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Self-scattering in Monte Carlo calculations of transient dynamic response in semiconductors
Volume 78A, number 4
PHYSICS LETTERS
18 August 1980
SELF-SCATTERING IN MONTE CARLO CALCULATIONS OF TRANSIENT DYNAMIC RESPONSE IN SEMICONDUCTORS * D.K. FERRY Department of Electrical Engineering, Colorado State University, Fort Collins, CO 80523, USA
Received 3 April 1980
We have calculated the transient dynamic response in GaAs by a Monte Carlo approach. We find that the results are extremely sensitive to the value of r, the self-scattering parameter, which differs from results previously reported in the literature. However, when 1’ is selected so that (at) 1/F, comparable results are obtained to those calculated without utilizing a self-scattering assumption. -~
The Monte Carlo (MC) method is a stochastic method in which sequences of random numbers are used in computer simulations of physical systems with many degrees of freedom [1]. In application, the MC method yields information on model systems, which it is hoped mirror the true system, and which in principle are exact in that the results can be made extremely accurate, apart from statistical errors if sufficient computer time is invested. The MC method has found considerable application in semiconductor transport problems [2—6],due to the fact that these methods
bution function were observed to last for a considerably shorter period of time than had been indicated by calculations using a MC technique. At that time, it was felt that this difference arose from limitations in the MC technique in evaluating energy relaxation processes. The decay of velocity overshoot, arising in transient dynamic response of electrons in many semiconductors, is largely a consequence of energy relaxation, rather than momentum relaxation, and thus is sensitive to details of the energy distribution function f0(E),just as is repopulation in a many-
are exact in the sense that no approximations to the physics, other than in the material model itself, are introduced and arbitrarily high accuracy is obtainable. It is especially useful in studying transport problems when details of the band structure parameters are unknown and must be varied [5]. It is important to note, however, that the great majority of applications of MC techniques have been to steady-state transport problems. Only recently have MC techniques been utilized for calculations of the dynamic response of electrons in semiconductors [6—11],and to small signal a.c. response of the conductivity in semiconductors [12]. In a recent calculation of the dynamic response of electrons in GaAs [131, however, transient velocity effects determined using a drifted maxwellian distri-
valleyed band structure. Thus it is important that any simulation technique treat this portion of the distribution function accurately. This, of course, is the case for the drifted maxwellian. In the MC technique, the velocity is found to converge in a few thousand samples [3] but the distribution function is slow to converge and may require many tens of thousands of iterations [4,14,15]. In this paper, we investigate the use of MC techniques to calculate the dynamic response of electrons in semiconductors. We find that contrary to the above remarks, MC can be used to give a good description of the dynamic response, and that when this is correctly done, the results agree with the drifted maxwellian as expected. Contrary to previous results, we find that the calculations are sensitive to the value of r, the self-scattering parameter. However, when F is selected so that (z~t)= 1/F .c~((r)), the results converge to a transient dynamic
*
Work supported by the US Office of Naval Research.
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18 August 1980
response comparable to that obtained earlier by the drifted maxwellian approach [13]. These results are confirmed by recent calculations in which self-scattering is not utilized [6]. In the following, we first review the transient dynamic response and then describe the ensemble Monte Carlo technique used, If a distribution of electrons is suddenly subjected to a high, spatially homogeneous electric field, the ensemble accelerates eventually to a new distribution characteristic of the value of the high electric field [16]. The manner in which the ensemble achieves this new distribution, on a temporal scale, is referred to as the transient dynamic response (TDR). The TDR is important for semiconductor devices operating at very high fields or for those which exist in a physically small space [171,since for these situations the ensemble may never exist in an equilibrium steadystate. Two characteristic times generally dominate the TDR: the energy relaxation time i-a, and the momentum relaxation time T~ [181. If these times are of comparable size, or if T~
momentum distribution corresponding to the lattice temperature. The distribution function can also be calculated by histograms that record the trajectory of the electron in a three-dimensional k-space. When a large number of these histories have been calculated, the TDR of a typical electron is determined by averaging over the ensemble of individual histories. This approach is limited by the mathematical convergence problems discussed in the introduction and by the use of running estimators to determine v(t) from the time sequence of values generated during the generation of the history [4]. The use of estimators in TDR is quite questionable as the ensemble is not stationary and the response is, in general, expected to be nonmarkovian [20—22]in this region, an effect which will especially show up in MC simulations. Many of the problems arising from the use of MC techniques to calculate the TDR in semiconductors were apparently anticipated by Lebwohl and Price [151.They suggested a hybrid method which we refer to here as the ensemble Monte Carlo (EMC). An ensemble of electrons is adopted. This ensemble is composed of N electrons, with variables R~,i = 1 N, where R1 = {ki, x1, bi, . ..}, with b1 the band index,
