Finite element analysis of planar electron waveguides using a PML-Like spatial mapping

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This work presents an approach for the analysis of guided and quasi-guided modes in planar electron waveguides with arbitrarily shaped potential and arbitrary effective mass profiles. The Finite Element Method (FEM) allied to a Perfectly Matched
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  Finite Element Analysis of Planar ElectronWaveguides Using a PML-Like Spatial Mapping Marcos A. R. Franco, Valdir A. Serrão, Francisco Sircilli, Angelo Passaro and Nancy M. Abe  Abstract       This work presents an approach for the analysis of guided and quasi-guided modes in planar electron waveguideswith arbitrarily shaped potential and arbitrary effective massprofiles. The Finite Element Method (FEM) allied to a PerfectlyMatched Layer-Like (PML-Like) spatial mapping is used for thephase constant and the mode confinement computations. Theadopted spatial transformation maps an infinitely large region,surrounding an open waveguide, to a layer of finite thickness.  Index Terms       Finite element method, PML, spatialmapping, quantum well, electron wave, electron waveguide. I. I NTRODUCTION Electron waveguides have widespread use in thedevelopment of quantum-optic devices. The study of guidedelectron waves presented in literature has been carried outusing several different approaches, such as, the hard wallapproximation, in which the electrons are trapped completelywithin the quantum well [1]-[5] and the analysis considering arbitrary potential profiles [6]-[9]. Some studies on guided electron waves in an open systemin which both sides of the quantum well are finite, as occursin the trapping of electrons by heterostructure, are based onthe Finite Element Method. The FEM is applied to the thin-film region (trapping region) while the solution in thesubstrate and in the cladding (semi-infinite regions) areobtained by analytical techniques or by imposing artificialboundaries far from the trapping region in order to apply theFEM in the entire region [9], [10].In this work, the FEM allied to a PML-Like spatialmapping is used for the analysis of guided and quasi-guidedmodes in a planar electron waveguide with arbitrary potentialprofile and arbitrary effective mass profile.   Solutions for theone-dimensional Schödinger equation within the masseffective approximation are computed considering threedifferent electron waveguides defined by a rectangular, aparabolic and a perturbed-rectangular potential profile,respectively.II. F INITE E LEMENT M ODEL We consider a planar electron waveguide and a wavepropagating in  z -direction with phase constant  . The time-independent Schrödinger equation under theeffective mass approximation is given by [3], [9]:        E  yU  y yd d  yd d   222 22  (1) where   is the wave function,  is Planck’s constant dividedby 2  ,  E   is the electron energy, U   is the potential,    ym 1   and m  is the effective mass.Applying the Weighted Residual Method to (1) yields:          T T   M F   2  , (2) The matrices [ F  ] and [  M  ] for each line finite element aregiven by:              yT   yd  N  N  y M   2 2  , (3)         321  F F F F   , (4) where              yT   yd  yd  N d  yd  N d  yF   2 21  , (5)              yT   yd  N  N  yU F  2 , (6)            yT   yd  N  N  E F  3 , (7) {  N  } is a row matrix representing a complete set of basefunctions, and { } T   stands for a transposed matrix.In this work, both the potential energy U (  y ) and theparameter   (  y ), which represents the inverse of the