- Email: [email protected]
Analysis of electron transfer between parallel quantum wires
Superlattices
and Microstructures,
509
Vol. 7.2, No. 4, 1992
ANALYSIS OF ELECTRON TRANSFER BETWEEN PARALLEL QUANTUM WIRES M. Macucci(‘), U. Ravaioli(‘*‘) and T. Kerkhoven(3) (‘)Be&rnan Institute, (*)Coordinated Science Laboratory and c3)Department of Computer Science University of Illinois at Urbana-Champaign Urbana, Illinois, 6 180 1 (Received 4 August 1992)
Coupled quantum wire structures, created by applying various potentials to properly shaped gates, have been recently demonstrated. These experiments indicate that directional couplers for electron waveguides, in which the rate of lateral tunneling is controlled by means of external potentials, can be conceived. Such devices could play an important role as switching elements in future logic circuits based on quantum effects. The exact 3-D numerical simulation of these structures is a formidable problem. However, if we can assume that the variation of the potential along the quantum wires is quasi-adiabatic, the transverse wavefunctions and the potential for each crosssection along the wire can be obtained from independent solutions of 2-D selfconsistent problems, for which efficient numerical techniques exist. A quasi 3-D transport model can then be obtained by proper matching of the solutions for the transverse sections. This work describes an adaptation of a recursive Green’s function technique, based on the tight-binding formalism, which is suitable for a fast analysis of coupled quantum wires. The transmission probability and the conductance are reported for some representative structures at liquid helium temperature.
I. INTRODUCTION Coupled quantum wire structures have been proposed for the realization of directional couplers for electron waveguides [ 1,2] and have been recently demonstrated [3]. A detailed self-consistent simulation of these structures requires the solution of an extremely large 3-D numerical problem. A possible approximation is to solve a 2-D transport problem, ignoring the variations normal to the 2-D electron gas plane and using a reasonable 1-D approximation for the transverse wavefunctions [4]. Conversely, techniques have been developed to obtain the detailed 2-D transverse wavefunction by solving Schriidinger’s equation and Poisson’s equation self-consistently [5,6]. If we can assume that the variation of the potential along the quantum wires is quasi-adiabatic, it is possible, in principle, to combine the two approaches, by subdividing the structure into small sections of constant longitudinal potential, where 2-D self-consistent wavefunctions are
0749-6036/92/080509+04$08.00/0
calculated on the cross-section. A quasi 3-D transport model can then be obtained by proper matching of the solutions for the transverse sections. II. NUMERICAL APPROACH We have developed an improved version of the recursive Green’s functions technique discussed in [4], obtaining a significant reduction of the CPU time required for the solutions, in order to make quasi-3-D analysis possible. In the conventional recursive Green’s function technique, the structure to be studied is filled with a regular tight-binding grid on which the Schrodinger equation can be easily discretized. The method is based on the recursive addition of sections of constant transverse width (when a simple flat potential delimited by hard walls is considered) or, more generally, sections with a constant transverse potential profile. The method in [4] is based on considering the coupling V between two sections as a perturbation of
0 1992 Academic
Press Limited
510
Superlattices and Microstructures, Vol. 12, No. 4, 1992
the unperturbed Hamiltonian HO of the uncoupled sectionsas H =H, +?. By means of the Dyson equation, the Green’s function for the perturbed system (G) can be related to the one for the unperturbed system (GO) as G = GO + G, P G. This is an implicit equation, but by performing some relatively straightforward manipulations [4], it is possible to derive explicit expressions for the Green’s function connecting the adjacent lattice sites of the transverse sections and connecting a transverse section to itself, after coupling. The overall goal is to obtain, by adding one section at a time, the Green’s functions connecting the input to itself and the input to the output for the complete systen. These Green’s functions can then be related to the reflection and transmission coefficients of the structure. A mixed representation is used for the Green’s function: in real space for the longitudinal direction (where the propagation occurs) and in the space of transverse eigenmodes for the transverse planes. This is particularly convenient for the solution of our problem, since within each of the uniform sections the transverse modes are orthogonal and therefore uncoupled. As a consequence, the matrices representing the Green’s function between two transverse lattice chains of a given section, before it is coupled to the rest of the structure, are diagonal, and a simple analytic expression is available for the matrix elements. In this_representation, the elements of the coupling matrices V are.simply proportional to the overlap integrals between the transverse modes of the sections which are being co~ected. In the method discussed in [4] the number of transverse modes corresponds to the number of points used for the transverse discretization, in order to have always a complete basis (every grid point of the discretization corresponds to a distinct eigenvalue of the problem). In a