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1. oh 5 epp E P 7 5 ET 6 8 The default is P P 0 6 3 MODELS SECTION 67 6 3 3 Mobility models Mobility models have currently to be defined for electrons and holes independently The corresponding keywords are electron mobility and hole mobility i e physical model electron mobility d model physical model hole mobility d model The default model is the constant mobility model The parameters for the different mobility models are needed for both electrons and holes In the material files they are specified with a common keyword in arrays e g mobility constant electrons holes mu max 1400 0 250 0 exponent 1 0 s 2 1 Constant mobility model The constant mobility model identifier constant assumes a mobility which depends only on temperature by means of the following formula Hconst Lo T 10 6 9 In the material data file uo and y have to be specified with the keywords mu max and exponent uy can be ovverridden from the physical model section using the keyword mu or from the Region sections using the keywords mu e and mu h Doping dependent mobility model The doping dependent mobility model identifier doping dependent implements two models for mobility depending on the total doping density and the temperature The model that is used depends on the value of the mobility formula parameter 68CHAPTER 6 DRIFT DIFFUSION SIMULATION OF ELECTRONS AND HOLES
2. 80 CHAPTER 7 HEAT BALANCE SIMULATION where g surf indicates and temperature stands for Tert Alternatively it is possible to indicate the R 1 G_s quantity i e the thermal surface resistance The notation is BC Regions 1 BC Region name BC region d type thermal surface resistance BC reg numb 3 r surf 100 temperature 300 Furthermore it is possibile to fix the normal thermal power density to a given value Fent Le EN dos 7 6 This condition is set with the BC type thermal_flux BC_Regions 1 BC Region name BC region 1 type thermal flux BC reg numb power density I wl 100 7 3 Output data The variable labels are listed in the tables 7 3 7 5 It is also possible to identify all heat sources with the keyword HeatSource and all power fluxes with PowerFlux Finally with the keyword thermal all quantities concern ing the thermal simulation will be stored 7 3 OUTPUT DATA 8l description type parameters units Ideal insulating interface Default No parameters Ideal conducting interface heat reservoir temperature K Resistive interface thermal surface resistance temperature K r_surf cm KW Resistive interface thermal_surface_conductance temperature g_surf VR cem Power density condition thermal flux power density Wem Table 7 2 Thermal boundary conditions Nodal scalar quantities LatticeTemp Temperature K Table 7 3 Nodal scalar quantities
3. 1 material Si02 Models 1 1 2 DEFINITION OF PHYSICAL AND BOUNDARY REGIONS IN TIBERCAD 3 model driftdiffusion 1 simulation name dd Simulation E solve dd The section Device is used to associate or more mesh regions to a material and to a set of physiscal properties such as doping concentrations doping levels etc The section Model is used to define the physical models to be solved Each physical model can be applied to the whole device or to a set of regions of the device defined in Device section Finally Simulation section states the type of calculation to be executed that is the simulation to be solved These sections will be described in full details in the following Besides these main sections there other other two called Solver and Physics where some parameters can be set respectively for the numerical solvers and for the physical models The aim of these sections is to give the user the maximum of flexibility to tune his her simulation Especially regarding the Solver case values of the numerical parameters have been already tuned for each application and should be modified only by an advanced user 1 2 Definition of physical and boundary regions in TiberCAD Let see now in more detail how to associate physical information to the regions of the device model and how to define the boundary regions that is contacts or in general regions where some kind of boundary condition
4. Model by Masetti et al 4 The model by Masetti et al is identified by mobility formula 1 It uses the following formula Po N Lconst 2 Hi 6 10 H Umin 1 1 N C 2 14 C N where is the total doping density and const the mobility obtained from the constant mobility model The parameters are specified in the material file as given in table 6 6 parameter keyword Hmin 1 mumini Mmin 2 mumin2 pa mul P Pc Cr C Cs Q alpha B beta Table 6 6 Data file parameters for the mobility model by Masetti et al Model by Arora 5 The model by Arora is identified by mobility_formula 2 It reads Hd HT mi AE QNIN with Hmin Amin T To pa Aaf dal No Au T To AT To 6 11 The parameters are given in table 6 7 Field dependent mobility model The field dependent mobility model describes the degradation of mobility at high driving fields It is identified by the identifier field dependent The electric field component in directon of the current flow or the gradient of the electro chemical potential can be chosen as driving force driving force efield grad fermi 6 3 MODELS SECTION 69 parameter keyword Amin mumin Aq mud An NO Aa A Am am ad ad ON aN Qa aA Table 6 7 Data file parameters for the mobility model by Arora The default driving force is the gradient of the electro chemical potential The mod
5. barrier 1 QWell barrier 2 Atomistic TB 2 physical regions 49 EXAMPLE OF INPUT FILE Definition of Simulation Models Models 1 model driftdiffusion d options d simulation name ddl physical regions all physical model recombination model srh physical_model recombination model direct C 1 1e 8 BC Regions d BC Region cathode t BC reg numb 1 type ohmic voltage 0 0 and associated Boundary Conditions 45 46 CHAPTER 4 INPUT FOR TIBERCAD BC Region anode 1 BC reg numb 2 type ohmic voltage 0 0 model macrostrain 1 options 1 simulation name strain physical regions all BC_Regions BC_Region substr BC_reg_numb 1 type substrate material GaN structure 2 y growth direction 1 0 1 0 z growth direction 1 2 1 0 x growth direction 0 0 0 1 49 EXAMPLE OF INPUT FILE 47 model efaschroedinger 1 options d simulation name quantum electrons Syntax physical regions list Tiber region Tiber cluster physical regions Quantum 1 model efaschroedinger 1 options 1 simulation name quantum holes physical regions Quantum 1 Definition of Model dependent Solver parameters Solver 48 CHAPTER 4 INPUT FOR TIBERCAD driftdiffusion d coupling poisson ksp type bcgsl nonlin abs tol 1e 10 nonlin step tol 1e 2 tnonlin rel tol 1e
6. through the keyword BC_Region BC_Regions BC_Region cathode 1 3 SIMULATION ENVIRONMENTS 7 As before it is possible group more boundary regions in a single BC_Region with the keyword BC_reg_numb BC_Regions BC_Region cathode d BC reg numb contacti contact2 Again in this second case cathode is a label chosen by user while contact and contact2 are labels previously defined inside GMSH In 2D case a set of Physical Surface will be defined to be used as mesh regions while Physical Line is used for boundary regions Finally in 3D case Physical Volume is used to define mesh regions while Phys ical Surface is used to define boundary regions 1 3 Simulation environments TiperCAD allows to compute different physical models in different parts of a device or nanostructure by coupling in a general way different simulation environments A simula tion environment is composed by all the physical regions to which a particular model is assigned A simulation environment is therefore defined by the mesh elements belonging to its physical regions and by the simulation model which has been associated to these regions This association is made possible by the definition of TIBERCAD Regions and Clusters Different simulation environments can have physical regions in common In this way each simulation is run on a subset of the device and can be possibly coupled even self consistently with a simulation run on a different subs
7. 0 124 0 187 0 188 0 0688 0 0 0688 0 188 Figure 3 2 2D Mesh for the mosfet device obtained vvith GMSH 21 1 Definition of Device Regions Three TrBERCAD regions are defined to each of them one mesh region is associated that is the Phisical Surfaces 1 2 and 3 defined in Step 1 However in general more than one mesh_ region can be associate to a single TIBERCAD region if this is convenient Region substrate 1 mesh regions 1 material Si doping 1e18 doping type acceptor Region contact mesh_regions 2 material Si doping 5e19 doping_type donor Region oxide mesh_regions 3 material 5102 2 Definition of Simulation Now we define the Simulation dd it belongs to the class driftdiffusion model driftdiffusion options simulation_name dd physical_regions all 22 CHAPTER 3 GETTING STARTED 2D We declare two driftdiffusion physical models the first defines a srh recom bination model see 6 3 1 the second defines a field dependent mobility model for electrons which implements a doping dependence for the low field mobility see 6 3 3 physical model recombination 1 model srh physical model electron mobility d model field dependent low field model doping dependent 3 Definition of Boundary Conditions The source drain and gate contacts of the Mosfet device are defined as Boundary conditions regions BC Region source BC Reg
8. Strain in transistor d substrate layer of Al 0 3 Ga O 7 N 5 3 2 Structure without a substrate freestanding In this case the parameter substrate should not be present instead the following pa rameters should be defined e The reference lattice material is defined by the coordinates of a point belonging to this material using the parameter reference material point 56 CHAPTER 5 SIMULATION OF STRAIN e As follows from Ref 1 additional geometrical points have to be specified accord ing to the device dimensionality The corresponding parameters are fixed point fixed point2 and fixed point3 Since the elasticity energy is invariant with respect to translations and rotations of the structure then for the sake of uniqueness of solutions of the equations another set of constraints is required Hereafter we assume that a mesh is defined over the simulation domain and the displacement field u r is defined at the mesh nodes Let D be the dimensionality of the structure minus the number of directions along which the structure is periodic If D gt 0 then a Dirichlet boundary condition is applied at an arbitrarily chosen node 44 0 5 2 in order to prevent the structure from undesirable translations In the case of D gt 1 another node 4 is chosen and a constraint u ri u ti ri 0 5 3 is applied in order to keep the direction between the nodes 71 22 unchanged If D 3 another
9. V eVy P e n p Nf V unn Von PVT R 6 1 V us V PVT R P is the electric polarization due to e g piezoelectric effects and R is the net recombi nation rate ie recombination rate minus generation rate P and P are the electron and hole thermoelectric power respectively The models for the mobilities and the net recombination rates can be specified in the physical model sections as described in the following 6 2 Plot variables See tables 6 1 6 2 and 6 3 6 3 Models section The Models section looks as given in Listing 3 61 62CHAPTER 6 DRIFT DIFFUSION SIMULATION OF ELECTRONS AND HOLES model driftdiffusion 1 options simulation_name whatever_you_want physical_regions 2 3 4 physical_model recombination model model_to_be_used physical_model electron_mobility model model_to_be_used J physical_model hole_mobility model model_to_be_used J physical_model thermoelectric_power model model_to_be_used Listing 3 Models section for drift diffusion 6 3 MODELS SECTION Nodal quantities 63 Ec Conduction band edge eV EcO Conduction band edge without electric po eV tential Ev Valence band edge eV EvO Valence band edge without electric potential eV Eg Band gap eV QFermi e Electro chemical potential of electrons eV e QFermi h Electro chemical potential of holes eV ElPotential Electric potential V eDensity E
10. be separated by comma and enclosed between and parenthesis 34 CHAPTER 4 INPUT FOR TIBERCAD 4 4 2 physical model block physical model recombination model SRH The following options can be applied to any physical model name a user defined name for the model for unique identification restrict to region a single region name or a list of regions where the model should be used This allows to use different implementation of the same type of model in different regions 4 4 3 BC region block BC Region anode 1 BC reg numb 2 type ohmic voltage 0 0 voltage Vb 1 5 BC region name of the present boundary region it can be the name of a boundary physical region specified in the meshing program GMSH or ISE TCAD BC reg numb BC region ID s as specified in the meshing program GMSH If this keyword is not present it is assumed that the mesh region associated to this TiberCAD BC region is given by the name assigned to BC region type type of boundary condition ohmic schottky substrate for strain calcula tions voltage value of voltage V applied to the present BC region for ohmic and schottky BCs it can be the value of a sweep variable Vb indicated with Vb A possible default value can be indicated in parenthesis Vb 1 5 zero grad fermi h zero_grad_fermi_e if true set Neumann b c to the fermi level in the b c region If type is substrate for strain calculations
