NanoTCAD ViDES User`s Manual
Contents
1. comment command it has to be followed by a space prev the non linear solver starts from the solution provided in the file Phi out flat the non linear solver starts from the flat band solution default This works both for the SNWT and CNT GNR simulations In the case of CNT GNR the flat band potential computed in correspon dence of the drain is increased by its electrochemical potential This can be useful in order to start with a drain to source voltage different to zero as suggested in the golden rule section gridonly code generates only the grid and then exits poiss the Poisson equation is solved default ps the self consistent Poisson Schr dinger equations in the effective mass approximation are solved This flag has to be set if SWNT simulations want to be performend Please refer to the region section for more details CNT the open boundary condition Schrodinger equation within the tight binding NEGF formalism is solved self consistently with the Poisson equa tion in order to compute transport in CNT FETs As stated in section 2 1 ViDES can simulate n 0 zig zag nanotubes In order to specify the chiral ity and the length of the nanotube the command CNT has to be followed 15 16 by two values the chirality and the length expressed in nanometers re spectively For example if a 13 0 CNT 15 nm long wants to be defined then you have to specify CNT 13 15 Pay attention that the CNT a2. If declared it has to be followed by six numbers corresponding to the electric field at each surface By default the electric field is imposed to zero on each surface Null Neumann Boundary condition maxwell Maxwell Boltzmann statistics for the semiclassical electron den sity default 3 2 Grid definition The mesh generator within the ViDES simulator defines non uniform rectangu lar grids along the three spatial direction In order to define the grid the gridx gridy gridz commands have to be used in the input deck Let s now focus our attention on the discretization along the x axis The general statement is the following gridx mesh x spacing Ay mesh x spacing An The mesh command followed by a number x defines the position in nm of the grid point along the x direction while the number A after the spacing command defines the spacing between two user s defined grid points The same considerations follow for the definition of the grid along the y and z directions where the gridx command has to be replaced by gridy and gridz commands respectively In Fig 3 1 is shown an example of generated grid along the x y plane when the input file is the following gridx mesh 2 spacing 0 3 mesh 0 spacing 0 1 mesh 2 spacing 0 3 18 3 ee 0 5 0 5 1 5 Dr 1 2 x axis nm Figure 3 1 Example of generated non uniform grid along the x y plane gridy mesh 2 spacing 0 3
3. NanoTCAD ViDES User s Manual by G Fiori and G Iannaccone Copyright 2004 2009 Gianluca Fiori Giuseppe Iannaccone University of Pisa Contents 1 Getting started 7 2 Implemented Physical Models 9 2 1 Transport in zig zag Carbon Nanotubes and Arm chair Graphene Nanoribbon Transistors 9 2 2 Transport in Silicon Nanowire Transistors 11 2 3 Semiclassical Models 0 o e 12 2 4 Numerical Implementation i a 111 13 3 Definition of the Input Deck 15 3 1 Main commands o 15 3 2 Grid definition LL LL eM 17 3 8 Solid definition 1 42 sr wer wo mod w owa w ers 18 33 1 Soldexz imple 224 2 8 car Kiwi x 21 3 4 Gate definition 2 sgn a A men Den 21 3 5 Region command LL Lena eee 22 3 0 Output fil s y 2 5 240826 ran EW eo he ria de 23 3 7 Golden rules for the simulation 1 144420211 25 3 00 plot kb oto ued ted wota MIE ead Ete ie oh Gee 25 4 Examples 29 4 1 Carbon Nanotube Field Effect Transistors 29 4 2 Silicon Nanowire Transistors 22 22 222mm 31 4 3 Graphene Nanoribbon Field Effect Transistors 34 5 Inside the code 37 5 1 Discretization of the three dimensional domain 37 5 2 The m in body s l ao tke oe A en 39 5 2 1 physical quantitiesh 2 2 2 1 39 5 2 2 devicemapping h 2004 40 9 2 3 38oliduzze h 1 4 2 22 2 ed ehr ns 42 9 2
4. So positive values are for electrons negative for holes pcar out hole density Nam out lonized acceptors concentration Ndp out lonized donors concentration jayn out the current obtained in CNT GNR simulations bandent the mean intrinsic Fermi level along the nanotube ring as a function of the z coordinate along the nanotube This applies only when the CNT GNR command is specified coordent gnu coordinates of the atoms of the CNT Atom positions can be plotted by means of newplot with the command splot coordcnt gnu w p coordent jmol coordinates of the atoms of the CNT Atom positions can be visualized by means of the jmol code available at http jmol sourceforge net download coordGNR gnu coordinates of the atoms of the GNR Atom positions can be plotted by means of newplot with the command splot coordent gnu w p coordGNR jmol coordinates of the atoms of the GNR Atom positions can be visualized by means of the jmol code available at http jmol sourceforge net download Tn transmission coefficient in CNT GNR simulations jay out the current obtained in SNWT simulations T1 transmission coefficient in SWNT simulations for the first couple of minima 3 7 Golden rules for the simulation 25 e T2 transmission coefficient in SWNT simulations for the first couple of minima e T3 transmi
5. after the gate statement 22 3 5 xmin define the minimum coordinate nm in the x direction of the rectangular region It has to be followed by a number default xmin minimum z coordinate of the grid xmax define the maximum coordinate nm in the x direction of the rectangular region It has to be followed by a number default xmax maximum z coordinate of the grid ymin define the minimum coordinate nm in the y direction of the rectangular region It has to be followed by a number default ymin minimum y coordinate of the grid ymax define the maximum coordinate nm in the y direction of the rectangular region It has to be followed by a number default ymax maximum y coordinate of the grid zmin define the minimum coordinate nm in the z direction of the rectangular region It has to be followed by a number default zmin minimum z coordinate of the grid zmax define the maximum coordinate nm in the z direction of the rectangular region It has to be followed by a number default zmax maximum z coordinate of the grid Ef defines the Fermi energy of the gate eV default Ef 0 It is equal to the gate voltage changed by sign workf defines the work function of the gate eV default workf 4 1 cylinder imposes the shape of the region to be a cylindrical shell The axis of the cylinder is parallel to the z axis Once the cylinder command has been specified th