quasi-3-D approach, the transverse sections are 2-D, and the large number of points necessary to resolve accurately the details of the wavefunctions would lead to unmanageable problems, since inversion of matrices (with size equal to the number of modes) must be performed at each interface to obtain the Green’s function [4]. However, we observe that for the energies of interest in our problem, only very few modes are actually propagating, and just a small number of the evanescent modes really inlluences the coupling at the interfaces. It is important to mention that we are interested in the energy range over which the derivative of the Fermi-Dirac distribution function has non-negligible values. This is because the conductance at a finite temperature (4.2 K in our calculations) is obtained by averaging me total transmission coefficient over the energy using the derivative of the Fermi function as the weighing factor [7]:
For the simulation of the coupled waveguides shown later, at 4.2 K we need only to go up to the threshold for propagation of the second mode in the input leads when injection in the tirst mode is considered, since the occupation of the higher subbands is negligible. A total number of six transverse modes is sufficient for performing the calculation in the structures considered here. Since the modes propagate independently within the uniform sections, each mode is described by a onedimensional Green’s function in space. The spacing for the discretization step (a) along the longitudinal directo have a hopping potential tion is chosen V = li2/(2ma2)between the lattice sites, which is much larger than the maximum energy considered in the problem. This is necessary to avoid inconsistencies in the energy range under study, since, while the original Schr&ger equation assumes a parabolic band, the tight-binding discretization corresponds to a cosinusoidal band. If the energy is much smaller than V, the energy-wave vector relationship is quasi-parabolic. In the present approach, the discretixation on the transverse sections is independent of the choice of the longitudinal mesh, and is only relevant for the calculation of the transverse modes of the overlap integrals between adjacent sections. We have also considered mode matching techniques for the calculation of the transmission coefficients, starting from the knowledge of the transverse eigenvectors in the sections of the structure. Mode-matching can be accomplished either with a transfer matrix formulation or with the enforcement of the matching conditions at discontinuities, which entails a large linear system of equations characterized by a very sparse coefficient matrix. Although the transfer matrix formulation is very attractive from the computational point of view, for realistic structures we experienced severe overflow problems related to the real exponentials characterizing the evanescent modes. The other matching technique does not present such problems, but requires increasing computational resources as the size and the number of transverse sections are increased. The recursive Green’s function approach appears to be the least expensive, provided that the number of transverse modes can be limited as in the present approach.
The calculations of transverse mode eigenvalues and eigenfunctions are performed with an efficient selfconsistent S&&linger/Poisson solver described in [6]. The iteration procedure follows two steps. Starting
Superlattices
and Microstructures,
511
Vol. 72, No. 4, 1992
from an initial condition, the two equations are solved alternatively and are coupled by using an underrelaxation scheme. When the convergence of this procedure becomes too slow, a Jacobian-free Newton iteration (Generalized Miniium Residual method 181) is used, which converges very rapidly, provided the solution has been approached sufficiently by the underrelaxation The eigenvalue problem for the discretized steps. S&&linger equation is solved using a subspace iteration method [9], which allows one to select only the desired subset of eigenvalues, with great reduction of the computational burden. The application of this selfconsistent method to the analysis of adjacent quantum wires was presented in [lo].
“g
“,
“I A’o.3 @0.7
“,
NA = 6 x Id7cwa”
As
Alo,3 Gao,7
“*
As
Na = 101"crn"
GaAS
N A = 10'"cm3
Alo.s Ga,., As
NA = 10'4cm~' T
III.
SIMULATION
RESULTS
The numerical techniques was tested on the coupled waveguide geometry shown in Fig. 1. The symmetric structure is subdivided into 19 longitudinally uniform sections. The first 6 sections from the right and from the left are 10 run long, and the remaining sections are in the order 5, 4, and 3 WI. The length L,,, of the middle section, where the coupling between the two waveguides is strongest, varies in length from 0 to 80 WI in the examples presented here.
b
72
_,_ LIII J_
72
4
0 W
Fig. 2 - Topology of contacts and layers for the coupled wire structure, with L, = 12 - 120 rrm , L = 36 nm , and s=12nm.
simulation conditions, the extra gates add obvious complications for the construction of the structure. The numerical technique presented here is however independent of this choice. The middle gate determines the coupling between the two adjacent quantum wires, and varies in width between 120 run at the right and left boundaries, down to 12 nm in the middle section, decreasing by steps of 12 nm moving from one section to another towards the center. The gap between the contacts is s = 12 nm. Assuming a Schottky barrier of about -1 V, the applied biases are VI = 0.2 V, Vs = 0.3 V, and V, = 0.75 V.
