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Polarised hot electron luminescence in p-GaAs: Electric field effects
Physica E 2 (1998) 478—482
Polarised hot electron luminescence in p-GaAs: Electric field effects V. Saxena*, J.P. Evans, H.P. Hughes Optoelectronics Group, Cavendish Laboratory, Madingley Road, Cambridge, CB3 0HE, UK
Abstract Electrons photo-excited to high-energy conduction band states of GaAs exhibit complex energy and momentum distributions determined by the anisotropic valence band structure and the optical matrix elements. In p-type GaAs a fraction of these hot electrons combine with localised acceptor states, producing a hot electron luminescence (HEL) spectrum with a cascade of peaks corresponding to discrete energy losses resulting from LO-phonon emission. The highest peak involves unscattered electrons, and their energy distribution is due to warping of the initial heavy-hole (HH) bands. We report measurements of the line shape of this 0-HH peak, and its polarisation profile which identifies emission from electrons along particular directions. An applied electric field of 1 kV cm~1 distorts the hot electron momentum distribution, and this is reflected in the polarisation profile. These line shapes and profiles, with and without field, are calculated using a computer model incorporating a k·p band structure and optical matrix elements, the effect of electric field being included using a k-broadening model. The data and model are in good quantitative agreement assuming an electron lifetime of &100 fs, and confirm the expected differences in the profiles for different excitation polarisation states and applied field directions. ( 1998 Elsevier Science B.V. All rights reserved. Keywords: Hot electron luminescence; GaAs
1. Introduction: Hot electron to neutral acceptor continuouswave (CW) spectroscopy in III—V semiconductors has recently revealed interesting polarisation phenomena [1—3] which provide insights into ultrafast scattering processes and the band structures of III—V materials. We have studied the polarised luminescence from a steady-state distribution of
* Corresponding author. Fax: #44 1223 353397; e-mail: [email protected].
electrons which have been photoexcited by polarised CW laser light into the conduction band of p-type GaAs. A cascade of peaks can be observed, and meV resolution reveals fine detail in this hot electron luminescence (HEL). The highest energy unscattered peak and its polarisation properties with and without an applied electric field are investigated here. A typical measured HEL spectrum is shown in Fig. 1. The leading 0-HH peak corresponds to the recombination of unrelaxed electrons photoexcited from the heavy hole (HH) valence band. This 0-HH peak is highly polarised for linearly polarised light
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excitation due to the anisotropic momentum distribution of the photoelectrons [1—3]. Here we discuss the 0-HH HEL peak only, and in particular its degree of linear polarisation (DoLP) profile, with and without an applied electric field E of 1 kV cm~1 along the [1 1 0] and [1 0 0] directions, and with the exciting light polarised either parallel or orthogonal to the applied field. A computer model incorporating a 16]16 k · p matrix to calculate the band structure and transition matrix elements from the dipole model [3] was also developed to calculate the 0HH peak line shape and DoLP profile with and without field.
2. Experimental setup and measurements HEL spectra were recorded using a DILOR-XY 0.5 m triple monochromator with a resolution of 0.67 meV. The excitation energy E was 1.6774 eV, %9 supplied by a low power (&9 mW) titanium-sapphire laser to keep the photoelectron density low (&2]1015 cm~3) to eliminate electron—electron scattering which becomes significant at densities around 1017 cm~3 [4,5]. The samples were lightly doped p-type (Be: N "1017 cm~3) bulk GaAs A grown using molecular beam epitaxy, and were cooled to 6.2 K. The low doping density ensured that the acceptor sites are localised and have a negligible energy dispersion. An electric field of 1 kV cm~1 was applied along ohmically contacted channels oriented along the [1 1 0] or [1 0 0] directions, and the sample was illuminated by linearly polarised light with its electric field vector e such that eEE or eoE. The DoLP was obtained from HEL spectra polarised parallel and perpendicular to the excitation, i.e. e@Ee and e@oe, where e@ is the electric field vector of the HEL.
