Dynamical Coulomb blockade theory of plasmon-mediated light emission from a tunnel junction
Fei Xu, Cecilia Holmqvist, Gianluca Rastelli, Wolfgang Belzig
DDynamical Coulomb blockade theory of plasmon-mediated light emission from atunnel junction
F. Xu, C. Holmqvist, G. Rastelli, and W. Belzig Fachbereich Physik, Universit¨at Konstanz, D-78457 Konstanz, Germany Department of Physics and Electrical Engineering, Linnaeus University, 39182 Kalmar, Sweden (Dated: December 6, 2016)Inelastic tunneling of electrons can generate the emission of photons with energies intuitivelylimited by the applied bias voltage. However, experiments indicate that more complex processesinvolving the interaction of electrons with plasmon polaritons lead to photon emission with overbiasenergies. We recently proposed a model of this observation in Phys. Rev. Lett. , 066801 (2014),in analogy to the dynamical Coulomb blockade, originally developed for treating the electromagneticenvironment in mesoscopic circuits. This model describes the correlated tunneling of two electronsinteracting with a local plasmon-polariton mode, represented by a resonant circuit, and showsthat the overbias emission is due to the non-Gaussian fluctuations. Here we extend our modelto study the overbias emission at finite temperature. We find that the thermal smearing stronglymasks the overbias emission. Hence, the detection of the correlated tunneling processes requirestemperatures k B T much lower than the bias energy eV and the plasmon energy (cid:126) ω , a conditionwhich is fortunately realized experimentally. I. INTRODUCTION
Electron transport through a nano system displays,due to the quantum nature of the underlying elementaryprocesses, a current that exhibits quantum noise withzero-point fluctuations . As a quantum object, the cur-rent is associated to a time-dependent operator ˆ I ( t ) inthe Heisenberg representation. Hence, the noise spectraldensity S ( ω ) = (cid:82) dte iωt (cid:104) ˆ I (0) ˆ I ( t ) (cid:105) acquires a frequency-antisymmetric component S ( ω ) (cid:54) = S ( − ω ) because of thenoncommuting current operators at different times. Thisasymmetry can actually be accessed by coupling the sys-tem to a detector . The result is that the positive andnegative branches of S ( ω ) are related to the emission andabsorption spectrum, respectively. Concerning the emis-sion processes, if the source of noise is the system biasedby a voltage V , intuitively one expects that the maxi-mum energy available for the tunneling electron is eV ,and, thus, the energy of an emitted photon is limited to eV as well, as shown by several experiments and theoret-ical investigations . Such inelastic effects in tunnelingjunctions are interesting because they can help to revealunusual phenomena like electron-electron correlation andelectron-plasmon effects.In regard to experimental measurements and realiza-tions of current noise detection, one of the first proposalswas a quantum tunneling detector consisting of a doublequantum dot (DQD) coupled to the leads of a nearbymesoscopic conductor , in which the inelastic currentthrough the DQD measures the equilibrium and nonequi-librium fluctuations in the conductor .Additionally, the light emission of electrons tunnelingfrom a scanning tunneling microscope (STM) to a metal-lic surface has already been studied and used as a probe ofthe shot noise at optical frequencies for many years .Using a single electron scattering picture and at zero b) | ✏ | < eV a) SPP
SPP
FIG. 1. Sketch of electron tunneling processes. (a) Oneelectron tunnels through the barrier and excites the surfaceplasmon-polariton (SPP), which eventually emits a photonwith energy | (cid:15) | < eV . (b) The two coherent electrons tunnelthrough the barrier, creating an overbias SPP excitation andleading to the overbias light emission with energy | (cid:15) | > eV . temperature of the system, the Pauli principle blocks in-elastic tunneling transitions with energy exchange largerthan the energy difference between the two Fermi seas,consisting of noninteracting electrons in the leads. Theemitted light spectrum is then limited in frequency bythe bias voltage according to (cid:126) ω < eV . i.e., the detec-tor signals are in the sub-bias energy range E < eV ; seeFig. 1(a).However, some experiments reveal the unexpectedfeature of light emitted at energy exceeding the biasvoltage (cid:126) ω > eV . Such an overbias spectrum appearsas reminiscent of the surface plasmon-polariton (SPP)modes which can be also observed via other methods.Using essentially energy considerations, such a processcan be attributed to two simultaneously tunneling elec-trons providing enough energy to explain the overbiasemission . Similar findings have also been reportedin photon emission from Josephson junctions andmolecular films with fluorescent emission of pho-tons with energies above the threshold energy. In orderto understand these diverse systems, a detailed under-standing of the electron tunneling processes involved is a r X i v : . [ c ond - m a t . m e s - h a ll ] D ec necessary .In a previous Letter [30], we developed a theoreticalframework for the description of the plasmon-mediatedlight emission by a tunnel contact based on dynamicalCoulomb blockade. In qualitative terms, in an elemen-tal tunneling event, an electron gains energy eV at biasvoltage V but must pay a charging cost of E c ∼ e /C with C the junction capacitance. Hence, after tunnel-ing, a nonequilibrium situation occurs since the chargeon the junction and the charge imposed by the voltagesource are different. Now, when an impedance is con-nected in series to the tunnel junction, it allows us todischarge and dissipate energy, thus, reducing Coulombcharging effects. In other words, the electromagnetic en-vironment of the junction crucially affects the charge tun-neling events. The effect on the tunneling is captured bythe probability P ( E ) of emitting an energy amount E to the electromagnetic environment. The so-called P ( E )function is related to the spectral density of voltage fluc-tuations, which in turn is determined by the impedanceof the environment. Going beyond the simple tunneling events, this frame-work captures the coherent two-electron tunneling pro-cesses, in which each electron contributes an energy E i (cid:46) eV ( i = 1 ,
2) but the overall process creates anexcitation in the broadened SPP spectrum with an en-ergy exceeding the bias voltage E + E > eV , as shownin Fig. 1(b). Afterwards, the relaxation of the SPP’senergy finally leads to the overbias light emission. Bymodeling the SPP as a broadened, damped resonator, atzero temperature we have quantitatively reproduced theexperimentally observed bias-voltage-dependent emissionspectrum [29].Here, we extend our model described in Ref. [30] toinclude a finite temperature in the general expressionfor the tunneling rate. First, we confirm that the non-Gaussian voltage fluctuations in the tunnel junction ex-plain the light emission with energy above the bias volt-age, (cid:126) ω > eV , in the limit of low temperature. Sec-ond, we provide a quantitative estimation for the typicaltemperature above which overbias emission is masked bythermal effects.Indeed, finite temperature affects either the rate asso-ciated to the Gaussian voltage fluctuations or the rateassociated to the non-Gaussian voltage fluctuations. Forthe Gaussian rate, we find that increasing the tempera-ture gradually smears out the sharp boundary at emis-sion energy E = eV which occurs in the limit of vanish-ing temperature. For the non-Gaussian rate, finite tem-perature smooths the characteristic cusp of the overbiasemission which is obtained at zero temperature. Sucheffects are prominent even in the relatively low temper-ature regime, namely k B T ∼ − (cid:126) Ω with Ω ∼ ω , theaverage position of the SPP spectrum, or Ω ∼ η , itsbroadening. These results point out that the overbiasemission spectrum is sensitive to finite temperature ef-fects. However, remarkably, the non-Gaussian rate canstill represent the leading term in the overbias range E > eV for sufficiently low temperatures. Hence, byanalyzing the temperature dependence, the bias voltagedependence and their interplay for the individual rates,i.e. the Gaussian and the non-Gaussian one, we discusshow to distinguish finite temperature effects from the ex-pected “zero-temperature” overbias emission.The structure of the paper is as follows. We describethe model and the theoretical methods based on theKeldysh action in Sec. II as well as the expression forthe detection rate. In Sec. III, we calculate the total rateformed by two separate contributions, i.e., the Gaussianpart and the non-Gaussian part, and analyze the ratebehavior as a function of temperature and voltage bias.We discuss our conclusions in Sec. IV. Details of the ratederivation are given in the Appendix.
