aestimo 1D User Guide and Tutorial
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1. Add resulting potential to band edge There are a few main pa rameters that control loop calcu Table 3 Parameters used to control convergence of self consistent lations and convergence At the solving of coupled Schrodinger Poisson equations by using shoot moment of this writing in case of ing method aestimo version 0 8 these param eters are located in aestimo py Parameter Description and aestimo_numpy py files Ta convergence is reached when the ble 3 max_iterations ground state energy meV is stable to within this number between iterations Schr dinger nonparabolicity Energy step for initial search Energy to start shooting method from Averaging factor between iterations to smooth convergence convergence_test delta E E start 2 3 Nonparabolicity of E k Nonparabolicity parameter is in troduced in order to take into damping account dependence of effective mass on particle energy Without that correction effective masses from the bottom of the conduction band is considered The approxi mation in bulk semiconductors can be described by the dispersion relation Harrison 1 p 105 Nelson 3 Ak 3 where E k and m are the energy wave number and effective mass of the charge carrier and y is the nonparabolicity parameter In database py the nonparabolicity parameter for electrons is represented by m e alpha It is important to account for this effect in particular in case of narrow QWs or in2. 0 n The meaning of it is as follows Our device consists of three regions layers The first region has the width 20 nm second 10 nm and the third one 20 nm The first and third regions are of AlGaAs while the second is GaAs In case of AlGaAs we assume Al concentration x 0 3 Only the second region is dopped with concentration of 2 1015cm and doping type is n The grid a descrite set of points on which computation is performed is controled by parameter gridf actor in this file which sets the maximum value in nm of distance between grid points Additionaly we have there a parameter mazgridpoints that sets a limit on the number of allowed grid points That last one can be safely set to a large number a something like 10e5 though for simple structures we use only the number of grid points of the order of 10 103 when gridfactor is set to to lnm or 0 1nm which gives good accuracy of computation 3 2 How the material database is implemented As of version 0 8 of aestimo the following binary compounds and alloys have entries in material database AlGaAs InGaAs InGaP AllnP GaSb AlSb InSb AlGaAs InGaAs InGaP AllnP Have a look to database py file to find out how the database have been implemented in python We use a very quick and dirty so lution to code it Its advantage is however that is is simple and could be understood modified and new compounds and parameters can be added there by anyone All parameters values l
3. 3 title Symbol Y 2 exit 0 15 0 1 0 05 0 15 0 1e 08 2e 08 3e 08 4e 08 5e 08 x m Figure 7 Example figure created in Gnuplot by using code as in this section Results show wavefucions for the first and second bound states in a simple one QW heterostructure 17 5 2 2 PYTHON Some people have aversion to commercial software others to software where intensive use of mouse and clicking is needed Often these two sets of people overlap and belong to that common part of them pylab provides a procedural interface to the matplotlib object oriented plotting library It is modeled closely af ter Matlab TM There fore the majority of plotting commands in pylab have Matlab TM 0 15 analogs with similar ar 0 05 guments Important commands are explained with interactive exam amp 0 00 ples iPython tutorial for beginners http www loria fr rougi In interactive mode ipython pylab r teaching matplotlib This opens a shall 0 10 where commands can be typed in Or you may run atag 1 2 3 5 examples by typing in x m 1e 8 terminal window python example file pnare gy The same data as these in Figure 7 but this figure was created by eol Bust ipart a using python example code from this section import numpy x wavel wave2 numpy loadtxt outputs wavefunctions dat skiprows 0 unpack True plot x wavei color red linewidth 2 5 linestyle la
