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Electron–phonon interaction in nanowires: A Monte Carlo study of the effect of the field
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Mathematics and Computers in Simulation 81 (2010) 515–521
Electron–phonon interaction in nanowires: A Monte Carlo study of the effect of the field E. Atanassov a , T. Gurov a , A. Karaivanova a , M. Nedjalkov a,b,∗ , D. Vasileska c , K. Raleva d a
IPP, Bulgarian Academy of Sciences, Sofia, Bulgaria Institute for Microelectronics, TU-Vienna, Austria c Dept. of Electrical Engineering, Arizona State University, USA d Faculty of Electrical Engineering, Univ. of Skopje, Macedonia b
Received 2 October 2007; accepted 7 September 2009 Available online 16 September 2009
Abstract The femtosecond dynamics of highly non-equilibrium, confined carriers is analyzed within a Monte Carlo approach. The physical process considered corresponds to a locally excited or injected into a semiconductor nanowire distribution of heated carriers, which evolve under the action of an applied electric field. The carriers are cooled down by dissipation processes caused by phonons. The process is described by a quantum-kinetic equation which generalizes the classical Boltzmann equation with respect to two classical assumptions, namely for temporal and spatial locality of the carrier–phonon interaction. We investigate the effect of the field on the electron–phonon interaction—the intra-collisional field effect (ICFE). A Monte Carlo method for simulation of the considered process has been utilized. Simulation results for carrier evolution in a GaAs nanowire are obtained and analyzed for phenomena related to the ICFE. © 2009 IMACS. Published by Elsevier B.V. All rights reserved. Keywords: Monte Carlo; Quantum transport; Semiconductor carriers; Electron–phonon interaction
1. Introduction Core elements in modern electronic devices as ultrafast optical switchers, highly sensitive photo-FET’s and ultradense memories are semiconductor quantum wires. The electronic characteristics of these devices are basically determined by the carrier transport in the wire. The transport picture corresponds to a locally excited or injected carrier distribution, e.g. an impulse, which presents the signal evolving in the wire, which evolves under the action of an applied electric field. The carrier distribution is highly non-equlibrium so that besides the usual spatial evolution a fast process of thermal relaxation occurs, caused by energy dissipation from phonons. According the classical Boltzmann description of the transport, point-like carriers move along Newton’s trajectories accelerated by the electric field. Their movement is interrupted by scattering events with the phonons, which are assumed of having no duration: the carrier momentum is changed by a phonon instantaneously, while the position ∗
Corresponding author at: Institute for Microelectronics, TU-Vienna, Austria. E-mail address: [email protected] (M. Nedjalkov).
0378-4754/$36.00 © 2009 IMACS. Published by Elsevier B.V. All rights reserved. doi:10.1016/j.matcom.2009.09.006
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E. Atanassov et al. / Mathematics and Computers in Simulation 81 (2010) 515–521
coordinate remains unchanged. The theoretical foundations of this picture require large transport domains, which for typical semiconductors are of above 102−3 nm, as well as evolution times above several tens of picoseconds. On the contrary, the transport regime in a wire is determined by the nanometer dimensions in the normal to the direction of the wire plane, the energy of the initial carriers, and the electron–phonon coupling element, which furthermore depends on the temperature and the shape of the wire. A highly non-equilibrium semiconductor carrier can cover a distance of 100 nm for 100 fs. Hence characteristic for the evolution process under consideration are the sub-picosecond and nanometer scales, which requires a quantum mechanical description. In what follows we introduce a quantum-kinetic equation, which has been recently derived from a first-principle approach to the carrier–phonon system in a quantum wire [3]. A hierarchy of approximations are utilized, aiming to obtain a closed form for the reduced electron Wigner function fw : t dt {S fw − Sfw }; f = fw (Z(t ), kz (t ), t ) (1) Lfw (z, kz , t) = dkz 0
Here L is the Liouville operator, the Wigner function on the right hand side depends on the initialized at z, kz , t Newton trajectory
t
z(t ) = z − t
h ¯ kz (τ)dτ, m
t
kz (t ) = kz −