II. MODEL
We model the tunneling from the STM tip to thesurface in an electromagnetic environment, according tostandard DCB theory , as the circuit diagram de-picted in Fig. 2.The tunneling is described by a tunnel conductorthat has a dimensionless conductance g c = R Q /R c with R Q = h/ e and R c being the quantum and tunnel-ing resistances, respectively. The junction is coupledto a damped LC circuit, modeled by an impedance z ω = iz ωω / ( ω − ω + iωη ), where ω = 1 / √ LC isthe resonance frequency of the SPP mode, η = 1 /RC models the damping, and z = (cid:112) L/C/R Q is the scaledcharacteristic impedance. The interaction between thetunnel junction and the SPP occurs in this model viathe dynamical voltage fluctuations on the node betweenthe tunnel junction and the LRC circuit. These volt-age fluctuations can be expressed by the phase variable ϕ ( t ) = e (cid:126) (cid:82) t −∞ dtV ( t (cid:48) ).For the photon detection, we choose a simple two-levelsystem with energy difference (cid:15) and transition probability T to absorb or emit photons. Formally the system canbe described by a Hamiltonian H detec = ( (cid:15) + αV ) σ z / T σ x with the unperturbed eigenstates |±(cid:105) with energies ± (cid:15)/
2, respectively. The coupling α between the STMjunction and the photon detector is set to be weak, sincethe photon detectors in a typical experiment are far awayfrom the junction. We can calculate the transition ratefrom the transition probability P −→ + ( t ) = |(cid:104)− ( t ) | + (cid:105)| to lowest order in the coupling T . Using Fermi’s goldenrule and setting (cid:126) = 1, the transition rate at energy (cid:15) inthe detector due to the fluctuations of ϕ ( t ) readsΓ( (cid:15) ) = |T | (cid:90) dt (cid:104) e iαϕ ( t ) e − iαϕ (0) (cid:105) e i(cid:15)t . (1)This rate formula corresponds to emission or absorptionfor (cid:15) > (cid:15) <
0, respectively. In this work we studyonly the absorption rate, and therefore we have to con-sider only negative energies (cid:15) <
0. Our central theoretical T ✏ ↵ L C R R c ' FIG. 2. Sketch of a STM contact with bias voltage V . Theelectrons interact via the SPP mode that is mimicked by theLRC resonant circuit. Photons emitted from this tunnel junc-tion are absorbed by the detector, i.e. a two-level system withenergy separation (cid:15) , leading to the absorption process char-acterized by the transition probability T . The coupling α between the detector and the tunnel system is weak in con-cordance with the experiment . task is the calculation of Γ( (cid:15) <
0) to the lowest order inthe detector coupling constant, i.e., α .We employ the path integral method to evaluate (cid:104) e iαϕ ( t ) e − iαϕ (0) (cid:105) . Using the Keldysh actions of the con-ductor, S c , and the circuit, S e , the correlator can be rep-resented as (cid:104) e iαϕ ( t ) e − iαϕ (0) (cid:105) = (cid:82) D [Φ] exp {− i S e [Φ] − i S c [Φ]+ iα [ − ϕ + (0) + ϕ − ( t )] } , (2)where the two-component phase Φ = ( φ, χ ) T with φ =( ϕ + + ϕ − ) / χ = ϕ + − ϕ − , and the real fields ϕ ± ( t )are defined on the forward and backward Keldysh con-tours, respectively. Later in the Keldysh action, the realfields can be written as ϕ ± ω = φ ω ± χ ω in frequencyspace.The action of the damped LC oscillator acting as theenvironment on the tunnel conductor, is quadratic in thefields and given by S e = (cid:90) dω π Φ T − ω A ω Φ ω , A ω = − i (cid:18) − ωz − ω ωz ω W ( ω ) (cid:60){ z ω } (cid:19) , with W ( ω ) = ω coth( ω/ T ). Here T denotes the tem-perature and we have set k B = 1. The action S c can beexpressed in terms of Keldysh Green’s functions ˇ G L,R forthe free electrons on the left ( L ) and right ( R ) sides ofthe tunneling barrier : S c = i g c (cid:90) dtdt (cid:48) Tr { ˇ G L ( t, t (cid:48) ) , ˇ G R ( t (cid:48) − t ) } (3)in the tunneling limit g c (cid:28) . With the help of the equilibrium Keldysh Green’s functionˇ G ( ω ) = (cid:18) − f ( ω ) 2 f ( ω )2[1 − f ( ω )] 2 f ( ω ) − (cid:19) , (4)containing the Fermi function f ( ω ) = [exp( ω/T ) + 1] − ,we can write ˇ G R ( ω ) = ˇ G ( ω − eV ) and hence ˇ G ( t ) = (cid:82) dω exp( − iωt ) ˇ G ( ω ) / π . Again using the Fourier trans-formation, we write ˇ G L ( t, t (cid:48) ) = ˇ U † ( t ) ˇ G ( t − t (cid:48) ) ˇ U ( t (cid:48) ) withthe counting fields introduced asˇ U ( t ) = (cid:32) e − iϕ + ( t ) e − iϕ − ( t ) (cid:33) . (5)Due to the nonquadratic contribution to the action ofthe conductor S c in Eq. (3), the correlator cannot becalculated exactly and we need an approximation scheme.Here, we use the cumulant expansion for the action S c bywhich we obtain the resultΓ( (cid:15) ) = Γ G ( (cid:15) ) + Γ nG ( (cid:15) ) + O ( λ ) . (6)The first Gaussian term scales as Γ G ( (cid:15) ) ∼ Γ =(2 π ) α |T | g c z /ω whereas the second non-Gaussianterms scales as Γ nG ∼ g c z Γ pointing out that the va-lidity of our expansion is based on the smallness of theexpansion parameter λ = g c z . III. RESULTSA. Gaussian contribution