11. n Si 46 8102 Physical Surface 1 Physical Surface 2 Physical Surface 3 e Definition of the Phisical Lines In this 2D simulation 1D physical regions are used to carry information about boundary condition regions In other word each Phisical Line corresponds to a boundary condition a contact in the case of a driftdiffusion calculation thus Physical Line 1 refers to source contact P L 2 to gate contact P L 3 to drain contact The numerical identifications of these Phisi cal Lines will be asigned to TIBERCAD BC regions by means of the BC reg numb instruction Physical Line 1 Physical Line 2 Physical Line 3 13 source 39 38 gate 19 drain In fig 3 1 the obtained geometrical model is shown Step 2 Meshing the device The geo script file with the geometrical description can be run in GMSH to display the modelled device and to mesh it through the GMSH graphical interface see fig 3 2 Alternatively a non interactive mode is also available in GMSH without graphical user interface For example to mesh this 2D tutorial in non interactive mode just type gmsh mosfet geo 2 o mosfet msh Step 3 TiberCAD Input file Now we have to write down the TIBERCAD input file see mosfet tib in the Tutorials 20 CHAPTER 3 GETTING STARTED 2D 0 0025 0 0606 0 124 0 187 0 138 0 0688 0 0 0688 0 138 Figure 3 1 Geometrical structure as defined by GMSH modeller 0 0025 0 0606
12. current at the contacts of a diode or quantized energy levels in a quantum well These Integrated quantities are displayed in separated files with the format simname ext e g quantum electrons dat where simname is the name of the model simulation associated to the results If a sweep is performed the output file gets the format sweep simname varname ext where varname is the variable with respect to which the sweep is performed for example sweep driftdiffusion Vb dat Inside the file output values for all the steps of calculation are shown Finally a last class of output files is the Materials output These files contain the information about the physical regions of the device for each class of simulation a different material file is produced containing all and only the mesh regions associated to that simulation model The file has the format simulationname materials ext e g driftdiffusion materials dat 4 9 Example of Input file Here is an example of the input file template Description of the device physical regions Device 1 Syntax Region Tiber_region 42 CHAPTER 4 INPUT FOR TIBERCAD mesh regions list gmsh region ID names list ISE TCAD region names Hif mesh regions is empty gt mesh regions Tiber region Region buffer 1 mesh regions 1 structure wz y growth direction 1 0 1 0 z growth direction 1 2 1 0 x growth direction 0 0 0 1 material GaN dopi
13. for the line search 1 step in units of eV nonlin max it maximum number of nonlinear iterations 20 Table 6 12 Parameters for the PETSc nonlinear solver keyword description default ksp type the linear solver type bcgsl pc_type the preconditioner type ilu lin_rel_tol relative tolerance for the linear solver le 6 lin abs tol absolute tolerance for the linear solver 10e 50 lin_max_it maximum number of linear iterations 500 The linear tolerance gets automatically decreased after each nonlinear step Table 6 13 Parameters for the PETSc linear solver The ksp_type specifies the type of Krylov subspace method to be used The mostly used methods are bcgs A stabilized version of the biconjugate gradient method This one works better in 1D than bcgs1 bcgsl default A modified version of bcgs gmres Generalized minimal residual method The pc type specifies the type of preconditioner to be used The most useful ones are 76CHAPTER 6 DRIFT DIFFUSION SIMULATION OF ELECTRONS AND HOLES ilu default Incomplete LU factorization Does not work for materials with high band gap jacobi Jacobi preconditioning diagonal scaling composite Combination of ilu and jacobi The 1s max step parameter defines an upper bound of the lo norm of the nonlinear line search step It should be not too big to prevent the algorithm from diverging but also not too small to minimize the number of iterations Values between 1 and 10
14. in that part cutoff is specified as a percentage of the embracing length and should therefore be between 0 0 and 1 0 plot embracing regions bool Whereas the automatic creation of the embracing region in 1D is a very simple task it is a more difficult one in higher dimensions By setting this flag to true the embracing region and the mixing coefficient x will be plotted for a visual control of the quality of the embracing region The default is false 6 5 Solver section Many of parameters for the numerical solver depend on the type of solver being used and on the device to be simulated Table 6 11 lists the options that are independent on the type of solver used The linear and nonlinear solvers to be used can be chosen using the keywords linear_solver and nonlinear_solver respectively For the nonlinear solver one can chose between the PETSc implementation petsc and the TIBERCAD implementation tiber of line search When using the TIBERCAD nonlinear solver one can additionally chose between TACHAPTER 6 DRIFT DIFFUSION SIMULATION OF ELECTRONS AND HOLES keyword description coupling defines which equations to couple together poisson solve only poisson eq electrons electrons and poisson holes holes and pois son current only electron and hole cur rents full the fully coupled system el qfermi level the spatially constant electrochemical poten tial for the electrons for quasi equilibrium calculations hl
15. node 73 is chosen and additional constraint is set D s Ti Ti Ti uj 0 5 4 so the node 3 has to belong to the r r plane Example for a 2D simulation Strain in transistor 1 reference material point 0 100 0 fixed pointi 0 0 0 fixed point2 10 0 O 5 3 3 Additional parameters Numercal solver parameters tolerance relative tolerance of the iterative solver e g tolerance 1e 10 The default value is 1e 10 Esp type type of solver gmres bcgsl bcgs cg richardson Default is 5 3 SOLVER PARAMETERS 57 e gmres for 1D e bcgsl for 2D and 3D pc type of pre conditioner ilu composite jacobi lu cholesky eisenstat Default is e ilu for 1D e jacobi for 2D and 3D maa iterations max number of iterations default 1000 monitor monitor if true textual or graphical monitor of convergence process is enabled default false Periodic boundary conditions It is possible to specify periodic boundary conditions along the coordinate axes The relative parameters are periodicity_z true false periodicity_y true false periodicity_z true false The default value is false Mesh refinement For the details about mesh refinement refer to the Libmesh library documentation The parameter refinement_steps defines the number of the refinement steps to be done with default value equal to zero The parameter uniform refinement tru
16. not provided the default value of 300 K is used 6 4 1 Simple semiconductor model When specifying model simple a very simple semiconductor model is used For this model one has to provide conduction and valence band edges and the effective density of states masses in the Region sections The corresponding keywords are given in table 6 10 keyword description EG conduction band edge eV Ev valence band edge eV m dos e conduction band effective DOS mass me m dos h valence band effective DOS mass me Table 6 10 Parameters for the simple semiconductor model 6 4 2 Default semiconductor model The default semiconductor model uses a bulk k p model to calculate the band parameters It can be chosen explicitly by model default The model reads all needed parameters from the material data file The band parameters are calculated considering locally strain and lattice temperature as obtained from the corresponding simulations specified using the strain simulation and thermal simulation keywords 6 4 3 Special options for Schr dinger Poisson calculations TrBERCAD is able to do selfconsistent Schr dinger Poisson or Schr dinger Drift Diffusion calculations For this purpose electron quantum density or hole quantum density has to be specified for at least one region and a selfconsistent simulation should be defined in the Selfconsistent block The following options to be specified in the Physics sectio
17. physical regions are created which associate together geomet rical regions and then the related mesh elements which share some common physical properties It s only these physical regions which can be referred to outside GMSH In TrBERCAD this is done by associating one or more physical regions to a TiberCAD region through the keyword mesh regions see in the following e Definition of two physical entities Physical Point Physical Point 1 1 Physical Point 2 2 Beginning from GMSH v 2 it is possible alternatively to assign more convenient Physical Names to the Physical entities instead of numerical IDs Physical Names consist of strings enclosed between quotation marks The syntax is the following 11 Physical Line bulk 1 Physical Point Anode 1 N B In general in a n Dimension nD simulation n 1 D physical regions points in 1D lines in 2D surfaces in 3D are used by TIBERCAD to impose the required boundary conditions Each n 1 D physical region defined in this way in GMSH will be associated in TrBERCAD to a boundary condition region through the keyword BC reg numb Thus in this case Physical points 1 and 2 will be associated respectively to two BC regions see in the following Step 2 Meshing the device The geo script file with the geometrical description can be run in GMSH to display the modelled device and to mesh it through the GMSH graphical interface Alternatively a non inter
18. should be a good choice The nonlin step tol defines at which line search step size in l3 norm the algorithm stops ie assumes to have reached convergence nonlin step tol is measured in eV 6 5 2 Parameters for the TrserCAD nonlinear solver Table 6 14 summarizes the parameters used for the TrBERCAD implementation of the line search algorithm keyword description default nonlin_rel_tol relative tolerance for the residual lo norm 10e 9 with respect to first nonlinear step nonlin abs tol absolute tolerance for the residual 5 norm 10e 15 nonlin step tol tolerance for the maximum norm of the non 10e 3 linear step eV nonlin max it maximum number of nonlinear iterations 20 Table 6 14 Parameters for the TIBERCAD line search The stopping criterion based on the line search step uses the maximum norm of the nonlinear step i e convergence is controlled locally In addition to the parameters in table 6 14 one has to provide also parameters for the linear solver 6 5 3 Parameters for the PARDISO linear solver NOTE the PARDISO linear solver is currently not included in the distribution Currently the PARDISO interface has no adjustable parameters Chapter 7 Heat Balance simulation The theoretical model of the heat balance problem can be found in Ref 7 7 1 Heat equation The steady state heat equation with the continuity equation reads as where T is the temperature k is the thermal conductivit
19. simulations 2 for 1D simulations k space basis is true if the k space is defined by means of k vectors if false vectors are expressed in real space If refine E space true that is adaptive k mesh refinement is enabled all the el ements whose error is greater than the value 1 refine fraction maximum error are going to be refined In this case Error is just the integrated quantity The refinement will end when the relative accuracy is obtained number of nodes numb of elememts in k mesh along each direction wedge half quarter to reduce calculation time by exploiting symmetry optical matr elem model name of the opticskp model associated polarization light polarization vector Emin Emaz dE energy range and step of spectrum calculation 10 1 Output The output variables for optics calculations are e optical spectrum k space integrated optical emission spectrum calculated by opticalspectrum model Chapter 11 Quantum dispersion There is a possibility to calculate the dependence of quantum eigenstates on k vector Such dependence is called dispersion The simulation name is quantumdispersion Example model quantumdispersion 1 options 1 simulation name dispersioniD el physical regions all 11 1 Solver options The dispersion of quantum states is calculate at k points that are nodes of the mesh in k space e quantum simulation name of the Schr dinger equation simulatio
20. tagname is a string e tagvalue is a single numerical or string item or a list of items between and parenthesis and separated by commas e g cathode anode 27 28 CHAPTER 4 INPUT FOR TIBERCAD Format is free for the parameter assignements provided that they are separated by spaces Everything which follows a is considered as a comment and is disregarded For example driftdiffusion 1 coupling poisson nonlin max it 70 nonlin rel tol 1e 10 ls max step 2 ksp type bcgs Here and in the whole input file a string item can include a combination of characters special characters and numbers but not spaces if a space is found the string item is taken as terminated The input file is composed by the following sections Device Scale Models Physics Solver Simulation which will be described in the following 4 2 Device section Device 1 Region buffer In Device section two kinds of block can be present the Region block and the Cluster block 42 DEVICE SECTION 29 The Region blocks contain the description of the device in continuous media ap proach the Cluster blocks define each a group of regions mesh regions even with different physical properties but to be treated together somewhere in the simulation e g quantum calculation In this way it is possible to refer to the set of these regions simply by the Cluster name Each Region block must be preceded by the keyword Regi
