Controlling spontaneous emission via electronic correlations in transparent metals

TL;DR

This paper explores how electronic correlations in transparent metals can be used to control and enhance spontaneous emission from embedded quantum emitters, paving the way for tunable solid-state single-photon sources.

Contribution

It introduces a novel mechanism linking electronic correlations to light emission control in correlated transparent metals, with potential applications in quantum photonics.

Findings

Existence of a critical correlation strength for transparency in visible light.

Strong correlations can significantly enhance light-matter coupling.

Electronic correlations enable tunable control of spontaneous emission.

Abstract

We study the spontaneous emission of agglomerates of two-level quantum emitters embedded in a correlated transparent metal. The characteristic emission energy corresponds to the splitting between ground and excited states of a neutral, nonmagnetic molecular impurity (F color center), while correlations are due to the existence of narrow bands in the metal. This is the case of transition metal oxides with an ABO3 Perovskite structure, such as SrVO3 and CaVO3, where oxygen vacancies are responsible for the emission of visible light, while the correlated metallic nature arises from the partial filling of a band with mostly d-orbital character. For these systems we put forward an interdisciplinary, tunable mechanism to control light emission governed by electronic correlations. We show that not only there exists a critical value for the correlation strength above which the metal becomes transparent in the visible, but also that strong correlations can lead to a remarkable enhancement of the light-matter coupling. By unveiling the role of electronic correlations in spontaneous emission, our findings set the basis for the design of controllable, solid-state, single-photon sources in correlated transparent metals.

Equations

$${\omega }_{pl}^2\left(U\right)={\omega }_{pl}^2\left(0\right)\left[1-{\left(U/U_c\right)}^2\right].$$

$$H$$

$$E_{\sigma }^{\pm }=\frac{{\mathcal{E}}_{1,\sigma }+{\mathcal{E}}_{2,\sigma }\pm \sqrt{({\mathcal{E}}_{1,\sigma }-{\mathcal{E}}_{2,\sigma }{)}^2+4{\mathcal{T}}_{\sigma }^2}}{2}.$$

$$\hslash {\omega }_0=2t-V+\frac{U}{2}.$$

$$\left|⟨g\left|H_{int}\right|he⟩\right|=\sqrt{\frac{\hslash {\omega }_0}{2{\epsilon }_0{ϵ}_r\left({\omega }_0,U\right)V}}{\widehat{ϵ}}_{\lambda }\cdot ⟨g|\vec{\mu }|he⟩,$$

$${ϵ}_1\left(\omega ,U\right)$$

$${ϵ}_2\left(\omega ,U\right)$$

$$G_U^R\left(\omega ,\mathbf{k}\right)=\frac{1}{{ϵ}_1\left(\omega ,U\right){\omega }^2-c^2|\mathbf{k}{|}^2+i{ϵ}_2\left(\omega ,U\right){\omega }^2},$$

$${\Gamma }_i(U)=\left(\frac{2\pi }{{\hslash }^2}\right)\left[\frac{g^2}{{ϵ}_r\left({\omega }_0,U\right)}\right]{\int }_0^{\infty }\frac{dk}{{\pi }^2}{k}^2\left\{\frac{2ck{ϵ}_2\left(\omega ,U\right){\omega }^2}{{\left[{ϵ}_1\left(\omega ,U\right){\omega }^2-c^2{k}^2\right]}^2+{ϵ}_2^2\left(\omega ,U\right){\omega }^4}\right\}=\eta ({\omega }_0,U){\Gamma }_0.$$

$${\Gamma }_{cav}={\Gamma }_0{\int }_0^{\infty }d\omega F_{cav}(\omega ){\left|\frac{{\epsilon }_{in}(\omega )+2}{3}\right|}^2{\eta }_{in}(\omega )\delta (\omega -{\omega }_0),$$

$$F_{cav}(\omega )=\frac{3\pi c^3}{\omega }\sum _{q=1}^{\infty }\mathcal{I}m\left[\frac{1}{V_{1,0,q}{\omega }_{1,0,q}({\omega }_{1,0,q}-\omega )}\right],$$

$$\sqrt{{\epsilon }_{in}}{\psi }_1(k_{in}a){\xi }_1^{\prime }(k_{out}a)-\sqrt{{\epsilon }_{out}}{\psi }_1^{\prime }(k_{in}a){\xi }_1(k_{out}a)=0,$$