A first approximation is obtained by considering onlythe quadratic part of the conductor action, in which thewhole path integral becomes Gaussian and, in the limit ofvanishing voltage V = 0, corresponds to the well-knownresults from P(E) theory. The quadratic part of the con-ductor action reads S G c = (cid:90) dω π Φ T − ω B ω Φ ω , B ω = − i (cid:18) − ωg c ωg c S c ( ω ) (cid:19) , with the symmetrized quantum noise of a tunnel con-tact S c ( ω ) = g c ( W ( ω + eV ) + W ( ω − eV )) ≡ g c (cid:99) W ( ω ).At T = 0 temperature, the symmetrized quantum noisevanishes for | ω | > eV thus we can already conclude that,even if just the Gaussian part of the conductor action isincluded, Eq. (1) can only describe photon emission withenergies limited by the bias voltage.Combining all the quadratic parts from both the LRCcircuit and the conductor in a single matrix, D ω ≡ π ( A ω + B ω ) = − i π (cid:18) − ω ˜ z − ω ω ˜ z ω S ( ω ) (cid:19) , with S ( ω ) = S c ( ω ) + W ( ω ) (cid:60){ /z ω } . Then, the corre-lation function (cid:104) e iαϕ ( t ) e − iαϕ (0) (cid:105) ≡ e α J ( t ) can be calcu-lated. As a result, one finds J ( t ) = (cid:90) dω | ˜ z ω | ω S t ( ω )( e − iωt − , (7)where S t ( ω ) = 2 π ( S ( ω ) + ω (cid:60){ / ˜ z ω } )= 2 πg c [ (cid:99) W ( ω ) + ω ] + 2 π [ W ( ω ) + ω ] (cid:60){ /z ω } is the total nonsymmetrized noise spectral density. Theimpedance ˜ z ω = z ω / (1 + z ω g c ) is the parallel connectionof the tunnel junction and the environmental impedanceplaying the role of the “effective environment” to the de-tector. This means that the factor g c leads to an in-creased damping of the resonator, which can be absorbedin a renormalized η → η + 1 /R c C and will be ignoredhenceforth. From Eq. (1), in the lowest order in α , weobtain the rate in scaled units,Γ G ( (cid:15) ) = 2 πα |T | | ˜ z (cid:15) | (cid:15) S t ( (cid:15) ) , (8)which is consistent with the known emission rate at finitetemperature.In the limiting case T → W ( ω ) → | ω | and the result(8) reduces to the one obtained in Ref. [30], namelyΓ G ( (cid:15) ) = Γ R η ( (cid:15) ) θ ( eV + (cid:15) ) (cid:18) eV + (cid:15)ω (cid:19) ( T = 0) , (9)in which we set the dimensionless resonance shape func-tion R η ( (cid:15) ) = 1 / [( (cid:15) /ω − + (cid:15) η /ω ]. In this limitthe maximum energy eV for a photon emission due toinelastic transitions is eV as a consequence of the sharpFermi surfaces on both sides of the tunnel junction, andthe emission spectrum has indeed a cutoff at | (cid:15) | = eV .In Fig. 3 we give an example of the Gaussian emissionspectrum at zero temperature for three different valuesof the bias voltage at different damping parameters η .At a voltage below the resonance eV < ω in Fig. 3(a),the broadening has only a small influence on the emissionspectrum and no peak occurs in the spectrum. The SPPresonance is visible only when the bias voltage becomescomparable or larger than the resonance ω , as shownin Figs. 3(b) and 3(c). For instance, in Fig. 3(b), closeto the threshold eV the emission is enhanced, but thethreshold remains clearly visible. In the limit of largebias voltage eV > ω [Fig. 3(c)], the full resonance isreflected in the emission spectrum and its shape is essen-tially determined by the resonance function appearing asa prefactor to the noise in Eq. (9). Hence, the maximumis ∼ /η and can be strongly increased in high-qualityresonators or well-defined plasmonic modes.At finite temperature we can cast the Gaussian rate asΓ G ( (cid:15) ) = Γ R η ( (cid:15) ) (cid:34)(cid:99) W ( (cid:15) ) + (cid:15)ω + (cid:18) g c z (cid:19) W ( (cid:15) ) + (cid:15)ω (cid:35) , (10)and the clear cutoff at T = 0 due to the Fermi distribu-tion is smoothed out.Figure 4(a) shows the emission rate for different tem-peratures at a voltage below the resonance eV = 0 . ω and for g c z = 1. Different values of the ratio g c z do not
0. 0.5 1. 1.50.20.40.60.8 0.90.70.50.30.2 Η (cid:144) Ω (cid:45)Ε (cid:144) Ω (cid:71) G (cid:144) (cid:71) a)
0. 0.5 1. 1.50.20.61.1.4 0.90.70.50.30.2 Η (cid:144) Ω (cid:45)Ε (cid:144) Ω (cid:71) G (cid:144) (cid:71) b)
0. 0.5 1. 1.51.55.8.512. 0.90.70.50.30.2 Η (cid:144) Ω (cid:45)Ε (cid:144) Ω (cid:71) G (cid:144) (cid:71) c) FIG. 3. The Gaussian contribution to the emission spectrumfor different broadenings, at zero temperature for three differ-ent values of the bias voltage: (a) voltage below the resonance eV = 0 . ω , (b) voltage at the resonance eV = ω , (c) voltageabove the resonance eV = 1 . ω . In all cases, the thresholdoccurs at | (cid:15) | = eV . The SPP resonance becomes visible oncethe threshold is larger than ω . The smaller the broadening η is, the sharper the SPP peak becomes. change the result significantly provided that eV (cid:29) T be-cause the noise of the intrinsic thermal contribution of theplasmon - corresponding to the second term in Eq. (10) -scales as exp[ − eV /T ] around the cutoff | (cid:15) | = eV and itis hence exponentially small. Since a finite temperaturesoftens the sharp cutoff at | (cid:15) | = eV that exists at zerotemperature, the SPP resonance can come into play evenat an energy lower than the bias voltage, thus contribut-ing an overbias emission as well. The resonance stronglyenhances the thermally excited overbias emission. It isremarkable that the step at eV is already almost invisibleat a small temperature of just a few % of ω . This can betraced back to the thermally excited quasiparticles in thelead with the higher chemical potential - correspondingto the first term in Eq. (10) - so that the thermal tail atthe resonance is ∼ exp[ − ( ω − eV ) /T ] with ω ∼ eV andtherefore exponentially larger than the intrinsic thermal
