Electron pairing in nanostructures driven by an oscillating field

TL;DR

This paper theoretically demonstrates that strong high-frequency electromagnetic fields can induce electron confinement and pairing in nanostructures with repulsive potentials, revealing a novel mechanism for electron pairing.

Contribution

It introduces a new theoretical mechanism for electron pairing in nanostructures driven by oscillating fields, expanding understanding of electron behavior under high-frequency electromagnetic influence.

Findings

Electron confinement can occur in repulsive potentials under high-frequency fields.

Metastable bound electron states can form in such nanostructures.

Electron pairing may be facilitated by this confinement mechanism.

Abstract

It is shown theoretically that the confinement of an electron at a repulsive potential can exist in nanostructures subjected to a strong high-frequency electromagnetic field. As a result of the confinement, the metastable bound electron state of the repulsive potential appears. This effect can lead, particularly, to electron pairing in nanostructures containing conduction electrons with different effective masses.

Equations

$$\mathcal{H}(x,p)=\frac{1}{2m}{\left[p-\frac{e}{c}A(t)\right]}^2+U(x),$$

$$\mathcal{H}(x^{\prime },p^{\prime })=\frac{p^{\prime 2}}{2m}+U_0(x^{\prime })+\left[\sum _{n=1}^{\infty }{U}_n(x^{\prime })e^{in\omega t}+c.c.\right],$$

$$U_0(x^{\prime })=\frac{1}{2\pi }{\int }_{-\pi }^{\pi }U(x^{\prime }-x_0\sin \omega t)d(\omega t),$$

$$U_0(x^{\prime })=\frac{2e^2}{\pi \sqrt{R(x^{\prime })}}K\left(\sqrt{\frac{1}{2}-\frac{a^2+x^{\prime 2}-x_0^2}{2R(x^{\prime })}}\right),$$

$${\ddot{x}}^{\prime }+{\omega }_0^2{x}^{\prime }=\sum _{n=1}^{\infty }[F_n(x^{\prime })/m]e^{in\omega t}+c.c.,$$

$$\hat{U}(t)=\exp \left\{\frac{i}{\hslash }{\int }_{-\infty }^t\left[\frac{e}{mc}\mathbf{A}(\tau )\hat{\mathbf{p}}-\frac{e^2}{2m{c}^2}{A}^2(\tau )\right]d\tau \right\}.$$

$${\hat{\mathcal{H}}}^{\prime }=\frac{{\hat{\mathbf{p}}}^2}{2m}+U_0(\mathbf{r})+\left[\sum _{n=1}^{\infty }{U}_n(\mathbf{r})e^{in\omega t}+c.c.\right],$$

$$U_0(\mathbf{r})=\frac{1}{2\pi }{\int }_{-\pi }^{\pi }U\left(\mathbf{r}-\frac{e}{mc}{\int }_{-\infty }^t\mathbf{A}(\tau )d\tau \right)d(\omega t),$$

$$U_{0}(\rho)=\left\{\begin{array}[]{rl}({2e^{2}}/{\pi\rho_{0}})K\left({\rho}/{\rho_{0}}\right),&{\rho}/{\rho_{0}}\leq 1\\ \\ ({2e^{2}}/{\pi\rho})K\left({\rho_{0}}/{\rho}\right),&{\rho}/{\rho_{0}}\geq 1\end{array}\right.,$$

$$U_0({\mathbf{r}}_1,{\mathbf{r}}_2)={\int }_{-\pi }^{\pi }U\left({\mathbf{r}}_1-{\mathbf{r}}_2-\frac{e}{\overline{m}c}{\int }_{-\infty }^t\mathbf{A}(\tau )d\tau \right)\frac{d(\omega t)}{2\pi },$$

$$U_0(\overline{\rho })=\frac{2e^2}{\pi \sqrt{(\overline{\rho }+{\overline{\rho }}_0{)}^2+d^2}}K\left(\sqrt{\frac{4{\overline{\rho }}_0\overline{\rho }}{(\overline{\rho }+{\overline{\rho }}_0{)}^2+d^2}}\right),$$

