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Diffstat (limited to 'Costin_Costin_Jauslin_Lebowitz_2018.tex')
| -rw-r--r-- | Costin_Costin_Jauslin_Lebowitz_2018.tex | 4 |
1 files changed, 2 insertions, 2 deletions
diff --git a/Costin_Costin_Jauslin_Lebowitz_2018.tex b/Costin_Costin_Jauslin_Lebowitz_2018.tex index e0ccaab..11486e4 100644 --- a/Costin_Costin_Jauslin_Lebowitz_2018.tex +++ b/Costin_Costin_Jauslin_Lebowitz_2018.tex @@ -324,7 +324,7 @@ and integrated current (the current integrated over the supply function at 0 tem \begin{equation} J_{k_{\mathrm F}}(x,t):=\int_0^{k_{\mathrm F}}dk\ j_k(x,t) \end{equation} -as a function of time at two different values of $x$: $x_0:=\frac{2U-k_{\mathrm F}^2}{2E}\approx 11\ \mathrm{nm}$ and $10x_0$ ($x_0$ is the point at which $V(x_0)=\frac{k_{\mathrm F}^2}2$), and at two different values of $E$: $4$ and $8\ \mathrm{V}\cdot\mathrm{nm}^{-1}$. We have normalized the current $j$ by $2k$, which is the current of the incoming wave $e^{ikx}$, and the integrated current $J$ by $k_{\mathrm F}^2$, which is the current of the incoming wave integrated over the supply function. We find that there is a transient regime that lasts a few femtoseconds before the system stabilizes to the FN value. Note that the approach to the FN regime has some ripples, which come from the imaginary parts of the poles in the $p$-plane (see Fig.\-~\ref{fig:contour}). There is a delay before the signal reaches $x_0$, and between $x_0$ and $10x_0$. As expected, the asymptotic value of the current is independent of $x$. Note that the current and density depend strongly on the field $E$. +as a function of time at two different values of $x$: $x_0:=\frac{2U-k_{\mathrm F}^2}{2E}$ and $10x_0$ ($x_0$ is the point at which $V(x_0)=\frac{k_{\mathrm F}^2}2$), and at two different values of $E$: $4$ and $8\ \mathrm{V}\cdot\mathrm{nm}^{-1}$. We have normalized the current $j$ by $2k$, which is the current of the incoming wave $e^{ikx}$, and the integrated current $J$ by $k_{\mathrm F}^2$, which is the current of the incoming wave integrated over the supply function. We find that there is a transient regime that lasts a few femtoseconds before the system stabilizes to the FN value. Note that the approach to the FN regime has some ripples, which come from the imaginary parts of the poles in the $p$-plane (see Fig.\-~\ref{fig:contour}). There is a delay before the signal reaches $x_0$, and between $x_0$ and $10x_0$. As expected, the asymptotic value of the current is independent of $x$. Note that the current and density depend strongly on the field $E$. \bigskip \begin{figure} @@ -338,7 +338,7 @@ as a function of time at two different values of $x$: $x_0:=\frac{2U-k_{\mathrm \includegraphics[width=7cm]{integrated_current-4.pdf}{\scriptsize({\bf e})}& \includegraphics[width=7cm]{integrated_current-8.pdf}{\scriptsize({\bf f})} \end{tabular} - \caption{The density ({\bf a}),({\bf b}), current ({\bf c}),({\bf d}) and integrated current ({\bf e}),({\bf f}) as a function of time at $x=x_0\equiv\frac{2U-k_{\mathrm F}^2}{2E}$ and $x=10x_0$. We have taken $U=9\ \mathrm{eV}$ and $k^2/2=k_{\mathrm F}^2/2\equiv E_{\mathrm F}=4.5\ \mathrm{eV}$. In ({\bf a}),({\bf c}),({\bf e}), the field is $E=4\ \mathrm{V}\cdot \mathrm{nm}^{-1}$ and $x_0\approx11\ \mathrm{nm}$. In ({\bf b}),({\bf d}),({\bf f}), $E=8\ \mathrm{V}\cdot\mathrm{nm}^{-1}$ and $x_0\approx22\ \mathrm{nm}$. In ({\bf a}),({\bf c}),({\bf e}), the plots seem to indicate that the curves converge to 0, but they actually tend to small finite values.} + \caption{The density ({\bf a}),({\bf b}), current ({\bf c}),({\bf d}) and integrated current ({\bf e}),({\bf f}) as a function of time at $x=x_0\equiv\frac{2U-k_{\mathrm F}^2}{2E}$ and $x=10x_0$. We have taken $U=9\ \mathrm{eV}$ and $k^2/2=k_{\mathrm F}^2/2\equiv E_{\mathrm F}=4.5\ \mathrm{eV}$. In ({\bf a}),({\bf c}),({\bf e}), the field is $E=4\ \mathrm{V}\cdot \mathrm{nm}^{-1}$ and $x_0\approx1.1\ \mathrm{nm}$. In ({\bf b}),({\bf d}),({\bf f}), $E=8\ \mathrm{V}\cdot\mathrm{nm}^{-1}$ and $x_0\approx0.56\ \mathrm{nm}$. In ({\bf a}),({\bf c}),({\bf e}), the plots seem to indicate that the curves converge to 0, but they actually tend to small finite values.} \label{current_density} \end{figure} |