such as InSb or InAs, where scattering is dominated by intravalley processes, especially in the case of the polar LO phonon, which is relatively inefficient in momentum relaxation [19]. If, however, T~> Tm, the velocity response can rise to a value characteristic of the initial energy of the distribution, then decay to its steady-state response as the energy rises to its steady-state value. This is often the case for semiconductors that exhibit intervalley transfer, especially if nonequivalent sets of valleys are involved, Such overshoot velocity effects are seen, for example, in Si and GaAs. The TDR is significant when considering size effects in semiconductor devices. In GaAs, for example, when the spatial extent of the high-field region drops below 0.25 jim, we expect the performance to be dominated by TDR, as electrons in this region may never achieve their steady-state velocity, The traditional manner of calculating the TDR of electrons by a MC procedure is to generate a series of electron histories by following single electrons through their k-space flights using a MC procedure [4]. A large number of these histories are generated, with the initial states selected from the equilibrium
and the set R, includes all necessary descriptors of the electron’s state. At each time step, all R1 are calculated by a MC process, and the set {R1 } is treated as an ensemble evolving in time. The EMC method meets both criteria discussed in the previous section. A distribution function exists, and evolves with R~,and no estimators (in the sense of Fawcett et al. [4]) are used to calculate variables such as velocity and position. The velocity, for example, is calculated not from an estimator but rather is calculated by computing an ensemble average over {R1} at each time step, so that v(t) is therefore easily determined. The variance in n (t) is controlled by a sufficiently large value for N. We have used the EMC technique to investigate TDR in semiconductors. We use an ensemble with N 10~and find acceptable variance in v(t) when the self-scattering parameter, when used, is chosen sufficiently large. In fig. 1, the TDR for electrons in GaAs at 300 K, subjected to a static homogeneous field of 10kV/cm is shown. Three EMC results are shown for comparison, each the result of a single calculation for a single seed in the pseudo-random number generator. Each calculation is for a different value of F, the self-scattering parameter. In curve (a),
376
Volume 78A, number 4
PHYSICS LETTERS
GaAs 300K 10KV/cm
18 August 1980
°~oo
b
0
G
000 0
0 0
0
00
~L;1~H I
0
-~+ + 0
0
14 b-3x10 c- 5z10 4
4’
I
I
2
I 5
3 time
(10
6
I 7
sec.)
Fig. 1. The transient dynamic response (TDR) in GaAs for a field of 10 kV/cm. The curves labeled (a), (b) and (c) are results using the EMC technique for F = I x 1014, 3 x 1014, and 5 x 10~, respectively. The solid curve arises from a calculation in which self-scattering was not used and is due to Littlejohn et al. [6].
r
= 1 X 1014 chosen for F ~X~(k). If several different values are used for the seed, and the different results averaged and smoothed, a result is obtained that compares well with the MC calculated earlier [7,9,10]. However, these curves all show indications of what may be numerical instability, as indicated by the large jumps in u(t) (at 0.5 Ps for example) and by the very large scatter in the overshoot region. For larger values of I’, the TDR is quickened and the velocity response, shown by curves (b) and (c), converges to a faster TDR characteristic. To check this result, equivalent calculations which were performed by the more difficult method in which the energy dependent timepropagator integrals were kept in their full form [61, i.e., no simplifying self-scattering is introduced, are shown as well. The results of this calculation, due to Littlejohn et al. [61,are shown by the solid curve in the figure, confirming our results by the EMC. The apparent dependence of the TDR on 1/F arises from the fact that the latter quantity represents a pseudorandom average time step for the process and, as in
any numerical simulation, must be chosen sufficiently small. This will be discussed in a separate publication. It should be added that the results shown for the fast responses in fig. I agree well with the TDR calculated by a drifted maxwellian approach [13], thus establishing a sounder basis for the latter approach to this particular problem. In summary, we have shown that MC techniques can be extended to accurate calculations of the TDR of carriers in semiconductors. The use of EMC avoids problems that may arise from the use of estimators as well as allowing accurate ensemble estimates to be made. Moreover, when the MC technique is accurately applied, it agrees well with calculations performed with the drifted maxwellian techniques. As a consequence, it is now clear that TDR can be treated by either of these techniques which will yield comparable results.
