Effect of quantized conductivity on the anomalous photon emission radiated from atomic-size point contacts

TL;DR

This study investigates how quantized conductance in atomic-size point contacts influences anomalous photon emission, revealing a strong dependence on conductance steps and an unexpected exponential decay with electrical power.

Contribution

It introduces an analytical model linking quantized conductance to photon emission via bremsstrahlung from hot electrons, aligning well with experimental observations.

Findings

Photon emission depends on conductance quantization.

Light intensity decreases exponentially with electrical power.

Model explains emission through hot electron bremsstrahlung.

Abstract

We observe anomalous visible to near-infrared electromagnetic radiation emitted from electrically driven atomic-size point contacts. We show that the number of photons released strongly depends on the quantized conductance steps of the contact. Counter-intuitively, the light intensity features an exponential decay dependence with the injected electrical power. We propose an analytical model for the light emission considering an out-of-equilibrium electron distribution. We treat photon emission as bremsstrahlung process resulting from hot electrons colliding with the metal boundary and a find qualitative accord with the experimental data.

Equations

$$C_e\frac{\partial T_e}{\partial t}=\nabla ({\kappa }_e\nabla T_e)-g(T_e-T_L)=0,$$

$$T_e^2=T_L^2+T_0^2\sum _{n=0}^{+\infty }{a}_n{J}_0({\mu }_{n}r/R_0)\sinh ({\lambda }_n(L_c-z)/R_0)$$

$$a_n$$

$$T_{e[h\to +\infty ]}^2\approx T_L^2+T_0^2\sum _{n=0}^{+\infty }\frac{J_1({\mu }_n^{\prime }{\zeta }_N)}{[{\mu }_n^{\prime }{J}_1({\mu }_n^{\prime }){]}^2}\tanh \left(\frac{{\mu }_n^{\prime }{L}_c}{R_0}\right)$$

$$T_{e[h\to 0]}^2\approx T_L^2+T_0^2\left[\frac{{\zeta }_N{L}_c}{R_0}+\sum _{n=0}^{+\infty }\frac{J_1({\mu }_n^{\prime \prime }{\zeta }_N)}{[{\mu }_n^{\prime \prime }{J}_0({\mu }_n^{\prime \prime }){]}^2}\tanh \left(\frac{{\mu }_n^{\prime \prime }{L}_c}{R_0}\right)\right]$$

$$\frac{d{I}_{\omega }}{dS}=-{\alpha }_{\omega }{I}_{\omega }+{\alpha }_{\omega }{B}_{\omega }(T_e)$$

$$\frac{d{N}_{ph}(\omega )}{dSd\omega dt}\approx \frac{4e^2{\epsilon }_F^2}{3{\pi }^3{\hslash }^3{c}^3}\frac{1}{\exp (\hslash \omega /k_B{T}_e)-1}$$

Full text

An atomic-scale contact formed between two macroscopic electrodes has been a canonical testbed for understanding the quantum nature of electron and heat transport at this ultimate length scale Agraït et al. (2003); Cui et al. (2017). Central to the discussion is the role of dissipation, which must be taken into account in any finite conductance externally driven electrical device. In the phenomenological treatment of quantum transport of a one-dimensional conductor Datta (1995); van Houten and Beenaker (1996), the collision-free transmission imposes the dissipation to occur away from the ballistic channel, i.e., in the reservoirs contacting the conductor in a distance equals to the inelastic electron mean free path. Even when describing electron flow from first-principle quantum kinetics Green and Das (2005), inelastic coupling to the interface region guarantees the conservation of the charge required for any open geometry Das and Green (2009). It is generally understood that the main channel for energy dissipation in a out-of-equilibrium ballistic contact occurs via a coupling to the phonon bath and the local generation of heat Todorov (1998). Population of the phonon distribution has been confirmed in voltage-dependent conductance spectroscopies Agraït et al. (2002); Frederiksen et al. (2004) and in weak-field current fluctuations analysis Reznikov et al. (1995); Kumar et al. (2012). Such inherent fluctuations of the charge current is necessarily accompanied by the emission of transverse electromagnetic field. For low driving voltages, i.e., in the linear regime, the radio-frequency photons may feature nonclassical statistics depending on the voltage applied Beenakker and Schomerus (2004); Lebedev et al. (2010) and the temperature Fulga et al. (2010). This has been experimentally measured on tunnel junctions at cryogenic temperature and emitting in the GHz frequency range Zakka-Bajjani et al. (2010); Forgues et al. (2016). For larger driving bias, the situation complicates and the standard fluctuation-dissipation theory is no longer applicable Rogovin and Scalapino (1974). Electron-electron scattering must be included in the dissipation as it contributes to elevate the temperature of the Fermi-Dirac distribution. In turns, the electron and the phonon subsystems are not longer thermalized Fedorovich et al. (2000); Green et al. (2004); D’Agosta et al. (2006); Chen et al. (2014). Here, we identify the presence of a corollary dissipation mechanism. We show that the high-temperature nonequilibrium electron gas formed in an externally-driven atomic-scale contact is dissipating energy by emitting electromagnetic radiation tailing in the visible part of the spectrum. We observe an increase of the photon rate every time a transmission channel governing the electronic transport closes. Opposite to the conventional exchange of energy to a thermal bath and to standard electroluminescence, the light intensity emitted by the contact inversely scales with the electrical power dissipated nearby the ballistic conductor. We treat photon emission as spontaneous bremsstrahlung radiation emerging when hot electrons collide with the metal wall to explain the experimental results.

