For the density-gradient formulation, the additional dependent variables φn and φp (Equation 3-78) usually obey simple natural boundary conditions (Ref. 47):

In the case of a boundary representing an abrupt potential barrier, for example, a silicon-oxide interface, Jin et. al. in Ref. 48 suggested using the WKB approximation to obtain the boundary condition:

where bn,ox is the density-gradient coefficient in the oxide (SI unit: V m^2), and dn is the penetration depth into the oxide (m), given by

where m*n,ox and mn,ox are the effective masses in the oxide (SI unit: kg) and Φn,ox is the potential barrier height (SI unit: V). For silicon-silicon dioxide interfaces, Jin et. al. suggested the values of 0.22 m0, 0.5 m0, and 3.15 V, where m0 is the electron mass.