Model for tunneling transport

To calculate the tunneling probability P(E) through a trapezoidal barrier, we used the following expression³⁶:

This equation is valid only for the case when the tunneling occurs between the edges of the barrier, i.e., the tunneling length is equal to the geometrical length of the barrier d. This is the case if Φ_b − E > 0 and Φ_b − E − eV > 0. Depending on the sign of V_d, the barrier can either decrease or increase by the amount ∣eV∣ along the tunneling path. Since the situation is symmetrical with respect to the direction of the applied voltage, we will consider in the following the case of V ≥ 0. In the case Φ_b − E > 0 and Φ_b − E − eV < 0, the carrier has to tunnel under the triangular barrier and the transmission coefficient is described by the Fowler–Nordheim theory⁵⁶. In Eq. (⁴), d indicates the barrier length, m is the charge carrier effective mass inside the barrier, φ(x,V) = Φ_b + (x/d) ⋅ (−eV) is the barrier height at coordinate x, V is the applied voltage, and Φ_b denotes the barrier height at x = 0. In the case of a triangular barrier, the integration over x in the exponent of Eq. (³) should be performed from zero till the value x_c determined by the condition φ(x_c, V) − E = 0. The expression for P(E) then reads,

in agreement with the exponent in Eq. (4) of ref. ³². The integral over E in Eq. (¹) is to be calculated in the range E ≥ 0, E ≥ −eV. This integration was performed numerically. The position of the Fermi level E_F in Eq. (¹) is determined by the carrier concentration in the channel, which in turn is controlled by V_g. The carrier concentration n in the channel of the heterojunction FET is given by n = n₀ + V_gC_g/e. Approximating the density of states by a constant ρ, we have n−n0=ρ(EF−EF0) where n − n₀ is the change in the carrier concentration induced by the gate voltage and EF−EF0 is the Fermi level shift induced by the gate voltage. Rearranging the above equation yields EF=EF0+(n−n0)/ρ. Substituting n = n₀ + V_gC_g, we get EF=EF0+CgVg/(eρ). We set α = C_g/(eρ) and take into account that the source–drain voltage also affects the gate potential at a given position along the channel. For instance, a barrier close to the source contact experiences a different potential than a barrier close to the drain contact. Thus, the gate voltage dependence is modeled in our fit as EF(Vg,Vd)=EF0(T)+α(T)(Vg+βVd). This relation assumes that the E_F is a linear function of carrier concentration, which is always true for a sufficiently small range of E_F, e.g., for a small gate voltage range. In all our fits of I_d – V_d curves, the variation of EF0 was within 100 meV relative to band edge, i.e., small as compared to Φ_b. The product αβ accounts for the asymmetry between the source and drain contacts, with the former taken as the reference potential against which both gate potential V_g and drain potential V_d are applied.