Network for Computational Nanotechnology (NCN) Purdue, Norfolk State, Northwestern, UC Berkeley, Univ.
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Network for Computational Nanotechnology (NCN) Purdue, Norfolk State, Northwestern, UC Berkeley, Univ. of Illinois, UTEP nanoMOS 4.0: A Tool to Explore Ultimate Si Transistors and Beyond Xufeng Wang School of Electrical and Computer Engineering Purdue University West Lafayette, IN 47906 Outline • Why nanoMOS simulator? • Device geometries in nanoMOS. • nanoMOS development history and my involvement • Overview of nanoMOS code structure • Overview of nanoMOS software development • Conclusion • Acknowledgement Xufeng Wang Introduction Xufeng Wang Featured devices Si/III-V double gate MOSFET SOI MOSFET HEMT spinFET Flexible and efficient modeling is needed to explore these device proposals. Xufeng Wang Why nanoMOS? • It studies a very general structure: double gate, thin body, nMOSFET with fully depleted channel. • It features several transport models: drift-diffusion, semiclassical ballistic, quantum ballistic, and quantum dissipative. • It is computationally efficient, easily modified, written in MATLAB, and freely available on nanohub.org with Rappture interface • Well documented on various thesis and papers. Xufeng Wang Development history Zhibin Ren Kurtis Cantley S. Clark, S. Ahmed Creation Quantum transport for 2008 III-V material nanoMOS 2.0 nanoMOS 3.0 Asymmetrical gate Code restructure: configuration modulation, testing suite Rappture interface nanoMOS 1.0 2000 Himadri Pal nanoMOS HEMT Yang Liu X. Wang, D. Nikonov Xufeng Wang Unification of branches nanoMOS phonon scattering Himadri Pal Xufeng Wang Xufeng Wang Drift-diffusion transport for III-V material Parallel support, Rappture interface nanoMOS 3.5 nanoMOS spinFET nanoMOS 4.0 Yunfei Gao Today Xufeng Wang Inside thesis Unification of branches Drift-diffusion transport for III-V material Models and techniques • Various transport models • Non-linear damping • Boundary conditions • Recursive Green’s function • Scharfetter and Gummel method • nanoMOS applications Drift-diffusion transport …… for III-V material Code restructure: modulation, testing suite Parallel support, Rappture interface Code restructure: modulation, testing suite Software development • Rappture interface • nanoMOS Unification parallelization of branches • Benchmark and testing suite Parallel support, Rappture interface GOAL: Deliver a comprehensive documentation and understanding of nanoMOS, physics and software wise. Xufeng Wang Numerical Approach Gummel’s Method Initial Guess for carrier density No n(new ) n(old) Yes Converge! n(old) n(new ) tolerance ? Solve Poisson’s Equation 1 2 E ( p n Nd Na ) n(old) Solve Transport Equations Jn qn(x) qDn n n 1 J Gn t q n n(new ) Regardless how we start, all equations must be self-consistently satisfied at the same time Xufeng Wang Solving the Transport Equations “Straight-forward” Method of Solving Transport Equations n(x) d d Jn (n(x) (x) n(x)) kT dx dx • In order to solve this equation, we first need to find a linear approximation to turn the differential equation into a discretized linear equation. x i1 x 1 i 2 x xi x i1 1 i 2 First step is to use the mesh point variables to interpolate the midpoint variables Xufeng Wang Solving the Transport Equations “Straight-forward” Method of Solving Transport Equations Substitute the approximated variables back to transport equations x i1 x i 1 2 Continuity Equation tells us: Xufeng Wang x i i x i1 1 2 Ji-1/2 x i1 x xi Ji+1/2 1 2 xi x i x i1 1 2 Solving the Transport Equations Stability