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Optical spectra of ultrathin II–VI quantum wells
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0038-1098/93 $6 00+ 00 Pergamon Press Ltd
Solid State Communications, Vol. 88, No. 9, pp. 687-691, 1993. Printed in Great Britain
OPTICAL SPECTRA OF ULTRATHIN II-VI QUANTUM WELLS Fang Yang*, B. Henderson and K.P.O'Donnell Department of Physics and Applied Physics, University of Strathclyde, Glasgow G40NG, Scotland, UK
(Received 9 July 1993, acceptedfor publication 15 October 1993 by G. Bastard) Abstract Ultrathin wide band gap II-VI quantum wells (QW's) feature a large inhomogeneous broadening of exciton transitions in the optical absorption and photoluminescence (PL) spectra. When the photon energy of the excitation laser approaches, or lies within the broadened PL band, a series of additional sharp peaks appear at low energy. The laser induced linear polarisation of the sharp peaks enables the clear demonstration of evenly spaced phonon-assisted transitions. The relative intensity of the phonon-assisted transitions and the mean phonon energy depend strongly on the photon energy of the exciting light. A new model of phonon-assisted relaxation between classically localised states explains these results well.
1.Introduction
has been described previously O. All ultra thin MQWs of the Zn(Cd)S(Se) demonstrate similar spectra as discussed later. Here we present a study of a CdSe MQW (as an example), which consists of 100 alternate layers of CdSe (lnm) and ZnSe (5nm) grown at 300°C, on top of a ZnSe buffer layer 0.8~m thick, on a (100) GaAs substrate.
Thin II-VI semiconductor multiple quantum wells (MQW) are particularly interesting materials in which to study the relaxation of the exciton and the interaction of the exciton with optical phononsl,2,3. Because of the large electron and hole masses, excitons which penetrate the barrier region in very thin quantum wells of III-V materials would be confined in similar structures based on II-VI materials. In thin II-VI quantum wells, the exciton energy depends strongly on the well thickness. Samples with fair structural quality feature strongly broadened optical spectra. For example, in a MQW containing periods of 1.2nm ZnSe and 4.6nm ZnS, a one monolayer fluctuation in well thickness induces a spread of about 100meV in the excitonic absorption4 and the photoluminescence (PL) 5 lines. This energy spread is larger than both the exciton binding energy and the longitudinal optical (LO) phonon energy. Recently, the energy relaxation of the exciton between localised states was found in CdSe-ZnSe MQWs 1 by measurement of the steady and near resonant-excited PL spectra and in CdTe-ZnTe single quantum wells by using time-resolved photoluminescence spectroscopy2, 3. However, the mechanism of the relaxation has not been understood completely. In this paper, we analyse resonantly excited PL spectra of ultra thin II-VI MQWs to reveal the mechanism of the energy relaxation of the exciton between localised states. The growth of binary wide band gap MQWs of the Zn(Cd)S(Se) family by metal organic vapour phase epitaxy
2.Experiments When the sample was cooled to 10K and excited with the multiline UV output of an Ar+ laser (351.1363.6nm), a single luminescence band is observed which peaks near 520nm (Fig.l). Its large spectral bandwidth (-0.2eV) points to gross inhomogeneous broadening. The absorption spectrum of the sample peaks at 480nm and has a line width consistent with an estimated well width variation of about one monolayer. Exciting the MQW with the individual visible lines of the Ar+ laser, at wavelengths close to or within the luminescence envelope produces the spectra in Fig.2, which feature a series of additional sharp lines. The energy separation of the sharp lines is of the order of magnitude expected for optical phonons of CdSe or ZnSe. It is interesting that the sharp lines appear only at wavelengths where the broad PL band appears. For the 457.9nm excitation, the barely resolved peak marked '3' in Fig.2 corresponds to a displacement from the laser line equal to the energy of 3 LO phonons. The first and second order sharp lines can not be detected at all, as is shown more clearly in the derivative form of the spectrum in the inset of Fig.2. The sharp lines are more intense when detected with the polarisation aligned with that of the exciting light, as shown (//) in Fig.3. This property allows the polarisation difference spectrum to be obtained by subtracting the 'orthogonar (_1_) spectrum from the 'aligned' (//) one. An
*Present address: Department of Physics and Astronomy, University of Wales College of Cardiff, Cardiff CF1 3TH,UK. 687
ULTRATHIN
450
500 WAVELENGTH
550
II-VI QUANTUM WELLS
600
(nm)
and UV laser Fig.1 Absorption luminescence spectra of a CdSe-ZnSe 10K.