effectivemass, are expanded in terms of the base functions {  N  } in eachfinite element:       niT ii  N  N  1  ,      niT ii  U  N U  N U  1 ,where   ii  ym 1   ,   ii  yU U    and n  is the number of nodal points of each finite element.III. O PEN B OUNDARY P ROBLEM – PML-L IKESPATIAL MAPPING The application of the spatial mapping schema to a planarelectron waveguide, in which the cladding and the substrate Manuscript received on February 02, 2001. This work was supported inpart by FAPESP – Fundação de Amparo à Pesquisa do Estado de São Paulo,process No. 98/07789-7. M.A.R. Franco, V.A. Serrão, F. Sircilli, A. Passaro and N.M. Abe are withthe Centro Técnico Aeroespacial CTA/IEAv, Brasil (email:marcos@ieav.cta.br).  regions are very large with respect to the film region, isrepresented in Fig. 1. The following relation represents thekind of spatial mapping adopted in this work for both thecladding and the substrate regions:  ybas  t t    ,(8)the film region remains unchanged.The parameters a t   and b t    , for each transformed region, areobtained by imposing that the mapping brings the coordinates  y i      s i  and  y e      s e , resulting: eieeiit   y y ysssa   and eiieeit   y y y yssb  .(9) Film U   f  (  y ),  m  f  (  y ) Cladding U   c (  y ),  m  c (  y )  y i_c  y e_c  y e_s  y i_s  d   f  Substrate U   s (  y ),  m  s (  y )  d   c  d   s (a) Film U   f  (  s ),  m  f  (  s ) Mapped claddingMapped substrate  s i_c  s e_c  s e_s  s i_s  s  d   f  (b) Fig. 1. Schematic view of the planar electron waveguide. (a) electronwaveguide in a open boundary domain (  y  coordinates) , (b) electronwaveguide with spatial mapping ( s  coordinates). The parameters d  s , d   f  , and d  c  are the substrate, film and cladding thicknesses, respectively. In the mapped regions, (2) is still valid, provided thematrices [  M  ] and [ F  ] be rewritten as follows:           sT  sd  N  N  M   A 2 2  ,(10)           sT  sd sd  N d sd  N d F   B  21  ,(11)           sT  sd  N  N F   C  2 ,(12)           sT  sd  N  N  E F   D  3 ,(13)where,     ss  I  T A  ,     ss  D  T B   ,     sU s I  T C   ,   s I  T D   ,    T   N   ,    T  U  N U   ,    T  D  N  D  T T   , and    T   N  I I   T T   .These new matrices are obtained by applying the followingtransformation relations:    sd d ssd d bassd d  yd sd  yd d  t t  D  T   2 (14)               sss t t  y y sd s f ssd s f  asb yd  y f  I  T  2121 2 (15)In the non-transformed region the parameters T  D  , T  I   assumethe value 1.IV. N UMERICAL R ESULTS GaAs/Ga 1   x  Al  x  As compound semiconductor  In the GaAs/Ga 1   x Al  x As compound semiconductor, thepotential U  , the effective mass m  and the film thickness d   f   aregiven as [6]-[9]:(eV)7730  x.U     (kg)08300670 0 m x..m     nm282670  N .d   f   where  x  is the aluminium content ratio, m 0  is the rest mass of electron,  N   is the number of monoatomic layers. Themonoatomic layer thickness is 0.28267 nm.The results obtained in this work for three waveguidesdefined by different quantum wells are presented in the nexttwo sections. The phase constant (   ) and the confinementfactor (  ) for the fundamental mode are shown as functions of the electron energy. The factor   is computed as follows: Γ =              −+−∞+∞ ∫ ∫  φ φ  d d   f  f  dy dy 2222 .(16)A.  