In this work, atomic-size electron channels are formed by electromigrating Au constrictions Park et al. (1999). Quantized steps of the conductance in units of the quantum of conductance $G_{0}=2e^{2}/h$ is the signature of a ballistic transport, where $e$ is the electron charge and $h$ is Planck’s constant Strachan et al. (2005); Hoffmann et al. (2008). We detect the light activity during the electromigration process by capturing photon emission with two cross-polarized single photon counting avalanche photodiodes (APD). The quantum efficiency of the APDs sets the detected spectral range to high energy photons spanning the visible and near-infrared region (ca. 1.2 eV-3.1 eV). The detail of the experimental methodology is provided in Supplemental Material sup . Figure 1 shows two examples of time traces recorded towards the end of the electromigration process leading to the electrical failure of the devices. The applied bias is constant at $V_{\rm b}=0.8$ V in (a) and $V_{\rm b}=0.7$ V in (b). The step-like evolution of the normalized conductance $G/G_{0}$ suggests that the devices undergo change of transport mechanism from ballistic to tunnel; the abrupt passage is taking place at $t=27.8$ s in Fig. 1(a) and at $t=64.2$ s in Fig. 1(b). The relative large values of $V_{\rm b}$ reduced the probability of $G$ to explore the smallest integer numbers $N\times G_{0}$ Fujii et al. (2004), and the last measured steps are at $N=4$ in both cases. Figure 1 also displays the simultaneously acquired photon counts measured by the APDs. The graphs show an unambiguous correlation between the conductance steps and the light emission. Photons emitted in the detected spectral window are measured as soon as $G\sim 5G_{0}$ with a constant rate during the conductance plateaus. A ten fold increase of the number of photons is concomitant to the closing of an electron transmission channel identified by the short excursion of $G$ at $4G_{0}$ in both examples. Immediately after the rupture of the device, the tunneling junctions in Fig. 1(a) and Fig. 1(b) have conductances of $G=6\times 10^{-3}G_{0}$ and $G=1.4\times 10^{-3}G_{0}$, respectively. In both cases, the photon rate drops when transport changes from ballistic to tunnel. During the entire time traces and the excursion of $G$ in the different transport regimes, the photon energy is always greater than the bias energy. The quantum inequality $h\nu\leq eV_{\rm b}$ is systematically violated, where $\nu$ is the frequency of the photon. We can thus exclude emission processes akin to inelastic tunneling Parzefall et al. (2017) to explain the light activity. This is further confirmed by the similarity of the signals detected by the two cross-polarized APDs. Inelastic coupling to raditaive surface plasmon modes are expected to show a polarization anisotropy Parzefall et al. (2015); Uskov et al. (2017).

Early observations of an overbias emission in an atomic contact has been shown to follow a power law relationship with the electrical power injected in the device. For a given value of the conductance, and regardless of the emission mechanism at play, increasing the current by changing the electrical bias drastically boosted the detected photon counts Schull et al. (2009); Malinowski et al. (2016). In the present experiment, the voltage bias is maintained at a constant value during the last moment of electromigration. The excursion of the conductance in the ballistic regime allows us to monitor the evolution of the photon counts with the electrical power dissipated in the contact without changing the driving conditions, and to obtain a deeper insight on the emission mechanism. When transport channels are closing, the electrical power dissipated in the contact reduces concomitantly. Figure 2 shows a semi-logarithmic plot of the measured light intensity (red circles) versus the electrical power $P$ inferred from Fig. 1(a) before the electrical failure using the relation $P=I_{\rm b}^{2}\times G^{-1}$, where $I_{\rm b}$ is the current flowing through the contact. The graphs unequivocally demonstrates that the photon counts is maximum at lower electrical power and features an exponential decay with $P$ before saturating aroudn the lowest electrical power. This trend is opposite to measurements performed at constant $G$ Schull et al. (2009); Malinowski et al. (2016). This trend has been consistently confirmed with other devices (see Supplemental Material sup )