377
Volume 78A, number 4
PHYSICS LETTERS
References [1] J.M. Hammersley and D.C. Hanscomb, Monte Carlo methods (Methuen, London, 1964); I.M. Sobol, The Monte Carlo method (Univ. of Chicago Press, Chicago, 1974); Yu A. Shreider, The Monte Carlo method (Pergamon, New York, 1966). [2] T. Kurosawa, J. Phys. Soc. Japan 21 (Suppl.) (1966) 424. [3] A.I~.Boardman, W. Fawcett and H.D. Rees, Solid State Commun. 6 (1968) 305. [41 W. Fawcett, A.D. Boardman and S. Swain, J. Phys. Chem. Solids 31(1970)1963. [5] M.A. Littlejohn, J.R. Hauser and T.H. Glisson, AppI. Phys. Lett. 30 (1977) 242. [6] M.A. Littlejohn, L.A. Arledge, T.H. Glisson and J.R. Hauser, Electron. Lett. 15 (1979) 586. [7] J.G. Ruch, IEEE Trans. Electron. Dev. ED-19 (1972) 652. [8] M. Brauer, Phys. Stat. Sol. (b) 81(1977)147. [9] T.J. Maloney and J. Frey, J. Appi. Phys. 48(1977) 781.
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[10] S. Kratzer and J. Frey, J. Appl. Phys. 49 (1978) 4064. [11] G. Hill, P.N. Robson, A. Majerfeld and W. Fawcett, Electron. Lett. 13(1977)235. [121 P.A. Lebwohl, J. Appl. Phys. 44(1973)1744. [13] D.K. Ferry and J.R. Barker, Solid State Electron., to be published. [141 E.G.S. Paine. IBM J. Res. Dev. 13(1969) 562. [15] P.A. Lebwohl and P.J. Price, Solid State Commun. 9 (1970) 1221. [16] See, e.g., P. Price, Solid State Electron. 21(1978) 9; D.K. Ferry, in: Handbook of semiconductors, Vol. I, ed. W. Paul (North-Holland, Amsterdam), to be published. [171 J.R. Barker and D.K. Ferry, Solid State Electron., to be published. [18] P. Das and D.K. Ferry, Solid State Electron. 19 (1976) 851. 1191 E. Conwell, High field transport in semiconductors (Academic Press, New York, 1967). [20] J.G. Kirkwood, J. Chem. Phys. 14 (1946) 180. [21] R. Zwanzig, Phys. Rev. 124 (1961) 983. [22] R. Kubo, Rep. Frog. Phys. 29 (1966) 255.
PHYSICS LETTERS
18 August 1980
SELF-SCATTERING IN MONTE CARLO CALCULATIONS OF TRANSIENT DYNAMIC RESPONSE IN SEMICONDUCTORS * D.K. FERRY Department of Electrical Engineering, Colorado State University, Fort Collins, CO 80523, USA
Received 3 April 1980
We have calculated the transient dynamic response in GaAs by a Monte Carlo approach. We find that the results are extremely sensitive to the value of r, the self-scattering parameter, which differs from results previously reported in the literature. However, when 1’ is selected so that (at) 1/F
The Monte Carlo (MC) method is a stochastic method in which sequences of random numbers are used in computer simulations of physical systems with many degrees of freedom [1]. In application, the MC method yields information on model systems, which it is hoped mirror the true system, and which in principle are exact in that the results can be made extremely accurate, apart from statistical errors if sufficient computer time is invested. The MC method has found considerable application in semiconductor transport problems [2—6],due to the fact that these methods
bution function were observed to last for a considerably shorter period of time than had been indicated by calculations using a MC technique. At that time, it was felt that this difference arose from limitations in the MC technique in evaluating energy relaxation processes. The decay of velocity overshoot, arising in transient dynamic response of electrons in many semiconductors, is largely a consequence of energy relaxation, rather than momentum relaxation, and thus is sensitive to details of the energy distribution function f0(E),just as is repopulation in a many-
are exact in the sense that no approximations to the physics, other than in the material model itself, are introduced and arbitrarily high accuracy is obtainable. It is especially useful in studying transport problems when details of the band structure parameters are unknown and must be varied [5]. It is important to note, however, that the great majority of applications of MC techniques have been to steady-state transport problems. Only recently have MC techniques been utilized for calculations of the dynamic response of electrons in semiconductors [6—11],and to small signal a.c. response of the conductivity in semiconductors [12]. In a recent calculation of the dynamic response of electrons in GaAs [131, however, transient velocity effects determined using a drifted maxwellian distri-
valleyed band structure. Thus it is important that any simulation technique treat this portion of the distribution function accurately. This, of course, is the case for the drifted maxwellian. In the MC technique, the velocity is found to converge in a few thousand samples [3] but the distribution function is slow to converge and may require many tens of thousands of iterations [4,14,15]. In this paper, we investigate the use of MC techniques to calculate the dynamic response of electrons in semiconductors. We find that contrary to the above remarks, MC can be used to give a good description of the dynamic response, and that when this is correctly done, the results agree with the drifted maxwellian as expected. Contrary to previous results, we find that the calculations are sensitive to the value of r, the self-scattering parameter. However, when F is selected so that (z~t)= 1/F .c~((r)), the results converge to a transient dynamic
*
Work supported by the US Office of Naval Research.