21. the definition of a geometrical Line the two expressions inside the braces on the right hand side give the identification numbers of the start and end Points of the line Then two physical regions are defined each associated to one of the two geometrical entities Physical Line n_side and Physical Line p_side The expression s inside the braces on the right hand side give in general the identification numbers of all the geometrical lines that need to be grouped inside the Physical Line In this way these physical regions are made available for TrBERCAD and will be used to associate them to a TiBERCAD region through the keyword Region as follows Region n side Region p side 6 CHAPTER 1 OVERVIEW It is also possible to group more than one physical region in a single Device Region with the keyword mesh regions as follows Region reg 1 mesh regions regioni region2 Region reg_2 1 mesh regions region3 region4 Note that in this case region1 region2 region region4 are the labels previously defined inside GMSH while reg_1 and reg_2 are names chosen by user for his convenience Then in the GMSH script two Physical Point are defined anode and cathode and associated to the first and to the last point of our 1D device These points are needed to impose some boundary conditions and in this way they are made available for TiberCAD and will be used to associate each of them to a boundary condition region
22. 12 lin rel tol 1e 6 nonlin max it 30 Hlocal density scaling true Hls type none discretization fem ls max step 1 pc_type lu pc_type composite integration_order 2 relaxation_factor 0 5 macrostrain x substrate substr efaschroedinger 1 X periodicity false Dirichlet bc everywhere true particle hl number_of_eigenstates 30 model conduction band eff mass cb poisson_model_name driftdiffusion potential from driftdiffusion strain_model_name macrostrain convergent_density true 49 EXAMPLE OF INPUT FILE 49 quantum electrons particle el quantum holes t particle hl Definition of Model dependent physical parameters Physics driftdiffusion statistics FD strain_simulation macrostrain default driftdiffusion model including local strain obtained from macrostrain quantum_electrons particle el model conduction_band eff mass cb d quantum holes 1 50 CHAPTER 4 INPUT FOR TIBERCAD particle hl model kp k p for valence band kp_model 6x6 Definition of model indipendent parameters of the Simulation Simulation searchpath materials mesh_units ie 9 Hom dimension 1 meshfile test msh temperature 300 solve strain driftdiffusion quantum electrons quantum holes resultpath output plot Ec Ev QFermi e QFermi h EField eDensity hDensity eCurrent hCurrent Current NetRecombin
23. 7 Simulation section In this section one can specify several general parameters and settings for the actual calculation to be run such as the mesh file to be used the dimension of simulation the process flow of simulation etc searchpath path for material files 40 CHAPTER 4 INPUT FOR TIBERCAD driftdiffusion 1 Strain simulation strain meshfile name of mesh file N B the extension is mandatory grd for ISE TCAD msh for GMSH mesh file v 1 and v 2 0 mesh units units of measurements used in the meshing relative to meters e g 1079 for um dimension dimension of simulation 1 2 3 temperature temperature of the system K solve list of simulations to be executed in the order of execution if the list contains sweep a sweep is performed as specified in sweep block in the Solver section solve strain driftdiffusion quantum electrons quantum holes resultpath path for output directory output format format of the output data gmv for GMV ise for Tecplot grace for xmgr ascii data column type vtk for Paraview plot list of output variables which are calculated and available in output files See the corresponding chapters for the available output variables for each model 4 8 Output description At the end of the execution the program will write the results of the simulation in the directory specified by resultpath with the format specified by output format The output variables are speci
24. 83 N D Arora J R Hauser and D J Roulston Electron and Hole Mobilities in Silicon as a Function of Concentration and Temperature IEEE Trans on Electron Devices vol 29 pp 292 295 1982 C Canali G Majni R Minder and G Ottaviani Electron and hole drift velocity measurements in Silicon and their empirical relation to electric field and temperature IEEE Trans on Electron Devices vol 22 pp 1045 1047 1975 Gerhard K Wachutka Rigorous thermodynamic treatment of heat generation and conduction in semiconductor device modeling IEEE Transaction on Computer aided Design vol 9 pp 11 1990 95
25. AD gets installed Version 2 3 0 of GMSH http www geuz org gmsh will be installed together with TiperCAD For the Linux version of GMSH you need OpenGL libraries installed on your system Prerequisites Get the installer package for your OS architecture from http www tibercad org or by contacting support tibercad org Table 1 lists the packages available for download run TrBERCAD you will also need a license file that you will have to copy into the installation directory of TiBERCAD In the Windows version some graphical features such as graphical convergence mon itors are only available if an X Window server is installed and running Windows installation procedure To install TrBERCAD in Windows run the setup program tibercad version Setup ere During the installation you can choose the installation directory After finishing installa tion copy your license file tibercad lic into the license subdirectory of the TIBERCAD installation directory INSTALLPATH license without changing its filename installer package name Target architecture tibercad version_setup exe Windows 32 bit tibercad version_installer bin Linux 32 bit self extracting installer Table 1 Installer packages V VI CONTENTS Linux installation procedure To install TiBERCAD in Linux download and run the self extracting installer tibercad version_installer bin and follow the installation instructions After installation copy your licens
26. BC Regions 1 BC Region cathode In Models section one or more model blocks must be present each model block must be preceded by the keyword model followed by the single word model name This must be the name of one of the TIBERCAD simulation models Here are the simulation models implemented until now e driftdiffusion Poisson driftdiffusion transport of electrons and holes e thermal Heat balance simulation e excitontransport Exciton transport model e macrostrain Calculation of Elastic deformations in heterostructures e efaschroedinger Envelop Function Approximation EFA solution of single particle Schr dinger equation for electrons and holes e quantumdensity Calculation of quantum density of electrons and holes e quantumdispersion Dispersion of quantized states in k space e opticskp Optical properties optical kp matrix elements e opticalspectrum Emission spectrum with k space integration 44 MODELS SECTION 33 For a complete description of these simulation models see the next chapters Each model block can contain some optional blocks to be written in the following order e one options block preceded by the keyword options This block can contain general options for the present model e one or more physical model blocks each physical model block must be preceded by the keyword physical model followed by the single word name of the phys ical model Each physical model block can contain paramete
27. Elemental scalar quantities eJoule Electron Joule effect Wem hJoule Hole Joule effect VVem RecHeat Recombination heat Wem ePelTh Electron Peltier Thomson effect Vem hPelTh Hole Peltier Thomson effect TotalHeat TotalHeat VVem LatticeThermalCond Lattice thermal conductivity Wem RE Table 7 4 Elemental scalar quantities Elemental vector quantities Wq Thermal flux Wem 2 Wn Electron power flux Wem Wp Hole power flux Wem W Total power flux Wem Table 7 5 Elemental vector quantities 82 CHAPTER 7 HEAT BALANCE SIMULATION Chapter 8 Envelope Function Approximation The envelope function appriximation EFA simulation tool of TiBERCAD is developed in order to solve a single particle Schr dinger equation for electrons and holes in a semi conductor crystal This problem is an eigenvalue problem that is treated as a generalized complex eigenvalue problem Sy 8 1 where and S are the Hamiltonian and S matrix respectively 8 1 Models section parameters The Models section looks like follows model efaschroedinger 1 options i simulation name quantum welli physical regions 1 2 The default boundary conditions of the simulation domain are open that is zero flux for single band calculation It is possible to specify Dirichlet boundary conditions BC Region infinite barrierl d BC reg numb 12 83 84 CHAPTER 8 ENVELOPE FUNCTION APPROXIMATION type D
28. TIBERCAD User Manual Fabio Sacconi Matthias Auf der Maur Michael Povolotskyi Giuseppe Romano Alessandro Pecchia Gabriele Penazzi Stefano Bellocchio Aldo Di Carlo May 12 2009 TIBERCAD User Manual TiberCAD authors F Sacconi M Auf der Maur M Povolotskyi G Romano A Pecchia G Penazzi S Bellocchio A Di Carlo 2008 Document revision 1 2 2 1446 Contents Contents I Installation instructions V 1 Overview 1 1 1 Intoduction to numerical simulation with TiberCAD 1 LLI input a uus e erie o x 2 1 2 Definition of physical and boundary regions in TiberCAD 3 Lei Usm EE oe a sar n E enu 4 LAS Une DN uA Mr Rex URGES GR RECORD Qe REO GS 4 1 3 T 2 Getting started 1D 9 3 Getting started 2D 17 4 Input for TiberCAD 27 4 1 Description of Input file structure 2T E A ew Eee ea See ee Pee ee Kok 28 da OPO eh k eo lores MK e eh oe ea lora 30 Zo ADOS NES silicosis kee eo Ree ee Ro wor cho 31 Adl options Doek lt cossos PP 33 142 physicalmmodel bloke lt c s cos Ee Rx 34 Ads BC mion Doti io gie we ew Glew ee ee Shae ew Ol 34 io RINES SOCIO e RBA AR AAA Peu URL GRON erra 95 A6 PCs SOC aule eee eRe aa eee 39 Ar SECHOD e saca EN ER aa A S 39 As Output description 2 rack x 3 3x aa A A EN 40 a Example of Input ile e s reps
29. VK em electric power Table 6 2 Elemental quantities Scalar quantities ContactCurrents Contact currents a depends on dimension and symmetry Table 6 3 Scalar quantities SRH recombination is defined as follows ni n nie sie p me El Erap Ec E 2 is the trap level with respect to the midband energy n is the intrinsic carrier density and 7 are the recombination times The parameters are taken from the material database The recombination times are dependent on temperature and doping density e g par mm eer 6 3 To P Tmaz n dd Tmin n 6 4 Tmin n 7 LE N Nreg where T is the reference temperature 300 Table 7 2 shows the corresponding pa rameters for the material data files The parameters for holes and electrons have to be specified in an array e g taumin 1e 5 3e 6 6 3 MODELS SECTION 65 parameter keyword Tom taumin Tmar taumax Nref y gamma E Etrap Q Talpha Tcoeff Table 6 4 SRH material data file parameters The recombination times and trap level can be overridden from the input file by using the keywords of table 6 5 in the appropriate physical model section or in the Region section the latter overrides the former tau tau n tau tau p E Et Table 6 5 SRH input file parameters Direct radiative recombination The direct recombination model can be enab
30. active mode is also available in GMSH without graphical user interface For example to mesh this 1D tutorial in non interactive mode just type gmsh bulk geo 1 o bulk msh where bulk geo is the geometrical description of the device with GMSH syntax 1 means 1D mesh generation some command line options are 1 2 8 to perform 1D 2D or 3D mesh generation o mesh file msh to specify the name of the mesh file to be generated In this way a msh has been generated and is ready to be read in TrBERCAD Step 3 TiberCAD Input file Now we have to write down the TIBERCAD input file see bulk tib in the Tutorials 1 Definition of Device Regions First we have to list all the TIBERCAD Regions present in our device a TIBERCAD Region is usually a somehow physically homogeneous region of the device or the nanos tructure we are going to model featuring the same material and possibly the same doping Device Region bulk 12 CHAPTER 2 GETTING STARTED 1D 1 mesh regions 1 material Si doping 1e16 doping type donor In this example the TIBERCAD Region bulk is made of Silicon and n doped with a concentration 10 3 Through the keyword mesh regions one or more of physical regions Physical Lines in 1D Physical Surfaces in 2D Physical Volumes in 3D previously defined in the GMSH mesh can be associated to the present TrBERCAD Region With mesh regions 1 we associate the Physical Line 1 d