Problem of “Straight-forward” Method • Observe the equation: i i1 i i1 (i i1 2)(i1 i 2) n) ( i1 2) 0) i1( i i1 i i1 i ni1(i1 i 2)ni ( i1i If both i i1> 2 Then, at least 1 carrier density is forced to be negative • This means if electric potential difference between any two neighboring nodes is greater than 2kT/q, the “straight-forward” method might get negative non-physical carrier density solutions. • Therefore, a finer grid is required at regions that the rate of change of electric potential is high. This may lead to a huge number of grid nodes, thus increasing the computational cost dramatically. Xufeng Wang Solving the Transport Equations Scharfetter and Gummel Method • We will attempt a direct integration by introducing the following factor: (x) n(x)e u(x) • Carrier density • Exponential of electric potential • An unknown function of x Find the derivative of carrier density (n) d d (x) d (x) n(x)e u(x)e u(x) (x) dx dx dx Substitute the introduced factor into transport equation J 1 i 2 i i 1 Xufeng Wang 2k d d d (e ( x )u( x) (x ) e ( x ) u(x) e ( x ) u(x) (x) T d x d x d x J 1 i 2 i i1 2k T e(x) d u(x) dx ) Solving the Transport Equations Scharfetter and Gummel Method Recast the equation J 1 i i1 i 2 2k T e(x) d u(x) dx (x) J 1e i 2 i i1 d u(x) 2k T d x Attempt a direct integration on both sides of the equation xi x i 1 (J i ( x ) )d x 1e 2 xi x i 1 ( i i1 d 2k u( x ) d ) x T d x Now, let’s look at this equation’s left and right hand side separately. Xufeng Wang Solving the Transport Equations Scharfetter and Gummel Method Join the left and right hand side together xi (J x i 1 i ( x ) )d x 1e xi ( x i 1 2 i i1 d 2k u( x ) d ) x T d x x i1 i i1 i i1 i J1 (e e ) (nie ni1e ) i i1 2k T 2 i i i1 ni ni1ei i1 J 1 B(i i1)e i 2k T x 2 B(z) Xufeng Wang i z is the Bernoulli ez 1 Function Solving the Transport Equations Scharfetter and Gummel Method i i1 i i1 n n e J 1 B(i i1)ei i i1 i 2k T x 2 at node xi-1/2 This is the 1-D electron Transport Equation via finite difference with Scharfetter and Gummel Method at node xi-1/2 • Similarly, one can write down the transport equation at node xi+1/2 i i1 i1i n n e J 1 B(i1i)ei1 i1 i i 2k T x 2 at node xi+1/2 • Now, just as what we did in “straight forward” method, we can use the relationship establish by Continuity Equations to solve the problem Xufeng Wang Solving the Transport Equations Scharfetter and Gummel Method • How can the stability of transport equation be guaranteed by Scharfetter and Gummerl Method? (i i 1 )B(i i 1 )ei i1 ni1 (B(i i 1 ) B(i1 i ))(i i 1 )ni (i i1 )B(i 1 i )ei i1 ni 1 0 (i i 1 )B(i i 1 )ei i1 ni1 (B(i i 1 ) B(i1 i ))(i i 1 )ni (i i1 )B(i 1 i )ei i1 ni 1 0 • Notice that the Bernoulli Function is ALWAYS positive. z B(z) z e 1 • One coefficient is always negative, so the carrier densities are no longer forced to be negative. Xufeng Wang Numerical Approach Gummel’s Method Initial Guess for carrier density No n(new ) n(old) Yes Converge! n(old) n(new ) tolerance ? Solve Poisson’s Equation 1 2 E ( p n Nd Na ) n(old) Solve Transport Equations Jn qn(x) qDn n n 1 J Gn t q n n(new ) Regardless how we start, all equations must be self-consistently satisfied at the same time Xufeng Wang Solving the Poisson Equation Boundary Conditions for Poisson Equation • Although source and drain bias are given as inputs, we still use Neumann boundary for source and drain ends to avoid convergence problem. • Source and drain bias are used to calculate electron density, thus indirectly influence the potential at ends. Xufeng Wang Between 2D Poisson solver and 1D transport • Effective mass Schrodinger equation is solved