We develop a fundamental formulation for electrically-pumped plasmonic semiconductor nanolasers based on a metallic bowtie structure. Because of the negative dielectric constant of the metal at optical frequencies, the effective modal volume of the plasmonic mode can be compressed to the nanometer scale. In addition, the curvature effect of the bowtie tips provides additional field enhancement in the bowtie gap and further reduces the modal volume. With this small modal volume, the required volume of the active region is reduced correspondingly, which significantly decreases the threshold current. The huge field enhancement due to the small modal volume at the gap of the bowtie may overcome the material and radiation losses by increasing both the spontaneous and stimulated emission rates, and it makes the lasing action possible.
©2008 Optical Society of America
As the size of lasers scales down, the wave nature of photons becomes one of the most critical challenges in the realization of a tiny coherent photon source. The localization of the wave is difficult when wavelengths of photons become much larger than the spatial variation of the confinement structure. A rule of thumb for spatial structures based on a typical dielectric is that they can at most support the optical field with an effective wavelength in the dielectric as small as its characteristic length. This restriction leads to a lower bound for the effective modal volume, V eff, of a 3D confined mode with free space wavelength λ and effective refractive index in the dielectric n: V eff>[λ/(2n)]3, which is the minimal standing-wave condition in each dimension, or the so-called diffraction limit. Most of the dielectric cavity structure such as the quarter-λ distributed feedback structure, [1, 2] defects in photonic band-gap structure, [3, 4, 5, 6], or microdisk lasers [7, 8, 9] are mostly limited by this restriction, and thus the smallest dimension can only be as low as the submicrometer scale in the optical frequency range.
On the other hand, opticalmodeswhich are coupled to metallic surface plasmons and become the constituent of the surface plasmon polaritons are less restricted by the diffraction limit. The negative dielectric constant of the metal below the plasma frequency is the key element for the compression of the effective modal volume. The optical mode whose main polarization component is perpendicular to the metal-dielectrics interface has a modal profile localized at the interface . The optical field decays faster into the metal side than into the dielectric side. Thus, if a dielectric is placed between two metallic objects, the much faster decay of the optical field into metals ensures that the effective volume is bounded by the metallic objects instead of spreading outside the dielectric. This kind of surface wave is not accessible to the usual dielectric structure because a negative dielectric constant is needed. Plasmoinc surface waves have been applied to active devices such as infrared and terahertz quantum cascade lasers [11, 12, 13, 14] and plasmonic antenna lasers . At optical frequencies, the plasmonic cavity has also been successfully implemented for submicrometer semiconductor lasers .
Accompanied by the small effective volume of the optical field is the enhanced spontaneous emission [17, 18, 19, 20]. The increase is not only restricted to the spontaneous emission but also applies to the stimulated emission, which is implied by the Einstein’s A and B coefficients. Therefore, the small effective volume of the optical mode in the plasmonic nano cavity implies that the stimulated emission rate can surpass other nonradiative recombination rates more easily than conventional lasers. Thus, the field enhancement increases the intrinsic quantumefficiency. However, one of the drawbacks in plasmonics is that metallic surface plasmons are lossy, and the photon part of the generated plasmon-polaritons are easily damped out via the decay of the plasmon part into heat. Although the conversion efficiency of the electron-hole recombinations into photons can be potentially high, the loss rate of the photons can also be significant. In fact, this point is the main difference between a conventional laser and a plasmonic nanolaser. Take photonic-crystal lasers as an example. The main effort on photonic-crystal lasers [4, 6] is focused on the construction of a high-quality (Q) cavity (to decrease the radiation loss) so that the photon storage in a cavity is readily achievable for the initiation of the stimulated emission. For a plasmonic nanolaser, another alternative distinct from the high-Q cavity is adopted. Despite the presence of the material and radiation loss, we increase the stimulated emission to compensate other inevitable loss mechanisms. Indeed, the coherent amplification of surface plasmons  and a dipolar nanolaser  have been proposed.