Full text

I Introduction

Achievements in laser physics and microwave technique have made possible the use of an oscillating field as a tool to control physical properties of various systems GoldMan_14 ; Bukov_15 . Particularly, conduction electrons driven by a strong high-frequency field are actively studied to exploit features of composite field-matter states in various nanostructures, including superlattices Holthaus_92 ; Holthaus_95 , semiconductor quantum wells Kibis_12 ; Kibis_14 , quantum rings Kibis_11 ; Koshelev_2015 , graphene Kibis_10 ; Glazov_14 , etc. Among effects caused by an oscillating field, the field-induced stabilization of unstable systems (the dynamical stabilization) should be noted especially (see, e.g., Ref. Bukov_15, ). This effect is of fundamental nature and manifests itself in different areas of physics — from mechanical systems (the Kapitza pendulum Kapitza_51 ) to the trapping of particles (the Paul trap Paul_90 ). Although the dynamical stabilization has been known for a long time, related electronic phenomena in nanostructures still await for detailed analysis. Filling this gap, it was found that an oscillating field can induce the metastable bound electron states of various repulsive potentials in nanostructures. Particularly, the field-mediated electron pairing appears in nanostructures containing conduction electrons with different effective masses. The present article is devoted to the theoretical analysis of this phenomenon.

The article is organized as follows. In Sec. II, the one-dimensional (1D) classical dynamics of an electron in the presence of a repulsive potential and an oscillating field is considered. In Sec. III, the 1D theory is extended for a quantum multidimensional case and the field-induced electron pairing is discussed. The last section contains the conclusion and acknowledgments.

II Model

To clarify physical origin of the claimed effect, let us consider the classical dynamics of an electron confined inside a 1D nanostructure (quantum wire) irradiated by an electromagnetic wave linearly polarized along the wire. The classical Hamilton function of the electron in the presence of the repulsive potential $U(x)$ reads

[TABLE]

where $x$ is the electron coordinate along the quantum wire, $p$ is the generalized momentum of the electron, $e$ is the electron charge, $m$ is the effective electron mass in the nanostructure (which can differ from the electron mass in vacuum, $m_{e}$), $A(t)=(cE_{x}/\omega)\cos\omega t$ is the vector potential of the wave, $E_{x}$ is the electric field amplitude, and $\omega$ is the wave frequency. In the particular case of $U(x)=0$, which corresponds to the free electron driven by the oscillating field $A(t)$, the electron coordinate oscillates harmonically, $x(t)=x(0)-x_{0}\sin\omega t$, where $x_{0}=eE_{x}/m\omega^{2}$ is the amplitude of the oscillations. To solve the Hamilton problem in the general case of $U(x)\neq 0$, it is convenient to rewrite the Hamiltonian (1) in the reference frame of the oscillating electron, where its coordinate is $x^{\prime}=x+x_{0}\sin\omega t$. Following the conventional procedure of canonical transformation of the Hamiltonian (1) to the new variables $x^{\prime}$ and $p^{\,\prime}$ (see, e.g., Ref. Landau_01, ), one has to start from the Lagrangian ${\cal L}(x,\dot{x})=m\dot{x}^{2}/2+eA(t)\dot{x}/c-U(x)$ which is physically equivalent to the Hamiltonian (1). Substituting the electron coordinate $x=x^{\prime}-x_{0}\sin\omega t$ into the Lagrangian, it can be rewritten as ${\cal L}(x^{\prime},\dot{x}^{\prime})=m[\dot{x}^{\prime}-x_{0}\omega\cos\omega t]^{2}/2+eA(t)[\dot{x}^{\prime}-x_{0}\omega\cos\omega t]/c-U(x^{\prime}-x_{0}\sin\omega t)$. Therefore, the new generalized momentum $p^{\,\prime}=\partial{\cal L}(x^{\prime},\dot{x}^{\prime})/\partial\dot{x}^{\prime}$ corresponding to the new coordinate $x^{\prime}$ is $p^{\,\prime}=m\dot{x}^{\prime}$. As a result, the Hamiltonian (1) written in the new canonical variables $x^{\prime}$ and $p^{\,\prime}$ is ${\cal H}(x^{\prime},p^{\,\prime})=p^{\,\prime}\dot{x}^{\prime}-{\cal L}(x^{\prime},\dot{x}^{\prime})={p^{\,\prime\,2}}/{2m}+U(x^{\prime}-x_{0}\sin\omega t)+{m(x_{0}\omega\cos\omega t)^{2}}/{2}$. It should be noted that the last term of this Hamiltonian describes the energy shift arising from oscillations of the new reference frame. Since this term does not depend on the canonical variables $x^{\prime}$ and $p^{\,\prime}$, it does not effect on dynamics of the electron and will be omitted in what follows. The kinetic energy of the electron in the new reference frame, ${p^{\,\prime\,2}}/{2m}$, does not depend on the oscillating field, whereas the repulsive potential, $U(x^{\prime}-x_{0}\sin\omega t)$, oscillates with the field frequency $\omega$. Expanding this oscillating potential into the Fourier series, one can rewrite the Hamiltonian ${\cal H}(x^{\prime},p^{\,\prime})$ as