31. and used to solve a simulation problem with an atomistic approach The association to the physical macroscopic regions of the device allows the implementation of multi scale calculations Each Atomistic block must be preceded by the keyword Atomistic followed by the single word name of the atomistic region Atomistic TB 1 1 physical regions barrier 1 qwell barrier 2 physical regions to be described with atomistic model 4 4 MODELS SECTION 31 Here are the description of the available keywords for an Atomistic block physical_regions mandatory list of the physical regions TiberCAD Regions or Clusters of the device associated to an atomistic description all default is used to specify all the physical regions path optional path for importing an atomistic structure from an external file and gen formats are supported and are automatically recognized by file extension Each of the atom positions is imported as is so the atom coordinates must be consistent with the geometry of the device If no path is specified the TiberCAD Atomistic Generator builds the atomistic structure it is constructed as a bulk crystal structure covering with proper atomic species the physical regions and taking in account the dimension of the problem up to now 1D structures are supported Atomistic Generator options to be put in the Atomistic section are described in the following reference region mandatory t
32. aracteristic length associated to that point this is an essential feature to control the mesh density and refine it where necessary usually n the channel region N B In a 2D simulation it is assumed that the geometrical model is restricted to the xy plane z 0 Any other geometrical orientation could give impredictable results Point 1 0 h 0 lsub Point 2 10 0 0 1 Point 3 xmax h 0 0 lsub Point 4 xmax h 0 0 lsub Point 5 xmax 0 0 0 lh Point 6 xmax 0 0 0 lh Line 1 4 1 Line 2 3 13 Line 6 4 14 Line 7 10 9 Line 8 12 2 Line 9 8 7 Line 10 11 8 Line 11 9 12 Line 13 7 6 e Definition of a surface first a line loop is composed listing all the lines con stituting the boundary of the surface then this line loop is assigned to a Plane Surface object this procedure can be alternatively performed throgh the graphical interface Line Loop 40 28 2 34 33 8 29 31 30 6 1 Plane Surface 41 40 19 e Definition of the Physical surfaces each of them is composed by one or more geometrical Plane Surface For example Physical surface 2 comprises the two separated contact regions while Physical surface 3 corresponds to the oxide region The Physical surfaces are the 2D Physical regions of the mesh and will be as signed to the related TIBERCAD regions through the keyword mesh regions see Step 41 n Si 44 47
33. ation sweep 1 simulation driftdiffusion 5 Definition of Execution parameters In the Simulation section we decide the simulation dimension dimension 2 then which simulations to perform and in which order we set solve sweep_2 to execute the external gate voltage sweep sweep_2 which in its turn call the sweep sweep 1 where drain current is calculated for all the chosen drain voltage steps by running dd simulation Simulation 3 meshfile mosfet msh dimension 2 temperature 300 solve sweep 2 resultpath output IV char output format vtk plot Ec Ev QFermi e QFermi h eDensity hDensity eCurrent hCurrent NetRecombination EField ElPotential ContactCurrents Output files with conduction and valence band profiles quasi fermi levels electron and hole density recombination electric field and potential plot Ec Ev will be generated together ContactCurrents with a file with all the calculated values of the drain current at the contacts for each gate bias step the IV characteristics Step 4 Run TiberCAD Now we can run TiberCAD tibercad mosfet tib The generated Output files are 25 driftdiffusion materials vtk information about the material regions of the device driftdiffusion nodal vtk output for the nodal quantities which have been calcu lated e g conduction and valence bands quasi fermi levels electron and hole density and mobility driftdiffusion elemental vtk output for t
34. ation eMob hMob T strain polarization xEffPot xDensity xMob ExcitonRecombination EigenFunctions EigenEnergy EnergyLevels xCurrent output format grace 49 EXAMPLE OF INPUT FILE 51 52 CHAPTER 4 INPUT FOR TIBERCAD Chapter 5 Simulation of strain 5 1 Theory The theoretical model of strain simulation can be found in Refs 1 2 The code can compute elastic deformations in a heterostructure and can calculate the deformed shape of the structure The heterostructure can be either grown on a substrate or not External pressure may be applied to a structure as well 5 2 Models section parameters The Models section looks like follows model macrostrain 1 options d simulation name strain in transistor physical regions 2 3 4 BC Regions d 53 54 CHAPTER 5 SIMULATION OF STRAIN There are three possible kinds of boundary conditions The mandatory keyword type substrate pressure extended material specifies the boundary condition type 5 2 1 Substrate boundary condition In this case the boundary condition region see 1 2 is the boundary between the device and the substrate The substrate does not belong to the device Therefore it is necessary to define both the boundary region number and the substrate material Role of substrate In general the substrate is a material that defines the lattice matching conditions and not necessarily a real solid body om which the device is gr
35. cal Entities allow to associate one or more geometrical entities to a single numerical ID or better to a string label so that several mesh_regions and boundary 1 2 DEFINITION OF PHYSICAL AND BOUNDARY REGIONS IN TIBERCAD 5 regions can be defined and referred to by TiberCAD Here is a simple example of a script to generate a 1D geometrical model geo file in GMSH see also chapter 2 Point 1 25 0 0 0 5 Point 2 0 0 0 0 002 Point 3 25 0 0 0 5 Line i 1 2 Line 2 2 3 Physical Line p_side 2 Physical Line n_side 1 Physical Point cathode 3 Physical Point anode 1 Here first the geometrical entities Points and Lines are defined In the definition of the geometrical Points the three first expressions inside the braces on the right hand side give the three X Y and Z coordinates of the point the last expression 0 5 or 0 002 in this example sets the characteristic mesh length at that point that is the size of a mesh element defined as the length of the segment for a line mesh element the radius of the circumscribed circle for a triangle mesh element and the radius of the circumscribed sphere for a tetrahedron mesh element Thus the smaller is the value of the characteristic mesh length the greater is the mesh density close to that point The size of the mesh elements will then be computed in GMSH by linearly interpolating these characteristic lengths in the whole mesh In
36. d by its model name Concerning electron and hole dissipations the model is named drift diffusion dissipation as reported below physical model heat source 1 model drift diffusion dissipation drift diffusion simulation dd simul name In order to include such a heat source we have to use drift diffusion simulation The syntax drift diffusion simulation dd simul name allows this connection 7 2 PHYSICAL MODEL 79 7 2 2 Boundary conditions By default thermally insulating surfaces are considered i e Jq N 0 7 3 On the opposite side it is possible to include an ideal thermally conducting interface by fixing the temperature to the external one i e T Text 7 4 This condition can be imposed with the following notation BC_Regions BC Region name BC region 1 type heat reservoir BC reg numb 3 temperature 300 Once a BC Region is inserted in the thermal section heat reservoir is the default type T he default temperature is the one indicated in the solve section Between such extreme situations it is possible to take into account a thermally re sistive interface i e Ja NS DAT Tal 7 5 where G is the thermal surface conductance and Tert is the external temperature One can include this condition with the BC type thermal_surface_conductance BC_Regions BC_Region name_BC_region type thermal_surface_conductance BC_reg_numb 3 g_surf 0 01 temperature 300
37. direction Bravais vectors with Miller indexes for wurtzite crystal 4 element vectors or zincblende crystal 3 element vectors doping doping concentration doping_type donor or acceptor 30 CHAPTER 4 INPUT FOR TIBERCAD doping level energy level of the dopant eV Each Cluster block must be preceded by the keyword Cluster followed by the single word name of the Cluster Cluster Quantum 1 1 mesh regions 3 4 5 mesh regions mandatory list of physical regions region IDs or physical names or TIBERCAD region names as specified in the meshing program to be grouped in the cluster Regions and Clusters represent the macroscopical description of the device or struc ture to be be simulated in TiberCAD In the rest of the input file the physical regions associated to Models or Atomistic descriptions will be indicated by means of the TiberCAD Region and Cluster names 4 3 Scale section The section Scale is dedicated to the optional definition of Non Continuous Media regions for the device these regions will be described and studied at a different scale e g atomistic circuit level lumped model etc As for now just the atomistic description is implemented Atomistic blocks if present specify a possible atomistic description associated to one or to a group of physical regions described by Region blocks So for each Atomistic block defined in Input file an atomic structure description will be generated
38. e false is used to choose between uniform and adaptive refinement The default value is false i e adaptive refinement Example refinement_steps 4 refine_fraction 0 25 coarsen_fraction 0 max_refinement_level 10 Deformed shape calculation The displacement field and lattice matching parameters that are found from master equations can be used in order to define a new shape of the heterostructure This new shape is the first approximation to the equilibrium one The next approximations are 58 CHAPTER 5 SIMULATION OF STRAIN obtained iteratively by the following steps at the n th iteration the master equations are solved using the lattice matching deformation 7 which is defined as 1 dur n Um 5 5 2 Ox T Ox H Ei i where the displacement field vu has been taken from the iteration n 1 Then the new shape is defined by using the displacements from the last step solution and the iterative process is repeated until the displacement field vanishes and additional lattice parameters stabilise The iterative cycle usually converges after 3 4 iterations The only parameter that controls shape calculation is number shape steps The value defines number of iterations The default value is zero that means no shape deformation calculation 5 4 Physics section parameters There is a possibility to consider converse piezoelectric effect For this it is necessary to specify a name of another sim
39. e file tibercad lic into the license subdirectory of the TIBERCAD installation directory INSTALLPATH license without changing the file name You can also provide the license file during installation The standard method to launch TiBERCAD is by means of a shell script that is installed alongside the TiberCAD executable It takes care of setting all necessary envi ronment variables If for some reason you have to run the executable directly remember to set TIBERCADROOT to the TiberCAD installation directory INSTALLPATH Quick start guide In the examples subdirectory you can find several examples ready to run They are the same as the tutorials on http www tibercad org documentation tutorial list Windows Open Windows Explorer and go to the TIBERCAD installation directory If you have write permission in the installation directory you can browse to an examples directory and start the simulation by double clicking the input file e g bulk tib in Example 0 If not copy the whole directory to a location in your personal area and run the examples from there If you cannot run TrBERCAD by double clicking an input file tib then the input files are probably not correctly associated with the TrBERCAD executable In this case try to establish the association by right clicking the input file choosing open with Choose Program Browse browsing to the TIBERCAD installation directory and choosing the TIBERCAD e
40. e generated Step 4 Run TiberCAD Now we can run TiberCAD 15 tibercad bulk tib The generated Output files are driftdiffusion materials dat material mesh regions in this case just region 1 driftdiffusion nodal dat nodal quantities here conduction and valence band sweep driftdiffusion Vb dat integrated current at the two contacts for each sweep step 16 CHAPTER 2 GETTING STARTED 1D Chapter 3 Getting started 2D In this second example we will refer to the Tutorial 4 Si n Mosfet that you can find in the Tutorials directory Step 1 Modeling the device Again as a first step we have to model the device We ll see in some details how to design and mesh a mosfet device with GMSH e In the GMSH script mosfet geo several variables are defined and given a value in this way 1sub 0 03 lacc 0 002 1ct 0 0005 1g 0 0015 1h 0 01 1c 0 0005 these variables are used in the script to assign proper values to the mesh charac teristic lengh of the defined Points Lg_2 0 0375 d 0 01 Ls 0 1 h 0 25 b 0 0025 o 0 005 17 18 CHAPTER 3 GETTING STARTED 2D xd Lg_2 d xd2 Lg_2 d 2 xmax xd Ls d These other convenient variables are used to parametrize the most relevant geometrical features such as channel length oxide thickness and so on e Geometrical Points and Lines are defined to design the device structure the fourth parameter in Point assignement is the ch