Full text

Progresses in the understanding of light-matter interactions have enabled new mechanisms of controlling light emission, propagation and extraction on chip lipson ; lodahl . Since the pioneering work of Purcell it has been known that spontaneous emission is not only determined by the emitters’ (atoms, molecules, and quantum dots) intrinsic electronic levels but is also influenced by the surrounding electromagnetic environment  Purcell-46 . This discovery has paved the way to the development of several different strategies to modify and tailor the spontaneous emission of quantum emitters, typically relying on different choices of material platforms and/or physical mechanisms that affect spontaneous emission. Photonic crystals goban , dielectric microcavities senellart , nanophotonic waveguides arcari , graphene-based structures tielrooij ; wilton2015 , dielectric regmi and plasmonic tanaka ; muskens nanoantennas, and metamaterials noginov ; roth ; liu are examples of different material platforms that have been explored to that purpose. In particular, metamaterials and 2D metasurfaces are versatile systems to achieving far-field emission patterns with desired properties, such as divergence and directionality bohn , and even completely suppressing light emission wilton2013 . Metallic structures have also been employed to modify spontaneous emission, with plasmonic resonances being designed to increase the electric field and enhance the local density of states at the emitters location pelton ; makarova . Regarding the different physical mechanisms to modify light emission one may cite the Fano effect livro-tiago ; tiago2017 and critical phenomena. Indeed there are evidences, both theoretical juanjo1 ; juanjo2 ; szilard and experimental carminati , that phase transitions affect spontaneous emission in a crucial way, thus allowing for the optical determination of critical exponents via the Purcell factor mbsn . Altogether these different routes have allowed for many technological applications, such as solar cells oregan , molecular imaging moerner ; vallee , and single-photon sources michler .

In the present letter we propose a twofold, interdisciplinary strategy to control spontaneous emission that combines: i) an alternative material platform, transition metal oxides; and ii) a novel physical mechanism, electronic correlations. More specifically, we consider color center agglomerates incorporated into a correlated transparent metal, namely ABO3 Perovskites, such as SrVO3 and CaVO3, where oxygen vacancies are responsible for emission of visible light zhang2016 and electronic correlations arise from the narrowness of the partially filled B- ion, $d$-bands. Our choice for this particular material platform is motivated by the fact that transition metal oxides exhibit unique electrical and optical properties, such as excellent carrier mobilities, mechanical stress tolerance, compatibility with organic dielectric and photoactive materials, and high optical transparency xu . These materials are also versatile and cost-effective to many applications in optoelectronics, such as electronic circuits, flexible organic light-emitting diode (OLED) displays, and solar cells xu .

A key issue while engineering metallic materials for optoelectronic applications relies on an adequate combination of high optical transparency and high electrical conductivity. This is usually challenging since the plasmon frequency of the best conducting metals is typically of the order of $\hbar\omega_{pl}\sim 10eV$, well above the visible transparency window $1.8eV\leq\hbar\omega\leq 3.1eV$, see Fig. 1. Hence achieving transparency in good conductors requires minimizing the plasmon frequency $\omega_{pl}^{2}=e^{2}n/\varepsilon_{0}m^{*}$ (given in terms of the electric charge $e$, the dielectric constant in free space $\varepsilon_{0}$, the electronic density in the metal $n$, of the effective mass $m^{*}$), while keeping the electrical conductivity $\sigma=e^{2}\tau(n/m^{*})$ (with $\tau$ the scattering time) large enough by a judicious choice of the ratio $n/m^{*}$. The tradicional strategy to adjust the ratio $n/m^{*}$ relies on a trade-off between increasing $n$, as in wide-bandgap semiconductors, and/or decreasing $m^{*}$ via heavy doping, as in the case of tin-doped indium oxide (ITO). Nevertheless, although ITO exhibits the largest conductivity for materials of its class (horizontal dashed line in Fig. 1), it is still much smaller than the conductivities of transition metals oxides, such as SrVO3 and CaVO3. These materials have very good transmission efficiency of the order of $80\%$, except in the blue region zhang2016 , besides being excellent conductors (see Fig. 1), remarkable properties on which we capitalize to achieve enhanced and controllable spontaneous emission.

