crystal in the ground subband is therefore accompanied by the onset of a periodic perturbation experienced by electrons confined in higher subbands of the potential well. More remote carriers in continuum states, including all valence band states, will be more weakly perturbed. Pho-
toluminescence spectroscopy is sensitive to the densities of states for both electrons and holes and, in the range of filling factors in which crystallisation is expected, luminescence from nonequilibrium electrons in the first excited subband is observed. Spatial order in the ground subband might therefore be detected through its effect on this component of the luminescence spectrum. In a strong magnetic field the effect of a periodic potential is to broaden the Landau level density of states and split it into bands in a pattern that depends on the symmetry and on the number of flux quanta penetrating a unit cell of the potential [16,17]. (The term Hofstadter band will be used to distinguish these fragments of a Landau level, corresponding to in-plane motion, from the subbands of the potential well confining the electron system.) When the filling factor for a crystalline 2DES in the ground subband is a rational fraction v =p/q then q/p flux quanta penetrate a unit cell and the density of states for electrons in the first excited subband is split into q bands. Relatively large gaps in the density of states occur at simple filling factors, where there are few bands, and persist over a range of magnetic fields [18]. To confirm the presence of spatial order in the ground subband it is this characteristic pattern of gaps that must be detected. In the crystalline phase we expect the spectrum of luminescence from photoexcited electrons in the first excited subband to broaden and split into a field-dependent pattern of transitions from Hofstadter bands. It is reasonable to assume that all bands are populated by photoexcitation. The most easily identified feature in this spectrum should be the principal gap which, for l/5 < v < l/10, is approximately half the total width of the spectrum in magnitude and divides it into a narrow high-energy line and a broader collection of transitions from bands at lower energies. The characteristic energy scale for this structure, the magnitude of the periodic potential, depends on the in-plane charge distribution in the ground subband and on the form of both ground and excited subband wavefunctions normal to the interface; we estimate this to be of order 1 meV. The excitonic character of luminescence from the upper subband [l] will reduce the broadening of
1.521
1.520
T<
1.519
,Y 2 i : w
1.518
1.516
I-1. 6
.7
6
9 B
10
11
12
CT)
Fig. 2. Upper subband (E, ) luminescence (3648. Representative spectra are inset.
peak
energies
for
the spectrum as the sign of the perturbation experienced by a valence hole is reversed. Fig. 2 shows the magnetic-field-dependent energy of E, luminescence peaks (corresponding to recombination from states in the lowest Landau level of the first excited subband) for sample G648. Inset spectra show the full luminescence lineshape with E, transitions marked; the peak - 0.5 meV lower in energy corresponds to E,, recombination from the degenerate 2DES in the lower subband. At filling factors v > l/5, where the 2DES is liquid, the E, peak is featureless (spectrum a). Between v = l/4 and v = l/5 the E, peak broadens from 0.3 meV FWHM to 0.6 meV and at v = l/5, the suggested threshold for crystallisation, a high-energy shoulder appears (spectrum b); the transition develops into a wellresolved doublet at lower filling factors (spectra c and e). The lower-energy E, peak loses intensity as the filling factor is reduced below v = l/7 and at v = l/10 the higher-energy E, peak dominates
I.N. Harris et al. /Surface
the spectrum. This characteristic development of the luminescence spectrum correlates with changes in transport properties associated with crystallisation and has been observed previously over a smaller range of filling factors in the higher-density heterojunction G641 [3]. Over a range of filling factors around v = l/7 the E, spectrum reverts to the single line (spectrum d) characteristic of higher filling factors v > l/5. Changes in the width and structure of the E, luminescence transition for v _< l/5 suggest corresponding changes in the density of states of electrons in the upper subband that are consistent with the effect of crystallisation of the 2DES in the ground subband. The width of the E, luminescence transition for v < l/5 is consistent with our estimate of the periodic perturbation due to the electron crystal. The minimum luminescence linewidth measured from this sample is N 0.3 meV so we cannot expect to resolve small gaps in the density of states; the shape of the spectrum for v < l/5 is consistent with the calculated density of states if only the principal gap is resolved. The reversion of the spectrum to its low-field form around u = l/7 may indicate a reversion to a liquid ground state of the 2DES: a re-entrant liquid phase at this filling factor has been inferred from measurements of a pXX minimum [13,19]; luminescence spectra of an acceptor-doped system have been interpreted as showing coexisting solid and liquid phases over all filling factors v < l/5 [14]. Changes in the luminescence spectrum are not due solely to the orbital wavefunction: the spin states of the E, electrons respond differently to spin and spatial ordering of the 2DES. There may be enhancement of the spin splitting in the upper subband due to exchange with the degenerate 2DES in the ground subband; if this were responsible for the structure observed at low filling factors then Fig. 2 shows that there must be a rapid threshold for this interaction at v = l/5. Measurements of the polarisation of the luminescence do not support this interpretation: at v = l/5 both E, and E, are predominantly left circularly polarised (LCP) with ZJZ, = 3 [2]; the higher-energy E, peak that develops for v < l/5 becomes more strongly LCP at lower filling fac-
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tors whereas the lower-energy E, peak becomes distincly unpolarised. A detailed interpretation of the measurements requires a complete model for electron-valence hole recombination, but the magnitude of the splitting and the polarisation data are not consistent with a simple spin-splitting picture. It would be interesting, however, to investigate whether a transition from spinpolarised liquid to a solid could enhance the magnitude of the exchange interaction with upper subband electrons.