3. Experimental results and calculations Fig. 1 shows HEL spectra recorded for e@Ee and e@oe with eE[1 1 0]. We will discuss only the leading peak in the cascade which arises from the HH band and which corresponds to electrons that have
Fig. 1. Typical HEL spectra from p-type GaAs for e@Ee (solid lines) and e@oe (dashed lines) without an applied electric field and for an eE[1 1 0]. The spectra show a series of peaks corresponding to electrons from the heavy-hole band (0-HH, 1-HH, 2-HH) and the start of a second series due to electrons from the light-hole band (0-LH). The numbering of the peaks corresponds to the number of LO phonons emitted.
not undergone relaxation by LO-phonon emission. The spectral width of this 0-HH peak arises from three sources; the anisotropic HH valence band structure, the electron lifetime due to LO phonon emission, and the broadening of the acceptor level [6]. The acceptor broadening is estimated to be 5.7$0.6 meV from the bandedge luminescence. The expected luminescence spectra of the 0-HH peak for a given excitation energy and polarisation is calculated using Fermi’s Golden Rule [3] and including direct transitions at k only:
P
DM (k, e)D2DM (k, e)D2d(E (k)!E (k) 7# #! # 7 BZ !E )d(E (k)#E !E !E ) dk, (1) %9 # 0 ! where the integration is carried out over the first Brillouin zone. E (k) and E (k) are the conduction # 7 and valence band energies both measured from the conduction band minimum (i.e. E (k) are positive # and E (k) are negative). E is the band-gap energy, 7 0 E the acceptor level binding energy (28 meV for ! I(E )" l
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V. Saxena et al. / Physica E 2 (1998) 478—482
Be), E the excitation energy and E the lumines%9 cence energy. The two d functions conserve energy for the excitation and recombination transitions. M (k, e) and M (k, e) are the angular parts of the 7# #! matrix elements for excitation from the valence to the conduction band, and for recombination from the conduction band to the acceptor level, respectively. The combined transition probability for excitation and recombination is proportional to [3]
G A B HG A B H 1!
k·e 2 k
1!
k · e@ 2 . k
(2)
The computer model evaluates Eq. (1) over first 10% of the Brillouin zone via the tetrahedron method of Allen [7] to produce a calculated 0-HH peak profile; because of the HH valence band warping, the low-energy side of the peak represents emission from electrons with kES1 0 0T and the high-energy side from electrons with kES1 1 1T, while electrons from S1 1 0T states appear near the peak maximum. The calculations are then convolved with a Voigt function representing the Lorentzian broadening due to the finite electron lifetime and the Gaussian broadening of the acceptor level. An LO-phonon scattering lifetime q "100 fs gave a good fit beLO tween the model and experiment. The line shapes depend critically on the choice of Luttinger parameters, and the commonly adopted set c "6.85, 1 c "2.1 and c "2.9 [8] were used here. 2 3 The calculated and experimental line shapes for eE[1 1 0] and eE[1 11 0] (for e@Ee only) are shown in Fig. 2a and b, and the corresponding DoLP profiles in Fig. 2c and d, for situations with and without an applied field E along [1 1 0]; the DoLP profiles for eE[1 0 0] and eE[0 1 0] are shown in Fig. 2e and f. For E"0, the calculated DoLP profiles are in fair qualitative agreement with the measurements: for eE[1 1 0] the calculated maximum DoLP ("0.18) lies on the high-energy side of the peak corresponding to the S1 1 1T directions, but for eE[1 0 0] the calculated maximum DoLP ("0.22) lies on the low energy side corresponding to the S1 0 0T directions. When E is applied along [1 1 0], the peaks for both e@Ee and e@oe (Fig. 2a and b) are broadened, more prominently when eoE, as can be understood