In this paper, we use semiconductor quantum dots (QDs) and multiple quantum wells (MQWs) as the gain medium for an electrical-injection, plasmonic nanoloaser based on the bowtie cavity. Such a structure has been experimentally demonstrated as a photoluminescence super radiant center  or a field-enhanced light scatter . As shown in Fig. 1(a), we consider an open cavity in which a few QDs exist in the semiconductor post at the center of bowtie gap. The advantage of the metallic bowtie structure originates from the plasmonic confinement and curvature effect of the two bowtie tips, which localize the mode in the gap, as indicated by the field distribution in Fig. 1(a). For MQWs, they are placed below the bowtie cavity [Fig. 1(b)]. Electrical injection of the carriers into QDs and QWs provides the gain for lasing. The plasmonic cavity is an open cavity with significant radiation and material losses. We show that for uniform QDs, it is possible to overcome the losses for lasing. Meanwhile, due to a reduced volume of the active region, a much lower threshold current than that of a conventional semiconductor laser is expected.
2. Spontaneous and stimulated emission rates
We consider first a lossless but dispersive system. For a given lossless cavity structure, there are optical modes which satisfy the boundary conditions imposed by the cavity. The mode is labeled with index n, and the corresponding modal profile is written as
where ωn is the eigenfrequency of the mode; 𝓔n(r) and ℋn(r) are the modal profiles of the electric and magnetic fields, respectively; and f n(r) is the normalized modal function. The normalized modal function satisfies the eigen equation in the dispersive medium:
where R(r,ωn) is the real part of the relative dielectric response function (r,ωn), which includes the contribution from the relative dielectric response function ε(r,ωn) and conductivity σ f(r,ωn):
For a narrow band signal which are composed of modes with eigen frequencies around ω 0, we adopt the following normalization condition of the modal function f n(r) in the dispersive medium:
In Eq. (4), the first term containing the partial derivative with respect to ω originates from the electric energy , and the second term comes from the magnetic energy. Equation (4) is also the quasi-orthonormalization condition of the optical modes in a dispersive medium.
We use the dipole interaction Hamiltonian H int=-e d·E, where E is the electric field composed of various optical modes, and d is the position operator of an electron. The Fermi’s golden rule states that the spontaneous emission rate due to the interaction between mode n and a single dipole e d cv (c and v are initial and final states, respectively) at position r is
where Ecv=Ec-Ev is the transition energy between initial state in the conduction band c and final state in the valence band v. Similarly, the stimulated emission rate due to the dipole at position r is
where Nph,n is the photon number of mode n. In a real material, the energy conservation due to the delta function is not strict. Therefore, one replaces the delta function with a Lorentzian lineshape function
where Γcv is the half-width-at-half-maximum (HWHM) linewidth of the transition. For the spontaneous emission, we also need to consider the broadening of the modal photon density (cm-3) of states
where ħΔωn is the HWHMlinewidth of the density of states and is related to the quality factor of the mode. The modal spontaneous emission rate per unit volume R̂ sp,n is calculated by summing the contribution fromall the dipolemoments in the active region and the availablemodal density of states ρn(ħω), taking into account the occupation of various states:
where V a is the volume of the active region; fc,i and fv,i are the occupation probabilities of the initial and final states of dipole i at position r i, which will be modeled as the Fermi-Dirac distribution; and we have parameterized various quantities with the label i. The total spontaneous emission rate will be the sum of all the modal contributions: R̂ sp=∑n R̂ sp,n.
