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Ensemble Monte Carlo simulation of optical excitation of AlGaAs
Physica B 272 (1999) 419 } 421
Ensemble Monte Carlo simulation of optical excitation of AlGaAs L. Shifren!,*, D.K. Ferry!, K.T. Tsen" !Department of Electrical Engineering and Center for Solid State Electronics Research, Arizona State University, Tempe, AR 85287-5706, USA "Department of Physics and Astronomy, Arizona State University, Tempe, AR 85287-1504, USA
Abstract Ternary semiconductor alloys such as Al Ga As provide a challenge to our understanding of the physics of alloy x 1~x e!ects in semiconductors. It is well known that Al Ga As is a two-mode alloy, even though the material is supposed to x 1~x be a smooth random alloy. Recent experimental work has suggested that there is a non-monotonic behavior to the relative scattering strength of the Al mode. We have studied the transport of a range of alloys using the ensemble Monte Carlo (EMC) approach, incorporating the two LO modes, non-equilibrium phonons, and the full electron}electron interaction. We "nd that anomalies in the ratio of scattering strength, as measured by Raman scattering, are sensitive to the details of the phonon distribution function and the shape of the Raman signal with time. Di!erences in various reported results arise primarily from di!erences in measurement technique. ( 1999 Elsevier Science B.V. All rights reserved. Keywords: AlGaAs; Raman scattering; Electron}phonon interaction
Ternary semiconductor alloys, such as Al Ga As, have important device applications, x 1~x but also provide a challenge to our understanding of the physics behind alloy-related e!ects in semiconductors. It is well known that Al Ga As is x 1~x a two-mode alloy, in that two longitudinal-optical (LO) modes of lattice vibration exist, even though the material is supposed to be a smooth random alloy. While early studies of Raman scattering of Kash et al. [1] suggested that the relative fraction of scattering by the Al modes varied linearly across the alloy, more recent work has suggested that
* Corresponding author. Tel.: #1-480-965-3452; fax: #1480-965-8058. E-mail address: [email protected] (L. Shifren)
there is a non-monotonic behavior to this quantity [2]. Consequently, we have studied the transport in bulk Al Ga As using the ensemble Monte Carlo x 1~x (EMC) approach. We have incorporated the two LO modes with matrix elements determined by the measured frequencies of the two modes, which is consistent with a linear theory of lattice vibrations [3,4]. In addition, we incorporate the non-equilibrium phonons, by which the scattering strengths are measured in Raman scattering, and the full electron}electron interaction through the molecular dynamics approach [5]. The role of intervalley scattering has been evaluated carefully, as has the e!ect of the band-gap widening with alloy composition. We introduce non-equilibrium phonons via the method of Lugli et al. [6]. This involves using a
0921-4526/99/$ - see front matter ( 1999 Elsevier Science B.V. All rights reserved. PII: S 0 9 2 1 - 4 5 2 6 ( 9 9 ) 0 0 3 1 4 - 2
420
L. Shifren et al. / Physica B 272 (1999) 419 }421
phonon distribution that is updated as the simulation progresses. The emission and absorption of phonons is governed by this distribution through the use of a rejection technique. The presence of the non-equilibrium phonon distribution a!ects the relative scattering rate of the polar modes as these are dependent on the phonon distribution. The two polar modes are included in the code as separate polar optical scattering events. We use the measured phonon dispersions [7] of the two modes to calculate the coupling strengths via [3,4]
K A K A
BA BA
B B
u2 !u2 u2 !u2 L01 T01 T02 L01 , (1) e u2 !u2 L01 L02 = G! u2 !u2 u2 !u2 1 T02 T01 . L02 " L02 (2) e u2 !u2 c L01 L02 = AThe C valley is treated as non-parabolic, while the X and L valleys are treated as parabolic. We incorporate the full range of scattering events, including both equivalent and non-equivalent intervalley scattering, which plays a large role in cooling the hot electrons, but has little e!ect on the relative strength of the polar modes. When electrons are scattered by optical phonons, they can only emit (or absorb) a narrow range of phonon wave vectors. It is important to note that the Raman signal senses a di!erent wave vector. The maximum and minimum phonon wave vectors for the scattering and Raman processes at an energy corresponding to injection into the conduction band are obtained by [8]
1 c
"
S S
2mu q "k# k2! , .!9 +
(3)
2mu q "k! k2! . .*/ +
(4)
The Raman wave vector changes little with changing concentration, while the upper and lower limits of phonon emission change substantially. This effect is primarily due to the increase of the band gap as the Al concentration increases. The Raman q vector begins to move out of the main part of the phonon distribution as the value of x increases. This e!ect is a contributor to the non-monotonic behavior that is seen in the relative coupling
Fig. 1. Phonon distribution of the Al and Ga type modes in Al Ga As plotted versus wave number and time. Brightness 0.3 0.7 corresponds to amplitude of the distribution. The solid line is the Raman q vector, which lies far from the distribution peak, especially in the Al mode.