0. 0.5 1. 1.50.51.1.5 00.020.050.10.2 k B T (cid:144) Ω (cid:45)Ε (cid:144) Ω (cid:71) G (cid:144) (cid:71) a)
0. 0.5 1. 1.50.51.1.5 0.30.50.70.91.1eV (cid:144) Ω (cid:45)Ε (cid:144) Ω (cid:71) G (cid:144) (cid:71) b) FIG. 4. (a) The Gaussian contribution to the emissionspectrum for different temperatures with the bias voltage eV /ω = 0 .
8. As the temperature is increased, the cutoffat the bias eV is washed out. In addition, more electrons areinvolved in the tunneling processes, leading to an increasedrate. (b) The Gaussian contribution to the emission spec-trum for different bias voltages. The SPP peak is more pro-nounced when the bias voltage exceeds the resonance energy ω . In all cases, the sharp threshold for − (cid:15) > eV that ex-ists at zero temperature is smoothened at finite temperature,which is already achieved at the surprisingly small but finitetemperature T = ω /
30. The broadening parameter in bothfigures is chosen as η = 0 . ω , whereas the product of thetunneling conductance and the characteristic impedance ofthe resonator is set to g c z = 1. contribution of the plasmon.Figure 4(b) shows the emission rate for different biasvoltages at low temperature T = ω /
30. In all cases,from bias voltages below the resonance eV < ω to biasvoltages above the resonance eV > ω , we have the dis-appearance of the zero temperature cutoff at | (cid:15) | = eV .As long as the voltage becomes larger than eV > ω , theSPP resonance becomes visible in the emission spectrumin a similar way to the case of vanishing temperature T = 0. In other words, at finite and small tempera-tures T (cid:28) ω , we have substantial corrections to thezero-temperature result for the Gaussian rate around thecutoff at | (cid:15) | = eV . B. Non-Gaussian contribution
Although single-electron tunneling events produce sig-natures of the overbias SPP peak at finite temperature,we will now turn to the nonquadratic part of the ac-tion S c describing the electron-electron correlation thatgives contributions to the overbias emission. As pointedout in Ref. [30], comparing the absolute orders of mag-nitude, the non-Gaussian phase fluctuations are smallerthan the dominating Gaussian fluctuations due to thesmall environmental impedance g c z ω (cid:28)
1. However, thenon-Gaussian rate represents the only one contributionto the total rate in the overbias region | (cid:15) | (cid:29) eV at T = 0.We aim to understand in which range of parameters, forsufficiently low temperature and well inside the overbiasregion | (cid:15) | > eV , the non-Gaussian rate can continue todominate over the thermal Gaussian rate. Before dis-cussing the results for the non-Gaussian rates, we reportthe main steps for calculating such a rate. Further detailsare given in the Appendix.First, from Eqs. (3)-(5), we expand the action of thecoherent conductor to the fourth order of ϕ while thehigher-order terms can be neglected due to the factor g c z ω (cid:28)
1, yielding S = S e + S Gc + S (3) c + S (4) c + O (Φ ).Second, using exp[ − i S (3) c − i S (4) c ] ≈ − i S (3) c − i S (4) c , inaccordance with the approximation above, we can writethe path integral as (cid:104) e iαϕ ( t ) e − iαϕ (0) (cid:105) (cid:39) e α J ( t ) − i (cid:104)(cid:104) S (3) c (cid:105)(cid:105) − i (cid:104)(cid:104) S (4) c (cid:105)(cid:105) , (11)in which we used the Gaussian average (cid:104)(cid:104)· · · (cid:105)(cid:105) ≡ (cid:82) D [Φ]( · · · ) e (cid:82) dω {− i Φ T − ω D ω Φ ω + iαb Tω ( t )Φ ω } and b ω ( t ) =( e − iωt − , − ( e − iωt + 1) / T . After expanding for small α , the first term in Eq. (11) yields the Gaussian rate dis-cussed in the previous section. Concerning S (3) c , it is anodd term which gives a nonvanishing result only to theorder α and we neglect it for α (cid:28)
1. Thus, we focus onthe fourth term which is given in frequency space by S (4) c = 112 1(2 π ) i g c (cid:90) dωdω (cid:48) dω (cid:48)(cid:48) { (cid:16) F ( ω ) + F ( − ω )] − F ( − ω − ω (cid:48) )+ F ( − ω − ω (cid:48) )] (cid:17) [ ϕ + ( ω (cid:48) ) ϕ + ( ω ) ϕ + ( ω (cid:48)(cid:48) ) ϕ + ( − ω − ω (cid:48) − ω (cid:48)(cid:48) )+ ϕ − ( ω (cid:48) ) ϕ − ( ω ) ϕ − ( ω (cid:48)(cid:48) ) ϕ − ( − ω − ω (cid:48) − ω (cid:48)(cid:48) )] − F ( − ω ) ϕ + ( ω ) ϕ − ( ω (cid:48) ) ϕ − ( ω (cid:48)(cid:48) ) ϕ − ( − ω − ω (cid:48) − ω (cid:48)(cid:48) ) − F ( ω ) ϕ − ( ω ) ϕ + ( ω (cid:48) ) ϕ + ( ω (cid:48)(cid:48) ) ϕ + ( − ω − ω (cid:48) − ω (cid:48)(cid:48) )+6 F ( − ω − ω (cid:48) ) ϕ + ( ω ) ϕ + ( ω (cid:48) ) ϕ − ( ω (cid:48)(cid:48) ) ϕ − ( − ω − ω (cid:48) − ω (cid:48)(cid:48) )+6 F ( − ω − ω (cid:48) ) ϕ − ( ω ) ϕ − ( ω (cid:48) ) ϕ + ( ω (cid:48)(cid:48) ) ϕ + ( − ω − ω (cid:48) − ω (cid:48)(cid:48) ) } , with F ( ω ) = ( − ω − eV ) + W ( − ω − eV ) ,F ( ω ) = ( ω + eV ) + W ( ω + eV ) ,F ( ω ) = F ( ω ) + F ( − ω ) . For the field ϕ ± ω , we list the results: (cid:104)(cid:104) Φ ω (cid:105)(cid:105) = (cid:104)(cid:104) φ ω χ ω (cid:105)(cid:105) = 12 D − ω b − ω (1 + O [ α z ])= 2 πiα (cid:20) S ( ω ) | ˜ z ω | ω −