F (τ)dτ
(2)
t
as follows: Z(t ) = z(t ) +
h ¯ (kz − kz ) c ; 2m
c = t − t
(3)
We note that the electric force F (t) = qE(t)/¯h can be time dependent but does not depend on the position z along the wire. Furthermore S(kz , kz , t, t ) is obtained from the Boltzmann scattering rate by replacing the energy conserving δ-function δ((kz ) − (kz ) − h ¯ ω) (where (kz ) and (kz ) are the carrier energies before and after the scattering by a phonon with energy h ¯ ω) by ⎛ t ⎞ ⎜ − (kz (τ) + k (τ)) ⎟ ⎞ ⎛t z ⎜ ⎟
⎜ ⎟ ¯ω (kz (τ)) − (kz (τ)) − h 1 ⎜ ⎟ D = exp ⎜ t dτ ⎠ ⎟ cos ⎝ ⎜ ⎟ 2 h ¯ 2 ⎜ ⎟ t ⎝ ⎠
(4)
is the total Boltzmann out-scattering rate, S and fw are obtained from S and f by exchanging of the primed and unprimed momentum variables. The main factors which outline the transport model (1) from the classical Boltzmann equation are: (a) the existence of the time integral on the right hand side of (1); (b) the function (4) which replaces the classical energy conserving delta function in the scattering operator S; (c) the time interval c identified as the collision duration time. The later is zero in the classical case so that the quantum trajectory Z(t ) reduces to the classical Newton’s trajectory z(t ). These factors introduce a number of quantum phenomena in the carrier–phonon kinetics, demonstrated by the evolution of the Wigner function and its first moments—the carrier density, energy and wave vector distributions. Here we summarize the introduced effects: (i) the non-Markovian character of the equation introduced by the time integral leads to retardation of the quantum evolution. (ii) The lack of energy conservation causes collisional broadening and appearance of carriers in the classically forbidden energy regions. The spatial manifestation of the broadening is ultrafast spatial transfer (UST): existence of quantum carriers which reach larger distances than the fastest classical carriers. (iii) Collisions with phonons have finite duration c which affects the carrier trajectories (3). It can be shown that if this time is neglected, the evolution becomes unphysical. (iv) The action of the electric field during the collision process—the ICFE.
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517
These phenomena must be taken into account in the next generation simulators of transport processes occurring on femtosecond and nanometer scales. While the effects (i)–(iii) have been studied in details for the filed-less case, [2,4,5], here we focus on the ICFE (iv). 2. Integral form In the case of applied electric field Eq. (1) becomes very inconvenient for a numerical treatment. According to (2) the domain of integration becomes dependent on the evolution time t. This problem is avoided with the help of a coordinate transform as follows: t kz1 = kz1 − Ft;
t kz1 (τ) = kz1 + Fτ;
def
f (z, kz , t) = f (z, kzt + Ft, t) =f t (z, kzt , t),
Here we assumed a constant electric field. A generalization for the time-dependent case is similar. All terms in the equation are now expressed as functions of kzt , kzt . By omitting the superscripts of the arguments and using qz = kz − kz = kzt − kzt , the quantum-kinetic equation obtains the integral form: t t h ¯ kz h ¯F 2 dq⊥ dkz × f (z, kz , t) = f (z − dt dt t− t , kz , 0) + m 2m 0 t
h ¯ q h ¯ kz h ¯F 2 2 (t − t ) − (t − t ) + z (t − t ), kz , t S(kz , kz , t , t , q⊥ )f z − (5) m 2m 2m
h ¯ q h ¯ kz h ¯F 2 2 −S(kz , kz , t , t q⊥ )f z − (t − t ) − (t − t ) + z (t − t ), kz , t m 2m 2m Eq. (4) is written explicitely in the kernel S 2V S(kz , kz , t , t , q⊥ ) = C(q⊥ , qz , F )× (2π)3
¯ ωq (kz ) − (kz ) + h h ¯ 2 2 (n(q ) + 1)cos (t − t ) + F.qz (t − t ) h ¯ 2m
) −h ) − (k ¯ ω (k h ¯ z q z 2 2 +n(q )cos (t − t ) + F.q (t − t ) h ¯ 2m z
(6)
where the factor C depends on the electron–phonon coupling, the electron ground state in the plane normal to the wire and the exponential factor of (4) [3]. We note that the time dependence of kz in (5) conveniently disappears. The equation formally resembles the zero field counterpart, which allows to implement the zero field algorithm with minor modifications. The Monte Carlo algorithm that is used for this problem is described in details in [1]. The numerical procedure is characterized by heavy computational demands which require application of variance reduction techniques and a Grid computing paradigm, Here we omit the discussions of the numerical aspects in order to focus on the analysis of the effects caused by the field. From physical considerations we expect the solution along one of the directions of the wire to be the same as the solution in the other if the field direction is switched: f (z, k, t)|F = f (−z, −k, t)|−F