intuitively as follows. The overall effect of the field during the lifetime of the photoelectron is to shift the momentum distribution along the field direction by *k"!eEq /+. When eEE the distribuLO tion is such that most of the electrons have their k oriented perpendicular to E; this means that DkD changes relatively little under the effect of the field, producing only a small energy broadening. When eoE most photoelectrons have k oriented along E, so DkD changes significantly with field producing more significant energy broadening. This shift can be included in the calculation of the HEL line shape with M (k)PM (k#*k), #! #! and E (k)PE (k#*k) in the d-function in Eq. (1), # # but agreement with experiment is then poor. The more acurate calculated line shapes presented here are based on the field broadening model which supposes that the electrons photo-injected at k achieve a steady-state population of N(k ) 0 0 through scattering and CW excitation; when E is applied the electrons drift in k while undergoing LO-phonon scattering, resulting in a steady-state distribution given by n(k)"N(k ) exp (!+Dk!k D/eDEDq ). (3) 0 0 LO This broadened k distribution is easily incorporated into the line shape calculation via a series of simple shifts in k. Line shapes calculated using q "100 fs and an LO acceptor broadening of 5.7 meV are in good agreement with experiment (Fig. 2a and b) for different configurations of the ingoing and outgoing polarisations. The DoLP profiles of the 0-HH peak with EE[1 1 0] are shown in Fig. 2c and d and it can be seen that the calculated and experimentally obtained profiles are in fair quantitative and good qualitative agreement. However, when E is applied along [1 0 0] the calculated DoLP profiles agree less well with the experimental profiles as shown in Fig. 2e and f.
4. Conclusion We have presented new experimental results which show the HEL line shape and the variation in the DoLP across the 0-HH peak under different configurations of exciting light and applied electric
V. Saxena et al. / Physica E 2 (1998) 478—482
481
Fig. 2. HEL measurements (grey lines) and calculations (black lines) of the 0-HH peak for different excitation configurations with (dashed line) and without (solid line) an electric field. (a) and (c) are the line shapes and the DoLP profiles for eEEE[1 1 0]; (b) and (d) are line shapes and DoLP profiles for excitation eoEE[1 1 0]; (e) and (f ) are the DoLP profiles for e E and o to EE[1 0 0]. The energies corresponding to the high symmetry directions are shown in (c), (d), (e), and (f ).
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V. Saxena et al. / Physica E 2 (1998) 478—482
field. We have considered linearly polarised excitation and fields applied along the [1 1 0] and [1 0 0] directions, and have found that the DoLP profiles change differently when the exciting light is polarised either parallel or perpendicular to the applied field. The calculated line shapes and DoLP profiles based on the field broadening model using an electron lifetime of 100 fs and an acceptor broadening of 5.7 meV are in fair quantitative and good qualitative agreement with the observations.
Acknowledgements The assistance of Dr. D.R. Richards is gratefully acknowledged. We also thank Dr. G.W. Smith (DRA) for providing the MBE samples and Mr. G. Wiheicki for the subsequent processing.
References [1] B.P. Zakharchenya, D.N. Mirlin, V.I. Perel, I.I. Reshina, Sov. Phys. Usp 25 (3) (1982) 143. [2] M.A. Alekseev, I.A. Karlik, D.N. Mirlin, V.F. Sapega, Sov. Phys. Semicond. 23 (5) (1989) 479. [3] G. Fasol, W. Hackenberg, H.P. Hughes, K. Ploog, E. Bauser, H. Kano, Phys. Rev. B 41 (1990) 1461. [4] W. Hackenberg, G. Fasol, Appl. Phys. Lett. 57 (1990) 174. [5] G. Fasol, H.P. Hughes, Phys. Rev. B 33 (1986) 2953. [6] M.A. Alekseev, I.Ya. Kirklik, I.A. Merkulov, D.N. Mirlin, L.P. Nikitin, V.F. Sapega, Sov. Phys. Solid State 26 (11) (1984) 2025. [7] P.B. Allen, Phys. Stat. Sol. B 120 (1983) 529. [8] M. Cardona, N.E. Christensen, G. Fasol, Phys. Rev. B 38 (1988) 1806.