In what follows we propose for the first time an alternative physical mechanism, based on electronic correlations, for: i) reducing the plasmon frequency of transition metal oxides, making them transparent in the visible; and ii) controlling and enhancing the spontaneous emission of visible light. We start by recalling that moderate to strong electronic correlations, represented by the Hubbard interaction $U$, renormalizes the effective mass of carriers according to $m^{*}(U)/m^{*}(0)=1/[1-\left(U/U_{c}\right)^{2}]$, where $U_{c}\approx 5$eV is the critical value for the metal-to-Mott-insulator transition in these systems Brinkmann-Rice . The plasmon frequency then modifies as

[TABLE]

Such renormalization has indeed been observed in SrVO3 by ARPES SrVO3-ARPES , where $U\approx 3.2$eV, and $\hbar\omega_{pl}(U)<1.33$eV, i.e. below the lower visible edge zhang2016 , see Fig. 1.

Having established the transparency of our material platform, in the following we specify the properties of the two-level quantum emitter. The crystal structure of ABO3 perovskites consists of corner shared BO6 octahedra, with the transition metal B, inside each octahedron, and with the cation A, at the center of a unit cell of coordination 12 ( Fig. 2). The removal of one oxygen atom from the structure, i.e. the introduction of an oxygen vacancy (O-vac), causes the trapping of two electrons, at the vacancy, each coming from a neaby B-ion. This scenario is consistent with LDA+DMFT calculations on oxygen deficient supercells in SrVO3, and has been confirmed by ARPES for the case of UV irradiated SrVO3 crystals SrVO3-ARPES-II . The large Coulomb repulsion imposes that the spins of these electrons be anti-parallel due to virtual tunneling processes. As a result, the ground state of an O-vac is a neutral, nonmagnetic spin-single state (an F color center), where the two electrons fill up the two molecular orbitals (bonding and anti-bonding) obtained from the hybridization of the original $e_{g}$ orbitals of the neighboring B ions, Fig. 2.

The electronic structure at the O-vac discussed above can be calculated by the following two-site Hamiltonian

[TABLE]

where ${\cal E}_{0}$ is the onsite energy, $t$ is the direct tunnelling, $U$ is the onsite Coulomb repulsion that opposes double occupancy, $V$ is the nearest neighbour Coulomb repulsion that opposes direct tunneling, and $n_{i,\sigma}=c^{\dagger}_{i,\sigma}c_{i,\sigma}$ with $c^{\dagger}_{i,\sigma}$ and $c_{i,\sigma}$ corresponding to the creation and anihilation operators for the electrons at the two B-ion, $d$-orbitals. A mean field treatment of the Coulomb interactions allows us to replace $\sum_{\sigma=\uparrow,\downarrow}\left[{\cal E}_{0}n_{i,\sigma}+Un_{i,\sigma}n_{i,-\sigma}\right]$ by $\sum_{\sigma=\uparrow,\downarrow}{\cal E}_{i,\sigma}n_{i,\sigma}$, in terms of ${\cal E}_{i,\sigma}={\cal E}_{0}+U\langle n_{i,-\sigma}\rangle$, and also to replace $\sum_{\sigma,\sigma^{\prime}}[t\delta_{\sigma,\sigma^{\prime}}c^{\dagger}_{1,\sigma}c_{2,\sigma^{\prime}}-Vn_{1,\sigma}n_{2,\sigma^{\prime}}]$ by $\sum_{\sigma=\uparrow,\downarrow}{\cal T}_{\sigma}c^{\dagger}_{1,\sigma}c_{2,\sigma}$, in terms of ${\cal T}_{\sigma}=t-V\sum_{\sigma^{\prime}}\langle c_{1,\sigma}c^{\dagger}_{2,\sigma^{\prime}}\rangle$. The Hamiltonian is now quadratic and can be diagonalized providing eigenvalues

[TABLE]