41. efined in the Step 1 to the TIBERCAD Region bulk 2 Definition of Simulation Now we define the Simulation driftdiffusion 1 it belongs to the class driftdiffusion Models 1 model driftdiffusion t options t simulation name driftdiffusion 1 physical regions all The TrBERCAD simulation driftdiffusion_1 belonging to the model driftdiffusion will be applied to the whole device structure physical regions all 3 Definition of Boundary Conditions The anode and cathode contacts of our 1D Si sample are defined as Boundary condi tions regions BC Region anode BC Region cathode in the following way BC Region anode d BC reg numb 1 13 type ohmic voltage Vb BC Region cathode 1 BC reg numb 2 type ohmic voltage 0 0 Both contacts are defined as ohmic cathod is assigned a fixed voltage 0 0 while anode voltage is given by the value of the variable Vb voltage Vb Through the keyword BC_reg_numb one or more of the n 1 Dimension physical regions Physical Points in 1D Physical Lines in 2D Physical Surfaces in 3D previously defined in the GMSH mesh can be associated to the present TIBERCAD BC Region With BC_reg_numb 1 we associate the Physical Point 1 defined in the Step 1 to the TIBERCAD BC Region anode with 2 we associate the Physical Point 2 defined in the Step 1 to the TIBERCAD BC Region cathode Alternatively one can make use of the phy
42. el is based on the Caughey Thomas model refined by Canali 6 Hlow field gv LUS How field El 1 Usat B Bo T Toy is the modulus of the driving field Howfiela is the low field mobility For the latter one can specify the model to be used using the parameter lowfield_model As default the doping dependent model is used There are two models for vsat identified with Vsat_Formula 1 and 2 Formula 1 reads 6 12 with Usat Vsat ol T T Formula 2 reads Usat max Avsat Bosal T To Uni The parameters for the field dependent mobility model are summarized in table 6 8 6 3 4 Boundary conditions Boundary conditions are implemented for ohmic contacts Schottky contacts free sur faces and interfaces Contacts are boundary models that allow a nonzero normal elec trical current For this type of boundaries one can define a contact resistance using the contact resistance option The contact resistance has units Qcm The applied volt age is specified with the option voltage A variable can be assigned to this using the syntax TOCHAPTER 6 DRIFT DIFFUSION SIMULATION OF ELECTRONS AND HOLES parameter keyword Bo beta0 b betaexp Usat 0 vsatO y vsatexp A vsat Basat B vsat Ur n vsat_min Table 6 8 Data file parameters for the mobility model by Arora For a finer control of the behaviour at electrical contacts the options zero_field zero_grad_fermi_e and zero_grad_
43. erata y RS ee ed eee a 41 CONTENTS Simulation of strain 53 MESI o eae eS a Be A QS ee 53 5 2 Models section parameters a gt a sr eco eto e dk Re ere 53 5 2 1 Substrate boundary condition 54 5 2 2 External pressure boundary condition 54 5 2 3 Extended device boundary condition 55 5 3 Solyer parameters LL Lou c2 uw diag xa XR 55 5 3 1 Structure with a substrate 55 5 3 2 Structure without a substrate freestanding 55 5 3 9 Additional parameters ss od s ar ad R 56 5 4 Physics section 58 dr OUP e e CT 58 Drift diffusion simulation of electrons and holes 61 DI THEON oe cros LO ox ub ee ERASE ES d X ded x wed 61 p Plot VADE r pos wori ex AE HS Rr ATE BH Rd 61 Go Magele BESO oko ce ES ee RY ag CR 61 Dat Recombination models s uu 44 46 44048 28204646 4 63 6 3 2 Thermoelectric power models 66 Go meds ice bebe owe o aa AR 67 D Boundary condone 2 R9 x E Rd modd ia BRERA 69 bl PUPILOS SOC elo 6E E 71 641 Simple semiconductor model lt lt kk xs 72 6 4 2 Default semiconductor model 72 6 4 3 Special options for Schr dinger Poisson calculations 72 Co Olver elem colo oo Runden due O at a Der 73 6 5 1 Parameters for PETSc s
44. ermal_simulation name the thermal simulation to be used electron_quantum_density name the quantum density simulation to be used for the electron den sity hole_quantum_density name the quantum density simulation to be used for the hole density Table 6 9 Common options for the drift diffusion semiconductor models not specified the default semiconductor model based on bulk p theory is used The electron quantum density and hole quantum density will use the parti cle densities calculated from the corresponding quantum density simulation In re gions where no quantum density is available the classical density will be used The electron quantum density and hole quantum quantum density keywords can be used also in the Region sections to be able to use different quantum density simulations in different regions For further options regarding selfconsistent Schr dinger Poisson Drift Diffusion calculations see Sec 6 4 3 72CHAPTER 6 DRIFT DIFFUSION SIMULATION OF ELECTRONS AND HOLES The strain_simulation option is used to specify the simulation that provides strain in the case of strained systems If it is omitted an unstrained system is assumed for the drift diffusion calculation The thermal_simulation option is used to specify the simulation that provides the lattice temperature for non isothermal simulations If it is omitted the simulation tem perature as provided in the Simulation section of the input file or if
45. es to this value eV are found spectrum inversion tolerance tolerance used for linear solver Table 8 1 Iterative eigensolver parameters If the spectral_shift is not defined then it will be calculated internally from the band edges For the iterative solvers the important parameters that may significantly change the performance are the Krylov subspace method type and the preconditioner type The Krylov method is defined as follows Esp type bcgsl gmres begs cg richardson preonly 8 3 PHYSICAL MODELS PARAMETERS 85 The preconditioner type is defined as follows pc_type cholesky jacobi ilu composite Other options e r periodicity true false e y periodicity true false e z periodicity true false e number_of_eigenstates number of eigenstates to be computed 8 2 2 Schr dinger equation parameters e particle el hl Y specifies a particle type according to each the eigenvalues are sorted e strain model name name of the simulation that can provide elastic strain e poisson model name name of the simulation that can provide electric and elec trochemical potential e heat model name of the simulation that can provide e relative density tolerance relative tolerance for charge density calculation e initial eigenstates number initial eigenstates number for charge density calcula tion e convergent density true false if true than number of eigen
46. et of the device corre sponding to a simulation environment with a non void intersection with the first one The coupling of the two simulations is performed by means of appropriate Boundary Conditions e g Current Density Voltage In principle the two simulation environ ments can refer to two simulations at different scale e g atomistic tight binding and macroscopic drift diffusion This allows an effective multi scale simulation of the device to be studied CHAPTER 1 OVERVIEW Chapter 2 Getting started 1D In this section we will see step by step how to use TIBERCAD to simulate numerically a semiconductor device As a very simple example we will refer to the Tutorial 0 Si bulk that you can find in the Tutorials directory Step 1 Modeling the device As a first step we have to model the device To do so you can use DEVISE module of ISE TCAD 9 5 software package or GMSH program Here we ll see in details the procedure for GMSH There are two possible ways to use GMSH 1 Interactive using the graphical interface 2 Using a script file In the following we ll see how to write a basic GMSH script bulk geo for any details please refer to GMSH manual GMSH http geuz org gmsh In a GMSH script several variables can be defined and given a value in this way Eom d 0 01 these are valid GMSH variables L is just the length of the Si sample d is the value of a characteristic mesh length see below e Defi
47. fermi_h can be used which when set to true will impose zero normal electric field and zero normal gradient of the electron and hole electro chemical potential respectively The ohmic contact identifier ohmic has no further parameters A Schottky contact identifier schottky has the additional parameter barrier which signifies the energy difference between the semiconductor band edge and the fermi energy in the metal As default the barrier is taken with respect to the conduction band By specifying band v the barrier can be imposed with respect to the valence band p type contact Alternatively the metal work function can be defined using the keyword work_function Note however that its value has to be aligned with the band energies given in the material files for the other materials If a contact touches several different regions with different materials it can be neces sary to provide a reference material using the keyword reference_material and spec ifying the name of a TIBERCAD region TiBERCAD will prefer semiconductors over dielectrics if no reference material is specified The type of boundary model is chosen by the parameter type e g type schottky The free surface or interface model identifier interface models surface charges and surface recombination Two modes are possible for the surface charge model constant charge a constant charge can be assigned by specifying only the sheet carrier density Ns in cm 2 The sheet c
48. fied in the list plot TiberCAD output is divided in three classes nodal elemental and integrated quantities Nodal quantities are all the quantities associated with the nodes of the mesh such as Fermi level electron and hole density conduction and valence band etc The output values for these quantities are reported in the files modelname nodal ext where model name is the simulation model used for the calculations and ext is the extension of the chosen file format 49 EXAMPLE OF INPUT FILE 41 In the case a sweep calculation is performed and the plot_data keyword is set to true the output files are of the kind modelname_nodal_sweepvariable_step ext where sweep variable is the variable with respect to which the sweep is performed e g gate voltage and step is the value of this variable at that step e g driftdiffusion nodal Vb 0 000 dat for the result at the step Vb 0 0 Elemental quantities are all the quantities associated with the elements of the mesh such as current density The output values for these quantities are reported in the files modelname elemental ext In the case a sweep calculation is performed and the plot data keyword is set to true the output files are of the kind modelname elemental sweepvariable sten ert e g driftdiffusion elemental Vb 1 150 dat Integrated quantities are the quantities which are not associated to the mesh but are obtained by an integration on real or reciprocal space for example
49. harge density will then equal Ns multiplied by the elementary charge e A positive Ns produces a positive surface charge electronic surface states in this case the surface charge is produced by electrons oc cupying a surface state with a density of states in form of a delta function The density of occupied states then reads Ns E AEs ep ebn kgT Ns 1 expl 6 4 PHYSICS SECTION 71 The density of states N is specified by Ns the energy of the state with respect to the conduction band AF by Es g denotes the multiplicity of the state and defaults to 2 It can be changed by assigning a value to g Surface recombination is switched on by setting the keyword surface_rec to true The model used for surface recombination is formally the same as the bulk SRH recom bination However the only parameters are the recombination velocities which have to be specified in cm s in the input file using the keywords rec_vel_e and rec_vel_h 6 4 Physics section Options for controlling the drift diffusion semiconductor models can be specified in the Physics section The corresponding paramaters are given in table 6 9 When model is keyword possible val description ues model see following the model to use for the descrip subsections tion of the conduction and va lence band properties statistics B FD Boltzmann default or Fermi Dirac statistics strain_simulation name the strain simulation to be used th
50. he Atomistic Generator can only build pseudomor phical heterostructures A reference region must be defined to specify from which region TiberCAD Regions to get structure parameters such as lattice constants which depend on the material defined in the reference region passivation optional no is default option If set to no no passivation is performed If yes is specified a hydrogenation of the structure is performed taking into account the structure periodicity Up to now hydrogenation is supported fo Silicon structures preserve optional Default is none In some cases it is requested to build a structure in which the atom basis or the conventional cell has be preserved regardless to mesh geometry If none is specified no conservation is performed and only atoms strictly be longing to geometrical regions are put in the atomic structure If lattice is specified atom basis is preserved e g to preserve anion cation couples If conventional is specified conventional cell is preserved y_lenght optional Atomistic Generator builds the minimum periodical structure along y and z directions If y lenght is specified the structure will be at least y_lenght sized along y growth direction Exact lenght is internally defined in order to keep struc ture periodicity z lenght optional same as above for the z direction 4 4 Models section Models 1 model driftdiffusion 1 32 CHAPTER 4 INPUT FOR TIBERCAD