(7)
For this the initial condition must be symmetric and the phonon interaction to be isotropic with respect to the direction of the incident carrier. As a rule this is the case since the electron–phonon coupling depends on |q|. The property (7) can be proven in a formal way by switching the sign of the involved variables in (5) and using the fact that in (4) and thus C are invariant for a simultaneous switch of the signs of qz and F. Finally we note that the main dependence on the electric force is via the cosine function, where the times t and t are identified as the initial and final time of the interaction, c = t − t . 3. The effect of the field The transport equation (1) has been solved for the following physical conditions: a GaAs wire with 10 nm square cross section has been chosen. The carriers are assumed in the Gamma valley, a single optical phonon with a constant
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E. Atanassov et al. / Mathematics and Computers in Simulation 81 (2010) 515–521
energy is considered in the semiconductor model. The initial carrier distribution corresponds to a Gaussian distribution both in energy and space that is generated by a laser pulse with 150 meV excess energy. As the shape of the wire affects the carrier–phonon coupling via the carrier state and band in the plane normal to the wire, we filter out the effects of the inter-band transitions by a choice of a very low temperature, which fixes the carriers in the ground state in the plane. This is in accordance with (4), where the inter-band energies are excluded. The evolution of the carriers takes place in both, the real z and the momentum pz = h ¯ kz subspaces. The carrier distribution initially centered at z = 0 splits into two parts which move in positive and negative directions respectively. In the momentum subspace occur processes of energy transfer: both the applied electric field and the phonon scattering change the wave vector kz of the carriers. We first discuss the carrier behavior provided by the classical model, which will be used as a reference to outline the quantum effects. At very low temperature classical carriers can only emit phonons and since the chosen constant polar optical phonon energy give rise to replicas of the initial condition. The consecutive replicas correspond to emission of one phonon (first replica), a second phonon (second replica), etc. The highest energies have the carriers of the initial condition: the energy region above remains classically forbidden as only phonon emission is possible. According the physical model, the mean time between the successive collisions is 50 fs, so that at least three well formed replicas are expected above 150 fs of evolution. This gives a measure of the thermal energy transferred to the lattice. The more is the number of the replicas, the smaller is the averaged kinetic energy of the carrier system. In contrast to the classical case, the quantum-kinetic equation (1) is non-Markovian: to obtain the solution at time t one needs the solutions at all times t < t due to the time integral on the right. The non-Markovian evolution gives rise to a significant retardation of the processes of electron cooling. The momentum distributions f (kz ) = fw (z, kz )dz shown in Figs. 1 and 2 for three values of the electric field reveal only one well formed replica for evolution time of 180 fs. The zero field curves at the two figures are symmetric, which is consistent with the fact that without electric field the two directions of the wire are equivalent. The field, breaks the symmetry. It is chosen so that the vector of F is positive, pointing to the right. In the spirit of (7) the behavior of the carriers with negative z, Fig. 1, resembles the carriers with positive z, Fig. 2, provided that the field direction is inverted. Thus the major reason for the difference between the two cases is in the sign in front of F in (6). If the field is against the direction of the carriers, Fig. 1, the field effectively increases the phonon energy as well as causes an additional broadening which is proportional to the field magnitude. We note the existence of finite number of quantum carriers in the classically forbidden region kz ∈ (−60 : −55) × 107 /m. Just the opposite is the situation if the carriers move along the field, Fig. 2. The phonon
Fig. 1. Momentum distribution of the moving along the negative direction carriers after 180 fs of evolution.