In the following, we develop a theoretical framework to understand the relationship between the number of channels opened for electron transmission and the optical activity emitted at an overbiased photon energy. The delivered electric power scales with the number of transport channels $N$ as $P_{N}=V_{\rm b}^{2}NG_{0}$. The radius of the $N$-th channel can be estimated as $r_{N}\approx Nr_{1}$, where $r_{1}\approx\lambda_{F}/4$ is the characteristic radius of the first quantum channel Datta (1995) and $\lambda_{F}$ is the Fermi wavelength of the ballistic electrons. As a result, both current and power densities are increasing in proportion of $1/N$ when channels are closing and results in a rise of the peak electron temperature within an area located at the end of the transport channel. Photons may be emitted by such a nonequilibrium distribution if electrons interact with the surface Fedorovich et al. (2000). Qualitatively, this is expected to be the origin of the measured increasing photon yield at energies higher than the bias when the constriction explores the lower values of conductance quanta.

Below, we present a simple qualitative model which illustrates the above consideration. We assume that the electric current is transported by a channel connected to the drain contact through an interconnection region which we model by a cylinder of radius $R_{0}$ and finite length $L_{c}$ along the $z$ axis. The electron subsystem in this interconnection region is out of equilibrium due to the fast heating with arriving and colliding quasi-ballistic electrons. We assume a local electron temperature $T_{e}$, which is well above the homogeneous lattice temperature $T_{L}$ Chen et al. (2014). We treat the heat transport problem in this interconnection region in the frame of the two-temperature model, assuming the lattice temperature $T_{L}$ does not change significantly along $L_{c}$.

In accordance with the experimental conditions, we seek a steady state temperature distribution

[TABLE]

where $g$ is the electron-lattice coupling constant, $\kappa_{e}$ and $C_{e}$ are the electron thermal conductivity and heat capacity. As the natural boundary conditions, we assume the electron temperature in the drain electrode far from the contact to be at the equilibrium with the lattice temperature, so that $T_{e}(z=L_{c})=T_{L}$. At the front end of the contact $z=0$, the electric power is assumed to be homogeneously deposited in a spot with the radius $r_{N}$ of opened quantum transport channels $N$, so that the boundary heat flux is $\kappa_{e}(\partial T_{e}/\partial z)_{z=0}=-p_{N}\Theta(r-r_{N})$, where $p_{N}=P_{N}/\pi r_{N}^{2}=(G_{0}V_{\rm b}^{2}/\pi r_{1}^{2})N^{-1}$, and $\Theta(u)$ is the step function defined as $\Theta(u>0)=0$ and $\Theta(u<0)=1$. At the side wall of the cylinder $r=R_{0}$, we assume that the heat flux is determined by the energy loss of the electrons in collisions with the metal boundary, in analogy with the Fedorovich-Tomchuk mechanism Belotskii and Tomchuk (1990) (See Supplemental Material sup for the discussion). In this framework, the heat flux at the side wall is proportional to the squared temperature $\kappa_{e}(\partial T_{e}/\partial r)=-B(T_{e}^{2}-T_{L}^{2})$. Here B is a proportionality coefficient. As far as $\kappa_{e}=b\times T_{e}$, we find $(b/2)(\partial T_{e}^{2}/\partial r)=-B(T_{e}^{2}-T_{L}^{2})$. Finally, we set $h=2BR_{0}/b$ and find $\partial T_{e}^{2}/\partial r+(T_{e}^{2}-T_{L}^{2})h/R_{0}=0$ with $h$ is the dimensionless parameter characterizing the electron energy exchange rate at the side wall. The limit $h=0$ corresponds to zero heat flux at the side wall and $\partial T_{e}^{2}/\partial r$ vanishes. Large values of the parameter $h>>1$ describe a fast energy exchange leading to rapid establishment of the equilibrium between electrons and the lattice, i.e. $T_{e}=T_{\rm L}$.