In Fig. 2(a), we plot the transition lineshape Lcv(ħω) and the photon density of states ρn(ħω) from Eq. (9) as a function of the photon energy ħω (the relative magnitude is up to a scaling factor). Intuitively, if the peaks of the two lineshapes coincide with each other, one expects that the spontaneous emission rate will be enhanced, which is known as the Purcell effect . The integration over the modal photon density of states ρn(ħω) and Lorentzian lineshape Lcv,i(ħω) is proportional to the spontaneous-emission efficiency of a dipole with a transition energy Ecv. Since the convolution of the two Lorentzians is still a Lorentzian, we can carry out the integration analytically:
The integration in Eq. (10) is plotted as a function of the transition energy Ecv in Fig. 2(b). From this Loretntzian curve, we see that spontaneous emission is efficient if the energy difference Ecv-ħωn is within the HWHM linwidth Γcv+ħΔωn and reaches a maximum when the transition energy Ecv is the same as that of the cavity mode ħωn, which is exactly the statement of the Purcell effect.
Unlike the spontaneous emission, the sum over the photon density of states for the stimulated emission is not needed since the lasing mode will have a linewidth much narrower than the cavity linewidth when lasing. After including the occupation numbers and a scaling factor resulted from different volume factors which define the photon and carrier densities, the stimulated emission rate R̂ st,n per photon for mode n becomes
where V eff is the effective modal volume  which contains most of the field strength in the real space; |f(r)|2 max is the maximum of |f(r)|2, and n̄ is the refractive index at the position of field maximum. An alternative definition in Ref.  can also be used. The ratio V eff/V a plays the role of the confinement factor in the rate equations for typical semiconductor lasers. Equation (11) is most general for any configurations of dipole moments, which are different from samples to samples. We are interested in the statistical behavior and will denote R st,n and R sp,n as the counterparts of R̂ st,n and R̂ sp,n after taking the ensemble average. The details are presented in the Appendix. The expressions of Eqs. (9) and (11) for homogeneous QDs (uniform two-level system) and QWs under the parabolic approximation are also obtained in the Appendix.
Usually, one defines the modal spontaneous emission factor β sp,n=R sp,n/R sp as the ratio between themodal spontaneous emission rate R sp,n and total spontaneous emission rate R sp. The spontaneous emission factors β sp,n’s satisfy the sum rule ∑n β sp,n=1. In practical calculations, we divide the various modes into two categories: cavity modes (C) and radiation modes (R). Roughly speaking, the cavity modes are quasi-bound modes of the cavity structure while the radiation modes correspond to those photons which immediately leak out of the cavity structure once emitted. The radiation modes form the background part of the total emission spectrum. The effect of the radiation modes are difficult to calculate because there are infinitely many of them. In the Appendix, we will use an empirical approach to include the effect of the radiation modes.
3. Rate equations
We use the rate equations to calculate the light-versus-current (LI) curve. Under a single-mode operation, the rate equations for a nanolaser can be written as
where N and n are the total carrier number (dimensionless) and carrier density (cm-3) in the active region; nc and nh are carrier densities (cm-3) of electrons and holes, respectively; N ph and S are the photon number (dimensionless) and photon density (cm-3) of the lasing mode; ηi is the external quantum efficiency; I is the injection current (mA); R nr(n) is the nonradiative recombination rate for carriers (cm-3s-1); τ p is the photon lifetime (s) of the lasing mode with τ p,mat and τ p,rad as its components from material loss and radiation loss (coupling out of cavity); and V a/V eff is a scaling factor analogous to the confinement factor in typical rate equations.
