## Abstract

We develop a theoretical model for the semiconductor generator of the sub-wavelength surface plasmons, operating on a single mode and often referred to as a spaser. We show that input-output characteristics of the single mode spaser does not exhibit nonlinearity inherent in most lasers, but the linewidth of the emission collapses, as in any laser which allows us to define the threshold. Our rigorous derivations show that as long as the mode remains substantially sub-wavelength in all three dimensions, the threshold current (power) shows virtually no dependence on the gain material and geometry of the active layer and is determined solely by the intrinsic loss of the metal in the device. For the semiconductor single mode surface plasmon generators operating in the telecommunication range the threshold current is on the order of 10-20 µA, and the threshold current density grows fast with the decrease of the device size reaching 100’s of kA/cm^{2} or more. This fact makes coherent sources of sub-wavelength SP’s unattainable from our point of view, but there exists a room for efficient broad-band incoherent SP sources either optically or electrically pumped.

©2012 Optical Society of America

## 1. Introduction

There has always existed a keen interest in developing miniaturized sources of coherent radiation, and by now steady advances in nanofabrication have shrunk the size of conventional nanolasers to just above the diffraction limit. The next logical step looming at the horizon is how to advance beyond the diffraction limit and develop truly-sub-wavelength sources, i.e. measuring less than $\lambda /2n$ in all three dimensions. Clearly, such shrinkage is unattainable in all-dielectric structures [1,2] hence the research has shifted to metal-dielectric structures where existence of surface-plasmon (SP) modes in which the electric field is coupled to the charge oscillations of the free carriers in the metal does permit sub-wavelength confinement. The research has been given a great impetus by the seminal proposal by Bergman et al [3] for a coherent emitter of surface plasmons (SPASER) consisting of a metal nanoparticle surrounded by a semiconductor gain material and the first estimate given in that work has shown that the population inversion required to compensate the loss in the metal was realistic. Since then, a multi-pronged effort, both theoretical [4–8] and experimental [9–21] devoted to spasing has blossomed, and yet, there had been but a single demonstration of “laser-like” behavior in gold-dye media consisting of a linewidth narrowing and somewhat nonlinear output behavior under intense pulsed optical pumping [9]. When it comes to developing a practical semiconductor spaser, a great number of diverse schemes had been successfully demonstrated, including mostly optical but also injection [12] pumping. Various geometries, such as nanowires [10], double heterostructure mesas [11,12], disc structures [13], nano-patches [14], and coaxial pillars [15–17] have been used, but they all have one feature in common: the device was larger than a wavelength in the material in at least one dimension. The threshold has been universally high and with a few exceptions, the pumping was either optical or required low temperature. A number of review works [4,5,18–21] recently published confirm the fact that truly sub-wavelength electrically pumped spaser is not anywhere on the horizon yet.

One obstacle encountered on the pathway to a spaser has been expected – high optical losses in the metal. Numerous attempts to engineer lower loss structures by altering geometry have not yielded desired results. In fact, it has been shown as early as 2006 [22] that the loss for a truly sub-wavelength resonant plasmonic structure does not depend on its geometry and more recently our detailed analysis [23] have shown this loss to be always equal to one half of the loss in pure metal. Thus in sub-wavelength plasmonic structures based on noble metals [24] the damping rate is always on the scale of *𝛾* ~10^{14} s^{−1}. To compensate for such a strong energy loss mechanism the material gain of the order of *𝛾*/*c* ~10^{3}-10^{4} cm^{−1} would be required. This value is definitely quite high, yet it is not unattainable in a typical semiconductor media with the injected carrier density on the order of a few times 10^{18} cm^{−3}, and such densities are routinely attained in semiconductor lasers. This discovery has prompted the researchers to embark on a quest to first compensate the loss and eventually achieve spasing in sub-wavelength structures. What most of the early proposals neglected is the fact that what matters is not just the value of gain and carrier density associated with it, but first and foremost the pumping current density that is required to maintain this carrier density. And in the sub-wavelength structures this current density can quickly reach entirely unsustainable values.