Rectangular and parabolic quantum well  The Al content ratios of the substrate, film and claddingare  x s =0.2,  x  f  =0,  x c =0.45 (Fig. 2), respectively, for therectangular quantum well. The same values for  x s  and  x c  are assumed in the substrateand in the cladding for the parabolic quantum well. The Alcontent ratio in the film is given by:         2 112    sc f  f s f   x xd d  y x y x , the resulting profile is shown in Fig. 3. In both cases, the srcinal domain dimension considered isfrom s_e  y =  5.0 10 +8  nm to c_e  y = +1.0 10 +4  nm and thedimension of the transformed domain is from s_e s =   10 nmto c_e s = +10 nm . The film thickness is d   f   = 7.34942 nm(674713 .s y s_is_i  nm and 674713 .s y c_ic_i   nm),corresponding to N = 26 monoatomic layers. The number of finite elements used in the substrate, in the film and in thecladding are 700, 200 and 300, respectively. Fig. 4 shows the phase constant (    ) and the confinementfactor (   ) of the fundamental mode for both the rectangularand the parabolic quantum well waveguides. The results arein good agreement with those reported in [9].     x  f = 0   d   f     y  x    x s    x c   Fig. 2 . Aluminium content ratio (  x ) profile of the rectangular quantum well.The potential U   is directly proportional to  x.    y  x    x s    x c   d   f       x c   Fig. 3. Al content ratio (  x ) profile of the parabolic quantum well. Thepotential U   is directly proportional to  x.   0.00.20.40.60.81.0 Electron energy, E (eV) 0.00.40.81.21.6    P   h   a   s   e   c   o   n   s   t   a   n   t ,    β    (   n   m   -   1    ) 0.00.20.40.60.81.0    C   o   n   f   i   n   e   m   e   n   t   f   a   c   t   o   r ,    Γ parabolic-profilerectangular-profileand- Ref. [9]  Fig. 4. Phase constant (   ) and confinement factor (    ) of the fundamentalmode for the rectangular (solid lines) and the parabolic (dotted lines)quantum well electron waveguides.  The wave functions (   ) of the electrons in the parabolicquantum well as a function of the coordinates s  (withtransformation) and  y  (without transformation) are shown inthe Fig. 5 and Fig. 6, respectively. For electrons with  E   0.65 eV the wave functions decay slowly and becomezero for  y  coordinates far from 50 nm.  -10.0-5.00.05.010.0 Axis s   (nm) -0.020.000.020.040.060.080.10    W   a   v   e   F   u   n   c   t   i   o   n   (    φ    ) E = 0.10 eVE = 0.50 eVE = 0.65 eVE = 0.68 eVE = 0.70 eV  Fig.   5. Wave function (   ) versus the transformed coordinate s  for severalelectron energies in the parabolic quantum well.   -80-60-40-200 Axis y   (nm) 0.000.020.040.060.08    W   a   v   e   F   u   n   c   t   i   o   n   (    φ    ) E = 0.10 eVE = 0.50 eVE = 0.65 eVE = 0.68 eVE = 0.70 eV  Fig. 6. Wave function (   ) versus the real coordinate  y  for several electronenergies in the parabolic quantum well. The srcinal range of coordinate  y  is  5.0 10 +8  nm up to +1.0 10 +4  nm. B. Perturbed-rectangular quantum well The electron waveguide with a perturbed-rectangularquantum well is shown in the Fig. 7. The Al content ratios inthe substrate and in the cladding, the total film thickness, thesrcinal domain size (coordinate  y ), the transformed domain(coordinate s ) and the number of finite elements are the sameof the previous cases.  x q  d   f   y  x    x s    x c a   a   a   a   a   bb Fig. 7. Al content ratio (  x ) profile of the perturbed-rectangular quantum well.The parameter a = four monoatomic layers, b = three monoatomic layers and  x q  is the perturbed wall height. The potential U   is directly proportional to  x.  Fig. 8 shows the phase constant (     ) and the confinementfactor (   ) of the fundamental mode for the perturbed-rectangular quantum well for different values of the perturbedwall height   (  x q ). As  x q  increases, both   and    decrease for agiven value of electron energy. 