In Eq. (16), the electron density, nc, and hole density, nh, are related to the respective quasi Fermi levels Efc and Efv. We assume the charge-neutrality condition (nc=nh), and the two quasi Fermi levels have to be solved self-consistently. The nonradiative transition rate is modeled by the relaxation-time approximation and Auger recombination:
where C is the Auger coefficient. The photon lifetime τ p and the term R st(n) are related to the linewidth Δω of the density of states ρ(ħω) and total quality factor Q t of the lasing mode:
where Q rad=ω L τ p,rad and Q mat=ω L τ p,mat are the quality factors due to radiation loss and material loss of the lasing mode (excluding the contribution from laser medium); Q is the corresponding quality factor when the gain medium is absent; and ωL is the lasing frequency. The term R st(n) is not always positive. Before population inversion is reached, R st(n)S is the stimulated absorption rate. Thus, before population inversion, the term R st(n) also reduces the total quality factor Q. The observed power P is the radiative leakage of the photons in the cavity:
At steady state, the photon density S becomes
The threshold density n th can be obtained by looking into the denominator of S and P in Eq. (21). When the denominator is close to zero, the photon density and output power will be huge. In reality, the denominator cannot be exactly zero due to spontaneous emission. Other definitions of the threshold condition may be used . However, setting the denominator to zero is usually a good estimation. In general, this condition has to be solved numerically. For the special case of a uniform 2-level system such as uniform QDs in the Appendix, we can utilize the expression of the stimulated emission rate in Eq. (32) and obtain the threshold carrier density:
which can be rewritten as
where s is the state degeneracy; nd is the QD density; and is a ratio related to the squared magnitude of the dipole-field inner product, which is independent of the normalization of the modal profile. The threshold carrier density n th is proportional to the ratio V eff/Q. At resonance (ħ ωL=Ecv), the threshold carrier density becomes
The effective modal volume V eff will be smaller if the field is more confined in a small volume. Decreasing this ratio can effectively reduce the threshold carrier density n th of a plasmonic nanolaser, which originates from the enhanced stimulated emission rate. Another factor which affects the threshold density n th is . A significant overlap between the active region and field as well as the alignment of the dipole moment and field can both increase and reduce the threshold carrier density n th. All of these effects can be characterized by a single integral in Eq. (22), which simultaneously includes the effects of the overlap between the field and active region (restricted integration over the active region V a only), small modal volume [through the normalization condition of f(r) in Eq. (4)], and the alignment of the field with dipole moments (the dipole-field inner product).
4. The bowtie structure and gain materials
The parameters of the device have been labeled in Fig. 3 and listed in Table 1. Gold is used as the bowtie material. We denote the tip-to-tip direction as the x direction and the growth axis as the z direction. The three dimensional (3D) field distribution is obtained using the finitedifference- time-domain (FDTD) method in which an external plane wave is used to excite the cavity mode . Since the main component of the plasmonc cavity mode within the bowtie gap shall be perpendicular to the metal surface, the main polarization direction of the modal profile is tip-to-tip (x-polarized). Thus, we use the x-polarized plane wave to excite the cavity mode. The computation domain is enclosed by perfect-matched layers (PMLs) to reduce the effect of boundary reflections. The modal profile also decays exponentially toward the air and substrate. A rule of thumb for nanolasers is that the gain medium has to be placed in the region where a significant field strength of the modal profile occurs. In this way, the advantage of the field enhancement can be fully exploited. Therefore, the QDs are located in the bowtie gap, and for MQWs, they have to be as close to the bottom of the metallic bowtie as possible.
The amplitude of the field at the center of the bowtie gap is calculated as the wavelength of the external plane wave is varied. The resonance wavelength of the cavity mode corresponds to the maximumamplitude at the gap center. We design the size and geometry of the bowtie so that the resonance is around 1550 nm. The field distribution of the resonant mode is obtained from an FDTD calculation after the external plane wave is turned off for several optical periods. The modal profile f(r) resembles the near-field pattern. We renormalize the field pattern via Eq. (4) to obtain f(r) for the calculations of the spontaneous and stimulated emission rates.
In the solution of the rate equations, the required parameters related to the optical field are the modal profile f(r) and quality factor Q. The quality factor Q is a measure of how long a photon can stay in the cavity if no gain medium is present. We obtain the quality factor by observing the ratio between the stored energy and power loss after the source is slowly turned off. Due to the nature of the open cavity, small size of the device, and material loss frommetallic surface plasmons, the quality factor is about 3–10, depending on the design of the cavity. In our calculations, we set the temperature at 77 K so that the narrow transition linewidth of QDs [29, 30] at low temperature can further increase the stimulated emission transition rate. Lowering the temperature also reduces the material loss from the metallic surface plasmons  even though from the FDTD calculation, the radiation loss is the dominant loss. For completeness, we assume that the imaginary part of the metallic dielectric is reduced to one tenth of the roomtemperature value. The FDTD calculation also indicates that most optical power leaks out the the cavity from the -z direction (substrate, 37.5%) and ±y directions (55.3%), along which no confinement structure is present. In the future, feedback structures can be designed along those directions of leakage to block the radiation loss, for example, a cross-bowtie structure with an extra pair of metallic triangles along the y direction (tip-to-tip direction).
