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Theory of collision broadening through a sequence of scattering events with applications to semiclassical Monte Carlo
Microelectronic Engineering 47 (1999) 353-355
Theory of collision broadening through a sequence of scattering events with applications to semiclassical Monte Carlo Leonard
F. Register
Beckman Institute Urbana-Champaign,
and Kafl Hess and Coordinated Science Laboratory, University of Illinois at 405 N. Mathews Av., Urbana, IL 61801, U.S.A.
Insights $re presented into the modeling of collision broadening within semiclassical Monte Car10simulatlrin of carrier transport. The effects of collision broadening through a sequence of scattering events are analyzed via both basic theory and first-principles simulation using SchrGdinger Equation Monte Carlo (SEMC), a simulation method designed to bridge the gap from quantum to clasleical transport. Based on this analysis, an improved treatment of collision broadening for semiclassical Monte Carlo is suggested.
Of importance in semiclassical Monte Carlo (SCMC) simulation of energetic carrier transport effects such as impact ionization, hot-carrier degradation and transport over heterointerfaces, is the treatment of the well recognized high-order quantum process of collision broadening (CB) that allows for nominally non-energy conserving transitions between carrier states. However, there is no consensus opinion on the proper treatment of CB in SCMC, and the high-energy tail of the carrier distribution is sensitive to the specific model of CB used [l-3]; even seemingly rigorous analysis of CB can lead to seemingly implausible SCMC simulation results [2]. A common approach to introducing CB into SCMC is to allow transitions between carrier states within a non-zero range about the energy conserving transition such that E n+l
=
En
T
hWn+l,n
+
(1)
AEn+l,n
for phonon emission or absorption, respectively, where h~,.,+~,~ is the phonon energy [l-3]. The distribution for stochastically selecting the CB contribution AEn+l,n depends on the scattering rates R of, in principle, both the initial n and final n + 1 states [2]. These scattering rates can be taken to be a function of energy only, R, =
R(&) ,
(2)
if directional dependence is ignored for simplicity. Simulations results produced using the quantum transport simulation method Schriidinger Equation Monte Carlo (SEMC) [4], h owever, reveal significantly different CB effects, not for individual scattering events, but for a sequence of scattering events. In This work was supported
by the US AR0
0167-9317/99/$ - see front matter PII: SO167-9317(99)00232-4
0
under
Con. Nos. DAAH0495-1-0362
1999 Elsevier Science B.V. All rights twmved.
and DAAG55-98-1-0306
354
L.F. Register, K. Hess I Microelectronic
normally incident electron phonon,emission (35 meV) ,\
Engineering
47 (1999) 353-355 I
10’
(a) 5
(b)
i
-
h”
hw-_l-
after emission
I I I
1
-I
n ,
initial state
262 meV 231 meV
energy 4
,k,,,
1
_-____..
cI_ position
__
I.
75
rim-Y
bare-electron
energy (meV)
Figure 1: (a) Electron capture by a quantum well via phonon emission. The bare electron effective mass, m* = O.O57m,, is modeled as position independent so that the well-normal component of bare-electron energy El in the single bound state subband is independent of transverse momentum ,$I. (b) Transition probability to bare-electron states as a function of energy El + tL21ii112/2m* for emission of one phonon, and for sequential emission of two phonons via two different intermediate bare-electron energies. The slight offsets of the peaks from integer multiples of fiw are due to real self-energy differences. contrast to the case for SCMC, CB is an inherent feature of SEMC calculations, an unavoidable by-product of allowing real scattering that cannot be independently adjusted. Consider the system of Fig. l(a) where an electron of well defined initial energy Eo, 25 meV for the bare electron relative to the band edge (24.6 meV for the polaron), is incident on a quantum well with one bound state subband; only emission of 35 meV polar-optical phonons is allowed (0 K lattice temperature) such that the carrier cannot undergo real scattering until it reaches the vicinity of the well. Upon reaching the well, the carrier can emit a sequence of phonons as it is first captured into the well and then scatters within the well