375
Volume 78A, number 4
PHYSICS LETTERS
18 August 1980
response comparable to that obtained earlier by the drifted maxwellian approach [13]. These results are confirmed by recent calculations in which self-scattering is not utilized [6]. In the following, we first review the transient dynamic response and then describe the ensemble Monte Carlo technique used, If a distribution of electrons is suddenly subjected to a high, spatially homogeneous electric field, the ensemble accelerates eventually to a new distribution characteristic of the value of the high electric field [16]. The manner in which the ensemble achieves this new distribution, on a temporal scale, is referred to as the transient dynamic response (TDR). The TDR is important for semiconductor devices operating at very high fields or for those which exist in a physically small space [171,since for these situations the ensemble may never exist in an equilibrium steadystate. Two characteristic times generally dominate the TDR: the energy relaxation time i-a, and the momentum relaxation time T~ [181. If these times are of comparable size, or if T~
momentum distribution corresponding to the lattice temperature. The distribution function can also be calculated by histograms that record the trajectory of the electron in a three-dimensional k-space. When a large number of these histories have been calculated, the TDR of a typical electron is determined by averaging over the ensemble of individual histories. This approach is limited by the mathematical convergence problems discussed in the introduction and by the use of running estimators to determine v(t) from the time sequence of values generated during the generation of the history [4]. The use of estimators in TDR is quite questionable as the ensemble is not stationary and the response is, in general, expected to be nonmarkovian [20—22]in this region, an effect which will especially show up in MC simulations. Many of the problems arising from the use of MC techniques to calculate the TDR in semiconductors were apparently anticipated by Lebwohl and Price [151.They suggested a hybrid method which we refer to here as the ensemble Monte Carlo (EMC). An ensemble of electrons is adopted. This ensemble is composed of N electrons, with variables R~,i = 1 N, where R1 = {ki, x1, bi, . ..}, with b1 the band index,
such as InSb or InAs, where scattering is dominated by intravalley processes, especially in the case of the polar LO phonon, which is relatively inefficient in momentum relaxation [19]. If, however, T~> Tm, the velocity response can rise to a value characteristic of the initial energy of the distribution, then decay to its steady-state response as the energy rises to its steady-state value. This is often the case for semiconductors that exhibit intervalley transfer, especially if nonequivalent sets of valleys are involved, Such overshoot velocity effects are seen, for example, in Si and GaAs. The TDR is significant when considering size effects in semiconductor devices. In GaAs, for example, when the spatial extent of the high-field region drops below 0.25 jim, we expect the performance to be dominated by TDR, as electrons in this region may never achieve their steady-state velocity, The traditional manner of calculating the TDR of electrons by a MC procedure is to generate a series of electron histories by following single electrons through their k-space flights using a MC procedure [4]. A large number of these histories are generated, with the initial states selected from the equilibrium
and the set R, includes all necessary descriptors of the electron’s state. At each time step, all R1 are calculated by a MC process, and the set {R1 } is treated as an ensemble evolving in time. The EMC method meets both criteria discussed in the previous section. A distribution function exists, and evolves with R~,and no estimators (in the sense of Fawcett et al. [4]) are used to calculate variables such as velocity and position. The velocity, for example, is calculated not from an estimator but rather is calculated by computing an ensemble average over {R1} at each time step, so that v(t) is therefore easily determined. The variance in n (t) is controlled by a sufficiently large value for N. We have used the EMC technique to investigate TDR in semiconductors. We use an ensemble with N 10~and find acceptable variance in v(t) when the self-scattering parameter, when used, is chosen sufficiently large. In fig. 1, the TDR for electrons in GaAs at 300 K, subjected to a static homogeneous field of 10kV/cm is shown. Three EMC results are shown for comparison, each the result of a single calculation for a single seed in the pseudo-random number generator. Each calculation is for a different value of F, the self-scattering parameter. In curve (a),
376
Volume 78A, number 4
PHYSICS LETTERS
GaAs 300K 10KV/cm
18 August 1980
°~oo
b
0
G
000 0
0 0
0
00
~L;1~H I
0
-~+ + 0
0
14 b-3x10 c- 5z10 4
4’
I
I
2
I 5
3 time
(10
6
I 7
sec.)