4. for a 001 oriented zinc blende ZB crystal by using k P theory Finite Difference Method Description We consider the upper 3x3 Hamiltonian for a 001 oriented zinc blende ZB crystal see ref 1 after block diagonalization the KP FDM k P theory Finite Difference Method method is explained in ref 2 to ensure the Hermitian property of Hamiltonian you have to apply the we have to write all operators of the form presented in ref 3 to understand Hermiticity property see ref 2 page 110 same code in fortran language presented in the index of ref 1 Dirichlet boundary conditions were applied 1 1 D Ahn amp S H Park ENGINEERING QUANTUM MECHANICS P 238 2 P Harrison QUANTUM WELLS WIRES AND DOTS P 357 362 3 S L CHUANG physics of Optoelectronic De vices P 183 14 4 3 Modulation doped AlGaAs GaAs heterostructure 4 4 Shooting method 4 5 Barrier doped AlGaAs GaAs heterostructure 4 6 Double Quantum well doped AlGaAs GaAs heterostructure Aestimo Results 1250 1200r 1150r 1100r 1050 A 4 4 Energy meV 1000r Position m 1e 8 Figure 6 The first 2 localized wave functions solid lines and their cor responding energies broken lines for a simple 2 QWs structure obtained with sample double qw py sample input file 15 4 7 Calculations for many changing parameters main iterating py In this example calculations are perfo
5. potential is computed by procedure calc Vxc sigma eps cb meff which is a function of dielectric constant effective mass and charge density It is based on the equation 1 3 Vee A T 1 0 0545 r log 1 11 4 7 3 where A q 4n 3 7 n is electron density per square meter and r is average distance between charges in units of effective Bohr radius 2 5 k p method for valance band splitting In semiconductors valence bands are well char acterized by 3 Luttinger parameters At the T point in the band structure p3 2 and p orbitals form valence bands But we have there also a spin orbit coupling due to nonzero angular mo mentum state electrons i e other than s type wave functions generate a magnetic field through which they interact with the spin of the electrons This is particulary important for the valence band p like states Because of this coupling neither spin nor orbital angular momentum but the to tal angular momentum becomes a good quantum Heavy hole number The spin orbit coupling splits sixfold de Eig ale generacy into high energy 4 fold and lower energy 2 fold bands Again 4 fold degeneracy is lifted into heavy and light hole bands by phenomenological Hamiltonian by J M Luttinger see Wikipedia ar ticle on luttinger parameters Calculations for holes are performed by us ing aestimo_numpy_h py file That file includes Figure 2 Kane model Only a conduction band a VBHM py
6. 7 Calculations for many changing parameters main iterating py softquakeQgmail com tWith numerous contributions of Aestimo development team www ON O0 ov wh 5 Working with Datafiles 16 5 1 Structure of Datafiles ooo o m Rom ee dm SO ee 16 5 2 Examples of Working with Datafiles llle 17 521 GNUPLOT x i added bho Wai E iai bd AA Spe de ded A Ba ier 17 Be aue DC wee a we I to oh wee ete Se nae woe Ue dee 18 6 Appendix Additional example how to control computation with Perl 19 1 AESTIMO WHAT IS THAT Aestimo is a self consistent Schr dinger Poisson solver written for simulating basic physical phenomena of 1 dimensional 1 D semiconductor heterostructures Aestimo was started as a hobby by Professor Sefer Bora Lisesivdin from Gazi University Ankara Turkey at the beginning of 2012 and become a usable tool which can be used as a co tool in an educational and scientific work 1 1 Who can use it Aestimo is aimed to be both educational and research tool For educational purposes it must be easy to understand For research it must have some point of computational correctness and sensitivity Written in Python scripting language broadly used by scientific community combines easy access to external quality libraries and source code with speedy and accurate simulation extensibility and community technical support The very basic experience with computer programming is needed if at a