excited photoMQW sample at
Vol. 88, No. 9
example of the narrowed fluorescence line obtained in this way is reproduced in the inset of Fig.3. The unpolarised background is eliminated to reveal a series of lines which are equally spaced in energy. In addition, the polarisation difference spectrum is found to be independent of the orientation of the excitation polarisation with the respect to the crystal axes. At elevated temperature, the sharp lines decrease in intensity and are barely resolved above 1OOK. The mean spacing of the sharp lines varies with the photon energy of excitation as shown in Fig.4. The energy of the l-1 LO phonon for ZnSe, 32meV, is indicated on the figure. Although the phonon energies of cubic CdSe are still to be determined, the energy of the l-1 axial LO phonon of hexagonal CdSe, 26 meV7, is also shown on the figure as a reference. The phonon energies measured from the energy separation of sharp peaks in spectra of the ultrathin CdSe wells in ZnSe barrier almost exactly span the range between 32 and 26 meV. Since ZnCdSe exhibits one mode behaviour8, apparently the phonons which take part in the production of the sharp lines may represent an average taken by an exciton over a region containing both Zn and Cd ions. These phonons may be excitations of a quantum alloy which has larger Cd content in the thicker wells selectively excited by the laser lines of longer wavelength. In this model we might expect to observe a saturation of the phonon energy at the CdSe value in the thicker wells. This has not been observed, suggesting that the LO phonon energy in cubic CdSe may in fact be rather less than our estimate. A detailed study of phonon modes in ultrathin II-VI quantum wells is underway.
3.Discussion From the spectrum shown in Fig. 1, the inhomogeneous broadening of the absorption and PL are -
10
II ASERr’l
650 WAVELENGTH (nm)
Fig.2 Resonantly excited photoluminescence spectra of the CdSe-ZnSe MQW sample. Upwards arrows indicate the laser lines. The feature marked ‘3’ is described in the text. Inset A reproduction of the spectrum of 457.9nm excited in derivative form illustrates the photon coupling in detail.
480
510
540
WAVELENGTH
570
600
(nm)
Fig.3 Polarisation anisotropy of the photoluminescence with 488nm excitation. Shown in the inset is the difference spectrum obtained by subtracting the ‘orthogonal’ from the ‘aligned’ spectrum.
ULTRATHIN II-VI QUANTUM WELLS
Vol. 88, No. 9
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PHOTON ENERGY (eV) Fig.4 Variation of the mean peak separation with photon energy. The straight line is a least-square fit to the data.