::
1 8
i
Fig. 1 - Schematic diagram of the simulated structure. is the length of the middle coupling section in correspondence of the minimum distance between the wires. All lengths are in nanometers.
L,,,
0.8
.! 0.6 2 ._ i 0.4 c=
0.2
i
0 -1
For the formation of the quantum wires, we have used for convenience a contact topology as presented in [6, lo], where an additional gate is introduced between the laterally confining gates, in order to have an additional means of controlling the electron concentration in the channel, as shown in Fig. 2. Whiie such an arrangement is very useful in order to quickly select suitable
0
1 Energy
2
3
(meV)
Fig. 3 - Transmission probability of the fundamental mode from input lead A to output lead A’ (solid line) and to output lead B’ (dashed line) as a function of energy, for a structure with L,,, = 5 nm. The Fermi level (E = IIF = 0) is the reference energy.
4
512
Superlattices
and Microstructures,
Vol. 72, No. 4, 7992
calculations were implemented on a CRAY YMP/48. The selfconsistent calculations of wavefunctions on each cross-section, performed on a nonuniform rectangular grid with 76x48 nodes, require less than 30 CPU second with 6 eigenvalues. The remaining calculations of transmission coefficients and conductance require well below one minute of CPU time per data point. The
0’
I_____-
0
--
10
_-
,
I
1
20
1
30 L,
I
1 40
(nm)
Fig. 4 - Normdizcd conductance between input lead A and output lead A’ (GAA’- solid line ) and output lead B’ (GM - dashed line ), for a middle coupling section L, =o-4Onm.
Acknowledgements - This work was supported National Science Foundation, grant EET-9120641. putations on the CRAY Y-MP/48 of the National for Supercomputing Applications (NCSA) were possible by a grant from CRAY Research, Inc.
by the ComCenter made
REFERENCES 1. J.A. de1 Alamo and C.C. Eugster, Appl. Phys. Lett. 56,78 (1990).
Figure 3 shows the transmission probability as a function of electron energy, assuming injection in the fundamental mode and L, = 5 WI. The solid line indicates transmission along the direct path AA’, while the broken line corresponds to transmission into output B’. The reference zero energy corresponds to the position of the Fermi Level in the structure. In the input lead (A), the threshold for the fundamental transverse mode is at about -0.85 meV and for the second mode 5.4 meV, with respect to the Fermi level. Note that at the simulation temperature of 4.2 we have kBT = 0.362 meV. In Figs. 4, we show the normalized conductance obtained from Eq. (l), for the direct output branch AA’ and the coupled output branch AB’, changing the length of the middle section Lm in the range 0 - 40 run. In conclusion, we have demonstrated a quasi-3-D technique which couples the iterative Green’s functions method to 2-D self-consistent calculations for transverse wavefunctions. This approach is suitable for the study of coupled waveguides and is valid as long as transitions between adjacent sections are smooth and in conditions near equilibrium, where Eq. (4) is valid and one can define a flat Fermi level for the entire structure.
2. N. Dagli, G. Snider, J. Waldman Appl. Phys. 69, 1047 (1991).
and E. Hu, J.
3. C.C. Eugster and J.A. de1 Alamo, 67, 3856 (1991).
Phys. Rev. Lett.
4.
F. Sols, M. Macucci, U. Ravaioli, and K. Hess, J. Appl. Phys. 66, 3892 (1989).
5.
SE. Laux, Proc. NASECODE V, 270 (1987).
6. T. Kerkhoven, A. Galick, U. Ravaioli, J.A. Arends and Y. Saad, J. Appl. Phys. 67, 3461 (1990). 7.
B.J. van Wees, L.P. Kouwenhoven, E.M.M. Willems, C.J.P.M. Harmans, J.E. Mooji, H. van Houten, C.W.J. Beenakker, J.G. Williamson, and CT. Foxon Phys. Rev. B 43, 12431 (1991).