6. GNR Once the length L and n are defined the coordinates in the three dimensional domain of each carbon atom are computed and the three dimensional domain is discretized so that a grid point is defined in correspondence of each atom while a user specified grid is defined in regions not including the CNT GNR A point charge approximation is assumed i e all the free charge around each carbon atom is spread with a uniform concentration in the elementary cell including the atom Assuming that the chemical potentials of the reservoirs are aligned at the equilibrium with the Fermi level of the CNT GNR and given that there are no fully confined states the electron concentration is 00 n 2 f dE Ws B P E Ers bv E f f E a Er 2 3 while the hole concentration is Ei pP 2 dE Jus E DP 1 FE Ep E 7 1 f E Em 2 4 where r is the coordinate of the carbon site f is the Fermi Dirac occupation factor and s Vpl is the probability that states injected by the source drain reach the carbon site 7 and Er Ep is the Fermi level of the source drain The current has been computed as 2q pre gt dET E f E Ers E Er 2 5 where q is the electron charge h is Planck s constant and T is the transmis sion coefficient T Tr Es z G Eo x Eb ei l 2 6 2 2 Transport in Silicon Nanowire Transistors 11 Figure 2 1 In the SNWT mod
7. INCLUDING BUT NOT LIMITED TO PRO CUREMENT OF SUBSTITUTE GOODS OR SERVICES LOSS OF USE DATA OR PROFITS OR BUSINESS INTERRUPTION HOWEVER CAUSED AND ON ANY THEORY OF LIABILITY WHETHER IN CONTRACT STRICT LIABILITY OR TORT INCLUDING NEGLIGENCE OR OTHERWISE ARIS ING IN ANY WAY OUT OF THE USE OF THIS SOFTWARE EVEN IF ADVISED OF THE POSSIBILITY OF SUCH DAMAGE CONTENTS Chapter 1 Getting started The provided NanoTCADViDES tgz archive contains the NanoTCAD ViDES source code the User s manual and the post processing utility fplot which allow to elaborate in a user friendly way the data provided by NanoTCAD ViDES In order to install the software you have first to accomplish the following software requirements In particular you need e fortran 77 compiler e C compiler e make e gnuplot e python e python tcl tk libraries To install simply run the tcsh script install sh or just go in the sre di rectory and type make and then make install Please note that the Makefile included in the src directory and the install sh script are meant to be used on a machine in which g77 gcc compilers are installed The optimization flag O3 has also been set If you want to use other compilers or different flag options please make changes in the Makefile in the src directory If compilation has been succesfull you will find in the bin directory the com piled ViDES code the utility fplot and its file sectionx sectiony
8. electron and hole concentrations n and p respectively instead they are computed by means of the different physical models which will be described in the following sections 2 1 Transport in zig zag Carbon Nanotubes and Arm chair Graphene Nanoribbon Transis tors The simulations are based on the self consistent solution of the 3D Poisson Schr dinger equation with open boundary conditions within the Non Equilibrium 9 10 2 Green s Function formalism where arbitrary gate geometry and device archi tecture can be considered 1 The electron and hole concentrations are computed by solving the Schrodinger equation with open boundary conditions by means of the NEGF formalism A tight binding Hamiltonian with an atomistic p orbitals basis has been used both in the real and the mode space approach 2 The Green s function can be expressed as G E El H YEs5 Yp 2 2 where E is the energy 7 the identity matrix H the Hamiltonian of the CNT GNR and s and p are the self energies of the source and drain respectively As can be seen transport is here assumed to be completely ballistic In the released version of the code the considered Carbon Nanotubes CNTs are ALL n 0 zig zag nanotube where n is the chirality while the considered Graphene Nanoribbons GNRs are ALL ARMCHAIR In particular the n Armchair GNR is simply an unrolled n 0 zig zag nanotube i e n is the number of dimmer lines along the unrolled
9. mesh 0 spacing 0 1 mesh 2 spacing 0 3 The grid generator yields as output the files points out in which the number of points along the x y z directions are stored and gridx out gridy out gridz out which contain the points of the mesh along x y and z respectively NOTE THAT If a CNT GNR is simulated the grid along the z axis is self computed so that the gridz command has not to be specified 3 3 Solid definition The solid command defines in the three dimensional domain the regions of different materials that belong to the device to be simulated As for the the grid command its subset of commands has to be included between parenthesis and In particular within the solid statement we have 3 3 Solid definition 19 e rect impose the shape of the region to be rectangular default e xmin define the minimum coordinate nm in the x direction of the rectangular region It has to be followed by a number default xmin minimum z coordinate of the grid e xmax define the maximum coordinate nm in the z direction of the rectangular region It has to be followed by a number default xmax maximum z coordinate of the grid e ymin define the minimum coordinate nm in the y direction of the rectangular region It has to be followed by a number default ymin minimum y coordinate of the grid e ymax define the maximum coordinate nm in the y direction of the rectangular region It has to be f
10. sectionz as well as the file input material and dimension max SUCH FILES HAVE TO RESIDE IN THE SAME DIRECTORY YOU LAUNCH THE CODE AS WELL AS THE INPUT DECK So if you copy the code into another directory different from the bin directory take care of copying all the files contained in the bin directory To run the code just simply type ViDES inputdeck where inputdeck is the input file in which the nanoscale device has been defined Chapter 2 Implemented Physical Models The implemented code is a three dimensional Poisson solver in which different physical models for the simulation of nanoscale devices have been included In particular ViDES is able to e compute transport in n 0 zig zag carbon nanotubes with general geome tries by means of atomistic p tight binding Hamiltonian within the real and mode space approximation e compute transport in Silicon Nanowire Transistors within the effective mass approximation e solve the 3D Poisson Laplace equation at the equilibrium assuming semi classical models for the electron and hole densities In particular the potential profile in the three dimensional simulation do main obeys the Poisson equation VAVA q p r nF NB NI F Pfie 2 1 where 7 is the electrostatic potential e r is the dielectric constant Nj and N are the concentration of ionized donors and acceptors respectively and Pfia is the fixed charge For what concerns the