strength measured by Raman scattering. However, the main e!ect lies in the Raman pulse shape, as discussed further below. Two separate distributions were kept, one for the Ga mode and one for the Al mode. Phonons could be emitted or absorbed, as well as being allowed to decay via the phonon lifetime, whereby the polar optical phonons decay via an anharmonic process. The phonon distribution for each mode is plotted in Fig. 1. This plot is for the two modes at a concentration x"0.3. As can be seen, the Raman q vector lies far from the peak and any measurements would be on a shoulder of the distribution, which drops o! quickly on the lower sides of the distribution. This e!ect becomes more exaggerated for larger Al concentrations. Moreover, during the scattering
L. Shifren et al. / Physica B 272 (1999) 419 }421
421
where *n is the change in the phonon distribution q (the non-equilibrium phonons) of the particular mode. Here, we evaluate the time dependence of the phonon population at the Raman wave number. In Fig. 2(a), the composition dependence of the peak of the phonon pulse, for a 0.5 ps excitation pulse, is plotted along with the data from Kash et al. [1]. It may be seen that this method, which is used in the experiment, gives the linear increase found by these authors. In Fig. 2(b), the composition dependence of the integrated phonon pulse and the data of Tsen et al. are plotted. The integrated data is appropriate to the long pulse length used in this latter experiment, and again agreement is found with the experiment. In conclusion, there are two factors which a!ect these results. First, the Raman probe does not measure the major part of the disturbed phonon distribution for either mode. Secondly, the spread of the distribution through continuous absorption and emission of phonons leads to a complicated temporal behavior of the phonon Raman signal, which causes the integrated behavior to di!er from the simple peak behavior. As a result, the manner in which the measurement is carried out will directly a!ect the results obtained. Acknowledgements
Fig. 2. The compositional dependence of the fractional Al scattering. (a) Measured from the peak of the phonon Raman signal (open circles) and the data of Kash et al. [1] (solid circles). (b) Measured from the integrated phonon Raman signal (open circles) and the data of Tsen et al. [2] (solid circles). The di!erences in these two results arise from complicated temporal behavior of the phonon Raman signal, rather than a simple exponential decay.
process, phonons can be shifted from the high population region to the shoulders, which results in a complicated variation of the Raman phonon intensity with time. In addition, the Al distribution tends to shift further than the Ga distribution, due to the modes larger phonon wave vector. The relative strength of the Al mode is described by *n q Af" , *n #*n q Aq G!
(5)
This work is supported in part by the O$ce of Naval Research. The authors wish to express their appreciation to S. M. Goodnick, S. Wigger, and R. Akis for helpful discussions. References [1] J.A. Kash, S.S. Jha, J.C. Tsang, Phys. Rev. Lett. 58 (1987) 1869. [2] K.T. Tsen, D.K. Ferry, A. Salvador, H. Morkoc, Phys. Rev. Lett. 80 (1998) 4807. [3] U. Bockelmann, G. Bastard, Phys. Rev. B 42 (1990) 8947. [4] L. Swierkowski, W. Zawadzki, Y. Guldner, C. Rigaux, Solid State Commun. 27 (1978) 1245. [5] A.M. Kriman, M.J. Kann, D.K. Ferry, R. Joshi, Phys. Rev. Lett. 65 (1990) 1619. [6] P. Lugli, C. Jacoboni, L. Reggiani, P. Kocevar, Appl. Phys. Lett. 50 (1987) 1251. [7] S. Adachi, GaAs and Related Material, World Scienti"c, New York, 1994, p. 95 [8] D.K. Ferry, Semiconductors, Macmillan, New York, 1991, p. 212.