[TABLE]

where

[TABLE]

is the stationary part of the oscillating potential, and $U_{n}(x^{\prime})$ are the Fourier series coefficients of the potential. First of all, let us discuss qualitatively appearance of the stationary potential (3) which is responsible for the smooth motion of the electron. The most of physically relevant repulsive potentials — including, particularly, the Coulomb repulsion — are described by a barrier-like function $U(x)$, which has a maximum at the coordinate $x=0$ and decreases with increasing $|x|$. Correspondingly, the time-dependent function $U(x^{\prime}-x_{0}\sin\omega t)$ describes the moving potential barrier which is centered at the oscillating coordinate $x^{\prime}=x_{0}\sin\omega t$. During the period of the oscillations, this oscillating potential barrier spends most time around the two points, $x^{\prime}=\pm x_{0}$, where the oscillator’s velocity, $x_{0}\omega\cos\omega t$, is zero [see the green dashed lines in Fig. 1(a)]. As a consequence, the time-averaged potential (3) acquires a two-barrier structure with a local minimum at $x^{\prime}=0$ [see the blue solid line in Fig. 1(a)]. It should be noted that the local minimum of the integral (3) appears for any integrable barrier-like potential $U(x^{\prime}-x_{0}\sin\omega t)$ if the oscillation amplitude $x_{0}$ is large enough compared with the typical scale of the potential. In order to illustrate this, one can consider the model repulsive potential $U(x)={e^{2}}/{\sqrt{a^{2}+x^{2}}}$ which can be used, particularly, to describe the Coulomb interaction of an electron confined in a quantum wire with the effective thickness $a$. In this case, the potential (3) reads

[TABLE]

where $R(x^{\prime})=\sqrt{a^{2}+(x^{\prime}+x_{0})^{2}}\sqrt{a^{2}+(x^{\prime}-x_{0})^{2}}$, and $K(\xi)$ is the complete elliptical integral of the first kind. In accordance with the aforesaid, the potential (4) has a local minimum at $x^{\prime}=0$ if the ratio $\lambda=x_{0}/a$ is greater than the critical value $\lambda_{0}\approx 2.1$. The local minimum for $\lambda>\lambda_{0}$ is clearly seen in Fig. 1(b), where the potential (4) is plotted for different $\lambda$. As a consequence of the local minimum, the domain of attraction appears in the core of the repulsive potential.

To complete the analysis of the classical Hamilton problem (1), we have to consider the effect of the oscillating terms of the Hamiltonian (2) on the electron dynamics in the domain of attraction. Near the point of local minimum, $x^{\prime}=0$, the stationary potential (3) can be written as $U(x^{\prime})\approx m\omega^{2}_{0}{x}^{\prime 2}/2$, where $\omega_{0}$ is the eigenfrequency of free electron oscillations near the local minimum. Then the Hamilton equations corresponding to the Hamiltonian (2), $\dot{p}^{\,\prime}=-\partial{\cal{H}}/\partial x^{\prime}$ and $\dot{x}^{\prime}=\partial{\cal{H}}/\partial{p}^{\,\prime}$, lead to the equation of electron motion near the local minimum:

[TABLE]

where $F_{n}(x^{\prime})=-\partial U_{n}(x^{\prime})/\partial x^{\prime}$ are the Fourier components of the periodical force arising from the oscillating terms of the Hamiltonian (2). Physically, the dynamic equation (5) describes the usual forced oscillations of an electron under a periodical force with the harmonics $F_{n}(x^{\prime})$. If the frequency of the force is far from the resonance, $\omega\gg\omega_{0}$, the amplitudes of harmonics of the forced oscillations are substantially smaller than the typical length of the domain of attraction, $x_{0}$. Therefore, the high-frequency periodical force cannot expel the electron from the local minimum. It should be noted also that the coordinate dependence of the oscillating terms $F_{n}(x^{\prime})$ leads to the stationary ponderomotive force. However, the corresponding ponderomotive addition to the stationary potential (3), $\Delta U_{0}(x^{\prime})=\sum_{n=1}^{\infty}|F_{n}(x^{\prime})|^{2}/mn^{2}\omega^{2}$, is negligibly small for large frequencies $\omega$. Thus, the oscillating terms of the Hamiltonian (2) can be neglected if the frequency of the driving field, $\omega$, is large enough. In the high-frequency limit, the dynamics of an electron within the domain of attraction can be described solely by the stationary potential (3) which should be treated as an effective potential renormalized by the rapidly oscillating field. It follows from the aforesaid that the local minimum of the potential at $x^{\prime}=0$ corresponds to an electron bound at the repulsive potential with the binding energy $\Delta$ [see Fig. 1(a)]. In the laboratory reference frame, the bound electron state corresponds to the rapidly oscillating finite movement of the electron along the stable trajectory $x(t)=-x_{0}\sin\omega t$ which is confined near the potential peak [see Fig. 1(c)].

It follows from the aforesaid that the (meta)stable trajectory confined in the core of a 1D repulsive potential driven by an oscillating field (the bound electron state of the repulsive potential) exists if the oscillation amplitude $x_{0}$ is large enough compared with the typical scale of the repulsive potential $a$. This is why the bound state cannot be described directly within the established theory of motion in a rapidly oscillating field, which was elaborated before in the most general form for small oscillation amplitudes (see, e.g., Sec. 30 of Ref. Landau_01, ). As a consequence, the present analysis of the Hamilton problem (1) — which formally looks like an exercise in classical mechanics — contains nontrivial physics under consideration.

III Results and discussion

To go from the classical 1D problem (1) to the quantum multidimensional case, let us start from the Hamiltonian $\hat{\cal H}=[\hat{\mathbf{p}}-e\mathbf{A}(t)/c]^{2}/2m+U(\mathbf{r})$, where $\hat{\mathbf{p}}=(\hat{p}_{x},\hat{p}_{y},\hat{p}_{z})$ is the momentum operator, $U(\mathbf{r})$ is the repulsive potential, $\mathbf{r}=(x,y,z)$ is the radius vector of an electron, and $\mathbf{A}(t)=(A_{x},A_{y},A_{z})$ is the vector potential of a homogeneous field which oscillates with the period $T=2\pi/\omega$. In order to transform the laboratory reference frame into the rest frame of the oscillating electron, we have to apply the Kramers-Henneberger unitary transformation Kramers_52 ; Henneberger_68 ,

[TABLE]

Then the transformed Hamiltonian reads $\hat{\cal H}^{\prime}=\hat{U}^{\dagger}(t)\hat{\cal H}\hat{U}(t)-i\hbar\hat{U}^{\dagger}(t)\partial_{t}\hat{U}(t)=\hat{\mathbf{p}}^{2}/2m+U(\mathbf{r}-[{e}/{mc}]\int^{\,t}_{-\infty}\mathbf{A}(\tau)d\tau)$. Expanding the oscillating potential $U(\mathbf{r},t)$ into a Fourier series, the transformed Hamiltonian can be written as