12 ˜ z ω ω (cid:21) e iωt − (cid:20) S ( ω ) | ˜ z ω | ω +
12 ˜ z ω ω (cid:21) − ˜ z − ω ω e iωt + ˜ z − ω ω and (cid:104)(cid:104) Φ ω Φ T − ω (cid:105)(cid:105) = (cid:18) (cid:104)(cid:104) φ ω φ − ω (cid:105)(cid:105) (cid:104)(cid:104) φ ω χ − ω (cid:105)(cid:105)(cid:104)(cid:104) χ ω φ − ω (cid:105)(cid:105) (cid:104)(cid:104) χ ω χ − ω (cid:105)(cid:105) (cid:19) = − i D − ω = 2 π (cid:32) S ( ω ) | ˜ z ω | ω ˜ z ω ω − ˜ z − ω ω (cid:33) . In the weak coupling limit, α (cid:28)
1, corresponding to weak detection that is the experimentally relevant regime, themain order pairings of averages appearing in S (4) c areof the type (cid:104)(cid:104) ϕ ω (cid:105)(cid:105)(cid:104)(cid:104) ϕ − ω (cid:105)(cid:105)(cid:104)(cid:104) ϕ ω (cid:48) ϕ − ω (cid:48) (cid:105)(cid:105) and they are pro-portional to ∼ α . Such terms can be calculated usingWick’s theorem to find all possible pairings of single anddouble averages. Finally, we consider only the lowestorder terms in ∼ g c in order to obtain the following ex-pression for the non-Gaussian contribution:Γ nG = πg c α |T | | ˜ z (cid:15) | (cid:15) (cid:90) ∞ dω (cid:40) | ˜ z ω | ω (cid:16)(cid:99) W ( ω ) − W ( ω ) (cid:17) (cid:104) − (cid:99) W ( (cid:15) ) + (cid:16)(cid:99) W ( ω + (cid:15) ) + (cid:99) W ( ω − (cid:15) ) (cid:17)(cid:105) +2 (cid:16)(cid:99) W ( (cid:15) ) − W ( (cid:15) ) (cid:17) Re { ˜ z (cid:15) } (cid:15) Re { ˜ z ω } ω (cid:104)(cid:99) W ( ω + (cid:15) ) − (cid:99) W ( ω − (cid:15) ) (cid:105) +2 (cid:16)(cid:99) W ( (cid:15) ) − W ( (cid:15) ) (cid:17) Im { ˜ z (cid:15) } (cid:15) Im { ˜ z ω } ω (cid:104) W ( eV ) − (cid:99) W ( ω ) − (cid:99) W ( (cid:15) ) + (cid:99) W ( ω + (cid:15) ) + (cid:99) W ( ω − (cid:15) ) (cid:105)(cid:41) . (12)More details on the deriviation of this expression can befound in the Appendix.Similarly as for the Gaussian fluctuations, we recoverour former results [30] in the limit T →
0. Examples ofthe non-Gaussian rate at zero temperature are given inFig. 5 scaled with λ Γ and with λ = g c z , our expansionparameter. The non-Gaussian rate yields a contributionin the underbias as well as in the overbias regime. More-over, the non-Gaussian rate here calculated to lowest or-der in α and g c has also a high-energy cutoff at | (cid:15) | = 2 eV above which Γ nG = 0. The latter result is in agreementwith the picture of two correlated electrons involved ina single photon emission whose energy is now limited by (cid:126) ω < eV . Such a cutoff is less pronounced than thesharp cutoff of the Gaussian rate at | (cid:15) | = eV although itis evident in the experimental data (see next section andFig. 11).As for the Gaussian case in Fig. 3, we plot in Fig. 5the three different cases corresponding to bias voltagesbelow or above the resonance eV < ω or eV > ω , andthe resonant case eV = ω .In the first case eV < ω , Fig. 5(a), the curve forthe non-Gaussian rate shows a characteristic cusp at thethreshold | (cid:15) | = eV . Such a curve has also peaks in boththe underbias region | (cid:15) | < eV as well as in the overbiasregion | (cid:15) | > eV in correspondence with the resonanceof the SPP mode at | (cid:15) | = ω . The overbias emissionat T = 0 corresponds to the first line of Eq. 12. How-ever, in the underbias region | (cid:15) | < eV , the non-Gaussianrate is dominated by the leading Gaussian contribution so that the first peak hardly can be distinguished andone expects that the overbias emission rate is distinctlyresolved around the resonance ω > eV only.For bias voltages at the resonance eV = ω , Fig. 5(b),the two peaks associated with the non-Gaussian ratemerges into a single peak and the curve shows a kinkat the threshold | (cid:15) | = eV . In this case the non-Gaussianrate has still a noticeable contribution in the overbiasregime | (cid:15) | > eV in terms of the tail of the resonancepeak centered at the threshold.Then, for the last case, eV > ω , shown in Fig. 5(c),the non-Gaussian rate behaves in a way similar to theGaussian rate in Fig. 5(c) with a single peak at theresonance | (cid:15) | = ω . Such a peak is now located well insidethe underbias region in which the non-Gaussian rate isdominated by the Gaussian rate.Finally, we consider the case that when the SPP reso-nance ω is quite close to the two-electron energy cutoff2 eV , which is shown in Fig. 6. Here we can see, unlikeFig. 5(a) where the SPP resonance ω is far away fromthe 2 eV cutoff, that the overbias peak can still be presentalthough strongly weakened.Thus we can conclude that overbias photon emissiondue to the non-Gaussian voltage fluctuations in meso-scopic tunnel junctions is, a priori , always a possible ef-fect even far away from the resonance of the plasma-polariton modes, but the effect’s magnitude can besmaller than the limit of a photon detector. On the con-trary, the overbias photon emission becomes a substantialeffect provided that the system has a resonant plasmonic