7. aestimo 1D User Guide and Tutorial Zbigniew Koziol State University Education Science Production Complex Russia September 12 2013 Version 0 4 Preliminary CONTENTS Aestimo What is that dal WhorCamUSeit 113 dumme usu eon E ER A d aa a wan saa aa dod S 4 4 wate 1 2 Download and installation o aooaa oaa a a cce 1 9 XGODYIOHE s arso e a ogy EUE Eum EeM Es mo ue el a NE X x E E A 4 PEDE E ees What does it solve 2 44 Schr dinger equation lll 2 2 Coupled Schr dinger Poisson equations 00 a ee ee 2 9 Nonparabolicity of E k s sss m om oko m m m m m mm m mm ee 24 Exchange interaction uus cae BA ERA RSE RD ee X de ecd Rs eR RR 2 5 k p method for valance band splitting len How the material structure and grid are created 3 1 How the device structure and grid are created ee 3 3 Bands alignement and initial potential sn er sq px 3 4 1 Bowing parameter Bowing param 2 sss ses Overview of Examples 4 1 Quantum well doped AlGaAs GaAs heterostructure llle 4 2 Barrier doped AlGaAs GaAs heterostructure 00 cee eee eee 4 3 Modulation doped AlGaAs GaAs heterostructure 2 ee ee 44 Shooting method eu nuu a sor ada RR a ee ae eee a RR d 4 5 Barrier doped AlGaAs GaAs heterostructure 2 0 ee ee 4 6 Double Quantum well doped AlGaAs GaAs heterostructure sn 4
8. ample showing how to perform computation for a changing parameter It is used as an include file by aestimo numpy h py It contains procedures for solving 3x3 Hamiltonian for a 001 oriented zinc blende ZB crystal by using k P theory Finite Difference Method The name can be anything These are sample input files with structure description Put a file name name into config py 42 CETORI FF wea 1 Many example solutions of simple heterostructures and quantum wells are discussed in details by Harrison 1 Aestimo solves 1 D Schr dinger equation by using shooting method In numerical analysis the shooting method is a method for solving a boundary value problem by reducing it to the solution of an initial value problem For Schr dinger equation it is described in details by Harrison 1 When Schr dinger equation alone is solved also with nonparabolicity scheme 0 and 1 in Table 1 than a one loop calculations are performed only When Poisson solving is included into computational scheme than self consistent calculations in a loop are performed 2 2 Coupled Schr dinger Poisson equations Solving Schr dinger equation adding there nonparabolicity as well goes in one iteration step We have an input potential V x and just find numericaly eigenvalues and corresponding wave functions The problem becomes more complex when we would like to take into account other effects Have a look on this schematic drawing Input pote
9. bel Psi_1 plot x wave2 color blue linewidth 2 5 linestyle label Psi_2 legend ylabel Psi xlabel x m savefig figures python example00 eps dpi 72 show 18 6 APPENDIX ADDITIONAL EXAMPLE HOW TO CONTROL COMPUTATION WITH PERL Actually come through these problems when attempting to use perl for control of computational pro cess Yes know you are not enthusiasts of using perl for that But anyway if python would be used then you probably will encounter similar problems with calculation flow as did perl was supposed to change for instance Al concentration in regions left and right of QW in steps controlled by an index wanted to have a separate log file for every index and separate output file names Right now that requires changes during the run of input file config py and even aestimo py or aestimo numpy py If changes were done as in points 1 and 2 than only config py needs to be changed on the run In file sample config perl control template py we would like to have different file names for input We put there inputfilename qw perl MY INDEX and MY INDEX will determine the name of input filename We would like also to have a different log file name also determined by loop index hence we put there as well logfile MY INDEX log In file sample qw perl control template py we need to replace WAVEGUIDE AL q with Al con centration This
10. ced by a quatratic one A bowing parameter b denoted in database by Bowing param is added Eg InPAs Egar 1 ze x s Ej in s b a 1 x 7 11 4 OVERVIEW OF EXAMPLES Table 6 Examples No compulation scheme 1 sample qw qwdope aestimo 2 2 sample qw barrierdope aestimo numpy 6 3 sample moddop aestimo 2 4 sample qw HarrisonCh3 3 aestimo 2 5 sample qw barrierdope p aestimo 2 6 sample double qw aestimo 2 7 any main_iterating py any 12 Aestimo Results Sigma 151814 g 1 0 N lt E 05 E m m E 0 0 o i 0 5 i 1 i 10 1 2 3 5 Position m 1e 8 Potential 2 018719 1 9 J 1 8 J 1 7 J Ww 1 6 J 1 5 J 1 4 0 1 2 3 5 Position m 1e 8 Figure 5 Output results obtained by running sample qw qwdope py input file for a single quantum well doped of AlGaAs Figures show charge Electric Field strength V m 2000000 1500000 1000000 500000 0 500000 1000000 1500000 20000005 0 15 0 10 0 05 0 05 0 10 Quantum well doped AlGaAs GaAs heterostructure Electric Field 2 3 4 5 Position m 1e 8 First state 0 15 2 3 4 5 Position m distribution electric field final potential and two first wave functions 13 4 2 Barrier doped AlGaAs GaAs heterostructure VBHM py It is used as an include file by aestimo numpy h py It contains procedures for solving 3x3 Hamiltonian