200meV in the CdSe MQW, corresponding to spatial variations of the QW width of one monolayer. The overlap of the absorption and PL bands is of the same order. Previous studies on the relation of optical spectra and the quantum well width fluctuation9,10 reveal that the line shape of the exciton absorption peak mirrors the distribution of the optical band gap of the QW, which accurately reflects the distribution of the exciton transition energy if we assume the exciton binding energy is independent of QW width. The line shape of the exciton luminescence mirrors the energy distribution of the local minima of the band gap, which is approximately equal to the energy distribution of classically localised excitons. We note that the line shape of the exciton absorption peak contains information about both delocalised and localised excitons, while the line shape of PL contains only the information on the localised exciton. Hence, the energy overlap of localised and delocalised excitons is represented by the overlap of the absorption and the PL spectra, -200meV in this CdSe MQW sample. Illumination at fixed photon energy may produce localised excitons in one part of the MQW and delocalised excitons in another. Consider the events which follow the incidence of the exciting light on the sample and which result in the emission of light via exciton decay. Excitation at energies higher than the band gap of the barrier material (EgZnSe = 2.82eV at 10 K) generates electron-hole pairs throughout the sample. Energy and momentum relaxation via phonon emission favours the formation of excitons in the CdSe QWs. Within the wells, excitons further lose potential energy by emitting phonons and eventually occupy localised states, which subsequently recombine radiatively. No resolvable structure is observed in the PL spectrum because an exciton may occupy any localised state in a
689
band which has a large and continuous range of energies due to the QW width fluctuation. Excitation at lower photon energies directly creates quantum well excitons with zero momentum, which may be localised or delocalised depending on the environment at the point of exciton formation. The delocalised excitons relax in a similar manner to that described above. Their annilailation results in a broad structureless background emission which is however truncated at the exciting photon energy, as shown in Fig.2. The localised excitons created by resonant excitation cannot relax in the same manner since they are already at local minima. Instead, they recombine directly producing resonant fluorescence (RF). The sharp lines shown in Fig.2, with energy lower than the RF by an integer number of optical phonons, indicate that the RF is not the only recombination process, and suggest that energy transfer between localised exciton states, i.e. tunnelling, must occur via optical phonon emission. Recombination of the resulting localised excitons produces the sharp lines at fi ¢0-nhf~, where h co is the energy of exciting photon (and the primary exciton), h ~ is the energy of the phonon coupled to the primary exciton, and n is the coupling number. Obviously a thermally-activated hopping process can be eliminated, because it produces delocalised excitons first and hence does not lead to a series of sharp lines. In principle, the probability of phonon-assisted exciton tunnelling between localised states in this CdSeZnSe MQW can be calculated, by considering the Schr'odinger equation of the electron and phonon in a 2dimensional disordered potential with an electron-phonon interaction Hamiltanian. This topic is itself equivalent to the content of a paper, and thus beyond the concern here. In what follows, we use our knowledge on exciton localisation to estimate qualitatively the phonon assisted tunnelling probability between localised states and compare this estimate with the experimental results. For a multi-phonon process, the tunnelling probability P, is approximated by the product of the densities of the primary (absorbing) Na(hto ) and the final (emitting) Ne(hC0-nhf~) localised exciton states and a phonon coupling factor g
ILo(hO~-nh~)= gn(ho~)Na(hOONe(hO)-nhf~)
(1)
where h 0~ and h f~ are the absorbing photon and the phonon energy respectively, and, n is the phonon coupling order. The strongest phonon-assisted line appears where the product of Na(t~ co) and Ne(h o)-nti f~) have the largest value. The coupling of a localised electronic state to the three dimensional phonons of the host material is well known in solid state physics 11. In polar semiconductor, the Fr61ich interaction is the most important, so that the dominant electron-phonon coupling is to the LO-phonons. According to Ref.12, the coupling constant g is a function of the electronic localisation length L0. For CdSe, g=ll.54/L0(/~); and for ZnSe, g=7.25/L0(/~). L0 is determined by the smallest value of those under quantum
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Fig.5 Calculated relative intensity of sharp lines. Value are normalised to a maximum of 1 and zeros are shifted for clarity.
Fig.6 The energy separation between the sharp line and laser are illustrated as a function of the coupling order n. The energy corresponding to each step of n represent the LO phonon energy for the nth order coupling.