Under the above assumptions, the steady-state temperature distribution is written as

[TABLE]

The coefficients $a_{n}$ are

[TABLE]

Here $\zeta_{N}=r_{N}/R_{0}$, $\mu_{n}$ is the root of the equation $\mu_{n}J_{1}(\mu_{n})/J_{0}(\mu_{n})=h$ with $n=0,1,2$, …and the eigen values of the problem along the $z$ axis are $\lambda_{n}=(\mu_{n}^{2}+R_{0}^{2}/L_{0}^{2})^{1/2}$. $L_{0}$ is a characteristic length defined in the Supplemental Material sup . The coefficient $T_{0}^{2}$ before the sign of the sum in Eq. 2 doesn’t depend on the channel’s number, $k_{B}^{2}T_{0}^{2}=96(eV_{\rm b})^{2}E_{F}/\pi^{4}\hbar N_{e}v_{F}^{2}\lambda_{F}\tau$. Within the accepted above approximation for the electron transport relaxation time, $\tau\sim R_{0}/v_{F}$, using the well-known relation $N_{e}=k_{F}^{3}/3\pi^{2}$, we find $k_{B}T_{0}=(6/\pi^{2})(eV_{\rm b})(\lambda_{F}/R_{0})^{1/2}$, which for the applied voltage $V_{\rm b}=0.7$ V results in $T_{0}[{\rm 10^{3}~{}K}]\approx 4.92\sqrt{\lambda_{F}/R_{0}}$.

According to Eq. 2, the maximum temperature is in the center of a hot spot situated at the front end of the cylinder ($z=0$, $r=0$). When the side wall heat transfer is fast ($h\rightarrow+\infty$), the boundary condition reads $T_{e}(r=R_{0})=T_{L}$, and $\mu_{n}\rightarrow\mu^{\prime}_{n}$ becomes zeros of the Bessel function $J_{0}(\mu^{\prime}_{n})=0$ with $n=1,2~{}\ldots$. Because $L_{0}/R_{0}\gg 1$ sup , $\lambda_{n}\approx\mu^{\prime}_{n}$ and the maximum temperature in this limiting case is

[TABLE]

In the opposite case of vanishing energy exchange at the side wall, $h<<1$ or even $h=0$, the roots $\mu_{n}\rightarrow\mu^{\prime\prime}_{n}$ consist in a zeroth root $\mu^{\prime\prime}_{0}\approx\sqrt{2h}$ (for which the corresponding eigen value is $\lambda_{0}\approx(2h+R_{0}^{2}/L_{0}^{2})^{1/2}\rightarrow R_{0}/L_{0}$), and the sequence of the roots of the first-order Bessel function $J_{1}(\mu^{\prime\prime}_{n})\approx 0$, with $n=1,2\ldots$, and the maximum temperature can be estimated as

[TABLE]

Figure 3(a) and (b) displays the dependence of $T_{e}$ versus the number of quantum transport channels at the fixed length $L_{c}=R_{0}/3$ and various radii $R_{0}$ ranging from $R_{0}=2\lambda_{F}$ (8 quantum channels) to $R_{0}=4\lambda_{F}$ (16 quantum channels available) and for the two heat exchange scenarios. Clearly, $T_{e}$ drops when more channels are available for electron conduction. For each channel number $N$, the peak temperature is greater with a decrease of the interconnection radius. The dependencies are more pronounced when $h<<1$ and for smaller radii and become smoother with an increase in $R_{0}$. The dependence of $T_{e}$ with fixed $R_{0}$ and varying $L_{c}$ is treated in the Supplemental Material sup . We can draw a first important counter intuitive conclusion: regardless of the mechanism dictating the inelastic energy loss at the wall of the constriction, the electronic temperature drops when increasing the electrical power dissipated ($P_{N}=V_{\rm b}^{2}G_{0}N$). Figure 3 agrees with experimental trend of Fig. 2(b) provided we can link the electron temperature $T_{e}$ to the number of photons emitted by the contact.

In a bulk metal, non-equilibrium electrons loose their energy mostly during non-radiative collisions with phonons or impurity atoms. Primary photons are emitted as a result of correspondent bremsstrahlung processes. Establishment of thermal equilibrium of photons is a result of complicated kinetics of free-free electron transitions consisting in emission and absorption bremsstrahlung processes as well as Compton effect Kompaneets (1957); Zeldovich (1975). In a simplified diffusion approximation, photon emission can be treated through the radiation transfer equation