[TABLE]

where the stationary part of the potential is

[TABLE]

and $U_{n}(\mathbf{r})$ are the Fourier coefficients of the oscillating potential. In the theory of laser-atom interaction, the renormalized potentials of kind (7) are known as the Kramers-Henneberger (KH) potentials Henneberger_68 ; Delone_2000 . Certainly, the KH potential (7) turns into the 1D potential (3) if $\mathbf{r}=(x,0,0)$ and $\mathbf{A}(t)=([cE_{x}/\omega]\cos\omega t,0,0)$. Solving the 1D Schrödinger problem with the two-barrier potential $U_{0}(x)$, one can found the bound state confined between the two barriers [see Fig. 1(d)]. Within the quantum description, the bound state is nonstationary and can decay in two ways: (i) the electron tunneling through the potential barriers; (ii) the photon absorption arising from the oscillating terms of the Hamiltonian (6) [the corresponding electron transitions from the bound state are marked by the arrows in Fig. 1(d)]. However, the increase of the oscillation amplitude $x_{0}$ increases both the barrier height and the barrier width [see Fig. 1(b)]. Therefore, the probability of the tunnel decay can be very small if the oscillation amplitude $x_{0}$ is large enough. In this case, the ground bound state confined near the local minimum of the potential $U_{0}(x)$ can be described approximately by the stationary wave function $\psi_{0}(x)$ with the energy $\varepsilon_{0}\sim\hbar\omega_{0}$ [see Fig. 1(d)]. Regarding the decay caused by absorption of the field energy, it corresponds to the electron transitions from the ground bound state with matrix elements that are very small under the condition $\omega\gg\omega_{0}$. As a result, the two mentioned decay mechanisms can be neglected if both the oscillation amplitude $x_{0}=eE_{x}/m\omega^{2}$ and the field frequency $\omega$ are large enough. These two conditions can always be satisfied simultaneously since one can vary the field strength $E_{x}$ and the field frequency $\omega$ independently. Thus, both the classical description of the problem and the quantum one lead to the same effect: The metastable bound states of a repulsive potential driven by an oscillating field appear if the field is both strong and high-frequency.

Let us extend the consideration for the 2D case corresponding to a conduction electron confined inside a quantum well irradiated by an electromagnetic wave. If the quantum well lies in the $(x,y)$ plane and the wave propagates along the $z$ axis, the vector potential of the wave inside the well can be written as $\mathbf{A}(t)=([cE_{x}/\omega]\cos\omega t,\,[cE_{y}/\omega]\cos[\omega t-\phi],\,0)$, where $E_{x,y}$ are amplitudes of the wave along the $x,y$ axes, and $\phi$ is the wave phase. For definiteness, let us restrict the consideration by the 2D Coulomb repulsive potential, $U(\mathbf{r})=e^{2}/\sqrt{x^{2}+y^{2}}$. Substituting this 2D potential into Eq. (7), we arrive at the repulsive potential renormalized by the oscillating field, $U_{0}(\mathbf{r})$. If the oscillating field is linearly polarized along the $x$ axis ($E_{y}=0$), the KH potential (7) takes the form (4) with the replacement $a\rightarrow y$. Since this potential has a two-peak structure without a local minimum, it cannot confine movement of an electron along the $y$ axis [see Fig. 2(a)]. As a consequence, the 2D bound states are absent in the case of a linearly polarized field. However, they can be induced by a circularly polarized field. Indeed, if the wave is circularly polarized (the amplitudes are $E_{x}=E_{y}=E_{0}$ and the phase is $\phi=\pi/2$), the KH potential (7) reads

[TABLE]

where $\rho=\sqrt{x^{2}+y^{2}}$ is the polar radius vector, and $\rho_{0}=|e|E_{0}/m\omega^{2}$ is the radius of the circular trajectory of a free electron induced by the circularly polarized field (the oscillation amplitude). Since the potential (8) has a crater-like structure with a local minimum at $\rho=0$ [see Fig. 2(b)], it results in the total confinement of the electron along the $x,y$ axes and, therefore, can produce the bound states. It should be noted that the bound states of the potential (8) exist for any oscillation amplitude $\rho_{0}$, since the pure 2D Coulomb potential has singularity at $\rho=0$ (i.e., its typical scale is $a=0$). The renormalized potentials plotted in Fig. 2 also have weak singularities which look smoothed in the plots. Namely, Eqs. (4) and (8) involve the elliptical integral $K(\xi)$, which has the logarithmic singularity at $\xi=1$. In addition, there are the root singularities at the potential peaks plotted in Fig. 2(a).