For QD cases, three identical QDs are present in the semiconductor post at the center of the bowtie gap. We assume that the transition frequency of QDs is exactly the same as the resonance frequency of the bowtie structure so that the lasing occurs at peak gain. For MQWs, we consider five In0.53Ga0.47As/Ga0.2In0.8As0.43P0.57 quantum wells (lattice-matched to InP substrate) with a well width of 7 nm and a barrier width of 6 nm. The area of the active region is set to the cross section of a 350 nm×200 nm rectangle which contains the bowtie. The center of the first QW is positioned 9.5 nm below the metallic bowtie. For a nanolasaer, the field-dipole inner product also determines the threshold carrier density n th, as indicated in Eq. (22). In order to reduce the threshold carrier density, it is better to have transition dipoles aligned with the modal field in most of the active region. For QDs, the dipole moment of the ground-state transition is usually TE-polarized (perpendicular to the growth axis) and coincides with the polarization of the cavity mode within the gap (x polarization). On the other hand, typical QWs also exhibit dominant TE-polrized gain, but such a polarization component is only significant around the portion of the substrate just below the bowtie gap. Away from the metal-substrate interface, the field decays exponentially. That is the reason why MQWs have to be as close to the bottom of the bowtie as possible. Our FDTD calculation indicates that the overlap integral for MQW case is larger than that of QDs due to a larger cross section below the bowtie. Nevertheless, due to the momentum distribution of the carriers in QWs, the material gain is not spectrally peaked at the cavity frequency, implying the gain spectrum is not fully utilized. Therefore, the required quality factor for lasing is higher in the MQW case.
5. Transition rates and LI curves
Figure 4(a) shows the squared magnitude of the x modal profile |fx(r)|2 (obtained from FDTD) along the x axis at one half of the bowtie thickness (12 nm above the metal-semiconductor interface) in the logarithmic scale. The discontinuities along the x direction are consequences of the air-metal and air-semiconductor interfaces. The contour plot of |fx(r)|2 in the x-y plane near the tips is also shown in Fig. 4(b). As expected, the surface plasmons significantly increase the field within the gap. The maximum of the field occurs at the surface of the metallic tip, and the field decays exponentially into the inner of the metallic bowtie. In the gap, the field is symmetrically distributed along the x direction, and one can regard the field within the gap as the symmetrically-coupled mode of the respective plasmonic modes at the two bowtie tips. With this field pattern, we carry out the field normalization and obtain the modal profile using Eq. (4). The key parameters in calculating the LI curve is the quality factor Q and the overlap integral of the squared magnitude of the dipole-field inner product. From FDTD calculations, we obtain a quality factor of about 4.5 and an effective volume V eff=6.937×104 nm3.
For the QD case, the Auger nonradiative recombination is negligible. The nonradiative recombination rate is modeled by the non-radiative lifetime τ nr=0.1 ns. The homogeneous linewidth Γcv of a QD transition can be as small as several to hundreds of micro-electronvolts (µeV) at low temperatures [29, 30]. Due to electrical injection, we expect that the linewidth will be higher than those obtained via optical excitation. We thus set the homogeneous linewidth Γcv of a QD to 0.5meV at 77K. The QDs aremodeled as quantumdisks with 4 nmheight and a 7 nm radius, and the corresponding transition dipole moment |e d cv| is set to 11 eÅ. In addition to the ground states, there are other excited states which will be included in the calculation of the rate equations. The overlap integral I dp related to the dipole-field inner product is 0.55 (eÅ)2. The background radiative constant R rad sp is 10 ns-1. The radiative part of the photon lifetime τ p,rad is set to the photon lifetime τ p since the radiation dominates.