11. f xe x and E x for heterostructure alloys we may find out Band of f set parameter a Let us assume also that compound on right is the same as on left but differes only in concentration of one component z with E a b x and x c d x Hence we search a value of AE AE on the right side of heterointerface AE AET Ay AE or AE AE d Ax b Ax d b Simple and nice right As an example let us take Al Gai_ As Al Gai_z As interface where x and zz are less than 0 45 For this compund d 3 575 4 11826 0 45eV and b 1 98515 1 42248 0 45eV the parameters were taken from Synopsus Sentaurus database Hence we obtain AE AE 0 54 which is close to empirical value 3 4 Alloys 3 4 1 Bowing parameter Bowing param In metallurgy Vegard s law is an approximate empirical rule which holds that a linear relation exists at constant temperature between the crystal lattice parameter of an alloy a and the concentrations of the constituent elements For example consider the semiconductor compound n P As A relation exists then such that QInPAs 2 ampt 1 2x GIn As 6 One can also extend this relation to determine semiconductor band gap energies Using InP As as before one can find an expression that relates the band gap energies to the ratio of the con stituents In that case however and also in case of other physical quantities a linear relation is usually better repla
12. higher lying subbands for wide QWs E 2 4 Exchange interaction Electrons holes are Fermions Two of them can not occupy the same state when their spins are the same At the same time electrostatic interaction exists between them regardless how far apart they are A very introductory text on this matter is for instance in this PDF lecture We may say that their quantum mechanical state depends on an additional potential which is called exchange correlation potential Vz in Aestimo There is no a theory that would allow to compute exactly how strong are these interactions There are though approximations used that are in practice sufficiently accurate even though V correction to potential energy may be significant That additional potential ought also be included into a self consistent loop of iterations Exchange interaction plays more significant role in cases of strong doping at low temperatures and very low doping levels it may be ignored An effective field describing the exchange correlation between the electrons is derived from Kohn Sham density functional theory This formula is given in many papers for example see Gunnarsson et al 1976 12 Ando et al 2003 13 or Ch 1 in the book Intersubband transitions in quantum wells 14 Exchange interaction is the main contribution to the effective narrowing of band gap Band Gap Narrowing BNG when a large number of carriers is present there In Aestimo that correction to
13. is the part of the file that interests us material 20 0 AlGaAs 0 __WAVEGUIDE_AL__ 0 n 10 0 AlGaAs 0 08 1e16 n 20 0 AlGaAs 0 __WAVEGUIDE_AL__ 0 n Finaly we would like to have calculation results saved to filenames that are also indexed by loop index For instance if we want to save data about potential and energy of states than we need to have something like this in file aestimo perl template py potn__MY_INDEX__ dat states__MY_INDEX__ dat instead of potn dat states dat Here is a perl example usr bin perl my CONFIG QWPERL AESTIMO 19 open CONFIG sample config perl control template py while my 1 lt CONFIG gt 1 CONFIG 1 J close CONFIG open QWPERL lt sample qw perl control template py while my 1 lt QWPERL gt QWPERL 1 close QWPERL open AESTIMO lt aestimo perl template py while my 1 lt AESTIMO gt AESTIMO 1 close AESTIMO waveguide is Al concentration in regions on left and right of QW In this case we change it from 0 081 to 0 400 in steps of 0 002 for my waveguide 79 waveguide lt 400 waveguide waveguidet my idx_replacement waveguide if waveguide 100 1 idx replacement O waveguide else idx replacement waveguide my myOut CONFIG myOut s MY INDEX idx replacement g open CONFIG gt config