The Hilbert space for the O-vac problem is spanned by four basis states $\left|1,\uparrow\downarrow;2,0\right>,\left|1,\uparrow;2,\downarrow\right>,\left|1,\downarrow;2,\uparrow\right>,\left|1,0;2,\uparrow\downarrow\right>$, in terms of which $\langle n_{i,-\sigma}\rangle$ and $\langle c_{1,\sigma}c^{\dagger}_{2,\sigma^{\prime}}\rangle$ must be calculated. If the two electrons occupy the same molecular orbital, $\langle n_{i,-\sigma}\rangle=1/2$ (projected subspace of doubly-occupied states), whereas if the two electrons occupy a different molecular orbital each, $\langle n_{i,-\sigma}\rangle=0$ (projected subspace of singly-occupied states). Analogously, $\langle c_{1,\sigma}c^{\dagger}_{2,\sigma^{\prime}}\rangle=\delta_{\sigma,\sigma^{\prime}}/2$ (projected subspace of mixed singly- and double-occupied states). For $U>2(2t-V)$ the ground state $|g\rangle$, corresponds to a singlet configuration with the two electrons occupying a different molecular orbital each. There is a low lying singlet exited state $|le\rangle$, corresponding to a doubly-occupied bonding orbital, and a high energy singlet excited state $|he\rangle$, corresponding to a doubly-occupied anti-bonding orbital. The $|g\rangle$ and $|he\rangle$ states are shown in Fig. 2 and their splitting is

[TABLE]

By using typical values for the hopping, $t\approx 1.0$eV, and Hubbard parameters, $U\approx 3.2$eV, and $V\approx 0.8$eV, consistent with GW+DMFT and LDA+DMFT calculations for SrVO3 LDA-DMFT-SrVO3 , we obtain $\hbar\omega_{0}\approx 2.8$eV for the characteristic emitter’s energy, which corresponds to light emission/absorption in the blue, exactly where SrVO3 transmits poorly zhang2016 . Similar values for $\hbar\omega_{0}$ can also be found for other ABO3 Perovskites, using similar values for $t,U,V$, satisfying $V<t<U$, all within the visible, as observed in photoluminescence experiments in a variety of disordered Perovskites such as BaTiO3, CaTiO3, PbTiO3, LiNbO3, SrWO4, besides SrVO3 itself PL-Disordered-ABO3 .

We are now ready to calculate the spontaneous emission rate $\Gamma$ for a dilute and homogeneous collection of O-vac impurity states embedded in a correlated metal whose optical properties depend on the correlation strength $U$ through the dielectric function $\varepsilon(\omega,U)=\varepsilon_{0}\epsilon_{r}(\omega,U)$. To this end we calculate the electric-dipole matrix element between the singlet, ground $|g\rangle$, and highest excited $|he\rangle$ states

[TABLE]

describing the coupling between an O-vacancy, with nonzero electric-dipole moment $\vec{\mu}$ between the bonding and anti-bonding orbitals, and radiation with polarization $\hat{\epsilon_{\lambda}}$. Here $V$ is the volume and $\epsilon_{r}\left(\omega,U\right)=\epsilon_{1}\left(\omega,U\right)+i\epsilon_{2}\left(\omega,U\right)$, with

[TABLE]

is the relative permitivity of a lossy, $\gamma\neq 0$, medium. For isotropic systems with electric dipole moment $\vec{\mu}=\mu\hat{z}$, we can write $\left|\hat{\epsilon}_{\lambda}\cdot\langle g|\vec{\mu}|he\rangle\right|^{2}=\mu^{2}\left|\left\langle\hat{\epsilon}_{\lambda}\cdot\hat{z}\right\rangle\right|^{2}=\mu^{2}/3$. By defining the quantity $g^{2}=\hbar\omega_{0}\mu^{2}/6\varepsilon_{0}$, and recalling the definition of the spectral distribution of electromagnetic modes in the medium, ${\cal{A}}_{U}(\omega,|{\bf k}|)=-(2c|{\bf k}|/\pi){\cal{I}}m[G^{R}_{U}\left(\omega,{\bf k}\right)]$, where retarded photon propagator is AGD