Indeed, as the volume of plasmonic mode decreases the spontaneous emission rate into that mode increases manifold due to the Purcell effect, which reduces the effective lifetime of the upper state of the gain medium (radiative recombination rate in semiconductor) by a Purcell factor that can be as high as few 100’s and thus causes the commensurate rise of threshold pumping rate (current density) from far less than 1 kA/cm^{2} in a typical double heterostructure laser to up to a MA/cm^{2}. In our recent work [25] we have shown how Purcell effect makes loss compensation in truly sub-wavelength propagating surface plasmon polaritons (SPPs) a completely unrealistic proposition.

In that work, however, we have not addressed the threshold of localized SP laser (spaser) which has characteristics that are distinct from any other laser due to the fact that spaser by definition has just one (or perhaps a few) mode resonant with the gain. As a result, the threshold of such a laser becomes ill-defined, and, in the absence of any directivity of emission, one has to revert to the dynamics of the linewidth in order to characterize whether one deals with a true spaser or simply a Purcell-enhanced spontaneous source, such as LED.

The goal of this work is to clarify all these issues, and in striving for this goal we obtain a simple expression for the critical pump rate at which one can characterize the emission as that of the spaser, show that as long as dimensions are substantially sub-wavelength this critical rate does not depend on the geometry, size, and, above all, even the nature of gain medium, and estimate that if the gain medium is a semiconductor (the most and perhaps the only practical of spaser’s implementations), the critical current density is expected to be on the order of MA/cm^{2}, making the spaser of truly sub-wavelength in all three dimensions a rather difficult feat to accomplish, while the incoherent source of sub-wavelength SPs is definitely within the reach.

## 2. Surface plasmon modes

While most of our results are to be relevant for wide range of spasers, as an example we consider a spaser designed to operate in the telecommunication window, i.e. near 1320 nm. For spherical noble metal nanoparticles the SP resonances are all in the visible range, hence one shall revert to a nanoparticle of different shape with a smaller fraction of electric field contained in the metal. This can be either a spherical nanoshell [26] or a prolate spheroid [27]. A spherical nanoshell would have three nearly degenerate modes which would increase the threshold by a factor of three, while in the prolate spheroid, the degeneracy is lifted and one has just one lowest order (dipole) mode in the desired window. Our calculations have shown that a prolate spheroid of Fig. 1
with the half- axis ratio *b/a* ~0.425 in case of Au and 0.385 in case of Ag surrounded by a In_{.53}Ga_{.47}As (lattice-matched to InP) gain medium (n ~3.63) would have its lowest SP resonance with dipole moment directed along the long axis *a* at 1320 nm, which is about 200 meV above the 740 meV bandgap of In_{.53}Ga_{.47}As thus assuring us that the density of states (DOS) at the SP frequency in In_{.53}Ga_{.47}As is sufficient to obtain required gain. The next two dipole modes with dipole moment normal to axis *a* have much shorter wavelength and thus can be excluded from consideration.

In this work it is not our goal to delve into the specifics of injection pumping and we simply assume that somehow the undoped gain region is placed between n- and p-doped regions of semiconductor thus providing carrier confinement and assuring that recombination takes place in the undoped region, as in any modern double-heterojunction laser. In other words, we are considering by far the best case scenario and trying to avoid such obvious issues as carrier depletion near the metal interface. We simply assume that a thin spacing layer (we avoid using the word “spacer” in order to avoid confusion with ‘spaser”) of wider bandgap between the active region and metal should be sufficient to keep all the carriers from the metal surface where their nonradiative recombination may be quite strong.

We characterize the mode by the effective volume *V _{eff}* which we introduce as

_{.53}Ga

_{.47}As is related to its index of refraction as ${\epsilon}_{s}={n}^{2}$,

*E*is the maximum field in the mode as shown in Fig. 1, and

_{max}*U*is the total energy of the mode

_{t}From the same energy conservation argument we can determine the kinetic energy of electrons as ${U}_{kin}={U}_{t}/2-{U}_{mag}$ and thus effective non-radiative energy loss rate in the metal as

^{13}s

^{−1}for Ag and 1.3 × 10

^{14}s

^{−1}for Au at 1320 nm. As magnetic energy decreases with the reduction of particle size, the kinetic energy fraction increases to 1/2 and the nonradiative decay rate approaches

_{$\gamma $}as indicated in Fig. 3 . At the same time, the energy also decays via dipole radiation which is proportional to the volume of the particle

_{$\omega $}is significantly larger than$\gamma $, i.e. covering entire optical and mid-IR ranges.