Ensemble Monte Carlo simulation of optical excitation of AlGaAs L. Shifren!,*, D.K. Ferry!, K.T. Tsen" !Department of Electrical Engineering and Center for Solid State Electronics Research, Arizona State University, Tempe, AR 85287-5706, USA "Department of Physics and Astronomy, Arizona State University, Tempe, AR 85287-1504, USA
Abstract Ternary semiconductor alloys such as Al Ga As provide a challenge to our understanding of the physics of alloy x 1~x e!ects in semiconductors. It is well known that Al Ga As is a two-mode alloy, even though the material is supposed to x 1~x be a smooth random alloy. Recent experimental work has suggested that there is a non-monotonic behavior to the relative scattering strength of the Al mode. We have studied the transport of a range of alloys using the ensemble Monte Carlo (EMC) approach, incorporating the two LO modes, non-equilibrium phonons, and the full electron}electron interaction. We "nd that anomalies in the ratio of scattering strength, as measured by Raman scattering, are sensitive to the details of the phonon distribution function and the shape of the Raman signal with time. Di!erences in various reported results arise primarily from di!erences in measurement technique. ( 1999 Elsevier Science B.V. All rights reserved. Keywords: AlGaAs; Raman scattering; Electron}phonon interaction
Ternary semiconductor alloys, such as Al Ga As, have important device applications, x 1~x but also provide a challenge to our understanding of the physics behind alloy-related e!ects in semiconductors. It is well known that Al Ga As is x 1~x a two-mode alloy, in that two longitudinal-optical (LO) modes of lattice vibration exist, even though the material is supposed to be a smooth random alloy. While early studies of Raman scattering of Kash et al. [1] suggested that the relative fraction of scattering by the Al modes varied linearly across the alloy, more recent work has suggested that
* Corresponding author. Tel.: #1-480-965-3452; fax: #1480-965-8058. E-mail address: [email protected] (L. Shifren)
there is a non-monotonic behavior to this quantity [2]. Consequently, we have studied the transport in bulk Al Ga As using the ensemble Monte Carlo x 1~x (EMC) approach. We have incorporated the two LO modes with matrix elements determined by the measured frequencies of the two modes, which is consistent with a linear theory of lattice vibrations [3,4]. In addition, we incorporate the non-equilibrium phonons, by which the scattering strengths are measured in Raman scattering, and the full electron}electron interaction through the molecular dynamics approach [5]. The role of intervalley scattering has been evaluated carefully, as has the e!ect of the band-gap widening with alloy composition. We introduce non-equilibrium phonons via the method of Lugli et al. [6]. This involves using a
0921-4526/99/$ - see front matter ( 1999 Elsevier Science B.V. All rights reserved. PII: S 0 9 2 1 - 4 5 2 6 ( 9 9 ) 0 0 3 1 4 - 2
420
L. Shifren et al. / Physica B 272 (1999) 419 }421
phonon distribution that is updated as the simulation progresses. The emission and absorption of phonons is governed by this distribution through the use of a rejection technique. The presence of the non-equilibrium phonon distribution a!ects the relative scattering rate of the polar modes as these are dependent on the phonon distribution. The two polar modes are included in the code as separate polar optical scattering events. We use the measured phonon dispersions [7] of the two modes to calculate the coupling strengths via [3,4]
K A K A
BA BA
B B
u2 !u2 u2 !u2 L01 T01 T02 L01 , (1) e u2 !u2 L01 L02 = G! u2 !u2 u2 !u2 1 T02 T01 . L02 " L02 (2) e u2 !u2 c L01 L02 = AThe C valley is treated as non-parabolic, while the X and L valleys are treated as parabolic. We incorporate the full range of scattering events, including both equivalent and non-equivalent intervalley scattering, which plays a large role in cooling the hot electrons, but has little e!ect on the relative strength of the polar modes. When electrons are scattered by optical phonons, they can only emit (or absorb) a narrow range of phonon wave vectors. It is important to note that the Raman signal senses a di!erent wave vector. The maximum and minimum phonon wave vectors for the scattering and Raman processes at an energy corresponding to injection into the conduction band are obtained by [8]
1 c
"
S S
2mu q "k# k2! , .!9 +
(3)
2mu q "k! k2! . .*/ +
(4)
The Raman wave vector changes little with changing concentration, while the upper and lower limits of phonon emission change substantially. This effect is primarily due to the increase of the band gap as the Al concentration increases. The Raman q vector begins to move out of the main part of the phonon distribution as the value of x increases. This e!ect is a contributor to the non-monotonic behavior that is seen in the relative coupling
Fig. 1. Phonon distribution of the Al and Ga type modes in Al Ga As plotted versus wave number and time. Brightness 0.3 0.7 corresponds to amplitude of the distribution. The solid line is the Raman q vector, which lies far from the distribution peak, especially in the Al mode.