in confinement direction Xufeng Wang Solving the Transport Equations Complete Scheme of Drift-Diffusion Modeling Initial Guess for carrier density No Yes Converge! n(new ) n(old) Solve Poisson’s Equation n(old) n(new ) tolerance ? n(new ) Solve Transport Equations n 1 J Gn t q n Jn qn(x) qDn n Xufeng Wang Newton Iteration Converge? Yes n(old) Schrodinger Equation Solver _ 2D( poisson) _1D(subbands) (subband, x) 1 2 E ( p n Nd Na ) No Other available transport models Drift-diffusion computationally efficient mobility difficult to determine Semiclassical ballistic evaluates device ballistic limit may be too optimistic Quantum ballistic RGF based; quantum effects no scattering; longer run time Quantum dissipative with phonon scattering Phonon scattering Xufeng Wang longest run time Software development: Overview test & benchmark Developer SVN User Rappture on nanoHUB parallel job submitter Xufeng Wang Software development: Rappture interface Xufeng Wang Conclusion • Overviewed nanoMOS development history • Demonstrated Scharfetter and Gummel method as numerical sample • Demonstrated Rappture interface as software sample GOAL: Deliver a comprehensive documentation and understanding of nanoMOS, physics and software wise. Xufeng Wang Acknowledgement • Committee members: Professor Klimeck, Professor Lundstrom, and Professor Strachan. • Funding and support from my advisors. • Encouragement and help when needed from my colleagues. • Mrs. Cheryl Haines and Mrs. Vicki Johnson for scheduling the examination and being the most helpful secretaries. • As always, thank and love to my entire family. Xufeng Wang Now, welcome the questions…… Xufeng Wang Xufeng Wang Device geometry #1: Si/III-V double gate MOSFETs Sample double gate MOSFET geometry 3D electron density Conduction band profile • Si/III-V as channel material • Thin body (< 10nm). Single channel conduction if thin enough. • Double gates can be biased separately • Source/drain can be metallic and turn into Schottky barrier FET Xufeng Wang Device geometry #2: SOI MOSFET 3D conduction band near front gate Sample SOI geometry Conduction band in transverse direction • Si/III-V as channel material. • Similar to previous structure, except the bottom oxide layer is thick. • Back gate can be biased to push channel electron toward front gate. Xufeng Wang Device geometry #3: HEMT Charge and conduction band profile from Yang Liu * Sample HEMT geometry • Intrinsic III-V material as channel = high mobility. • Delta-doped layer controls threshold voltage. * Y. Liu, M. Lundstrom, “Simulation-Based Study of III-V HEMTs Device Physics for High-Speed Low-Power Logic Applications”, ECS meeting, 2009 Xufeng Wang Device geometry #4: spinFET Sample spinFET geometry • Device structure suggested by Sugahara & Tanaka • Controls current by manipulating electron spin Xufeng Wang Problem Statement and the Semiconductor Equations What are we trying to solve? Top Gate Source Drain Buttom Gate • Given device geometry and material parameters (such as gate length, dielectric constant, mobility) • Look for solution for: carrier density electric potential • Both carrier density and electric potential solutions must satisfy all the equations. Regardless how we start, all equations must be self-consistently satisfied at the same time Xufeng Wang Transport model #1: drift-diffusion • Computationally efficient • Account scattering via mobility, thus suitable for long channel devices • Do not consider quantum effects such as tunneling and interference. Xufeng Wang Scharfetter and Gummel Method • If apply finite difference method directly: i i1 