11. the number of modes to be used the maximum number of modes is equal to chirality As shown in 2 however from a computational point of view it is nonsense to use a large number of modes since real space simulation will be more efficient THE MODE SPACE COMMAND IS NOT DEFINED FOR GNRs mul this defines the electrochemical potential for the source It has to be followed by a value expressed in eV mu2 this defines the electrochemical potential for the drain It has to be followed by a value expressed in eV underel under relaxation coefficient for the potential at each NR cycle default underel 0 This is e in eq 2 14 potsottoril under relaxation coefficient for the potential at the end of NR cycle default potsottoril 0 This is e in eq 2 15 carsottoril under relaxation coefficient for the charge at the end of NR cycle default carsottoril 0 This is e in eq 2 16 tolldomn inner tollerance of the linear solver default tolldomn 0 1 3 2 Grid definition 17 poissnorm tollerance inside the NR cycle default poissnorm 10 normad tollerance of the outer cycle default normad 10 This is the e in Fig 2 2 e temp temperature K default temp 300 complete complete ionization of donors and acceptors incompl incomplete ionization of donors and acceptors default This refers to eq 2 12 field the electric field on the six lateral surfaces of the simulation domain V m
12. 4 SCIUTA Ch 0 02 is r ne LILO sg eal 43 5 2 0 _ s lveSystem 6 i zawory wa awa men OSS 43 CONTENTS License term Copyright 2004 2009 G Fiori G lannaccone University of Pisa All rights reserved Redistribution and use in source and binary forms with or without modifi cation are permitted provided that the following conditions are met e Redistributions of source code must retain the above copyright notice this list of conditions and the following disclaimer Redistributions in binary form must reproduce the above copyright notice this list of conditions and the following disclaimer in the documentation and or other materials provided with the distribution e All advertising materials mentioning features or use of this software must display the following acknowledgement This product includes software developed by G Fiori and G lannaccone at University of Pisa Neither the name of the University of Pisa nor the names of its contributors may be used to endorse or promote products derived from this software without specific prior written permission THIS SOFTWARE IS PROVIDED BY G FIORI AND G IANNACCONE AS IS AND ANY EXPRESS OR IMPLIED WARRANTIES INCLUDING BUT NOT LIMITED TO THE IMPLIED WARRANTIES OF MERCHANTABIL ITY AND FITNESS FOR A PARTICULAR PURPOSE ARE DISCLAIMED IN NO EVENT SHALL G FIORI AND G IANNACCONE BE LIABLE FOR ANY DIRECT INDIRECT INCIDENTAL SPECIAL EXEMPLARY OR CONSEQUENTIAL DAMAGES
13. EE Transaction on Nanotechnology Vol 6 Issue 4 pp 475 480 2007 3 J Guo S Datta M Lundstrom and M P Anantram Multi scale modeling of carbon nanotube transistors Int J Multiscale Comput Eng Vol 2 pp 257260 2004 4 A Trellakis A T Galick A Pacelli and U Ravaioli Iteration scheme for the solution of the two dimensional Schrdinger Poisson equations in quan tum structures J Appl Phys Vol 81 p 7880 7884 1997 5 R Lake G Klimeck R C Bowen and D Jovanovic Single and Multi band modeling of quantum electron transport through layered semiconduc tors devices J Appl Phys Vol 81 pp 7845 7869 Feb 1997 A Svizhenko M P Anantram T R Govindam and B Biegel Two dimensional quantum mechanical modeling of nanotransistors J Appl Phys Vol 91 pp 2343 2354 Nov 2001 45
14. a 3D system of equations can be easily derived Let us consider Fig 5 1 which shows the Voronoi cell around rq the position of the l th point of the mesh We want to discretize around point l a generic equation V F o 5 1 on a Delauney mesh using box integration If we integrate 5 1 on the surface 2 we obtain V fdr gdr 5 2 Q Q and applying the Gauss Green theorem on the left hand side F dr gu 5 3 a where the surface integral becomes a flux integral over the boundary of the Voronoi cell o while the right hand term has been simplified bringing out of the integral the argument since 2 is small and the argument can be considered constant 37 38 5 Figure 5 1 Voronoi cell associated to the l th point of the mesh at position r The discretized equation can then be written as NR o 54 gt J where the sum is performed all over the nearest neighbor j dj is the distance between j and l and o is the boundary of the Voronoi cell which cuts the dij segment In three dimensions Q is the volume of the Voronoi cell dj is still the distance between and its nearest neighbors j while oj is the surface of the Voronoi cell cut by the segment dj Since it is easier to work with vectors rather than with matrices the discretized quantities function of the point are mapped in ViDES as in Fig 5 2 points are ordered first along x then along y and z This means that in the 3D domain if the ge
15. ated by fixed charge where the right hand term of the Poisson equation is considerably large An under relaxation of the potential and of the charge can also be performed in order to help convergence In particular three different under relaxations can be performed inside ViDES e relaxation on the potential at each NR cycle prew pre e Mold _ new 2 14 e relaxation on the potential at the end of each NR cycle 9 b e b 4 2 15 14 2 C sm gt V Initial potential y NEGF y np Poisson solution Newton Raphson 9 Figure 2 2 Flow chart of the self consistent 3D Poisson Schr dinger solver e relaxation of the charge density pngar computed by the NEGF modules PNEGF PNEGF tE PNEGF PNEGr 2 16 Chapter 3 Definition of the Input Deck Let s now define the input deck of ViDES simulator ViDES has its own lan guage composed by a limited set of commands as well as its own syntax through which it is possible to define both the grid the structure of the nanoscale device and the physical models to be used in the simulations Along ViDES commands are also present some instances deliminated by the parenthesis and which have other subset of commands We refer in particular to gridx gridy gridz solid 3 1 gate region whose meaning will be explained in the following sections Main commands Let s first focus on the list of main commands