[TABLE]

we can use Fermi’s golden rule to calculate the spontaneous emission rate for the $i$-th isolated, single O-vac, in terms of the value in free space, $\Gamma_{0}=(2\pi/\hbar^{2})g^{2}[\omega_{0}^{2}/\pi^{2}c^{3}]$, as

[TABLE]

This result generalizes the one obtained for the decay rate of excited atoms in absorbing dielectric insulators Barnett , to the case of correlated metallic systems. The real, $\epsilon_{1}(\omega_{0},U)=\eta^{2}(\omega_{0},U)-\kappa^{2}(\omega_{0},U)$, and imaginary, $\epsilon_{2}(\omega_{0},U)=2\eta(\omega_{0},U)\kappa(\omega_{0},U)$, parts of the relative permitivity are given in terms of the refraction index, $\eta(\omega,U)$, and extinction coefficient, $\kappa(\omega,U)$, of the correlated metal. The integral over $k$ was done by extending the integration from $[0,\infty)$ to $(-\infty,\infty)$, and closing the contour of integration in the upper-half of the complex $k$ plane. Note that, since $\eta(\omega_{0},U)<1$ inside the metal, the spontaneous emission rate $\Gamma_{i}$ for the $i$-th isolated O-vac is smaller than the result in free space $\Gamma_{0}$ and the role of $\eta(\omega_{0},U)\neq 0$, which encodes electronic correlations, is to allow for transmission in the visible, forbidden for $\omega_{0}<\omega_{pl}(0)$, i.e., in the absence of correlations.

The highly diffusive character of O-vacs in oxides, which naturally occur even in the purest ABO3 samples, allows for their migration and easy accumulation near grain boundaries Schie2012 , their binding to foreign dopants Arora2017 , or their segregation at other defects Nyman2012 . Alternatively, O-vac rich regions can be engineered in a controlled way via microwave irradiation Microwave-Irradiation or pulsed laser deposition PLD , forming thermodynamically stable agglomerates of micrometer sizes containing a large number of color centers. In order to model such agglomerates we consider a collection of electric-dipoles, $\vec{\mu}$, enclosed by a spherical cavity of radius $a$, which acts as a boundary between a region rich in O-vacs and the bulk metal, Fig. 3. The local fields on a given dipole, due to the presence of all other dipoles, is accounted for by the usual Lorentz local-field factor, ${\cal{L}}_{cav}=\left|(\varepsilon(\omega_{0})+2)/3\right|$ Lorentz . Most importantly, spontaneous emission will now occur due to the coupling to the cavity’s electromagnetic modes. The dipole-rich region can be seen as an inclusion, characterized by a dielectric constant, $\varepsilon_{in}$, and the enclosing metal as a host, with dielectric constant, $\varepsilon_{out}$. The mismatch $\varepsilon_{in}\neq\varepsilon_{out}$, allows for the reflection of the emitted radiation back into the center of the cavity, transforming the dipole-rich region into a Mie resonator. In this case, the SE rate corresponds to the convolution Haroche

[TABLE]

where the Purcell cavity-enhancement factor $F_{cav}(\omega)$ is Bonod

[TABLE]

and $V_{j,m_{z},q}$ are the mode volumes for the $(j,m_{z},q)$ cavity modes corresponding to the complex valued eigenfrequencies $\omega_{j,m_{z},q}=\omega^{\prime}_{j,m_{z},q}+i\omega^{\prime\prime}_{j,m_{z},q}$. For dipoles along the $\hat{z}$ direction only the $j=1$ and $m_{z}=0$ contributions are relevant and the associated frequencies are solutions to the equation

[TABLE]

where $k_{in,out}=\omega/c_{in,out}$, with $c_{in,out}$ being the speed of light inside and outside the cavity, and $\psi_{1}(x)=xj_{1}(x)$ and $\xi_{1}=xh_{1}^{(1)}(x)$, are written in terms of the $j_{1}(x)$ spherical Bessel function and the $h_{1}^{(1)}=j_{1}(x)+iy_{1}(x)$ spherical Hankel funcion, with the prime indicating derivative with respect to its argument. The mode volumes, $V_{1,0,q}$, need not be identical to the physical cavity volume, $V_{cav}=4\pi a^{3}/3$, and, in fact, are a decreasing function of $q$ Bonod . The cavity modes are, in turn, characterized by a discrete set of frequencies, $\omega^{\prime}_{1,0,q}$, that increase with $q=1,2,\dots$, and inverse lifetimes, $\omega^{\prime\prime}_{1,0,q}$, that decrease with $q=1,2,\dots$ Bonod .