## 3. Gain and mode confinement

According to the Fermi’s Golden rule the stimulated radiative recombination rate of electron-hole pairs can be found as

To evaluate the gain (absorption) per second, i.e. the total number of carriers recombined (generated) per second in the whole active layer we first integrate (6) over the volume of gain medium surrounding the nanoparticle. Then we invoke our definition of the effective volume Eq. (1) and relate the total energy ${U}_{t}$ of the mode to the number of SPs in it, ${N}_{SP}$ as ${U}_{t}=\hslash \omega {N}_{SP},$ and obtain the rate equation for SPs and carriers

Since the electric field of the dipole SP mode decays as ${r}^{-3}$, the value of confinement factor depends strongly on the thickness *d* of the spacing layer that is used to separate the gain region from the metal surface where such deleterious effects as carrier depletion, surface recombination and gain quenching via emission into the higher order modes all take place. We consider two rather extreme cases - a somewhat unrealistic scenario of zero spacing thickness and the case of 2 nm thick spacing which makes high confinement impossible for small nano-particles. The results are shown in Fig. 4
where we have plotted the thickness of the active gain layer${d}_{a}$ required to achieve confinement factors of 0.4 and 0.8.

As one can see, without a spacing layer a very large confinement factor is easily attainable with a relatively thin active layer because the field is so tightly concentrated. But once even a very thin, 2 nm thick spacing layer is introduced, most of the field gets concentrated in the spacing layer, rather than in the active layer and large confinement is attainable only in larger spheroids. Luckily, as we shall see further on, the gain confinement factor does not affect threshold of a single mode spaser nearly as much as in the conventional double heterostructure lasers.

Equation (9) for the gain can be simplified significantly by invoking the relation [28], $\frac{1}{{m}_{e}}~\frac{1}{{m}_{0}}+\frac{2{P}^{2}}{{m}_{0}^{2}{E}_{g}}$, where${E}_{g}~\u0127\omega $ is the bandgap energy and introducing the fine structure constant ${\alpha}_{0}={e}^{2}/4\pi {\epsilon}_{0}\u0127c$ to obtain

^{7}cm/s for GaAs to 5.65 × 10

^{7}cm/s for InAs, i.e. is practically material-independent. It is instructive to quickly consider$T=0$K case in order to get the order of magnitude of carrier density ${n}_{c}$ required to achieve the gain commensurate with the modal loss ${\gamma}_{m0}$, i.e. on the order of 10

^{14}s

^{−1}. For $T=0$K, ${k}_{\omega}$is just below the Fermi level, hence ${k}_{\omega}^{3}=3{\pi}^{2}{n}_{c}$and we obtain from Eq. (11), $g\left(\omega \right)=1.75\times {10}^{8}{n}^{1/3}\u0413,$ indicating that for $\u0413~0.5$, carrier density that is on the order of 10

^{18}cm

^{−3}can provide sufficient gain ($g\left(\omega \right)~{10}^{14}$/s) at $T=0$K entirely independent of which semiconductor is used. But attaining that density in the presence of rapid Purcell-enhanced recombination is an entirely different story as show below.

## 4. Modal gain

The expression for gain in Eq. (11) is derived under the assumption that the SP field $E\left(r\right)$is monochromatic with frequency $\omega $. In reality though, finite net width of SP even in the presence of gain remains to be only a few times less than the linewidth of the SP mode ${\gamma}_{m0}={\gamma}_{nrad}+{\gamma}_{rad}$ evaluated without gain or loss in semiconductor. When measured in units of energy, this linewidth is not far from 25 meV, i.e. commensurate with room temperature thermal energy. Thus the Fermi-Dirac function and gain $g\left(\omega \right)$ vary over the linewidth of the SP mode. The modal gain ${g}_{m}$ therefore has to be evaluated by taking into account the fact that the mode with a resonant frequency ${\omega}_{0}$ has a finite net linewidth

Hence the modal gain is found by convolving the gain profile with the Lorentzian line shape of width ${\gamma}_{m}$,## 5. Spontaneous emission