16. belongs solido returns the solid number to which the element belongs The solid is defined in str3d in file by means of the solid command ele vector which returns the elements to which the generic points I belong dist the matrix which implements dy i e the distances between the I and j points as explained in the first section of the manual In particular working with rectangular grid the generic point l has six prime neighbors the first index runs over the points of the grid while the second index over the six prime neighbors As in Fig 5 2 index 0 refers to 1 1 point index 1 tol 1 index 2 to l ng index 3 to l ny index 4 to l n Ny and index 5 to I nany surf isa matrix which implements the surfaces o of the Voronoi volume multiplied by the dielectric constant as explained in the first section of the manual Its indexes are ordered in the way as the dist matrix dVe is a matrix which implements the Q volume of the Voronoi cell While the first index runs over the points of the grid as in dist and in surf the second index runs over the eight sub volumes belonging to the Voronoi elements corrSP as explained in the ViDES User Manual the user can specify regions belonging to the three dimensional domain in which quantum analysis is performed It can be useful to reorder points belonging to these regions identifying them with a relative index r It is however necessary to define also a corresponde
17. dfunpoint is defined as typedef double dfunpoint int i physical_quantities p device_mapping d soliduzzo mat and e nf vector of pointer to functions associated to the electron charge density 44 pf vector of pointer to functions associated to the hole charge density accf vector of pointer to functions associated to ionized acceptors den sity donf vector of pointer to functions associated to ionized donors density dnf vector of pointer to functions associated to the derivative of the electron charge density with respect to the electrostatic potential dpf vector of pointer to functions associated to the derivative of the hole charge density with respect to the electrostatic potential daccf vector of pointer to functions associated to the derivative of the ionized acceptors density with respect to the electrostatic potential ddonf vector of pointer to functions associated to the derivative of the ionized donors density with respect to the electrostatic potential Bibliography 1 G Fiori G Iannaccone G Klimeck A Three Dimensional Simulation Study of the Performance of Carbon Nanotube Field Effect Transistors With Doped Reservoirs and Realistic Geometry IEEE Transaction on Electron Devices Vol 53 Issue 8 pp 1782 1788 2006 2 G Fiori G Iannaccone G Klimeck Coupled Mode Space Approach for the Simulation of Realistic Carbon Nanotube Field Effect Transistors E
18. e axis and the radius of the inner and outer cylinder are specified by the following commands xcenter specify the x coordinate of the axis ycente specify the y coordinate of the axis radiusl specify the radius of the inner cylinder radius2 specify the radius of the outer cylinder Region command The region command defines where and which kind of quantum analysis has to be performed The following ensemble of commands has to be declared and included in brackets after the region statement xmin define the minimum coordinate nm in the x direction of the rectangular region It has to be followed by a number default xmin minimum z coordinate of the grid xmax define the maximum coordinate nm in the x direction of the rectangular region It has to be followed by a number default xmax maximum z coordinate of the grid 3 6 Output files 23 e ymin define the minimum coordinate nm in the y direction of the rectangular region It has to be followed by a number default ymin minimum y coordinate of the grid e ymax define the maximum coordinate nm in the y direction of the rectangular region It has to be followed by a number default ymax maximum y coordinate of the grid e zmin define the minimum coordinate nm in the z direction of the rectangular region It has to be followed by a number default zmin minimum z coordinate of the grid e zmax define th
19. e defined CNT FET in the inputdeckCNT file CNT 13 15 Here it starts the definition of the structure tolldomn le 1 normad 5e 2 poissnorm le 1 mul 0 mu2 0 1 mode 2 flat potsottoril 0 2 solid field oxide sio2 nulln nullp solid source sio2 nulln nullp zmax 5 fraction 5e 3 solid drain sio2 nulln nullp zmin 10 fraction 5e 3 4 2 Silicon Nanowire Transistors 31 metal gate tox 1nm tn 4nm 1 ELI Hal 4nm tox 1nm metal gate Figure 4 2 Example of the defined SNW FET in the inputdeckSNWT file gate out xmin 190 xmax 200 gate bottom xmax 1 7 zmin 5 zmax 10 Ef 0 0 gate top xmin 1 7 zmin 5 zmax 10 Ef 0 1 4 2 Silicon Nanowire Transistors To run the example in the bin directory type the command ViDES inputdeck SNWT The simulated device is a double gate SNWT FET shown in Fig 4 2 The device cross section is rectangular 4x4 nm the channel length is 10 nm and the source and drain reservoirs are 10 nm long The oxide thickness is equal to 1 nm 4 modes have been taken into account Here follows the structure definition gridx mesh 3 spacing 0 1 32 mesh 2 spacing 0 05 mesh 0 spacing 0 2 mesh 2 spacing 0 05 mesh 3 spacing 0 1 gridy mesh 3 spacing 0 1 mesh 2 spacing 0 05 mesh 0 spacing 0 2 mesh 2 spacing 0 05 mesh 3 spacing 0 1 gridz mesh 0 1 spacing 0 1 mesh 0 spacing 0 2 me
20. e maximum coordinate nm in the z direction of the rectangular region It has to be followed by a number default zmax maximum z coordinate of the grid e negf2d solve the transport along coupled one dimensional subbands by means of the NEGF formalism and solve the 2D Schr dinger equation in the x y plane The physical model is described in section 2 2 e nauto specifies the maximum number of eigenvalues i e the number of one dimensional subbands computed by the 2D Schr dinger equation 3 6 Output files ViDES produces the following output files e solido out the defined solids along the structure It can be usesful in order to understand if the defined structure is the one expected In particular solids are numbered with an increasing number as they appear in the input deck e tipopunto out specify the type of points along the defined 3D domain Even in this case post processing of this file can reveal to be usesful in order to understand if the defined structure is the one expected In particular it specifies which kind of boundary conditions have to be applied in each point of the domain If tipopunto is smaller than 100 it specifies that the considered point is a point belonging to a gate and it returns the gate number If tipopunto 101 1 belongs to the surface with z Imin Where Lmin is the least abscissa along x and where Neumann boundary conditions are imposed the derivative of the elec