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Fig. 2. Momentum distribution of the moving along the positive direction carriers after 180 fs of evolution.
energy effectively decreases with the magnitude of the field so that the coherence of the initial signal is conserved to a large extend. The carrier density n(z) = fw (z, kz )dkz after 180 fs of evolution is shown in Fig. 3 in moving with the field frame. The distance covered by the signal at positive z increases with the field magnitude. For these carriers the field causes an additional retardation of the interaction process: the largest number of ballistic carriers corresponds to the highest peak at 30 kV/cm and decreases with the decrease of the field magnitude. This is confirmed by Fig. 4 presenting the field in a logarithmic scale. At the left hand side (z < 0) the broadening increases with the field, while on the right (z > 0) the 30 kV/cm curve presents the most coherent signal. Finally a comparison of the zero field curve with it’s ballistic counterpart (obtained by turning off the phonon interaction) demonstrates the effect of UST: a finite density of quantum carriers reach distances further away from the origin, in the regions above 200 (below −200) nm. As the front ends of the ballistic peaks are comprised by the
Fig. 3. Carrier density for zero field (continuous line), 15 kV/cm (dashed line) and 30 kV/cm (dot-dashed line).
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Fig. 4. Logarithm of the carrier density for zero field (continuous line), 15 kV/cm (dashed line) and 30 kV/cm (dot-dashed line). The dotted line presents the ballistic curve at zero field.
fastest classical carriers, this is an entirely quantum phenomena. The UST effect is associated with the lack of energy conservation in the interaction process: some carriers gain energy from the collisions and thus increase the velocity due to the scattering. The effect is expected to be inverse proportional to the field magnitude since the retardation increases with the field. 4. Conclusions Nanowires are used a building blocks in some modern nanoelectronic devices. The electronic characteristics of these systems depend on the carrier transport in the nanowire, which is characterized by the ultimate nanometer and femtosecond scales. Some of the basic notions of the classical transport picture fail at these scales so that a quantum description is necessary. A quantum-kinetic equation, derived to provide such description is now used to explore the effect of the applied electric field. The equation is solved for an initial condition—a packet of hot carriers which can be associated with an impulse initiated in the wire. The ICFE affects of retardation and collision broadening in a directionally dependent way. It effectively reduces the phonon energy in the direction of the field, while in the opposite direction the phonon energy is increased. Additionally, In the former case the carrier package is accelerated by the field further away from the origin, while the transfer of the carrier energy to the lattice is slowed down. These phenomena work in parallel to preserve for longer time the initial coherence of the signal. The above typically quantum effects must be taken into account in the next generation device simulators. As the computational problem is very intensive, in this research we have used the powerful Monte Carlo approach and computing capacity of the SEEGRID infrastructure. Acknowledgments This work is partially supported by EU through Project BIS 21++ , by Österreichische Forschungsgemeinschaft (ÖFG), Project MOEL239. and by the Bulgarian Ministry of Science and Education, grant I1405/2004. References [1] T.V. Gurov, E. Atanasov, I. Dimov, V. Palankovski, Femtosecond Evolution of Spatially Inhomogeneous Carrier Excitations: Part II: Stochastic Approach and GRID Implementation, Lecture Notes in Computer Science, vol. 3743, Springer-Verlag, 2006, pp. 157–163, ISBN 3-540-31994-8, 3743. [2] M. Herbst, M. Glanemann, V. Axt, T. Kuhn, Electron–phonon quantum kinetics for spatially inhomogeneous excitations, Physical Review B 67 (2003) 195305-1–119530515-119530515.
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[3] M. Nedjalkov, D. Vasileska, D.K. Ferry, C. Jacoboni, C. Ringhofer, I. Dimov, V. Palankovski, Wigner transport models of the electron–phonon kinetics in quantum wires, Physical Review B 74 (2006) 035311-1–1353118-1353118. [4] M. Nedjalkov, T.V. Gurov, H. Kosina, D. Vasileska, V. Palankovski, Femtosecond Evolution of Spatially Inhomogeneous Carrier Excitations: Part I: Kinetic Approach, Lecture Notes in Computer Science, vol. 3743, Springer-Verlag, 2006, pp. 149–156, ISBN 3-540-31994-8, 3743. [5] M. Nedjalkov, D. Vasileska, E. Atanassov, V. Palankovski, Ultrafast Wigner transport in quantum wires, Journal of Computational Electronics 6 (2007) 235–238.