0. 0.5 1. 1.50.020.040.060.08 0.90.70.50.30.2 Η (cid:144) Ω (cid:45)Ε (cid:144) Ω (cid:71) n G (cid:144) Λ (cid:71) a)
0. 0.5 1. 1.50.30.60.9 0.90.70.50.30.2 Η (cid:144) Ω (cid:45)Ε (cid:144) Ω (cid:71) n G (cid:144) Λ (cid:71) b)
0. 0.5 1. 1.536912 0.90.70.50.30.2 Η (cid:144) Ω (cid:45)Ε (cid:144) Ω (cid:71) n G (cid:144) Λ (cid:71) c) FIG. 5. The non-Gaussian contribution at zero temperatureto the emission spectrum for different broadenings. There is akink for | (cid:15) | = eV whereas the resonance peak appears alwaysat | (cid:15) | = ω . Parameters are the same as in Fig. 3. mode at a frequency in the overbias range eV ≥ ω andbelow the cutoff for the two electrons emission ω < eV .We discuss now the effects of a finite temperature forthe non-Gaussian rate for the case eV < ω . Some ex-amples are shown in Fig. 7 with an intrinsic broadeningof the SPP mode η = 0 . ω .In order to distinguish between the low and high tem-perature regimes, a priori we can compare the broad-ening η with the thermal smearing expected at finitetemperature ∼ k B T . Then one expects that the non-Gaussian rate continues to exhibit sharp features in thelow temperature range, defined by k B T < η and thatit becomes a smooth, smeared function as the temper-ature approaches the broadening k B T (cid:46) η . In Fig. 7,we can see that, increasing the temperature, the two dis-tinct peaks merge into a single peak and the kink at thebias voltage | (cid:15) | = eV is weakened concealing any over-bias signatures. Remarkably, this merging occurs even atrelatively low temperature T ∼ − ω compared to thebroadening of the mode η ∼ − ω pointing out that
0. 0.3 0.6 0.9 1.20.0020.0040.0060.008 0.50.40.30.20.1 Η (cid:144) Ω (cid:45)Ε (cid:144) Ω (cid:71) n G (cid:144) Λ (cid:71) FIG. 6. The non-Gaussian contribution at zero temperatureto the emission spectrum for different broadenings. The biasvoltage is eV = 0 . ω such that the two-electron energycutoff is at 2 eV = 1 . ω .
0. 0.5 1. 1.50.10.30.50.7 00.020.050.10.2 k B T (cid:144) Ω (cid:45)Ε (cid:144) Ω (cid:71) n G (cid:144) Λ (cid:71) FIG. 7. The non-Gaussian contribution to the emission spec-trum for different temperatures at the bias voltage eV /ω =0 . η = 0 . ω . Due to the increased temperature, thekink at the bias disappears and the two peaks are mergedinto a single peak. the overbias is highly sensitive to finite temperature.On the other hand, increasing the temperature en-hances the height of the peak in a similar way as theGaussian rate, as discussed in the previous section. Inother words, above the threshold | ε | > eV and at fi-nite temperature, one can not discriminate the overbiasemission due to the Gaussian fluctuations - associatedto single electron processes - from the overbias emissiondue to the non-Gaussian fluctuations - associated to two-electron processes. In order to resolve such processes, wehave to consider the low temperature range.In Fig. 8, we discuss the behavior of the non-Gaussianrate at low temperature, T = ω /
30, as varying thedamping η when the resonance is close to the two-electroncutoff 2 eV = 1 . ω . By comparing with the Gaussianpart under the same condition - the inset of Fig. 8 - wenotice that at finite temperature, since the bias voltage eV , that is important for the single electron emission, isfar away from the resonance, the Gaussian part aroundthe resonance is small as it is due to the temperaturesmearing of the Fermi distribution. Meanwhile, the non-Gaussian part can represent the larger contribution inthe case of a sharp resonance. Η (cid:144) Ω
0. 0.2 0.4 0.6 0.8 1. 1.2 1.40.0020.0040.0060.0080.010.0120.014 (cid:45)Ε (cid:144) Ω (cid:71) n G (cid:144) Λ (cid:71) (cid:71) G (cid:144) (cid:71) FIG. 8. The non-Gaussian contribution at finite temperature T = ω /
30 to the emission spectrum for different broadenings,at 2 eV = 1 . ω , viz. the SPP resonance dominates near the2 eV cutoff. The inset shows the Gaussian contribution aroundthe SPP resonance. Thus, in this case, with proper λ , evenat finite temperature, the overbias due to the two-electronemission (non-Gaussian part) can dominate the Gaussian one. C. Total rate and Comparison with theexperiments
For the total tunneling rate, we have to take the Gaus-sian as well as the non-Gaussian rates into account. Inorder to compare the theoretical results with the exper-imental data of G. Schull and co-workers [29], in thissection we plot the rate explicitly as a function of en-ergy ( eV ) for a SPP mode centered at ω = 1 . η = 0 . λ = g c z . Then as λ increases, thenon-Gaussian rate gradually gives the dominant contri-bution to the total emission rate in the overbias energyregime, leading to the overbias emission peak becomingmore visible (see Fig. 9). However, for small λ , withinthe validity of our expansion, the non-Gaussian featuresare weak and smeared out by the Gaussian propertiesdue to the finite temperature. Λ - Ε [eV] (cid:71) (cid:144) (cid:71) FIG. 9. The total rate for different dimensionless factors λ = g c z at the bias voltage V = 1 .