4. Conclusions Structure in the spectrum of luminescence from an incompressible liquid reveals multiple recombination channels which involve different numbers of quasi-electrons and quasi-holes and allow measurement of the quasi-particle gap A. The spectral lineshape at Y = l/2 is anomalous and may provide information about the compressible ground state at even fractions. Spatial order in the 2DES may be detected by measuring luminescence from photoexcited electrons in a higher subband. Our measurements are consistent with crystallisation at filling factor v N l/5 and suggest a reentrant liquid phase around u = l/7.
5. References [l] A.J. Turberfield, S.R. Haynes, P.A. Wright, R.A. Ford, R.G. Clark, J.F. Ryan, J.J. Harris and C.T. Foxon, Phys. Rev. Lett. 65 (1990) 637. 121 A.J. Turberfield, R.A. Ford, I.N. Harris, J.F. Ryan, C.T. Foxon and J.J. Harris, Phys. Rev. B 47 (1993) 4794. 131 R.G. Clark, R.A. Ford, S.R. Haynes, J.F. Ryan, A.J. Turberfield, P.A. Wright, C.T. Foxon and J.J. Harris, in: High Magnetic Fields in Semiconductor Physics III, Ed. G. Landwehr (Springer, Berlin, 19921 p. 231; A.J. Turberfield, S.R. Haynes, P.A. Wright, R.A. Ford, R.G. Clark, J.F. Ryan, J.J. Harris and CT. Foxon, Surf. Sci. 263 (1992) 1. [4] H. Buhmann, W. Joss, K. von Klitzing, I.V. Kukushkin, G. Martinez, A.S. Plaut, K. Ploog and V.B. Timofeev, Phys. Rev. Lett. 65 (1990) 1056. [5] I.V. Kukushkin, N.J. Pulsford, K. von Klitzing, K. Ploog and V.B. Timofeev, Surf. Sci. 263 (1992) 30. [6] A.H. MacDonald, E.H. Rezayi and D. Keller, Phys. Rev. Lett. 68 (1992) 1939.
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[7] .I. Wakabayashi. S. Kawaji, J. Yoshino and H. Sakaki, J. Phys. Sot. Jpn. 55 (1986) 1319. [8] V.M. Apal’kov and E.I. Rashba, Pis’ma Zh. Eksp. Tear. Fiz. 55 (1992) 38 [JETP Lett. 55 (1992) 371. [9] B.I. Halperin. P.A. Lee and N. Read, Phys. Rev. B 47 (1993) 7312. [IO] J.K. Jain, Adv. Phys. 41 (1992) 105. [II] H.W. Jiang, R.L. Willett, H.L. Stormer, D.C. Tsui, L.N. Pfeiffer and K.W. West, Phys. Rev. Lett. 65 (1990) 633. [12] V.J. Goldman, M. Santos, M. Shayegan and J.W. Cunningham, Phys. Rev. Lett. 65 (1990) 2189. [I31 F.I.B. Williams. P.A. Wright, R.G. Clark, E.Y. Andrei. G. Deville. D.C. Glattli. 0. Prohst, B. Etienne, C. Dorin. C.T. Foxon and J.J Harris. Phys. Rev. Lett. 66 (1991) 3285.
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[I41 H. Buhmann. W. Joss. K. von Klitzing. 1.V. Kukushkin. AS. Plaut, G. Martinez. K. Ploog and V.B. Timofcev. Phys. Rev. Lett. 66 (1991) 92h. [15] M.A. Paalanen, R.L. Willett, P.B. Littlewood. K.W. West, L.N. Pfeiffer and D.J. Bishop. Phys. Rev. B 45 (1992) I 1342. [lo] D.R. Hofstadter, Phys. Rev. B 14 (1976) 2239; A.H. MacDonald, Phys. Rev. B 28 (1983) 6713. [I71 F.H. Clara and G.H. Wannier. Phys. Rev. B I9 (1979) 6068. [1X] Eq. (5) and Fig. 1 of Ref. [I71 give the expected result for a hexagonal potential. [IY] V.J. Goldman, M. Shayegan and D.C. Tsui, Phys. Rev. Lett. 61 (1988) 881.