confinement for the electron, hole and exciton. For the widest QW which is CdSe rich, L 0 is -12/~, g ~1. The narrowest QW is ZnSe rich, but the localisation length is > 12,~ because the QW is too narrow to confine the electron, hole and exciton in 2D, hence we estimate g --0.5. The relative density of localised and delocalised states in QWs has been described in detail 9. For simplicity, we present here in short form the results of the semiempirical consideration in Ref.10. In a QW sample, the random fluctuations of well width and alloy composition lead to an approximately Gaussian distribution of the exciton energy at k--0. We assume that the total exciton density N(h co) o~ exp[-4ln2(E-Eo)2/l-'], where E o is the mean energy and F is the full width at half maximum of the distribution. Resonantly excited excitons can be locallsed or delocalised in the random potential of which N(E) is the distribution function. An exciton is defined to be localised if it is found at a local minimum of the potential, which has a density N 1 given approximately by
Fig.2 for the 457.9 nm excitation, the missing phonon orders in the experiment are explained by the lack of the 'receiving' density of localised exciton states in this model. The photon energy of the strongest phonon-assisted line shifts gradually as the excitation moves into the broad PL band. These predictions agree qualitatively well with the experimental results. Hence we believe that the sharp lines are due to the exciton tunnelling between localised states. A prediction of the absolute intensity requires not only a full quantum mechanical calculation of the probability of the phonon assisted-exciton tunnelling, but also a more precise measurement of the distribution of the localised states since it is not as ideal as predicted by Eq.(2) in a real sample. Finally, one may ask if the LO-phonon-assisted tunnelling is a cascade process. If this is the case then instead of using Eq.(1), the sharp line intensity would be proportional to
I LO ( hco - n h ~ ) = fi gi N~ (hco - ntif~) N(E) N (E)=
(2) E-E
1
1 + exp (
(3)
i=1 O
/ F/4
+F )
Substituting Eq.(2) into Eq.(1) for E=h co and h c0-nh ~, we may obtain the relative intensities of phonon-assisted lines for all excitation energies as shown in Fig.5, with g(h co) set from 0.5 to 1 changing linearly with h co. In Fig.5, the calculated intensities are normalised to the maximum value of each excitation. Compared to those spectra in
where the coupling in Eq.(1) is replaced by the multiple product of the densities of intermediate exciton states. Eq.(3) is different from that for the cascade momentum relaxation of hot excifons 13. For the energy distribution of Eq.(2), the ILO calculated by Eq.(3) decreases so fast with increasing coupling order n that no high order sharp line should be observed. Furthermore, for a multi-phonon process, the phonon coupled to the exciton represents the lattice vibration of the QW which has the band gap energy of h co; whereas for a cascade process, the phonon coupled
Vol. 88, No 9
ULTRATHIN II-VI QUANTUM WELLS
to the exciton represents the lattice vibration of the QW which has the band gap t~¢x~-(n-1)hf~ for the nth order coupling. Fig.4 shows that the mean LO phonon energy coupled for each excitation is a function of the exciting photon energy (the band gap), so that a cascade process (Eq.3) should induce a coupling order (n) dependence of the LO phonon energy for certain excitation photon energies, i.e. h ~(n). Fig.6 presents the LO phonon energy as a function of the coupling order for excitation h co= 2.47 and 2.41eV, i.e. 496.5 and 514.5nm laser lines, respectively. It shows that the phonon energy does not
691
depend on the coupling order. This result also eliminates the cascade tunnelling process. 4.Conclusion
From an analysis of the energy relaxation of Iocalised excitons, the sharp lines shown in the resonantlyexcited PL spectra is shown to be due to a LO phononassisted tunnelling of exciton between classically localised states in QWs. Acknowledgement--We would like to thank P.J.Parbrook, P.J.Wright and B.Cockayne for providing the samples.
References
1.
2.
3.
4.
5.
F.Yang, P.J.Parbrook, B.Henderson, K.P. O'Donnell, P.J.Wright and B.Cockayne, J.Lum. 53, 427(1992), Proceeding of the 8th Int.Conf. on Dynamical Processes in Excited States of Solids, Leiden, the Netherlands, 28-31 August, 1991. J.H.Collet, H.Kalt, L.S.Dang, J.Cibert, K. Saminadayer and S.Tatarenko, Phys. Rev. B43, 6843 (1991). H.Kalt, J.H.Collet, S.D.Baranovskii, R.Saleh, P.Thomas, L.S.Dang, and J.Cibert, Phys. Rev. B45, 4253(1992). F. Yang, P.J.Parbrook, B.Henderson, K.P.O~Donnell, P.J.Wright and B.Cockayne, Appl. Phys. Lett. 59, 2142(1991). K.P.O'Donnell, P.J.Parbrook, B.Henderson, C. Trager-Cowan, X.Chen, F.Yang, M.P.Hallsall, P.J.Wright and B.Cockayne, J.Crystal Growth 101, 554 (1990).