8. Y. Saad and M.H. Schultz, SIAM .I. Sci. Stat. Comput. 7, 856 (1986). 9. H. Rutishauser, in Handbook for Automatic Computation, vol. II of Linear Algebra, J.K. Wilkinson and C.H. Reinsch, eds. (Springer, New York, 1971). 10. U. Ravaioli, T. Kerkhoven, M. Raschke and A.T. Galick, Superlattices and Microstructures 11, 343 (1992).
and Microstructures,
509
Vol. 7.2, No. 4, 1992
ANALYSIS OF ELECTRON TRANSFER BETWEEN PARALLEL QUANTUM WIRES M. Macucci(‘), U. Ravaioli(‘*‘) and T. Kerkhoven(3) (‘)Be&rnan Institute, (*)Coordinated Science Laboratory and c3)Department of Computer Science University of Illinois at Urbana-Champaign Urbana, Illinois, 6 180 1 (Received 4 August 1992)
Coupled quantum wire structures, created by applying various potentials to properly shaped gates, have been recently demonstrated. These experiments indicate that directional couplers for electron waveguides, in which the rate of lateral tunneling is controlled by means of external potentials, can be conceived. Such devices could play an important role as switching elements in future logic circuits based on quantum effects. The exact 3-D numerical simulation of these structures is a formidable problem. However, if we can assume that the variation of the potential along the quantum wires is quasi-adiabatic, the transverse wavefunctions and the potential for each crosssection along the wire can be obtained from independent solutions of 2-D selfconsistent problems, for which efficient numerical techniques exist. A quasi 3-D transport model can then be obtained by proper matching of the solutions for the transverse sections. This work describes an adaptation of a recursive Green’s function technique, based on the tight-binding formalism, which is suitable for a fast analysis of coupled quantum wires. The transmission probability and the conductance are reported for some representative structures at liquid helium temperature.
I. INTRODUCTION Coupled quantum wire structures have been proposed for the realization of directional couplers for electron waveguides [ 1,2] and have been recently demonstrated [3]. A detailed self-consistent simulation of these structures requires the solution of an extremely large 3-D numerical problem. A possible approximation is to solve a 2-D transport problem, ignoring the variations normal to the 2-D electron gas plane and using a reasonable 1-D approximation for the transverse wavefunctions [4]. Conversely, techniques have been developed to obtain the detailed 2-D transverse wavefunction by solving Schriidinger’s equation and Poisson’s equation self-consistently [5,6]. If we can assume that the variation of the potential along the quantum wires is quasi-adiabatic, it is possible, in principle, to combine the two approaches, by subdividing the structure into small sections of constant longitudinal potential, where 2-D self-consistent wavefunctions are
0749-6036/92/080509+04$08.00/0
calculated on the cross-section. A quasi 3-D transport model can then be obtained by proper matching of the solutions for the transverse sections. II. NUMERICAL APPROACH We have developed an improved version of the recursive Green’s functions technique discussed in [4], obtaining a significant reduction of the CPU time required for the solutions, in order to make quasi-3-D analysis possible. In the conventional recursive Green’s function technique, the structure to be studied is filled with a regular tight-binding grid on which the Schrodinger equation can be easily discretized. The method is based on the recursive addition of sections of constant transverse width (when a simple flat potential delimited by hard walls is considered) or, more generally, sections with a constant transverse potential profile. The method in [4] is based on considering the coupling V between two sections as a perturbation of
0 1992 Academic
Press Limited
510
Superlattices and Microstructures, Vol. 12, No. 4, 1992
the unperturbed Hamiltonian HO of the uncoupled sectionsas H =H, +?. By means of the Dyson equation, the Green’s function for the perturbed system (G) can be related to the one for the unperturbed system (GO) as G = GO + G, P G. This is an implicit equation, but by performing some relatively straightforward manipulations [4], it is possible to derive explicit expressions for the Green’s function connecting the adjacent lattice sites of the transverse sections and connecting a transverse section to itself, after coupling. The overall goal is to obtain, by adding one section at a time, the Green’s functions connecting the input to itself and the input to the output for the complete systen. These Green’s functions can then be related to the reflection and transmission coefficients of the structure. A mixed representation is used for the Green’s function: in real space for the longitudinal direction (where the propagation occurs) and in the space of transverse eigenmodes for the transverse planes. This is particularly convenient for the solution of our problem, since within each of the uniform sections the transverse modes are orthogonal and therefore uncoupled. As a consequence, the matrices representing the Green’s function between two transverse lattice chains of a given section, before it is coupled to the rest of the structure, are diagonal, and a simple analytic expression is available for the matrix elements. In this_representation, the elements of the coupling matrices V are.simply proportional to the overlap integrals between the transverse modes of the sections which are being co~ected. In the method discussed in [4] the number of transverse modes corresponds to the number of points used for the transverse discretization, in order to have always