Polarised hot electron luminescence in p-GaAs: Electric field effects V. Saxena*, J.P. Evans, H.P. Hughes Optoelectronics Group, Cavendish Laboratory, Madingley Road, Cambridge, CB3 0HE, UK
Abstract Electrons photo-excited to high-energy conduction band states of GaAs exhibit complex energy and momentum distributions determined by the anisotropic valence band structure and the optical matrix elements. In p-type GaAs a fraction of these hot electrons combine with localised acceptor states, producing a hot electron luminescence (HEL) spectrum with a cascade of peaks corresponding to discrete energy losses resulting from LO-phonon emission. The highest peak involves unscattered electrons, and their energy distribution is due to warping of the initial heavy-hole (HH) bands. We report measurements of the line shape of this 0-HH peak, and its polarisation profile which identifies emission from electrons along particular directions. An applied electric field of 1 kV cm~1 distorts the hot electron momentum distribution, and this is reflected in the polarisation profile. These line shapes and profiles, with and without field, are calculated using a computer model incorporating a k·p band structure and optical matrix elements, the effect of electric field being included using a k-broadening model. The data and model are in good quantitative agreement assuming an electron lifetime of &100 fs, and confirm the expected differences in the profiles for different excitation polarisation states and applied field directions. ( 1998 Elsevier Science B.V. All rights reserved. Keywords: Hot electron luminescence; GaAs
1. Introduction: Hot electron to neutral acceptor continuouswave (CW) spectroscopy in III—V semiconductors has recently revealed interesting polarisation phenomena [1—3] which provide insights into ultrafast scattering processes and the band structures of III—V materials. We have studied the polarised luminescence from a steady-state distribution of
* Corresponding author. Fax: #44 1223 353397; e-mail: [email protected].
electrons which have been photoexcited by polarised CW laser light into the conduction band of p-type GaAs. A cascade of peaks can be observed, and meV resolution reveals fine detail in this hot electron luminescence (HEL). The highest energy unscattered peak and its polarisation properties with and without an applied electric field are investigated here. A typical measured HEL spectrum is shown in Fig. 1. The leading 0-HH peak corresponds to the recombination of unrelaxed electrons photoexcited from the heavy hole (HH) valence band. This 0-HH peak is highly polarised for linearly polarised light
1386-9477/98/$19.00 ( 1998 Elsevier Science B.V. All rights reserved PII: S 1 3 8 6 - 9 4 7 7 ( 9 8 ) 0 0 0 9 8 - 8
V. Saxena et al. / Physica E 2 (1998) 478—482
479
excitation due to the anisotropic momentum distribution of the photoelectrons [1—3]. Here we discuss the 0-HH HEL peak only, and in particular its degree of linear polarisation (DoLP) profile, with and without an applied electric field E of 1 kV cm~1 along the [1 1 0] and [1 0 0] directions, and with the exciting light polarised either parallel or orthogonal to the applied field. A computer model incorporating a 16]16 k · p matrix to calculate the band structure and transition matrix elements from the dipole model [3] was also developed to calculate the 0HH peak line shape and DoLP profile with and without field.
2. Experimental setup and measurements HEL spectra were recorded using a DILOR-XY 0.5 m triple monochromator with a resolution of 0.67 meV. The excitation energy E was 1.6774 eV, %9 supplied by a low power (&9 mW) titanium-sapphire laser to keep the photoelectron density low (&2]1015 cm~3) to eliminate electron—electron scattering which becomes significant at densities around 1017 cm~3 [4,5]. The samples were lightly doped p-type (Be: N "1017 cm~3) bulk GaAs A grown using molecular beam epitaxy, and were cooled to 6.2 K. The low doping density ensured that the acceptor sites are localised and have a negligible energy dispersion. An electric field of 1 kV cm~1 was applied along ohmically contacted channels oriented along the [1 1 0] or [1 0 0] directions, and the sample was illuminated by linearly polarised light with its electric field vector e such that eEE or eoE. The DoLP was obtained from HEL spectra polarised parallel and perpendicular to the excitation, i.e. e@Ee and e@oe, where e@ is the electric field vector of the HEL.