The one-electron theory developed above can be easily generalized to describe the interaction of two electrons. In this case, the KH potential (7) reads

[TABLE]

where $U(\mathbf{r}_{1}-\mathbf{r}_{2})$ is the initial potential of repulsive electron-electron interaction, $\overline{m}=m_{1}m_{2}/(m_{1}-m_{2})$ is the reduced electron mass, and $\mathbf{r}_{j}=(x_{j},y_{j},z_{j})$ and $m_{j}$ with $j=1,2$ are the radius vectors and effective masses of the interacting electrons, respectively. It follows from Eq. (9) that the oscillating field renormalizes the interaction potential $U(\mathbf{r}_{1}-\mathbf{r}_{2})$ if the electron masses are not equal, $m_{1}\neq m_{2}$. Otherwise, the field does not change distance between the electrons and, correspondingly, cannot modify their interaction. Thus, the field-induced electron-electron attraction can appear in nanostructures containing conduction electrons with different effective masses. As an example of such a nanostructure, let us consider a quantum well consisting of two layers ($1$ and $2$) filled with a 2D electron gas (2DEG), which are fabricated using different semiconductor materials and isolated from each other by a buffer layer with thickness $d$ [see Fig. 3(a)]. Then the Coulomb interaction of two electrons from the separated 2D layers $1$ and $2$ can be described by the potential $U(\mathbf{r}_{1}-\mathbf{r}_{2})=e^{2}/\sqrt{(x_{1}-x_{2})^{2}+(y_{1}-y_{2})^{2}+d^{2}}$. If the quantum well is irradiated by a circularly polarized electromagnetic wave [see Fig. 3(a)], the substitution of this initial potential into Eq. (9) yields the renormalized Coulomb potential

[TABLE]

where $\overline{\rho}=\sqrt{(x_{1}-x_{2})^{2}+(y_{1}-y_{2})^{2}}$ and $\overline{\rho}_{0}=|e/\overline{m}\,|(E_{0}/\omega^{2})$. The potential (10) has a local minimum at $\overline{\rho}=0$ if the ratio $\lambda=\overline{\rho}_{0}/d$ is greater than the critical value $\lambda_{0}\approx 1.4$. Physically, this minimum corresponds to the paired electrons from layers $1$ and $2$ with binding energy $\Delta$ (see Fig. 3).

The scattering of electrons can destroy the field-induced electron oscillations which are responsible for the considered effect. Therefore, the condition $\omega\tau\gg 1$ is crucial for the pairing, where $\tau$ is the mean free time of conduction electrons. In state-of-the-art semiconductor quantum wells, this condition can be satisfied for field frequencies of the microwave range, $\omega\sim 10^{11}$ rad/s. Assuming that the reduced mass is $\overline{m}\sim 0.1\,m_{e}$ and the buffer thickness is $d\sim$ nm, the electron pair corresponding to the local minimum of the renormalized potential (10) has the binding energy of room-temperature scale, $\Delta\sim 10^{-2}$ eV, and the typical size $\overline{\rho}_{0}\sim 10$ nm for irradiation intensity $I\sim 10^{-2}$ W/cm2. Thus, the electron pairing and related phenomena can be high-temperature for relatively weak irradiation.

IV Conclusion

It is demonstrated that a strong high-frequency electromagnetic field can induce the metastable bound states of various repulsive potentials in nanostructures. This leads, particularly, to the electron pairing in nanostructures containing electrons with different effective masses. The discussed effect strongly depends on the field polarization. Namely, a linearly polarized field can induce electron pairs only in 1D nanostructures, whereas a circularly polarized field induces them also in 2D nanostructures. Therefore, semiconductor quantum wells irradiated by a circularly polarized field look most appropriate for experimental observation of the pairs. Among a variety of possible effects caused by the electron pairing, superconductivity mediated by an oscillating field should be noted especially. To describe this prospective phenomenon correctly, the solved two-electron problem should be generalized for the many-electron case. However, such an extension of the developed theory goes beyond the scope of the present article and will be done elsewhere.

The work was partially supported by Russian Foundation for Basic Research (project 17-02-00053) and Ministry of Science and High Education of Russian Federation (project 3.4573.2017/6.7).