a complete basis (every grid point of the discretization corresponds to a distinct eigenvalue of the problem). In a quasi-3-D approach, the transverse sections are 2-D, and the large number of points necessary to resolve accurately the details of the wavefunctions would lead to unmanageable problems, since inversion of matrices (with size equal to the number of modes) must be performed at each interface to obtain the Green’s function [4]. However, we observe that for the energies of interest in our problem, only very few modes are actually propagating, and just a small number of the evanescent modes really inlluences the coupling at the interfaces. It is important to mention that we are interested in the energy range over which the derivative of the Fermi-Dirac distribution function has non-negligible values. This is because the conductance at a finite temperature (4.2 K in our calculations) is obtained by averaging me total transmission coefficient over the energy using the derivative of the Fermi function as the weighing factor [7]:
For the simulation of the coupled waveguides shown later, at 4.2 K we need only to go up to the threshold for propagation of the second mode in the input leads when injection in the tirst mode is considered, since the occupation of the higher subbands is negligible. A total number of six transverse modes is sufficient for performing the calculation in the structures considered here. Since the modes propagate independently within the uniform sections, each mode is described by a onedimensional Green’s function in space. The spacing for the discretization step (a) along the longitudinal directo have a hopping potential tion is chosen V = li2/(2ma2)between the lattice sites, which is much larger than the maximum energy considered in the problem. This is necessary to avoid inconsistencies in the energy range under study, since, while the original Schr&ger equation assumes a parabolic band, the tight-binding discretization corresponds to a cosinusoidal band. If the energy is much smaller than V, the energy-wave vector relationship is quasi-parabolic. In the present approach, the discretixation on the transverse sections is independent of the choice of the longitudinal mesh, and is only relevant for the calculation of the transverse modes of the overlap integrals between adjacent sections. We have also considered mode matching techniques for the calculation of the transmission coefficients, starting from the knowledge of the transverse eigenvectors in the sections of the structure. Mode-matching can be accomplished either with a transfer matrix formulation or with the enforcement of the matching conditions at discontinuities, which entails a large linear system of equations characterized by a very sparse coefficient matrix. Although the transfer matrix formulation is very attractive from the computational point of view, for realistic structures we experienced severe overflow problems related to the real exponentials characterizing the evanescent modes. The other matching technique does not present such problems, but requires increasing computational resources as the size and the number of transverse sections are increased. The recursive Green’s function approach appears to be the least expensive, provided that the number of transverse modes can be limited as in the present approach.
The calculations of transverse mode eigenvalues and eigenfunctions are performed with an efficient selfconsistent S&&linger/Poisson solver described in [6]. The iteration procedure follows two steps. Starting
Superlattices
and Microstructures,
511
Vol. 72, No. 4, 1992
from an initial condition, the two equations are solved alternatively and are coupled by using an underrelaxation scheme. When the convergence of this procedure becomes too slow, a Jacobian-free Newton iteration (Generalized Miniium Residual method 181) is used, which converges very rapidly, provided the solution has been approached sufficiently by the underrelaxation The eigenvalue problem for the discretized steps. S&&linger equation is solved using a subspace iteration method [9], which allows one to select only the desired subset of eigenvalues, with great reduction of the computational burden. The application of this selfconsistent method to the analysis of adjacent quantum wires was presented in [lo].
“g
“,
“I A’o.3 @0.7
“,
NA = 6 x Id7cwa”
As
Alo,3 Gao,7
“*
As
Na = 101"crn"
GaAS
N A = 10'"cm3
Alo.s Ga,., As
NA = 10'4cm~' T
III.
SIMULATION
RESULTS
The numerical techniques was tested on the coupled waveguide geometry shown in Fig. 1. The symmetric structure is subdivided into 19 longitudinally uniform sections. The first 6 sections from the right and from the left are 10 run long, and the remaining sections are in the order 5, 4, and 3 WI. The length L,,, of the middle section, where the coupling between the two waveguides is strongest, varies in length from 0 to 80 WI in the examples presented here.
b
72
_,_ LIII J_
72
4
0 W
Fig. 2 - Topology of contacts and layers for the coupled wire structure, with L, = 12 - 120 rrm , L = 36 nm , and s=12nm.
simulation conditions, the extra gates add obvious complications for the construction of the structure. The numerical technique presented here is however independent of this choice. The middle gate determines the coupling between the two adjacent quantum wires, and varies in width between 120 run at the right and left boundaries, down to 12 nm in the middle section, decreasing by steps of 12 nm moving from one section to another towards the center. The gap between the contacts is s = 12 nm. Assuming a Schottky barrier of about -1 V, the applied biases are VI = 0.2 V, Vs = 0.3 V, and V, = 0.75 V.