Mathematics and Computers in Simulation 81 (2010) 515–521
Electron–phonon interaction in nanowires: A Monte Carlo study of the effect of the field E. Atanassov a , T. Gurov a , A. Karaivanova a , M. Nedjalkov a,b,∗ , D. Vasileska c , K. Raleva d a
IPP, Bulgarian Academy of Sciences, Sofia, Bulgaria Institute for Microelectronics, TU-Vienna, Austria c Dept. of Electrical Engineering, Arizona State University, USA d Faculty of Electrical Engineering, Univ. of Skopje, Macedonia b
Received 2 October 2007; accepted 7 September 2009 Available online 16 September 2009
Abstract The femtosecond dynamics of highly non-equilibrium, confined carriers is analyzed within a Monte Carlo approach. The physical process considered corresponds to a locally excited or injected into a semiconductor nanowire distribution of heated carriers, which evolve under the action of an applied electric field. The carriers are cooled down by dissipation processes caused by phonons. The process is described by a quantum-kinetic equation which generalizes the classical Boltzmann equation with respect to two classical assumptions, namely for temporal and spatial locality of the carrier–phonon interaction. We investigate the effect of the field on the electron–phonon interaction—the intra-collisional field effect (ICFE). A Monte Carlo method for simulation of the considered process has been utilized. Simulation results for carrier evolution in a GaAs nanowire are obtained and analyzed for phenomena related to the ICFE. © 2009 IMACS. Published by Elsevier B.V. All rights reserved. Keywords: Monte Carlo; Quantum transport; Semiconductor carriers; Electron–phonon interaction
1. Introduction Core elements in modern electronic devices as ultrafast optical switchers, highly sensitive photo-FET’s and ultradense memories are semiconductor quantum wires. The electronic characteristics of these devices are basically determined by the carrier transport in the wire. The transport picture corresponds to a locally excited or injected carrier distribution, e.g. an impulse, which presents the signal evolving in the wire, which evolves under the action of an applied electric field. The carrier distribution is highly non-equlibrium so that besides the usual spatial evolution a fast process of thermal relaxation occurs, caused by energy dissipation from phonons. According the classical Boltzmann description of the transport, point-like carriers move along Newton’s trajectories accelerated by the electric field. Their movement is interrupted by scattering events with the phonons, which are assumed of having no duration: the carrier momentum is changed by a phonon instantaneously, while the position ∗
Corresponding author at: Institute for Microelectronics, TU-Vienna, Austria. E-mail address: [email protected] (M. Nedjalkov).
0378-4754/$36.00 © 2009 IMACS. Published by Elsevier B.V. All rights reserved. doi:10.1016/j.matcom.2009.09.006
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E. Atanassov et al. / Mathematics and Computers in Simulation 81 (2010) 515–521
coordinate remains unchanged. The theoretical foundations of this picture require large transport domains, which for typical semiconductors are of above 102−3 nm, as well as evolution times above several tens of picoseconds. On the contrary, the transport regime in a wire is determined by the nanometer dimensions in the normal to the direction of the wire plane, the energy of the initial carriers, and the electron–phonon coupling element, which furthermore depends on the temperature and the shape of the wire. A highly non-equilibrium semiconductor carrier can cover a distance of 100 nm for 100 fs. Hence characteristic for the evolution process under consideration are the sub-picosecond and nanometer scales, which requires a quantum mechanical description. In what follows we introduce a quantum-kinetic equation, which has been recently derived from a first-principle approach to the carrier–phonon system in a quantum wire [3]. A hierarchy of approximations are utilized, aiming to obtain a closed form for the reduced electron Wigner function fw : t dt {S fw − Sfw }; f = fw (Z(t ), kz (t ), t ) (1) Lfw (z, kz , t) = dkz 0
Here L is the Liouville operator, the Wigner function on the right hand side depends on the initialized at z, kz , t Newton trajectory
t
z(t ) = z − t
h ¯ kz (τ)dτ, m
t
kz (t ) = kz −
F (τ)dτ
(2)
t
as follows: Z(t ) = z(t ) +
h ¯ (kz − kz ) c ; 2m
c = t − t
(3)