51. he elemental quantities e g electric field current density sweep 2 driftdiffusion Vg 0 000 Vd dat and similar for all the Vg steps drain current characteristics for each Vg bias 26 CHAPTER 3 GETTING STARTED 2D Chapter 4 Input for TiberCAD Input for TrBERCAD is composed by an input file e g input tib and a mesh file gen erated by a mesher software as for now mesh files from GMSH msh v 1 and v 2 0 and from ISE TCAD grd are supported Be sure that the material files are in the correct directory as specified in 4 7 To run the program type tibercad input file name 4 1 Description of Input file structure A valid input file for TIBERCAD is a text file with the structure described in the following In the whole input file everything following a Z is considered as a comment and is disregarded blank lines can be present anywhere and are disregarded too Input file is composed by several sections each section begins with a section name preceded by e g Physics A section is enclosed between 4 and brackets and is possibly composed de pending on the section by a variable number of blocks enclosed between 7 and brackets Each block can be possibly composed by one or more blocks each preceded by a block name The elementary block parameters block is a block which contain zero or any number of parameter assignements in the form tagname tagvalue where e
52. ion approach 4 6 PHYSICS SECTION 39 Once a selfconsistent calculation has been defined it may be executed as an usual simulation by adding it in the solve list e g solve dd excitons see 4 7 or even in the sweep section with for example simulation converse_piezo In both cases the specified simulations will be executed in a self consistent way In the case of a single self consistent calculation block for backward compatibility it is also possible to define the self consistent feature by means of the keyword selfconsistent lower case in the following way selfconsistent X flavour relaxation simulations driftdiffusion excitons 4 6 Physics section In this section several physical parameters can be entered in addition to or overwriting the material parameters present in the material files The section is organized in blocks each one preceded by a block name This block name may be 1 the name of one of the user defined simulations in this case the physical param eters defined in each block will refer only to that particular simulation 2 the name of a TiBERCADModel only in this case the settings will be applied to ALL the simulations belonging to that Model For details on the available parameters for each Model see the relevant chapter in this Guide For example in this case strain simulation is used to specify the simulation that provides strain data in case of strained systems see 6 4 4
53. ion drain BC Region gate in the following way BC Region gate d BC reg numb 2 type schottky barrier_height 3 0 voltage Vg 0 0 BC_Region source BC_reg_numb 1 type ohmic 23 voltage 0 0 BC Region drain BC_reg_numb 3 type ohmic voltage Vd 0 5 j To each of the BC regions one BC reg numb is assigned that is one of the Physical Lines 1 2 3 defined in Step 1 which represent the contact regions Note that while source and drain are defined as type ohmic gate BC region is defined as type schottky barrier height 3 0 specifes the metal oxide interface barrier and depends on the contact metal workfunction Drain voltage is defined as Vd 0 5 and gate voltage as 9Vg 0 0 This specifies that the value of the voltage will be determined at each moment of the simulation by the value of the two variables Vd and Vg which will be assigned in the sweep definition 4 Definition of Simulation parameters Two sweeps are requested for this simulation that is an external loop on Vg the gate voltage and an internal loop on Vd the drain voltage for each value of Vg in this way the IV drain characteristics for a series of gate biases are obtained in output sweep 1 simulation dd variable Vd start 0 0 stop 2 0 0 1 steps 200 200 1 plot data true p sweep 2 d variable Vg 24 CHAPTER 3 GETTING STARTED 2D start 0 1 stop 0 5 steps 6 simul
54. irichlet There is a way to impose automatically the Dirichlet boundary conditions over all the boundary of the simulation region This is done by the parameter Dirichlet_bc_everywhere true false in Solver section The default value for EFA problem is true 8 2 Solver parameters There are two groups of parameters The first group is related to the general eigensolver problem the second one is related to the Schr dinger equation 8 2 1 Eigenvalue problem parameters These parameters are common for all eigenvalue problems Their default values may be different for different eigenvalue problems for example for the Schroedinger equation and for the electromagnetic eigenvalue problem The eigenvalue problem can be solved by the solvers that are implemented into the SLEPc library The relative parameter is solver arnoldi lapack krylovshur The default value is krylovshur In the case of the lapack solver all the eigenvalues are computed In the case of arnoldi or krylovshur solver it is necessary to specify which and how many eigenvalues have to be computed The idea is that the iterative solver calculates several eigenvalues that a close to a specific number reffered to as the spectral_shift The relative parameters are max_iteration_number maximum number of iteration used as a stop condition eigen_solver_tolerance numerical eigensolver tolerance used as a convergence criteria spectral shift the closest eigenvalu
55. is to be applied When TiBERCAD is run it reads the mesh file which contains the finite element grid which meshes the geometrical description of the device or nanostructure and which will be the basis of PDE discretization As we have seen before to execute the proper simulations TIBERCA D needs some in formation about the physical and boundary regions associated with the mesh A physical 4 CHAPTER 1 OVERVIEW region associates all the elements corresponding to an homogeneous part of the device usually related to the same material or doping In TrBERCAD these regions are referred to as mesh regions As for boundary regions they are needed to specify boundary conditions b c for the solution of the PDEs of our simulation By default to all the external boundary of the device a Neumann b c is imposed meaning null derivative of electric field and zero flux of current normal to the boundary These are the usual b c applied in the simulation of electronic devices in particular these conditions are implicitly satisfied by using the finite element formulation Usually however one needs to impose also specifical b c to the device relative most often to contacts of some kind ohmic schottky but also heat and temperature b c or reference substrates for strain calculations These regions constituted by surfaces lines or points respectively for 3D 2D and 1D simulations are called in TiBERCAD boundary regions It is important t
56. itcal calculation of density is done in the following way For each eigenstate we calculate the effective mass assuming quadratic dispersion Then the charge density is calculate as follows prole sie Pia 1 exp 7 Sail 12 1 kT E pap r dote For Fan Lie 12 2 where and are the 1D and 2D charge densities m is the averaged mass the mass is different for each quantized state and is position independent g is the degeneracy of the states The sign is for electrons the sign is for holes Numerical calculation is done by the following formula 1 1 OX LO E 12 3 n The integration is performed on a mesh in the k space 12 1 Output The output parameter is quantum density Bibliography H Michael Povolotskyi and Aldo Di Carlo Elasticity theory of pseudomorphic het erostructures grown on substrates of arbitrary thickness Journal of Applied Physics vol 100 pp 063514 2006 Matthias Auf der Maur Michael Povolotskyi Fabio Sacconi and Aldo Di Carlo Simulation of piezoresistivity effect in FETs J Comp Electronics vol 5 pp 323 2006 Siegfried Selberherr Analysis and Simulation of Semiconductor Devices Springer Verlag Wien New York 1st edition 1984 Masetti M Severi and 5 Solmi Modeling of carrier mobility against carrier concentration in Arsenic Phosphorus and Boron doped Silicon IEEE Trans on Electron Devices vol 30 pp 764 769 19
57. lectron density eic hDensity Hole density em eMob Electron mobility cm Vts hMob Hole mobility cm 18 1 Nd Ionized donor density ene Na Ionized acceptor density cm charge_density Total charge density Gm Pn Electron thermoelectric power VE Pp Hole thermoelectric power VK NetRecombination The net recombination rate for each recom em s bination model and the total rate Table 6 1 Nodal quantities The physical model sections can be omitted In this case default models are used namely no recombination generation for the recombination model and constant mobility for the mobility models There can be more than one recombination model 6 3 1 Recombination models This section describes the currently available generation recombination models Shockley Read Hall SRH recombination The SRH recombination model can be enabled in the input file by physical_model recombination E model srh 64CHAPTER 6 DRIFT DIFFUSION SIMULATION OF ELECTRONS AND HOLES Elemental quantities EField Electric Field Vem GradFermiE Gradient of the electron electro chemical potential GradFermiH Gradient of the hole electro Vem chemical potential Current Total current density eCurrent Electron current density Acm 7 hCurrent Hole current density Acm 2 Polarization Electric Polarization Cm GradPn Gradient of electron ther VK cm moelectric power GradPp Gradient of hole thermo
58. led in the input file by physical model recombination S model direct Direct recombination is modeled as follows Rirect C np n2 6 5 The material data file and the input file use the same keyword C for the parameter C The database value can be overridden from the input file as described for SRH recombination Auger recombination The Auger recombination model can be enabled in the input file by 66CHAPTER 6 DRIFT DIFFUSION SIMULATION OF ELECTRONS AND HOLES physical model recombination 1 model auger Auger recombination is modeled by the following equation Riquer Can np ee nz 6 6 with temperature dependent parameters T TX Pm C 1 The parameters A B C and are taken exclusively from the database They are different for and and have to be specified as arrays with keywords A B NO e g 1 31 1e 32 The calculated values for and C can be overridden from the input file by specifying values for the keywords Cn and Cp 6 3 2 Thermoelectric power models The thermoelectric power models are the same for electrons and holes The keyword is thermoelectric_power i e physical_model thermoelectric_power model The model keyword can be constant i e the thermoelectric powers are read from the database or diffusivity model where the thermoelectric powers are computed by _ R 5 Ec
59. material name of material in the substrate region structure crystal structure wz wurtzite zb zincblend x growth direction y growth direction z growth direction Bravais vectors with Miller indexes for wurtzite 4 5 SOLVER SECTION 35 4 5 Solver section Solver driftdiffusion 1 nonlinear solver tiber ksp type bcgsl nonlin rel tol 1e 12 nonlin abs tol 1e 15 nonlin step tol 1e 2 1 rel tol 1e 12 lin rel tol 1e 6 nonlin max it 30 ls max step 1 pc_type lu pc_type composite In this section one may choose and define the setting parameters for the numerical solvers to be applied to the specified simulations the section is organized in blocks each one preceded by a block name This block name may be 1 the name of one of the user defined simulations in this case the solver parameters defined in each block will refer only to that particular simulation 2 the name of a TiBERCADModel only in this case the settings will be applied to all the simulations belonging to that Model For details on the available parameters for each Model see the relevant chapter in this Guide 36 CHAPTER 4 INPUT FOR TIBERCAD In addition two special optional blocks may be present the Sweep block and the Selfconsistent block e Sweep block preceded by the keyword Sweep This block may contain one or more subblocks each one defining a set of calculations applied to a bounda