6.
7. 8. 9.
10. 11. 12. 13.
P.J.Parbrook, P.J.Wright, B.Cockayne, A.G. CuUis, B.Henderson and K.P.O'Donnell, J. Crystal Growth 106, 503 (1990). Landolt-Bornstein New_Series 17B, edited by K. H. Hellwege (Springer Verlag, Berlin, 1982). O.Brafman, Solid State Comm. 11, 447(1972). M.Wilkinson, F.Yang, E.J.Austin and K.P.O~Donnell, J.Phys.: Condens. Matter 4, 8863 (1992). F.Yang, M.Wilkinson, E.J.Austin and K.P.O~Donnell, Phys.Rev.Lett. 70, 323(1993). F.Yang, B.Henderson and K.P.O'Donnell, Physica B 185, 362(1993). G.D.Mahan, Many-Particle Physics (Plenum, New York, 1981), pp.36-38. N.S.Weingreen, K.W.Jacobsen and J.W.Wilkins, Phys.Rev.Lett. 61, 1396 (1988). S.Permogorov, Phys.Status Solidi (b) 68, 9(1975).
0038-1098/93 $6 00+ 00 Pergamon Press Ltd
Solid State Communications, Vol. 88, No. 9, pp. 687-691, 1993. Printed in Great Britain
OPTICAL SPECTRA OF ULTRATHIN II-VI QUANTUM WELLS Fang Yang*, B. Henderson and K.P.O'Donnell Department of Physics and Applied Physics, University of Strathclyde, Glasgow G40NG, Scotland, UK
(Received 9 July 1993, acceptedfor publication 15 October 1993 by G. Bastard) Abstract Ultrathin wide band gap II-VI quantum wells (QW's) feature a large inhomogeneous broadening of exciton transitions in the optical absorption and photoluminescence (PL) spectra. When the photon energy of the excitation laser approaches, or lies within the broadened PL band, a series of additional sharp peaks appear at low energy. The laser induced linear polarisation of the sharp peaks enables the clear demonstration of evenly spaced phonon-assisted transitions. The relative intensity of the phonon-assisted transitions and the mean phonon energy depend strongly on the photon energy of the exciting light. A new model of phonon-assisted relaxation between classically localised states explains these results well.
1.Introduction
has been described previously O. All ultra thin MQWs of the Zn(Cd)S(Se) demonstrate similar spectra as discussed later. Here we present a study of a CdSe MQW (as an example), which consists of 100 alternate layers of CdSe (lnm) and ZnSe (5nm) grown at 300°C, on top of a ZnSe buffer layer 0.8~m thick, on a (100) GaAs substrate.
Thin II-VI semiconductor multiple quantum wells (MQW) are particularly interesting materials in which to study the relaxation of the exciton and the interaction of the exciton with optical phononsl,2,3. Because of the large electron and hole masses, excitons which penetrate the barrier region in very thin quantum wells of III-V materials would be confined in similar structures based on II-VI materials. In thin II-VI quantum wells, the exciton energy depends strongly on the well thickness. Samples with fair structural quality feature strongly broadened optical spectra. For example, in a MQW containing periods of 1.2nm ZnSe and 4.6nm ZnS, a one monolayer fluctuation in well thickness induces a spread of about 100meV in the excitonic absorption4 and the photoluminescence (PL) 5 lines. This energy spread is larger than both the exciton binding energy and the longitudinal optical (LO) phonon energy. Recently, the energy relaxation of the exciton between localised states was found in CdSe-ZnSe MQWs 1 by measurement of the steady and near resonant-excited PL spectra and in CdTe-ZnTe single quantum wells by using time-resolved photoluminescence spectroscopy2, 3. However, the mechanism of the relaxation has not been understood completely. In this paper, we analyse resonantly excited PL spectra of ultra thin II-VI MQWs to reveal the mechanism of the energy relaxation of the exciton between localised states. The growth of binary wide band gap MQWs of the Zn(Cd)S(Se) family by metal organic vapour phase epitaxy