We note that the electric force F (t) = qE(t)/¯h can be time dependent but does not depend on the position z along the wire. Furthermore S(kz , kz , t, t ) is obtained from the Boltzmann scattering rate by replacing the energy conserving δ-function δ((kz ) − (kz ) − h ¯ ω) (where (kz ) and (kz ) are the carrier energies before and after the scattering by a phonon with energy h ¯ ω) by ⎛ t ⎞ ⎜ − (kz (τ) + k (τ)) ⎟ ⎞ ⎛t z ⎜ ⎟
⎜ ⎟ ¯ω (kz (τ)) − (kz (τ)) − h 1 ⎜ ⎟ D = exp ⎜ t dτ ⎠ ⎟ cos ⎝ ⎜ ⎟ 2 h ¯ 2 ⎜ ⎟ t ⎝ ⎠
(4)
is the total Boltzmann out-scattering rate, S and fw are obtained from S and f by exchanging of the primed and unprimed momentum variables. The main factors which outline the transport model (1) from the classical Boltzmann equation are: (a) the existence of the time integral on the right hand side of (1); (b) the function (4) which replaces the classical energy conserving delta function in the scattering operator S; (c) the time interval c identified as the collision duration time. The later is zero in the classical case so that the quantum trajectory Z(t ) reduces to the classical Newton’s trajectory z(t ). These factors introduce a number of quantum phenomena in the carrier–phonon kinetics, demonstrated by the evolution of the Wigner function and its first moments—the carrier density, energy and wave vector distributions. Here we summarize the introduced effects: (i) the non-Markovian character of the equation introduced by the time integral leads to retardation of the quantum evolution. (ii) The lack of energy conservation causes collisional broadening and appearance of carriers in the classically forbidden energy regions. The spatial manifestation of the broadening is ultrafast spatial transfer (UST): existence of quantum carriers which reach larger distances than the fastest classical carriers. (iii) Collisions with phonons have finite duration c which affects the carrier trajectories (3). It can be shown that if this time is neglected, the evolution becomes unphysical. (iv) The action of the electric field during the collision process—the ICFE.
E. Atanassov et al. / Mathematics and Computers in Simulation 81 (2010) 515–521
517
These phenomena must be taken into account in the next generation simulators of transport processes occurring on femtosecond and nanometer scales. While the effects (i)–(iii) have been studied in details for the filed-less case, [2,4,5], here we focus on the ICFE (iv). 2. Integral form In the case of applied electric field Eq. (1) becomes very inconvenient for a numerical treatment. According to (2) the domain of integration becomes dependent on the evolution time t. This problem is avoided with the help of a coordinate transform as follows: t kz1 = kz1 − Ft;
t kz1 (τ) = kz1 + Fτ;
def
f (z, kz , t) = f (z, kzt + Ft, t) =f t (z, kzt , t),
Here we assumed a constant electric field. A generalization for the time-dependent case is similar. All terms in the equation are now expressed as functions of kzt , kzt . By omitting the superscripts of the arguments and using qz = kz − kz = kzt − kzt , the quantum-kinetic equation obtains the integral form: t t h ¯ kz h ¯F 2 dq⊥ dkz × f (z, kz , t) = f (z − dt dt t− t , kz , 0) + m 2m 0 t
h ¯ q h ¯ kz h ¯F 2 2 (t − t ) − (t − t ) + z (t − t ), kz , t S(kz , kz , t , t , q⊥ )f z − (5) m 2m 2m
h ¯ q h ¯ kz h ¯F 2 2 −S(kz , kz , t , t q⊥ )f z − (t − t ) − (t − t ) + z (t − t ), kz , t m 2m 2m Eq. (4) is written explicitely in the kernel S 2V S(kz , kz , t , t , q⊥ ) = C(q⊥ , qz , F )× (2π)3
¯ ωq (kz ) − (kz ) + h h ¯ 2 2 (n(q ) + 1)cos (t − t ) + F.qz (t − t ) h ¯ 2m
) −h ) − (k ¯ ω (k h ¯ z q z 2 2 +n(q )cos (t − t ) + F.q (t − t ) h ¯ 2m z
(6)
where the factor C depends on the electron–phonon coupling, the electron ground state in the plane normal to the wire and the exponential factor of (4) [3]. We note that the time dependence of kz in (5) conveniently disappears. The equation formally resembles the zero field counterpart, which allows to implement the zero field algorithm with minor modifications. The Monte Carlo algorithm that is used for this problem is described in details in [1]. The numerical procedure is characterized by heavy computational demands which require application of variance reduction techniques and a Grid computing paradigm, Here we omit the discussions of the numerical aspects in order to focus on the analysis of the effects caused by the field. From physical considerations we expect the solution along one of the directions of the wire to be the same as the solution in the other if the field direction is switched: f (z, k, t)|F = f (−z, −k, t)|−F