subband. Figure l(b) displays the SEMC calculated transition probabilities to electron states via emission of one phonon, and via the sequential emission of two phonons, as a function of the electron bare energies El and E2, respectively. The results for emission of two phonons are subdivided by the energy of the intermediate state El with two cases shown, Er = Eo - Iiw and El = Eo - 1.5tiw. Figure 1 (b) clearly exhibits CB effects, but in contrast to Eq. (I), the most probable value of E2, E2 21 E. - 2fiw, is essentially independent of the value of El. The behavior displayed in the SEMC simulations is traceable to the Schrodinger equation of the true coupled carrier-phonon system that SEMC mimics, [H< + Hd +
ff&&G, 9’)= W(~, 8,,
(3)
where & is the total energy of the coupled carrier-phonon system, Hz is the carrier Hamiltonian, Hd is the Hamiltonian of the phonons (lattice vibrational modes), and Hza provides coupling b_etwe_enthe two systems. Fzr energy eigenstates of the uncoupled system, [Hi + H&b(k)$(Q) = [Ei + fd]$(k)$(Q), it is Hz,4 that allows scattering among
L.F. Register, K. Hess I Microelectronic Engineering 47 (1999) 353-355
355
states and inherently broadens the states in energy; much like the coupling of the quasibound state(s) of a resonant tunneling diode (RTD) to the adjacent carrier reservoirs both allows transport through the RTD and broadens the quasi-bound state(s) through which the carriers pass. (As in the SEMC simulations, the bare carrier eigenstates $(ic’) may not have well defined momentum.) For coupling to/through any particular carrierphonon quasi/bare energy eigenstate $(&)$(Qn) th a t is only weakly broadened, energy conservation in Eq. (3) requires &, + EQ~ z ,!?‘,(with allowances for real energy shifts) independent of the perhaps strong broadening of any previous state $(&_,)$(o,_,). For coupling through strongly broadened bare states n, it is the overlap of the broadened energy spectrum of that state with f that is of importance, not its overlap with the bare energies of previous states &i _ t-E4 _ . In brief, energy conservation within the coupled carrier-phonon system of El r3) d o&mnot allow for the accumulation of uncertainty in the bare-carrier energy through a sequence of scattering events as produced by Eq. (1). Although any treatment of CB in SCMC is inherently approximate, a simple alternative to Eq. (1) that avoids such artificial accumulation of energy uncertainty is available, E n+1 - E, - AE, F fiww,n + AEn+ >
(4
where the A_&, AE,+1 distributions are associated with the real scattering rates I&, R n+l of the individual states n, n + 1, respectively. Note that the net allowed CB for individual scattering events still depends on the scattering rates in both the initial and final states consistent with Ref. [2]. Also, because the range of allowed final states depends on E, - AE, rather than just E,, the scattering rates take the form R, = R(\&J, E, - AEn). For momentum randomizing scattering events such as those that dominate in silicon, the scattering rates reduce to a function of only one variable again, & = R(En - AEn). Eqs. (3) and (4) could have significant implications for SCMC simulations of hot carrier transport effects such as impact ionization [1,2] or even for just the calculation of the impact ionization rates [5]. Also, the tail of the broadening distribution becomes less critical; even fully Lorentzian broadening (justified or not) and effectively small high energy tails of the carrier distribution should not be mutually exclusive. In summary, a modified treatment of CB for SCMC has been suggested that eliminates the accumulation of uncertainty in the carrier energy and related artifacts of past treatments, yet retains the essential physics of CB.
1. Y.-C. Chang, 2. L. Reggiani,
D. Z.-Y. Ting, J. Y. Tang and K. Hess, Appl. Phys. Lett. 42, 76 (1983). P. Lugli and A. P. Jauho,
Phys. Rev. B 36, 6602 (1987).
3. K. Kim, B. A. Mason, and K. Hess, Phys. Rev. B 36, 6547 (1987). 4. L. F. Register, in Selected Topics in Electronics and Systems, Vol. 14: QuantumBased Electronic Devices and Systems, vol. ed.: M. Dutta and M. A. Stroscio, series ed.: P. K. Tien (World Scientific, Singapore, 1998) pp. 251-279, or L. F. Register, International Journal of High Speed Electronics and Systems 9 251 (1998). 5. J. Bude, K. Hess and G. J. Iafrate,
Phys. Rev. B 45, 10958 (1992).