2.Experiments When the sample was cooled to 10K and excited with the multiline UV output of an Ar+ laser (351.1363.6nm), a single luminescence band is observed which peaks near 520nm (Fig.l). Its large spectral bandwidth (-0.2eV) points to gross inhomogeneous broadening. The absorption spectrum of the sample peaks at 480nm and has a line width consistent with an estimated well width variation of about one monolayer. Exciting the MQW with the individual visible lines of the Ar+ laser, at wavelengths close to or within the luminescence envelope produces the spectra in Fig.2, which feature a series of additional sharp lines. The energy separation of the sharp lines is of the order of magnitude expected for optical phonons of CdSe or ZnSe. It is interesting that the sharp lines appear only at wavelengths where the broad PL band appears. For the 457.9nm excitation, the barely resolved peak marked '3' in Fig.2 corresponds to a displacement from the laser line equal to the energy of 3 LO phonons. The first and second order sharp lines can not be detected at all, as is shown more clearly in the derivative form of the spectrum in the inset of Fig.2. The sharp lines are more intense when detected with the polarisation aligned with that of the exciting light, as shown (//) in Fig.3. This property allows the polarisation difference spectrum to be obtained by subtracting the 'orthogonar (_1_) spectrum from the 'aligned' (//) one. An
*Present address: Department of Physics and Astronomy, University of Wales College of Cardiff, Cardiff CF1 3TH,UK. 687
ULTRATHIN
450
500 WAVELENGTH
550
II-VI QUANTUM WELLS
600
(nm)
and UV laser Fig.1 Absorption luminescence spectra of a CdSe-ZnSe 10K.
excited photoMQW sample at
Vol. 88, No. 9
example of the narrowed fluorescence line obtained in this way is reproduced in the inset of Fig.3. The unpolarised background is eliminated to reveal a series of lines which are equally spaced in energy. In addition, the polarisation difference spectrum is found to be independent of the orientation of the excitation polarisation with the respect to the crystal axes. At elevated temperature, the sharp lines decrease in intensity and are barely resolved above 1OOK. The mean spacing of the sharp lines varies with the photon energy of excitation as shown in Fig.4. The energy of the l-1 LO phonon for ZnSe, 32meV, is indicated on the figure. Although the phonon energies of cubic CdSe are still to be determined, the energy of the l-1 axial LO phonon of hexagonal CdSe, 26 meV7, is also shown on the figure as a reference. The phonon energies measured from the energy separation of sharp peaks in spectra of the ultrathin CdSe wells in ZnSe barrier almost exactly span the range between 32 and 26 meV. Since ZnCdSe exhibits one mode behaviour8, apparently the phonons which take part in the production of the sharp lines may represent an average taken by an exciton over a region containing both Zn and Cd ions. These phonons may be excitations of a quantum alloy which has larger Cd content in the thicker wells selectively excited by the laser lines of longer wavelength. In this model we might expect to observe a saturation of the phonon energy at the CdSe value in the thicker wells. This has not been observed, suggesting that the LO phonon energy in cubic CdSe may in fact be rather less than our estimate. A detailed study of phonon modes in ultrathin II-VI quantum wells is underway.
3.Discussion From the spectrum shown in Fig. 1, the inhomogeneous broadening of the absorption and PL are -
10
II ASERr’l
650 WAVELENGTH (nm)
Fig.2 Resonantly excited photoluminescence spectra of the CdSe-ZnSe MQW sample. Upwards arrows indicate the laser lines. The feature marked ‘3’ is described in the text. Inset A reproduction of the spectrum of 457.9nm excited in derivative form illustrates the photon coupling in detail.
480
510
540
WAVELENGTH
570
600
(nm)
Fig.3 Polarisation anisotropy of the photoluminescence with 488nm excitation. Shown in the inset is the difference spectrum obtained by subtracting the ‘orthogonal’ from the ‘aligned’ spectrum.
ULTRATHIN II-VI QUANTUM WELLS
Vol. 88, No. 9
33
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PHOTON ENERGY (eV) Fig.4 Variation of the mean peak separation with photon energy. The straight line is a least-square fit to the data.