Figure 5(a) shows various transition rates as a function of injected current for the QD case. There is a maximally available stimulated emission rate (or maximal modal gain) for dipole transitions because each QD state can accommodate only two electrons or holes (including spin degeneracy). If the maximally stimulated emission cannot overcome the material and radiation loss, it is usually difficult for the device to lase. Whether this condition can be reached is one of the most important issues for a working laser. For plasmonic nanolasers, the enhanced stimulated emission rate due to the small effective volume V eff plays the role of fulfilling the lasing condition. This is in contrast to the case of photonic-crystal lasers with a high-Q cavity, which reduces the radiation loss. A key feature of lasing is the pinning of the carrier density above the threshold. The carrier density does not increase with the injection current because carriers are immediately converted into photons by the stimulated emission once brought into the cavity. The accompanying phenomenon is the pinning of the nonradiative and spontaneous emission rates above threshold, as indicated by the saturation of R sp in Fig. 5(a). Figure 5(b) shows that at a small current before the population inversion is reached, the stimulated emission rate is negative, indicating the stimulated absorption. Above this point, the stimulated emission still has to overcome the material and radiative loss so that a significant amount of photons can be present in the cavity for the lasing action to take place. Note that the nonradiative transition rate is much smaller than the spontnaeous emission rate because of the field enhancement. For uniform QDs, we can use Eq. (22) to estimate the threshold carrier density n th=1.18×1018 cm-3, which is close to the occupation density of the ground states 1.24×1018 cm-3 after lasing. The range of the injection current is in the order of micro-Amperes. Such a small injection current compared with the milli-Ampere counterpart of a traditional semiconductor Fabry-Perot lasers is due to the nanometer size of the active region. The required current is significantly reduced because fewer carriers are required to reach the lasing condition.
The light output power vs. injection current of the same device is shown in Fig. 5(c). The output power is of the order of micro-Watts. The enhanced spontaneous emission into the cavity mode is much larger than the emission into the radiation modes modeled by the constant R rad sp. Effectively, the spontaneous emission factor β sp is close to unity, and it leads to the thresholdless behavior on the LI curve . However, the lack of turn-on behavior does not imply that the stimulated emission takes over immediately under the current injection. In Fig. 5(b), we can also find the current (0.5 µA) at which the stimulated emission rate is the same as the spontaneous emission rate and take the corresponding carrier density (1.22×1018 cm-3) as an alternative definition of nth. Well below the threshold, the spontaneously emitted photons dominate, and far above the threshold, the stimulated emitted photons dominate.
For the MQW case, the nonradiative recombination lifetime τnr is set to 0.1 ns while the Auger recombination constant C is set to 10-33 cm6s-1. The dipolemoment ed 0 is 11.1 eÅ, and the overlap integral I dp is 1.756 (eÅ)2 for the HH subband and 2.83 (eÅ)2 for the LH subband. The radiative constant describing the background spontaneous emission is set to 10 ns-1. The corresponding transition rates as a function of the injection current are shown in Fig. 6(a). From Fig. 6(a), the stimulated transition rate is too low compared with the nonradiative and spontaneous emission rates. We limit the calculation to 15 µA because further increase of the current will make the conduction band quasi Fermi level exceed the barrier potential. In other words, the MQW structure cannot function as a laser. The difficulty of lasing for the MQW case is that the stimulated emission rate is not spectrally peaked at the lasing wavelength. Injected carriers are not fully utilized because of their momentum distribution. Therefore, there is not enough gain to overcome the loss, and the accumulation of the photons becomes difficult due to the material loss from the metallic plasmons. Figure 6(b) shows the corresponding LI curve. The output power is mostly contributed from the spontaneous emission and is low compared with that of the QD case. The device works as a light-emitting diode (LED) rather than a laser.