::
1 8
i
Fig. 1 - Schematic diagram of the simulated structure. is the length of the middle coupling section in correspondence of the minimum distance between the wires. All lengths are in nanometers.
L,,,
0.8
.! 0.6 2 ._ i 0.4 c=
0.2
i
0 -1
For the formation of the quantum wires, we have used for convenience a contact topology as presented in [6, lo], where an additional gate is introduced between the laterally confining gates, in order to have an additional means of controlling the electron concentration in the channel, as shown in Fig. 2. Whiie such an arrangement is very useful in order to quickly select suitable
0
1 Energy
2
3
(meV)
Fig. 3 - Transmission probability of the fundamental mode from input lead A to output lead A’ (solid line) and to output lead B’ (dashed line) as a function of energy, for a structure with L,,, = 5 nm. The Fermi level (E = IIF = 0) is the reference energy.
4
512
Superlattices
and Microstructures,
Vol. 72, No. 4, 7992
calculations were implemented on a CRAY YMP/48. The selfconsistent calculations of wavefunctions on each cross-section, performed on a nonuniform rectangular grid with 76x48 nodes, require less than 30 CPU second with 6 eigenvalues. The remaining calculations of transmission coefficients and conductance require well below one minute of CPU time per data point. The
0’
I_____-
0
--
10
_-
,
I
1
20
1
30 L,
I
1 40
(nm)
Fig. 4 - Normdizcd conductance between input lead A and output lead A’ (GAA’- solid line ) and output lead B’ (GM - dashed line ), for a middle coupling section L, =o-4Onm.
Acknowledgements - This work was supported National Science Foundation, grant EET-9120641. putations on the CRAY Y-MP/48 of the National for Supercomputing Applications (NCSA) were possible by a grant from CRAY Research, Inc.
by the ComCenter made
REFERENCES 1. J.A. de1 Alamo and C.C. Eugster, Appl. Phys. Lett. 56,78 (1990).
Figure 3 shows the transmission probability as a function of electron energy, assuming injection in the fundamental mode and L, = 5 WI. The solid line indicates transmission along the direct path AA’, while the broken line corresponds to transmission into output B’. The reference zero energy corresponds to the position of the Fermi Level in the structure. In the input lead (A), the threshold for the fundamental transverse mode is at about -0.85 meV and for the second mode 5.4 meV, with respect to the Fermi level. Note that at the simulation temperature of 4.2 we have kBT = 0.362 meV. In Figs. 4, we show the normalized conductance obtained from Eq. (l), for the direct output branch AA’ and the coupled output branch AB’, changing the length of the middle section Lm in the range 0 - 40 run. In conclusion, we have demonstrated a quasi-3-D technique which couples the iterative Green’s functions method to 2-D self-consistent calculations for transverse wavefunctions. This approach is suitable for the study of coupled waveguides and is valid as long as transitions between adjacent sections are smooth and in conditions near equilibrium, where Eq. (4) is valid and one can define a flat Fermi level for the entire structure.
2. N. Dagli, G. Snider, J. Waldman Appl. Phys. 69, 1047 (1991).
and E. Hu, J.
3. C.C. Eugster and J.A. de1 Alamo, 67, 3856 (1991).
Phys. Rev. Lett.
4.
F. Sols, M. Macucci, U. Ravaioli, and K. Hess, J. Appl. Phys. 66, 3892 (1989).
5.
SE. Laux, Proc. NASECODE V, 270 (1987).
6. T. Kerkhoven, A. Galick, U. Ravaioli, J.A. Arends and Y. Saad, J. Appl. Phys. 67, 3461 (1990). 7.
B.J. van Wees, L.P. Kouwenhoven, E.M.M. Willems, C.J.P.M. Harmans, J.E. Mooji, H. van Houten, C.W.J. Beenakker, J.G. Williamson, and CT. Foxon Phys. Rev. B 43, 12431 (1991).
8. Y. Saad and M.H. Schultz, SIAM .I. Sci. Stat. Comput. 7, 856 (1986). 9. H. Rutishauser, in Handbook for Automatic Computation, vol. II of Linear Algebra, J.K. Wilkinson and C.H. Reinsch, eds. (Springer, New York, 1971). 10. U. Ravaioli, T. Kerkhoven, M. Raschke and A.T. Galick, Superlattices and Microstructures 11, 343 (1992).