60. mesh regions defines the physical properties and physical models to be 1 2 CHAPTER 1 OVERVIEW applied and the type of calculations to be performed Details about the modeler and mesher tools can be found in the specific user manuals Here we deal primarily with the TiberCAD input file However in discussing examples of 1D 2D and 3D simulations we will also describe in some detail the geometry input files used to run gmsh 1 1 1 Input file structure TiberCAD input file is a text file which includes a description of the device structure the definition of the solvers to be executed with all the relevant physical and numerical parameters for each of them The input file is organized into several sections each describing a different aspect of the problem to be solved The strategy employed is similar to other commercial T CAD tools and requires some practice to reach a good level of familiarity We strongly suggest to read first the following chapters Getting Started 1 and 2 in this manual and then study the input files provided in the examples directory and try to modify them before writing your own input file from scratch The example files touch all current features implemented in the code Let s see an overview of the main features of the input file The core of the input file comprises three sections called Device Models and Simulation Device Region Si channel d material Si doping 1e16 Region gate oxide
61. n control the behaviour of the selfconsistent simulation 6 5 SOLVER SECTION 73 use density predictor bool When set to true a predictor corrector scheme will be adopted in the selfconsistent cycle The Poisson Drift Diffusion solver does not just take the particle densities as given by the Schr dinger calculation but it will assume a dependency of the density on the potentials of the form ple Dquantum 00 bn Pp ur Dclassica 4 9 62 62 where y dp are the potentials for which the quantum density was calcu lated use density predictor true is the preferred method for selfconsistent Schr dinger Poisson Drift Diffusion calculations however it is not enabled by de fault Polassical O Qn 6 13 embracing length double When the domain of the quantum simulation is smaller than the domain of the full simulation the boundary conditions for the Schr dinger equation will disturb the transfer from classical to quantum density By defining an embracing region of a certain extension specified in meters a gradual transition from classical to quantum density will be done instead of an abrupt one using as effective density Pquantum 1 2 Pelassical The default is no embracing region at all zero extension cutoff double if an embracing region is used a part of this region near the boundary of the quantum region can be cut off so that only the classical density is considered
62. n e min eigenvalue number max eigenvalue number the dispersion is calcu lated for the states number 7 where max eigenvalue number gt 7 gt min eigenvalue number The rest of the parameters wedge k space dimension etc define the k space 91 92 CHAPTER 11 QUANTUM DISPERSION dispersioniD el 1 quantum simulation quantum el min eigenvalue number 0 max eigenvalue number 5 wedge half k space dimension 1 k1 0 0 1 0 number_of_nodes 10 output_format grace 11 2 Output The output variable name is k space_dispersion The output format for the dispersion can be controlled independently of the general specification in the Simulation section by redefining the output_format keyword Chapter 12 Quantum Density dens el 1 k space dimension 2 k space basis true ki 0 0 0 1 k2 0 0 1 0 number_of_nodes wedge quarter refine_fraction 0 20 relative_accuracy 0 01 refine_k_space true uniform_refinement false mesh_order FIRST 4 4 quantum_simulation quantum_el degeneracy 2 initial_eigenstates_number 10 analitic false e quantum_simulation name of the Schr dinger simulation e degeneracy degeneracy of the quantum state e initial eigenstates number initial number of eigenstates for the Schr dinger equation e analytic true false If true then the density is calculated analytically or numerically 93 94 CHAPTER 12 QUANTUM DENSITY Anal
63. ng 1e15 doping type donor Hdoping level 0 025 Region barrier 1 1 mesh regions 2 3 structure wz y growth direction 1 0 1 0 z growth direction 1 2 1 0 x growth direction 0 0 0 1 material AlInN 0 80 doping 1e15 doping_type donor doping_level 0 025 Region QWell mesh regions 4 49 EXAMPLE OF INPUT FILE 43 structure 2 y growth direction 1 0 1 0 z growth direction 1 2 1 0 0 0 0 1 x growth direction material GaN doping 1e15 doping type donor doping level 0 025 Region barrier 2 1 mesh regions 5 6 structure wz y growth direction 1 0 1 0 z growth direction 1 2 1 0 x growth direction 0 0 0 1 material AlInN x 0 80 t doping 1e15 doping type donor doping level 0 025 d Cluster group of mesh regions with DIFFERENT material in general Syntax Cluster Tiber cluster mesh regions list gmsh region ID names list ISE TCAD region names Tiber cluster to be used in Models section Cluster Quantum 1 1 mesh regions 3 4 5 44 CHAPTER 4 INPUT FOR TIBERCAD Definition of the description scale only for not Continuous Media regions Syntax level scale cluster level Atomistic physical regions list Tiber region Tiber cluster Scale section is optional Scale Atomistic TB 1 physical regions
64. nger model associated respectively to the initial state of optical transition e g electron and to the final state of optical transition e g hole initial eigenstates and final eigenstates refer to the range of eigenstates to be taken in account for optical calculations By specifying in Solver section a range of energy values in this way Emin 3 0 Emax 5 0 dE 0 001 the emission optical spectrum for k 0 is calculated The spectrum is calculated in the following way 1 we a P hw 2 3 Miel RE 5 0E d T 2 wij hw 1 2 dQ 9 1 where f and f are the Fermi distributions 8T 88 CHAPTER 9 SIMULATION OPTICSKP 9 1 Output The output variables for optics calculations are e optical spectrum k 0 optical emission spectrum fork 0 calculated through opticskp model Chapter 10 Simulation opticalspectrum By defining the model opticalspectrum optical matrix elements are used to calculate the associated emission spectrum with a k space integration In Solver section opticalspectrum 1 k space dimension 2 k space basis true ki 0 0 0 1 k2 0 0 1 0 refine fraction 0 30 relative accuracy 0 01 refine k space true number of nodes 2 2 wedge quarter optical matr elem model opticskp polarization 0 0 1 Emin 3 0 Emax 5 0 dE 0 001 The parameters 89 90 CHAPTER 10 SIMULATION OPTICALSPECTRUM k space dimension 1 for 2D
65. nition of geometrical entities Points Point 1 Point 2 mol AA t o oo oo a Q Gei Gei 10 CHAPTER 2 GETTING STARTED 1D In the definition of a geometrical point the three first expressions inside the braces on the right hand side give the three X Y and Z coordinates of the point the last expression d sets the characteristic mesh length at that point that is the size of a mesh element defined as the length of the segment for a line mesh element the radius of the circumscribed circle for a triangle mesh element and the radius of the circumscribed sphere for a tetrahedron mesh element Thus the smaller is the value of d the greater is the mesh density close to that point The size of the mesh elements will then be computed in GMSH by linearly interpolating these characteristic lengths in the whole mesh N B In a 1D simulation it is assumed that the geometrical model is restricted to the 2 axis Any other geometrical orientation could give impredictable results e Definition of geometrical entity Line Line 1 1 2 The two expressions inside the braces on the right hand side give the identification numbers of the start and end points of the line e Definition of the physical entity Physical Line 1 Physical Line 1 1 The expression s inside the braces on the right hand side give the identification numbers of all the geometrical lines that need to be grouped inside the physical line In this way in general
66. o know that the information about the physical and boundary regions must be present in the mesh file before it is read by TrBERCAD and thus have to be produced by making use of the modeling mesher software As for now TIBERCAD supports the mesh output of the following software tools GMSH v 2 and ISE TCAD v 9 5 1 2 1 Using ISE TCAD By means of the utility DEVISE of ISE TCAD v 9 5 it is possible to design and mesh a device after the meshing has been successfully performed an output file is produced with the extension grd This file contains the description of the mesh and also the list of the user defined material regions and contact regions By reading this grd file in TIBERCAD one can refer to the ISE TCAD material regions simply with the user defined name which is present in the grd file This name should be unique in the whole device In the same way ISE TCAD Boundary regions Contacts can be referred to in TrsERCAD by means of their user defined name present in the grd output file too 1 2 2 Using GMSH If GMSH program is used to model and mesh the device a bit more care has to be taken Here we introduce the procedure to be followed in the following tutorials Getting started the subject will be considered in detail with a step by step description See also the GMSH user manual http geuz org gmsh for further details In the context of GMSH it is possible to define several 1 2 and 3D Physical Entities These Physi
67. oisson and DriftDiffusion equations EFASchroedinger to solve Schroedinger equation in envelope function approximation Macrostrain to calculate macroscopical strain with an elastic model and others Similarly to other device CADs TiberCAD requires that one follows a three step procedure In a first step the device geometry must be sketched giving all the geometrical in formation needed by the simulations This can be performed by means of a text file or with the help of a graphical tool During this procedure one or more mesh regions and boundary regions have to be defined in a following stage the mesh regions will be associated to materials and boundary regions to device contacts boundary conditions in general The second step consists in running a mesher tool which reads the geometry file and sets up the computational mesh used to discretize the partial differential equations representing the physical models to be solved For this procedure TIBERCADmaRtes use of the GPL software gmsh Optionally mesh outout provided by other meshers such as the one included in ISE TCAD tool are also supported The output of the meshing procedure is a mesh file that contains information about the space discretization as well as the mesh regions and the boundary regions In the last step the actual simulations are performed Together with the mesh infor mation comprised in the mesh file TiberCAD requires an input file which associates materials to
68. olvers 75 6 5 2 Parameters for the TrBERCAD nonlinear solver 76 6 5 3 Parameters for the PARDISO linear solver 76 Heat Balance simulation TT Heat o oe be ty Re d ae Tf Lo Physwalmodel 1 131 ew oe hee ee Eee Quee d 77 T2 Electron and hole diseipatione ccoo iaa Pee s 78 74 2 ODONIS s scs och oo ko Saw OH A 79 d QUU A med uev Be UR Sree ad See UE Pre E RUP er 80 Envelope Function Approximation 83 8 1 Models section parameters 83 8 2 Solver parameters 2 s co doo redes da ales aa 84 CONTENTS 8 2 1 Eigenvalue problem parameters 8 2 2 Schr dinger equation parameters 8 3 Physical Models parameters Bao XM ru bo x49 uk ua Re3 ck x Rem ck x RUM eo eds 9 Simulation opticskp BI GERD sa eg etes ales 9o wis Ole Wo ira od wird 10 Simulation opticalspectrum 10 1 Output 11 Quantum dispersion lll DUNE ophion 2425 55 4 pg aa d EROR A EE EEO 11 2 Output 12 Quantum Density 12 1 Output Bibliography 84 85 85 86 87 88 89 90 91 91 92 93 94 95 IV CONTENTS Installation instructions In the following VERSION denotes the version number of the TIBERCAD release you downloaded and INSTALLPATH denotes the directory where TIBERC
69. on followed by the single word name of the TiberCAD Region The name of the TiberCAD Region can coincide with the name of a mesh region as defined during the modeling of the device in this case if the keyword mesh regions is absent the TiberCAD Region will be associated to the mesh region identified by the name assigned to the TiberCAD Region Otherwise the TiberCAD Region will be associated to the mesh regions specified by the keyword mesh regions Region Well 1 mesh regions 4 5 mesh regions 4 structure 2 y growth direction 1 0 1 0 z growth direction 1 2 1 0 x growth direction 0 0 0 1 material GaN doping 1 17 doping type donor doping level 0 025 Here are the description of the available keywords for a Region block material mandatory name of the material associated to the present region e g Si it may be a ternary alloy e g AlGaAs in this case keyword z described in the following has to be present x alloy concentration expressed as the molar fraction of the first component of the alloy e g to express an alloy Al Ga As with molar fraction x 0 2 that is Alo Gays As we select AlGaAs for the keyword material and 0 2 for the keyword thus we write z 0 2 mesh regions a list of region ID s or physical name s as specified in the meshing program structure crystal structure wz wurtzite zb zincblend x growth direction y growth direction z growth