32 V. The overbias peakincreases with increasing λ , which determines the weight ofthe non-Gaussian part to the total rate. The temperature ischosen to be the room temperature βω = ω /k B T = 72 andthe SPP resonance energy is taken to be ω = 1 . η = 0 . k B T (cid:144) Ω
1. 1.2 1.4 1.6 1.8 2. 2.2 (cid:45) (cid:45) (cid:45) (cid:45) (cid:45)Ε (cid:64) eV] L og [ (cid:71) (cid:144) (cid:71) ] FIG. 10. The logarithmic total emission rate at the bias volt-age V = 1 .
32 V for different temperatures. The kink at thebias voltage becomes more distinct at lower temperature. TheSPP resonance energy is taken to be ω = 1 . λ = 0 . η = 0 . We investigate the temperature dependence of the to-tal rate in Fig. 10 in logarithmic scale, in which theblack line shows the zero temperature case, giving theclear kink at the bias voltage eV , described in Ref. 30.Figure 10 shows how the rate sensitively depends on thetemperature; the clear kink at the bias voltage is quicklysoftened even at small finite temperatures, and the strongeffect of the temperature appears when the temperaturehas the same order of the factor eV − ω , leading to thesingle overbias peak as the temperature is increased.Moreover, for comparison with the experimental re-sults obtained by G. Schull and co-workers [29], we needfirst to determine the coupling parameter λ , which de-termines the weight between the Gaussian and non-Gaussian contributions and the width of the SPP res-onance η .The width can be directly obtained from the experi-mental results in Ref. 29, resulting in η ≈ . ω . Thecoupling parameter is determined by scaling the peakvalue at − (cid:15) = ω for the low bias V = 1 .
32 V by a fac-tor of 300 versus the peak at the bias V = 2 .
15 V, andthis yields λ = 0 . T (cid:39) T (cid:39)
300 K for λ = 0 . ω (cid:39) . K , the rate in Fig. 11(a) exhibits a distinctthreshold at − (cid:15) = eV , and the clear overbias peaks atthe SPP resonance due to the non-Gaussian contribu-tions, which gives a good explanation and agreement withRef. 29. By contrast, at room temperature [Fig. 11(b)],we find that the sharp threshold behavior at − (cid:15) = eV has been weakened and is relaxing into the overbias SPPresonance due to the smoothed distribution function un-der the temperature effect. Meanwhile, the temperatureeffect has also sensitively hidden the two-electron energycutoff line − (cid:15) = 2 eV , leading to the long and small tail = eV ✏ [eV] V [ V ] = 2 eV a)
2 2.5 1.5 = eV ✏ [eV] V [ V ] = 2 eV b) FIG. 11. The light emission spectrum on a logarithmic scaleas a function of bias voltage at (a) the low temperature βω = 3000 and (b) room temperature βω = 72. In panel(a), the clear threshold behavior reproduces the experimen-tal findings for the parameter λ = 0 . − (cid:15) = eV is less evident. Thisbehavior depends sensitively on the temperature. Here, weuse the parameter λ = 0 .
2. In both cases, the SPP resonanceenergy is taken to be ω = 1 . η = 0 . into the energy larger than 2 eV .Furthermore, we study the properties of the intensityof the overbias light emission as a function of the tem-perature. Since the non-Gaussian part has the prefactor λ = g c z compared to the Gaussian part, we considerthe Gaussian and non-Gaussian emission separately anddefine their intensities as I e G , nG = (cid:82) eVeV Γ G , nG d(cid:15) , respec-tively. In Fig. 12, we observe that both the Gaussianand non-Gaussian intensities increase with temperaturein the temperature range shown in the figure. It is in-teresting to note that for high enough temperature theheating effect smears out the Fermi edge and leads to asaturation of the non-Gaussian emission. Furthermore,we find that the intensities do not increase monotonically k B T (cid:144) Ω I G e (cid:145) (cid:71) Ω , I n G e (cid:145) Λ (cid:71) Ω I nG2 e I G e FIG. 12. The temperature dependence of the scaled inten-sity for the Gaussian and non-Gaussian contribution. TheSPP resonance energy is taken to be ω = 1 . η = 0 . with the bias voltages. Hence, it would be interesting tostudy the temperature dependence of the overbias lightemission, in order to distinguish thermally induced emis-sion from the pure quantum effect at low temperatures. IV. CONCLUSION
To summarize, motivated by the experimental obser-vation of photons emitted by tunnel junctions carryingthe energy larger than the bias voltage | (cid:15) | > eV , we havedeveloped a theoretical model to describe the electron-SPP mode interaction based on the dynamical Coulombblockade theory.In combination with the Keldysh path integral formal-ism, by treating the Gaussian and non-Gaussian contri-butions separately, our theory has shown that the non-Gaussian fluctuations give rise to the overbias photonemission, which can explain and reproduce the experi-mentally observed photon emission with energies largerthan the single-particle energy limit eV . Furthermore,due to the smeared edge of the Fermi distribution func-tion at finite temperature, our result also shows that theelectron tunneling is sensitively affected by the tempera-ture, thus influencing the overbias emission. The criticalpoint at the bias voltage − (cid:15) = eV is strongly weakened,and the overbias peak becomes a mixture of the Gaussianand non-Gaussian noise. In addition, we also considerthe interesting case when the bias voltage is far from theSPP resonance; here we set the resonance close to thetwo-electron energy limit, and we argue that this regimeis suitable to distinguish the Gaussian and non-Gaussiancontributions even at finite temperature and in the caseof sharp resonance. Finally, we investigate the tempera-ture dependence of the photon intensities in the overbiasregion at different bias voltages and show that it allows usto distinguish the quantum emission from a pure heatingeffect.0In conclusion, our work enables us to model the lightemission due to the electron-SPP mode interaction innanosize contacts and can be applied to more complexjunctions. ACKNOWLEDGMENTS
This work was supported by the DFG through SFB767, the Center of Applied Photonics (CAP), the KurtLion Foundation, and the Zukunftskolleg of the Univer-sity of Konstanz.