[TABLE]

where $I_{\omega}$ is the radiation intensity spectrum, $\alpha_{\omega}$ is the absorption coefficient at given frequency, and $B_{\omega}(T_{e})$ is the equilibrium radiation intensity given by Plank’s law. In a bulk metal, when the optical skin depth $\alpha_{\omega}^{-1}$ is much smaller than the characteristic dimension, Eq.(6) results in the Kirchhoff’s law, the emissivity is given by $j_{\omega}=\alpha_{\omega}B_{\omega}(T_{e})$. For the interconnection considered here, the region of elevated electron temperature is approximately $r_{N}\approx N\lambda_{F}/4<<\alpha_{\omega}^{-1}$. As a result, an equilibrium photon distribution cannot be established within the interconnection region, and the thermal emission of photons is primary guided by a bremsstrahlung process rather than the Kirchhoff’s law. The derivation of this bremsstrahlung mechansim is detailed in the Supplemental Material sup . We finally find the total photon rate

[TABLE]

One can easily check that for the temperature domain of interest, i.e. for peak temperatures below $3.5\times 10^{3}$K (see Fig. 3), the integrated bremsstrahlung photon yield is well approximated by the following relation, $dN_{ph}/dSdt\approx 6.13\times 10^{22}\times T^{2}\exp(-1/T)$ cm*-2s-1*, where the normalized temperature $T=k_{B}T_{e}/\hbar\omega$.

The results of our calculation of the bremsstrahlung photon yield are shown in Fig. 3(c) and (d) as a function of $N$ and for both limiting values of the parameter $h$ governing the heat transfer at the side wall of the system. The data correspond to the calculated peak temperatures shown in Fig. 3(a) and (b). The photon yield is calculated by taking into account the APD spectral efficiency (See Supplemental Material sup ). One can find that these dependencies at sufficiently small values of radius $R_{0}$ qualitatively recover the experimental data shown in Fig. 2 notably the exponential decay of the photon counts with $N$. The dependence of the photon yield with fixed $R_{0}$ and varying $L_{c}$ is treated in the Supplemental Material sup . We use the model described above to match the experimental dependence of the photon counts versus electrical power delivered in the contact again considering the two extreme heat exchange scenarii at the side wall. The open blue and magenta triangles in Fig. 2 are the results of the model considering a short cylinder length $L_{c}=\lambda_{F}/4$ and $R_{0}=2\lambda_{F}$. We estimate the total radiation area as $S\approx\pi R_{0}^{2}+2\pi R_{0}L_{c}=4.84\times 10^{-14}$ cm2. The overall detection efficiency is experimentally unknown, and we leave this as free parameter $\eta$. To fit the maximum calculated yield with the experimental value for the forth quantum channel, we set $\eta\simeq 0.43$, which means the collection efficiency of the microscope is about 43%. This is a reasonable value considering the objective’s numerical aperture and the presence of a glass substrate concentrating the emitted photons in the high index medium. The measured photon counts are bounded by the two limiting cases of the model indicating the qualitative agreement with the model (see dotted curves in Fig. 2). Hence electron thermalization at the side wall is an important process to consider.

The past research in atomic-size point contacts has provided a firm understanding of the radiofrequency electromagnetic response occurring when the system is driven in the linear regime of low bias voltages (e.g. mV range). Recent reports suggested that electrons transported through the contact with a large kinetic energy ($\sim$eV) may unveil new nonlinear mechanisms of light emission. Our findings showed that photons with energies much higher than the kinetic energy of the electron are emitted during the formation of the atomic contact when the transport becomes ballistic. By assuming an non-equilibrium electron distribution near the contact, we derive a model relating the electron temperature and the photon yield to the number of quantum channel. Within this model, we assume the presence of a small interconnection region where energy exchange is mainly guided by electrons colliding at the side wall. An anomalous electromagnetic response is emitted in an over-bias spectral domain as a result of a bremsstrahlung process occurring at the boundary of the interconnection region. We derive the quantum-mechanical formula for the rate of this bremsstrahlung photon emission, which in the limit $\hbar\omega\rightarrow 0$ coincides with the classical relation. We find a qualitative agreement between the estimated emission rates and the results of our measurements. Currently, the dynamic leading to the formation of the contact remains too rapid to interrogate the spectrum of the emitted photons. Once we have a reliable strategy to stabilize the number of transport channels, these findings will contribute to the development of integrated electrically-driven optical light sources at atomic length scales.

The work was funded by the European Research Council Grant Agreement 306772, the CNRS/RFBR collaborative research program number 1493 (RFBR-17-58-150007), the COST Action MP1403 “Nanoscale Quantum Optics”, the Regional Excellence funding scheme (project APEX). A.U. and I.S are thankful to Russian Science Foundation (Grant 17-19-01532) and A.B. for access to the nanofabrication facility ARCEN Carnot financed by the Regional council of Burgundy and la Délégation Régionale à la Recherche et à la Technologie.