i i1 ( i1 2)(i1 i 2) n) ( i1 2) 0) i1( i i1 i i i1 i ni1(i1 i 2)ni ( If both i1i i i1 > 2 Then, at least 1 carrier density is forced to be negative • Introduce Scharfetter and Gummel method n(x)e(x)u(x) • Carrier density • Exponential of electric potential • An unknown function of x (i i 1 )B(i i 1 )ei i1 ni1 (B(i i 1 ) B(i1 i ))(i i 1 )ni (i i1 )B(i 1 i )ei i1 ni 1 0 SG method ensures stability of carrier density solutions. Xufeng Wang Transport model #2: Semiclassical ballistic n(E) D(E) f (E) J qvn 1 dE 2E v dk m* • Simple model exploring device behavior at ballistic limit • Do not consider quantum effects such as tunneling and interferences. Xufeng Wang Transport model #3 & #4 : Quantum ballistic & dissipative S 0 G(E) [E l I H[E i (x)] ]1 H(E1 (x)) 0 0 H(E 2 (x)) H 0 0 H(E i (x)) 0 0 2t E i (1) 0 0 2t E i (2) H(E i (x)) 0 0 t 2m * a 2 2t E i (N x ) x 0 0 Xufeng Wang S (E) teikl a 0 D E E i (1) 2t(1 cos kl a) Transport model #3 & #4 : Quantum ballistic & dissipative An GnG n(E l ) S i(S S ) D i(D D ) 1 ( ( E l )AS 1/ 2 (D E l )AD ) 2 a 1/ 2 S TSD (E l ) Trace(S GDG ) q ISD (E l ) (1/ 2 (S E l ) 1/ 2 (D E l ))TSD (E l ) For dissipative transport, nanoMOS can treat phonon scattering, or general scattering via Buttiker probe approach (now obsolete). Xufeng Wang Development history nanoMOS 1.0 (Published in 2000) • Developer: Zhibin Ren • Original nanoMOS code for silicon MOSFETs is written in MATLAB. nanoMOS 2.0 (Published in 2005) • Developer: Steve Clark, Shaikh S. Ahmed • Rappture interface is added to nanoMOS, and the code becomes avaliable on nanoHUB.org. nanoMOS 3.0 (Published in 2007) • Developer: Kurtis Cantley • Support for III-V materials in semi-classical ballistic and quantum ballistic transport models is added. Rappture interface is updated to reflect the III-V implementation. nanoMOS 3.0 (Published in 2007) • Developer: Himadri Pal • Top and bottom gate can now have asymmetric configurations with different gate dielectrics and capping layers. nanoMOS 3.5 (Published in 2008) • Developer: Xufeng Wang • Support for III-V materials in drift-diffusion transport is added. Additional mobilities models are added. nanoMOS 3.5 (Published in 2009) • Developer: Xufeng Wang, Dmitri Nikonov • nanoMOS source code is restructured and modularized. Material parameters are separated out as a mini-library. Debugging functions are planted within source code to assist code developments. Benchmark and testing suite is created based on a script from Dmitri Nikonov. Xufeng Wang nanoMOS 4.0 (Developed in 2009) • Developer: Himadri Pal • Support for Schottky FET is added. NanoMOS now has the ability to simulate a double gate MOSFETs structure with metallic source/drain via NEGF\ formalism. nanoMOS 4.0 (Developed in 2009) • Developer: Yang Liu • Support for HEMT is added. NanoMOS now has the ability to simulate a III-V HEMT structure via NEGF formalism. nanoMOS 4.0 (Developed in 2009) • Developer: Xufeng Wang • Parallel Jobs Submitter (PJS) is added. PJS allows nanoMOS to sweep gate/source bias and run each bias on a cluster node. It supports only clusters with Portable Batch System (PBS) installed such at steele (steele.rcac.purdue.edu) or coates (coates.rcac.purdue.edu). nanoMOS 4.0 (Developed in 2009) • Developer: Yunfei Gao • Support for SpinFET is added. NanoMOS now has the ability to simulate a SpinFET structure via NEGF formalism. nanoMOS 4.0 (To be published in 2010) • Developer: Xufeng Wang • Merge working branches of Schottky FET, HEMT, and SpinFET modules. Code is restructrued. Rappture interface is updated to accommodate the newly published features.