21. el the 3D Poisson the 2D Schr dinger and 1D transport equations are solved self consistently where Tr is the trace operator 2 2 Transport in Silicon Nanowire Transistors Simulations of Silicon Nanowire Transistors are based on the self consistent solution of Poisson Schr dinger and continuity equations on a generic three dimensional domain Hole acceptor and donor densities are computed in the whole domain with the semiclassical approximation while the electron con centration in strongly confined regions needs to be computed by solving the Schr dinger equation within the effective mass approximation We assume that in the considered devices the confinement is strong in the transversal plane x y and the potential is much smoother in the longitudinal direction z As a consequence the Schr dinger equation is adiabatically decoupled in a set of two dimensional equations with Dirichlet boundary conditions in the x y plane for each grid point along z and in a set of one dimensional equations with open boundary conditions in the longitudinal direction for each 1D subband Fig 2 1 In particular the two dimensional Schr dinger equation for each cross sec tion along z reads 2010 1010 20m dx 2 Oy My Oy V xl 2 Eiv 2 Xiv Y z 2 7 Nat Y z where runs over subbands y runs over the three pairs of minima of the con duction band Xiv and e are the 2D eigenfunctions and the corresponding 12 2 eigenener
22. gies computed in the transversal plane at a given z while m and my are the effective masses along the k and ky axis respectively on the conduction band pair of minima v The main assumption based on the very small nanowire cross section con sidered is that an adiabatic approximation can be applied to the Schr dinger equation so that transport occurs along one dimensional subbands Transport is computed by means of the NEGF formalism in one dimensional subbands within the mode space approach taking into account inter and intra subbands scattering and the one dimensional electron density n1p z is there fore obtained The three dimensional electron density n to be inserted into the right hand side of the Poisson equation reads M n z y z EG gt Xiv y z n n 2 2 8 where the sum runs over the M considered modes Even in this case a point charge approximation has been assumed 2 3 Semiclassical Models No transport is computed within this approximation while only a simple semi classical expression is assumed for the hole and electron concentrations Such a model can be used for example in order to simulate polysilicon gate when CNT GNR or SNWT transport simulations are performed In the non degenerate statistics we obtain Ec 7 Ep nF Nee ORTE 2 9 where InKT T N 2 gt M 2 10 For the holes _ Ep Ey 7 pH Ny Men TERE 2 11 where Ey is the valence band and Ny is the equivalent densit
23. lassical electron concentration to zero Since in general CNTs GNRs are embedded in dielectrics when sim ulating CNTs GNRs and specifing the solid region nulln com mand has to be set 20 0 10 10 100 10 10 Figure 3 2 Two examples of solids a MOS structure b cylindrical nanowire The white dots univocally define the rectangular region nullp imposes the semiclassical hole concentration to zero Since in general CNTs GNRs are embedded in dielectrics when sim ulating CNTs GNRs and specifing the solid region nulln com mand has to be set In addition in SNWT simulations only electron transport is considered As a consequence impose nullp command Na defines the nominal acceptors concentration m 3 It has to be followed by a number default Na 0 Nd defines the nominal donors concentration m 3 It has to be followed by a number default Nd 0 rho fixed charge concentration m It has to be followed by a number default rho 0 Ef defines the Fermi level of the region eV It has to be followed by a number default Ef 0 fraction this command takes effect only if the CNT GNR command has been specified In particular it imposes the doping molar fraction of the nanotube If a doped nanotube wants to be defined a solid which contains the part of the nanotube the user wants to dope has to be defined NOTE THAT in case of overlapping solids the last defined shadows the pre
24. mesh 2 spacing 0 3 mesh 0 spacing 0 1 mesh 2 spacing 0 3 gridy mesh 1 spacing 0 3 mesh 0 spacing 0 1 mesh 1 62 spacing 0 1 4 3 Graphene Nanoribbon Field Effect Transistors 35 mesh 2 62 spacing 0 3 GNR 6 15 Here it starts the definition of the structure tolldomn 1e 1 normad 5e 2 poissnorm le 1 mul 0 mu2 0 0 flat potsottoril 0 2 solid field oxide sio2 nulln nullp solid source sio2 nulln nullp zmax 5 fraction 5e 3 solid drain sio2 nulln nullp zmin 10 fraction 5e 3 gate out xmin 190 xmax 200 gate bottom xmax 1 9 zmin 5 zmax 10 Ef 0 0 gate top 36 xmin 1 9 zmin 5 zmax 10 Ef 0 2 Chapter 5 Inside the code The following notes are a brief summary of what is inside the ViDES code Hopefully a more detailed Developer s guide will follow In the first section we will discuss how equations are discretized in the three dimensional domain while in the following sections we will take a look to the data structures as well as to the two subroutines which constitute the main body struttura and solvesystem 5 1 Discretization of the three dimensional do main The discretization method used in ViDES is the so called box integration method Let us suppose to discretize a 2D dimensional problem which is sim pler to be handled as compared to a 3D problem and from which general con siderations valid also for
25. nce between the relative index r and the absolute index l This is accomplished by means of the matrix corrSP r region which returns the absolute index l given the relative index r and the region to which r belongs 42 5 e corrPS corrPS 1 region returns instead the relative index r of the absolute index 1 which belongs to the region region e tipopunto it is a vector function of l which specifies which kind of boundary conditions have to be applied in the point J If tipopunto 1 is smaller than 100 it specifies that the considered point is a point belonging to a gate and it returns the gate number If tipopunto 1 101 1 belongs to the surface with z min Where Tmin is the least abscissa along x and where Neumann boundary conditions are imposed the derivative of the electric field is equal to E 0 If tipopunto 1 102 1 belongs to the surface with z maz where Lmaz is the largest abscissa along x and where Neumann boundary con ditions are imposed the derivative of the electric field is equal to E 1 If tipopunto 1 103 1 belongs to the surface with y ymin where Ymin is the least abscissa along y and where Neumann boundary conditions are imposed the derivative of the electric field is equal to E 2 If tipopunto 1 104 1 belongs to the surface with y ymax where Ymaz is the largest abscissa along y and where Neumann boundary conditions are imposed the derivative of the electric field i
