All estimators discussed so far depend on the injection coordinate
\(y_0\) through the term
\(\mathrm {e}^{\,\beta _D E_\mathrm {C}(\mathbf {r}_0)}/p_y(y_0)\). However, this dependence is weak and can even be eliminated from the estimator by choosing the injection distribution as
$$\begin{aligned} p_0(\mathbf {k}_0, y_0; x_0) = \frac{1}{A(T_\mathrm{D})} |v_x(\mathbf {k}_0)| \mathrm {e}^{-\beta _D H(\mathbf {k}_0,x_0, y_0)} \end{aligned}$$
(52)
Inserting (
22) in (
52) again yields a product of two independent PDFs.
$$\begin{aligned} p_0(\mathbf {k}_0,y_0) = f_0(\mathbf {k}_0)\, \tilde{p}_y(y_0) \end{aligned}$$
(53)
Here
\(f_0\) is given by (
41), and
\(\tilde{p}_y\) is defined as
$$\begin{aligned} \tilde{p}_y(y_0) = \frac{\tilde{n}(x_0, y_0)}{B(T_\mathrm{D})} \end{aligned}$$
(54)
with
\(\tilde{n}(x_0,y_0) = \mathrm {e}^{-\beta _D E_C(x_0, y_0)}\). This quantity is up to a constant, the equilibrium concentration, determined by the band edge energy
\(E_C\). On the other hand,
n in (
38) represents the actual carrier concentration as obtained from a device simulation. With the normalization integral in (
54) defined as
$$\begin{aligned} B(T_\mathrm{D}) = \int _0^{y_\mathrm {m}} \tilde{n}(x_0, y)\,\mathrm {d}y, \end{aligned}$$
(55)
the total normalization factor
A in (
52) becomes
$$\begin{aligned} A(T_\mathrm{D}) = V(T_\mathrm{D})\, B(T_\mathrm{D}). \end{aligned}$$
(56)
Using the boundary distribution (
31) and the injection distribution (
52), the current estimator (
30) can be reformulated as:
$$\begin{aligned} \mu = \mathrm {sign}(v_x(\mathbf {k}_0))\, v_\mathrm {inj}(T_\mathrm{D})\, B(T_\mathrm{D}) \,n(\mathbf {r}_b)\,\mathrm {e}^{\,\beta _D E_C(\mathbf {r}_b)} \end{aligned}$$
(57)
Other than the estimators discussed above, this estimator is independent of the injection coordinate
\(y_0\). A more transparent physical interpretation is achieved by expressing the equilibrium concentration
\(n(\mathbf {r}_b)\) as a function of the local quasi-Fermi level
\(F_n\).
$$\begin{aligned} n(\mathbf {r}_b) = N_C(T_\mathrm{D})\,\mathrm {e}^{\,\beta _D(F_n(\mathbf {r}_b) - E_C(\mathbf {r}_b)} \end{aligned}$$
(58)
Here the effective density of states
\(N_C\) is related to the partition function
Z by
\(N_C = Z/(4\pi ^3)\). Also, the normalization factor
B will be expressed through an energy
\(\bar{E}_C\) defined as
$$\begin{aligned} \bar{E}_C = -k_B T_\mathrm{D}\mathrm {ln}\left( \frac{B}{y_\mathrm {m}}\right) . \end{aligned}$$
(59)
\(\bar{E}_C\) has the meaning of an average of the band edge energy over the injection coordinate
\(y_0\):
$$\begin{aligned} \mathrm {e}^{-\beta _D\bar{E}_C(x_0)} = \frac{1}{y_\mathrm {m}} \int \limits _0^{y_\mathrm {m}} \,\mathrm {e}^{-\beta _D E_C(x_0, y_0)}\,\mathrm {d}y_0 \end{aligned}$$
(60)
Expressing the estimator (
57) in terms of the parameters
\(F_n\) and
\(\bar{E}_C\) gives
$$\begin{aligned} \mu = y_\mathrm {m} N_C(T_\mathrm{D})\, v_\mathrm {inj}(T_\mathrm{D})\,\mathrm {sign}(v_x(\mathbf {k}_0)) \,\mathrm {e}^{\,\beta _D(F_n(\mathbf {r}_b) - \bar{E}_C)}. \end{aligned}$$
(61)
This equation states that a backward trajectory represents an elementary particle flux
\(N_C\, v_\mathrm {inj}\). This flux is multiplied by a statistical weight given by the e-function. The higher the energy of the starting point (
\(\bar{E}_C\)) with respect to the Fermi level at the trajectory end point (
\(F_n\)), the lower is the statistical weight. If a constant
\(\varDelta \mathcal {E}\) were added to
\(\bar{E}_C\), the estimator
\(\mu \) and subsequently also the current
I would be scaled by the factor
\(\mathrm {e}^{-\beta _D \varDelta \mathcal {E}}\). In other words, increasing the barrier height by some energy increment will result in an exponential decrease in current. This means that the exponential dependence of the thermionic current on the barrier height can be directly deduced from the current estimator (
61).