The spontaneous emission in spaser has two components – first of all there is a spontaneous emission into the spaser mode at resonant frequency ${\omega}_{0}$ and then there is emission into the free-space propagating modes. The free space modes do get affected by the presence of the resonant spaser mode – and for simplicity we may assume that since the resonant mode is linearly polarized at each point, the free space modes of that polarization get “pulled” into the resonance mode while the other two polarizations remain unaffected, hence the density of free space modes near the spheroid is

Then the effective field density of vacuum fluctuations in the energy interval $d\left(\u0127\omega \right)$ can be found asOr, if using Eq. (11) in Eq. (18), we arrive at

Now for the spontaneous emission into the SP mode the effective DOS at a given point can be estimated as

Result Eq. (23) is important because it shows that in the single mode laser the rate of spontaneous emission into the mode is almost equal to the gain and their ratio depends on neither geometry nor the nature of gain medium. This result is of course a simple consequence of the most fundamental fact that the ratio of the first and second Einstein’s coefficients are closely related and their ratio depends only on the DOS at a given frequency. For the case of a single mode this density of states and the spectral density of SPs causing stimulated emission are the same which immediately leads one to Eq. (23). One most common consequence of Eq. (23) familiar to anyone in optical communications field is the fact that the noise figure of optical amplifier does not depend on either gain medium or geometry [32], another one is the ultimate, Schawlow-Townes linewidth of lasers [33] which will be considered further along.

## 6. Rate equations and threshold

With all the rates introduced, the coupled rate equations for the SPP in the mode and carriers in the gain region then become

^{−6}s

^{−1}for In

_{0.53}Ga

_{0.47}As at room temperature. These equations are in every way identical to the standard Statz-de-Mars [34,35] balance equations for laser, except for the fact that there is a very simple connection (23) between the modal gain and spontaneous emission.

Both carrier and SP numbers are the intrinsic characteristic of the spaser. Defining the extrinsic characteristics is a bit tricky since, as evident from Fig. 2, the SPs are mostly decaying non-radiatively. Therefore we first define another intrinsic output parameter, SP generation rate, which for the steady state regime is equal to the total SP decay rate, both radiative and non-radiative, i.e.

The extrinsic output parameter would then be a photon generation rate, defined as the fraction of SP generation rate associated with radiative decayi.e. SP generation rate multiplied by the output efficiency. In this example we considered simple radiative decay as the only out-coupling channel, but of course, one can envision the energy being coupled out of SP mode via dipole-dipole coupling to other SP modes or sub-wavelength plasmonic waveguides with efficiencies that can be higher than this. Therefore, to keep our analysis general we shall consider SP generation rate as the main output parameter of spaser, keeping in mind that only a fraction of this output will be a useful output rather than simple decay in the metal.Now, for small modal volume the last two terms in Eq. (25) can be neglected and the steady state solution is simply

or, in other words, the rate of SP generation is equal to the excitation ratei.e. the spaser is threshold-less from the point of view of the input-output characteristics. The result Eq. (29) is not surprising – it simply states the fact that the energy has nowhere to go but into the SP mode.Yet the super linear increase in generated power is not the most important distinctive feature of lasing – the increase in coherence or linewidth collapse is a much more definitive trait characterizing onset of lasing. To define some type of critical condition we thus turn our attention to the effective linewidth Eq. (13) and demand for example that it is reduced by a factor of two in comparison to the cavity without gain, i.e.