3. Experimental results and calculations Fig. 1 shows HEL spectra recorded for e@Ee and e@oe with eE[1 1 0]. We will discuss only the leading peak in the cascade which arises from the HH band and which corresponds to electrons that have
Fig. 1. Typical HEL spectra from p-type GaAs for e@Ee (solid lines) and e@oe (dashed lines) without an applied electric field and for an eE[1 1 0]. The spectra show a series of peaks corresponding to electrons from the heavy-hole band (0-HH, 1-HH, 2-HH) and the start of a second series due to electrons from the light-hole band (0-LH). The numbering of the peaks corresponds to the number of LO phonons emitted.
not undergone relaxation by LO-phonon emission. The spectral width of this 0-HH peak arises from three sources; the anisotropic HH valence band structure, the electron lifetime due to LO phonon emission, and the broadening of the acceptor level [6]. The acceptor broadening is estimated to be 5.7$0.6 meV from the bandedge luminescence. The expected luminescence spectra of the 0-HH peak for a given excitation energy and polarisation is calculated using Fermi’s Golden Rule [3] and including direct transitions at k only:
P
DM (k, e)D2DM (k, e)D2d(E (k)!E (k) 7# #! # 7 BZ !E )d(E (k)#E !E !E ) dk, (1) %9 # 0 ! where the integration is carried out over the first Brillouin zone. E (k) and E (k) are the conduction # 7 and valence band energies both measured from the conduction band minimum (i.e. E (k) are positive # and E (k) are negative). E is the band-gap energy, 7 0 E the acceptor level binding energy (28 meV for ! I(E )" l
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V. Saxena et al. / Physica E 2 (1998) 478—482
Be), E the excitation energy and E the lumines%9 cence energy. The two d functions conserve energy for the excitation and recombination transitions. M (k, e) and M (k, e) are the angular parts of the 7# #! matrix elements for excitation from the valence to the conduction band, and for recombination from the conduction band to the acceptor level, respectively. The combined transition probability for excitation and recombination is proportional to [3]
G A B HG A B H 1!
k·e 2 k
1!
k · e@ 2 . k
(2)
The computer model evaluates Eq. (1) over first 10% of the Brillouin zone via the tetrahedron method of Allen [7] to produce a calculated 0-HH peak profile; because of the HH valence band warping, the low-energy side of the peak represents emission from electrons with kES1 0 0T and the high-energy side from electrons with kES1 1 1T, while electrons from S1 1 0T states appear near the peak maximum. The calculations are then convolved with a Voigt function representing the Lorentzian broadening due to the finite electron lifetime and the Gaussian broadening of the acceptor level. An LO-phonon scattering lifetime q "100 fs gave a good fit beLO tween the model and experiment. The line shapes depend critically on the choice of Luttinger parameters, and the commonly adopted set c "6.85, 1 c "2.1 and c "2.9 [8] were used here. 2 3 The calculated and experimental line shapes for eE[1 1 0] and eE[1 11 0] (for e@Ee only) are shown in Fig. 2a and b, and the corresponding DoLP profiles in Fig. 2c and d, for situations with and without an applied field E along [1 1 0]; the DoLP profiles for eE[1 0 0] and eE[0 1 0] are shown in Fig. 2e and f. For E"0, the calculated DoLP profiles are in fair qualitative agreement with the measurements: for eE[1 1 0] the calculated maximum DoLP ("0.18) lies on the high-energy side of the peak corresponding to the S1 1 1T directions, but for eE[1 0 0] the calculated maximum DoLP ("0.22) lies on the low energy side corresponding to the S1 0 0T directions. When E is applied along [1 1 0], the peaks for both e@Ee and e@oe (Fig. 2a and b) are broadened, more prominently when eoE, as can be understood