14. isted there are for 300K The main sources of the data are works of Adachi 15 Hamaguchi 16 Chuang 17 Zhang and Razeghi 18 etc Table 4 Database parameter names used in Aestimo for binary and tertiary compounds Parameter me m hh m Ih epsilonStatic Eg mo Bowing_param Band offset m_e_alpha 1 eV G1 G2 G3 h 2mo C11 12 10 dyne cm a0 Ac eV Av eV B eV delta Material Material2 Description electron effective mass heavy hole effective mass light hole effective mass static electrical permittivity Energy gap Determines E change with alloy composition in ternary compounds AE AE see Eq 4 non parabolicity parameter for electrons see Eq 2 Luttinger Parameters 71 72 ya Elastic constants Lattice constant ao deformation potential for conduction band see Chuang Table 10 1 deformation potential for valence band see Chuang Table 10 1 shear deformation potenial see Chuang Table 10 1 spin orbit split of valence bands Binary compound No 1 in an tertiary alloy Binary compound No 2 in an tertiary alloy 3 9 Bands alignement and initial potential The very first step before any computation can be performed is creating the initial potential of the struc ture the conduction band and valence band There are two main approaches used for that by tools similar to Aestimo Some prefer to use so called band offset parameters to align energy gaps at het erostructure junction nextna
15. ll and very basic physics understanding to start manipulating existing examples or even to design new models This document aims at explaining the use of Aestimo at a students level and should be also useful for advanced researches 1 2 Download and installation Aestimo can be downloaded for free from Aestimo site where additional documentation may be found You will need to have a recent version of Python installed on your computer on Linux almost certainly you have it already For this please refer to Python website where binary packages for most platforms can be found Additionally you need libraries called numpy and pylab Aestimo itself does not need compilation Packages may need If you are on Linux Ubuntu than installation of packages is straightforward from Ubuntu Software Center or by using apt get from terminal console 1 8 Copyright Aestimo is copyrighted by Sefer Bora Lisesivdin under GNU GPL v 3 This document is copyrighted by Zbigniew Koziol under Creative Commons License CC BY NC 2 WHAT DOES IT SOLVE The following computational schemes are implemented these are determined by the parameter com putation scheme in model definition file Table 1 The computational schemes implemented computation scheme oci cnm o Solving Schr dinger equa tion is the primary step in finding physical phenom ena in heterostructures and quantum wells In the first approximation the poten tial in eq 1 origins fr
16. nds on how the initial potential was constructed refer to section Bands alignement and initial potential The file wavefunctions dat contains results of computation of wavefunctions solutions of Schr dinger equation or self consistent Schr dinger Poisson equations depending on computation scheme The first column there is coordinate in sample starting from it s left edge It is given in meters Next columns contain results for w s while their number depends on the number of states computed which is set in input file by parameter subnumber e in case of electrons Table 7 Structure of output datafiles File name da next columns efield x m Electrical field V m potn x m Electrical potencial J sigma x m Charge density e m states state No Energy meV Density of states 1 m eff mass m wavefunctions x m 1st v amplitude 2nd v amplitude etc 16 5 2 Examples of Working with Datafiles 5 2 1 GNUPLOT set term x11 set term postscript eps enhanced color Times 22 set output figures gnuplot example00 eps set xlabel x m font Helvetica 18 set ylabel Symbol Y font Helvetica 18 Symbol Y produces Greek letter Psi in postscript set xtics 1e 8 to draw xtics every 1e 8 if not specified the default is used which is too dense plot outputs wavefunctions dat u 1 2 with lines linewidth 3 title Symbol Y _1 outputs wavefunctions dat u 1 3 with lines linewidth