which according to the first equation in Eq. (25) and Eq. (23), corresponds to the number of SPs at threshold roughlyi.e., on average, just about one SP in the mode, which is the threshold definition according to [36].There can be many different interpretations of this condition, for instance, one in which coherence time is increased by a factor of two, or the one in which for each spontaneously-emitted SP another one is generated by stimulated emission, but all of these interpretations confirm that defining Eq. (30) as the onset of “spasing” appears to be a proper choice. On the most simplistic level, one can argue that when on average there is less than one SP in the mode at a given time, this SP has nothing to be coherent with, and the device cannot be considered to be source of coherent SP’s, or spaser. For the lack of better terms we shall refer to Eq. (30) as a threshold condition.Now, substituting Eq. (31) into the second equation in Eq. (25) with two last terms still neglected we immediately obtain for the threshold excitation rate ${R}_{ex,t}\approx {\gamma}_{m0}$ and the threshold current is thus${I}_{ex,t}\approx e{\gamma}_{m0}\approx 16$μA for gold and ${I}_{ex,t}\approx 6$μA for silver. In the rest of this work we test and confirm this most important, yet strikingly simple conclusion. This conclusion states that the threshold current (or power in case of optical pumping) of spaser depends neither on the shape and volume (as long as it is substantially sub-wavelength) of the metal nanoparticle, nor on the shape, volume, and the nature of the gain material. The one and practically only parameter defining the threshold current is the optical loss in the metal. The lack of the dependence on volume is the most consequential of the results – it indicates that the threshold current density is expected to increase dramatically for small volumes making development of the truly sub-wavelength spaser a truly daunting task.

## 7. Results

To illustrate all the processes occurring in the spaser near the threshold we first present detailed results for one particular case, namely Au elliptical nanoparticle with half-axis $a=20$nm, no spacing layer and confinement factor $\u0413=0.8$. Shown in Fig. 5(a) are the dependence of SP generation rate, ${R}_{SP}$, and the effective linewidth of the mode ${\gamma}_{m}$ on input current $I=e{R}_{ex}$. As expected, the first curve is essentially a straight line with no pronounced threshold, while the linewidth dependence starts with ${\gamma}_{m}>{\gamma}_{m0}$ due to additional losses caused by unexcited semiconductor, reaches ${\gamma}_{m}={\gamma}_{m0}$ when the “transparency current” ${I}_{tr}\approx 7.56$ μA becomes sufficient to bleach the semiconductor at the SP mode frequency, and finally falls down another 3dB to the threshold value ${\gamma}_{m}={\gamma}_{m0}/2$ as the current passes through the threshold value ${I}_{t}=22$μA, which is not far from the predicted material/geometry independent value of 16 μA. Following that, the linewidth continues to shrink inversely proportional to the pump current, in rough agreement with Shawlow-Townes formula [33].

In Fig. 5(b) we plot the dependence of the carrier density on the current and find that while at first it grows linearly with the current the slope gets reduced rather sharply near threshold and eventually the carrier density appears to get clamped near $1\times {10}^{19}$cm^{−3}. In this respect our single-mode spaser behaves no different form a conventional laser, albeit the knee of the curve is not as sharply pronounced and the clamping is not fully complete.

More insight can be gained from the evolution of the material gain $g\left(\omega \right)/\u0413$ and the shape spectral density of the modal gain following Eq. (14)

At the first point [Fig. 6(a)] the pump current of just 1.5 μA is capable of maintaining the carrier density of only about $4\times {10}^{18}$ cm^{−3} and the material gain at the SP energy of 0.9 eV is actually negative leading to a very broad curve of the modal gain spectral density that contains both positive and negative segments. When the transparency condition is reached [Fig. 6(b)] the carrier density is $6.5\times {10}^{18}$cm^{−3}, the material gain becomes positive at the SP energies, but the modal gain density curve still contains a negative segment at higher energies, so that the modal gain is actually equal to 0, meaning that absorption at higher energies cancels the gain at lower frequencies. Next, as the threshold condition is approached [Fig. 6(c)] and carrier density rises to$8.2\times {10}^{18}$cm^{−3} the material gain increases somewhat further and the density curve is now mostly positive. Finally [Fig. 6(d)] at high current of 150 μA the linewidth becomes narrow enough, so that the spectral density looks as a nice Lorentzian curve as it should be in a laser operating way above the threshold.

In Fig. 7(a)
we plot the number of SPs in the mode as a function of carrier concentration – indeed near the threshold ${N}_{SP,t}={\overline{n}}_{sp,m}$, i.e. slightly more than one SP in mode, but past threshold the number goes up rather fast. Finally in Fig. 7(b) we show the modal gain and all three components of recombination rate versus carrier density. First of all, the radiative emission into free space${r}_{sp,fs}$is negligibly small at all current densities. The Auger recombination rate _{${r}_{A}$} reaches the same order as spontaneous recombination into the mode ${r}_{sp,m}$, at the very high carrier concentration, past the threshold, but at that point stimulated recombination becomes dominant and Auger recombination simply decreases efficiency by a small amount. Also one can see that past threshold modal gain closely traces ${r}_{sp,m}$, indicating that spontaneous emission factor ${\overline{n}}_{sp,m}$ is not far from unity.