A higher quality factor is necessary to make a MQW nanolaser. A key issue is how high a quality factor has to be so that a MQW nanolaser can lase with the corresponding quasi-Fermi levels falling below the barrier potentials. We thus change the quality factor to see when the stimulated emission rate can exceed the spontaneous emission and nonradiative recombination rates. Since most of the loss comes from the radiation, changing the quality factor effectively means designing a feedback structure to block the radiation loss. For this particular structure, a quality factor around 100 is necessary to pin the quasi Fermi level of the conduction band below the potential barrier at T=77 K. A quality factor of about a few hundreds is the typical value for plasmonic lasers at optical frequencies to lase . Figure 7 shows various transition rates and LI curve after the quality factor Q is set to 100. Compared with Fig. 5(a), a higher current is required to change the stimulated absorption into the stimulated emission. Such a higher current is due to the larger active region and a gain spectrum which is inhomogeneously broadened by the momentum distribution. These two factors reflect the inefficient carrier usage both spatially and spectrally. Spatially, since carriers are efficiently used only when they are in the region just below the bowtie gap, those which are distributed in the remaining of the active region do not contribute much to the stimulated emission. Spectrally, the momentum distribution of the carriers means that many electron-hole pairs do not participate in the stimulated emission because the corresponding transition energies do not match the lasing energy. Therefore, more carriers have to be pumped into the active region so that the population inversion at the lasing wavelength can be achieved. Also, the stimulated emission rate grows more slowly than the counterpart of QDs, and it takes a much higher current for the stimulated emission rate to surpass the nonradiative recombination rate and spontaneous emission rate. The nonradiative recombination is not negligible compared with other transition rates, which reflects a much smaller spontaneous and stimulated emission rates compared to the QD case. In Fig. 7(b), the LI curve of MQW case is plotted. It takes a higher current in the MQW case for the stimulated emission rate to become the dominant transition rate. Due to many carriers which are not efficiently coupled to the lasing mode, the spontaneous emission factor β sp is not close to unity. Above the lasing threshold, the spontaneous emission factor β sp⋍0.3, and one can see a more significant turn-on behavior on the LI curve than the counterpart of the QD case. Comparing Fig. 7(b) with Fig. 6(b), we can see a large difference on the output power for cavities with different quality factors, which distinguish a laser from an LED.
We have modeled the plasmonic nanolasers based on the metallic bowtie structure. The plasmonic and curvature effects of the metallic bowtie tips confine the modal profile to the nanometer scale within the bowtie gap, which substantially reduces the volume of active region and the threshold current density. In contrast to other lasers such as photonic-crystal lasers, which require a high-Q cavity to improve the laser threshold, the enhanced stimulated emission rate from a small effective volume is utilized for plasmonic nanolasers. Despite significant material and radiation losses in the bowtie stucture, the enhanced stimulated emission rate can overcome these obstacles for lasing. We have investigated both uniform QDs and MQWs as gain media. The bowtie plasmonc nanolaser using uniform QDs shows a better performance because of the small active region in the bowtie gap (spatially) and peaked stimulated emission spectrum (spectrally). On the other hand, a higher quality factor is needed for the MQW structure.
Spontaneous and stimulated emission rates with inhomogeneous broadening
In Eq. (9), we replace the summation over label i with an integration over the coordinate space r in the active region. In this case, the modal spontaneous emission rate R̂ sp,n becomes
where D(r) is the number of dipole moments per unit volume (cm-3); and we have reparameterized various quantities with the position r. The excited energy level E c,r at location r has a finite energy distribution, and so does the ground energy level E v,r. Such an energy distribution originates from the inhomogeneity of the system, for examples, size and composition fluctuations of nonuniform QDs, or the distribution of the carriers in the momentum space of QWs. The energies E c,r and E v,r at the same position r are related to each other. One way to incorporate this correlation is to parameterize E c,r and E v,r by their difference E r=E c,r-E v,r, i.e., E c,r=Ê c(E r) and E v,r=Ê v(E r). We treat E r as a random variables with a probability distribution 𝒢cv(E r), which does not depend on the position r explicitly within the active region. For QWs, the probability distribution 𝒢cv(E r) is closely related to the joint density of states between conduction and valence subbands under the assumption of negligible photon momentum. The physical modal spontaneous emission rate R sp,n=〈R̂ sp,n〉 is the expectation value of R̂ sp,n:
where we do not show the dependence of Êc and Êv on variable E explicitly.