strength measured by Raman scattering. However, the main e!ect lies in the Raman pulse shape, as discussed further below. Two separate distributions were kept, one for the Ga mode and one for the Al mode. Phonons could be emitted or absorbed, as well as being allowed to decay via the phonon lifetime, whereby the polar optical phonons decay via an anharmonic process. The phonon distribution for each mode is plotted in Fig. 1. This plot is for the two modes at a concentration x"0.3. As can be seen, the Raman q vector lies far from the peak and any measurements would be on a shoulder of the distribution, which drops o! quickly on the lower sides of the distribution. This e!ect becomes more exaggerated for larger Al concentrations. Moreover, during the scattering
L. Shifren et al. / Physica B 272 (1999) 419 }421
421
where *n is the change in the phonon distribution q (the non-equilibrium phonons) of the particular mode. Here, we evaluate the time dependence of the phonon population at the Raman wave number. In Fig. 2(a), the composition dependence of the peak of the phonon pulse, for a 0.5 ps excitation pulse, is plotted along with the data from Kash et al. [1]. It may be seen that this method, which is used in the experiment, gives the linear increase found by these authors. In Fig. 2(b), the composition dependence of the integrated phonon pulse and the data of Tsen et al. are plotted. The integrated data is appropriate to the long pulse length used in this latter experiment, and again agreement is found with the experiment. In conclusion, there are two factors which a!ect these results. First, the Raman probe does not measure the major part of the disturbed phonon distribution for either mode. Secondly, the spread of the distribution through continuous absorption and emission of phonons leads to a complicated temporal behavior of the phonon Raman signal, which causes the integrated behavior to di!er from the simple peak behavior. As a result, the manner in which the measurement is carried out will directly a!ect the results obtained. Acknowledgements
Fig. 2. The compositional dependence of the fractional Al scattering. (a) Measured from the peak of the phonon Raman signal (open circles) and the data of Kash et al. [1] (solid circles). (b) Measured from the integrated phonon Raman signal (open circles) and the data of Tsen et al. [2] (solid circles). The di!erences in these two results arise from complicated temporal behavior of the phonon Raman signal, rather than a simple exponential decay.
process, phonons can be shifted from the high population region to the shoulders, which results in a complicated variation of the Raman phonon intensity with time. In addition, the Al distribution tends to shift further than the Ga distribution, due to the modes larger phonon wave vector. The relative strength of the Al mode is described by *n q Af" , *n #*n q Aq G!
(5)
This work is supported in part by the O$ce of Naval Research. The authors wish to express their appreciation to S. M. Goodnick, S. Wigger, and R. Akis for helpful discussions. References [1] J.A. Kash, S.S. Jha, J.C. Tsang, Phys. Rev. Lett. 58 (1987) 1869. [2] K.T. Tsen, D.K. Ferry, A. Salvador, H. Morkoc, Phys. Rev. Lett. 80 (1998) 4807. [3] U. Bockelmann, G. Bastard, Phys. Rev. B 42 (1990) 8947. [4] L. Swierkowski, W. Zawadzki, Y. Guldner, C. Rigaux, Solid State Commun. 27 (1978) 1245. [5] A.M. Kriman, M.J. Kann, D.K. Ferry, R. Joshi, Phys. Rev. Lett. 65 (1990) 1619. [6] P. Lugli, C. Jacoboni, L. Reggiani, P. Kocevar, Appl. Phys. Lett. 50 (1987) 1251. [7] S. Adachi, GaAs and Related Material, World Scienti"c, New York, 1994, p. 95 [8] D.K. Ferry, Semiconductors, Macmillan, New York, 1991, p. 212.