17. ne energy levels bro ken green lines and Fermi energy red broken line in GaSb AlSb In As heterostructure materials are GaSb AlSb In As etc starting from the left region 1 for a compund in this case for ALSb we implicitely select zero energy value at the top of valence band for that compund see Figure 4 If affinity energy Xe scheme is used for band allignement than conduction band is given by E Xe 5 while obviously E E Ey It is easy to check by using data in Table 5 that results displayed in Figure 4 are in a very good agreement with these that would be ob Table 5 Electron Affinity energy x and energy tained by using affinity energy scheme of compu gap E in eV for selected binary compounds at tation T 300K Compound Ret 3 5 AlAs AIP AlSb GaAs GaN GaP GaSb InAs InP InSb 10 18 There is however an important advantage by using the last method In case of heterostruc tures with alloys ternary and tertiary compunds it may become difficult to guess what Band of f set values should be used While at the same time we usually know how E and x change with alloy composition in most cases a linear interpolation between parameters of AB and BC compounds for A B1 C offers a good approximation excep tions are for instance AlGaAs and In ALAs where two linear interpolations must be used depending on z range studied On another hand once we know depen dences o
18. no for instance while others use the concept of affinity energy Synopsys Sentaurus TCAD Both have pros and cons Band offset pa rameters are somewhat more of empirical origin but can not be used in case of arbitrary hetero junctions band offset parameters refer to spe Conduction band offset cific pairs of compounds only Otherwise with the concept of affinity energy which is energy of an electron state at the bottom of conduc tion band needed to remove it from bulk mate rial into vacuum From the point of physics the concept is nicely defined However it appears that in practical experiments using electron affini ties does not always work very well for predicting Figure 3 Illustration of band offset parameter at a Measured heterostructure offsets see Wikipedia heterostructure junction Ps about the Anderson model of heterostruc tures For these reasons we allow users of Aes timo to decide themselves which method to use for aligning bandgaps This is done with param eter use bandgap of f set in config py If it is set True than the band offset method is used with Band of f set parameter values from database If use bandgap of f set is set False initial heterostruc ture potential is computed by using af f nity energy parameter from database In case of conduction band the following line of code is used in Aestimo enumerates grid points fi i matprops Band_offset matprops Eg q Joule where
19. ntial V x Schr dinger Equation gt Wave functions Charge disttribution E Poisson Equation eo oo Once we solve Schr dinger Solve Poisson s equation equation and get wavefunctions we have at the same time charge Solve Schrodinger s equation for new distribution since absolute value Solve oae potential equation for band Has energy converged Yes of wavefunctions is related to charge distribution However the resulting charge distribution will modify potential V x for which we solved the original Schr dinger equation A new value of poten tial is found by solving Poisson equation Therefore we ought to solve Schr dinger equation again end in a new potential and find again a new charge distribution We Figure 1 Block diagram illustrating the process of self consistent should repeat these steps over iteration Figure 3 35 in Harrison and over again until the results of the before the last solution are reasonably close to the results of outcome from the last solution This method of solving equations is called a self consistent method Harrison draws the scheme of calculations as shown in Figure 1 The self consistent method is used whenever we solve Poisson equation or add any other correc tions to potential In these cases the calculations run in a loop until either the allowed loop number is exceeded or calculations become stable i e a convergence is reached edge potential