26. nction and the Fermi level respec tively of the first defined gate in the str3d in file which is the referring gate for the potential vt the thermal energy expressed in eV defined as KgT q where Kp is the Boltzmann constant and T the temperature flagmbfd a flag which specifies if a Maxwell Boltzmann or a Fermi Dirac statistics has to be used for the semiclassical expression of the electron and hole densities flagion a flag which specifies if a complete or incomplete ionization for the acceptor and donor concentrations has to be considered pot it is the vector function of the gate index which specifies the elec trostatic potential 1 at the gate Dirichlet boundary conditions E it is the electric field imposed on the six lateral surface of the three dimensional domain Neumann boundary conditions 5 2 2 device mapping h Let us consider the other data structure device_mapping h which instead passes to the subroutine all the quantities related to the grid typedef struct 5 2 The main body 41 int int int int int nx ny nz solido xele double dist double surf double dVe int int int int corrSP corrPS tipopunto tipopuntos device_mapping nx number of points along the x axis ny number of points along the y axis nz number of points along the z axis solido it is a vector function of element given the element to which the generic l point
27. neric point r has index 4 j and k along the three axis its index in the mapped vector is l i j na k na ny 5 5 with 0 7 1 5 6 j 0 ny 1 5 7 kalim 5 8 5 9 The grid used in ViDES is a non uniform rectangular grid so the Voronoi cell is a parallelepiped Fig 5 2 which does not represent the Voronoi cell shows 8 different parallepipeds each parallelepiped is the element associated with each grid point This distinction is necessary because as we will see later each element can be associated to different a material 5 2 The main body 39 Element associated to point 1 l4nx hy a a if 5 I nx i 112 je 1 1 WAS asia Mo ei ES ZM LEI ey STARA gt ZY er x 4 X Figure 5 2 Three dimensional grid index of the nearest neighbors 5 2 The main body ViDES is a three dimensional Poisson solver so 5 1 becomes V e 7 VO 7 a p 7 nF N5 F NA P pric 5 10 where amp is the electrostatic potential e is the dielectric constant q is the ele mentary charge p and n are the hole and electron densities respectively NE is the concentration of ionized donors and N is the concentration of ionized acceptors while pr is the fixed charge The main body of the code is constituted by two subroutines struttura which takes care of all the quantities needed to define the discretized Poisson equation like the volume and the surface
28. nt we have to define a structure soliduzzo mat and write in the code mat d solido d ele 1 eps where d is a device_mapping structure 5 2 4 struttura c This subroutine implements the parser which reads the input deck str3d in and returns all the quantities associated to the grid like for example the dist the surf and the dVe matrices and properly fills the structures explained above 5 2 5 solvesystem c This subroutine implements the three dimensional non linear Poisson solver by means of the Newton Raphson algorithm void solvesystem physical_quantities p device_mapping d int IERR double tolldomn int neq soliduzzo mat double normapoisson double sottoril dfunpoint nf dfunpoint pf dfunpoint accf dfunpoint donf dfunpoint dnf dfunpoint dpf dfunpoint daccf dfunpoint ddonf Apart from the structures already explained vectors of pointer to function are passed as argument of solvesystem which represents the terms belonging to the right hand term of 5 10 and their derivatives with respect to the elec trostatic potential Having defined vectors of pointer to function has the advantage of separating the physics from the numerics In this way indeed all the numerics concerned with the solution of the Poisson equation is included in solvesystem while all the physics is defined specifying for each point l which function i e which physical model has to be solved in that point In particular the structure
29. of the Voronoi cell as well as all the distances between the neighbors while solvesystem implements the 3D Poisson solver All the physics instead is included in separate modules which can be devel oped and passed to solvesystem as functions as we will see in the next sessions Before focusing our attention on the main subroutines we have to spend some comments on the data structures which are defined inside ViDES and which are the arguments of the subroutines of the code Passing all the data through structures has the benefits of leaving unchanged the header of the sub routines as changes to the code are concerned 5 2 1 physical quantities h This structure which contains all vectors associated with physical quantities which are generally function of the point i e vectors which depends only on the index I of the discretized domain 40 Here is the listing of the data structure typedef struct double Phi double Na double Nd double Ef double rhof double phim1 double ef1 double vt char flagmbfd char flagion double pot double E physical_quantities Phi vector associated with the electrostatic potential defined in the three dimensional domain Na vector associated with the acceptor concentration Nd vector associated with the donor concentration Ef the Fermi level defined in the considered domain rhof the fixed charge concentration phimi and ef1 they are the workfu
30. ollowed by a number default ymax maximum y coordinate of the grid e zmin define the minimum coordinate nm in the z direction of the rectangular region It has to be followed by a number default zmin minimum z coordinate of the grid e zmax define the maximum coordinate nm in the z direction of the rectangular region It has to be followed by a number default zmax maximum z coordinate of the grid e sphere impose the shape of the region to be a spherical shell It has to be followed by 5 numbers nm the z y z coordinates of the center nm and the inner and outer radius of the sphere e cylinder imposes the shape of the region to be a cylindrical shell The axis of the cylinder is parallel to the z axis Once the cylinder commands has been specified the axis and the radius of the inner and outer cylinder are specified by the following commands xcenter specify the x coordinate of the axis ycenter specify the y coordinate of the axis radiusl specify the radius of the inner cylinder radius2 specify the radius of the outer cylinder e silicon declares that the material of the solid is Silicon e sio2 declares that the material of the solid is SiOz e air declares that the material of the solid is Air e AlGaAs declares that the material of the solid is AlGaAs e Al defines the molar fraction of Aluminium in case the AlGaAs material has been defined e nulln imposes the semic