Next we consider four rather extreme cases of the Au-based spaser and plot the dependence of SP generation rate and linewidth on pump current in Fig. 8(A)
and carrier density on pump current in Fig. 8(B). The cases (a) and (b) are both for the spaser having no spacing layer between the gain medium and metal and thus having high confinement factor $\u0413$ = 0.8. The case (a) is for a large spheroid with half-axis $a=40$ nm and case (b) for a much smaller spheroid with $a=$15 nm. Nevertheless the curves (a) and (b) in both Fig. 8(B) and 10 are very close to each other and the threshold currents in both cases are still roughly equal (${I}_{t}=$23 μA for a larger spheroid, ${I}_{t}=$ 31μA for smaller one). The carrier density is clamped at somewhat higher value for smaller ellipsoid (b) because of larger (by 1/3) loss in the smaller ellipsoid. The other two cases (c) and (d) are for the same metal ellipsoids as (a) and (b) but this time with 2 nm spacing layer separating the gain region from the metal surface and thus smaller confinement factor $\u0413=0.8$. Now the carrier concentration required to achieve the threshold reaches above 2 × 10^{19}cm^{−3} which causes rapid increase in the rate of Auger recombination, which is no longer negligible and causes increase in the threshold and also change in the slope of the input-output curve making it look a bit more like a conventional laser. Thus for the case (c) of a = 40nm spheroid with $\u0413=0.4$, ${I}_{t}=$38 μA and for the case (d) of $a=$45 nm spheroid with $\u0413=0.4$, ${I}_{t}=$50 μA. Once again we want to emphasize that this increase is primarily due to enormous Auger recombination rate.

If we now turn our attention to silver based spasers and consider the results for the same 4 cases shown in Figs. 9(A)
and 9(B), we can see that the curves are much closer to each other, and, in fact equal to the analytical results obtained earlier. Since the loss in Ag devices is a factor of 3 less than in Au, the threshold carrier density is on the order of ${N}_{t}~4-5\times {10}^{18}$cm^{−3} at which the Auger recombination does not present much of a factor and the spaser characteristics are very close to the ideal single-mode laser with radiative recombination into the mode being the dominant energy loss channel. The SP generation rate is a perfect, threshold-less linear function of current density and the linewidth gets to 1/2 of the empty cavity linewidth at about the same 6-8 μA value of threshold current.

## 8. Discussion

While the value of the threshold current itself does not appear to be outrageously high, to put the matters into perspective we shall evaluate the current density per unit area of active region. Assuming a hypothetical injection scheme in which we have a p-region on one side and n-region on the other sign, as shown in Fig. 1, one can evaluate the threshold current densities under the assumption that the each carrier current is uniformly distributed over the half of spherical surface. The results are shown in Figs. 10(A) and 10(B) for gold and silver respectively.

Then one can see that for the largest Au device which has a total length of about 120 nm, i.e. roughly 𝜆/3 in InGaAs the threshold current density is about 220 kA/cm^{2} while for Ag device it is 120 kA/cm^{2} which is orders of magnitude higher than threshold current densities in the conventional semiconductor diode lasers. One can hardly call 𝜆/3 size device a truly sub-wavelength one – and for smaller devices the threshold current density quickly surpasses a MA/cm^{2} i.e. becomes unsustainable in any conceivable semiconductor. Note that in our calculations we have considered “ideal” metals with loss parameters from Johnson-Christie [24] and disregarded inevitable surface scattering loss in small nanoparticles that can easily raise the threshold by a factor of a few. Also we have assumed a 100% efficient current injection – and assuming more realistic values of it will further increase the threshold beyond already unattainable values. And we should also remember that as a threshold we have taken the value of current at which roughly one half of the SPs are coherent – and the linewidth only narrows by a factor of two. Attaining more substantial narrowing requires enormous power densities, which is difficult to obtain even under optical pumping.