Appendix A: Expansion
Here, we describe some useful intermediate results forthe derivation of the non-Gaussian rate, Eq. (12), andthe expansion of the action of the tunnel conductor S c tothe fourth order in the fluctuating fields. According tothe Gaussian averages (cid:104)(cid:104) Φ ω (cid:105)(cid:105) and (cid:104)(cid:104) Φ ω Φ T − ω (cid:105)(cid:105) , we obtain (cid:104)(cid:104) ϕ + ( ω ) (cid:105)(cid:105) = iα [ Y ( ω ) e iωt − X ( ω )] , (cid:104)(cid:104) ϕ − ( ω ) (cid:105)(cid:105) = iα [ Q ( ω ) e iωt − P ( ω )] , (cid:104)(cid:104) ϕ + ω ϕ + ω (cid:48) (cid:105)(cid:105) = 2 πX ( ω ) δ ( ω + ω (cid:48) ) , (cid:104)(cid:104) ϕ + ω ϕ − ω (cid:48) (cid:105)(cid:105) = 2 πY ( ω ) δ ( ω + ω (cid:48) ) , (cid:104)(cid:104) ϕ − ω ϕ + ω (cid:48) (cid:105)(cid:105) = 2 πP ( ω ) δ ( ω + ω (cid:48) ) , (cid:104)(cid:104) ϕ − ω ϕ − ω (cid:48) (cid:105)(cid:105) = 2 πQ ( ω ) δ ( ω + ω (cid:48) ) . with X ( ω ) = S nc ( ω ) | ˜ z ω | ω + 1 ω [ W ( ω ) Re { ˜ z ω } + iωIm { ˜ z ω } ] ,Y ( ω ) = S nc ( ω ) | ˜ z ω | ω + 1 ω [ W ( ω ) − ω ] Re { ˜ z ω } ,P ( ω ) = S nc ( ω ) | ˜ z ω | ω + 1 ω [ W ( ω ) + ω ] Re { ˜ z ω } ,Q ( ω ) = S nc ( ω ) | ˜ z ω | ω + 1 ω [ W ( ω ) Re { ˜ z ω } − iωIm { ˜ z ω } ] , with S nc ( ω ) = g c [ W ( ω + eV ) + W ( ω − eV ) − W ( ω )].After performing the symmetrization over ω , we obtain (cid:104)(cid:104) S (4) c (cid:105)(cid:105) = − iπα g c (cid:90) (cid:90) dωdω (cid:48) (cid:40) [ Y ( ω (cid:48) ) e iω (cid:48) t − X ( ω (cid:48) )][ Y ( − ω (cid:48) ) e − iω (cid:48) t − X ( − ω (cid:48) )] (cid:104) X ( ω )[ − F (0)+2 F s ( ω )+2 F s ( ω (cid:48) ) − F ss ( − ω − ω (cid:48) )]+ Q ( ω ) F (0) − P ( ω ) F ( ω ) − Y ( ω ) F ( − ω ) (cid:105) + [ Q ( ω (cid:48) ) e iω (cid:48) t − P ( ω (cid:48) )][ Q ( − ω (cid:48) ) e − iω (cid:48) t − P ( − ω (cid:48) )] (cid:104) Q ( ω )[ − F (0)+2 F s ( ω )+2 F s ( ω (cid:48) ) − F ss ( − ω − ω (cid:48) )] − P ( ω ) F ( ω ) − Y ( ω ) F ( − ω ) + X ( ω ) F (0) (cid:105) + [ Y ( ω (cid:48) ) e iω (cid:48) t − X ( ω (cid:48) )][ Q ( − ω (cid:48) ) e − iω (cid:48) t − P ( − ω (cid:48) )] (cid:104) − [ Q ( ω ) + X ( ω )] F ( − ω (cid:48) ) + Y ( ω ) F ( − ω − ω (cid:48) ) + P ( ω ) F ( ω − ω (cid:48) ) (cid:105) + [ Q ( ω (cid:48) ) e iω (cid:48) t − P ( ω (cid:48) )][ Y ( − ω (cid:48) ) e − iω (cid:48) t − X ( − ω (cid:48) )] (cid:104) − [ Q ( ω ) + X ( ω )] F ( ω (cid:48) ) + Y ( ω ) F ( − ω + ω (cid:48) ) + P ( ω ) F ( ω + ω (cid:48) ) (cid:105)(cid:41) with the defined functions F s ( ω ) = [ F ( ω ) + F ( − ω )] / F ss ( − ω − ω (cid:48) ) = [ F ( − ω − ω (cid:48) ) + F ( − ω + ω (cid:48) ) + F ( ω − ω (cid:48) ) + F ( ω + ω (cid:48) )] /
4, in which F ( ω ) = F ( ω ) + F ( − ω ) =( − ω − eV ) + W ( − ω − eV ) + ( − ω + eV ) + W ( − ω + eV )as given in the text. One can show that the terms proportional to e iω (cid:48) t andthe ones proportional to e − iω (cid:48) t , are interchanged underthe operation ω (cid:48) → − ω (cid:48) . Using (cid:82) e iωt e i(cid:15)t dt = 2 πδ ( ω + (cid:15) )and keeping the terms in the lowest order of g c Z , thenon-Gaussian rate Eq. (12) can be expressed as1Γ (4)nG = π α |T | g c (cid:90) dω (cid:40) Y ( − (cid:15) ) X ( (cid:15) ) (cid:104) X ( ω )[ − F (0)+2 F s ( ω )+2 F s ( (cid:15) ) − F ss ( − ω + (cid:15) )] − P ( ω ) F ( ω ) − Y ( ω ) F ( − ω )+ Q ( ω ) F (0) (cid:105) + Q ( − (cid:15) ) P ( (cid:15) ) (cid:104) Q ( ω )[ − F (0)+2 F s ( ω )+2 F s ( (cid:15) ) − F ss ( − ω + (cid:15) )] − P ( ω ) F ( ω ) − Y ( ω ) F ( − ω )+ X ( ω ) F (0) (cid:105) + Y ( − (cid:15) ) P ( (cid:15) ) (cid:16) − [ Q ( ω ) + X ( ω )] F ( (cid:15) ) + Y ( ω ) F ( − ω + (cid:15) ) + P ( ω ) F ( ω + (cid:15) ) (cid:17) + Q ( − (cid:15) ) X ( (cid:15) ) (cid:16) − [ Q ( ω ) + X ( ω )] F ( − (cid:15) ) + Y ( ω ) F ( − ω − (cid:15) ) + P ( ω ) F ( ω − (cid:15) ) (cid:17)(cid:41) . This expression can be cast as Eq. (12) in the main text after replacing all the functions, i.e.,
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