Fig. 1. The transient dynamic response (TDR) in GaAs for a field of 10 kV/cm. The curves labeled (a), (b) and (c) are results using the EMC technique for F = I x 1014, 3 x 1014, and 5 x 10~, respectively. The solid curve arises from a calculation in which self-scattering was not used and is due to Littlejohn et al. [6].
r
= 1 X 1014 chosen for F ~X~(k). If several different values are used for the seed, and the different results averaged and smoothed, a result is obtained that compares well with the MC calculated earlier [7,9,10]. However, these curves all show indications of what may be numerical instability, as indicated by the large jumps in u(t) (at 0.5 Ps for example) and by the very large scatter in the overshoot region. For larger values of I’, the TDR is quickened and the velocity response, shown by curves (b) and (c), converges to a faster TDR characteristic. To check this result, equivalent calculations which were performed by the more difficult method in which the energy dependent timepropagator integrals were kept in their full form [61, i.e., no simplifying self-scattering is introduced, are shown as well. The results of this calculation, due to Littlejohn et al. [61,are shown by the solid curve in the figure, confirming our results by the EMC. The apparent dependence of the TDR on 1/F arises from the fact that the latter quantity represents a pseudorandom average time step for the process and, as in
any numerical simulation, must be chosen sufficiently small. This will be discussed in a separate publication. It should be added that the results shown for the fast responses in fig. I agree well with the TDR calculated by a drifted maxwellian approach [13], thus establishing a sounder basis for the latter approach to this particular problem. In summary, we have shown that MC techniques can be extended to accurate calculations of the TDR of carriers in semiconductors. The use of EMC avoids problems that may arise from the use of estimators as well as allowing accurate ensemble estimates to be made. Moreover, when the MC technique is accurately applied, it agrees well with calculations performed with the drifted maxwellian techniques. As a consequence, it is now clear that TDR can be treated by either of these techniques which will yield comparable results.
377
Volume 78A, number 4
PHYSICS LETTERS
References [1] J.M. Hammersley and D.C. Hanscomb, Monte Carlo methods (Methuen, London, 1964); I.M. Sobol, The Monte Carlo method (Univ. of Chicago Press, Chicago, 1974); Yu A. Shreider, The Monte Carlo method (Pergamon, New York, 1966). [2] T. Kurosawa, J. Phys. Soc. Japan 21 (Suppl.) (1966) 424. [3] A.I~.Boardman, W. Fawcett and H.D. Rees, Solid State Commun. 6 (1968) 305. [41 W. Fawcett, A.D. Boardman and S. Swain, J. Phys. Chem. Solids 31(1970)1963. [5] M.A. Littlejohn, J.R. Hauser and T.H. Glisson, AppI. Phys. Lett. 30 (1977) 242. [6] M.A. Littlejohn, L.A. Arledge, T.H. Glisson and J.R. Hauser, Electron. Lett. 15 (1979) 586. [7] J.G. Ruch, IEEE Trans. Electron. Dev. ED-19 (1972) 652. [8] M. Brauer, Phys. Stat. Sol. (b) 81(1977)147. [9] T.J. Maloney and J. Frey, J. Appi. Phys. 48(1977) 781.
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18 August 1980
[10] S. Kratzer and J. Frey, J. Appl. Phys. 49 (1978) 4064. [11] G. Hill, P.N. Robson, A. Majerfeld and W. Fawcett, Electron. Lett. 13(1977)235. [121 P.A. Lebwohl, J. Appl. Phys. 44(1973)1744. [13] D.K. Ferry and J.R. Barker, Solid State Electron., to be published. [141 E.G.S. Paine. IBM J. Res. Dev. 13(1969) 562. [15] P.A. Lebwohl and P.J. Price, Solid State Commun. 9 (1970) 1221. [16] See, e.g., P. Price, Solid State Electron. 21(1978) 9; D.K. Ferry, in: Handbook of semiconductors, Vol. I, ed. W. Paul (North-Holland, Amsterdam), to be published. [171 J.R. Barker and D.K. Ferry, Solid State Electron., to be published. [18] P. Das and D.K. Ferry, Solid State Electron. 19 (1976) 851. 1191 E. Conwell, High field transport in semiconductors (Academic Press, New York, 1967). [20] J.G. Kirkwood, J. Chem. Phys. 14 (1946) 180. [21] R. Zwanzig, Phys. Rev. 124 (1961) 983. [22] R. Kubo, Rep. Frog. Phys. 29 (1966) 255.