0.00.20.40.60.81.0 Electron energy, E (eV) 0.00.40.81.2    P   h   a   s   e   c   o   n   s   t   a   n   t ,    β    (   n   m   -   1    ) 0.00.20.40.60.81.0    C   o   n   f   i   n   e   m   e   n   t   f   a   c   t   o   r ,    Γ x q  = 0.10x q  = 0.05x q  = 0.00x q  = 0.15 Fig. 8. Phase constant (    ) and confinement factor (    ) of the fundamentalmode for the perturbed rectangular quantum well electron waveguide fordifferent perturbed wall height (  x q ). V. C ONCLUSIONS The FEM with a PML-Like spatial mapping was used forguided and quasi-guided mode analysis of a planar electronwaveguide with an arbitrary potential profile and an arbitraryeffective mass profile.   This spatial mapping technique allowsan efficient solution of open boundary domain problems. Theresults obtained by applying this method to the rectangularand to the parabolic electron waveguides agree well withthose reported in the literature. Additionally, a perturbed-rectangular profile electron waveguide was analyzed.A PPENDIX The element matrices were obtained by analyticalintegration. Below, an example for the first order Lagrangetype line elements is presented. The shape functions are givenby: 1 2121  y y y y N   , 2 1212  y y y y N   .In this work the finite element matrices for the nonmapping region are:     2121 2121 2 3324    L M   ,        11114 2121   LF   ,    2121 2121 2 3312 U U U U  U U U U   LF  ,     21126 3  L E F  .For the PML-Like spatial mapping region, the matrices aregiven by:            123323223 322323312 120 2  , , , Z  , , , Z   , , , Z  , , , Z   L M   ,    111112 21  X  LF   ,            123323223 322323312 60 2  , , , R , , , R  , , , R , , , R  LF  ,     2121 2121 3 3312  I I I I  I I I I  T T T T  T T T T   L E F  ,where:   22211211 22 D D D D   T T T T     X  ,   22211211 U lU k U  jU il ,k  , j ,i R I I I I   T T T T   ,   22211211  I I I I   T T T T   lk  jil ,k  , j ,i Z   ,   ii sm 1  ,   ii sU U   ,   t t ii bas 2  D  T   and ii  D I   T T  1  .R EFERENCES[1] H. R. Frohne, M. J. McLennan, and S. Datta, “An efficient method forthe analysis of electron wave-guides”,  J. Appl. Phys ., vol. 66, n. 6, pp.2699-2705, September 1989.[2] A. Weisshaar, J. Lary, S. M. Goodnick, and V. K. Kirkner, “Analysisof discontinuities in quantum waveguide structures”,  Appl. Phys .  Lett. ,vol. 55, n. 20, pp. 2114-2116, November 1989.[3] C. S. Lent, and D. J. Kirkner, “The quantum transmitting boundarymethod”,  J. Appl. Phys ., vol. 67, n. 10, pp. 6353-6359, May 1989.[4] C. S. Lent, “Transmission through a bend in an electron waveguide”,  Appl. Phys .  Lett. , vol. 56, n. 25, pp. 2554-2556, June 1990.[5] C. S. Lent, “Ballistic current vortex excitations in electron waveguide”,  Appl. Phys .  Lett. , vol. 57, n. 16, pp. 1678-1680, October 1990.[6] T. K. Gaylord, E. N. Glytsis, and K. F. Brennan, “Semiconductorelectron-wave slab waveguides ”,  J. Appl. Phys ., vol. 66, n. 3, pp.1483-1485, August 1989.[7] T. K. Gaylord, E. N. Glytsis, and K. F. Brennan, “Semiconductorquantum well as electron wave slab waveguides ”,  J. Appl. Phys ., vol.66, n. 4, pp. 1842-1848, August 1989.[8] D. W. Wilson, E. N. Glytsis, and T. K. Gaylord, “Quantum well, voltage-induced quantum well, and quantum barrier electron waveguides: modecharacteristics and maximum current”,  Appl. Phys. Lett  ., vol. 59, n. 15,pp. 1855-1857, October 1991.[9] R. Kaji, K. Hayata, and M. Koshiba,   “Finite-Element analysis of planarelectron waveguides”,  Electronics and Communications in Japan,  part.2, vol. 76, n. 5, pp. 56-62, 1993.[10] M. Koshiba, Optical Waveguide Theory by the Finite Element Method, KTM Scientific Publishers, Tokio, 1992.  L
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