(7)
For this the initial condition must be symmetric and the phonon interaction to be isotropic with respect to the direction of the incident carrier. As a rule this is the case since the electron–phonon coupling depends on |q|. The property (7) can be proven in a formal way by switching the sign of the involved variables in (5) and using the fact that in (4) and thus C are invariant for a simultaneous switch of the signs of qz and F. Finally we note that the main dependence on the electric force is via the cosine function, where the times t and t are identified as the initial and final time of the interaction, c = t − t . 3. The effect of the field The transport equation (1) has been solved for the following physical conditions: a GaAs wire with 10 nm square cross section has been chosen. The carriers are assumed in the Gamma valley, a single optical phonon with a constant
518
E. Atanassov et al. / Mathematics and Computers in Simulation 81 (2010) 515–521
energy is considered in the semiconductor model. The initial carrier distribution corresponds to a Gaussian distribution both in energy and space that is generated by a laser pulse with 150 meV excess energy. As the shape of the wire affects the carrier–phonon coupling via the carrier state and band in the plane normal to the wire, we filter out the effects of the inter-band transitions by a choice of a very low temperature, which fixes the carriers in the ground state in the plane. This is in accordance with (4), where the inter-band energies are excluded. The evolution of the carriers takes place in both, the real z and the momentum pz = h ¯ kz subspaces. The carrier distribution initially centered at z = 0 splits into two parts which move in positive and negative directions respectively. In the momentum subspace occur processes of energy transfer: both the applied electric field and the phonon scattering change the wave vector kz of the carriers. We first discuss the carrier behavior provided by the classical model, which will be used as a reference to outline the quantum effects. At very low temperature classical carriers can only emit phonons and since the chosen constant polar optical phonon energy give rise to replicas of the initial condition. The consecutive replicas correspond to emission of one phonon (first replica), a second phonon (second replica), etc. The highest energies have the carriers of the initial condition: the energy region above remains classically forbidden as only phonon emission is possible. According the physical model, the mean time between the successive collisions is 50 fs, so that at least three well formed replicas are expected above 150 fs of evolution. This gives a measure of the thermal energy transferred to the lattice. The more is the number of the replicas, the smaller is the averaged kinetic energy of the carrier system. In contrast to the classical case, the quantum-kinetic equation (1) is non-Markovian: to obtain the solution at time t one needs the solutions at all times t < t due to the time integral on the right. The non-Markovian evolution gives rise to a significant retardation of the processes of electron cooling. The momentum distributions f (kz ) = fw (z, kz )dz shown in Figs. 1 and 2 for three values of the electric field reveal only one well formed replica for evolution time of 180 fs. The zero field curves at the two figures are symmetric, which is consistent with the fact that without electric field the two directions of the wire are equivalent. The field, breaks the symmetry. It is chosen so that the vector of F is positive, pointing to the right. In the spirit of (7) the behavior of the carriers with negative z, Fig. 1, resembles the carriers with positive z, Fig. 2, provided that the field direction is inverted. Thus the major reason for the difference between the two cases is in the sign in front of F in (6). If the field is against the direction of the carriers, Fig. 1, the field effectively increases the phonon energy as well as causes an additional broadening which is proportional to the field magnitude. We note the existence of finite number of quantum carriers in the classically forbidden region kz ∈ (−60 : −55) × 107 /m. Just the opposite is the situation if the carriers move along the field, Fig. 2. The phonon
Fig. 1. Momentum distribution of the moving along the negative direction carriers after 180 fs of evolution.