70. own BC Region layer of Al 0 3 Ga 0 7 N BC reg numb 14 type substrate mat AlGaN x 0 3 structure wz y growth direction 1 0 1 0 z growth direction 1 2 1 0 x growth direction 0 0 0 1 5 2 2 External pressure boundary condition The parameter pressure specifies the value in GPa of the normal pressure applied to the boundary region BC reg numb BC Region tip upon a surface d BC reg numb 12 type pressure pressure 12 3 Sign of pressure The value of the pressure has a positive sign if the external force acts towards the surface which in general has to be the boundary of a simulation environment 5 3 SOLVER PARAMETERS 55 5 2 3 Extended device boundary condition If device that is grown on a substrate is very large we may want to simulate a part of it only In this case the simulation domain boundary is not a free surface any more The boundary conditions are as follows Ox The syntax is as follows BC Region boundaryl 1 BC reg numb 12 type extended material 5 3 Solver parameters The choice of the necessary parameters to be put in the Solver section depends on the type of the strain boundary condition for the structure namely if it is considered as grown on a substrate or not 5 3 1 Structure with a substrate The only mandatory parameter is substrate to which a name of a substrate boundary condition region has to be assigned e g referring to the previous example
71. qfermi level the spatially constant electrochemical poten tial for the holes for quasi equilibrium cal culations integration order order of the numerical gauss integration De fault is 2 current integration method method for the calculation of the contact cutrrents surfint integrate the local cur rent density over the contact surface rstf use the Ramo Shockley test functions gives better results Local scaling apply a local scaling scheme which leads to better scaled matrices true or false exact newton use exact or approximate without some parts in the jacobian Newton true or false the default is given in brackets Table 6 11 Solver independent parameters the PETSc petsc or PARDISO pardiso linear solvers The possible combinations are nonlinear solver petsc Or nonlinear solver tiber linear solver petsc pardiso 6 5 SOLVER SECTION 6 5 1 Parameters for PETSc solvers 75 Tables 6 12 and 6 13 list all solver parameters significant for the PETSc linear and nonlinear solvers A more detailed description of the most important parameters follows keyword description default nonlin_rel_tol relative tolerance for the residual lo norm 10e 9 with respect to first nonlinear step nonlin abs tol absolute tolerance for the residual 5 norm 10e 15 nonlin step tol tolerance for the lo norm of the nonlinear 10e 3 step ls max step the maximum
72. rs relevant to a specific model of a physical property or quantity related to the present model e one or more Boundary Condition regions blocks BC regions block The BC regions block must be preceded by the keyword BC Regions and it is com posed by one or more parameters blocks each preceded by the keyword BC Region followed by the single word name of the boundary condition region This parameters block can contain the possible description of the boundary region These optional blocks must be strictly in this order first the options then the physical model and finally the BC regions blocks A detailed description of the possible parameters for these blocks follows 4 4 1 options block options 1 simulation name dd1 physical regions all physical regions channel 1 channel 2 simulation name user defined name of the particular instance of the simulation model defined for this block More than one simulation with different name and prop erties can be defined in separated model blocks which refer to a same TiBERCAD simulation model If simulation name is not assigned by default the TrBERCAD model name is taken as current simulation name physical_regions list of physical region s to which the present simulation model will be applied Physical region s are the TiberCAD regions or clusters as defined in Device section Default value is all all physical regions of the device In a list the names must
73. ry region e g a set of bias values to be assigned to a drain contact of a MOSFET for the calculation of an output drain IV characteristic in this Guide referred to as sweep calculation Each sweep definition must be preceded by its user defined name e g sweep_1 see Listing 1 In the case of a single sweep subblock for backward compatibility it is also allowed to define the sweep feature by means of the keyword sweep lower case in the following way Sweep 1 simulation driftdiffusion variable Vb start 0 0 stop 4 0 steps 80 plot_data true plotvariable current The following keywords are defined for this feature variable name of the variable to which the sweep is applied its value is assigned to a quantity e g voltage in a BC Region section to perform the sweep calculation see 4 4 3 start stop steps sweep starts from start value is repeated steps times and stops in stop simulation name of the simulation model associated to the sweep calculation it may be the name of another sweep defined in the same block plotvariable obsolete specify the integrated quantity to be calculated during the sweep and that will be shown in the output file sweep_modelname_sweepvariable dat eg sweep_driftdiffusion_Vb dat for a sweep of current calculation on the variable Vb typically a contact voltage 4 5 SOLVER SECTION Sweep t sweep 1 4 simulation driftdiffusion variable Vd star
74. sical names associated to the physical regions in the meshing tool In this case we simply associate the n 1 D physical region respectively anode and cathode by means of the TIBERCAD BC Region name BC_Region anode type ohmic voltage Vb BC Region cathode 1 type ohmic voltage 0 0 Note that in this case the TIBERCAD BC Region name needs to be identical to one of the physical names defined during the modeling of the device with GMSH 14 CHAPTER 2 GETTING STARTED 1D 4 Definition of Simulation parameters The variable Vb is specified in the sweep block in the Solver section sweep Simulation driftdiffusion 1 variable Vb start 0 0 stop 1 steps 10 In this way the simulation driftdiffusion_1 is performed for 10 steps 10 values of the anode voltage variable Vb between 0 and 1 5 Definition of Execution parameters In the Simulation section we decide which simulations to perform and in which order we set solve sweep to execute the sweep which run driftdiffusion_1 simulation for the specified loop Simulation d searchpath meshfile bulk msh dimension 1 temperature 300 solve sweep resultpath output output_format grace plot Ec Ev ContactCurrents Output files with conduction and valence band profiles plot Ec Ev and all the calculated values of the current at the contacts ContactCurrents the IV characteristic ar
75. states will be increased in oder to reach convergent density e cigen number increase factor factor to increase eigenvalues number for the next charge density calculation 8 3 Physical Models parameters e particle el hl e model conduction band kp single conduction band point or k p e kp model 6x6 8x8 86 CHAPTER 8 ENVELOPE FUNCTION APPROXIMATION Here the particle name is the name of a particle type electron or hole model kp conduction band k p or single conduction band model If k p model is applied specify kp model 6x6 8x8 8 4 Output e FigenEnergy Eigen energy in eV e EigenFunctions v r function of the eigenstate Occupation probability to find the state occupied It is calculated assuming Fermi distribution and mean electrochemical potential and temperature ES hjal 8 2 QT r v 8 3 EnergyLevels graphical output used for showing the energy level over the band diagram Chapter 9 Simulation opticskp By defining the opticskp model calculation of optical properties is enabled in particu lar the optical kp matrix elements are calculated from the quantum models specified in Solver section opticskp 1 initial state model Qui electrons quantum el final state model Wi holes quantum hl initial eigenstates 0 19 final eigenstates 0 19 Here initial state model and final state model are respectively the quantum models efaschroedi
76. t 0 0 stop 2 0 0 1 steps 20 50 plot_data true sweep 2 1 variable Vg start 0 1 stop 1 0 0 5 steps 11 6 simulation sweep 1 simulation driftdiffusion plot data true Listing 1 Example of Sweep section 37 38 CHAPTER 4 INPUT FOR TIBERCAD plot data default is false if it is set to true then output data will be written for each step of the sweep calculation otherwise just the results for the final step will be present in the output Once a sweep calculation has been defined it is treted as a special case of simulation and may be executed as an usual simulation by adding it in the solve list e g solue sweep drain see 4 7 e Selfconsistent block preceded by the keyword Selfconsistent This block may contain one or more subblocks each one defining a self consistent calculation based on two different simulation models e g driftdiffusion and excitontransport The definition of a selfconsistent calculation must be preceded by its user defined name e g converse piezo Selfconsistent 1 converse piezo 1 flavour relaxation simulations driftdiffusion strain dd excitons i flavour relaxation simulations driftdiffusion exciton Listing 2 Example of Selfconsistent section The following keywords are defined for this feature simulations the list of simulations to be performed self consistently flavour specifies broyden or relaxat
77. ulation that can provide electric field The parameter is poisson_equation Example macrostrain d poisson equation DriftDiffusion Interaction with other simulations n order to take into account the converse piezzo effect the poisson equation has to recalculate the necessary parameters after the strain simulation To do so the following parameters has to be set in the Physics section of the drift diffusion equation for detailes see Sec 6 4 driftdiffusion 1 model strained strain simulation str recompute band parameters true 5 5 Output The output variables are 5 5 OUTPUT 59 e strain strain tensor 6 components in calculation system e stress stress tensor 6 components in calculation system e polarization piezo polarization vector 3 components in calculation system Plot keyword label Units strain eps stress stress GPa polarization polarization C em Table 5 1 Elemental vector quantities By using StrainVariables as a plot keyword it s possible to include all quantities of the strain simulation 60 CHAPTER 5 SIMULATION OF STRAIN Chapter 6 Drift diffusion simulation of electrons and holes 6 1 Theory The semi classical transport simulation of electrons and holes is based on the drift diffusion approximation see e g 3 Beside the electric potential the electro chemical potentials are used as variables such that the system of PDEs to be solved reads as follows
78. xecutable tibercad exe A directory containing the simulation results will be created with the name provided in the input file Linux After the correct installation of TIBERCAD you should be able to run TIBERCAD from the command line using the command tibercad If not you probably have to add the bin subdirectory of the TiBERCAD installation directory to your PATH environment variable or start the TIBERCAD executable using the absolute path INSTALLPATH bin tibercad Copy the directory of the example you want to run e g bulk Si to your home directory or any place you have write permissions for Change to the newly created directory and run TiBERCAD by assuming Example 0 CONTENTS VII tibercad bulk tib A directory containing the simulation results will be created with the name provided in the input file Bug reports Feedback Please send bug reports feedback or suggestions to support tibercad org When sub mitting bug reports please always include the full version number of TiBERCAD you are running The full version number appears in the first line of output when running the program tibercad TiberCAD version 1 0 0 961 Usage tibercad lt inputfile gt Chapter 1 Overview 1 1 Intoduction to numerical simulation with Tiber CAD TiberCAD is a multiphysics software tool it comprises a set of solvers called simulation models each one describing a physical problem to be solved e g DriftDiffusion to solve P
79. y tensor and Ag is the total heat source The latter term is the sum of the heat sources specified by the submodels described below The term between the brackets represents the thermal flux 7 2 Physical model By neglecting particle effects the thermal conductivity is only due to the lattice contri bution The lattice thermal conductivity is read from the database The thermal model is tagged as thermal In options subsection we indicate the simulation name simulation name whatever_you_want and the simulation domain physical_regions wherever you want model thermal 1 options 1 simulation name whatever you want TT 78 CHAPTER 7 HEAT BALANCE SIMULATION physical regions wherever you want 7 2 1 Electron and hole dissipations Electron and hole dissipations give the following heat source Hs V PaT ar F P T F Pp p 7 2 where P and P are the thermoelectric power of electrons and holes respectively o and are the electro chemical potentials The equation 7 2 represents severals heat source contributions Their estimates are reported in table 7 1 Expression Heat source Bal Electron Joule effect ek Hole Joule effect ass dp En T P Recombination effect TJa4 VB Electron Peltier Thomson effect TJ VP Hole Peltier Thomson effect Table 7 1 Drift diffusion heat sources The physical model heat source includes a specific source identifie

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