The total spontaneous emission rate R sp is rewritten as a summation over the cavity mode C and radiation modes R:
Due to the continuum of the radiation modes, we expect that the integration over the active region in Eq. (28) is a slowly-varying function of variable E. Thus, we may replace the integration over the active region with a phenomenological constant, i.e.,
where […] is an effective average over the energy E; and is the average dipole density. Note that under full population inversion [fc(Êc)=1, fv(Êv)=0], the integral over the variable E in Eq. (29) is simply unity. In this way, the quantity R rad sp,cv can be interpreted as the spontaneous emission rate due to the radiation modes under full population inversion. Following a similar approach, the modal stimulated emission rate R st,n becomes:
In all the equations above, we model the occupation numbers fc(Êc) and fv(Êv) as Fermi-Dirac distributions with quasi-Fermi levels EFc and EFv, respectively:
The uniform two-level system
For a uniform two-level system such as quantum dots, we assign a single label for the excited state (c) with an energy Ec and another one for the ground state (v) with an energy Ev, but include possible degeneracy s, e.g., spin degeneracy (s=2). The dipole density profile D(r), or the density of QDs (cm-3) is set to a constant n d within the active region. The probability density function 𝒢cv(E) is replaced with an energy delta function δ(E-Ecv) (Ecv=Ec-Ev). In this case, the modal spontaneous and stimulated emission rates are
For the QW case, the dipole density D(r) is uniformin the active region since the in-plane eigen functions are 2D plane waves. In the following, the indices c and v refer to the subband labels, e.g., C1↑ (↓), HH1, LH1…etc. The probability density function multiplied by the dipole density plays the role of the joint density of states. The joint density of states comes from the integration over the equi-energy contour in the momentum space. Thus, one replaces the integration over E with the integration in the momentum space:
where k t is the in-plane wave vector; Lw is the thickness of the QW; and we have replaced the dependence on the variable E with that on k t. Note that in Eq. (33), transitions only take place between states with the same wave vector k t. This approximation corresponds to an electric field which varies slowly compared with the size of nanostructures, which is not always satisfied in our cases. However, the transitions with Δk t~0 remain the main contribution. In the future, if a more precise calculation is required, contributions with Δk t≠0 have to be considered.
Equation (33) is often the starting point of calculations. The QW band mixing leads to non-parabolic band structures and anisotropic dipole moments. However, under the simple approximation of parabolic band, we can obtain similar expressions with those in Eqs. (29) and (30). Assuming no band mixing, the k t dependence of the dipole moment is absent. Under the parabolic-band approximation, the energy and become
where mc and mv are the effective mass of the respective subbands; and mr,cv is the reduced effective mass of the transition. Let us define a new variable E and introduce the intrinsic joint density of states ρ cv,0(E) without any degeneracy (e.g., no spin degenracy)
After changing the integration variable from k t into E, the spontaneous emission rate R sp,n and stimulated emission rate R st,n become
The form of the transition dipole moment e d cv depends on different states c and v and can be written as e d cv,0 Tcv, where Tcv is the overlap integral between the conduction and valence band wave functions along the growth direction. The dipole moment e d cv is tabulated in Table 2.
This work was sponsored by the DARPA NACHOS Program. We thank many technical discussions with Professors Peidong Yang, Connie Chang-Hasnain, Ming Wu, and Dr. Jim Schuck at University of California at Berkeley, Professor Cun-Zheng Ning at Arizona State University, and Dr. Tzy-Rong Lin, and Chien-Yao Lu at University of Illinois at Urbana-Champaign.
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