It should be noted that the circumstances of semiconductor spaser development are different from the situations that took place with other types of lasers in their early stages of development. Early stage of development is always characterized by very high threshold. For instance, in early 1960s the room temperature threshold for the first injection laser was 100’s of kA/cm^{2} [37] but then rapid progress ensued and by 1970 room temperature double heterostructure lasers [38] have been operated with threshold current densities of less than 1kA/cm^{2}. The reason for such a rapid progress was that from the very inception of the idea of semiconductor lasers it has been well understood that the cause of high threshold was the absence of carrier and light confinement, and the method for overcoming this problem, double heterostructure had been identified [39,40] right away. Once inevitable material growth related problem had been solved the threshold had quickly come down by orders of magnitude. In contrast, our not-so-optimistic result for the single mode spaser is based entirely on the best case assumptions with all the carrier and plasmon confinement already ensured. The high threshold current density is the consequence of the two of the most basic laws of physics. First one is the Maxwell equation $\nabla \times H=j\omega D$stipulating [22,23] that in substantially sub-wavelength mode the magnetic field is weak and the losses are determined by the high loss of the metal alone. The second one is the relation between the first and second Einstein coefficients – the foundation of quantum electro-dynamics. Short of repealing these laws the only hope lies in reducing the metal loss, but this goal would essentially require engineering of new materials on the atomic level [41].

Therefore, if one sets the goal of achieving a coherent source of SPs in a sub-wavelength volume in all three dimensions, this goal does not appear to be remotely realistic in any conceivable combination of metal, gain material and geometry, primarily due to very low finesse resonance of the SP mode. Of course, one may achieve “spasing” using pulsed optical pumping [9] but the practicality (and repeatability) of such a scheme is dubious. The injection-pumped coherent spaser remains out of reach.

Yet if, on the other hand, one is simply looking for a powerful source capable of generating a lot of power in a small volume then the injection-pumped generation of SPs predominantly by spontaneous electron-hole recombination can be quite efficient due to large Purcell factor. It is quite conceivable that “surface plasmon emitting diodes” (SPEDs) with intrinsic efficiencies of 90% and more can be developed in place of SPASERS, offering most of the attractive characteristics of the latter without requirement of unsustainably high current densities. Our calculations show that at current densities as small as 100 A/cm^{2} the radiative recombination dominates and the internal efficiency is very high. Of course, the overall efficiency of the device will depend on what fraction SP energy can be coupled into the free space or waveguide for some useful purpose and due to high metal losses overall efficiency of SPED might not turn out to be all that spectacular. Optimization along the lines [42,43] needs to be performed before the efficiency and practicality of SPED can be determined.

## 9. Conclusions

In conclusion, we have developed a rigorous theory of the semiconductor source of coherent surface plasmons (SPASER) and have obtained a set of coupled rate equations that characterize a single mode laser. We have shown that input-output characteristics of the single mode SPASER does not exhibit nonlinearity inherent in most lasers, but the linewidth of the emission does collapse, as in any laser which has allowed us to define the threshold of a SPASER as a value of current (or current density) at which the linewidth is decreased by a factor of two. We have shown that as long as the mode is considerably sub-wavelength in all three dimensions the threshold does not depend on either shape, or size, or the nature of gain material used in the device, but is determined exclusively by the scattering rate of the metal. The threshold current densities evaluated by us are orders of magnitude higher than those in conventional in-plane or vertically emitting semiconductor lasers, which, in our view, make the idea of truly sub-wavelength sources of coherent SPs, appealing as it is, quite short on practicality.

At the same time, once the requirement of coherence is taken out of consideration, the goal of sub-wavelength injection-pumped source of SPs is no longer impeded by the basic laws of physics and becomes limited only by the engineering issues, because currently the requisite nano-scale injection heterostructures are not available. These engineering issues are complex but not irresolvable and SP injection diodes (SPED) may find applications, such as, single photon sources. And of course, once the SP source is allowed to extend to at least half wavelength in one of the dimensions, which is perfectly good for most applications, the loss is reduced dramatically and the coherent SP sources become practical.

## Acknowledgment

This work was supported in part by the Air Force Office of Scientific Research under grant FA9550-10-1-0417.

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