200meV in the CdSe MQW, corresponding to spatial variations of the QW width of one monolayer. The overlap of the absorption and PL bands is of the same order. Previous studies on the relation of optical spectra and the quantum well width fluctuation9,10 reveal that the line shape of the exciton absorption peak mirrors the distribution of the optical band gap of the QW, which accurately reflects the distribution of the exciton transition energy if we assume the exciton binding energy is independent of QW width. The line shape of the exciton luminescence mirrors the energy distribution of the local minima of the band gap, which is approximately equal to the energy distribution of classically localised excitons. We note that the line shape of the exciton absorption peak contains information about both delocalised and localised excitons, while the line shape of PL contains only the information on the localised exciton. Hence, the energy overlap of localised and delocalised excitons is represented by the overlap of the absorption and the PL spectra, -200meV in this CdSe MQW sample. Illumination at fixed photon energy may produce localised excitons in one part of the MQW and delocalised excitons in another. Consider the events which follow the incidence of the exciting light on the sample and which result in the emission of light via exciton decay. Excitation at energies higher than the band gap of the barrier material (EgZnSe = 2.82eV at 10 K) generates electron-hole pairs throughout the sample. Energy and momentum relaxation via phonon emission favours the formation of excitons in the CdSe QWs. Within the wells, excitons further lose potential energy by emitting phonons and eventually occupy localised states, which subsequently recombine radiatively. No resolvable structure is observed in the PL spectrum because an exciton may occupy any localised state in a
689
band which has a large and continuous range of energies due to the QW width fluctuation. Excitation at lower photon energies directly creates quantum well excitons with zero momentum, which may be localised or delocalised depending on the environment at the point of exciton formation. The delocalised excitons relax in a similar manner to that described above. Their annilailation results in a broad structureless background emission which is however truncated at the exciting photon energy, as shown in Fig.2. The localised excitons created by resonant excitation cannot relax in the same manner since they are already at local minima. Instead, they recombine directly producing resonant fluorescence (RF). The sharp lines shown in Fig.2, with energy lower than the RF by an integer number of optical phonons, indicate that the RF is not the only recombination process, and suggest that energy transfer between localised exciton states, i.e. tunnelling, must occur via optical phonon emission. Recombination of the resulting localised excitons produces the sharp lines at fi ¢0-nhf~, where h co is the energy of exciting photon (and the primary exciton), h ~ is the energy of the phonon coupled to the primary exciton, and n is the coupling number. Obviously a thermally-activated hopping process can be eliminated, because it produces delocalised excitons first and hence does not lead to a series of sharp lines. In principle, the probability of phonon-assisted exciton tunnelling between localised states in this CdSeZnSe MQW can be calculated, by considering the Schr'odinger equation of the electron and phonon in a 2dimensional disordered potential with an electron-phonon interaction Hamiltanian. This topic is itself equivalent to the content of a paper, and thus beyond the concern here. In what follows, we use our knowledge on exciton localisation to estimate qualitatively the phonon assisted tunnelling probability between localised states and compare this estimate with the experimental results. For a multi-phonon process, the tunnelling probability P, is approximated by the product of the densities of the primary (absorbing) Na(hto ) and the final (emitting) Ne(hC0-nhf~) localised exciton states and a phonon coupling factor g
ILo(hO~-nh~)= gn(ho~)Na(hOONe(hO)-nhf~)
(1)
where h 0~ and h f~ are the absorbing photon and the phonon energy respectively, and, n is the phonon coupling order. The strongest phonon-assisted line appears where the product of Na(t~ co) and Ne(h o)-nti f~) have the largest value. The coupling of a localised electronic state to the three dimensional phonons of the host material is well known in solid state physics 11. In polar semiconductor, the Fr61ich interaction is the most important, so that the dominant electron-phonon coupling is to the LO-phonons. According to Ref.12, the coupling constant g is a function of the electronic localisation length L0. For CdSe, g=ll.54/L0(/~); and for ZnSe, g=7.25/L0(/~). L0 is determined by the smallest value of those under quantum
690
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4
ORDER OF SHARPLINE
Fig.5 Calculated relative intensity of sharp lines. Value are normalised to a maximum of 1 and zeros are shifted for clarity.