Theory of collision broadening through a sequence of scattering events with applications to semiclassical Monte Carlo Leonard
F. Register
Beckman Institute Urbana-Champaign,
and Kafl Hess and Coordinated Science Laboratory, University of Illinois at 405 N. Mathews Av., Urbana, IL 61801, U.S.A.
Insights $re presented into the modeling of collision broadening within semiclassical Monte Car10simulatlrin of carrier transport. The effects of collision broadening through a sequence of scattering events are analyzed via both basic theory and first-principles simulation using SchrGdinger Equation Monte Carlo (SEMC), a simulation method designed to bridge the gap from quantum to clasleical transport. Based on this analysis, an improved treatment of collision broadening for semiclassical Monte Carlo is suggested.
Of importance in semiclassical Monte Carlo (SCMC) simulation of energetic carrier transport effects such as impact ionization, hot-carrier degradation and transport over heterointerfaces, is the treatment of the well recognized high-order quantum process of collision broadening (CB) that allows for nominally non-energy conserving transitions between carrier states. However, there is no consensus opinion on the proper treatment of CB in SCMC, and the high-energy tail of the carrier distribution is sensitive to the specific model of CB used [l-3]; even seemingly rigorous analysis of CB can lead to seemingly implausible SCMC simulation results [2]. A common approach to introducing CB into SCMC is to allow transitions between carrier states within a non-zero range about the energy conserving transition such that E n+l
=
En
T
hWn+l,n
+
(1)
AEn+l,n
for phonon emission or absorption, respectively, where h~,.,+~,~ is the phonon energy [l-3]. The distribution for stochastically selecting the CB contribution AEn+l,n depends on the scattering rates R of, in principle, both the initial n and final n + 1 states [2]. These scattering rates can be taken to be a function of energy only, R, =
R(&) ,
(2)
if directional dependence is ignored for simplicity. Simulations results produced using the quantum transport simulation method Schriidinger Equation Monte Carlo (SEMC) [4], h owever, reveal significantly different CB effects, not for individual scattering events, but for a sequence of scattering events. In This work was supported
by the US AR0
0167-9317/99/$ - see front matter PII: SO167-9317(99)00232-4
0
under
Con. Nos. DAAH0495-1-0362
1999 Elsevier Science B.V. All rights twmved.
and DAAG55-98-1-0306
354
L.F. Register, K. Hess I Microelectronic
normally incident electron phonon,emission (35 meV) ,\
Engineering
47 (1999) 353-355 I
10’
(a) 5
(b)
i
-
h”
hw-_l-
after emission
I I I
1
-I
n ,
initial state
262 meV 231 meV
energy 4
,k,,,
1
_-____..
cI_ position
__
I.
75
rim-Y
bare-electron
energy (meV)
Figure 1: (a) Electron capture by a quantum well via phonon emission. The bare electron effective mass, m* = O.O57m,, is modeled as position independent so that the well-normal component of bare-electron energy El in the single bound state subband is independent of transverse momentum ,$I. (b) Transition probability to bare-electron states as a function of energy El + tL21ii112/2m* for emission of one phonon, and for sequential emission of two phonons via two different intermediate bare-electron energies. The slight offsets of the peaks from integer multiples of fiw are due to real self-energy differences. contrast to the case for SCMC, CB is an inherent feature of SEMC calculations, an unavoidable by-product of allowing real scattering that cannot be independently adjusted. Consider the system of Fig. l(a) where an electron of well defined initial energy Eo, 25 meV for the bare electron relative to the band edge (24.6 meV for the polaron), is incident on a quantum well with one bound state subband; only emission of 35 meV polar-optical phonons is allowed (0 K lattice temperature) such that the carrier cannot undergo real scattering until it reaches the vicinity of the well. Upon reaching the well, the carrier can emit a sequence of phonons as it is first captured into the well and then scatters within the well subband. Figure l(b) displays the SEMC calculated transition probabilities to electron states via emission of one phonon, and via the sequential emission of two phonons, as a function of the electron bare energies El and E2, respectively. The results for emission of two phonons are subdivided by the energy of the intermediate state El with two cases shown, Er = Eo - Iiw and El = Eo - 1.5tiw. Figure 1 (b) clearly exhibits CB effects, but in contrast to Eq. (I), the most probable value of E2, E2 21 E. - 2fiw, is essentially independent of the value of El. The behavior displayed in the SEMC simulations is traceable to the Schrodinger equation of the true coupled carrier-phonon system that SEMC mimics, [H< + Hd +