31. s equal to E 3 If tipopunto 1 105 1 belongs to the surface with z Zmin Where zmin is the least abscissa along z and where Neumann boundary conditions are imposed the derivative of the electric field is equal to E 4 If tipopunto 1 106 1 belongs to the surface with z Zmaz where Zmaz is the largest abscissa along z and where Neumann boundary conditions are imposed the derivative of the electric field is equal to E 5 e tipopuntos is a vector function of which specifies if a point belongs to a region in which quantum analysis has to be performed 5 2 3 soliduzzo h This structure contains all the quantities which are related to the material like the effective mass the energy gap the dielectric constant etc and which are functions of the solid Here we will explain some quantities included in soliduzzo typedef struct double eps double chi double Egap double mel double me2 double me3 double Nc double Nv soliduzzo e eps dielectric constant e chi electron affinity 5 2 The main body 43 e eps energy gap e mel effective mass along the k direction e me2 effective mass along the ky direction e me3 effective mass along the k direction e Nc equivalent density of states in the conduction band e Nv equivalent density of states in the valence band If for example we want the value of the dielectric constant of the material in correspondence of the th poi
32. se of python tcl tk libraries Since output files are concerned with quantities defined in the three dimensional domain it is useful to cut such data in slices along the three different axis in order to be visualized fplot does exactly this or better given the input file the plane through which the cut has to be done and the coordinate of the plane it produces an output file which is read and visualized by gnuplot In Fig 3 3 it is shown the GUI as it appears once fplot is called The cut plane is specified by means of the radio buttom while the coordinate and the file name of the ViDES output file are specified in the input box To plot just push the plot button and a window like the one shown in Fig 3 4 will be displayed The image can also be rotated If this does not happen add the following line to the gnuplot file in the user s home set mouse on 26 e088 X fplot v1 0 section plane x w section plane y Sections Section plane z Insert filename Insert plane coordinate nm gnuplot command v ON OFF Figure 3 3 Picture of fplot as it appears once called SS Ay FE RL MMI LEE SOKO UHR YMM DHE a yp ey NYM WM UA NONI R cress oo ooo the electrostatic potential Figure 3 4 Example of the fplot displayed output in a CNT is shown 3 8 fplot 27 As can be noted there is also another input field which appears in the GUI In such a field gnuplo
33. sh 29 9 spacing 0 2 mesh 30 0 spacing 0 1 tolldomn 1e 1 normad le 2 poissnorm le 1 mul 0 0 mu2 0 5 flat ps prev potsottoril 0 2 complete solid field oxide i sio2 nulln nullp J solid field oxide silicon nullp xmin 2 xmax 2 ymin 2 ymax 2 solid field source silicon nullp 4 2 Silicon Nanowire Transistors 33 xmin 2 xmax 2 ymin 2 ymax 2 zmax 10 Nd le26 solid field drain silicon nullp xmin 2 xmax 2 ymin 2 ymax 2 zmin 20 Nd 1e26 gate out xmin 190 xmax 200 gate top xmin 3 zmin 10 zmax 20 Ef 0 5 workf 4 1 gate bottom xmax 3 zmin 10 zmax 20 Ef 0 5 workf 4 1 region negf2d xmin 2 5 xmax 2 5 ymin 2 5 ymax 2 5 nauto 4 34 4 Figure 4 3 Example of the defined GNR FET in the inputdeckGNR file 4 3 Graphene Nanoribbon Field Effect Transis tors To run the example in the bin directory type the command ViDES input deckGNR The simulated device is a double gate GNR FET shown in Fig 4 3 The GNR is a 6 Armchair GNR and 15 nm long The channel is 5 nm and the doped source and drain extension are each 5 nm Their molar fraction is equal to 5 x 107 The GNR width is equal to n 0 5 V3acc where a 0 144 nm is the Carbon Carbon distance and the oxide thickness is 2 nm The lateral spacing is equal to 1 nm The whole nanoribbon is embedded in SiOz dielectric Here follows the input file inputdeckGNR gridx
34. ssion coefficient in SWNT simulations for the first couple of minima 3 7 Golden rules for the simulation 1 When defining a grid pay attention in imposing a fine enough grid in correspondence of the interface e g Si SiOz interface A spacing equal to 0 05 nm should be fine enough 2 Every time you create a new grid which means that you do not have a previous initial potential from which to start you have to impose the flat band solution flat command In this way the initial solution is internally generated starting from the assumption of flat band in the whole semiconductor 3 When computing transfer characteristics for drain to souce voltages Vps different to zero first find a solution for Vps 0 and then starting from this solution prev command increase Vps up to the desired value Expe rience has tought that step voltages larger than 0 05 V are not extremely good 4 In CNT GNR simulations the following parameter has most of the time resulted to be usefull in order to achieve better convergence poissnorm le 1 normad 5e 2 Anyway in order to achieve better accuracy try to use a normad as small as possible 5 When solving the Schr dinger equation choose a region which includes the Si Si02 interface or other heterostructure interfaces 6 When simulating SNWT use complete ionization in order to avoid prob lems with acceptor donors level occupations 3 8 fplot fplot is a python script which makes u
35. t commands can be specified and take effect only when the ON radio button on the right is set If for example a colormap of the plot wants to be displayed the following commands have to be written set pm3d map unset surf rep The colormap will be shown after pressing the plot button 28 Chapter 4 Examples Here we give two examples of input files for two different nanoscale devices a CNT FET a SNWT device and a GNR FET 4 1 Carbon Nanotube Field Effect Transistors To run the example in the bin directory type the command ViDES input deckCNT The simulated device is a double gate CNT FET shown in Fig 4 1 The CNT is a 13 0 zig zag and 15 nm long The channel is 5 nm and the doped source and drain extension are each 5 nm Their molar fraction is equal to 5x 107 The nanotube diameter is 1 nm and the oxide thickness is 1 5 nm The lateral spacing is equal to 1 5 nm The whole nanotube is embedded in SiO2 dielectric A mode space approach is followed and 2 modes are considered Here follows the input file inputdeckCNT gridx mesh 2 spacing 0 3 mesh 0 spacing 0 1 mesh 2 spacing 0 3 gridy mesh 2 spacing 0 3 mesh 0 spacing 0 1 mesh 2 spacing 0 3 gridz mesh 0 1 spacing 0 1 mesh 0 spacing 0 1 mesh 10 05 spacing 0 1 mesh 10 2 spacing 0 1 29 30 S 1 5 nm tox G 5 E1 5n ME S 2 2 SiO2 2 1 5nm metal gate metal gate Figure 4 1 Example of th