In Fig. 4 the normalized SE rate $\Gamma/\Gamma_{0}$ is shown, as a function of $\omega_{0}/\omega_{pl}$ (left) and of $U/U_{c}$ (right) for: i) a single emitter in a nearly lossless transparent metal (red dashed line); and ii) an agglomerate of emitters confined to an O-vac rich spherical cavity, regarded as an optically active inclusion in an otherwise transparent metallic host (solid blue line). The case of SrVO3, with $U\approx 3.2$eV, $\hbar\omega_{pl}<1.33$eV and $\hbar\omega_{0}\approx 2.8$eV, is represented by black arrows at $\omega_{0}/\omega_{pl}=2.2$ (left) and $U/U_{c}=0.67$ (right). The real part of the cavity frequencies, $\omega^{\prime}_{1,0,q}$, are determined by the ratios: i) $\varepsilon_{in}/\varepsilon_{out}\neq 1$; and ii) $k_{in}a\sim 1$. If the emitter’s frequency, $\omega_{0}$, coincides with one of the cavity modes’ frequencies, $\omega^{\prime}_{1,0,q}$, i.e., if they are on-resonance, then spontaneous emission can be strongly enhanced. On the other hand, if the emitter and the cavity modes are off-resonance, then spontaneous emission is strongly suppressed to values even smaller than half the one in free space, see Fig. 4. If we recall that $\omega_{0}(t,V,U)$ and $\omega_{pl}(U)$ are functions of $t,V$ and $U$, it is clear that, even the smallest variations in any of such parameters, especially $U$, could switch the on- and off- resonance situations, causing the emitter to blink in a controlled way.

A number of strategies are known to dynamically vary the values of the parameters $t,V$ and $U$, thus allowing for the tuning of the resonances in a controlled way. For instance, the Hubbard $U$ parameter can be decreased by almost $10\%$ on femtosecond timescales via laser driving in correlated materials dejean . Dynamically decreasing $U$ would not only red-shift the emitter’s frequency $\omega_{0}(t,V,U)$, but would, at the same time, blue-shift the plasmon frequency $\omega_{pl}(U)$, producing, simultaneously, an off-resonance situation while reducing the refraction index of the material, $\eta(U)\rightarrow 0$. Alternatively, with $U$ fixed one could also use mechanical strain, normal, $\varepsilon_{ij}$, or shear, $\gamma_{ij}$, with $i,j=x,y,z$, to control spontaneous emission. In ABO3 oxides strain can be used to modify both the B-O-B and B-B bond angles, through the rotation of the oxygen octahedra Rotation-Octahedra , which leads to changes in the hopping parameter $t$ and positions of the $e_{g}$ levels. In this case, although the plasmon, $\omega_{pl}$, and cavity mode, $\omega_{1,0,q}$, frequencies are kept unchanged, the emitter’s frequency, $\omega_{0}(t,V,U)$, can be tuned via strain, which modifies $t(\varepsilon_{ij},\gamma_{ij})$.

In conclusion, we have investigated spontaneous emission in transparent metals subject to electronic correlations. We have demonstrated that there exists a critical value for the correlation strength $U$, above which spontaneous emission is not only allowed but is also strongly enhanced due to resonant coupling between the emitter’s electric dipole moment to long-lived electromagnetic modes inside an optically cavity filled up with two-level color centers (oxygen vacancies agglomerates). The situations on- and off- resonance can be tuned in a controlled way either through variations of the correlation strength and/or by applying mechanical strain/stress, thus strongly enhancing or suppressing spontaneous emission. Altogether, our results suggest concrete and feasible routes towards the external control of spontaneous emission in metallic systems that may be applied to solid state single photon sources to produce photons on demand.