Non-perturbative Solution of the 1d Schrödinger Equation Describing Photoemission from a Sommerfeld Model Metal by an Oscillating Field

Abstract

We analyze non-perturbatively the one-dimensional Schrödinger equation describing the emission of electrons from a model metal surface by a classical oscillating electric field. Placing the metal in the half-space $x\leqslant 0$, the Schrödinger equation of the system is $i\partial _t\psi =-\frac{1}{2}\partial _x^2\psi +\Theta (x) (U-E x \cos \omega t)\psi$, $t>0$, $x\in {\mathbb {R}}$, where $\Theta (x)$ is the Heaviside function and $U>0$ is the effective confining potential (we choose units so that $m=e=\hbar =1$). The amplitude E of the external electric field and the frequency $\omega$ are arbitrary. We prove existence and uniqueness of classical solutions of the Schrödinger equation for general initial conditions $\psi (x,0)=f(x)$, $x\in {\mathbb {R}}$. When the initial condition is in $L^2$ the evolution is unitary and the wave function goes to zero at any fixed x as $t\rightarrow \infty$. To show this we prove a RAGE type theorem and show that the discrete spectrum of the quasienergy operator is empty. To obtain positive electron current we consider non-$L^2$ initial conditions containing an incoming beam from the left. The beam is partially reflected and partially transmitted for all $t>0$. For these initial conditions we show that the solution approaches in the large t limit a periodic state that satisfies an infinite set of equations formally derived, under the assumption that the solution is periodic by Faisal et al. (Phys Rev A 72:023412, 2005). Due to a number of pathological features of the Hamiltonian (among which unboundedness in the physical as well as the spatial Fourier domain) the existing methods to prove such results do not apply, and we introduce new, more general ones. The actual solution exhibits a very complex behavior, as seen both analytically and numerically. It shows a steep increase in the current as the frequency passes a threshold value $\omega =\omega _c$, with $\omega _c$ depending on the strength of the electric field. For small E, $\omega _c$ represents the threshold in the classical photoelectric effect, as described by Einstein’s theory.

Appendices

Laplace Transform Versus Discrete-Laplace Transform

In a way similar to the classical Poisson summation formula approach, working in distributions, taking a Laplace transform, which we denote by ${\mathcal {L}}$, followed by a discrete Fourier transform is related to a discrete-Laplace transform in the original variable, as seen below.

$$\begin{aligned} \frac{1}{2\pi }\sum _{n\in \mathbb {Z}}({\mathcal {L}}\psi )(-i\sigma -in\omega )\textrm{e}^{-i(\sigma +n\omega )r}&=\frac{1}{\omega }\left[ \psi (r)\Theta (r)+\sum _{k=1}^\infty \textrm{e}^{i\sigma \frac{2k\pi }{\omega }}\psi \left( r+\frac{2k\pi }{\omega }\right) \right] \nonumber \\&:=({\mathcal {P}}_\sigma \psi )(r) \end{aligned}$$

(A.1)

where $-\tfrac{\pi }{\omega }\le r<\tfrac{\pi }{\omega }$ and $\sigma \in [0,\omega )$.

To deduce this formula, we calculate

$$\begin{aligned}{} & {} \frac{1}{2\pi }\sum _{n\in \mathbb {Z}}({\mathcal {L}}\psi )(-i\sigma -in\omega )\textrm{e}^{-in\omega r}=\frac{1}{2\pi }\sum _{n\in \mathbb {Z}}\int _0^\infty e^{(i\sigma +in\omega )t}\psi (t)\textrm{e}^{-in\omega r}\, dt\nonumber \\{} & {} \quad =\frac{1}{2\pi }\sum _{n\in \mathbb {Z}}\left[ \int _0^{\pi /\omega } e^{(i\sigma +in\omega )t}\psi (t)\textrm{e}^{-in\omega r}\, dt +\sum _{k=1}^\infty \int _{(2k-1)\pi /\omega }^{(2k+1)\pi /\omega } e^{(i\sigma +in\omega )t}\psi (t)\textrm{e}^{-in\omega r}\, dt \right] \nonumber \\{} & {} \quad =\frac{1}{2\pi }\sum _{n\in \mathbb {Z}}\int _{-{\pi /\omega }}^{\pi /\omega }e^{i\sigma t}\psi (t) \Theta (t)\,e^{in\omega (t-r)}\, dt \nonumber \\{} & {} \qquad +\frac{1}{2\pi }\sum _{n\in \mathbb {Z}}\sum _{k=1}^\infty \int _{-\pi /\omega }^{\pi /\omega } e^{i\sigma (s+\frac{2k\pi }{\omega })} \psi (s+\frac{2k\pi }{\omega })e^{in\omega (s-r+\frac{2k\pi }{\omega })} \end{aligned}$$

(A.2)

where we let $t=\tfrac{2k\pi }{\omega }+s$. Using the fact that $\tfrac{1}{2\pi }\sum _ne^{in\omega (s-r)}=\frac{1}{\omega }\delta _{s-r}$ formula (A.1) follows.

Fig. 2

The normalized current $\frac{j}{k}$ at the interface (in atomic units, so $\frac{j}{k}$ is dimensionless) as a function of $\frac{t\omega }{2\pi }$ for $\omega =1.55\ \textrm{eV}$ and for the electric field: $E=25\, \textrm{V}\,\textrm{nm}^{-1}$. The dotted line is the graph of $\cos (\omega t)$ (not to scale)

Fig. 3

An average of the current after a number of periods as a function of $\omega -\omega _c$, for various values of the field: $E=3\ \textrm{V}\,\textrm{nm}^{-1}$ (blue), $E=10\ \textrm{V}\,\textrm{nm}^{-1}$ (red). For the sake of comparison, we have also plotted the asymptotic current predicted in [28] as dotted lines: green for $E=3\ \textrm{V}\,\textrm{nm}^{-1}$ and purple for $E=10\ \textrm{V}\,\textrm{nm}^{-1}$. All four curves are almost on top of each other. We see a sharp transition as $\omega$ crosses the critical frequency $\omega _c=U-\frac{k^2}{2}+\frac{E^2}{4\omega _c^2}$

Figures

As already mentioned in the introduction Eq. (1.5) is the underlying basic model used for the interpretation of experiments of electron emission from a metal surface irradiated by lasers of different frequencies [1, 4, 22, 24, 27,28,29,30,31,32,33,34,35,36,37]. This is so despite the fact that the system described by (1.5) is very idealized, both in the description of the metal and in the use of a classical electric field. The literature therefore contains many approximate qualitative solutions of (1.5) or some modification of it. Our analysis which proves the existence of physical solutions to (1.5) does not give a visualization of the form of such solutions. To do that requires carefully controlled numerical solutions. Figure 2 shows the complex behavior of the current at early times for large fields. Figure 3 shows the steep rise of the current as the frequency of the applied field crosses the field dependent critical frequency, which is the energy that is necessary for an electron to absorb in order to be extracted from the metal: it is the real solution to the cubic equation $\omega _c=U-\frac{k^2}{2}+\frac{E^2}{4\omega ^2_c}$ (the term $\frac{E^2}{4\omega _c^2}$ comes from the “Zitterbewegung” [40]). For small E, this reproduces the usual physical picture of the photoelectric effect.

The figures are obtained by solving the integral equation numerically for $\psi (x,t)$ with controlled approximations [38].