intuitively as follows. The overall effect of the field during the lifetime of the photoelectron is to shift the momentum distribution along the field direction by *k"!eEq /+. When eEE the distribuLO tion is such that most of the electrons have their k oriented perpendicular to E; this means that DkD changes relatively little under the effect of the field, producing only a small energy broadening. When eoE most photoelectrons have k oriented along E, so DkD changes significantly with field producing more significant energy broadening. This shift can be included in the calculation of the HEL line shape with M (k)PM (k#*k), #! #! and E (k)PE (k#*k) in the d-function in Eq. (1), # # but agreement with experiment is then poor. The more acurate calculated line shapes presented here are based on the field broadening model which supposes that the electrons photo-injected at k achieve a steady-state population of N(k ) 0 0 through scattering and CW excitation; when E is applied the electrons drift in k while undergoing LO-phonon scattering, resulting in a steady-state distribution given by n(k)"N(k ) exp (!+Dk!k D/eDEDq ). (3) 0 0 LO This broadened k distribution is easily incorporated into the line shape calculation via a series of simple shifts in k. Line shapes calculated using q "100 fs and an LO acceptor broadening of 5.7 meV are in good agreement with experiment (Fig. 2a and b) for different configurations of the ingoing and outgoing polarisations. The DoLP profiles of the 0-HH peak with EE[1 1 0] are shown in Fig. 2c and d and it can be seen that the calculated and experimentally obtained profiles are in fair quantitative and good qualitative agreement. However, when E is applied along [1 0 0] the calculated DoLP profiles agree less well with the experimental profiles as shown in Fig. 2e and f.
4. Conclusion We have presented new experimental results which show the HEL line shape and the variation in the DoLP across the 0-HH peak under different configurations of exciting light and applied electric
V. Saxena et al. / Physica E 2 (1998) 478—482
481
Fig. 2. HEL measurements (grey lines) and calculations (black lines) of the 0-HH peak for different excitation configurations with (dashed line) and without (solid line) an electric field. (a) and (c) are the line shapes and the DoLP profiles for eEEE[1 1 0]; (b) and (d) are line shapes and DoLP profiles for excitation eoEE[1 1 0]; (e) and (f ) are the DoLP profiles for e E and o to EE[1 0 0]. The energies corresponding to the high symmetry directions are shown in (c), (d), (e), and (f ).
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V. Saxena et al. / Physica E 2 (1998) 478—482
field. We have considered linearly polarised excitation and fields applied along the [1 1 0] and [1 0 0] directions, and have found that the DoLP profiles change differently when the exciting light is polarised either parallel or perpendicular to the applied field. The calculated line shapes and DoLP profiles based on the field broadening model using an electron lifetime of 100 fs and an acceptor broadening of 5.7 meV are in fair quantitative and good qualitative agreement with the observations.
Acknowledgements The assistance of Dr. D.R. Richards is gratefully acknowledged. We also thank Dr. G.W. Smith (DRA) for providing the MBE samples and Mr. G. Wiheicki for the subsequent processing.
References [1] B.P. Zakharchenya, D.N. Mirlin, V.I. Perel, I.I. Reshina, Sov. Phys. Usp 25 (3) (1982) 143. [2] M.A. Alekseev, I.A. Karlik, D.N. Mirlin, V.F. Sapega, Sov. Phys. Semicond. 23 (5) (1989) 479. [3] G. Fasol, W. Hackenberg, H.P. Hughes, K. Ploog, E. Bauser, H. Kano, Phys. Rev. B 41 (1990) 1461. [4] W. Hackenberg, G. Fasol, Appl. Phys. Lett. 57 (1990) 174. [5] G. Fasol, H.P. Hughes, Phys. Rev. B 33 (1986) 2953. [6] M.A. Alekseev, I.Ya. Kirklik, I.A. Merkulov, D.N. Mirlin, L.P. Nikitin, V.F. Sapega, Sov. Phys. Solid State 26 (11) (1984) 2025. [7] P.B. Allen, Phys. Stat. Sol. B 120 (1983) 529. [8] M. Cardona, N.E. Christensen, G. Fasol, Phys. Rev. B 38 (1988) 1806.