20. om mismatch of energy bands at heterointerfaces One ought to pay attention only to proper differentiation there due to mismatch of effec tive masses of particles at in terface boundaries Hence the kinetic energy operator should not be used in the form 1 m y 27 but instead in this way 3r tnd Model Schr dinger Schr dinger nonparabolicity Schr dinger Poisson Schr dinger Poisson with nonparabolicity Schr dinger Exchange interaction Schr dinger Poisson Exchange interaction Schr dinger Poisson Exchange interaction with nonparabolicity 2 1 Schr dinger equation File name aestimo py aestimo numpy py aestimo numpy h py config py database py main py main iterating py VBHM py sample py Table 2 Aestimo files Description Main Aestimo file containing all procedures for some computational schemes Table 1 A more advanced version of aestimo py recommended for use It contains some procedures not available in aestimo py and it uses numpy python module for more efficient computation Likely will be renamed to aestimo py in the future A 3x3 k p aestimo calculator for valence band calculations Configuration file available to user There you put also the name of input file name with structure description you want to simulate Material parameters database Keep an original copy of it but entries there and parameters can be changed main file Run it as python main py An ex
21. py print CONFIG myOut close CONFIG my myOut QWPERL myOut s __MY_INDEX__ idx_replacement g myOut s __WAVEGUIDE_AL__ idx_replacement g open QWPERL gt qw perl idx_replacement py print QWPERL myOut close QWPERL my myOut AESTIMO myOut s __MY_INDEX__ idx_replacement g 20 myOut s WAVEGUIDE AL idx replacement g open AESTIMO gt aestimo py print AESTIMO myOut close AESTIMO system usr bin python aestimo py Now once we have these results we would like to draw dependence of bound QW energies as a function of Al content in regions surrounding QW We will use this Python script for that see Figure 9 for drawing results from pylab import import numpy CBO my_x my_state_0 my state 1 my state 2 for idx in range 80 400 2 if idx 100 myidx 0 idx else myidx idx my potential input file name output potn myidx dat my_states_input_file_name output states myidx dat xX potn numpy loadtxt my potential input file name skiprows 0 unpack True no states yl y2 numpy loadtxt my states input file name skiprows 1 unpack True CBO0 append potn 0 my x append idx 0 001 my state O append states O my state 1 append states 1 my state 2 append states 2 xlim 0 08 0 4 plot my_x CBO color red linewidth 2 0 linestyle label E_ CBO plot my_x my_
22. r Artech House Inc 8 www ioffe rssi ru 9 Schubert 10 SERGEY L RUMYANTSEV AND MICHAEL S SHUR MICHAEL E LEVINSHTEIN MATERIALS PROPERTIES OF NITRIDES SUMMARY 11 Wikipedia on Luttinger parameters 12 Gunnarsson and Lundquist 1976 O Gunnarsson M Jonson and B I Lundqvist Exchange and Correlation in Atoms Molecules and Solids Phys Lett 59A 177 1976 13 Taro Ando Hideaki Taniyama Naoki Ohtani Masaaki Nakayama and Makoto Hosoda Self consistent calculation of subband occupation and electron hole plasma effects Variational ap proach to quantum well states with Hartree and exchange correlation interactions J Appl Phys 94 4489 2003 14 M Helm in Intersubband transitions in quantum wells edited by Liu and Capasso 15 Sadao Adachi Properties of Semiconductor Alloys Group lV III V and II VI Semiconductors 2009 John Wiley amp Sons Ltd 16 Chihiro Hamaguchi Basic Semiconductor Physics Second Edition 2010 Springer 17 S L Chuang Physics of Optoelectronic Devices 1995 John Wiley amp Sons Inc 18 Wei Zhang and M Razeghi in Semiconductor Nanostructures for Optoelectronic Applications Todd Steiner Editor Artech House Inc 23
23. rmed for a set of values of thickness of the first region within one run Hence we do not need to create a separate input file for every value of the parameter that we want to have results for Have a look to the code of main iterating py for some explanations Of course if calculations for some another parameter need to be looped through than just create another version of main iterating py for your needs Mostly not many changes will be needed 5 WORKING WITH DATAFILES 5 1 Structure of Datafiles Computation results are stored in subfolder outputs or outputs numpy by default These names can be changed by user in model configuration files with parameter output directory For instance the following files may be found there efield dat potn dat sigma dat states dat wavefunctions dat Each of these files contains rows and columns of data Columns are separated by empty space 0x20 This is a convenient ASCII format of storing data since it allows for easy viewing the results by naked eye and also it is accessible by many software around for instance gnuplot as well it can be easy parsed by python scripts Table 7 describes details of output files structure The file states dat contains information about energy states in multi structure Most likely it will be energy of localized Quantum Well states The fourh column in that file is carriers effective mass in units of free electron mass Where the zero of energy is localized depe