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Fig. 2. Momentum distribution of the moving along the positive direction carriers after 180 fs of evolution.
energy effectively decreases with the magnitude of the field so that the coherence of the initial signal is conserved to a large extend. The carrier density n(z) = fw (z, kz )dkz after 180 fs of evolution is shown in Fig. 3 in moving with the field frame. The distance covered by the signal at positive z increases with the field magnitude. For these carriers the field causes an additional retardation of the interaction process: the largest number of ballistic carriers corresponds to the highest peak at 30 kV/cm and decreases with the decrease of the field magnitude. This is confirmed by Fig. 4 presenting the field in a logarithmic scale. At the left hand side (z < 0) the broadening increases with the field, while on the right (z > 0) the 30 kV/cm curve presents the most coherent signal. Finally a comparison of the zero field curve with it’s ballistic counterpart (obtained by turning off the phonon interaction) demonstrates the effect of UST: a finite density of quantum carriers reach distances further away from the origin, in the regions above 200 (below −200) nm. As the front ends of the ballistic peaks are comprised by the
Fig. 3. Carrier density for zero field (continuous line), 15 kV/cm (dashed line) and 30 kV/cm (dot-dashed line).
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Fig. 4. Logarithm of the carrier density for zero field (continuous line), 15 kV/cm (dashed line) and 30 kV/cm (dot-dashed line). The dotted line presents the ballistic curve at zero field.
fastest classical carriers, this is an entirely quantum phenomena. The UST effect is associated with the lack of energy conservation in the interaction process: some carriers gain energy from the collisions and thus increase the velocity due to the scattering. The effect is expected to be inverse proportional to the field magnitude since the retardation increases with the field. 4. Conclusions Nanowires are used a building blocks in some modern nanoelectronic devices. The electronic characteristics of these systems depend on the carrier transport in the nanowire, which is characterized by the ultimate nanometer and femtosecond scales. Some of the basic notions of the classical transport picture fail at these scales so that a quantum description is necessary. A quantum-kinetic equation, derived to provide such description is now used to explore the effect of the applied electric field. The equation is solved for an initial condition—a packet of hot carriers which can be associated with an impulse initiated in the wire. The ICFE affects of retardation and collision broadening in a directionally dependent way. It effectively reduces the phonon energy in the direction of the field, while in the opposite direction the phonon energy is increased. Additionally, In the former case the carrier package is accelerated by the field further away from the origin, while the transfer of the carrier energy to the lattice is slowed down. These phenomena work in parallel to preserve for longer time the initial coherence of the signal. The above typically quantum effects must be taken into account in the next generation device simulators. As the computational problem is very intensive, in this research we have used the powerful Monte Carlo approach and computing capacity of the SEEGRID infrastructure. Acknowledgments This work is partially supported by EU through Project BIS 21++ , by Österreichische Forschungsgemeinschaft (ÖFG), Project MOEL239. and by the Bulgarian Ministry of Science and Education, grant I1405/2004. References [1] T.V. Gurov, E. Atanasov, I. Dimov, V. Palankovski, Femtosecond Evolution of Spatially Inhomogeneous Carrier Excitations: Part II: Stochastic Approach and GRID Implementation, Lecture Notes in Computer Science, vol. 3743, Springer-Verlag, 2006, pp. 157–163, ISBN 3-540-31994-8, 3743. [2] M. Herbst, M. Glanemann, V. Axt, T. Kuhn, Electron–phonon quantum kinetics for spatially inhomogeneous excitations, Physical Review B 67 (2003) 195305-1–119530515-119530515.
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[3] M. Nedjalkov, D. Vasileska, D.K. Ferry, C. Jacoboni, C. Ringhofer, I. Dimov, V. Palankovski, Wigner transport models of the electron–phonon kinetics in quantum wires, Physical Review B 74 (2006) 035311-1–1353118-1353118. [4] M. Nedjalkov, T.V. Gurov, H. Kosina, D. Vasileska, V. Palankovski, Femtosecond Evolution of Spatially Inhomogeneous Carrier Excitations: Part I: Kinetic Approach, Lecture Notes in Computer Science, vol. 3743, Springer-Verlag, 2006, pp. 149–156, ISBN 3-540-31994-8, 3743. [5] M. Nedjalkov, D. Vasileska, E. Atanassov, V. Palankovski, Ultrafast Wigner transport in quantum wires, Journal of Computational Electronics 6 (2007) 235–238.