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## silicon_dg5

The ``silicon_dg5'' system solves for the five equation density gradient model.

The quantities in parentheses are negative fluxes. The solution variables for the above five equations are , , , , and , respectively. The physical constants and are defined as:

where l can be considered to be either the space dimensionality or a fitting parameter; ( ) is the electron (hole) effective mass.

Required Field Definitions

This system requires netdope to be defined after the structure is created with grid and deposit statements.

An implementation of the silicon_dg5 system is given below:

```system name=silicon_dg5
+   sysvars=psi,sqrt_n,sqrt_p,phi_n,phi_p
+   term0=ndiv_fbm.lapflux(psi|psi)@{silicon,poly,oxide}
+   term1=nodal.nscd(electrons,holes,netdope|psi)@{silicon,poly}
+   term2=constraint.continuity(psi|psi)@{silicon/oxide,poly/oxide,silicon/poly}
+   term3=dirichlet.dg_dirichlet(netdope|psi,phi_n,phi_p)@{silicon/gate,poly/gat
e,silicon/back,silicon/source,silicon/drain,oxide/gate,silicon/substrate}
+   term4=ndiv_fbm.diffusion(bn,sqrt_n|sqrt_n)@{silicon,poly}
+   term5=-0.5*nodal.prod(sqrt_n,dphi_n|sqrt_n)@{silicon,poly}
+   term6=dirichlet.default_dirichlet(0|sqrt_n,sqrt_p)@{silicon/oxide,poly/oxide}
+   term7=ndiv_fbm.diffusion(bp,sqrt_p|sqrt_p)@{silicon,poly}
+   term8=0.5*nodal.prod(sqrt_p,dphi_p|sqrt_p)@{silicon,poly}
+   term9=ndiv_fbm.nqfflux(psi,phi_n,tl,mobn,electrons,edge|phi_n)@{silicon,poly}
+   term10=ndiv_fbm.nqfflux(psi,phi_p,tl,mobp,holes,edge|phi_p)@{silicon,poly}
+   tmpvars=electrons,holes,phin0,phip0,dphi_n,dphi_p,bn,bp,mobn,mobp
+   func0=prod(sqrt_n,sqrt_n|electrons)@{silicon,poly}
+   func1=prod(sqrt_p,sqrt_p|holes)@{silicon,poly}
+   func2=phiMB(electrons,psi,tl|phin0)@{silicon,poly}
+   func3=phiMB(holes,psi,tl|phip0)@{silicon,poly}
+   func4=sub(phin0,phi_n|dphi_n)@{silicon,poly}
+   func5=sub(phip0,phi_p|dphi_p)@{silicon,poly}
+   func6=6.35e-9*inverse(mn|bn)@{silicon,poly}
+   func7=6.35e-9*inverse(mp|bp)@{silicon,poly}
+   func8=mob0(electrons,holes,tl,netdope|mobn,mobp)@{silicon,poly}
+   init0="tl:\${solve/bias/Tcelsius}+\${library/physics/celsius}"
+   init1="edge:1"
+   init2="sqrt_n:sqrt(electrons)"
+   init3="sqrt_p:sqrt(holes)"
+   init4="mn:0.19"
+   init5="mp:0.49"```

Prophet Development
Mon Jul 1 14:45:00 PDT 2002