24. state_0 color green linewidth 1 0 linestyle label E_ C1 plot my x my state 1 color blue linewidth 1 0 linestyle label E_ C2 plot my x my state 2 color F000F0 linewidth 1 0 linestyle label E_ C3 legend loc upper left 21 ylabel E meV xlabel x A1 content savefig tutorial figures python example10 eps dpi 72 show 1300 1250 1200 E meV 1150 1100 1050 1090 0 10 0 15 0 20 0 25 0 30 0 35 0 40 x Al content Figure 9 Dependence of bound QW energies for single QW heterostructure of AlGaAs AlGaAs AlGaAs as a function of Al content in regions surrounding QW Al in QW content is 0 08 Conduction Band Offset energy is drawn by thick red line Parts of curves that lye below Ecgo represent electron states localized in QW The figure is created with python code shown in this sub section 22 REFERENCES 1 Paul Harrison Quantum Wells Wires and Dots Theoretical and Computational Physics of Nanos tructures Wiley Interscience 2005 2 D Ahn S H Park ENGINEERING QUANTUM MECHANICS 3 Nelson D F R C Miller and D A Kleinman Band nonparabolicity effects in semiconductor quantum wells Physical Review B 35 no 14 1987 7770 4 aestimo ndct org 5 matplotlib org 6 matplotlib org contents html 7 Wei Zhang and M Razeghi in Semiconductor Nanostructures for Optoelectronic Applications Todd Steiner Edito
25. t3 is potential energy matprops Band offset is value of Band of f set from database and matprops Eg is energy gap of the material In other words in Asetimo when using band offset concept we assume that conduction band energy is given by E AE AE Es 4 where a AE AE is the value of Band of f set parameter from database After we have FE valence band energy is obtained in a straightforward way E E Eg Users may manipu late values of material pa rameters Here is an ex ample of how we were able to reproduce structure consisting of 3 compunds with the following layers 10 nm wide each of them GaSb AlSb InAs AlSb GaSb We did not had en tries of Band of f set for all heterointerfaces in such a sandwich Our aim was to reproduce diagram from Thesis of Sebastian Brosig from 2000 Transport Mea surements on InAs AISb Quantum Wells Fig 3 2 in that work In order to get con duction band alignement A 20 we used the following Band_of f set Position nm parameters GaSb 1 AlSb 1 InAs O 6 0 57 Actually any values are good there as long as the structure obtained cor responds well to that real one from measurements Let us notice also that by selecting Band offset Aestimo Results 1600 1400 1200 1000 Energy meV 800r 400r 200 50 Figure 4 Conduction band alignement black solid li
26. which contains procedures for solving heavy and light holes band and a spin orbit split off 3x3 Hamiltonian for a 001 oriented zinc blende band with double degeneracy are considered ZB crystal by using k p theory Finite Difference Method The acronym 323 means in this case that we are dealing with 3 valance bands This is a crude approximation as it is known All of three bands are double spin degenerate A better approxi mation would be to use 6x6 method but we have no explicite any magnetic interactions so far in our Aestimo Even better one would be to use a 4x4 method when conduction band is included And even better one would be to use 8r8 method especially in case of applied external magnetic field when 3 valance bands interact and one conduction band and all are spin degenerate The method we use is explained by Harrison 1 page 110 and also Chuang 17 p 183 For a same code in Fortran and Dirichlet boundary conditions see Ahn and Park 2 Conduction band soh Spin orbit split off 3 HOW THE MATERIAL STRUCTURE AND GRID ARE CREATED 3 1 How the device structure and grid are created The device structure and grid on which computation is performed are created in the simplest possible way As an example have a look to file sample qw qwdope py At the end we find there the following definition material 20 0 AlGaAs 0 3 O n 10 0 GaAs 0 2e18 n 20 0 AlGaAs 0 3
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