ff&&G, 9’)= W(~, 8,,
(3)
where & is the total energy of the coupled carrier-phonon system, Hz is the carrier Hamiltonian, Hd is the Hamiltonian of the phonons (lattice vibrational modes), and Hza provides coupling b_etwe_enthe two systems. Fzr energy eigenstates of the uncoupled system, [Hi + H&b(k)$(Q) = [Ei + fd]$(k)$(Q), it is Hz,4 that allows scattering among
L.F. Register, K. Hess I Microelectronic Engineering 47 (1999) 353-355
355
states and inherently broadens the states in energy; much like the coupling of the quasibound state(s) of a resonant tunneling diode (RTD) to the adjacent carrier reservoirs both allows transport through the RTD and broadens the quasi-bound state(s) through which the carriers pass. (As in the SEMC simulations, the bare carrier eigenstates $(ic’) may not have well defined momentum.) For coupling to/through any particular carrierphonon quasi/bare energy eigenstate $(&)$(Qn) th a t is only weakly broadened, energy conservation in Eq. (3) requires &, + EQ~ z ,!?‘,(with allowances for real energy shifts) independent of the perhaps strong broadening of any previous state $(&_,)$(o,_,). For coupling through strongly broadened bare states n, it is the overlap of the broadened energy spectrum of that state with f that is of importance, not its overlap with the bare energies of previous states &i _ t-E4 _ . In brief, energy conservation within the coupled carrier-phonon system of El r3) d o&mnot allow for the accumulation of uncertainty in the bare-carrier energy through a sequence of scattering events as produced by Eq. (1). Although any treatment of CB in SCMC is inherently approximate, a simple alternative to Eq. (1) that avoids such artificial accumulation of energy uncertainty is available, E n+1 - E, - AE, F fiww,n + AEn+ >
(4
where the A_&, AE,+1 distributions are associated with the real scattering rates I&, R n+l of the individual states n, n + 1, respectively. Note that the net allowed CB for individual scattering events still depends on the scattering rates in both the initial and final states consistent with Ref. [2]. Also, because the range of allowed final states depends on E, - AE, rather than just E,, the scattering rates take the form R, = R(\&J, E, - AEn). For momentum randomizing scattering events such as those that dominate in silicon, the scattering rates reduce to a function of only one variable again, & = R(En - AEn). Eqs. (3) and (4) could have significant implications for SCMC simulations of hot carrier transport effects such as impact ionization [1,2] or even for just the calculation of the impact ionization rates [5]. Also, the tail of the broadening distribution becomes less critical; even fully Lorentzian broadening (justified or not) and effectively small high energy tails of the carrier distribution should not be mutually exclusive. In summary, a modified treatment of CB for SCMC has been suggested that eliminates the accumulation of uncertainty in the carrier energy and related artifacts of past treatments, yet retains the essential physics of CB.
1. Y.-C. Chang, 2. L. Reggiani,
D. Z.-Y. Ting, J. Y. Tang and K. Hess, Appl. Phys. Lett. 42, 76 (1983). P. Lugli and A. P. Jauho,
Phys. Rev. B 36, 6602 (1987).
3. K. Kim, B. A. Mason, and K. Hess, Phys. Rev. B 36, 6547 (1987). 4. L. F. Register, in Selected Topics in Electronics and Systems, Vol. 14: QuantumBased Electronic Devices and Systems, vol. ed.: M. Dutta and M. A. Stroscio, series ed.: P. K. Tien (World Scientific, Singapore, 1998) pp. 251-279, or L. F. Register, International Journal of High Speed Electronics and Systems 9 251 (1998). 5. J. Bude, K. Hess and G. J. Iafrate,
Phys. Rev. B 45, 10958 (1992).