36. then repeated cyclically until the norm of the difference between the potential computed at the end of two subsequent NR cycles is smaller than a predetermined value Some convergence problems however may be encountered using this iterative scheme Indeed since the electron density is independent of the potential within a NR cycle the Jacobian is null for points of the domain including carbon atoms SNWT region losing control over the correction of the potential We have used a suitable expression for the charge predictor in order to give an approximate expression for the Jacobian at each step of the NR cycle To this purpose we have used an exponential function for the predictor In particular if n is the electron density as in 2 3 the electron density n at the i th step of the NR cycle can be expressed as ni nexp 2 2 13 Vr where b and d are the electrostatic potentials computed at the first and i th step of the NR cycle respectively and Vr is the thermal voltage Same considerations follow for the hole concentration Since the electron density n is extremely sensitive to small changes of the electrostatic potential between two NR cycles the exponential function acts in the overall procedure as a dumping factor for charge variations In this way convergence has been improved in the subthreshold regime and in the strong inversion regime Convergence is still difficult in regions of the device where the charge is not compens
37. tric field is equal to E 0 If tipopunto 102 1 belongs to the surface with Tmax Where Tmax is the largest abscissa along x and where Neumann boundary conditions are imposed the derivative of the electric field is equal to E 1 If tipopunto 103 1 belongs to the surface with y Ymin Where Ymin is the least abscissa along y and where Neumann boundary conditions are imposed the derivative of the electric field is equal to E 2 If tipopunto 104 1 belongs to the surface with y ymax where Ymaz is the largest abscissa along y and where Neumann boundary conditions are imposed the derivative of the electric field is equal to E 3 If tipopunto 105 1 belongs to the surface with z Zmin where Zmin is the least abscissa along z and where Neumann boundary conditions are imposed the derivative of the electric field is equal to E 4 If tipopunto 106 1 belongs to the surface with z Zimax where Zmaz is the largest abscissa along z and where Neumann boundary conditions are imposed the derivative of the electric field is equal to E 5 tipopuntos out if a point is in correspondence of a C atom or in cor respondence of the nanowire then tipopuntos is equal to 777 Phi temp Temporary electrostatic potential at each NR step Phi out Electrostatic potential Ec out Conduction Band ncar out electron density In the case of CNT GNR ncar gives the sum of holes and electrons
38. vious one 3 4 Gate definition 21 3 3 1 Solid examples In Figs 3 2a b we show two examples of structures which can be defined by the solid command a MOS Fig 3 2a and a cylindrical silicon nanowire Fig 3 2b Suppose that you have already defined the grid For the MOS case let s assume that xmin 2 nm xmax 2 nm ymin zmin 0 nm and ymax zmax 10 nm The MOS structure can then be defined as solid rectangular region which defines the SiO2 I took the whole domain SiO2 nulln nullp solid rectangular region which defines the doped Si Note the overlap Si xmin 0 Na 1e26 For the cylindrical case let s assume that xmin ymin zmin 0 nm xmax ymax 3 nm zmax 20 nm The SNW structure can then be defined as solid rectangular region which defines the SiO2 I took the whole domain SiO2 nulln nullp solid cylindrical nanowire Note the overlap Si xcenter 1 5 ycenter 1 5 radiusl 1 radius2 2 As you can see the definition is really straightforward if you take advantage of the default values for the solid coordinates and the overlap of solids 3 4 Gate definition The gate command defines the gate region in the three dimensional domain i e the point in which Dirichlet boundary conditions are imposed The first defined gate is the referring gate whose electrostatic potential is set to zero As for the solid command the following ensemble of commands has to be declared and included in brackets
39. xis is in correspondence of the z axis In addition the source is supposed to lay on the left of the structure while the drain on the right GNR the open boundary condition Schrodinger equation within the tight binding NEGF formalism is solved self consistently with the Poisson equa tion in order to compute transport in GNR FETs As stated in section 2 1 ViDES can simulate n Armchair GNRs In order to specify the number of dimmer lines along the unrolled carbon nanotube ring and the length of the nanoribbon the command GNR has to be followed by two values the number of dimmer lines and the length expressed in nanometers re spectively For example if a 6 GNR 15 nm long wants to be defined then you have to specify GNR 6 15 Pay attention that the z axis is in correspondence of the lower edge of the GNR In addition the source is supposed to lay on the left of the structure while the drain on the right Schottky impose Schottky contact at CNT GNR ends the Schottky barrier is imposed to be equal to Egap 2 Doped impose boundary conditions as if semi infinite CNTs GNRs were connected at device ends This is a condition to be imposed if doped source and drain contacts wants to be simulated real a real space basis set is used to compute transport in CNTs GNRs only if the CNT GNR command is specified mode a mode space approach is used to compute transport in CNTs only if the CNT command is specified It has to be followed by
40. y of states in the valence band For the non completely ionized impurities we have HA Np r N ge E po 002 2 12 Hio Na T Naf 1 ga Ar ga 4 where Ex and Ep are the acceptors and donors level N4 e Np are the acceptor and donors concentrations and ga e gp are the degeneration factors 2 4 Numerical Implementation 13 2 4 Numerical Implementation The Green s function is computed by means of the Recursive Green s Function RGF technique 5 6 Particular attention must be put in the definition of each self energy matrix which can be interpreted as a boundary condition of the Schr dinger equation In particular in our simulation we have considered a self energy for semi infinite leads as boundary conditions which enables to consider the CNT GNR as connected to infinitely long CNTs GNRs at its ends In addition Schottky contacts are considered for both CNT and GNR fol lowing a phenomenological approach described in 3 From a numerical point of view the code is based on the Newton Raphson NR method with a predictor corrector scheme 4 In Fig 2 2 we sketched a flow chart of the whole code In particular the Schr dinger NEGF equations are solved at the beginning of each NR cycle starting from an initial potential and the charge density in the CNT GNR and SNWT is kept constant until the NR cycle converges i e the correction on the potential is smaller than a predetermined value The algorithm is
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