Fig.6 The energy separation between the sharp line and laser are illustrated as a function of the coupling order n. The energy corresponding to each step of n represent the LO phonon energy for the nth order coupling.
confinement for the electron, hole and exciton. For the widest QW which is CdSe rich, L 0 is -12/~, g ~1. The narrowest QW is ZnSe rich, but the localisation length is > 12,~ because the QW is too narrow to confine the electron, hole and exciton in 2D, hence we estimate g --0.5. The relative density of localised and delocalised states in QWs has been described in detail 9. For simplicity, we present here in short form the results of the semiempirical consideration in Ref.10. In a QW sample, the random fluctuations of well width and alloy composition lead to an approximately Gaussian distribution of the exciton energy at k--0. We assume that the total exciton density N(h co) o~ exp[-4ln2(E-Eo)2/l-'], where E o is the mean energy and F is the full width at half maximum of the distribution. Resonantly excited excitons can be locallsed or delocalised in the random potential of which N(E) is the distribution function. An exciton is defined to be localised if it is found at a local minimum of the potential, which has a density N 1 given approximately by
Fig.2 for the 457.9 nm excitation, the missing phonon orders in the experiment are explained by the lack of the 'receiving' density of localised exciton states in this model. The photon energy of the strongest phonon-assisted line shifts gradually as the excitation moves into the broad PL band. These predictions agree qualitatively well with the experimental results. Hence we believe that the sharp lines are due to the exciton tunnelling between localised states. A prediction of the absolute intensity requires not only a full quantum mechanical calculation of the probability of the phonon assisted-exciton tunnelling, but also a more precise measurement of the distribution of the localised states since it is not as ideal as predicted by Eq.(2) in a real sample. Finally, one may ask if the LO-phonon-assisted tunnelling is a cascade process. If this is the case then instead of using Eq.(1), the sharp line intensity would be proportional to
I LO ( hco - n h ~ ) = fi gi N~ (hco - ntif~) N(E) N (E)=
(2) E-E
1
1 + exp (
(3)
i=1 O
/ F/4
+F )
Substituting Eq.(2) into Eq.(1) for E=h co and h c0-nh ~, we may obtain the relative intensities of phonon-assisted lines for all excitation energies as shown in Fig.5, with g(h co) set from 0.5 to 1 changing linearly with h co. In Fig.5, the calculated intensities are normalised to the maximum value of each excitation. Compared to those spectra in
where the coupling in Eq.(1) is replaced by the multiple product of the densities of intermediate exciton states. Eq.(3) is different from that for the cascade momentum relaxation of hot excifons 13. For the energy distribution of Eq.(2), the ILO calculated by Eq.(3) decreases so fast with increasing coupling order n that no high order sharp line should be observed. Furthermore, for a multi-phonon process, the phonon coupled to the exciton represents the lattice vibration of the QW which has the band gap energy of h co; whereas for a cascade process, the phonon coupled
Vol. 88, No 9
ULTRATHIN II-VI QUANTUM WELLS
to the exciton represents the lattice vibration of the QW which has the band gap t~¢x~-(n-1)hf~ for the nth order coupling. Fig.4 shows that the mean LO phonon energy coupled for each excitation is a function of the exciting photon energy (the band gap), so that a cascade process (Eq.3) should induce a coupling order (n) dependence of the LO phonon energy for certain excitation photon energies, i.e. h ~(n). Fig.6 presents the LO phonon energy as a function of the coupling order for excitation h co= 2.47 and 2.41eV, i.e. 496.5 and 514.5nm laser lines, respectively. It shows that the phonon energy does not
691
depend on the coupling order. This result also eliminates the cascade tunnelling process. 4.Conclusion
From an analysis of the energy relaxation of Iocalised excitons, the sharp lines shown in the resonantlyexcited PL spectra is shown to be due to a LO phononassisted tunnelling of exciton between classically localised states in QWs. Acknowledgement--We would like to thank P.J.Parbrook, P.J.Wright and B.Cockayne for providing the samples.
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